San Fe rnan d o Val l ey Stat e C o l l ege
THJt:
RESPONSE OF A LINEAR NETWORK TO A FM WAVE
,,
A t he s i s s ubm i t t e d i n part ial s at i sfac t ion o f t h e
re quirements £or t he d e gre� o f Mas t e r o f S c i en c e i n
Engi n e e r i ng
by
J ohn L o u i s Ovn i c k , Jr .
June , 1 9 6 9
The the s i s of ,John Louis Ovn i c k , J r . i s approve d :
San fe rnand o Va l ley State C o l l e ge
June , 1 9 6 9
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TABLE OF
CONTENTS
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1 . 1 The Prob l e m w i t h Fre quency M o d u la t i on . . . . . . . . .
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1 . 2 Re v i e\v o f Fl1 The ory . . . . . . . . , . . . . . . . . . . . . .. . . . . .
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ABSTR4Crr
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IN'fRODUCTION
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2 . 1 H i s t o r i c a l Re v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 . 2 T h e Four i e r or Spe c t ra l A pp r oa c h . . . . . . . . . . . . . .
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2 . 3 The Dyna m i c Approa ch . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 . 4 The Quasi- Sta t i onary Approx ima t i on . ; . . . . . . . . . .
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FOW.IULATION
OF
PROBLEM .
THE
SOLUTIONS. .
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3 . 1 The Approach t o Prob l e m S o l v i ng . . . . . . . . . . . . . ..
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3 . 2 TJ1e C omput e r . . . . . . . . . .
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COMPUTER
A IDED
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. . . . . . . . .· . . . . . . . . . . . . . .
3 . 3 Eva lua t i on o f t h e Ne t work Tra n s f e r I mpedanc e :
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ECAP .
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Gen e ra t i on o f t h e
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FH
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Spe c t rum : BESL . . . . . . . . . . .
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3 . 5 Eva lua t i on o f t he D i s c r im i na t or Output :
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FMDIST
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3 . 6 Harmon i c A na lysis o f t h e D i s c rimi nator Out put :
SERIES. ·.
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. . ... . . . .
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CIRCUIT INVESTIGATIONS..............................
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3 . 7 The
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Quasi-Stationary
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Response;
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TABLE OF CONTENTS
PAGE
4 . 1 Out l ine o f t h e St udy
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4 . 2 S i d e band El iminat i on :' Wan g ' s C r i t e r i on
4 . 3 S i n g l e -Po l e F i l t e r
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4 . 4 Und e r c o up l e d Two - P o l e F i l t e r .
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4 . 5 But t e rworth Two- P o l e F i l t e r . . . . . . . . . . . . . . . . . . .
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4 . 6 T c h e byshev Two - Po l e F i l t er . . . . . . . . . . . . . . . . . . . .
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4 . 7 Su1nmary o f Res u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5. C ONCLU S I ONS . . .
BI BLI OGP..A PIIY
APPENDICES
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1 15
120
A. ECAP S o l ut i on o f t h e But t e rworth Two-Pole F i l -
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B . Subr out i ne Subpro gram BESL
c. Program FMDI ST
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Subr out ine Subp r o gram SERI ES
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E . Err or in C omput ing F o ur i e r C o e f f i c i e�t s .
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F . Pr o gram QS
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LI ST OF TABLES
TABLE
1 . FM Spe c t rum f o r B = ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PAGE
64
2 . Four ie r C oe f f i� i e n t s o f Dis c r im i na t or Outp ut w i t h
Five Pa i rs o f S i d e bands . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 . Four i e r C oe f f i c i en t s o f D is c r im inat or Out put w i t h
Four Pa i rs o f S id e bands . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 . Four i e r C oe f f i c i en t s o f D is c r i mina t or Output w i t h
Thr e e Pa i rs o f S i de ba nds
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5. Four i e r C o e f f i c i ent s of- D i s cr i m i na t or Out put w i t h
Two Pa i rs o f S i d e ba n ds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 . Four i er C o e f f i c ie nt s o f D is c r im i na t or Out put w i t h
One Pa i r of S i d ebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 . Re l a t ive Level o f D i s t ort i on i n D i s c r i m i na t or Output
Due t o S i d e band El i m i nat ion .
Wan g ' s Resul t s C om-
pared w i t h Th os -e o f A u t h or . . . . . . . . . . . . . . . . . . . . . . . .
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8 . A na lys i s o f D i s t ort i on in the D is c r illl i na t ?r Output
Due t o the FM Re s p ons e of a S in g l e - P o l e F i l t er f o r
B = 2 a n d P = 0.2 ..
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9� A na lys i s o f D ist ort i on i n t h e D i s c r im i na t or Out put
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Due t o t he Ff.'i Response o f a S in gl e - P o l e F i l t e r f or
B = 4 a n d P = 0.4 . .
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LI ST Ol', TABLES
PAGE
TABLE
1 0 . Ana lys i s o f D i s t or t i on i n t h e D is c ri m i na t or Out put
Due t o the FM R e s p onse of a S i n gl e - Po l e F i l t er
f or B = 6 and P = 0 . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
1 1 . Ana lys i s of D i s t ort i on in t he D i s c r im ina t or Out put
Due t o t he FM Re sponse o f a S i ngl e - P o l e F i l t e r
f o r B = 8 and P = 0 . 8
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1 2 . Analys i s o f D i s t ort i on i n t he D i scr i m i na t or Out put
Due t o the FM Re s p onse o f a S i n gl e - P o l e F i l t er
f o r B = 1 0 and P = 1 . 0� . . . . . . . . . . . . . . . . . . . . . . . . . .
80
1 3 . Ana lys i s o f D i s t ort i on in the D i s c r i m i na t o r Out p u t
D u e t o t h e FH Re s p onse o f a S in g l e -Pol e F i l t e r
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f o r B = 1 2 and P = 1 . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
1 4 . Ana lys i s of D is t or t i on i n t h� D i s c r im i n a t o r Output
Due t o t he FM Re s ponse of a Two - P o l e Und e r c o up l e d
F i l t e r f or B = 2 a n d P = 0 . 333 . . . . . . . . . . . . . . . . . . .
87
1 5 . Ana l ys i s of D is t or t i on i n t he D i s c r i m i na t o r Out p u t
D u e t o the FH Response of a Two -Po l e Und er c o up l e d
F i l t e r f o r B = 4 a n d P = 0 . 666 . . . . . . . . . . . . . . . . . . .
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1 6 . Ana lys i s of D is t or t i o� in the D i s c r i � ina t o r Out p u t
D u e t o t h e FM Response o f a Two-Pol e Unde r c o up l e d
F i l t e r f or B = 6 and P = 1 . . . . . . . . . . . . . . . . . . . . . . .
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LI ST Ol'' TABLES
PAGE
TABLE
1 7 . A na lys i s of D is t ort i on in the D iscrimina t or Out put
Due t o the FM Res p onse of a Two -Pole Underc oupl e d
F i l t e r f or B = 8 a nd P = 1 . 333 . . . . . . . . . . . . . . . . . . .
88
1 8 . Ana lys i s of D i s t ort i o n in t h e D is c r im i na t or Out p u t
Due t o t h e fl1 Resp onse o f a Two -Po l e Underc oup l e d
F i l t e r f o r B = 1 0 and P = 1 . 66 6
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1 9 . Ana lys i s o f D i s t ort i on i n t h e D is c r i m i na t or Out put
Due t o t he FM Re sponse of a Two-Pol e Unde r c o up l e d
Fil te� for B = 1 2 and � = 2 . . . . . . . . . . . . . . . . . . . . . .
89
2 0 . Ana lys i s o f D i s t ort i on i n t h e D i s c r im i na t or Out p u t
D u e t o t h e FM Resp onse o f a But t e rworth Two - Pol e
F i l t e r f or B = 2 and P = 0 . 2 . . . . . . . . . . . . ... . . . . . .
96
2 1 . Ana lys i s o f D i s t ort i on i n t h� D i s c r im i na t or Out put
Due to the FM Resp onse o f a But t e rwo r t h Two-Pol e
F i l t e r f or B = 4 a nd P
=
0. 4 . . . . .. . . . . . . . . . . . . . . .
96
2 2 . Analys i s of D i st ort i on i n t h e D i s c r i m i na t o r Out put
Due t o t h e FM Resp onse o f a But t e rworth Two-P o l e
F i l t e r f or B = 6 a n d
P
= 0.6 . . . . . . . . . . . . . . . . . . . . .
97
2 3 . Ana lys i8 o f D i s t o rt i on· in the D i s c r i m ina t o r Out p u t
Due t o t he FH Response o f a But t e rworth Two-P o l e
Filt e r f or B = 8 a n d P = 0 . 8 . . . . . . . . . . . . . . . . . . . . .
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LI ST OF TABLES
PAGE
TABLE
24 . Analys i s o f D i s t ort i on i n t h e D i s cr im i na t or Out put
Due t o the FM Resp �_ n s e o f a But t e rwort h Two - P o l e
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F i l t e r f o r B = 1 0 and P = 1 . . . . . . . . . . . . . . . . . . . . . .
98
2 5 . Ana ly s i s of D i s t ort i on in t he D is c r i m i na t or Out put
Due t o t h e FM Response o f
a,
But t e rw or t h Two - P o l e
Fi l t e r f o r B = 1 2 and P = 1 . 2 . . . . . . . . . . . . . . . . . . . .
98
2 6 . A na lys i s o f D is t ort i on i n t h e D i s c r i m i na t or Output
Due t o t he Ft-1 Re s p onse of a T c h e byshev Two- P o l e
F'i l t e r f o r B = 2 and
p
= 0.2
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F i l t e r f or B · = 8 and p = 0 . 8 . . . . . . . . . . . . . . . . . . . . . .
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27 . Analys i s o f D i s t ort i on i n t h e D is c rim i n a t or Out put
Due t o t he .fi'l.t Re sponse of a T c h e byshev Two-Pol e
F i l t e r f or
B
= 4 and p = 0 . 4
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2 8 . Ana ly s i s o f D is t ort i on i n t h e D i s c r i m ina t or Output
Due t o t h e
Fr-1
Respons e o f a Tchebyshev Two - P o l e
FHt e r f o r B = 6 a n d
p
= 0 6
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2 9 . Analys i s of D is t o r t i on i n the D i s c r i m ina t or Out put
Due t o t h e FM Response of a T c h e bysh e v Two-Pol e
3 0 . Ana lys i s o f D is t ort i on i n t he D is c rim ina t or Output
Due t o t he FM Respons e o f a T c hcbyshev Two-P o l e
F i l t e r f o r B = 1 0 and p = 1
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1 05
LI ST Oli' TABLES
TABU�
PAGE
3 1 . A na lys i s of D i s t ort i on i n t he D i s c r im ina t or Out p u t
I
Due t o t he fl't Response o f a T c h e byshev Two - P o l e
F i l t e r f or B = 1 2 a n d P = 1 . 2
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lat i on I ndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 2 . Numbe r o f S ign i f i c a n t S ide ba nd Pa i r s Versus tw! odu-
3 3 . Err or i n C omput ing Ifurm on i c Le ve l s f o r a Saw-T o ot h
\Vave
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135
LI ST OF FI GURES
PAGE
FIGURE
l - 1 . F).-t S p e c t r um f o r B = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 - 2 . Pha s o r Repre senta t i<? � o f v ( t ) . . .
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2 - 1 . S umma t i on o f S i d e ba n d Ve c t ors . . . . . . . . . . . . . . . . . . . . .
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2-2 . I dea l D i s c r im i na t or Respons e . . . . . . . . . . . . . . . . . . . . . .
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3 - 1 . S impl i f i e d Fl owchart f o r Subp r ogram BESL . . . . . . . . . .
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3 - 2 . S imp l i f i e d Fl owchart f or fl!DI ST . . . . . . . . . . . . . . . . . . .
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3 - 3 . Samp l i ng of f ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 - 4 . S impl i f ied F l owc ha rt of S ubpr o gram SERIES . . . . . . . . .
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3 - 5 . Simp l i f i e d F l owc ha r t o f Program QS............
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and Only t h e F i r s t Pa i r of S id e bands Reta i ne d . . .
70
4 - 2 . Schema t i c D i a gram o f S ingl e - Pol e l<"' i l t er . . . . . . . . . . .
72
4 - 3 . Amp l i t u d e Response o f S in gl e - Po l e F i l t er . . . . . . . . . .
73
4 - 4 . Phas e Respons e o f Singl e -P o l e F i l t e r . . . . . . . . . . . . . .
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4 - 5 . Schema t i c D i agram o f Unde rcoup l e d Two-Pole F i l t e r .
84
4 - l . D i s c r i m i na t or Out p u t v
dN
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( t ) Versus T ime f o r B = l
4 - 6 . Amp l i t ud e Re s p onse o f Underc o up l e d Two - P o l e F i l ter
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4 - 7 . Phas e Response of Underc oupl e d Two - Po l e F i l t e r . . . .
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4 - 8 . D i s c r i m i na t or Out p u t v
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t he FM Response of a Two - Po l e Unde rc oup l eJ F i l t e r f o r B = 1 2 a nd P = 2 . . . . . . . . . . . . . . . . . . . . . . . .
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L I ST OF FIGURES
F I GUHE
4-9 .
PAGE
Schema t ic D ia gram of But t erworth Two - P o l e F i l t e r .
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4 - 1 0 . Amp l i t ud e Respons e .. of But t e rw o r t h Two-Pole F i l ­
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ter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 - 1 1 . Phase Resp onse of But t erworth T w o - P o l e F i l t e r
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4 - 1 2 . Schema t i c D iagram of Tchebys h e v Two - P o l e F i l t e r
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4 - 1 3 . Amp l i t ud e Response of Tch e byshev Two-Pole F i l t e r . 101
4 - 1 4 . Phas e Re spons e o f T c h e byshev Two-Pole F i l t e r
4 - 1 5 . D i s c r im ina t or Output v
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( t) V e rs us T ime Due t o
the FH Resp onse o f a Tcheby s h e v Two-Po l e F i l t e r f or B = 8 a n d P = 0 . 8
4 - 1 6 . D i s c r im i na t or Out put v
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( t) Ve r s us T ime D ue t o
the FM Resp ons e o f a Tchebys h e v 'fwo-Pole F i 1 =
t e r. f or B
1 0 a nd P = 1
4 - 1 7 . D i s c r im i na t or Output v
dN
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•
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•
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•
•
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•
•
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( t) Ve rsus T im� Due t o
t h e FM Respons e o f a T c h e by s h e v Two-P o l e F i l t e r f o r B = 1 2 and P = 1 . 2
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4 - 1 8 . Pe rcent D is t ort i on Ve rsus Peak- t o - Pe a k D e v ia t i on
t o Bandwi dth Ra t i o f o r f
A- 1 .
= 0 . 005 Hz
m
•
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112
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122
ECAP S c h e ma t i c D ia gram o f But t e rworth Two-Pol e
Filter
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xi
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LI ST OF FI GUHES
FIGURE
E - 1 . Samp l i ng of
;
s 1n
0
2
nx
PAGE
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xii
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136
ABSTRACT
THE RESPONSE OF A LI NJ.�H NETWOHK TO A FM WAVE
"'
by
:2
J ohn Louis Ovn i c k , J r .
Mas t e r of S c i ence in Eng i nee ring
June , 1 9 6 9
t-/
Se vera l t e chniques f or ca l culat ing t he r e s p ons e o f a ne t w ork t o a var iab l e - fre quency e x c i ta t i on a re examine d and t h e ir
s uitab i l i ty f or c omput e r-aut oma t e d p r o bl em s o lv ing a re d is c uss e d .
The dynam i c or ope ra t i ona l approach is sh own t o have
s e veral d i sadvantages t h a t seve rely l im i t its us e f ulness and
i t s f orm is f o und t o be d if f i c ul t t o impl ement int o a general
p r o gram t hat may be use d ori a c omput e r t o s o l ve a l arge var i e ty
o f probl ews .
The Fou r i e r or spe ctra l approach t o arr iv ing a t a s o l ut i on
is s h own t o p ossess s ev e ra l d e s ira b l e f e a t ures t ha t l en d i t
idea l ly t o c omput e r imp l e menta t i on .
A c omputer p r o gram i s
d e ve l op e d f r om t he F our i e r approa c h and i s used t o invest iga t e
t h e F M r e s p onse of a s ingl e -ryo]e fil t e r a n d t hree c ommon types
of two-p o l e f i l t e rs .
The d i s t ort i on in t h e FM wa ve p r o duced b y
t h e s e f i l t ers i s ca l cu l at�d f o r a w i d e r�nge o f m od u l a t i on ind e xe s .
.
The q ua s i-sta tiona ry approx ima t i on , a s pe c i a l c a s e of
t h e dynami c a pproach ,
is c ompared t o the F our i e r s o l u t i on f or
x iii
t h e case o f t h e s ingl e - p o l e f i l t e r .
The e ff e c ts of s id e band
c l i p p ing a re a l s o p r e s e nt e d .
A gen e ral t e chni qu e f o r c omput e r -a i d e d pr oblem s o l ving i s
d e s c r ib e d a n d t he r e q u i r e d programm i n g i s prese nted.
,
Th e
a pproa c h i s e xpla ined i n s u ff i c i ent d e ta i l t o a l l ow i t s u s e by
a nyone wh o is fam i l i ar w i t h t h e use o f a c ompu t e r and has a
fa i r kn owl e d ge of p r o gramm i n g .
xiv
l
1
1.1
•
I NTRODUCTI ON
THE PROBLEM WITH FREQUENCY r.IODULATI ON
Ove r t h e pas t t h i rty years t h e
ttse
of f r e quen cy-modulat i on
h a s b e c ome w i de - spread a s one means o f t ransm i t t i n g i nf orma t i on
b ot h over w i re and by ra d i o wav e s .
A l t h o u gh it i s only one of
many p o s s i b l e type s o f m o d ula t i on t e c hn iqu e s t ha t may b e us e d,
i t i s part i c u l a rly we l l s u i t e d f o r many a ppl i cat i ons where w i d e­
ban d n o i s e impr ovement i s d e s i r e d w i t h o ut res ort i n g t o t h e c om­
p l e x and e xp e ns ive e n c o d e r and de c od e r c ir c u i t ry t ha t i s re­
qui re d f or more e x o t i c m od u l a t i on s c h e me s .
Hence freque ncy
m o d ulat i on e nj oys w i d e usa ge in a re a s s u ch as m ob i l e c ommun ica ­
t i ons and a i rborne t e l em e t ry t o m e nt i on o nly t w o .
Desp i t e i t s wide sprt:ad u s e in the f i e l d o f c ommuni c a t i ons ,
ma ny e n g i n e e rs fam i l ia r w i t h FM from w o rk i ng w i t h it i n a
p ra c t i c a l s e nse und e rs t and l i t t l e of the p r o c e s s fr om
the ore t i c a l v i ewpoint .
a
st r i c t ly
The pr ima ry reas o n f or t h e c oncept ua l
d i f f icul ty i s t h a t t h e fre que ncy of a FM wave i s itse.if a func­
t i on of t ime a nd not a c ons tant .
The re f ore t h e t o ols of ord i n­
a ry AC c i rcu i t ana lys i s break d o\vn and t h e e n g i neer a t once
f i n d s h i ms e l f on unfami liar gr o und .
T o ove rc ome t h e pr oblem impose d by t h e var ia b l e fre quency
of t h e Ft-1 wa ve it is requ i r e d that one abandon c onvent i ona l AC
c i r c u i t a na ly s i s and re t urn t o f irst p r i n c ipl e s; name ly , t h e
2
i nt e gra-d i ff e r e nt i a l e qua t i ons o f t h e c ir c u i t t o b e a na lyz e d .
I f t he m o d ul a t i n g i n f orma t i on happens t o b e per i od i c , a grea t
s imp l i f i ca t i on o f t he p r ob l em may b e rea l iz e d by expa n d i n g t he
FM wa ve int o a n i nf i n i t e s e r i e s of s inus o idal wa ves o f c ons tant
f r e q uency .
Ea c h t e rm of t he s e r i e s may t h e n b e op e ra t e d upo n
u s i n g ord i na ry A C c ir c u i t a na ly s i s .
T h e r e s p onse o f a ne twork
t o a FM e x c i ta t i on is t heh f o und as t h e s um of t h e mod i f i e d
t e rms of t h e s e r i e s .
T h i s approa c h t o t h e p r o b l e m i s r e f erre d
t o a s t h e Four i e r or s pe c t ra l me t h od o f a na lys i s a nd e x t e ns i ve
u s e w i l l b e made o f t h is t e ch n i que t o s o l ve s e ve ra l s p e c i f i c
probl ems d e a l i n g w i t h FM d i s t or t i on i n Cha p t e r 4 .
A l though the
F o ur i e r me t h o d o f a na lys i s has been we l l known for many years ,
n o d i rect a pp l i c a t i on o f t he general f o rm of t he s ol ut i on t o
s pe c i f i c c irc u i t p r ob l ems i s t o b e f ound i n t he l it e ra t ure .
Anot h e r approach that i s m o re genera l but a t t h e same t ime
more d i f f i c ult t o a p p l y i s t he o p e ra t i o na l or dynami c m e t h o d of
a na lys i s .
Th i s approa c h empl oys c onvol ut i on o r t rans f orm t e ch ­
n i que s t o a rr i ve a t t h e ne twork r e s p o n s e and many s im i la r
t h e or i e s a r e d is c us s e d i n t h e l it e ra t ure .
Un f o r t una t e ly they
a l l suffer f r om a c ommon p r obl e m t ha t s e r i ously i mpa i rs t h e i r
p ra c t i ca l
us e fu l l ne s s : name ly , . t h e s olut i on t a k e s on t h e f o rm
o f a n asymp t ot i c s e r i e s wh o s e c onvergen c e propert i e s mu�t be
clos e ly e xam i n e d f o r e a ch pa rt i c ul a r p r ob l em in o rd e r t o d e t er­
m i ne t h e e rr o r i ncurred by a pproxima t i ng t h e s e r i e s w i t h only a
f e w of i t s l ea d ing t e rms .
Few hardwa r e o r i e nta t e d e ngin e e rs
3
w i t h the n e e d t o s o l v e spe c i f i c £11.1 p r obl ems have e ithe r t h e
;
t ime o r t h e a b i l i ty t o c op e w i t h t h e ma t hema t i c a l c ompl ex i t i es
o f t h e dynam i c t e ch n i qu e o f a na lys is .
C onse quent ly the f ina l
r e s ort is t o e xp e r ime n t a t i on wh i c h gen e ra l ly resul t s in a
I
great er expense o f t ime a n d money and e nds w i t h l i t t l e add i t i ona l understa nd i ng of t he pr obl em .
Hence t he r e �x ists a n e e d f or a n e asy- t o-apply m e t h od t ha t
may b e use d t o s o lve t h e t y p e o f probl ems t ha t a r ise e ve ry day
i n the d es i gn o f FM c ommun i ca t i on e qu i pment .
I t i s b e l i e ve d
t ha t t h e a p p r oa ch employed i n t hi s paper w i l l ful f i l l t ha t
need .
' l. 2
HEVIEW OF fi-t T HIWRY
Be fore i nvest i gRt ing t h e r esp onse o f a network t o a fre -
quenc .y -mod u la t ed exc i ta t i on , a bri e f re v i ew o f elementary lt�i
t h e ory i s i n o rd e r .
C ons i d e r t he s i nus o i da l wave given by
v(t)
=
0
A cos ( w t
+
a)
where A i s t he p e a k ampl i t ude o f t he wave , w
( 1.2 - 1 )
0
f r e quency , a n d a i s a n a rb i t rary phase angl e .
is t h e a ngul a r
The t ot a l quant i -
t y i ns i d e t h e bra cke t s i s de f in e d a s the instanta ne ous phase
�i ( t )
of t h e wave .
The i nstantane ous frequ en cy w ( t ) of the
i
wave is d e f i n e d as t h e t ime - ra t e - of - c hange of t he i nstantane ous
phase .
Thus we have
4
w. ( t ) =
�
and
t
�.� ( t )
ti
d
dt
=
�0 w. ( t ) d t
1
(t )
u
'
-
dt
( 1 . 2- 2 )
(w t + a )
=
0
( 1 . 2- 3 )
w-- o
s o t ha t t h e i ns tantane ous fre que ncy i s a c onstant e qua l t o w
0
a s expe ct ed .
Now s uppose t ha t t h e phas e a n gl e a i s made t o va ry w i t h
r e s p e c t t o t ime i n s ome manner a c c or d i n g t o a m o d ul a t i ng funct i on f ( t ) .
The f o rm o f
longe r be a c onstant .
�i ( t )
w i l l c ha n ge and w ( t ) wi l l no
i
The r e s ul tant wa v e is sa i d t o b e "angl e -
modula t e d " and may b e expre s s e d ge n e ra l ly as
v(t
) = A cOS
[
w t
0
+
a(t )
J.
{ 1. 2- 4 )
Two f orms of a ngl e -m o d ula t i on a re c ommonly e n c ount ered:
fre quency m o d ulat i on and p ha s e modulat i on .
The p r i ma ry d i s t i nc-
t i on be t we e n the t w o is the manner in which the instantaneous
pha s e
�i ( t )
o f t he wave is ma d e t o vary w i t h respe c t t o t ime .
I n fre quency modulat ion t h e ins tantane o us fre quency i s made t o
vary i n a manne r proport i ona l t o t h e m o d u la t i n g f u nc t i on f ( t ) .
I n pha s e m od u la t i on t h e i n stantane ous pha s e is ma de
to
a manner proport i ona l t o t h e modul a t i n g func t i on f ( t ) .
vary i n
H e nce ,
we-obt a i n f or t h e two c a s e s :
( 1 ) Fre que ncy Modula t i on
w (t )
i
�
=-
w
0
+
k f(t)
F
w1 ( t ) d t = w t + k
F
0
( 1 . 2 -5 )
�
0
f ( t ) dt
( 1 . 2- 6)
5
( 2) Phas e M odul at i on
w i ( t) = w
t1 ( t)
where k
F
t
d
f ( t)
+ k
p dt
0
� w 1 { t) d t
=
( 1 .2-7)
= w t + k f ( t)
p
0
0
a n d k a re a rb i t ra ry c onstant s .
p
. ( 1 . 2-8)
From Equat i ons ( 1 . 2-6)
and ( 1 . 2-8) i t i s obs e rve� t ha t t here is n o f undam e n t a l d i f ference b e t we e n pha s e a nd f � e que ncy modulat i on s in c e t h e i ns tant a n e ous pha s e i n each c a s e may b e expre s s e d a s
'I' 1.
�. ( t ) = w
0
t
+ a ( t)
a s was d on e in Equat i on ( 1 . 2 - 4) .
c hange the na t ure o f the wave .
(1 . 2 - 9)
How a ( t) c omes a b out d o e s not
H e nc e , the re s p onse o f a ne twork
t o a pha s e -mod u l a t e d exc i ta t i on i s e n t i r e ly t h e same a s i t s
r e s ponse t o a f r e quency-mod u l a t e d exc � ta t i on p r ov i d e d · that
k f ( t)
F 1
t
=
k
P
\ f 2 ( t) d t
( 1 . 2- 1 0)
0
where f ( t) a n d f ( t) a re n o t equa l .
2
1
Thus t h e analys i s per­
f orme d i n Cha p t e r 2 of t h i s pape r app l i e s e qua l ly a s we l l t o
b o t h fre que ncy a n d phas e modul a t i on a l t h o ugh only fre quency
m o d u la t i on w i l l b e ment i on e d .
·-·
T o pr oc e e d w i t h t h e re vi�w of FM t h e ori l�i us c ons i d e r the
case where f ( t) i s a s i nus o i dal modul a t i n g func t i on g i ve n by
f ( t)
wh e re w
m
=
c os w t
m
i s ca l l e d t h e modula t ing fre quency .
( 1 . 2- l l)
From Equat i ons
.
6
( 1 . 2- 4 ) ,
( 1 . 2 - 5 ) , a nd ( 1 . 2 - 6 ) we obta i n
w.
1
cj>i
( t ).
{t)
w
�
+
0
( 1 . 2- 1 2 )
c os w t ,
m
k
F
w t + - sin w t '
m
0
w
m
k
F
A c os ( w t +- s in w t ) .
w
0
m
m
{ 1. 2 -1 3 )
==
==
v{t )
and
==
{ 1 . 2- 1 4 )
From E qua t i on ( 1 . 2- 1 4 ) i t i s obse rved tha t the Fr-1 wave has a
c onstant pea k ampl i t ude of
a s d id the unmodula t e d wave .
A
S in c e
c o s w t var i e s b e t w e e n + l a nd - 1 i t i s n o t e d from Equa t i on
m
( 1 . 2 - 1 2 ) that the i ns t a n t a n e ous f r e qu e n cy va r i e s f r om w
to w +
0
�·
Henc e l e t us re d e f i n e k
F
0
- k
F
for the cas e o f s i nus o i da l
: f r e que ncy m odula t i on a s w d and ca l l i t the peak fre quency d e ­
viat i on .
Thus
k
F
and
v( t )
==
==
w
A c os ( w t
0
f or the case o f s inus o ida l �1.
{ 1 . 2- 1 5 )
d
w
d
sin w t )
m
w
m
+ -
The fac t or
( 1 . 2- 1 6 )
w
d /w i s c a l l e d the
m
m o d u la t i on i n d e x or d e v ia t i on rat i o a n d i s c ommonly d e n o t e d by
B,
thus
( 1 . 2- 1 7 )
B =
S ome a uthors re f e r t o
w
d /w
0
a s the m o dul a t i on index a n d care
must be exe r c i s e d when r ea d i n g t h e l i t e ra t ure t o b e c e rta i n o f
.
the qua n t ity b e in g d i s c uss ed .
I n t h i s pa p e r B w i l l be re ferre d
t o a s the m o d ulat i on i n d e x a n d i s d e f i n e d a s i n Equa t i on
7
( 1 . 2- 1 7 ) .
Now t ha t we have a n e xpress i on f or t h e FM wave, 1 · e t us
inve s t i ga t e t h e na t ure of i t s fre quency s p e c t rum .
I t is well
kn own tha t a s inus o i da l f unc t i on o f c onstant angular fre quency
is repre s e nt e d as a s i ngle spe c t ra l l ine in the p os i t i v e fre quency d oma i n .
T o d e t e rm ine t h e spe c t ra l repre s e nt a t i on of a
FM wave l e t us f irst wr i t e Equa t i on ( 1 . 2- 1 6 )
p onent i a l f orm a s
v ( t ) = Re
or
v ( t ) = Re
[
[
\
1e
j( w t + B sin w t
o
m
. B sin
w t
A e J o eJ
\V
.
m
t
From the Bes s e l fun c t i on i dent ity
e
+00
ja s i n x
we imme d ia t e ly obta i n
v ( t ) = Re
[
or
n=-oo
J (a)e
n
in c omplex ex-
�
J
( 1 . 2- 1 8 )
( 1 . 2- 1 9)
•
J·nx
( 1 . 2-20 )
w t
AeJ o
.
( 1 . 2- 2 1 )
+00
v( t )
( 1 . 2 -22 )
n= -oo
wh i ch be c omes
v(t ) = A
+00
L
n= -ro
J ( B ) cos
n
[(
w
0
+ nw ) t
m
J
.
( 1 . 2 -23 )
T h us f r om Equa t i on ( 1 . 2 -23 ) t h e �1 s p e c t r um i s o b s e rve d t o c onl..
'
s i s t o f a n inf i n i t e number of s i d eba nds o f c onstant fre quency
,.
. . ---- · ·---� ------
8
c e n t ered a round t h e carr i e r fre quency w
mul t iples o f w
m
f r om w
o
a nd spa c ed a t i nt e gral
o
Th is s i t uat i o n i s contra s t e d w i t h
•
' t ha t of amp l i t ude modul a t i on where only two s i de ba nds are prese nt f o r t h e cas e o f s inus o ida l m odula t i on .
A l t h ough i n t he ory t h e fl-1 spe c t rum c onta i ns a n i n f i n i t e
numbe r o f s i d e bands, i n pra c t i c e t h e ma gn i t ud e o f t he s i de bands
be c omes ne gl i gi b l e f or f �eque n c i e s s omewhat b eyond :!: w from
d
t h e carr i e r w
0
•
[
T h i s p r o p e rty i s due t o t he nat ure o f t h e
Be s s e l f un c t ions J ( B ) .
n
Fi gure 1 - l i l l us t ra t es a p ort i on o f the
FM spe c t rum f or t he c a s e whe re B.= 3 .
I t should b e empha s i zed
a t t h is p o i nt that F i gure l - 1 gives a t r ue p i c t ure o f the fl-1
s pe c t rum .
I t is a s i t w o u l d a ppear i n a s pe c t rum a na l yz e r d i s -
p l a y o r by p l o t t i n g t he s p e c t rum p o i nt by p o int us ing a t unab l e
re c e iver w i t h e n o u gh s e l e c t i v i t y t o r e s o l ve t h e s i deband s .
f r e quency spe c t rum is t h us d i s c r e t e .
The
The f orm o f Equa t i on
( 1 . 2- 1 2 ) ha s o f t e n l e d ma ny t o t h e f a l s e conc l us ion that t h e
fre quency m o v e s ba c k a n d f or t h be t we e n w
0
- w
d
and w
0
+
w
d
in a
c ont inuous fash i on .
I n t he t ime d oma i n v ( t ) may be t h o ught o f as t h e p r oje c t i on
on i:.he rea l ax is o f t he s um o f a n i n f i n i t e numbe r of r o ta t in g
pha s o rs .
Th i s con c e p t i s i l l us t ra t ed i n F i gure l - 2 f o r o n l y a
f e w o f the phas ors .
F i na l l y , a word about t he p ower c on t a ined i n a FM wave i s
i n order .
From Equa t i on (1 . 2- 2 3 ) we knew that t he fl-1 wave i s
c omposed o f a n i n f i n i t e n umb e r of s i n us o ida l c ompone nt s .
The
a v e ra ge p ower in t he wave is just the s um of t h e pow e r in t he
i
j
(
···�···
Fi gure l - 1 .
F!-1 Spe c t rum f o r B = 3 .
10
i ma g i nary
axis
(
w t + ( n- l ) w t
m
o
!
AJ
(B)
n+1
w t + ( n+ l ) w t
m
o
1
._--------------------------��------�:>�
·
v(t )
Fi gure 1-2.
Pha sor Repre s e n t at i on of v ( t ) .
real
axis
i nd i v i d ua l c omponent s .
=
p
av
+00
2:
n=-oo
T h us ,
From t h e Be s s e l i d e n t i t y
we obta i n
2
A
J ( B) =
2 n
2
A
2
2
+00
2:
n=-oo
2
J (B)
n
•
+00
n=-oo
=
p
av
wh i c h is jus t t he p ow e r i n t he unm o d u l a t e d carri e r .
;
j
( 1.2-24 )
( 1 . 2-25 )
( 1 . 2- 2 6 )
.
12
2.
2.1
1''0 ID-1ULATI ON OF THE PROBLEM
HI STORI CAL REVI EW
I n t he a na ly s i s o f t h e resp onse of a ne twork t o a frequen-
cy-modulat e d exc i ta t i on , two bas i c a pp r oa ch e s are t o be f ound
in t h e l it e ra t ure:
( 1 ) t he F o ur ie r o r s pe c t ra l a pp r oa c h ; and
( 2 ) t he dyna m i c or opera t i ona l ca l cu l us a p pr oa ch .
B o t h me t h ods
are c ommon i n that t he n e twork t rans fer f unc t i on i s mad e t o
o p e ra t e upon t h e FM exc i t a t i on t o p r od u c e the re sul tant output .
The Four i e r or spe c t ra l approa c h was f i rst emp l oy e d by
Roder
1
i n 1 937 t o s ol ve se vera l spe c i f i c probl e ms a n d was l a t e r
genera l i z e d b y S t umpers?
M e d hurst
3
and A s sad our ian
4
a pp l i ed t h e
t e ch n i que f o r t he c a s e o f sma l l - or d e r d ist ort i on .
1
H . Rod e r , "Ef f e c t s of Tune d C i rc u i t s Upon a FrequencyY.I o dulated S i gna l , " Pro c e e d i ngs IP.E, 25 , No . 1 2 ( De c embe r , 1 937 ) ,
1 6 1 7- 1 647 .
2
F . L . H.M . St umpers , " D i s t ort i on o f Fre quency-Mo d u l a t e d S i g­
na l s in E l e c t r i cal Ne tworks , 11 Commun i ca t i on News _ , ,g , ( Ap r i l ,
1 948 ) , 82-8 3 .
R.G . M e d h urst , " Ha rmon i c D i s t ort i on o f FH Waves by Linear
-Ne tw orks , 1 1 Pr o c e e d ing-s lEE ( London) , 101 , Pt . 3 , Pap e r 1650
( Ma� 1 954 ) , 1 7 1 - 1 8 1 .
3
4
F . Assad ouria n , " D i s t ort ion o f a Fre quency-Mod u l a t e d S ig­
na l by Sma l l Loss a nd Pha s e Va r i a t i ons , " Proce e d i ngs IHE, 40,
( Fe bruary, 1 952) , 1 7 2 - 1 7 6 .
13
I n t he s p e c t ra l ·me t h o d t h e FM s i gna l i s f i rs t d e c ompos e d
int o its s i nus o i da l c omponent s via t h e B e s s e l func t i ons .
Eac h
c omp onent o r s i d eband is t h e n m od i f i e d b y t he va l ue of t he ne t work trans f e r f unc t i on a t t he s i de band f r e que ncy .
The output of
t he ne twork is t h e n t he ve c t or s um o f t he m od i f i e d s i debands .
I t i s import a n t t o n ote t ha t t h is i s a s t eady - s t a t e s o l ut i on .
T h e Bess e l c omponents a re ,' c ont i nuous wave ( cw ) s i gna l s a n d t he
{
n e t w ork t ra n s f e r f unc t i on i s e va l uat e d us ing c onven t i ona l AC
c i r c u i t ana l ys i s .
t
'
The s p e c t ra l a pproach was n o t w id e l y us e d when i t was f irst
d e ve l ope d due to the e n o rm o us a mount o f c omputa t i on r e qui red to
a rr i ve a t a s o l ut i on if t he number o f s i gn i f i cant s i d e bands wa s
large .
Thus , m o re a t t ent i on was i n i t ia l l y given t o t he dyna m i c
o r opera t i ona l m e t h o d o f s ol ut i on .
A t p r e s e nt , w i t h t h e gene ra l
a va i lab i l i ty o f h igh -s p e e d d igital c omput ers , t h e s p e c tral
a pp r oa c h b e c omes more a t t ra c t ive s in c e i t y i e l ds an exa c t s olut i on t o a ny degre e o f a c c ura cy depend i n g upon t h e number of
s i d e bands c ons idere d .
The dynam i c m e t h od was f irst d e v e l oped by Cars on and F'ry
and l a t er b y Van d e r P o l
6
a nd St umpers .
7
5
S i nce t hat t ime many
5
J . R . Ca rson and T . C . Fry , "Va r iahle -Fre que n cy E l e c t r i c
C i rc uit The ory , " B e l l Svs t e ms T e ch n i crtl Journa l , 1 6 , ( Oc t ob e r ,
1 937 ) , 5 1 3 - 5 40 .
B . Van d e r Pol, "The Fundament a l Princ i p l es of Fre quency
M o du l a t i on , " J ournal lEE (Lond on ) , 93 , Pt . 3 ( Hay , l 9 46 ) , 1 56 - l58 .
6
7
St ump e rs , op. c i t . , 8 3 - 92 .
14
a r t i c l e s have b e e n p ub l i s h e d us ing s im i l a r a pproache s .
8
The f undame nta l c on c e pt b e h ind the dynami c m e t h o d is that
a n e t work , b y v i rt ue of its e n e r�y s t o rage e l ements , p os s e s s e s
a certa in i n e r t ia a n d cannot f o l l ow i ns ta ntane o us c ha nge s i n
t h e fre qu e n c y o f a n e xci t a t i on.
C ons e qu e nt l y , d i s t o rt i on i s
p r o d u c e d i n fll modula t i on s i nc e t h e r e s p onse o f the n e twork
w i l l not in general be a f a i t hf u l reprod u c t i on of t h e e x c i t a t .lOll.
9
Th i s a pp r oa ch t o t h e prob l em d o e s n o t re quire the d e c ompof
I
s i t i on o f t he exc i ta t i on int o its s i nus o i d al c omp on e n t s .
The
s o l ut i on is genera l ly obta i n e d a s t he c onvolut i on o f the exc i tat i on and the n e twork ' s un i t impu l s e resp ons e .
The f orm o f t he
s o l ut i on is genera l l y a n i n f i n i t e s e r i e s w i t h t h e s o - c a l l e d
" quas i -sta t i ona ry " r e s p ons e a s t h e l ead i ng t e rms .
T h e qua s i -
s t a t i onary r e s p onse r e p r e s e n t s a part o f t h e s ol u t i on that can
8
A . S. G l a dw i n , "The Dist ort i on o f Frequency-Mo dula t ed Waves
by Transm i s s i on Net works , " Pr o c e e d ings I HE , 35 , ( De c ember,
1 94 7 ) , 1 436-1 445 ; a n d A .A. Gerla c h , " D i s t orti on-Bandpass Cons i d ­
e ra t i ons i n A n gu l a r Modula t i on , " Pr o c e e d ings I HE , 38 , ( Oct obe r ,
1 950 ) , 1203-1 2 07 ; a n d J.J. Hupert , "Normal i zed Pha ;w- and Ga in
D e r i va t ives as a n A id in Eva l ua t i on o f FM D is t ort i on , " Pr o c e ed,­
i ngs IRE, 4 2 , ( Fe b ruary , 1 954 ) , 4 3 8 - 4 46; and E. J . Ha ghda dy ,
"The ory o f Low-D i s t ort i on Reproduc t i on o f FM S i gna l s i n Linear
Sys t e ms , " I HI<� Tra nsa c t i ons on C ir c u i t Th e ory , CT-5 , ( Se p t e mb er ,
1 958 ) , 202 - 21 4 ; and D . D. We i n e r a nd B.J. Le on , "The Qua s i­
Sta t i ona ry Re s p onse o f L i n e a r Sys t ems t o M odula t e d Wa vi.!f or-'ms , "
Pr oc e e d i ngs I EE E , 5 3 , No. 6 ( June , 1 965 ) , 564-575 ; a n d E .
B e d r o s ian and S . O . R i c e , "D i s t ort i on a n d Cros s t a l k of Linearly
F i l t e re d , A ngl e -M o d u la t e d S i gna ls , " Pr o c e e d i ngs I EEE , 56 , No . 1
-( Ja nuary , 1 968 ) , 2 - 13 ; a n d J .W. Bay l e s s a n d S . C . Gupta , "A New
Approa ch t o fll D i s t orti on , " IEEE Transa c t i ons on C ommunica t i on
: · Te chnol ogy , C OM- 1 6 , No. 2 ( Apr i l , 1 968 ) , 2 6 1 - 26 7 .
9
Ba gh d a d y , op . c 1. t
• ,
2 0?.
_
15
b e obta i n e d f r om c onve nt i ona l s t eady - s t a t e s inus oida l c ir c u i t
t h e ory by s ub s tit ut ing t h e var ia b l e i n s t a ntane ous fre quency f o r
t h e a ssume d c ons tant f r e quency .
T h e rema i n ing t e rms o f t he
s e r i e s mod i fy t h e qua s i - s t a t i onary response a nd a d d t o t h e
d i s t o rt i on c onta i n e d i n t h e qua s i - s t a t i onary t e rms .
These
s e r i e s a r e a sympt ot i c i n na t ur e and t h e i r c onve rgent propert i e s
mus t be e xa m ine d f o r e a c h part i c ular probl e m .
10
The dyna m i c approa c h t o t h e prob l em ha s one advantage ove r
t h e s p e c t ra l approach i n tha t much l es s c omputa t i on is re quire d
t o a rr i ve a t a s ol ut i on , part i c ularly i f only o n e or t wo t e rms
of t h e s e r i e s expans i on a r e s i gn i f i cant .
I t has t h e d isadvan-
: t a g e , h owev e r , t hat i t i s more d i f f i c u l t t o emp l oy s i nc e the
c onvergence p r o p e rt ies of the s e r i e s are o f t e n obs c ur e .
At
b e s t , t h e dyna m i c m e t h od y ie l d s a n appr oximat e s o l ut i on .
Its
ul t ima t e a c c ura cy i s l im i t e d by t he na ture o f t he pa r t i c ul a r
a symp t ot i c s e r i e s expans i on .
11
2 . 2 T H E I',OURIER OR S PE CT RA L APPHOACH
The f ol l ow i ng d�ve l opment of t h e F o u r i e r m e t h o d o f a na lys is
para ll e l s t h e a pproach orie;ina l ly pre s ent e d by S t umpers .
10
12
The
H . E . Rowe , " D i s t ort i on o f Angl e - Hod ula t e d Wa ve s by Linear
Ne tworks , " I RE Transa c t i ons on Circuit T h e or� , CT- 9 , ( Sept embe r ,
l 9 6 2 ), 289 .
ll
12
Ba gh d a dy , 02. c i t . , 206 .
S t ump e rs , op . c i t . , 8 2 - 8 3 .
16
o r i g i na l s i gnal i s d e c ompos e d int o i t s Four i e r c omp on e n t s wh ich
are t h e n p r o c e s s e d a c c o r ding to c o nv e n t i ona l AC c i r c u i t t he ory .
T h e response o f t h e ne twork i s t h e n c omput e d as t h e r e s ul tant
s um o f the Fo u r i e r c omponents a ft er h a v i n g pa s s e d t h rough t h e
n e twork .
T h e d is c us s i on w i l l b e r e s t r i c t e d t o t he c a s e o f
s i ngl e - t one s inus o i dal f r e quency m o d u la t i on .
I n a d d i t i on , t h e
I
n e twork i s a s s um e d t o b e l i near a n d t ime indepe ndent s o t ha t
t h e princ i p l e of sup e rp os i t i on i s va l i d .
By l inear i t is meant
t ha t the c on s t a n t s of t h e ne twork a re n o t f unct i ons of the
a p p l i e d v o l t age o r c urrent .
H owe ver the phase o f the n e t work is
in genera l n o n - l inear w i t h r e sp e c t t o fre que ncy .
Cons i d e r a syst em c ons i s t ing o f a c urre nt s o urc e c onn e c t e d
t o a ne twork w i t h a t ra n s f e r impe dance Z ( j w ) .
Furt h e r c ons i de r
t ha t t he c urre nt s o urc e i s fre quency-mod ula t e d a n d t ha t i ( t )
may b e expre s s e d as
i (t )
==
0
I e
j (w t + B s in w t )
o
m
( 2 . 2- 1 )
where B i s the i n d e x o f m o d u la t i on a nd i s given by
B =
whe re w
d
w
d/ w
m
i s t h e peak ca r r i e r d e via t i on and
( 2 . 2- 2 )
w
m
i s t h e m o d ula t ing
f r e quen cy .
Our goal is t6 f i nd a n express i on f o r t h e n e t work response
'
v ( t ) and s ub s e quent ly for the instantane ous phas e a n gl e o f v ( t ) .
T o proc e e d w e may expand i ( t ) i nt o an i n f i n i t e s e r i e s via t he
B e s s e l fun c t i ons a s f o l l ows
17
i ( t) = I
0
+00
( 2 . 2- 3 )
n=-oo
N o w , l e t us c ons i d e r t he k ' t h s i de band g i v e n by
( 2 . 2- 4 )
T h e c ompo�ent o f v ( t ) p r o d uc e d by t h e k ' t h s id e band i s t he n
( 2 . 2-5 )
whe r e
= w
0
+ kw
m
( 2 . 2-6 )
•
' , Thus , w e have
( 2 . 2- 7 )
The r e f ore , s i n c e t he princ ipl e o f sup e rp o s i t i on i s va l i d , the
r e s p onse o f t h e n e twork is s imply the s um of t h e c on t r ibut i ons
of a l l t h e s i de bands .
Thus we may wri t e
+00
( 2.2-8 )
n=-oo
or
_ _
v(t )
I
o
+ 00
L
n = - oo
Jn ( B ) e
j (w
o
+ nw ) t
m z
n
[j ( w o
J]
+ nw
n
•
( 2 . 2- 9 )-
This i s t h e s o l ution that we a r e s e ek i n g a l th ough i t d oe s not
i
q u i t e s u i t our purpose i n i t s p r e s e nt f o rm .
We d e s i r e a n ex-
p re s s i on that gives the ma gn i t ud e and the phas e a ngl e as s e pa ra t e t e rms .
T o d e ve l op e t h e d e s i r e d e xpre ss i on l e t u s f i rs t
18
n
n
s e para t e Z ( ,jw ) into its real a n d ima ginary part s a s f o l l ows
·
·
Z ( j w ) = A ( j w ) + j X ( ,jw )
·
n
n
n
n
( 2 . 2 -10 )
where
w
n
= w
o
+ nw
m
( 2 . 2 - 1 1)
and A ( j w ) a n d X ( j w ) a r� r e a l functions o f j w
n
n
n
'
n
n
n
n
n
n
•
T h e ma gnit ude
P ( j w ) and t h e p ha s e e ( j w ) o f Z ( j w ) a re t h e n give n by
and
P ( jw ) =
n
n
[
2
2
A ( jw ) + x ( j w
n
n
n
n
)J
�
( 2 . 2- 1 2 )
( 2 . 2-13 )
s o t ha t w e ma y e xp r e s s
Z
n
n
( j w ) in t h e f o l l owing form
( 2 . 2- 1 4 )
1
Using Equa tions ( 2 . 2 - 1 4 ) and ( 2 . 2 - 7 ) we may n ow expre s s t h e k ' th
sideband a s f o l l ows
( 2 . 2-15 )
( 2 . 2- 1 6 )
We may a ga in obtain t h e r e s pons e o f t h e n e t work t o i ( t ) by
s umming up t h e c on t ribut i ons o f a l l t he sid e bands.
Thus ,
19
v(t ) = I
0
( 2 . 2- 1 7 )
n = -oo
; · From Equat i on ( 2 . 2- 1 7 ) we s e e t ha t v ( t ) may be v i ew e d as a
r o t a t ing v e c t o r w i t h a va r ia b l e ma gn i t ud e a n d a var i ab l e angul ar
fre q u ency.
Thus , we may s imply wr i t e
( 2 . 2- 1 8 )
where
+ 00
R(t ) =
+
10
[
f[ L
n=-oo
J ( B ) Pn ( j w ) s in ( w t + Q ( j w ) )
n
n
n
n
n
+00
L
n=-oo
J ( B ) P ( j w ) c os ( w t
n
n
n
n
+
e n ( j wn ) )
J
2
and
w t
0
1
]
y,
2
( 2.2- 1 9 )
+ Q(t ) = w t
0
+00
+ tan
'
� J n ( B ) Pn ( j wn ) s in
[ nwmt
' J ( B ) P ( .iw ) c os
n
n
n
n=-oo
[ nwmt
- 1 n= -ro
+00
�
+
e ( j w )1
n
+ e ( jw
n
nj
( 2.2-20 )
)1
n:J
T h e rela t i on b e t w e e n Equa t i on ( 2 . 2- 1 7) a nd ( 2 . 2 - 1 8 ) i s s h own in
F i gure 2 - 1 .
I t can be s e e n from Equa t i on ( 2 . 2 - 1 8 ) t ha t t h e
o u t p ut from t he n e t work w i l l in genera l b e a " hybr i d " wave .
That i s t o say t h a t t h e wa ve i s b oth amp l i t ude and pha s e m o d ul a t e d w i t h harmo n i c s o f w .
m
20
R(t )
0
r e f e rence
axis
F i gure 2 - 1 .
Summa t i on o f S id e band Ve c t ors .
21
T o remove unwant e d amp l i t u d e modulat i on a fl.l sys t e m empl oys
a d e v i c e cal l e d a " l im i t e r " t ha t opera t e s upon t h e m od · u la t e d
c a rr i e r j us t p r i or t o d e t e c t i on .
I d e a l l y , a l im i t e r produces
a s inus o i da l output o f c onstant amp l i t ud e re gard l e s s of t he
a mp l i t ude o f t he input .
I n prac t i c e l im i t e r s may b e mad e t o
o p e ra t e ove r a w i d e range o f s i gnal l e ve ls a nd i t i s common
t o d e s i gn a FM re c e i v e r s o t ha t l im i t i n g be gins t o o c c ur on
j us t t he a mb i e nt front - end n o i s e of the re c e i ve r .
The re f ore ,
l im i t ing w i l l b e c e rta in t o o c c ur f o r a ny s i gna l l e v e l larger
t han t he front - e nd n o i s e .
The l im i t e r o p e ra t e s by c l i pp in g t h e p os i t ive and n e ga t ive
peaks of the s i gna l t o produce a s quare wav e of c onstant amp­
l i t ud e but of va ry i ng fre quency .
The c l i p p i n g a c t i on , h oweve r ,
d o e s not c ha n ge t h e z e r o c r o s s ings s o t ha t t h e d e s i r e d f r e ­
quency inf orma t i on i s pre s e rve d .
f i l t e re d t o r emove i t s harmon i c s .
The square wave i s f ina l ly
A
v e ry broa d f i l t e r ( i n
c ompa r i s on w i t h t h e int e rmedia t e fre que n cy f i l t e rs ) i s use d t o
m i n im i z e t h e i n t r o d uc t i on o f a dd i t i ona l d is t or t i on .
Th e output
of t h e lim i t e r i s then a s ine wave o f c ons tant amp l i t ude but
va ry ing f re quency .
I t i s i mp ortant t o n o t e tha t a l t h ough t h e l im i t e r removes
unwant e d a mp l i t ud e modul a t i on ,
it d oe s not a f f e ct the a mp l i t ude
_ o f t h e s i d e bands o f t he FM s p e c trum i n d i v i d ua l ly .
It only
affe c t s t he amp l i t ude o f the res ultant summa t i on o f s i de band
22
ve c t ors .
13
( 2 2 - 18 ) .
.
T h i s fa c t i s r ea l i z e d by e xam i n in g Equa t i on
T h e e ff e c t o f t he l im i t er i s t o remove t h e t e rm
R ( t ) and r e p l a c e i t w ith a c onstant wh i c h we s ha l l arb i t ra r i ly
s e t equa l t o uni ty i n t he f o l l owing d i s cuss i on .
T h e exponen-
t ia l t erm c on t a i n in g the f r e quency i n forma t i on is una f f e c t e d
by t h e a c t i on o f t h e l i m i t e r .
Ret ur n i n g t o t h e mat h ema t i c a l d e ve l opment we may now wri t e
t h e output o f t h e l im i t e r a s
( 2 . 2- 2 1 )
whe re the l im i t e r i nput was given by Equa t i on ( 2 . 2 - 1 8 ) .
The
pha s e a ngl e w t + Q ( t ) i s given by Equat i on ( 2 . 2-20 ) .
0
The l im i t e r output v ( t ) i s f ina l ly a ppl i e d t o t he FM
L
d e t e c t or i n o rde r t o e xtra c t t he d e s i r e d inf orma t i on .
The FM
d e t e c t or or fre quency d i s cr i m ina t or r e sp onds t o t h e ins tantane ous frequency w ( t ) o f v ( t ) and i t s out put i s g i v e n by ,
i
L
a s s uming un i ty s l ope , v ( t ) wh ere
d
v (t )
d
=
w. (t )
1
- w
0
•
( 2 . 2-22 )
T h i s idea l d i s c r imina t o r chara c t e r i s t i c i s s h own i n F i gure 2 - 2 .
I n prac t i c e t h is i d e a l l inear r e s p on s e may b e very c l o s e ly
rea l i z e d over large p e r c e n t a ge bandw i d t h s .
13
P . F . Pant e r , "A Re v i e w of t h e S i de band The o ry o f F!ll W i t h
Part i c ular R e f e r e n c e t o D i s t ort i on Pr o b l ems , " I TT Federa l Lab ­
orator i e s I nt e rna l Report , ( 1 947 ) , 1 8 2.
23
w.
1
Figure 2-2 .
Ideal Discriminator Response.
(t )
24
1
F i na l ly , t h e f o rm o f w. ( t ) mus t b e f ound i n o r d e r t o d e t e rm i n e t h e d e t e c t e d output v (t).
d
I t i s h op e d that
v
d
( t) w i l l b e
a fa i t hful reproduct i on o f t h e or i g i na l s inus o ida l m o d ul a t in g
funct i on .
Th i s , h oweve r , i s n o t the c a s e .
T h e dis c r i mina t o r
out put v ( t ) w i l l i n ge n e ra l c onta i n d i&t ort i on c omp one n t s i n
d
add i t i on t o t h e o r i g i na l modul a t i n g func t i on .
The d is t ort i on
c omponents w i l l b e in t h e f o rm o f harm on ics cf t h e s in us o i da l
modula t i n g f un c t i on .
The i ns tantane ous fre quency o f v ( t ) i s given by
L
( 2 . 2-23 )
whe re
�i ( t )
i s t he i ns t antane ous p ha s e o f v ( t ) a n d i s g i ven by
L
w t + e( t )
0
+ 00
'
=
0
w
t + tan
-
J ( B ) P ( j w ) sin
n
n
L n
1 n= m
+ 00
-
' J (B)P ( j w ) c os
n
n
n
�
n= -oo
[ nwmt
[ nw t
m
+
+
Q
n
( jw
9 ( jw
n
n
>]
n
>]
( 2.2-24 )
Be f ore p e r f o rm ing t h e d i fferent ia t i on ca l l e d f or i n Equa t i on
( 2 . 2- 23 ) , l e t us s impl i fy t he nota t i on a s f o l l ows
e< t )
D i f fe rent i at i n g ,
=
tan
-l
�A
�
_
___;n
""
_.:;.n:;_-'n;.;___
s in
L A c os
n n
�n
=
tan
-
1 u
v·
( 2 . 2 -25 )
25
d
t)
dt 9(
=
=
d
-1 u
tan
dt
v
=
=
vdu - udv
2
2
v
u
+
( 2 . 2-26 )
I, A n c os �n m
LA � · c os $
n
mm m n:£ An s i n �n mLAm$m• s i n $m
nI, An s i n �nmLAm s i n �m + nLAn c os �nnLAm c os $m
+
A <j>• t c os, ' �n c os $m sin $n s i n �mJ
n:£ mI, Anmm
nI. mI, AnAm [ s i n �n s i n �m c os �n c os �mJ
+
+
( 2 . 2- 2 7 )
( 2 . 2-28 )
T h e bra cke t e d t e rms are r e c o gn i z e d t o b e t he f ol l ow i n g t r igonamet r i c i d e nt i t y
c os ( x - y )
=
s in x s in y
+
c o s x c os y
( 2 . 2-2 9 )
so t hat Equa t i on ( 2 . 2-28 ) may b e expre s s e d a s
d
e( t )
dt
=
A $• c os ( �n �m)
nI mI Anmm
nI mI AnAm c os ( �n - ffim)
-
'
( 2 . 2-30 )
't'
Re t urning t o Equa t i on ( 2 . 2 - 2 3 ) a nd w i t h a s l i ght change i n notat i on we may wri t e
w. ( t )
1
=
+
w
0
+m
w JnJmPnpm c os [ ( n
L L nm
n , rn=-ro
+00
L L JnJmPnpm
n,m=-ro
.
c os
-
m)wmt
[ ( n - m) wmt
+ g
+ g
n
-
n
-
gm]
gmJ
( 2 . 2- 3 1 )
From Equa t i on ( 2 . 2- 2 2 ) we may now wr i t e t he f i nal express i on f o r
t h e d is c r i m i na t or output a s f o l l ows
26
+00
[ (n
L L nwmJ nJmpnpm
c os
LL
[ (n -
n , m= -oo
+<D
n , m= -co
J J p p c os
n m n m
+
- m )w t
m
m )w t
m
+
(9
n
- 9 )
m
(Q - Q )
n
m
]
]
( 2 . 2-32 )
Equa t i on ( 2 . 2-32 ) i s a g e n e ra l expre s s i on f o r t h e out p ut o f a
FM d i sc r i m i na t or for s i ngl e - t one fre quency modula t i on .
The
exp r es s i on is an exa c t s ol u t i on und e r the a ssump t i ons o f i d e a l
l im i t ing and a l in e a r d i s c r im i na t or .
I t c onta i n s b o t h t h e de -
s i re d s in us o i d a l modul a t i on p l us i t s harm o n i c d i s t or t i on t erms
' b r ought a b out by n on - l inea r i t i e s in phas e a n d amp l i tude of t h e
n e twork t ra n s f e r impeda nc e .
Unfort una t e ly , t h e harmon i c t erms a r e npt g i v e n expl i cit l y .
H owe v e r , Equa t i on ( 2 . 2 -32 ) may b e e va l ua t e d num e r i ca l ly on a
p o i n t by p o i n t bas i s over one p e r i o d o f w
m
•
A rium e r i ca l F o ur i e r
a na lys is may t he n b e p e r f ormed on t h e one - pe r i o d s e gm e n t o f
vd ( t ) t o c omput e t h e ma gn i t ud e of t he harm on i c s .
The grea t e r
t h e numb e r o f p o i nt s , or sampl e s , taken o v e r t h e fundamental
p e r i od , the greater wil l be t h e a c c ura cy of t h e s o l ut i on.
Be f ore Equa t i on ( 2 . 2 -32 ) may b e us e d t o y i e l d mean i ngful
l'e &ul ts a c r i t e r i on must be d e ve l op e d w i t h r e ga rd t o h ow many
s i d e bands mus t be inc l ud e d t o y i e l d an answer of a s pe c i f i e d
a c c ura cy .
T h e inc l us i on o f t o o many s id e bands w i l l t a ke an
unrea s ona b l e amount of c omputa t i on t ime , e v en o n a h i gh s p e e d
27
c omput er .
F o r t unat e ly , the magn i t ud e o f the Bes s e l func t i ons
fal l off rap id ly for fre quen c i es o u t s i d e t he range o f 'w
0
±
w
d
s o t ha t only a few s i de bands beyond t hat range n e e d be c ons i d I t has b e e n shown b y Wan g
e re d .
14
that the d is t ort i on - t o - s i gna l
power ra t i o D/S a t t h e output o f a perfe c t ly l in e a r d is c rim ina t or i s g i v e n by
D/S
=
7TB2
l
T[ 2: Jn
-k
-1T
+00
+
( B ) ( B c os
n=-ro
L Jn ( B ) ( B
n=+k
c os f) -
whe r e the s i de bands from
-ro
e
n) c os (B
- n ) c os ( B s in e
sin
-
ne)
( 2 . 2-33 )
t o -k and f rom + k t o +ro hav e b e e n
t runca t e d f r om t he in f i n i t e Bess e l s e rles e xpans i on.
The d is -
t or t i on i n Equa t i on ( 2 . 2- 3 3 ) i s d ue only t o t he e l imina t i on o f
s i d e bands a n d n o t t o a ny n e twork pha s e or amp l i t ude nonl i nea r i t�
Wan g has e va l ua t e d Equa t i on ( 2 . 2- 3 3 ) and has prepa r e d a n e x t e n s i ve graph o f d is t ort i on - t o - s i gna l-ra t i o ve rs us mod u la t i on index
and bandw i d t h .
15
The Fouri e r s olut i on d e ve l op e d earl i e r i n t h i s cha p t e r w i l l
be us e d i n Chapt er 4 t o ve r i fy Wan g ' s resul t s and t o d e ve l op a
-
c r i t e r i on f o r d e t erm in i ng t h e numb e r o f s i d e bands t ha t mus t b e
14
H . S . C . Wang , "Ba n d w i d t h Re qu i r e m e n t s f o r Fre quenc:y­
�!o d u l a t e d S i gna ls , " Pro c e e d i ng·s IEEJl:, 53 , No . 8 ( A ugus t , 1 965 ) ,
_
1 1 50 .
15
I n t h i s case "ba ndw i d th " i s us e d t o mean the range over
wh i ch s i d e bands a re inc l ude d or 2kw
m
•
28
r e ta i ned t o a c h i e ve a n a c c epta b l e l eve l o f a c c uracy i n the
c omputa t i on s .
From the a ppeara n c e o f E quati on ( 2 . 2- 3 2 ) i t i s e a sy t o
u nd e rsta nd why t h e a bove t e chn i que o f a na lys i s wa s not w i d e l y
us e d i n t h e e a r ly days o f t h e deve l opment o f FM e v e n t hough t h e
t e c hni que wa s we l l known i n 1 937 .
T h e p o int b y p o int e va l ua t i on
I
o f t h e expr e s s i on p l us t�e d o uble s umma t i ons ma d e f o r an unreas 1
o na b ly l ong and t e d i ous task i f many s i d e bands were c ons i de re d .
Eve n w i th a d e sk ca l cu la t o r i t was s t i l l qu i t e an e norm o us
unde rtaki n g .
H oweve r , i n l a t er years , t h e a va i la b i l ity o f h i gh
s p e e d d i gi ta l c ompu t e rs has ma de i t poss ible t o r e d uc e t h e t a sk
t o only a few m in u t e s o f c omputa t i on onc e a s u i t ab l e program
has b e e n d e ve l ope d .
2.3
THE DYNA�iiC APPROACH
The f o l l ow ing i s a br i e f d is c us s ion of t h e dynamic or
o p e ra t i ona l t e chn i que o f a na ly s i s us ed by Carson a n d Fry
b y Van d e r Pol
17
.
18
and St umpers .
17
18
and
The d e v e l opme nt i s pre s ent e d
i n i t s f orm a s c l ar i f i e d by Ba gh dady .
16
16
19
Ca rs on a nd Fry , o p . c i t . , 5 1 5 -5 1 8 .
Van d e r Pol , op . c i t . , 1 5 6 - 1 5 7 .
S t umpers , op . � i t . , 83-8 6 .
1 9 Ba ghdady , op . c i t . , 203- 205 .
29
Cons i d e r a l i near n e t work with a charac t er i s t i c un i t i m p ul s e resp ons e d e f i ne d by
h(t ) =
l
( 2 . 3- l )
- oo
whe r e Z ( j w ) i s t he n e twork t rans f e r impedanc e .
d e f i n e d a s t h e Four i e r t ra n s f orm of Z ( j w ) .
Thus h ( t ) i s
Le t t h i s n e twork be
dr i v e n a t i ts i np'..lt by the f o l l ow i n g F!vt c urrent s ourc e
1..
(t) = e
j
[ wot
+
e( t ) ]
= e
J
·
S
o
w. ( t )dt
1.
( 2 . 3- 2 )
whe re t he amp l i t ud e has b e e n a rb it ra r i ly c h os e n a s un i ty .
fa c t or
!
'
e( t )
is an a rb i t ra ry f unc t i on o f t ime a nd c onta ins the
i n f o rmat i on that i s t o b e t ra nsm i t t e d t hr ough t h e n e twork via
' t h e m odula t e d carr i e r .
1
The
The i ns tantan e ous frequency o f the
c ur rent is given by
( 2 . 3- 3 )
The s t ea dy- s ta t e r e s p onse of t h e n e t work a f t er i n i t i a l
t ra ns i en t s have d i e d out may b e express e d , very gen e ra l ly , a s
v(t )
( 2 . 3- 4 )
whe re t h e f orm o f E ( t ) i s y e t t o b e d e t e rm i ne d .
Let u s n ow i nve rt Equa t i on ( 2 . 3 - 4 ) a nd wri t e
·[
-J
E( t ) = v( t )e
w t
o
+
e( t ) ]
( 2 . 3-5 )
I t i s we l l kn own t ha t t h e r e s p onse o f a l inear ne twork t o an
e x c i t a t i on such as i ( t ) may be express e d as t he c onvol ut i on of
30
t h e exc i ta t i on w i t h t h e n e t work un i t impulse r esp onse .
v ( t)
=
m
� h(r)i(t
H e nc e ,
- r )dr
0
( 2 . 3- 6 )
and f r om Equa t i on ( 2 . 3- 2 ) we obta i n
v ( t)
=
00
� h( r )e j [ wo (t - r)
+ e( t
0
- r) ] r.
d
( 2 . 3-7 )
S ubst i t ut i n g Equat i on ( 2 . 3- 7 ) i nt o E q ua t i on ( 2 . 3- 5 ) we obta i n
e( t
E ( t)
j .
( 2 . 3-8 )
or simply
Now,
E ( t)
=
r h( r {e - j[O (t)
0
- 9( t -
r)J] - j wo r d
e
r .
( 2 . 3- 9 )
l e t us subst i t ut e Equa t i on ( 2 . 3 - 3 ) i nt o Equa t i on ( 2 . 3- 9 )
and obta i n
E ( t)
which,
E ( t)
The
=
r h( r ) [ . - j [o (t) 0
O( t -
r ) J ] . -jr[w1 (�)
upon rea rranging t e rms , b e c om e s
=
r h( r ) [ e j[e (t - r ) -
0
g(t)
+
( 2 . 3- 1 0 )
( ]J jr ( )
e' t )
e
-
w .
1
t
d r.
( 2 . 3- 1 1 )
i nt e gra t i on of e i t h e r Equa t i on ( 2 . 3- 9 ) or Equa t i on ( 2 . 3- l l )
may b e exp e d i t e d by expand i n g t h e quant i ty in b ra c es i n t o a
Ta yl or se r i e s i n po�ers o f
the
b ra c es a s g ( t ,
r) .
- r.
Le t us d e n o t e the qua n t ity i n
The n , we may wr i t e
31
00
1
g(t, r) = "'� am (t)( - r) m .
=o
m
At this point one must stop and examine the convergence of
Equation
for the specific Q(t) of interest. To proceed
with the analysis it is required that the infinite series be
uniformly convergent for all r . Thus, the remaining discussion
is restricted to the class of functions e (t) for which Equation
is uniformly convergent.
Under the condition of uniform convergence we may express
E(t) as follows
E(t) = r h( r { �am (t)(- r) m ] · - jwr
m
where the form of am (t) and w are yet to be specified.
Now, h( r ) is bounded for all r, then the series
-w
L am ( t ) ( - r) ( r ) e j .r
=o
m
is uniformly convergent for all r . The series may thus be integrated term by term to yield the following uniformly convergent series expansion for E(t)
( 2 . 3- 1 2 )
( 2 . 3- 1 2 )
( 2 . 3- 1 2 )
( 2 . 3- 1 3 )
if
00
mh
( 2 . 3- 1 4 )
( 2 . 3- 15 )
Now let us digress for a moment. Recalling that h( r) and Z( j w)
form a Fourier transform pair we may write
� h(r )e - j wr dr.
Z (.j w )
=
00
0
( 2 . 3- 1 6 )
32
Differentiating both sides of Equation (2.3-16) tim�s with
respect to j w we obtain
dm m Z(jw) � (- m r)e -j wr dr.
(2.3-17)
d(jw)
Equation (2.3-17) is recognized to be the quantity in brackets
in Equation (2.3-15) .
Finally, we may express E(t) as follows
(2.3-18)
E(t) 'a ( t) d( 'w) m Z ( j w) .
we allow that w w in Equation (2.3-18) we obtain the
' Carson and Fry expansion. 20 However, if we allow that w w. (t)
in Equation (2.3-18) we obtain the Van der Pol - Stumpers
. 21
expans1on.
his paper Baghdady continues on to develop the form of
(t) for both the Carson and Fry and the Van der Pol - Stumpers
expansions. He shows that the leading two terms of the Van der
Pol - Stumpers expansion represent the quasi-stationary approximation of the response. 22 He then examines the convergence
properties of the two expansions and develops error bounds that
be employed to estimate the error incurred in representing
E(t) by only a few of the leading terms of the series. However,
20Carson and Fry, op. cit . , 517.
21 Van dcr . Pol, op. cit. , 157; and Stumpers, op. cit. , 84.
22Baghdady, op. c:it. ,' 213.
m
=
m
=
If
=
0
�
m= o
m
m
r) h (
0
d
·
m
J
=
i
In
a
m
1
, may
.
1
33
as Rowe h@s po�n ed out in a later paper, the error bounds de­
veloped by Baghdady are not universally valid and in many common
instances lead to serious error . 23 Rowe supported his claim
with several examples for which the error bounds were invalid.
He concludes that the series expansion must be closely examined
for each specific case in order to determine how many terms
m ust be retained to yield an acceptable error in the solution.
Since is one of the goals of this study to develop a
straight-forward, universally applicable procedure to be used
in solving problems, the operational or dynamic method of ana­
lysis will be abandoned at this point in favor of the spectral
. or Fourier approach. The quasi-stationary approximation, however, will be briefly examined and in Chapter the solution
obtained for a single pole filter will be compared with the
solution obtained using the Fourier method.
2 . 4 THE QUASI-STATIONARY APPROXHIA TION
As stated earlier, the quasi-stationary response of a
network to a FM excitation is an approximate solution that may
be obtained from conventional AC circuit theory by substituting
the variable instantaneous frequency for the assumed constant
frequency. In order for the quasi-stationary response to closeapproximate the actual network response, the instantaneous
frequency must vary slowly enough that the network may follow
2 3Rowe, op. cit . , 286-290 .
�
it
4
·
ly
34
the excitation. In the limit as approaches zero, the quasi­
stationary response equals the actual response.
To proceed with the derivation of the quasi-stationary
response, consider again a network with a transfer impedance
Z jw driven by a current source given by
2.4-l
The instantaneous frequency of the excitation is then
w. t w0 t .
The response of the network may then be expressed as
w
m
(
F1-1
)
(
1
( .)
=
+
9( )
)
( 2 . 4- 2 )
( 2 . 4-3 )
where � w is the phase component of the transfer impedance . In
the quasi-stationary approximation the network is followed by a
limiter which removes any amplitude modulation so .that a constant amplitude wave is obtained. Consequently, the amplitude term of the network transfe impedance introduces no distortion as may be seen from Equation
This is not
strictly true in reality as may be noted in Equation
where the amplitude characteristic of the network still influences the instantaneous frequency of the response in spite of
the action of the limiter. This false asswnption is one of the
sources of error in the quasi-stationary ·approximation. Another
source of error is in assu ing that the carrier moves in
continuous n�nner between w0 - wd and �0 wd . This implies
( )
FM
i--
( 2 . 4-3 ) .
( 2 . 2- 3 1 )
�
a
+
35
that all the energy of the wave is contained within those
bounds . From the discussion in Section we know that this
is not the case.
Returning to Equation
we may express the instantaneous frequency of the network output response as
t w t �t p [w t J
so that the discriminator output becomes, assuming unity gain,
1.2
( 2 . 4- 3 )
w. (
1
)
=
0
f
l
+
,'
Q( )
+
i
(
)
( 2 . 4- 4 )
( 2 . 4- 5 )
Equation
is called the quasi-stationary response of the
network.
For the case of sinusoidal frequency modulation we obtain
wo wd cos w t
and thus
( 2 . 4 -5 )
1
w . (t )
=
+
m
( 2 . 4- 6 )
( 2 . 4- 7 )
i
where the �t � [wi t J term contains the distortion components.
Equation
may be evaluated on a point by point basis over
one period of w and a numerical Fourier analysis performed on
the result to obtain the individual magnitudes of the hcrJTionics.
consequence of the approximations made in arriving at the
quasi-stationary solution is that the phase relationships between the harmonics of w have been lost although the amplitudes
have not been greatly altered by comparison. This fact is
( 2 . 4-7 )
( )
m
·
A
m
36
brought out dramatically in the quasi-stationary response of
the single pole-filter considered in Chapter
To prove that the phase relationships of the harmonics are
lost we need only prove that :t � [wi (t)J is an odd function
and hence contains only sine terms in its Fourier expansion.
It is not generally true that all the harmonic terms are in
phase at the start of a ��cle of w as would be the case if
only sine terms were present.
That ! [wi (t)J is an even function follows directly from
the fact that w. (t) wo
w t is an even function.
Hence, 1 [ �i ( t )] may be represented by a Fourier expansion of
only cosine terms. If we take the time derivative of the
Fourier expansion of f [w i ( t ] on a term by term basis we will
obtain only sine terms. Therefore, :t f [wi ( t ] is an odd
function.
Normally one is interested only in the magnitude of the
harmonics, and the phase relative to the fundamental is of
little concern. However, if the phase information is required,
the quasi-stationary approximation may not be used.
4.
m
i
1
=
+
wd
)
CO$
m
)
37
3.
C 0)1 PlJTER A I
D
ED SOLUTIONS
P
The Fourier method was adopted as the basis for developing
the computer-automated te�hnique of problem solving to be dis­
cussed in this chapter. ' I n the following chapter the technique
will be employed to investigate the response of several typical
bandpass filters to a excitation. In addition, the quasi­
stationary response will be found for the case of single-pole
filter and compared with the solution obtained using the Fourier
m ethod.
In computing the response of a network using the Fourier
approach, four distinct steps are required :
Evaluation of the network transfer impedar.ce ;
Decomposition of the wave into its spectral com­
ponents ;
Evaluation of the discriminator output d (t) ; and
Harmonic analysis of t) .
Since the problem may be divided conveniently into the four
separate steps above, the computer program was likewise divided
into four parts consisting of the main program called
an
IB�! library program called ECAP, and two subroutine subprograms
Aside from the ECAP program, all
called
and
the programs discussed herein were developed by the author
3.1
THE
A
P R OACH
TO PROBLE�I SOLVING
FM
a
(1)
FM
(2)
(3)
v
vd (
(4)
FMDIST ,
SERIES
BJ�SL .
IBM
38
s p e c i f ical ly f o r t h e FM p r ob l em .
F'MD I ST i s t h e c ont r o l l ing program a n d pe-rf orms t h e e va l ua t i on o f t h e d is c r i m ina t or out put .
Pr i or t o t h i s e va l ua t i on ,
h ow e ve r , i t a c c e pt s ne t work t rans fe r impe danc e data f r om ECAP
a n d input da ta s p e c i fy i n g t he Ft--1 wave .
A f t e r r e c e iving the
n e t work da t a a n d t h e n.t wave s p e c i f i ca t i on FMDI ST c a l l s upon
t h e subrout i n e BESL to g e n e ra t e the FM s p e c tr um .
The next s t ep i s t he e va l ua t i on o f t h e d i s c r im i na t or output
v (t).
d
The d i s c r im i na t or output is e va l ua t e d f or k e ve nly
s pa c e d t im e inc rement s over one p e r i o d o f t he fundamenta l fre q u ency
\V
m
•
i s obta ine d .
'
Thus one c omp l e t e cyc l e o f t h e d is c r im i na t or output
S in c e t he o utput is pe r i o d i c f o� the case of
s i ngl e - t one s i nus o i d a l m o d u la t i on , only one cyc l e is r e qu i r e d t o
o b t a i n a l l t h e i n f orma t i on about v ( t ) a nd i t s harm on i c s .
d
A f t er v ( t ) i s f ound , �ID I ST ca l l s upon t h e subrout ine
d
SERI ES to p e r f o rm a n ha rm o n i c a na lys i s on the d i s c r im i na t o r
m
o utput t o f i nd t he ma gn i t ude a n d pha s e o f t he ha rm on i c s of w
•
I n orde r t o c ompa re t h e quas i - s ta t i ona ry a pproach t o t he
F o u r i e r app r oa c h , a s h or t program ca l l e d QS was d e ve l op e d .
This
p r ogram genera t e s t h e pha s e cha ra c t e r i s t i c of a s i ngl e .... p o l e
f i l t e r a n d t he n proc e e d s t o c ompute v ( t ) v i a t he qua s i - s tat i on ­
d
a ry approx ima t i on of Equa t i on ( 2 . 4 - 5 ) .
QS t h e n ca l l s upon
SERI ES to p e r f o rm a n ha rm o n i c a na lys is on the d i s c r i m i na t or out -
put a s was d on e i n t h e p r o gram FMD I ST .
The p r o gram QS may b e use d for n e tworks o t h e r t ha n the
s i ngl e -p o l e f i l t er by r e p l a c ing the pha s e s pe c i f i ca t i on s ta t e -
39
ment w it h a ny o t h e r d e s i r e d p ha s e s pe c i f i ca t i on .
One mus t b e
caut i one d , h oweve r , i n t he us e of QS . s in c e t h e c orre spondence o f
t h e s ol ut i on obta i n e d w i t h t he a c t ua l n e t work r e s p o n s e is n o t
e a s i ly d e t e rm i n e d i n ea c h pa rt i c ular c a s e .
Th e b es t m eans f or
d e t e rm i n i n g t he e rr o r i n t h e s o l ut i on p r o d u c e d by QS i s t o c ompare t h e c a s e o f max imum m o d u l at in g fre que ncy w and max i mum
m
d e v ia t i on w
d
o f i�t e r e s t w i t h a s im i l a r s o l ut i on obta i n e d us i n g
t he Four i e r m e t h o d .
Th i s c a s e w i l l g e n e ra l ly repre s e nt t h e
max imum e r r or i n c urred a nd l e s s e rr o r w i l l r e s ul t f or d e crea s ing
va l ue s o f w a nd/or w .
d
m
The d e ta i l s o f t h e p r o grams ment i on e d a b ov e w i l l be d is . cu s s e d i n grea t er d e ta i l i n t h e f ol l ow i n g para graphs a l ong w i t h
a bri e f d i s c us s i on o f t h e c omput e r fa c i l i ty use d .
3 .2
THE C OMPUTJ<:R
The c omput e r that was us e d i n t h e c i r c u i t inve s t i ga t i ons t o
b e d is cus s e d i n Cha p t e r 4 wa s manufa c t ur e d by t h e C on t r o l Data
C o rp o ra t i on and i t s us e was ma d e p os s ib l e t h rough t he s er v i c e s
of Tymshare , I nc orpora t e d .
The c omput e r a nd i t s s o ftware were
d e s i gn e d s p e c i f ical ly f or t im e - sharing a pp l icat i ons and t h e
sys t em may b e us e d s im u l t a n e ous ly by a s many a s f orty opera t ors .
The c omput e r i s c e n t ra l ly l oca t e d a n d e a c h opera t or c ommuni ca t e s
w i t h i t from a t e l etype t e rmi na l via s ta n dard t e l ephone l i nes .
Due t o t h e e x t r eme l y h i gh spe e d a n d l arge memory of t he
c omput e r , ea ch opera t or ' s work i s pro c e s s e d rap id ly s o t ha t h e
expe r ienc e s l i t t le d e lay i n runn ing h i s pro gram d e s p i t e t he fa c t
40
t ha t many o t h e r u s e rs may b e r unni ng pro grams a t t h e same t ime .
:
The work i s hand l ed on a r o ta t ing bas i s s o t ha t wh i l e t h e c omp u t e r i s p e r f orming c omputa t i on f or one u s e r ,
i t is pr int ing
out data a nd a c c e p t i n g c ommands from o t h e r users .
Genera l ly ,
e x c ept und e r c ond i t i ons o f p e a k usa ge , t h e s p e e d o f t he sys t em
i s l im i t e d only by t he t y p i n g s p e e d o f t he t e l e type mach ine .
I
The pr o grams d is c us � e d b e l ow a re wr i t t en i n F or t ran I V a nd
i
w e r e d e ve l op e d on t h e Tymsha re sys t e m a nd c onta i n a f e w p e c ul i a r i t i e s tha t s h o u l d b e ment i on e d i n . ord e r t o prevent a ny c on fus i on t ha t may a r i s e wh e n r e a d ing t h e programs l is t e d i n t h e
a pp e nd i c e s .
S i nc e punch cards a re n o t use d a n d t he programs a re s t ore d
o n a d is c f i l e a t t h e c omput e r ,
it is n e c e ss a ry t o have a means
o f address i n g a pa rt i c ul a r s t a t ement s o t ha t i t may b e c ha nged
if d e s i r e d by t he use r .
T o a c c ompl i s h t h is , e ach s t a t ement i n
a program ha s a l in e numbe r wh i ch is u n i que t o t hat s t a t ement .
Thus the us e r may a d d r e s s t ha t s ta t ement d i re c t ly f o r mod i f i ca t i on i f he s o d e s ir e s .
A n o t h e r c onse quenc e o f n ot u s i n g punch cards i s t ha t Tyms ha r e For t ra n I V c onta ins a f re e f orm i nput s t a t ement c a l l e d
ACCEPT .
The ACCEPT c omma nd a l l ows t h e c ompu t e r t o read da ta
f r om t he t e l e type in a ny f o rma t d e s i re d by t h e us e r .
'J'he ACCEPT
c ommand take s t h e p l a c e o f t h e READ a nd FORI>lAT s t a t e me nt s norma l ly r e q u i r e d to input data us ing Fortra n I V .
The READ and
FOTh"' AT s t a t erne n t s a r e a va i la b l e , h owe ve r , i f the us e r wishes t o
empl oy t h e m .
41
Fina l ly , . Tymshare Fort ra n I V us e s b o t h parent h e s e s and
s quare bra c k e t s wh e r e a s s t a ndard A SA F o r t ra n I V us e s only the
pa renthe s es .
� h e s qua r e b ra c k e t s a r e r e s e rved f or l is t s of
var ia b l e s in s ub rout ine a n d f un c t i on s pe c i f i ca t i ons .
In m o s t o t h e r r e s p e c t s t h e Tymshare F or t ra n I V i s i d e nt i c a l
t o s tandard A SA Fortran I V a nd t he p r o grams d is c uss e d b e l ow may
be q u i ckly c o nve r t e d t o s t a ndard F o r t ra n by c hanging a f ew
s t a t em e nt s w i t h r e gards t o t h e a bove i t em s .
3.3
EVALUAT I ON OF THE NETWORK T HANSFER D1 PEDANCE : ECAP
In o rd e r t o c omput e t h e r e sponse of a n e t w ork t o a FM
: ex c i t a t i on u s i n g t h e Four i e r m e t h od , t h e magn i t ude a nd phas e of
t h e ne twork t ra ns f e r i mp e da n c e must b e kn own a t e a c h s i d e band
f r e quency .
T h i s i n f orma t i on may be obta i n e d by d i r e c t measure -
m e n t s on t h e n e tw ork i n ques t i on or i t may b e c a l c �l a t e d us i ng
o r d ina ry AC c i rc u i t ana lys i s t e c h n i que s .
I f the ne t work i s
v e ry c ompl i cat e d , h owe v e r , t h e ca l c ula t i ons b e c ome t e d i ou s and
v e ry t ime c ons um in g .
For t una t e ly , I B�I has d e v e l oped a v c ry
p owerful p r o gram c a l l e d ECAP ( El e c t r o n i c C i rc u i t A na lys is Pro gram ) wh i c h ea s i ly p e r f orms a n analys i s o f a . c omp l i c a t e d n e twork
in j us t a few m in u t e s o f c omput e r t ime .
1
T h i s program was
a va i labl e on t h e Tymshare sys t em a nd wa s u s e d t o e va l ua t e t h e
1
Handa l l W . J e n s e n a n d �la rk D . L i e b e rman , The I BM E l e c troni c C ir c u i t A na lys is Pro gram ( ECAP ) : T e c h n i q ues a nd App l i ca t i ons
{ Engl ewo o d C l i f fs , N . J . : Prent i c e - Ha l l , I nc . , 1 96 7 ) .
42
t ra n s f e r imp e danc e s of t he n e t works i nv e s t i ga t e d i n Chap t e r 4 .
The ECA P program r ea l ly c on s i s t s o f t hre e ba s i c programs :
DC analys i s , AC a na l y s i s and t ra ns i ent a na lys i s .
T h e c ir c u i t
t o be ana ly z e d i s c od e d a c c ord ing t o a s impl e f ormat f o r u s e b y
t h e ECAP p r o gram .
s pe c i f i e d .
The type o f a na ly s i s t o be p e r f or m e d i s a l s o
T h e c od in g o f t h e c i rc ui t bas i ca l ly c ons i s t s of
num b e r ing t he nodes a n d branches o f t he c ir c u i t a nd t h e n spe c i ­
fy i n g t h e e l e m e n t i n e a c h b ra nc h .
Appe n d ix A c onta ins t h e c ir ­
c u i t s c h e ma t i c d i a gram a n d ECAP program f or t h e t wo -p o l e But t � r­
w o r t h f i l t e r d is c uss e d in Chapt er 4 .
The ECAP program takes t h e c od e d c ir c u i t s pe c i f i ca t i on and
• c om pu t e s t he c i r c u i t n o da l a dm i t tance ma t r ix .
It then p r oc e e d s
t o s o l ve f or t h e v o l ta g e s a t e a c h node .
I n t h e AC a na ly s i s , t h e f r e q uency may b e s pe c i f i e d t o b e
i n c rement e d by a n y d e s i r e d amo unt a n d a c ompl e t e · c ir c u i t analy­
s is p e r f orme d a t e a c h new fre quency .
Th i s f e a ture is part i c u­
larly us e ful in p l o t t i n g t he f r e quency r e sponse o f f i l t e rs a n d
i s o f gre a t importance t o t h e prob l e m a t hand .
T h e data produc·e d by t h e program i s t he ma g n i t ud e a nd
p ha s e a ngl e o f t h e vol t a g e a t e a c h n o d e o f i n t e r e s t a t e a c h
s pe c i f i e d f r e que ncy .
I f t h e input e x c i tat i on t o t h e n e twork i s
s pe c i f i e d a s a n A C c urre n t gene rat or w i th a
one ampere
and
Rl\iS
ampl i t ude o f
a n i n i t ia l pha s e o f z e ro d e gre es , t h e v o l t a ge
o u t p ut o f t h e ne twork w i l l only ne e d t o b e d i v i d e d by uni ty t o
obta i n t h e t rans f e r impe danc e .
Thus t h e da ta prov i d e d by t h e
ECA P pro gram is in e xa c t ly t h e f o rm re qu i red f o r use i n Equa t i on
43
( 2 . 2- 32 ) .
3.4
GENE.RAT I ON OF THE FM SPECTRUM : BESL
A f t e r t h e n e twork t ra ns fe r impe da n c e ha s b e e n e va l ua t e d a t
e a c h s id e band fre que ncy , t h e n e x t s t e p i s t o f ind t h e ma gn i t ud e
and pha s e o f e a c h s i gn i f i cant s id e band f or t h e spe c i f i e d F�l
'
wave .
I n ord e r t o a c c ompl i s h t h is s t e p a s ub rout ine s ubpr ogram
ca l l e d BESL wa s d e ve l op e d .
The input d a ta t o t h is subprogram
m
c on s i s t s only of t h e m o d u l a t ing fre quency w , t h e carr i e r fre ­
quency w , and t h e p e a k d e v ia t i on w .
d
0
T h e output of t h e program
c on s i s t s of t he numbe r of s i gn i f icant s i d e bands , t h e magn itude
a nd s i gn o f t he a pprop r ia t e B e ss e l f unc t i on f o r e a c h s i d e band ,
and t h e fre quency o f ea c h s id e band .
T o c omput e t h e va l ue o f Be s s e l f un c t ion f o r ea c h s id e band
.
2
t h e f o l l ow i n g we l l kn own p ower-s e r i e s expans i on was use d
J (x ) =
p
00
L-
n= o
� n ( 16
p
2n +
x) 1
n .. ( n + p ) .
' ( -1
( 3 . 4- 1 )
n
T h i s r e la t i on s h i p was us e d t o c omp u t e J ( B ) f o r t h e carr i e r and
t h e upp e r s id e bands .
The va l ue s f o r t h e l ower s i d e bands a re
t h e n s imply g i ve n by t he f o l l owing i d e n t i ty
J
2
-p
( x ) = ( - l ) PJ ( x ) .
p
.
3
( 3 . 4-2 )
1 . s . S ok o l n ik o f f a n d R . M . Re dh e f fe r , Ma t he ma t i c s of Phy s ­
i c s a nd M o d e rn En gin e e r i n g ( New York : M cGraw-H i l l , 1 958 ) , 1 6 1 .
3
Ibid . , 1 64 .
44
Eno ugh t e rm s o f t h e s e r i e s -e xpans.i on o f Equa t i on ( 3 . 4 - 1 ) we re
s umme d t o y i e l d a va l u e a c.c ura t e to s ix d e c imal p l a c e s .
I n ord e r t o d e c i d e t h e numbe r o f s i gn i f i ca nt s id e bands i n a
pa r t i cu l a r s p e c t rum , t h e f ol l ow ing expr e s s i on was us e d
+k
2:
PWR =
wh ich g i v e s
s id e band +k .
u,e
J�(B)
( 3 . 4- 3 )
n=-k
t ot a l p owe r i n t he s p e c t rum from s id e band -k t o
The p r ogram c ompu t e s t h e s id e bands by pa i rs pro-
c e e d ing o u t wa rd from t he carr i e r .
A ft e r each pa i r i s c omput e d ,
t h e f ol l ow i n g d e c is i on i s ma d e i n t h e p r o gram
1 0
•
-
PWR
<
( 3 . 4-4 ) .
TEST
where TEST i s d e f i n e d a s t he amount o f p ow e r t hat may b e d i s c a r d e d w i t h out caus i n g s i gn i f i cant d i s t or t i on .
It is t o be
remembere d t ha t t h e t ota l p ow e r i n t he s p e c t rum i s 1 . 0 f r om
Equa t i on ( 1 . 2 - 25 ) .
The p r o gram c ont i nue s t o c omp u t e s i d e band
pa i r s unt i l t h e a b o ve i n e qua l i ty b e c o1ne s t rue .
The c omputat i on
i s t h e n t e rm i na t e d .
The va l u e o f TEST wa s t entat i v e ly s e t a t 1 x 1 0
-6
i n the
p r o gram p e nd i n g i nve s t i ga t i on of Wan g ' s c r i t � r i on i n Chap t e r 4 .
Fina l ly , t h e pro gram p r i n t s out t h e va l ue of
Jn ( B )
and t h e
f r e que ncy f o r e a c h o f t he s i gn i f i cant s i de bands a l ong w i t h t he
a ppr opr ia t e s i d e band numb e rs .
4
se e Equa t i on ( 2 . 2 - 3 3 ) in Chapt e r 2 .
4
45
Re f e r t o F igure 3 � 1 f o r a s impl i f i e d f l owch�rt o f t h e
s ubprogram BESL .
The c omp l e t e s ubpro gram i s c o n t a i n e d i n Appen-
dix B a l ong w i t h a samp l e e x e c ut i on .
3.5
EVALUATI ON OF THE DI SCRD!I NAT OR OUTPUT : B-1DI ST
Aft e r t he ne twork t ra n s f e r imp e da nc e has b e e n e va l ua t e d
f o r a l l s i d e ba n d fre que n c i e s a n d t he Jt1.-t s p e c t rum h a s b e e n gen-
e ra t e d , the next s t e p i n the s o l ut i on o f t he prob l e m i s t o e va l ua t e t h e d i s c r im inat or out p ut v ( t ) a s g i ve n i n Equa t i on
d
( 2 . 2-32 ) .
The c on t r o l l ing program ca l l e d B-1DI ST wa s d e ve l oped
t o p e r f orm t h i s opera t i on as we l l a s t o a c c e pt data from ECAP ,
c a l l upon BESL t o g e n e ra t e t he F�l spe c t r um , and f ina l ly t o ca l l
upon SERIES t o p erform an harmon i c � na l y s is o n t h e d i s crim inat or
output t o f i n d t h e ma gn i t ud e a nd pha s e of t h e harm o n i c s r e l a t i ve
t o t h e fundam e n t a l w
m
•
Severa l t r i a l r uns of t h e program f o r large m o dul a t i on
i nd ex e s ma d e t h e na t ur e o f Equa t i on ( 2 . 2 - 32 ) pa inful l y e v i de n t .
F o r only t h i r ty s i gn i f i cant s i d e bands , t he c ompu t e t ime f o r
e a c h samp l e o f t h e d is c r im ina t or output ( of a t ota l of f i fty- one
s a mp l e s t a k e n over t he f undame n t a l pe r i od of 2 rr /w ) wa s a l ready
m
i n e x c e s s o f twe nty s e c onds .
Due t o t h e d o ub l e s umma t i on , t h e
c omput e t ime i n c r ea s e d a s t h e s qua re o f t h e numb e r o f s i d e bands .
C o ns e quent ly , hours o f c omput e r t im e wou l d have b e e n re q u i r e d t o
s o l ve only
a
few pro b l ems .
T h i s s i t ua t i on s e em e d unavo ida b l e
a n d w o u l d h a v e b e e n a s er i bus drawba c k t o t h e u s e o f t h e program .
46
From
Na i n Pr o gram
C ompu t e
B
C ompu t e
J for
0
Carr i e r
C omput e
n , J , fn
f or �ext S e t
o f S i de ba nd s
C ompu t e
T ot a l S i d e ba nd
Power
PWP
No
Wr i t e
n , ,J n , f 0 f o r
A l l. S i d e bands
Fi gure 3 - 1 .
Re t urn
to
a i n Program
S i mp l i f i e d Fl owchart f o r S ubpr o gram BESL .
47
A f t e � c ons i d e ra b l e t h o ugh t on ways t o incr ea s e t he s p e e d
e ff i c i e n cy o f t h e pro gram had fa i l e d t o y i e l d any � i gni f i cant red uc t i o n i n c ompu t e t ime , a c a s ua l r e - exami na t i on o f t he
d e r i va t i on o f Equa t i on ( 2 . 2 -32 ) prov i d e d a s tart l i n g d isc overy
t ha t r e s u l t e d i n i n c r ea s i n g t h e s p e e d o f exe c ut i on of the program n t im e s over the pre v i o us spe e d where n is the numbe r of
s id ebands .
T h i s grea t i n c r e a s e in s p e e d was a c h i e ve d s imply by
u s i n g Equa t i on ( 2 . 2-26 ) t o e va l ua t e t he d i s c r im i na t or out put
ra t h e r t ha n Equa t i on ( 2 . 2 - 32 ) .
The t w o e qua t i ons a re ident i c a l
a s proved i n Equa t i on s ( 2 . 2 -26 ) through ( 2 . 2- 32 ) .
H oweve r , t h e
f i rs t cont a ins only s i n g l e s umma t i ons whereas t h e s e c ond has
d oubl e s umma t i ons .
Thus , t h e c omput e t ime us ing Equa t i on
{ 2 . 2 -26 ) i n c re a se s l inearly w i t h t he numb e r o f s id e bands ra t h e r
t ha n a s t h e s quare o f t h e numbe r o f s id e bands .
F o r t h irty
s id e bands , t h e n , the r e la t iv e increa s e i n s pe e d is t h i rty t im e s .
T h e program t h us t ook 0 . 6 6 s e c on d s t o d o a c omput a t i on t ha t
pre vi o us ly r e qu i red 2 0 s e c on d s .
U s i n g t he n o ta t i on o f Equa t i o n ( 2 . 2- 32 ) , E q ua t i on ( 2 . 2 -26 )
may b e exp r e s s e d a s f o l l ows
d
vdu
e( t ) =
2
dt
u
whe re
v
=
+ 00
n= - ro
udv
2
+
v
J P c os ( nw t + e ) '
n n
m
. n
( 3 . 5-l )
{ 3 . 5- 2 )
·
48
dv =
2:
-
n= - oo
+m
{
v
2
+ 00
n= - oo
+m
u =
n= - m
nw J P s in ( nw t + Q ) ,
m n n
m
n
( 3 . 5- 3 )
Jn Pn
( 3 . 5- 4 )
' JnPn s i n ( nwm t
+ 00
du =
c os ( nw t +
m
m
nw J
n= - m
P
n n
+ Q )
n
,
c os ( nw t + Q )
n
m
( 3 . 5 -5 )
,
( 3 . 5- 6 )
and
+m
n = - ro
J npn s in ( nwmt
( 3 . 5- 7 )
I n p e rf orm i ng t h e s ummat i ons i n t h e program s e ve ra l s ub s t i t ut i ons a re mad e wh i c h grea t ly s imp l i fy Equa t i ons ( 3 . 5- 2 )
through ( 3 . 5 - 7 ) .
T h e p r o gram proc e e d s t h r o ugh t h e s t eps o f
c omput ing t h e va l ue s o f t h e s e e xp re s s i ons f or t h e k ' t h s i deband
as f o l l ows
(l)
X
(2)
y
(5)
(6)
dv
(4)
(7)
--·- - · · .
=
z
Jkpk
= kw t + Q
vk =
uk
(3)
-·------
=
X
X
m
c os
y
s in
y
= kw
m
k
�-�1<.
= -Zu
=
n
k
�-�!<_:
_ _
·-··- - · - - ··-
--- --
-- · -
.
49
The s e s t eps a re repea t e d n t im e s wh e r e n i s t he numb e r of s i d e Ea c h t im e t h e s e qu e n c e is r e p ea t e d the new va l ue s f or
bands .
v ' u , d v , a nd d u a r e a d d e d t o t h e p r e v i o us va l ue s .
k
k
k
k
I n t h is
manne r t h e s umma t i on i s p e r f orme d y i e l d i n g t h e f o l l ow i n g resul t s
(l)
A
=
(2 )
B
=
(3)
c
(4)
D
=
=
l:
u
k
v
l: k
l: d v
k
� du .
k
T h us , Equa t i on ( 3 . 5 - 1 ) i s s imply
g i ve n
by
BD - A C
2"
2
+
A
B
( 3 . 5-8 )
I n ord e r t o norma l i z e the d is c r im i na t or output and t o mah:e
it independ e nt of t h e d e via t i on w , Equa t i on ( 3 . 5 -8 ) is d iv i d e d
d
by w
d
to y i e l d
( 3 . 5- 9 ) .
T h i s i s t h e expr e s s i on f o r t h e re s p on s e e mp l oy e d in t h e program .
For t he c a s e o f no d is t ort i on , t h e out put b e c omes a c os ine wave
·
w i t h un i ty a mp l i t ud e r e ga r d l e s s of t h e d e v ia t i on .
The wh o l e p r o c edure d e s c r i b e d a b ove is p e r f o rm e d f i fty - one
t im e s f or f i fty- one va l ue s of
t .
The t ime is s t e p p e d e ve n ly
ove r t h e fundam e n t a l pe r i od o f 2 rr /w i n fi fty increment s .
m
The
r e a s on b e h ind t he cho i c e o f f i fty - one a s t h e numbe r o f samp l es
w i l l b e d i s c us s e d l a t e r i n t h e f ol l ow i n g s e c t i on c on c e rn e d w i t h
t h e ha rm o n i c a na lys i s of t h e d i s c r im i na t or output .
Re f e r t o Fi gure 3 - 2 f o r a s impl i f i e d fl owcha rt o f t h e
50
Start
P ,
n
e
1-lf------ ECAP Dat a
n
Read
Ca l l BESL
BESL
C omput e and
Pr i nt Out
n, J , f
n
n
C omput e
v ( t ) f or
d
T ime
51
I nc reme n t s
Cal l SERI ES
F i gure 3-2 .
SERI ES
C omput e and
Pr i n t Out
Four i e r
C o e f f i c i e nt s
a o , bn , e n
S i mp l i f i e d Fl owchart for Fr>fDI ST .
51
p r ogram
1'1-WI ST .
The c ompl e t e program is c onta ine d i n A ppendix
C a l ong w i t h a samp l e e xe c ut i on .
3.6
HAm,I ONI C A NALYSI S
OF
THE
D I SCRHi i NATOR OUTPUT :
SERIES
The f i na l s t e p i n t he s o l ut i on of the prob l e m is t o perform
i
A s ubrout ine
a n harm o n i c a na lys i s on t h e d i s c r i m i na t or output .
s ubpr ogram c a l l e d
SERI E S
was d e ve l op e d t o p e r f orm this f unc t i on .
T h e s ubpro gram takes t h e f i f ty- one samp l e s of v
dN
( t ) and f i nds
the Four i e r s e r i es expans i on through n ume r i ca l i nt e gra t i on
t e c h n i ques t o be d e s c r i b e d .
The o utput o f t he s ubprogram i s
t h e Four i e r c oe f f i c i e n t s o f t h e harm on i c s u p t o t h e t en t h harmoni c pl us t h e i r l ev e l , re l a t ive t o t h e f undament a l , i n
d e c ibe l s ( dB ) .
T h e f ol l ow i n g i s a d i s c uss i on o f t h e proce dure
us e d t o p e r f orm t h e c omputa t i on .
I t i s we l l known that a p e ri o d i c f unc t i on f ( x ) may b e
expanded i nt o a Four i e r s e r i e s a s f o l l ows
f(x ) = a
0
+
ro
L:
n= l
(b
n
s i n nx
+
c
5
n
c os nx )
( 3 . 6- 1 )
whe re
1
a =
0
2 11
5
2 11
�
0
f ( x ) dx
( 3 . 6-2 )
soko l n i k o ff and Re d h e f f e r , op . c i t . , 1 75 - 1 8 2 .
I t i s r e q u i r e d that f 1 x ) sa t i s fy t h � D ir i c h l e t c ond i t i ons .
Tha t v ( t ) sa t is f i e s t h e s � c ond i t i ons i s obvi ous s in c e i t i s
dN
c ompose � of a f in i t e numb e r of s ine and c os i ne funct i ons o f w
m
a n d i t s harmon i c s .
.
52
b
n
=
1
TT
=
1
TT
2 TT
�
0
and
n
c
( 3 . 6- 3 )
f ( x ) s in nxdx
2 TT
� f ( x ) c os
0
nxd x .
( 3 . 6-4 )
Suppo s e t ha t we d iv i d e one p er i od ·o f f ( x ) i n t o k e qua l
s l i ce s of w i d t h 2 TT /k a s s h own i n F i gure 3 - 3 and t hus samp l e
f ( x ) k+ l t imes over o n e c omp l e t e p e r i od .
Equa t i on ( 3 . 6 - 2 ) may
be rewri t t e n a s t h e s um o f k i nt e gra ls a s f o l l ows
a
0
=
k
:L
1
2 TT
i=l
2 rr i
k
�
2 TT
-
k
f ( x ) dx
( 3 . 6- 5 )
( i-1 )
wh i c h i s a c ompl e t e ly e qu i va l ent express i on .
Now we s ha l l e mp l oy t h e . Trape z o i da l Rul e o f n um e r i ca l
i nt e grat i on t o approx ima t e t h e m ' th i n t e gra l in the s e r i e s o f
E qua t i on ( 3 . 6 - 5 ) .
6
Thus ,
2 rr m
k
�
2 TT
k
f ( x ) dx
_
TT
k
(m-1 )
[
f
[ ?; m J
+ f
[ 2krr ( m- l )] J
( 3 . 6- 6 )
o r s imply
-
We may now express a
6
0
:.
{
f (m)
+
f (m + l )
J.
( 3 . 6-7 )
a s f o l l ows
G . B . Th omas , J r . , Cal c u l us and A na lyt i c Ge ome t ry ( Re a d i n g ,
1-ta s s . : Ad d i s on Wes l ey , 1 95 1 ) , 207 - 2 1 1 .
53
f(x)
0
1
2 77
2
3
4
Fi gure 3 - 3 .
5
6
k- 1
Samp l i ng o f f ( x ) .
k
k+l
X
54
a
k
=
0
L
1
2 11
i=1
or
a
0
=
k
L
1
2k
i=1
]
:: [ f ( i )
+ f( i + l )
[f
(
(i) + f i + 1
(3. 6-8 )
)J
(3.6-9)
I n a ma nne r s im i la r t o t h e a b ove we may arrive a t t h e
!
J
f o l l owing e x p r e s s i ons f o r b
b
n
k
1
= 2k L [ ( )
i=l
f i sin n
n
2 11
k
and c
(
i
-
n
1
)
)
+ f ( i + l s in n
2
:i
J
(3.6-1 0 )
a nd
k
c
n
= ;k i2:= l [ ( )
(3.6-9)
f i
Equa t i ons
C OS
n
2 11
k
( 1) f(i
t hr o ugh
i
-
+
(3.6-11)
+ 1 ) c os n
2 11 i
k
J
.
(3.6-11)
may b e us e d t o c omput e
t h e F o ur i e r c o e f f i c i en t s o f t h e ha rmon i c s of t h e d is c r im i na t or
output .
I t now rema i ns t o c h oo s e a va lue f or
Appendix E t ha t f or
k=50,
k.
I t i s shown i n
t he e rror i nc ur r e d i n c omput i n g t h e
Four i e r c oe f f i c i e nts f o r t h e t e nth harm o n i c i s only a b out
dB .
0 .28
I t i s r e qu i r e d t h a t s l i gh t ly grea t e r than t w o samp l e s per
cyc l e be ta ken i n order to p e r f orm the t rape z o i dal i n t e �ra t i on
w i t h a ny d e gre e o f · a c cura cy .
For
k=50
t h e re are f i ve sampl e s
p e r cyc l e a t t h e t en t h harm o n i c s o t ha t good a c c ura cy i s t o b e
expe c t e d .
I n a d d i t i on t o c omput i ng t h e Four i er c oe f f i c i e nt s , t h e
55
p r o gram a l s o c ompu t e s t he e qu i va l � n t peak amp l i t ud e o f t h e har:...
m o n i e s f r om t h e f o l l owing expre ss i on
A =
-J
c
2
( 3 . 6-12 )
n
S i n c e only e n ou gh s i d e bands a r e reta ined by s ub r out ine
BESL t o ke e p t he d i s t ort i on d ue t o d ropp i n g s id e bands be l ow
-60 dB , t h e h a rmon i c l e ve l i n d B i s n o t p r i n t e d out by s ubpr o-
gram SERIES i f it is l e ss t ha n that l e ve l .
I ns t ea d , a note i s
p r i n t e d out t o i n d i c a t e t ha t t h e l e ve l i s l e s s than - 6 0 d B .
Re f e r t o Fi gure 3 - 4 f or a s impl i f i e d f l owchart o f subprogram SERI ES .
T h e c omp l e t e program is c onta i n e d in A ppendix D
; a l on g w i t h a samp l e e x e c ut i on .
3.7
THE
QUA SI - STATI ONARY ill� SPONSE : QS
I n o r d e r t o c ompa re t h e quas i - s ta t i onary approxima t i on t o
t h e exa c t Four i e r s o l ut i on , a s h ort p r o gram ca l l e d Q S was d e v e l oped .
T h i s p r ogram s imply gene ra t e s t he phas e chara c t e r is t i c
f or a s ingl e -p o l e f i l t e r a nd t h e n proc e e d s t o c omp u t e t h e d i s c r i m inat or o u t p u t v N ( t ) g i ve n by
d
V
dN
(t )
=
!
d
V
d
( t ) = C os wmt +
!d :t l [w i ( t )J
f o r f i f ty- one t ime increments where v
( 2 . 4-7)
d(
( 3 . 7-1 )
t ) i s give n i n Equa t i on
and
T h e phase chara c t e r i st i c o f a s i ngl e p o l e -f i l t e r
t<w )
( 3 .7- 2 )
i s give n
56
f --�
I!
C omput e
Four i e r
C oe f f i c i e nt
C omput e
Four i e r
C o e f f i c i en t s
b nand e n
C omput e
Magn i t ud e s and
Re l a t i ve Leve l s
o f Ha rmoni c s
A a nd d B
Fi gure 3 - 4 .
S i mp l i f i e d Fl owchart o f S ubpr o gram SERI ES .
57
� (w )
. 2
1
- tan-
(3.7-3 )
R ( l - w LC )
wL
=
where R i s t h e r e s i s tanc e , L i s t h e induc tanc e , and C i s t h e
c a pa c itanc e .
I n t h e p r o gram , t h e w o f Equa t i on
w 1 ( t ) given by Equa t i on
i
f [ w i ( t )J
��.7-2) .
(3.7-3)
i s r e p l a c e d w it h
T h e t ime -ra t e - of - change o f
i s c omput e d num e r i ca l ly a t e a ch t ime i ncrement from
t h e f o l l ow i n g e xpress i on
d
dt
where t i s t h e part i c ul a r t ime s t e p i n t h e range 0 � t �
and T
0
0
rr / w
m
i s g i v e n by
T
T
(3.7-4)
2
0
=
1 0020 TI
(3. 7-5)
w
m
was c h os e n t o have t he a b ove va l ue b a s e d up on s e v e 1 � l samp l e
runs o f t h e p r o gram .
Furt h e r reduc t i on i n t h e va l u e o f T
p r o d u c e s l i t t l e increase i n the a c c ura cy o f c omput i n g
d
dt
A f t e r Equa t i on
(3.7-1)
0
has b e e n e va l ua t e d a t a l l f i fty- one
t ime s t e ps , QS ca l l s upon S ERI ES t o p e r f orm a n harmon i c a na lys i s
o f t h e d is c r im ina t or output .
The p r o gram QS was n o t i n t ended
fu
b e gen e ra l ly use f u l but
was prepa r e d p r ima r i ly for the purp ose o f mak in g t h e c ompa r i s on
b e twe e n t h e Four i e r and qua s i -stat i ona ry s o l ut i ons .
The c ompl e t e QS p r o gram i s l i s t e d i n Append ix F a l ong w i t h
58
a sampl e e x e c ut i on .
chart of QS .
F i gure 3 - 5 r e pre s e n t s a s impl i f i e d f l ow­
59
Start
C omput e
v
(
dN t )
Ca l l SERI ES
Fi gure 3 - 5 .
SERI ES
C omput e and
Print Out
Four i e r
C o e f f i c i ent s
S imp l i f i e d Fl owchart o f Pro gram QS .
60
4.
4.1
C I RCUIT I NVESTI GATI ONS
OUTLI NE OF THE STUDY
To i l l us t ra t e t he u s e f u l n e s s of t h e programs d i s c u s ::; e d i n
Cha p t e r 3 , t h e e ff e c t s o f s e ve ra l c ommon t y p e s o f bandpa ss
f i l t e rs upon a F�1 wave were i nves t i ga t e d .
I n addit ion , the
e f fe c t of d r op p i n g s id e bands was inve s t i ga t e d i n ord e r t o de ­
v e l op a c r i t e r i on t o d e t ermine the numbe r o f s i gn i f i cant s id e ­
bands i n a g i v e n FM spe c t rum c ons i s t e nt w i t h a s pe c i f i e d a mount
of a l l owab l e d is t ort i on .
For the c a s e of the s ingl e - p o l e f i l ­
t e r , t h e F o ur i e r a nd q ua s i - s t a t i onary s o l ut i ons a re c ompare d .
The bandpa ss f i l t e rs c ons i d e r e d i n t h e s t udy were as
f ol l ows :
(l)
s i n gl e- p o l e f i l t e r , 1 0 p e r c e n t b a nd w i d t h ;
l
(2)
t w o-p o l e f i l t e r , unde rc oup l e d , 6 p e rc e nt bandwi d t h ;
(3)
t w o - p o l e f i l t e r , But t e rwort h , 1 0 p e rc e nt bandwid t h ;
(4)
t w o - p o l e f i l t e r , T c h e bys he v , 1 0 p e r c e nt bandwi d t h .
T h e s e f i l t e rs w i l l be d i s c us s e d i n more d e ta i l i n t he f o l l ow i n g
s e c t i ons .
1
F o r informa t i on re gard ing t h e d e s i �n of such f i l t e rs ,
Ba ndwid t h r e f e rs t o t h e - 3 d B ba ndw i d t h .
61
t he rea d e r i s r e f e rr e d t o t h e two s ourc e s l i s te d b e l ow .
2
I n o rd e r t o make t he r e s u l t s of t h e i nves t i ga t i on a s
general a s p o s s i b l e , t h e f i l t e r pa rame t e rs have b e e n l e ft i n
n o rma l i z e d f orm .
Tha t i s t o say , t h e r e s ona nt fre que n cy o f the
f i l t e rs i s 1 Hz and t he i nput a nd out put i mpe dan c e s a re 1 ohm .
T h e fre quency may t h e n b e s c a l e d t o any va lue d e s i r e d t o s u i t
t h e n e e ds o f a pa r t i c u l a r prob l e m .
l
F or t he n o rma l i ze d ca s e , a
1 0 p e r c e nt bandw i d t h w o u l d c or r e s p o n d t o 0 . 1 Hz .
The a mp l i t ud e
a n d pha s e r e sp onse o f a 1 00 M H z t w o -p o l e But t e rwor t h f i l t e r w i t h
a 1 0 MHz bandw i d t h , f or e xamp l e , ·woul d c orre s p ond e xa c t ly t o a
1 H z two- p o l e But t e rw o r t h f i l t er w it h a 0 . 1 H z bandw i d t h .
The f i l t e rs d e s c r i b e d a b ove were exami n e d f o r m o d ula t i on
i nd exes ran g i n g f rom two t o twe l ve i n i ncrements o f two .
A
m o d u la t i n g fre quency o f 0 . 005 Hz was c h os e n s o t ha t a modula t i on
i n d ex of t e n woul d c or r e s p ond t o a p e a k - t o - p ea k d e v i a t i on o f
0 . 1 Hz ,
or e qu i va l e n t t o t h e - 3d B bandw i d t h .
T o obta i n t he n e t w ork t ra n s f e r imp e da n c e i n f orma t i on , an
ECAP AC c i rc u i t analys i s was p e r f orme d o n each f i l t e r ove r t h e
f r e que ncy ran g e f r om 0 . 9 Hz t o 1 . 1 Hz i n i n c re m e n t s o f 0 . 005 H z
c orresp ond i n g t o e a c h p oss i b l e s id e ba n d f r e quency .
The f i l t e r
r e s p onse s i l l us t ra t e d i n t h i s chapt e r were p l o t t e d f r om t h e ECAP
.
G . L . Ma t t ha e i , L . Youn g , and E . �1 . T . Jones , �! i c rowave F i l t e rs , I m p e d an c e -Ma t c h i n g N e t wo rks , a nd C o upl i n �; St ruc t ures
( New York : M c Graw-H i l l , 1 9 6 4 ) � and Re f e re n c e Da ta f or Ra d i o
Engine e rs , F o urt h Ed i t i on ( New York : I nt e rna t i ona l T e l e ph one a nd
. Te l e graph C orp ora t i o n , 1 9 5 6 ) .
2
62
The da ta obta i n e d f r om t h e
ECAP
a na lys i s w a s t he n u s e d i n
p r o gram 1'1-l- DI ST t o c ompu t e t h e f i l t e rs ' r e s pons e s t o t h e f o l l owing
FM
wa v e s
i (t )
where B
=
=
s in ( 2
TT
2 , 4 , 6 , 8 , 10 , 1 2 .
t
+
B
s i n 0 . 005 2
TT
t)
( 4 . 1-1 )
The output o f t h e program conta ins
t h e Fi\1 spe c t rum , t he d i s c r im ina t or o ut p u t versus t im e f or one
cy c l e o f the m o d ulat i ng f r e quency , and an harmon i c a na lys i s of
t he d is c r i m i na t or output .
Due to t he ext reme ly l a rge v o l ume of
d a t a c rea t e d , only the r e s ul t s are pre s e n t e d i n t h e f o l l owing
; s e c t i ons .
( Comp l e t e s amp l e program e x e c u t i ons may b e f ound i n
t he append i c e s . )
I n e a c h c a s e c ons i d e re d , t he ha rmon i c ana lys i s
a n d p e rc e n t d i s t ort i on a re g i v e n .
I n ca s e s where t h e d i s t ort i on
was large e no u gh t o s e e v i s ua l ly , a graph of t h e " d i s c r irn i na t or
o u tp ut v e r s us t i me i s p r e s e n t e d t o i l l us t ra t e t h e e f fe c ts of t h e
d i s t or t i on on t h e d e s i r e d wav e f o rm .
4.2
SIDEBA ND
ELHH NATI ON :
WANG ' S CRI TERI ON
I n c omput ing t he r e s p ons e s o f t h e f i l t ers t o a FM exc i t a _
t i on i t was d e s i re d t o k e e p a ny d i s t ort i on due t o n e g l e c t i n g
s i d e bands be l ow - 6 0 d B .
T o d e t e rm ine t h e a mount of s i d e band
p o w e r tha t ha d t o b e r e ta i n e d i n the
FM
s p e c t rum t o a c h i e ve t ha t
g oa l , t h e f ol l ow i ng i nv e s t i ga t i on was p e r f o rme d .
U s ing s ubpr o gram
BESL
t h e l'f.i spe c t rum was ge ne ra t e d f or a
m o d ula t i ng fre que ncy of 0 . 005 Hz , a d e v i a t i on o f 0 . 005 Hz , and
I
s
63
a carr i e r fre quency o f 1 Hz .
s i d e bands were r e t a i ne d .
s i d e bands . )
As a sta rt i n g point , f ive pa i rs of
( Re f e r t o Tab l e 1 f or a l i s t · o f t h e
Pro gram FMDI ST was t h e n us e d t o c omput e t h e d i s -
c r im i na t or ou t put .
A ne twork w i t h un i ty f l a t amp l i t ud e and
z e r o pha s e sh i ft wa s a s s ume d so that n o d i s t or t i on w o u l d be
c a us e d by any n e tw ork nonl inear i t i e s .
The res ul tant d i s t ort i on ,
i f any , wou l d b e due only t o t h e e l im i na t i o n of s i d e bands .
The s i d e ba nds were t h e n e l imina t e d one pa i r a t a t ime
s t a rt i ng w i t h n=±5 and a new a nalys i s was p e r form e d in e a c h
c a s e by Fr.IDI ST .
T h e va l u e s of t h e Four i e r s e r i e s c oe ff i c i e nts
( as d e f ine d i n Equa t i on ( 3 . 6 - 1 ) ) f or t he d i s c rimina t or output
f or each c a s e are g i ve n in Ta b l es 2 thro ugh 6 .
The r e s u l t s
obta ined a re c ompa r e d w i t h Wan g ' s r e s u l t s i n Tab l e 7 .
3
Fi gure
4 - 1 i l l us t ra t e s t h e d i s c r i m iria t or output f or t h e c a s e in wh i ch
only t h e f ir s t pa i r o f s i d e ba nds a re r e ta i ned .
S i n c e t h e F o ur i e r c oe f f i c i e nt a
i s n o t l i s t e d i n the t a b l e s .
0
wa s z e r o i n a l l ca se s , i t
A n a s t e r i s k ( * ) i n t h e c o l umns
lab e l l e d Re l a t i ve Jrnrm o n i c Leve l " i nd i ca t e s t ha t t he d i s t or t i on .
i s grea t e r t ha n - 6 0 d B d own re l a t ive t o t h e f undam e n t a l c omp o n e nt .
3
H . S . C . Wan g , " Ba n d w i d t h Re q u i rements f or Fre quency-�lodu­
l a t e d S i gna l s , " Proc e e d i n �s I EEE , 5 3 , No . 8 ( Augus t , 1 965 ) , 1 1 50 .
64
S i d e ba n d
Numbe r
n
n
J (B)
S i d e band
Fre quency
Hz
B
=
1
-5
0 . 975
-0 . 0002
-4
0 . 98 0
0 . 0025
-3
0 . 985
-0 . 01 96
-2
0 . 990
0 . 1149
-1
0 . 995
- 0 . 4401
0
1 . 000
0 . 7652
1
1 . 005
0 . 4401
2
1 . 01 0
0 . 1149
3
1 . 015
0 . 01 96
4
1 . 020
0 . 0025
5
1 . 025
0 . 0002
Tab l e 1
FM S p e c t rum f or B
=
1.
65
Harmoni c
Numbe r
n
1
2
::;
4
5
6
7
8
9
10
Ta b l e 2
Harmoni c
Numbe r
n
1
2
n
c ( c os )
o.o
0.0
0.0
o.o
o.o
.o . o
10 . o
0.0
o.o
o.o
0 . 99999
0.0
o.o
o.o
o.o
0.0
0 . 0.
o.o
o.o
o.o
b (sin)
::;
:3
n
Re l a t i v e
Harm o n i c Le·:e l
dB
o . oo
*
*
*
*
*
*
*
*
*
F o ur i e r C o e f f i c i en t s o f D i s c r im i na t or Output
w i t h F i ve Pa i rs of S i de ba nds .
Four i e r C oe f f i c i e n t s
b ( sin )
n
o.o
o.o
0.0
o.o
o.o
o.o
0.0
o.o
0.0
o.o
4
5
6
7
8
9
10
Tab l e
Four i e r C o e ff i c i e n t s
c ( cos )
n
0 . 99999
o.o
- 0 . 000 1 6
o.o
- 0 . 00182
o.o
0 . 00022
o.o
o.o
o.o
R e l a t i ve
Harm on i c Leve l
dB
o . oo
*
*
*
-54 . 7 9
*
*
*
*
*
F o ur i e r C o e f f i c i e n t s o f D i s c r imina t or Out put
wi t h Four Pa i r s of S i d e ba nd s .
66
·
Harm on i c
Num b e r
n
1
2
3
4
5
6
7
8
9
10
Tab l e 4
Harm on i c
Nwnbe r
n
1
2
3
4
5
6
7
8
9
10
Tab l e 5
Four i e r C oe f f i c i e nt s
n
b ( s in )
o.o
o.o
0.0
0.0
0.0
o. o
o.o
o.o
0.0
o.o
c ( c os )
n
0 . 9 9 9 99
o.o
- 0 . 00649
0.0
0 . 00809
o.o
0 . 00029
0.0
o.o
o . o.
Re la t i ve
Harm o n i c Leve l
dB
0 . 00
*
- 4 3 . 74
*
- 4 1 . 86
*
*
*
*
*
Four i e r C oe f f i c i e nt s o f D i s c r im i na t o r Out put
w i t h Thr e e Pa i r s of S i d e ba nds .
Four i e r C o e f f i c i en t s
n
b ( sin )
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
0.0
c ( c os )
n
0 . 99517
o.o
- 0 . 0 9 3 90
0.0
- 0 . 0 1 34 9
o.o
0 . 00083 .
0.0
0 . 0001 6
0.0
R e l a t i ve
Harm o n i c Leve l
dB
0 . 00
*
- 20 . 5 1
*
- 37 . 35
*
*
*
*
*
Four i e r C o e f f i c i e n t s o f D is cr i m i nat or Output
w i t h Two Pa i rs o f S i d e ba nds .
67
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Tab l e 6
F o ur i e r C o e f f i c i e nts
n
c ( c os )
o.o
o.o
o.o
o.o
o.o
o.o
o.o
0.0
o . o·
o.o
0 . 91144
o.o
0 . 1 8 332
o.o
0 . 03567
0.0
0 . 00671
o.o
0 . 00 1 2 1
o.o
b { sin )
n
Re l a t ive
Harm on i c Leve l
dB
0 . 00
•
- 1 3 . 93
•
-28 . 15
•
- 4 2 . 66
•
-57 . 52
•
Four i e r C o e f f i c i e nts of D i s c r i m i na t or Output
w i t h One Pa i r of S i d e bands .
68
Re la t i ve Le ve l o f D i s t ort i on
Numbe r of
Pa i rs of
S i d e ba nds
�1DI ST
* *
5
4
WANG
77
dB
4
54 . 8 d B
55 dB
3
39 . 8 dB
40 dB
2
20 . 5 d.H
21 d B
1
1 3 . 8 dB
13 dB
* * Great e r t ha n - 6 0 d B d own .
Tab l e
4
7
Re la t i ve Leve l o f D i s t ort i on i n D i s ­
c r iminat or Out put D u e t o S i de band
El iw i na t i on . Wan g ' s Resu l t s C ompared
with Th ose o f A u t h or .
Wang , o p . c i t . , 1 1 5 0 .
69
From Ta b l e s 3 through 6 i t ip a pparent t ha t t h e d i s t or t i on
produced b y - e l imina t i n g s id e bands by pa i r s c onta i ns only odd
orde r harmon i c s .
Furt h e r , i t is obse rve d tha t the d i s t ort i on
c ompone n t s c onta i n only c os i ne t e rms .
T h i s s i t ua t i on r e s ul t s
i n t h e symm e t r i ca l d i s t ort i on o f e a c h p e a k o f t h e d i s cr i m i na t or
o ut put a s sh own i n F i gure 4 - 1 .
Exc e l l e nt a gre ement wa s f ound b e twe e n t h e re l a t i ve d i s ­
t ort i on l e ve l s ca l c ula t e d b y �IDI ST a nd t h ose o f Wa n g a s s h own
in Tab l e
7.
T o ke e p t h e d i s t or t i on d ue t o d r op p i n g s i d e bands b e l ow
- 6 0 d B , e n ou gh s i de ba nds s h o u l d be c ons i d e re d s o t ha t t he t ot a l
; p ower in t he FM s p e c t rum i s grea t e r t ha n 0 . 9 9 9 9 9 9 t ime s t h e
p ower i n t h e unmodula t e d ca r r i e r wh i ch h a d b e e n a r b i t ra r i ly s e t
a t unity .
T h i s c a s e c orresp onds t o t h e s p e c t rum g i v e n i n
Ta b l e 1 a n d t h e F o ur i e r c o e f f i c i en t s of t h e d i s cr i m i na t or output
g i ve n in Tab l e 2 .
The a b ove c r i t e ri on wa s use d i n t h e f i l t e r i nve s t i gat i ons
d e s cr i be d in t h e f ol l ow i n g s e c t i ons of t h i s chap t e r .
70
I
I
I
I I
�++�-r+++1�rr-t1 i1-+-Hc-�--++-t--f
'
I
�
I I j_ JI
� .
I
I
I
I
i i I
-lr-+-��-+-++-+�1-+-rr-l-1-1-+-+ -�-r-++--+-H-+-+�1
-+-l- +-tl
l-+-+-+-lc-l-++-f-l-l--+-f-l-+-+-+l-t--f-l--+-+·+-+-+-l-1-+-+-+-+-l- +-+-+-++-+
I .
1.- f-.-u
-r 1
I
I
1
t
.......
z
'tl
>
0
+- -f-1-i1 +��-t-+--1-fr+-t-+ 1 1
,
c-1
f-H
--r--+-t--1--t-+-+--+--t
H-+�
-
-r\�H-M
•tt_j_ LLL -+
1 I JLW-L
· H--t--+--t--l-f---1H
, 1
. I
i
! l
; I . I
1 I
1-+-+++-l-t-+-+-+--t-t-+-++-+1-t-r+++-+-1-+-+-+--+-t�-+-+-t-
-1
•
-
I I ... ....
I -H-r1
l -++-t-· +-+t l�_;t_t_j-_=tl_:=-+I1---+t-:---+-+1 _j-_=t�_;_t-l_:t-+-j+-t-+-t--+t,l--lt-H-+-+-+-1�tt=t=1=t+�t-:t-=,�·\:t:_ t1�tI -�+-t..
dI
I
-t 1- � -l-f-l--l-t--t--ll-fl l--l-+
l--+1---1ic-l-+- ++-+_ 1-+-+1- -H
L I
1
ril l
0
I
I
=
I
I
I
I
I
I I
-1-+I
r +H-I Hi-
_
-n
_, 1
l
I l - --1_p
� r-J
±i'
I
I
+- !I
t -�>�
I I
I
I
I
-
++,
---1-:JI
t
--r-1
I I
:::0
1 f
D i s c r im ina t or Output v N ( t ) Versus T ime For
d
1 and Only the First Pa i r o f S id e bands R e t a i ned .
Fi gure 4 - l .
B
I
+H H�
m
71
4.3
SI NGLE- P OLE FILTER
I n t h is s e c t i on , t h e r e s p on s e of a s in gl e - p o l e f i l t e r t o a
FM exc i t a t i on i s pre s ent e d w i t h a c ompa r i s on b e twe e n t h e qua s i -
s t a t i onary a pproxima t i o n a n d t he Four i e r s o l ut i on .
d ia gram o f t h e f i l t e r i s s h own i n Fi gure 4 - 2 .
A s c h emat i c
F i gure s 4 - 3 and
4-4 i l l us t ra t e t he ampl i t ud e and pha se chara c t er i s t i c s o f the
f i l t er ' s s ta t i c r e s p ons e as c omput e d by ECAP .
The FM waves use d i n t he . c omputa t i ons a r e g iv e n by Equat i on
( 4 . 1-1 ) .
The modul a t i on i nd ex e s c ons i d e r e d c ove r e d t h e range
f r om t wo t o t w e l ve .
The F o ur i e r c oe f f i c i en t s of t h e d i s c rimin-
a t or output f o r each case a re l is t e d i n Tab l e s 8 t h r ou gh 1 3
a l ong w i t h t h e p e r c e n t d i s t ort i on i n e a c h case .
None o f t h e d i s c r im i na t or out put s h a ve b e e n p l o t t e d f o r t h e
c a s e o f t h e s in gl e - p o l e f i l t e r d u e t o t he l ow d is t ort i ons obt a i ned .
( I n genera l , i f t he d i s t ort i on i s grea t e r t han 35 dB
d own , i t i s d i f f i cul t t o s e e v i s ua l ly . )
The re l a t i v e amp l i t ude
of each ha rmo n i c i.n d B is a l s o g iv e n a l ong w it h t he e qu i va l e n t
p eak amp l i t ude A , whe r e
2 }�
+ c )
n
( 4 . 3- 1 )
•
I t should b e kept i n m i nd wh i l e r e a d i n g t he f ol l ow in g t a b l e s
t ha t t h e peak- t o -peak d e v ia t i on t o bandw i d t h ra t i o P i s given by
p
=
2 fd
BW
=
2f B
_!!!_
BW'
=
( 2 ) ( . 005 ) B
0.1
=
O . lB .
( 4 . 3-2 )
Thus , a m o d ul a t i on index o f t e n c orre s p onds t o a peak-t o-peak
72
O . Ol 5 9 H
1 . 5 9F
F i gure 4 - 2 .
H1
Schema t i c D i a gram o f S i n g l e - P o l e F i l t e r .
.
73
.....
-j-
!-•
r)""
t�
II
1--
�-
r-�
1--1--
!--
j-
I
Fi gure 4 - 3 .
I
I
Amp l i t ud e Response o f S in gl e -P o l e F i l t e r .
I
74
1
1
H-
H-
I
I II I ��--�
�' I I I
-, I I I ..
I : -f-i- ,I
I
I
'
I -',
!1
++-i 1 -+-+-++
_LLJ
,: �::· i
' ' , ' ' ,
J:LS.r-:±!± 1 1 1
I f
' ±t±+--++++ H-+---8--+t-t+
H-1-' LJ+
1-l+t:fH+l=m�+R:�f#
-H'
yt-:-:- +H+
++HI r :'I :I u: !Mi±t
. ·�:
: : :• : : < < <
t+� -tffi H+: t-H-j_ cH:t+fAA
: ::
< +-H
:
I
I H=Et ' ' w·-++t+
'
: h � JJ± :r
:
I' :I
r- . l- -+++- LL J_ft:t++LL
: : : �--;_J. +H-t-1-++,-t +W�++ .-=q:,q=
' I
�: �
I
1
+++· , fTi I : l t
' :- : ,·-LLL
,· , 1_l_ LL
,· ++l+: rt++: -; H-' : :, : %Y- ; : : �
-LL-t- · _ :
±i±t _U_l_�_ �'. _:__:_ 1- -�+, r-1,.
,
,TlTIT ,-,
Lt+�
:
,
: :: ,
,
� � . ., ,+rt+-.
r, '
, , , , , , , , , , :, ,
,
��-- I :, �, : ,, , , ' H'
, +
r +++
, --b , : j , , JJJ
t+
:
:
'
' : : +Y+ +f: :ft-i-: ' h � H±::t : r-r-i *H-ii:f:H- :1-h- i
+i
i:tcc ' ' W,J:
-L!. i
j 1
I I
I
,
'·
j
\
I
t-t-tt
1
_
I
I
r
,
I
1
0.9
Fi gure 4 - 4 .
! • ,
I
I
I
1
, I I
I
,
.
I
•
,
·
.
t
,
I i
I I
, .
l
I I
I
, I
•
I
' J
I
r
1
o I i
I I I I
I
r
1
·
I I I ·
_I
I
' ; I
,
r-F i-t+ I
H- : t++- H \+
, •
, •
I
· 1
I
I t I
I
I
P+r�fH-f-R=I4
++-m1!
;++1-�ffli ;�+�
+'
::_b±lfut tta··(
1 .0
Fre quency ( Hz )
I
I
I
I
I
,
r I
i I
• I •
I I I
: : ; s�
•
,
I
,
1 .1
Pha s e Re s p on s e of S in g l e - Po l e F i l t e r .
r�
· 1
75
d e vi a t i on e qua l t o t h e f i l t er � 3 d B ba ndwidth f o r t h i s pa rt i cu­
lar case .
76
FOURI ER SOLUTI ON
Harmon i c
Number
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C o e f f i c i ents
b (sin)
n
0 . 098005
- 0 . 00007 3
- 0 . 000020
0 . 000228
- 0 . 000028
0 . 000023
- 0 . 0001 4 0
- 0 . 000002
- 0 . 000 1 94
o.o
c ( c os )
n
Peak
A mp l i t ud e
A
0 . 986433
0 . 0001 99
0 . 00051 0
- 0 . 000087
- 0 . 000262
- 0 . 00001 6
- 0 . 0002 1 3
. p . 000003
- 0 . 0001 8 2
o.o
0 . 9 9 1 2 90
0 . 00021 2
0 . 0005 10
0 . 000224
0 . 000264
0 . 000028
0 . 000255
0 . 000004
0 . 000266
o.o
Per c e n t D i s t ort i on
=
R e l a t ive
Harm o n i c Leve l
dB
o . oo
•
•
*
•
•
•
•
•
•
0 . 06 %
QUASI - STAT I ONARY A PPlWXHIAT I ON
Harmonic
Number
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C oe f f i c i en t s
b ( sin )
n
0 . 098630
- 0 . 000580
- 0 . 000932
0 . 00001 1
0 . 000009
o.o
o.o
o.o
o.o
o.o
c ( c os )
n
Peak
A mp l i t ud e
A
0 . 9 96057
0.0
o.o
0.0
o.o
o.o
o.o
o.o
o.o
o.o
Pe r c e nt D i s t ort i on
Tab l e 8
j.
1 . 000929
0 . 00058 0
0 . 000932
0 . 0000 1 1
0 . 000009
o.o
o.n
o.o
o.o
0.0
=
Re l a t ive
Harm on i c Leve l
dB
o . oo
•
. .
•
•
•
•
•
•
•
0 . 1 1%
A na ly s i s o f D i s t ort i on i n t h e D i s c r i m i na t or Output
D ue t o the Jl�f R e s p onse o f a S in gl e -Po l e F i l t e r for
B = 2 and P = 0 . 2 .
77
.. �-�-
---- ----
---- - - - · · - · ·--- --------- - - - · ·
--- ·-···-··· - -
. - - ·-
'
----------- ··- ------- ---------- ------
-- --- -----
-
-- -· ·--
--
- - - -·
-----
---------····-· ---- - - ----- ·
--·-· ·
FOURIER SOLUT I ON
Harm on i c
Number
n
1
2
3
4
5
6
7
8
9
10
c ( c os )
Pea k
Amp l i t ud e
A
0 . 987201
0 . 000424
0 . 002424
- 0 . 000055
-0 . 000107
-0 . 000008
- 0 . 0001 7 3
- 0 . 000006
- 0 . 00001 6
0 . 000003
0 . 9 9 1 77 9
0 . 000959
0 . 003305
0 . 000057
0 . 00016 9
0 . 0000 1 6
0 . 0001 97
0 . 000009
0 . 000016
0 . 000003
F o ur i e r C o e f f i c i e n t s
n
b ( sin )
0 . 095182
- 0 . 000860
- 0 . 002247
- 0 . 00001 2
- 0 . 0001 3 1
0 . 0000 1 4
- 0 . 000094
0 . 000006
o.o
0 . 000001
n
Per c e nt D i s t ort i on
=
Re lat i ve
Ha rmoni c Leve l
dB
0 . 00
*
- 4 9 . 54
*
*
*
*
*
*
*
0 . 35 %
QUA S I - STATI ONARY APPROXIMATI ON
Harmonic
Numbe r
n
1
2
3
4
5
6
7
8
9
10
c ( c os )
Peak
Amp l i t ud e
A
0 . 9 96057
o.o
o.o
0.0
o.o
0.0
o.o
o.o
o.o
o.o
1 . 000666
0 . 00103 9
0 . 003432
0 . 000073
0 . 0001 1 9
0 . 000004
0 . 000004
0.0
0.0
o.o
Four i e r C o e ff i c i en t s
_
n
b ( sin )
0 . 09 5 92 3
-0 . 0010 3 9
- 0 . 0034 3 2
0 . 000073
0 . 0001 1 9
- 0 . 000004
- 0 . 000004
o.o
0.0
o.o
n
Per c e nt D i s t ort i on
Tab l e 9
=
R e l a t i ve
Harm o n i c Leve l
dB
0 . 00
- 5 9 . 67
- 4 9 . 30
*
*
*
*
*
*
*
0 . 36 %
A na lys i s o f D i s t or t i on i n t h e D i s c r i m i na t or Output
D ue to t h e Fr-·1 R e s p c ns e o f a S i ngl e -Po l e F i l t e r f or
B = 4 a nd P = 0 . 4 .
78
FOURI ER S OLUTI ON
Harm on i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
I<, o ur i e r C oe f f i c i e n t s
b ( sin )
n
0 . 091450
-0 . 001 1 3 2
-0 . 004757
0 . 000066
-0 . 0004 6 1
-0 . 000074
-0 . 0002 4 6
-0 . 000051
-0 . 0001 1 1
-0 . 000005
c ( c os )
n
0 . 988185
0 . 000496
0 . 004577
- 0 . 0002 1 4
- 0 . 000400
<;> . 00004 9
-.0 . 0002 1 7
'0 . 000027
- 0 . 000132
0 . 0000 1 6
Per c e n t D i s t ort i on
P e ak
Amp l i t ud e
A
0 . 992407
0 . 0012 3 6
0 . 006601,
0 . 000225
0 . 0006 1 0
0 . 00008 9
0 . 000328
0 . 000058
0 . 0001 7 2
0 . 0000 1 7
=
Re l a t i ve
Ha rm on i c Leve l
dB
o . oo
- 5 8 . 09
- 4 3 . 54
•
•
*
•
•
*
*
0 . 68 %
QUASI - STATI ONARY APPROXI MAT I ON
Harm o n i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C o e f f i c i e n t s
b ( sin )
n
0 . 09 1 98 9
- 0 . 001 3 2 4
-0 . 0068 1 2
0 . 0001 93
0 . 000487
-0 . 000020
-0 . 000034
0 . 000002
0 . 000002
0.0
c ( c os )
n
0 . 9 96057
0.0
o.o
0.0
0.0
0.0
o.o
o.o
0.0
o.o
P e r c e n t D i s t or t i on
Tab l e 1 0
Peak
Amp l i t ud e
A
1 . 0002 96
0 . 001 324
0 . 0068 1 2
0 . 0001 93
0 . 000487
0 . 000020
0 . 000034
0 . 000002
0 . 000002
0.0
=
Re l a t ive
Harmon i c Leve l
dB
o . oo
- 5 7 . 56
- 4 3 . 34
*
•
•
*
•
•
•
0 . 6 9%
Ana lys i s o f D i s t ort i on i n t he D is c r im i na t or Out put
Due to the FM R e s p o n s e o f a S ingl e - Po l e F i l t e r f or
B = 6 and P = O . G .
79
FOURI ER S OLUTI ON
Harm o n i c
Numbe r
c ( co s )
Peak
Amp l i t ud e
A
0 . 989234
0 . 000457
0 . 006 585
- 0 . 000305
-0 . 001 1 7 9
0 . 000050
-0 . 0002 5 3
-0 . 000032
- 0 . 000180
- 0 . 000023
0 . 9 93052
0 . 001 387
0 . 010185
0 . 000306
0 . 00 1 2 3 2
0 . 000056
0 . 000380
0 . 000053
0 . 0001 84
0 . 000028
Four i er C oe f f i c i en t s
n
b (sin)
1
2
3
4
5
6
7
8
9
10
0 . 087000
- 0 . 001 309
- 0 . 007770
- 0 . 000028
-0 . 000358
0 . 000025
0 . 000284
0 . 000042
0 . 0000 3 9
0 . 00001 7
n
n
Pe r c e n t D i s t ort i on
=
R e l a t ive
Harm on i c Leve l
dB
o . oo
- 5 7 . 10
- 3 9 . 78
*
-58 . 1 2
*
*
*
*
*
1 . 03%
QUAS I - STATI ONARY APPROXHtAT I ON
Ha rmon i c
Number
n
1
2
3
4
5
6
7
8
9
10
Four i e r C o e f f i c i e n t s
n
b ( s in )
0 . 087386
- 0 . 00 1 4 5 1
-0 . 01 0384
0 . 000340
0 . 001 1 92
- 0 . 000058
- 0 . 0001 32
0 . 000008
0 . 000014
- 0 . 000001
c ( c os )
Peak
Amp l i t ude
A
0 . 996057
0.0
o.o
0.0
o.o
o.o
o.o
0.0
o.o
o.o
0 . 9 9 98 8 3
0 . 00 1 4 5 1
0 . 0 1 0384
0 . 000340
0 . 00 1 1 !)2
0 . 000058
0 . 000132
0 . 000008
0 . 000014
0 . 000001
n
Perc e n t D i s t or t i on
Tab l e l l
=
Re l a t i v e
Harmon i c Leve l
dB
o . oo
- 5 6 . 77
- 3 9 . 67
*
- 5 8 . 47
*
*
*
*
*
1 . 05%
Ana lys i s o f D i s t ort i on i n t h e D i s c r imina t or Output
Due to t h e FM Resp onse o f a S ingl e -Pol e F i l t e r for
B = 8 a nd P = 0 . 8 .
80
FOURI ER SOLUTI ON
Ha rmon ic
Number
n
1
2
3
4
5
6
7
8
9
'10
Four i e r C o e f f i c i e n t s
b ( si n )
n
0 . 082298
- 0 . 001 4 1 3
-0 . 010864
0 . 0001 7 1
- 0 . 000420
0 . 00009 7
0 . 00042 2
- 0 . 0000 1 3
- 0 . 00001 9
0 . 0000 1 2
c ( c os )
Peak
Amp l i t ud e
A
0 . 9902 1 2
0 . 000391
0 . 008067
- 0 . 000463
-0 . 0022 5 3
0 . 000074
- 0 . 000 1 50
0 . 000020
- 0 . 000103
-0 . 000022
0 . 9 93626
0 . 00 1 4 6 6
0 . 0 1 3532
0 . 000493
0 .0022 9 1
0 . 000122
0 . 000488
0 . 000024
0 . 000 1 05
0 . 000025
n
Per c e n t D i s t ort i on
=
Re lat ive
Ha rm o n i c Leve l
dB
o . oo
- 5 6 . 62
- 37 . 32
*
- 5 2 . 74
*
*
*
*
*
1 . 38%
QUASI - STATI ONARY APPROXHlATI ON
Harmoni c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Four i e r C oe f f i c i en t s
n
b ( s in )
0 . 08 2 5 6 1
- 0 . 00 1 4 6 6
- 0 . 01 3686
0 . 00047 9
0 . 0021 9 3
- 0 . 0001 1 3
-0 . 0003 3 9
0 . 000023
0 . 000050
- 0 . 000004
c ( c os )
Peak
Amp l i t ud e
A
0 . 9 96057
0.0
0.0
0.0
0.0
0.0
0.0
o.o
0.0
0.0
0 . 9 9 9473
0 . 00 1 4 6 6
0 . 01 36 8 6
0 . 00047 9
0 . 002 1 93
0 . 0001 1 3
0 . 00033 9
0 . 00002 3
0 . 000050
0 . 000004
n
Pe r c e n t D i s t ort i on
Tab l e 1 2
=
R e l a t ive
Ha rm o n i c Leve l
dB
o . oo
- 5 6 . 67
- 3 7 . 27
*
-53 . 18
*
*
*
*
*
1 . 3 9%
Analys i s o f D i s t ort i on i n t h e D i s c rimina t or Output
D ue t o t h e FM He sp ons e of a S i n g l e -P o l e F i l t e r for
B = 10 and P = 1 . 0 .
81
FOURI ER S OLUTI ON
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C o e f f i c i e n t s
n
b ( sin )
0 . 07 7 6 2 4
- 0 . 001362
-0 . 01 37 5 1
0 . 000264
-0 . 000045
0 . 0001 4 1
0 . 000880
0 . 00002 3
- 0 . 000 1 3 2
-0 . 00001 0
c ( c os )
P e ak
Amp l i t ud e
A
0 . 9 9 1 06 7
0 . 0002 96
0 . 008 9 8 9
- 0 . 000522
-0 . 003562
0 . 0001 7 2
0 . 000095
-0 . 000009
0 . 0002 34
-0 . 00002 9
0 . 9 9 4102
0 . 00 1 3 94
0 . 0 1 6 428
0 . 000585
0 . 003562
0 . 000222
0 . 000885
0 . 000025
0 . 000268
0 . 000031
n
Pe r c e n t D i s t ort i on
=
Re l a t ive
Harm o n i c Leve l
dB
0 . 00
- 5 7 . 06
- 3 5 . 64
*
- 48 . 91
*
*
*
*
*
1 . 70%
QUASI - STATI ONAHY A PPHOXHI ATI ON
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Four i e r C o e f f i c i en t s
b ( s in )
c ( c os )
Peak
Amp l i t ude
A
0 . 077806
-0 . 00 1 4 1 5
-0 . 01 64 97
0 . 0005 90
0 . 003382
-0 . 0001 7 9
-0 . 000668
0 . 000046
0 . 000127
- O . OOOOl l
0 . 9 9605 7
0.0
0.0
0.0
0.0
0.0
o.o
o.o
o.o
0.0
0 . 9 99092
0 . 00 1 4 1 5
0 . 01 6497
0 . 0005 90
0 . 003382
0 . 0001 7 9
0 . 000668
0 . 000046
0 . 000127
0 . 00001 1
n
n
Pe rcent D i s t or t i on
Tab l e 1 3
=
Re l a t i ve
Harm o n i c Leve l
dB
0 . 00
-56 . 98
- 35 . 64
*
- 4 9 . 41
*
*
*
*
*
1 . 6 9%
A na lys is o f D i s t o rt i on i n t h e D i s c r i m i na t or Out put
Due t o t h e FM Re s p ons e of a S i ngl e -P o l e F i l t e r f o r
B = 1 2 and P = 1 . 2 .
82
From Tab l e s 8 t hrough 1 3 we obs e rve that f o r the s pe c i f i c
problem c on s i de r e d i n t h is s e c t i on t h e a greement b e t we e n th�
Fouri e r s o l ut i on and the qua s i - s ta t i on a ry approx ima t i on is
ve ry exc e l l en t with r e s pe c t t o the r e l a t ive l eve l s o f the har­
m o n i c s and t hus t he p e r c e n t d is t ort i on .
I t i s a l s o n ot e d ,
h oweve r , t ha t the quas i - s t a t i onary approx ima t i on fa i l s c ompl e t e ­
l y t o pre d i c t t h e pha s e o f t h e harmon i c s re l a t ive t o t h e f unda ­
menta l .
A l l harmon i c s ha ve b e e n expr e s s e d a s only s i n e t e rms
f or the cas e whe re t he f undamental is a pure c os ine in t h e
und i s t ort e d c ond i t i on .
Th i s s i t ua t i on i s e xp l a i n e d i n Se c t i on
2.4.
I n t h e Four i e r s o l ut i on for B
=
2 g i ve n i n Tab l e 8 i t
s h o u l d be n o t e d that t h e harmo n i c ampl i t ud e s A d i ff e r grea t ly
f r om t ho s e given by t h e qua si - s t a t i onary approxima t i on .
are s e ve ra l ,
in fa c t ,
(n
=
Th ere
7 , 9 f o r examp l e ) t ha t are zero in
t h e quas i - s t a t i onary a pproxima t i on a n d d e f in i t e ly not zero in
t he Four i e r s o l ut i on .
The la rge r amp l i t ud e s f or the F o ur i e r
s ol ut i on a re d u e t o t he e l im inat i on o f s i d ebands wi t h t h e
r e s ul tant d i s t ort i on .
I t i s n o t e d , h oweve r , t ha t t h e s e amp l i ­
t ud e s are m u c h be l ow - 6 0 d B from t h e f undamenta l amp l i t ude as
i nt e nd e d f rom t h e d i s cuss i on i n S e c t i on 4 . 2 .
As t h e modula t i on i nd e x B increa s e s , t he d i s c re pa ncy b e ­
twe en t h e amp l i t ud e s d e t e rm i n e d i n t h e Four i e r s ol u t i on a n d t he
q ua s i -s ta t i o nary approx ima t i on d e crea s e s f or the c a s e c ons i d ­
e re d .
D u e to t h e approx ima t i on of t h e Ft- 1 s p e c t rum b y only i ts
" s i gn i f i ca n t s id ebands , " l i t t l e importan c e s h o u l d b e a t tached
83
t o harm o n i c a mp l i t ud e s b e l ow -60 d B from t h e fundame n t a l in t he
F o ur i e r s o l ut i on .
4.4
UNDERCOUPLED TWO-POLE FI LTER
I n t h i s s e c t i on , t h e r e s p onse of a two - p o l e und erc oupl e d
f i l t e r t o a F}1 exc i ta t i on i s pre s e nt e d a s d e t e rm i n e d by t he
I
The f i l t e r i s e ss e n t ia l ly t he t w o - p o l e But t e r ­
;
Four i e r me t h o d .
w o r t h f i l t e r g i v e n i n Se c t i on 4 . 5 but w i t h t h e c oup l i n g capa c i t or re duc e d b y f i f ty p e r c e n t and t h e s h un t capa c i t ors i n each
r e s ona t or inc reas e d c or r e s p ond i ngly to k e e p t he c en t e r fre q u ency a t 1 Hz .
i n Fi gure 4 - 5 .
The s c h ema t i c d ia gram o f t h e f i l t e r i s give n
F i gures 4 - 6 and 4 - 7 i l l us t ra t e t he amp l i t ud e
and p ha s e chara c t e ri s t i c s o f the f i l t e r ' s s ta t i c r e s p onse as
c ompu t e d by ECAP .
The fl1 waves use d i n t he c omp uta t i ons a re given i n Equat i on ( 4 . 1 - 1 ) .
The m odul a t i on indexe s c ons i d e re d c ov e r e d t he
range f r om t w o t o t we l ve c o rr e s p ond ing t 6 a peak- t o -peak d e v i a t i on t o bandw i d t h ra t i o of 0 . 3 33 t o 2 .
The - 3 d B bandwi d t h of
t h e f i l t e r i s 0 . 06 Hz s o t ha t
p =
B
6
( 4 . 4- 1 )
f r om Equa t i on ( 4 . 3-2 ) .
The Four i e r c �e f f i c i e n t s of th e d i s c r imina t or o u t put are
l i s t e d i n Tab l e s 1 4 t h rough 1 9 f or e a c h c a s e c ons i d e re d .
p e rc e nt d i s t ort i on and e qu i va l e n t peak amp l i t ud e s are a l s o
g i ve n .
The
84
0 . 07 96F
O . O l l 2H
1 !1
2 . 1 7F
2 . 17F
O . Ol l2H
F igure 4 - 5 .
Schema t i c D ia gram of Und e r c oup l e d T w o - P o l e
Fi l t e r .
85
loo
'"'-
=
=
- -+--
!--
- t-
t-t - -
j--
1
Ji'i gure 4 - 6 .
Amp l i t ud e Re s p ons e of Unde rc oup l e d
Two - Po l e F i l t e r .
86
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: l__l_l _tl-H-, 1!-1-l-IT-!::±:
_L 1
.
lj_ iH±
�J,++H i i f-�K=H:=Ei
:t::t:J:
_t-1
R=f1- 1
- -�: 1
1
: : �-- I _ : . ' I H-::.�c::J'-Htlfffl=r=tr-t=f'JJ�
-�H--I+H- ;::q:±t l f l j rl-++�::::H-t-�- 1 tL 1 1 1 =-1
1 1
1 1+ -++�
r--m-ffifr-H-1-tJ±r±±i
_ -b
1 .0
1. 1
-i
-n+�+l r-1-t- .. t-- J J j i r !=" r-.l !
rr
f1 1 1 1
_
- 9 0°
i\
;_Lit-1I L l
.LLLL ti
l
'
I
\J
�W1 ++-
t- -
I
I
,
II
1
I
1
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11
T
LI._W
1
__
_
_
I
Fre quency ( Hz )
Phas e Re s p o n s e o f Underc o up l e d Two-Po l e F i l t e r .
87
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Four i e r C o e f f i c i � n t s
b ( s in )
n
0 . 220249
- 0 . 001 1 1 7
- 0 . 0006 6 9
- 0 . 0000 1 4
-0 . 0001 4 9
- 0 . 00001 3
-0 . 0001 93
0 . 000024
- 0 . 000 1 97
0 . 000005
Peak
Amp l i t ud e
A
c ( c os )
n
0 . 961664
0 . 001068
0 . 000848
-0 . 000072
-0 . 000095
-0 . 000021
-0 . 000027
0 . 000005
0 . 000067
- 0 . 000002
0 . 986564
0 . 001545
0 . 001080
0 . 000074
0 . 000 1 7 7
0 . 000025
0 . 000 1 95
0 . 000025
0 . 000208
0 . 000005
Per c e n t D i s t or t i on
Tab l e 1 4
=
Re l a t ive
Ha rm o n i c Leve l
dB
o . oo
-56 . 10
-59 . 21
*
•
•
*
*
*
*
0 . 1 9%
A na lys i s o f D i s t ort i on i n t h e D is c ri m i na t o r Outp ut
Due t o t he FM Re s ponse of a Two-Po l e Underc oup1 e d
F i l t e r f or B = 2 and P
0 . 333 .
-
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Four i e r C o e f f i c i en t s
b ( sin )
n
0 . 21 7242
- 0 . 00 2 1 5 2
-0 . 002061
- 0 . 0002 90
0 . 000120
0 . 00001 6
- 0 . 000055
-0 . 000008
0 . 00000'3
0 . 00001 0
c ( c os )
n
0 . 962730
0 . 001 581
0 . 004170
-0 . 000259
-0 . 00009 9
- 0 . 000050
0 . 00001 3
- 0 . 00001 0
0 . 000004
0.0
P e r � e n t D i s t ort i on
Tab l e 15
Peak
Amp l i t ud e
A
0 . 9 8 6 936
0 . 002670
0 . 004651
0 . 00038 9
0 . 000156
0 . 000052
0 . 000056
0 . 0000 1 3
0 . 000005
0 . 000010
=
R e l at i ve
Harm o n i c Leve l
dB
o . oo
- 5 1 . 36
- 4 6 . 53
*
*
*
*
*
*
*
0 . 54%
Ana lys i s o f D i s t o rt i on in t h e D i s c r im i im t o r Output
Due t o the FH Response of a Two-Po l e Und e r c oup 1 e d
F i l t er f o r B = 4 and P = 0 . 6 6 6 .
88
Harm on i c
Number
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C o e f f i c i en t �
b (sin)
n
0 . 211289
- 0 . 002777
- 0 . 004257
- 0 . 0006 2 7
0 . 000337
0 . 000053
0 . 0000 1 5
0 . 0000 1 0
- 0 . 000076
0 . 000003
c ( cos )
n
0 . 965285
0 . 001 305
0 . 010907
0 . 00077 5
-0 . 0000 1 4
- 0 . 0001 1 3
0 . 000088
0 . 000006
0 . 000002
0 . 00001 1
Pe ak
Amp l i t ude
A
0 . 988 1 3 9
0 . 003069
0 . 01 1 708
0 . 000997
0 . 000338
0 . 000152
0 . 00008 9
0 . 000012
0 . 000076
O . OOOO l l
Rel a t ive
Harmon i c Level
dB
o . oo
-50 . 16
- 3 8 . 53
- 5 9 . 93
*
*
*
•
•
*
Pe r c e n t D i s t ort i on = 1 . 24%
Tab l e 16
Ha rmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
A na ly s i s o f D i s t ort i on i n t h e D is c r im i na t or Output
D u e t o t h e FM Resp onse o f a Two-Po l e Und e r c oup l e d
F i l t e r for B = 6 and P = 1 .
F o ur i e r C oe f f i c i en t s
b ( sin )
n
0 . 2022 90
- 0 . 002927
- 0 . 008431
- 0 . 000670
- 0 . 000698
0 . 000344
- 0 . 000091
0 . 0001 07
- 0 . 00021.9
0 . 000004
c ( cos )
n
0 . 96 9024
0 . 0005 1 5
0 . 020034
- 0 . 00 1 5 4 3
0 . 000640
- 0 . 0003 96
0 . 000559
0 . 000006
0 . 000077
0 . 000007
Peak
Amp l i t ud e
A
0 . 989914
0 . 002 972
0 . 02 1 7 3 6
0 . 001682
0 . 00091 7
0 . 000525
0 . 000566
0 . 000 107
0 . 000232
0 . 000008
Re l a t ive
Harm o n i c Le ve l
dB
o . oo
-50 . 45
-33 . 17
- 5 5 . 39
•
•
•
•
•
•
Pe r c e nt D i s t ort i on = 2 . 22% .
Tab l e 1 7
A na lys is o f D i s t ort i on i n t h e D is c r im i na t or Output
D u e t o t h e 1'11 Resp onse o f a Two-Pol e Und e r c oup 1 e d
F i l t e r f or B = 8 a n d P = 1 . 3 33 .
89
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
F o ur i e r C o e f f i c i en t s
n
b ( s in )
0 . 1 91 25 3
-0 . 002682
- 0 . 01 5 3 1 6
-0 . 00009 9
-0 . 004 1 65
0 . 001 007
- 0 . 00 1 1 7 6
0 . 000334
-0 . 00071 7
0 . 000030
c ( c os )
n
0 . 973 1 22
-0 . 000345
0 . 028 930
-0 . 002 1 97
0 . 000254
-0 . 000476
0 . 0002<18
0 . 0001 8 4
.... 0 . 0001 5 1
0 . 000070
Per c e n t D i s t or t i on
'
Tab l e 1 8
Harmon i c
Numb e r
n
1
2
3
4
5
6
7
8
9
10
0 . 9 9 1 738
0 . 002704
0 . 032734
0 . 002 1 99
0 . 00 4 1 7 3
0 . 00 1 1 1 3
0 . 001202
0 . 00038 1
0 . 000732
0 . 000076
=
Re l a t i ve
Harm o n i c Leve l
dlf
o . oo
- 5 1 . 29
- 2 9 . 63
- 5 3 . 08
-47 . 52
-58 . 99
-58 . 33
*
*
*
3 . 35%
Analys i s o f D i s t ort i on i n t h e D is c r i m i na t or Output
Due t o the FH Re s p on s e of a Two - P o l e Und e rc oupl e d
F i l t e r f o r B = 1 0 and P = 1 . 6 66 .
F o ur i er C o e f f i c i en t s
b ( sin)
n
0 . 1 7 9474
- 0 . 002238
-0 . 02373R
0 . 000768
-0 . 00858 9
0 . 00 1 6 8 6
- 0 . 00 1 6 7 1
0 . 000401
- 0 . 000444
- 0 . 0001 77
c ( c os )
n
0 . 976869
-0 . 000 968
0 . 035158
-0 . 002288
- 0 . 003082
0 . 000076
- 0 . 002070
0 . 000801
- 0 . 001 2 4 9
0 . 000343
Pe r c e n t D i s t or t i on
Tabl e 1 9
Peak
Ampl i t ude
A
Peak
Amp l i t ud e
A
0 . 9932 1 9
0 . 002438
0 . 042421
0 . 002414
0 . 009125
0 . 00 1 688
0 . 002 6 6 1
0 . 0008 96
0 . 001 325
0 . 000385
=
R e l a t i ve
Harmon i c Leve l
dB
o . oo
�52 . 20
-27 . 3 9
-52 . 2 9
- 4 0 . 74
- 55 . 39
- 5 1 . 44
*
- 5 7 . 50
*
4 . 39%
Ana lys i s o f D is t o rt i on i n the D i s c r i m i na t or Output
D u e t o t h e FM Resp onse of a Two - P o l e Und e r c oupl e d
F i l t e r f or B = 1 2 and P = 2 .
90
The d is c r im ina t or output v
for t he case of B
=
1 2 and P
=
dN
2.
( t ) i s p l o t t e d i n Fi gure 4-8
91
-'-,-H-r D' ! +
. , -::�rr+ � �. H--�-++ f-·
!
1
. . =t,.+++t-H-t-t
r :+t-r'l
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F i l t e r f or B
=
12 and
P
=
2.
�
I
I
( t ) Versus T ime
dN
Due t o t h e FM R e s p o n s e of a Two - P o l e Und e rc oup l e d
D i s c r imina t or Output v
I
1
t ---)>
Fi gure 4 - 8 .
--r-
:
[__
0
L
i -
,
I 1I i
I I
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m
1/ f
92
4.5
BUTTERWORTH TWO-POLE FILTER
I n t h i s s e c t i on , t h e r es p on s e of a But t erwort h t wo - p o l e
f i l t er t o a FM e x c i t a t i on is p res ent e d .
o f t h e f i l t e r i s g i v e n i n F i gure 4 - 9 .
The s c h ema t i c d ia gram
Fi gur e s 4 - 1 0 and 4 - 1 1
i l l us t r a t e t h e amp l i t ud e a n d pha s e chara c t e r i s t i c s o f t h e
f i l t e r ' s s ta t i c r e s p ons e a s c omp ut e d by ECAP .
The FM wa ves us ed i n t he c omputa t i ons are g i v e n i n Equat i on
( 4 . 1-1 ) .
T h e modula t i on indexes c ons i d e re d c overe d t h e ran ge
from t w o t o twe l ve c orrespond i n g t o a p eak-t o-peak d e v ia t i on t o
bandwi d t h ra t i o o f 0 . 2 t o 1 . 2 .
The F o ur i e r c o e f f i c i e n t s o f t h e d is c r im i na t or output a r e
l is t e d in Ta b l e s 2 0 t hrough 25 f or e a c h c a s e c ons i d e re d .
p e r c e n t d i s t ort i on and e qu i va l e nt peak ampl i t ud e s a r e a l s o
g iven .
The
,
93
O . l 5 9F
2 . 09F
O . Oll2H
2 . 09F
10
O . Ol l 2 H
Fi gure 4 - 9 .
Schema t i c. D ia gram of But t e rworth Two - P o l e
Filter .
94
'
1/
f\
e-lf-
I
!/
r-
!ir .
·
Jt, i gure 4 - 1 0 .
�t:i:t . v. _;.
+-
Amp l i t ud e Re s p on s e o f But t e rworth
Two- Pol e Fi l t e r .
!=
95
270°�
i
I
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:- -4---W-I:=J=t
1
-r -
+-++++ +++-1-+-+-+--f--1
l±l±l:1 l 1 r
'
1
1 1
1
-i-1- r-+-r
1 1 , 1 I
- 90 °Fl=r-1- --1-H+
-+-+-1-4-�t-HI--+-1+-I-+++ I
.L..L =rn-f1-+++�+-+-+---�+-l--+-r f-+-+++,��--��-�H 1 1
. _!
L..J...
0.9
F i gure 4 - 1 1 .
1 .0
Fre que ncy ( Hz )
i i
i
i
l
1
i
·
I
J
1
H--r
H---n-
H-I f-----++
I r-
-r-ir-t--r-��--��--t1.1
Pha s e Re s p o n s e o f But t e rworth Two - P o l e F i l t er .
96
Harm on i c
Number
n
l
2
3
4
5
6
7
8
9
10
Fouri e r C o e f f i c i ent s
b ( sin )
n
0 . 142145
- 0 . 002086
0 . 0005 8 9
0 . 000028
- 0 . 000209
- 0 . 000004
- 0 . 000242
0 . 000006
- 0 . 000284
0 . 000002
c ( c os )
Peak
Amp l i t ud e
A
0 . 985481
0 . 00 1 042
- 0 . 00 1 2 6 7
- 0 . 000022
- 0 . 000155
-0 . 0000 3 3
- p . 000122
0 . 00002 9
J. o . oooo57
0 . 000004
0 . 9 95680
0 . 002332
0 . 0 0 1 3 97
0 . 000036
0 . 000260
0 . 000033
0 . 000271
0 . 00002 9
0 . 000290
0 . 000005
n
Per c e n t D i s t or t i on
Tab l e 20
Harmoni c
Numbe r
n
l
2
3
4
5
6
7
8
9
10
o . oo
- 5 2 . 61
- 5 7 . 06
*
*
*
*
•
*
*
0 . 27 %
Analys i s of D i s t ort i on i n t h e D i s crimi na t or Outp ut
D u e t o t h e FM Re sponse of a But t e rworth Two -Pol e
F i l t e r f or B = 2 and P = 0 . 2 .
Four i e r C oe f f i c i en t s
b ( s in )
n
0 . 1 45 5 95
- 0 . 003 9 1 8
0 . 001872
0 . 000289
- 0 . 000026
- 0 . 000059
- 0 . 000225
- 0 . 0000 1 6
- 0 . 000018
0 . 0000 1 3
c ( c os )
Peak
Amp l i t ud e
A
0 . 984093
0 . 002273
- 0 . 004425
-0 . 000096
0 . 000390
-0 . 000004
-0 . 0001 2 1
- 0 . 00003 3
0 . 000004
0 . 000009
0 . 9 94805
0 . 004530
0 . 004804
0 . 000305
0 . 00039 1
0 . 00005 9
0 . 000255
0 . 000036
0 . 000018
0 . 0000 1 6
n
Per c e n t D i s t ort i on
Tab l e 21
=
Re l a t i ve
Harm o n i c Leve l
dB
=
Re l a t ive
Harmon i c Leve l
dB
o . oo
-·4 6 . 8 3
- 4 6 . 32
*
*
•
*
*
•
*
0 . 66 %
Analys i s o f D i s t ort i o n i n t he D i s c r im i na t or Output
Due t o t he FM Response o f a But t erwo r t h Two-Pol e
Fi l t e r f or B = 4 and P = 0 . 4 .
97
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
c ( c os )
Peak
Amp l i t ud e
A
0 . 982 2 1 0
0 . 003354
- 0 . 007742
- 0 . 000521
0 . 00 1 5 6 6
0 . 000155
-0 . 000350
- 0 . 000080
- 0 . 000012
0 . 000005
0 . 9 9 3 5 92
0 . 006320
0 . 008223
0 . 000728
0 . 00 1 6 1 2
0 . 000245
0 . 0004 9 4
0 . 000081
0 . 0001 37
0 . 0000 1 3
Four i e r C oe f f i c i e nt�
n
b ( s in )
0 . 149961
- 0 . 005.356
0 . 002772
0 . 000509
0 . 000380
- 0 . 0001 90
- 0 . 0003 4 9
-0 . 00001 3
- 0 . 000 1 3 7
0 . 0000 1 2
n
Re l a t i ve
Harm o n i c Leve l
dB
0 . 00
- 4 3 . 93
- 4 1 . 64
•
-55 . 80
•
•
•
•
•
Percent D i s t ort i on = 1 . 05%
Tab l e 22
Harm on i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
A na lys i s o f D i s t ort i on i n t h e D is cr i m i na t or Output
Due to t h e FM R e s p ons e of a But t e rwor t h Two -Pol e
F i l t e r f or B = 6 and P = 0 . 6 .
Four i e r C oe f f i c i e nt s
n
b ( s in )
0 . 153477
-0 . 006 1 37
0 . 002856
0 . 000392
0 . 001723
- 0 . 000353
- 0 . 0005 9 9
- 0 . 000027
0 . 0001 5.7
0 . 000031
c ( c os )
Peak
Ampl i t ud e
A
0 . 980645
0 . 003735
-0 . 008 6 1 3
- 0 . 001456
0 . 0036 1 6
0 . 0001 2 3
- 0 . 000536
-0 . 000204
- 0 . 00001 9
0 . 000048
0 . 9 92583
0 . 007 184
0 . 009074
0 . 001 508
0 . 004005
0 . 000551
0 . 00080<1
0 . 000206
0 . 000158
0 . 000057
n
P e r.c e n t D i s t ort i on = 1 . 2 4%
Tab l e 23
R e l a t i ve
Ha rm on i c Lev e l
dB
o . oo
-42 . 81
- 40 . 78
- 5 6 . 37
- 4 7 . 88
•
•
•
•
•
·
Analy s i s of D i s t o rt i on i n t he D i s cr i m i na t or Out put
Due t o t h e Jt,l\i R e s p onse of a But t e rwor t h Two-Pole
F i l t e r f or B = 8 and P = 0 . 8 .
98
Harm on i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
c ( c os )
Peak
Amp l i t ud e
A
0 . 980160
0 . 003080
-0 . 0054 05
-0 . 002825
0 . 00 6 3 1 5
0 . 000557
- 0 . 000508
-0 . 000460
-0 . 000095
0 . 000049
0 . 9 92278
0 . 006 918
0 . 005 7 1 0
0 . 002826
0 . 007202
0 . 000626
0 . 001 4 9 4
0 . 000490
0 . 000475
0 . 0001 4 3
Four i e r C o e f f i c i e nt s
n
b ( s in )
0 . 15460 4
- 0 . 006 1 95
0 . 001840
- 0 . 0000 4 6
0 . 003464
- 0 . 0002 8 5
-0 . 001 4 05
- 0 . 0001 6 9
0 . 000466
0 . 0001 3 4
n
Re l a t i ve
Harm o n i c Leve l
dB
0 . 00
-43 . 13
- 44 . 80
-50 . 91
-42 . 78
*
- 5 6 . 45
*
*
*
Pe r c e nt D i s t ort i on = 1 . 2 1 %
Tab l e 24
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Ana ly s i s o f D is t ort i on i n t h e D is c rim ina t or Out put
Due t o the FM Uesp onse of a But t erworth Two-Pol e
F i l t e r f o r B = 1 0 a nd P = 1 .
F o ur i e r C oe f f i c i e n t s
n
b ( sin)
0 . 1 52787
- 0 . 005575
- 0 . 001 1 30
- 0 . 000021
0 . 003 948
0 . 0002 6 3
-0 . 00 2 9 3 3
- 0 . 000205
0 . 000727
0 . 0003 4 3
n
c ( cos )
0 . 980876
0 . 001 7 6 5
0 . 001 301
- 0 . 004331
0 . 009535
0 . 000294
0 . 000 1 3 6
- 0 . 000720
- 0 . 0001 4 5
o.o
Peak
Amp l i t ud e
A
0 . 9 9 2 70 4
0 . 005848
0 . 001723
0 . 00 4 3 3 1
0 . 0 10320
0 . 0003 95
0 . 0 02 936
0 . 000748
0 . 00074 1
0 . 000343
Re l a t ive
Ha rmon i c Leve l
dB
0 . 00
- 44 . 60
- 5 5 . 21
- 4 7 . 20
-39 . 66
*
-50 . 58
*
*
*
Per c e n t D i s t o rt i on = 1 . 31 %
Tab l e 25
A nalys i s o f D is t ort i on i n t h e D is c r i m i na t or Out put
Due t o t h e Ji'M Response o f a Butt e rw o r t h Two-Po l e
F i l t e r f o r B = 1 2 and P = 1 . 2 .
99
4.6
TCHEBYSHEV TWO- POLE FI L'l'ER
I n t h i s s e c t i on t h e resp onse of a T c h e byshev t w o - p o l e
f i l t e r t o a F}i e xc i t a t i on i s present ed .
c ons i de r e d h a s
F i gure
Fi gure
4-1 3 .
4- 1 2 .
3
T h e part i c ular f i l t e r
d B of r ip p l e i n t h e pas s band as s h own i n
The s c h ema t i c d ia gram o f t h e f i l t e r i s g iven i n
4-�3
F i gur e s
and
4-14
i l l u s t ra t e t h e amp l i t ud e
a n d p ha s e c h a ra c t e r i s t i c s o f the f i l t e r ' s s t a t i c r e s p onse as
c omput e d by ECAP .
The F}t wa v e s u s e d i n t h e c omputa t i on s a re g iven i n Equa t i on
( 4 . 1 -1 ) .
The modula t i on i ndexes c ons i d e re d c overed the range
f r om two t o t we l ve c or r e s p ond ing t o a peak-t o-pea k d e v iat i on t o
ba ndw i d t h ra t i o of
0.2
to
1.2.
The Four i e r c oe f f i c i e n t s of the d i s c r im i na t or output are
l i s t e d i n Tab l e s
26
t hr o ugh
31
f o r each C Ase c ons i d e re d .
The
p e rcent d i s t o rt i on and e qu i va l ent p eak amp l i t ud e s a r e a l s o
g i ven .
The d is c r imina t or output v
t hro ugh
s e vere •
4-17
f or B
=
8,
10,
and
dN
( t ) i s p l o t t ed in F i gure s
12
whe r e t h e d i s t or t i on i s
4-15
'
100
0 . 388F
0 . 00508 H
l!l
F i gure 4 - 12 .
4 . 5 91"
4 . 5 9F
0 . 00508H
H1
Sc hema t i c D ia gram of T c h e byshev Two - P o l e
F i l t er .
1 01
r-
- t-:-
f-
1/
\
f- t ­
r-rr- -
1/
1\
I
- I=
1--
l .
- r- f--
A mp l i t ud e Response o f T c h e by s h e v
Tw o - P o l e F i l t e r .
!=
1 02
Ht- tt--
�+ I
1�
I
I
I
I
�t-1 i-l
...
-
r
-
-
--
I
--
I
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L_ .
; ·
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+
I I
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Fi gure 4 - 1 4 .
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-
t-
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t-Et
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I I II
I I
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-+
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I I I
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1-- 1--
I
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II III
I
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I
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1 .0
Fre que ncy ( Hz )
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t-t-
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C±- -r-
Bili0 . 9
I I
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I
H+I
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I� I
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.
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t-tJ-l I
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I�
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r
I
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1 t-
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r,n-�,
h" f-
t--l-;=-!
-
I I
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""
I
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r-r-K=
I
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l
rTT
I
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I '
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!
.
H+
I
I I I
I t±
I , -l-1-H-h-
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'
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f--- +�tr-d+
-±
LlJ
r-- tfR+u
i-
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T
I
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J� :-++ I
_L "+ i I
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00' r-r�: ·�1=t : r-�
t--H-
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H+·
I
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I I ]II
H=f.-- t-r-
>
1.1
Pha s e Re sp onse of Tch e by s h e v Two-Po l e F i l t e r .
-
103
Harmon i c
Number
n
Four i e r C oe f f i c i e n t s
n
b (sin)
0 . 095152
-0 . 003055
0 . 004444
-0 . 000101
-0 . 00048 6
0 . 00008 6
-0 . 00034 9
-0 . 000050
-0 . 000427
- 0 . 000009
1
2
3
4
5
6
7
8
9
10
c ( cos )
Peak
Amp l i t ud e
A
1 . 001740
- 0 . 000063
-0 . 001 2 5 9
0 . 000321
-0 . 000468
- 0 . 0001 1 5
- 0 . 0001 1 1
0 . 000092
- 0 . 000 1 40
0 . 0000 1 3
1 . 006249
0 . 003055
0 . 004 6 1 9
0 . 000337
0 . 000675
0 . 000144
0 . 000366
0 . 000 105
0 . 000450
0 . 000015
n
P e r c e nt D i s t ort i on
Tabl e 2 6
Harmon i c
Numb e r
n
1
2
3
4
5
6
7
9
10
Four i e r C oe f f i c i en t s
27
*
*
*
*
*
*
*
0 . 56%
n
b ( sin )
c ( c os )
Peak
Amp l i t ud e
A
1 . 000355
0 . 00 1 435
- 0 . 009763
0 . 002585
-0 . 001 6 7 9
- 0 . 001294
0 . 000873
0 . 000�556
-0 . 000 1 2 6
- 0 . 000 1 4 9
1 . 00 6034
0 . 00757 9
0 . 0 1 9799
0 . 00 2 7 1 2
0 . 00421 9
0 . 00 1 308
0 . 00 1 2 3 9
0 . 000563
0 . 0004 34
0 . 0001 62
n
Pe r c e nt D i s t ort i on
Tab l e
o . oo
-50 . 35
- 4 6 . 76
A na lys i s o f D is t ort i on i n t h e D is cr im i na t or Output
Due to the fl1 Respons e o f a T ch e byshev T w o - P o l e
F i l t e r f o r B = 2 and P = 0 . 2 .
0 . 1 06744
- 0 . 007442
0 . 0 1 7 225
0 . 000822
- 0 . 003870
0 . 000154
0 . 00087 9
- 0 . 000087
-0 . 00041 5
-0 . 000062
8
=
Re l a t ive
Ha rm o n i c Leve l
dB
=
Re l a t ive
Harm o n i c Leve l
dB
0 . 00
-42 . 46
- 34 . 1 2
-51 . 3 9
- 47 . 55
-57 . 75
-58 . 1 9
*
*
*
2 . 1 8%
Analys i s o f D i s t ort i on i n t h e D i s c r im i na t or Output
Due to t he FM Response of a T ch e bysh e v Two - Po l e
F i l t e r for B = 4 a n d P = 0 . 4 .
1 04
Harmon 1 c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
Four i e r C o e f f i c i e n t s
b ( si n )
c ( c os )
n
n
0 . 1 27732
- 0 . 0 1 1 735
0 . 028727
0 . 00631 2
- 0 . 014066
- 0 . 0042 6 1
0 . 007285
0 . 003283
- 0 . 00341 3
- 0 . 002223
0 . 994888
0 . 00 6 7 1 2
- 0 . 03 2 3 92
0 . 002 957
0 . 004788
-0 . 0031 1 6
- 0 . 002 1 7 9
0 . 001 748
0 . 002432
0 . 000 1 2 6
Perce n t D is t ort i on
Tab l e 28
Harm o n i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
1 . 00305 4
0 . 0 1 35 1 8
0 . 04 3295
0 . 006 970
0 . 014859
0 .0 0527 9
0 . 007604
0 . 00371 9
0 . 00 4 1 9 1
0 . 002226
•
=
Re l a t i ve
Ha rm o n i c L e ve l
dB
o . oo
-37 . 41
- 2 7 . 30
-43 . 16
- 3 6 . 59
- 45 . 58
- 4 2 . 41
- 48 . 62
- 4 7 . 58
-53 . 07
4 . 93%
Analys i s o f D is t or t i on i n t h e D is c r im i n a t o r Out put
Due t o t he �I Response of a T c h e by s h e v Two-Pol e
F i l te r f o r B = 6 a n d P = 0 . 6 .
Four i e r C o e f f i c i e n t s
b ( sin )
c ( c os )
n
n
0 . 154315
- 0 . 01238 2
0 . 026 931
0 . 008209
- 0 . 01 3 340
- 0 . 008478
0 . 007 1 3 4
0 . 006650
-0 . 000238
-0 . 002 6 4 7
0 . 983923
0 . 0 1 2425
- 0 . 05745 1
- 0 . 003 1 8 1
0 . 022201
0 . 003575
- 0 . 0 1 5 933
- 0 . 004 9 1 4
0 . 01 1 48 9
0 . 006036
Perc e n t D is t ort i on
Tab l e 2 9
Peak
Amp l i t ud e
A
Peak
Amp l i t ud e
A
0 . 995951
0 . 01 7541
0 . 063450
0 . 008804
0 . 025 901
0 . 00 9201
0 . 0 17457
0 . 008268
0 . 0 1 1 4 92
0 . 00 6 5 9 1
=
Re l a t ive
Harm o n i c Leve l
dB
o . oo
- 3 5 . 08
- 23 . 92
- 4 1 . 07
- 3 1 . 70
-40 . 6 9
- 35 . 1 3
- 4 1 . 62
- 38 . 76
-43 . 59
7 . 60%
A nalys i s o f D i s t ort i on i n t h e D i s c r i m i na t or Out p u t
Due t o t h e F�l Respons e of a Tc h e by s h e v Two - P o l e
F i l t e r f or B
8 and P = 0 . 8 .
=
105
Ha rmon i c
Numb er
n
1
2
3
4
5
6
7
8
9
10
F o ur i er C o e f f i c i e n t s
b ( s in )
n
0 . 1 7 6 606
- 0 . 01 1 570
0 . 023898
0 . 000680
0 . 003583
-0 . 0039 1 7
- 0 . 004553
0 . 0006 6 6
0 . 00986 3
0 . 003952
c ( c os )
n
Peak
· Amp l i tude
A
0 . 973660
0 . 01 1 36 9
- 0 . 06 4 4 7 3
- 0 . 006 922
0 . 02 9243
0 . 009584
-0 . 023220
�t 0 . 008485
0 . 01 2 3 90
0 . 007048
Per c e nt D i s t ort i on
Tab l e 30
Harmon i c
Numbe r
n
1
2
3
4
5
6
7
8
9
10
=
o . oo
-35 . 7 1
- 2 3 . 16
- 4 3 . 06
- 30 . 52
-39 . 61
- 3 2 . 43
- 4 1 . 31
- 35 . 92
- 4 1 . 76
8 . 44%
Analys i s o f D i s t ort i on in t h e D is c r im i na t or Out put
Due to t h e FM Resp onse of a T c h e bysh e v Two-Pol e
F i l t er f or B = 10 a nd P = 1 .
Four i e r C o e f f i c i e n t s
b ( s in )
n
0 . 1 85474
- 0 . 009708
0 . 0304 6 6
- 0 . 008 1 3 8
0 . 028880
- 0 . 0053 6 2
- 0 . 006 180
- 0 . 006 1 3 3
0 . 01 7 7 1 4
0 . 005 640
c ( c os )
n
0 . 972252
0 . 00301 3
- 0 . 04 3 9 36
- 0 . 0 1 2 1 09
0 . 033341
0 . 007 642
-0 . 0228 1 9
- 0 . 005049
0 . 005 1 2 7
0 . 005 1 6 8
P e r c e nt D is t ort i on
Tab l e 3 1
0 . 98 9547
0 . 01 62 2 1
0 . 068760
0 . 006 955
0 . 029462
0 . 0 1 0354
0 . 023662
0 . 008 5 1 1
0 . 0 1 5836
0 . 008080
Re l a t ive
Ha rm on i c Leve l
dB
Peak
Amp l i t ude
A
0 . 9 8 9785
0 . 010165
0 . 0534 65
0 . 0 1 4 5 90
0 . 044 1 1 0
0 . 009336
0 . 023641
0 . 007 944
0 . 018441
0 . 007650
=
Re l a t i ve
Ha rm o n i c Leve l
dB
0 . 00
- 3 9 . 77
-25 . 35
-36 . 63
-27 . 02
-40 . 51
- 3 2 . 44
-41 . 91
- 3 6 . 60
-42 . 24
7 . 91%
A na lys i s of D is t or t i on i n t h e D is c r i m i na t o r Out put
Due to t h e FH R e s p onse o f a T c h e by s h e v Two - P o l e
F i l t er f o r B = 1 2 and P = 1 . 2 .
1 06
f-
-f- _L·t-++-1-t-+-+-+-t-+-f--t-+-H--t -
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ITI
J,-
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l-t-�{-+-��+- +-+++ +++++++
1
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4.7
Sml�iAHY OF RESULTS
I t wa s ve r i f i e d in Se c t i on 4 . 2 t ha t t h e amount o f s id e ba nd
p ower that must b e r e t a i n e d i n t h e 1'�1 s p e c t rum i s 0 . 9 9 9 9 9 9
t im e � t h e unmodul a t e d c a r r i er p owe r i f t h e d i s t ort i on due t o
n e g l e c t ing s id e bands i s t o b e b e l ow - 6 0 d B r e la t ive t o the
f undame nt a l m o d u la t i n g fre quency c omp one nt .
T h i s c r i t e r i on
r e s u l t e d i n t h e numbe r o f s i gn i f i cant s i d e band pa i rs a s a
f unc t i on o f t h e m o d u la t i on index s h own i n Ta b l e 32 .
These w e r e
t h e numbe rs a c t ua l ly us e d i n t h e p r o gram fl!DI ST t o p e r f orm t h e
c omputa t i ons .
The am ount o f d i s t ort i on p r oduc e d by t h e s u c c e s s i ve
d r oppi n g o f s id ebands wa s i nve s t i ga t e d and t h e r e s ul t s obta i n e d
we re f ound t o b e i n good a gr e ement w i t h t h o s e o f Wan g as s h own
i n Tab l e 7 .
I n S e c t i on 4 . 3 t h e quas i -s ta t i ona ry r e s p onse was c ompa r e d
w i t h t h e Four i e r s ol ut i on f or t h e c a s e o f t h e s ingl e - p o l e
fi lter .
Go o d a gr e eme nt was f o und b e t w e e n t h e t w o me t hods w i t h
r e s p e c t t o t h e amp l i t ud e s o f t he harm o n i c s f o r t h e c a s e s c on ­
s i de re d .
As p r e d i c t e d , t h e qua s i - s ta t i ona ry a pprox i ma t i on
fa i l e d t o g i v e t h e c or r e c t phas e s for t h e harm on i c c omponents .
I n S e c t i ons 4 . 4 t h r ough 4 . 6 t h r e e typ e s o f t wo - p o l e band ­
pass f i l t ers w e r e c ons i d e re d .
T h e d i s c r im i nat or o u t p u t v
dN (
t
wa s pl ot t e d f or s e v e ra l cas es where t he d is t ort i on was s e v e re
e n ough t o s e e v i s ua l ly .
)
Fbr t h e Tch e by sh e v t w o - p o l e f i l t e r , t h e
l a rge pha s e nonl i n ea r i ty s h own in F i gure 4 - 1 4 i s obs e rv e d t o
1 10
�lo d ula t i o n
I nd e x
Tab l e 32
S i gn i f i cant
S i de band Pa i rs
2
6
4
9
6
11
8
14
10
16
12
19
Numbe r o f S i gn i f i cant S i de band Pa i rs
Ve r s us M o d u la t i on I ndex .
lll
p roduc e
a
s i gn i f i ca n t amoun t o f dis t or t i on as sh own i n F i gure s
4 - 1 6 and 4 - 1 7 .
I n o rder t o m o r e e a s i ly c ompa r e t h e r e l a t ive p e r f ormance
of t h e four f i l t e rs c ons i d e re d w i t h r e s p e c t t o d is t ort i on , t h e
p e r c e n t d i s t ort i on i s p l o t t e d v e rs us t h e p eak- t o - p e a k d e v ia t i on
t o bandw i d t h r a t i o P i n F i gure 4 - 1 8 f o r t h e cases c ons i de re d .
I t i s obs e rv e d that f o r P l e ss than a b o ut 0 . 3 the s ingl e -p o l e
f i l t er produc e s t h e l ea s t amount o f d i s t or t i on .
For P great e r
t han 0 . 3 a n d up t o 1 . 0 t h e und e rc oup l ed t w o - p o l e f i l t e r pro­
d u c e s the l e a s t d i s t ort i on .
For P grea t e r t ha n 1 . 0 t h e But t e r­
w o r t h two-p o l e f i l t e r i s s up e r i or .
I t i s n o t e d t ha t for P
grea t er t h a n 0 . 4 t h e d i s t or t i o n prod u c e d by t h e s in gl e - p o l e
f i l t e r i s a l m o s t a l in e a r f unc t i on o f P .
Final l y , i t i s n o t e d w i t h l i t t l e s urpr i s e t ha t a t w o - po l e
T c he bysh e v f i l t er w i t h 3 d B o f r i pp l e has n o p l a c e i n a FM
sys t em .
The d is t or t i on p r o d u c e d by t h i s f i l t e r is approxima t e l y
s ix t ime s grea t e r t han t h e ot h e r f i l t e rs c ons i d e re d .
1 1 2
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Unde r c oup l e d 2-Pol e F i l t e r
S in gl e - P o l e Fi l t e r
But t e rw o r t h 2 - P o l e F i lt e �
Fi gure 4 - 1 8 .
Per c e nt D i s t or t i on V e r s us Peak- t o - Peak
D e v i a t i on t o Bandw i d t h Rat i o f or f = 0 . 005 H z .
m
113
5.
C ONCLUSI ONS
I t ha s b e e n s hown t ha t t h e e va l ua t i on o f t h e r e s ponse of a
n e t wo rk t o a FM e x c i ta t i on may b e p e r formed i n two d i f fe re n t
ways .
T h e dynam i c o r o p e ra t i onal approach s u f f e rs from a n
e x c e s s o f ma t h ema t i c a l c ompl e x i ty and i s c on s e qu e n t ly o f l im i t ­
e d use t o t h e p ra c t i ca l d e s ign e n g i n e e r .
The a c c ura cy o f the
results obta i n e d us ing t h e dynami c approach a re o f t e n i n d oub t
due t o t h e obs cure c on v e rge n c e prope r t i e s o f t h e a sympt o t i c
� e r i e s from w h i c h t h e s o l u t i o n i s f ormed .
A n o t h e r l im i ta t i on
t o t h e use o f t h e dynam i c m e t h od i s t ha t t h e form o f t h e ge n e ra l
s o l ut i on c on t a ins f i rs t a n d h i g h e r ord e r d e r iva t i v e s o f t h e
n e twork ' s t ra n s f e r func.t i o n .
The s e d e r i va t ives a r e d i ff i c u l t t o
m e a s ur e o r c omput e i n a l l b u t t h e m o s t i imp l e c a s e s .
This
l im i ta t i on ma kes i t imp ra c t i ca l t o deve l op a gen e r a l c omput e r
r out i n e t o s o l ve a w i d e var i e t y o f p r ob l e ms .
The Four i e r o r s p e c t ra l approach t o t he prob l em , o n t he
o t h e r han d , o f f e rs a s t ra i gh t - f orward a nd exa c t s o ] ut j on t 0
t h e prob l e m f o r t h e c a s e o f p e r i o d i c f r e que n cy m o d ul a t i o n .
ge n e ra l Four i e r s ol ut i on
may
The
b e us e d t o d e t e rm i n e the r esponse
o f a n e twork t o a F?-1 e x c i ta t i on t o any d e s i re d d e gr e e of a c c ura­
c y s i nc e t h e re e x i s t s a s impl e means o f � va l uat i n g the e rror
p r o d u c e d by not c on s id e r i �g a l l s i d e ba nds o f t h e i n f i n i t e
s e r i e s e xpa ns i on .
T h e F o u r i e r appr oa ch has t he f u r t h e r a d van-
1 14
t a ge i n t ha t i t may b e ea s i ly programme d f or s o l ut i on by c omput e r .
Onc e
a
s u i tab J,e p r o gram is a_va i l a bl e , a r out i n e prob l e m
may b e s o l v e d i n a f e w m inut es .
The c omput e r p r ograms present e d and d i s c us s e d i n t h is
pap e r may b e e mp l oye d t og e t h e r t o f orm a p owerful t e chn i que f or
t h e rap i d s o l v i n g o f »t p r o b l ems s im i l a r t o t h e type inve s t i ga t e d i n t h i s s t udy .
S in c e m o s t c ompa n i e s e nga ge d in t h e f ie l d o f d e s i gn i n g
c ommun i ca t i on e qu i pment ha v e s ome t y p e o f c omput e r fa c i l i ty ,
t h is t e c hn i qu e may b e e as i ly u s e d by anyone i n t h e f i e l d wh o i s
i n t e r e s t e d e n o ugh t o i n v e s t a f e w h ours i n s e t t i n g u p the
p r o gram on h i s par t i cu l a r mach i n e .
The ECAP pro gram d is c u s s e d
h e re i n i s n ow w i d e ly us e d in indus t ry a n d i s read i ly a va i la b l e
t o any one wh o d o e s n o t n ow have a c c e s s t o i t .
T h e use o f ECAP
is not man da t o ry , h ow e ve r , s in c e t h e ne twork t ra ns fe r imp e dance
may b e f ound by o t h e r means ; but it i s t h e fa s t e s t me t h od and
it l e nds i t s e l f we l l to the imp l e m enta t i dn of t h e p r ogram
FMDI ST d e s c r i b e d i n t h i s pa p e r .
115
BI BLI OGRAPHY
" Bandw i d t h a nd Spe c t ra o f Pha s e -and -Fre que ncy­
A b rams on , N .
Modula t e d Waves , " I EEE Transa c t i on on C ommun i ca t i ons Sys ­
t ems , C S - 1 1 ( De c emb e r , 1 96 3 ) , 407- 4 1 4 .
A s s a d o ur ia n , F .
''D i s t ort i on o f a Fre que n cy-Modulat e d S i gna l by
Sma l l Loss a nd Phas e Var ia t i ons , " -Proc e e d i n�s I RE , 40
( Fe br ua ry , 1 952 ) , 1 72 - 1 7 6 .
Ba ghdady , E . J .
"The ory o f Low-D i s t or t i on Reproduc t i on of FM
S i gna l s i n L i n e a r Sys t ems , 1 1 Hlli Tra nsa c t i ons on C i r c u i t
The ory , CT-5 ( Se p t embe r , 1 958 ) , 202 - 2 1 4 .
Bay l e s s , J . W . and S . C . Gup ta .
''A New Approach t o FM D is t or ­
t i on , ' 1 I EEE Transa c t i ons on C ommun i ca t i on T e chno l o (!,y ,
C Of.i- 1 6 , No . 2 ( Apr i l , 1 968 ) , 2 6 1 - 2 6 7 .
B e d r o s i a n , E . a nd S . O . R i c e .
" D i s t ort i o n a nd Cros s t a l k of L i n ­
e a r l y F i l t e re d , A n gl e -·Modula t e d S i gna l s , " Pr o c e e d i ngs
I EEE , 5 6 , No . 1 ( Ja nua ry , 1 968 ) , 2 - 1 3 .
Brown , R . F .
":F're que ncy-M o d u la t i on D i s t ort i on i n · L i ne a r Net ­
works , " Pr o c e e d i ngs I EE ( London ) , 1 04 , Pap e r 2 1 96R
( Ja nuary , 1 95 7 ) , 5 2 - 6 2 .
Buch e r , T . T . N . a nd E . B e d r o s ian and S . O . R i c e .
" C omme nts on
' D i s t o r t i on a nd Cros s ta l k of L i n e a r F i l t e re d , A n gl e 1-tod u l a t e d S i gna l s , 1 1 1 Pr o c e e d i ngs I EE� , 5 6 , No . 1 0 ( Oc t obe r ,
1 968 ) , 1 7 5 3 - 1 75 4 .
Ca rs on , J . R . a n d T . C . Fry .
"Variabl e -Fre que ncy E l e c t r i c C i r c u i t
The ory , " Be l l Sys t ems Te �h n i c a l J ourna l � 1 6 ( Oc t ob e r ,
1 937 ) , 5 1 3 - 54 0 .
Cart ianu , G . - - rrva l i d i ty-- o f S ome Approx i lii a.f .fon.S--Tis ed i n F1-'l
D i s t or t i on The ory , " Proc e e d i n ,gs I E EE , 54 , No . 1 0 ( Oc t ob e r ,
1 96 6 ) , 1 46 5 - 1 46 6 .
Che rry , E . C . and R . S . Rivl in .
" Non- L i n e a r D i s t ort i on , w it h
Part i c u l a r He f e re nc e t o t he The ory o f Fre q u e ncy :r.l o dula t i on
Wa ve s , " Ph i l . �la g; . , Pa rt I : 32 ( 1 91 1 ) , 2 6 5 ; Pa rt I I : 33
( 1 942 ) ' 272 .
1 16
"Appl i ca t i on o f Four i e r T ra nsforms t o Var ia b l e
C l a vi e r , A . G .
Fre quency C i r c u i t A na ly s i s , " Pr oc e e d i n!!;s I RE , 37 ( Novemb e r ,
1 94 9 ) , 1 28 7 - 1 2 90 .
C o l l ings , R . H . P . and J . K . Skwirzynski .
"The D i s t or t i on of FM
S i gna l s i n Pas s i ve Networks , " ?-1a r c c n i R e v i e w , 1 7 ( 1 954 ) ,
1 1 3- 1 36 .
Enl oe , L . H . and C . L . Hut h of f .
"A C ommon Err or i n FM D i s t or t i on
The ory , " Pro c e e d i n gs I EEE , 51 ( May , 1 96 3 ) , 8 4 6 .
"The Transm i ss i on o f a Fre que ncy-�!odulat e d Wave
Frant z , W . J .
Through a Ne twork , " Proc e e d ings I RE , 34 ( l\1arch , 1 946 ) ,
114-125 .
Ge rla ch , A . A .
" D i s t ort i on-Band - Pa s s C ons i d e ra t i ons i n Angular
Modula t i on , " Proc e ed i np:s I RE , 38 ( Oc t ob e r , 1 950 ) , 1 2031 207 .
G l a dwin , A . S .
" D i s c uss i on on ' Fre que ncy-Mod u la t i on D i s t ort i on
in L i n e a r Ne t w orks , ' "
Pr o c e e d i ngs l EE ( Lon d on ) , 1 04
( January , 1 95 7 ) , 264 .
"The D i s t or t i on o f Fre quency-M odula t e d Wa ves by
Transm i s s i on Networks , " Pro c e e d i ngs I RE , 35 ( D e c ember ,
1 947 ) , 1 4 3 6 - 1 4 45 .
, a nd R . G . M e d h urs t a nd L . H . Enl oe and C . L . HuthofL
"A
C ommon Error in Fl\1 D i s t ort i on The o ry , 11 Proc e e d i ngs lEE� ,
52 ( Fe bruary , 1 96 4 ) , 1 8 6 - 1 8 9 .
---=--
"The S o l ut i on o f S t e ady-S t a t e Probl ems i n Fl\1 ,
Gold , B .
c e e d i n gs I RE , 37 ( 1 949 ) , 1 2 6 4 - 1 26 9 .
11
Pro­
" Norma l i z e d Pha se a nd Ga i n D e r iva t i v e s a s an A id
Hupert , J . J .
in Eva l ua t i on of F'r<l D i s t or t i on , "
Proc e e d i ngs I EB , 42
( :F'e bruary , 1 954 ) , 4 3 8 - 44 6 .
11
"--,;�
;;_;. ___; · - - '�Response o_f Linear Ne tworks t o �l odula ted Wave f orms ,
Pro c e e d i ngs I EEl<� , 5 3 , No . 9 ( S e p t e mb e r , 1 965 ) , 1 2 6 7 - 1268 .
__
"A The ore t i c a l a nd Exp e r i m e n t a l I nves t i ga t i on of
Ja f f e , D . L .
Tune d - C i r c u i t D i s t ort i on i n :F're q u e ncy-Modula t i on Sys tems , "
Proc e e d i ngs I RE , 33 ( May , 1 945 ) , 3 1 8 -:333 .
Jense n , Ra nda l l W . and Ma rk D . Li e b e rma n .
The I BM El e c t ron i c
C i r c u i t A na lys i s Pro gram ( ECAP ) : T e c h n i q ue s a nd A pp l ica­
t i ons .
Engl ewood C l i i fs , N . J . : Pre n t i c e - Ha l l , I nc . , 1 96 7 .
117
Ma gnus s on , R . I . a n d L . H . Enl oe and C . L . Ru t h off .
"A C ommon
Error i n FM D i s t or t i on The o ry , ' 1 Pro c e e d i n gs I EEE , 52
( Septembe!'�f Ul64: ) , 1 0 8 2 - 1 08 4 .
__
.
Ma t t ha e i , G . L . and L . Young and E . ?>t . T . J ones
M i c rowa ve F i l ­
t e rs , I mpe dance -Ma t c h i n g Ne tworks , a n d C o up l i n g S t r uc t ure s .
New York : �Ic Graw- H i l l , 1 96 4 .
•
.
�tc Ge e , W . F .
" FM D i s t or t i on : A �·l od i f i e d F i rs t - Order T h e o ry , "
Proc e e d i ngs I EEE , 5 6 , . No . 1 0 ( Oc t ob e r , 1 968 ) , 1 7 2 3 .
Me dhurs t , R . G .
" Fundame ri t a l a n d Harmon i c D is t ort i on of Wa ves
Fre quency-Mod u l a t e d w i th a S ingl e T on e , " Proce e d i ngs l EE
( London ) , 1 07 , Pap e r 3 1 8 2 ( March , 1 96 0 ) , 1 5 2 - 1 64 .
" Ha rmon i c D i st ort i on o f fit Waves by L i ne a r Net works , "
Proc e e d i ngs l EE ( Lond on ) , 1 0 1 , Pap e r 1 650 ( �lay , 1 954 ) , 1 7 1 181 .
" D i s t ort i on i n Fre quency-r>iodula t i on
, a nd G . F . Sma l l .
Sys t ems Due t o Sma l l S in us o i da l Va r ia t i ons of T ra nsmi ss i on
Chara c t e r i s t i c s , " Proc e e d ings I Rl'� , 40 ( Novemb e r , 1 956 ) ,
1 608 - 1 6 1 2 .
----
M ircea , A .
" FM D i s t or t i on The ory , "
( Ap r i l , 1 96 6 ) , 705- 706 .
Pro c e e d ings I EEE , 5 4 , No . 4
Pant e r , P . F . �·t o d u l a t i on , No i s e , and Spe c t ra l Ana ly � i s . New
York : McGraw- H i l l Book C ompany , 1 96 5 .
"A Re v i ew of t h e S i deband The ory of F}t w i t h Part i c u l a r
Re f e r e n c e t o D i s t ort i on Probl ems , " I TT F e d e ra l La b ora ­
t or i e s I nt e rna l Rep o r t , ( 1 947 ) , 1 7 7 -206 .
P l o t ki n , S . C .
11�1 Ba ndwi d t h a s a Func t i on o f D i s t ort i on and
Modul a t i on I nd ex , " I EEE Transa c t i ons on C ommun i ca t i on
Te chn o l o gy , C OM - 1 5 , No . 3 ( J une, 1 9 6 7 ) , 4 6 7 - 4 7 0 .
P o l ge , R . J . and B . �! . Ada i r ,
11An Exp e r i mental D e t e rm i na t i on of
D i s t ort i on and D e lay in t h e FM-FM T e l em e t ry Sys t e m of t h e
Sa t urn Spa c e Veh i c l e , " I EEE Tra nsa c t i ons o n A e r ospace a nd
E l e c t r o n i c Sys t ems , A ES- 3 , No . 6 ( iovemb e r , 1 96 7 ) , 97 6 - 980 .
Re ference Da ta f or Ra d � o Engine e rs .
Fourth Ed i t i on ,
New York :
I nt e rna t i ona l T e l e phone a nd Te l e gra p h C orpora t i on , 1 95 6 .
"D i s t or t i on Pr o d u c e d i n a N o i s e M o d ul a t e d fl'l S i gna l
Rice , S . O.
by Nonl i n e a r A t t e n ua t i on a nd Pha se Sh i f t , 11 Be l l Sys tems
T e c hn i ca l J ourna � , 36 ( J uly , 1 95 7 ) , R 7 9-88 9 .
118
" S e c ond and Th ird Ord e r �1odul a t i on T e rms i n t h e D is ­
t or t i on Pr oduc e d Whe n N o i s e M odul a t e d FM Waves a r e F i l ­
t er e d , " Be l l Sys t ems Te chnical J ournal , 48 , No . l ( January ,
1 96 9 ) , 8 7 - 1 4 1 .
"Ef f e c t s o f Tune d C i rcu i t s U p on a Fre que ncy-M odula t e d
Rod e r , H .
l
,
" Proc e e d i n gs I RE , 25 , No . 1 2 ( De c e mb e r , 1 937 ) ,
S i gna
1617-1647 .
Rowe , H . E .
"D i s t ort i on o f A n g l e -M o du l a t e d Wav e s by L i near Ne t ­
works , " I RE Transa c t i ons o n C i r c u i t The ory , C'l'- 9 ( Septem­
b e r , 1 962 ) , 2 8 6 - 2 90 .
Rut h r o f f , C . L .
"Comput a t i on o f FM D is t or t i on i n L i ne a r Net works
f or Band l im i t e d Per i od i c S i gnal s , " Be l l Sys t ems T e chni cal
J ourna l , 47 , No . 6 ( July-A ugus t , 1 968 ) , 1 043 - 1 06 3 .
S c hwart z , �t .
I nf o rma t i on Transmi ss i on , r.f odula t i on , a nd Noi s e .
New York : M c Graw- H i l l B o ok C ompany , 1 95 9 .
Sha ft , P . D .
"D i s t or t i on o f �lu l t i t one I'� S i gnal s Due t o Phase
Nonl i nea r i ty , " I EEE Tra nsa c t i ons on S pa c e E l e c t r on i cs and
'l'e l eme t ry , SET- 9 ( Ma r c h , 1 96 3 ) , 2 5 - 0 5 .
Skw i r zynski , J . K .
"The L i near D i s t ort i on of FM S i gna l s i n
Ba n d - Pa s s Fi l t e rs f or Large M odula t i on Fre quen c i es , " Mar­
c on i Rev i e w , 1 7 , Four t h Qua r t e r ( 1 95 4 ) , 1 01 . ·
S o ko l n ikof f , I . S . and R . M . Re dh e f fe r . Ma thema t i c s of Phys i c s
and Modern Eng i ne e r i n g . N e w York : r.t c Graw-Hi l l B o ok C ompany ,
1 958 .
S t umpers , F . L . H . t-1 .
" D i s c uss i on on ' The D i s t ort i on o f Fre quency­
Modul a t e d Waves by Transmi ss i on Networks ' by A . S . Gladwin , "
Proc e e d ings I RE , · 36 ( Oc t ob e r , 1 948 ) , 1 2 5 7 - 1 2 5 9 .
"D i s t ort i on o f Fre quency-Mod ula t e d . S i gna l s i n El e c t r i ­
c a l Ne t works , " C ommun i c a t i on News , 9 ( Apr i l , 1 948 ) , 82-92 .
Thoma s , G . B . ,Jr .
Ca l c u l us a nd Ana lyt i c Ge ome t ry .
Ma ss . : A d d i s on Wes l ey , 1 95 1 .
�
Read i ng ,
Van d e r Pol , B .
"The Fundamental Pr inc i pl e s of I<,re que ncy M odu­
la t i on , " J ourna l I EE ( Lond on ) , 9 3 , Pt . 3 ( .May , 1 946 ) , 1 5 3 1 58 .
119
wan g , H . s . c .
" Ba ndw i d t h Requ ireme nt f or Frequency-M o dula t e d
S i gna l s , " Proc e e d i ngs I EEE , 5 3 , N o . 8 ( Aug� s t , 1 965 ) , 1 150 .
" D i s t ort i on o f 1<�1 S i gna l s Ca u s e d by Channe l Phas e
Nonl inea r i ty and Amp l i t ude li'l u c t ua t i on , " I EEE Transa ct i ons
on C ommuni c a t i on T e c hn o l o gy , C OM - 1 4 ( A ugust , 1 966 ) , 440- 448 .
We i ne r , D . D .
" Exp e r i m e n t a l St udy o f FM Trans i en t s and Qua s i ­
Sta t i c Re s p onse , " M . S . The s is , M I T Cambr i d ge , !\la ss . ,
January , 1 958 .
, a nd B . J . Le on .
" On the Qua s i -S ta t i onary Resp onse o f
L i near T ime - I nvar iant F i l t e rs t o A r b i t ra ry Fr.t S i gna l s , "
I EEE Transa c t i ons on C i r c u i t The ory , CT- 1 1 ( J une , 1 964 ) ,
308 -309 .
----
"The Qua s i -Sta t i onary Response o f Linea r Sys t ems t o
Modula t e d Wave f orms , " Pro c e e d ings I EEE , 53 , No . 6 ( June ,
1 965 ) , 564 -575 .
Yavuz , Y .
"A N o t e on Fre quency Modula t i on Spe c t ra , "
i n gs I EEE , 56 , No . 2 ( Februa ry , 1 968 ) , 220-22 1 .
Pr o c e e d ­
, and P . M . Ebe r t and J . E . �1az o ,
"Anothe r N o t e on Fre ­
__q u e n cy M od u l a t i on Spe c t ra , " Pro c e e d i n gs I Er�E , 5 6 , No . 7
( J uly , 1 968 ) , 1 24 1 .
_
_
_
1 20
A PPE:NDI X A
ECAP SOLUTI ON OF THE BUTTERWOHTH TWO- POLE F I LTEH
p l e e xe c ut i on of ECAP f or t he case of
The f ol l ow in g is a s�m
:
t h e But t e rwor t h t w o - p o l e f i l t e r d i s cuss e d i n S e c t i on 4 . 5 . ( Only
a p ort i on o f the data is g i ven . )
For t h e p urp os e s o f c od ing t he c i r c u i t f or s pe c i f i cat i on
t o ECAP , t h e branches a nd nodes a r e num b e r e d a s s h own i n
For a d e ta i l e d e xp lana t i on o f ECAP pro gra mm ing
F i gure A - 1 .
and appl icat i ons , the r e a d e r is re f e r r e d t o the s o u r c e l i s t e d
b e l ow .
1
C
Bl
B2
B3
B4
B5
B6
B7
2 POLE FI LTER , BUTTEHWORTH
AC
N ( l , O ) , I = l/O , R= l
N ( l , O ) , C= 2 . 09 1
N ( l , O ) , L= 0 . 01 1 25
N ( l , 2 ) , C= 0 . 1 5 92
N ( 2 , 0 ) , L= 0 . 01 1 25
N ( 2 , 0 ) , C= 2 . 09 1
N ( 2 , 0 ) , R= l
FR= 0 . 9 ( + 40 ) 1 . 1
PR , NV ( 2 )
EX
FREQ =
. 90000000E 0
NO
NO VOL .
PHA
2 = . 96 4 7 4 9 6 6E - 1
- . 1 2 9 92542E
1
3
Randa l l W . J e ns e n and Mark D . L i e b e rman , The I BM E l e c t r o�
.
i c C ir c u i t Ana lys i s Pro gram ( ECAP ) : Te c hn iques and A pp l i ca t i ons
( Englewood C l i f fs , N . J . : Prent i c e - Ha l l I nc . , 1 96 7 ) .
121
FREQ =
. 90500000E 0
NO
NO VOL
PHA
2 = . 1 0744 5 1 9E 0 - . l 32 3 3 1 8 0E
FREQ =
. 91 000000E 0
NO
NO VOL
PHA
2 = . 1 201 2767E 0 - . l 35 02748E
FHEQ =
. 91500000E 0
NO
NO VOL
PHA
2 = . 1 3484070E 0 - . 1 38 06406E
FREQ =
. 92000000E 0
NO
NO VOL
PHA
2 = . 1 51 95372E 0 - . l 4 1 50394E
3
3
3
3
FllEQ =
. 92500000E 0
NO
NO VOL
PHA
2 = . 1 71 8801 6E 0 - . 14 542185E , . 3
FREQ =
. 93000000E 0
NO
NO VOL
PHA
2 = . 1 9505188E 0 - . 1 4 9 90580E
3
FREQ =
. 93500000E 0
NO
. NO VOL
PHA
2 = . 22 1 8 5 955E 0 - . l 5505587E
3
FREQ =
. 94000000E 0
NO
NO VOL ·
PHA
2 = . 25253632E 0 - . l 6 0 97882E
3
·•
.
j
122
I
I
r
O . l 5 9F
B4
Nl
N2
O . Ol l 2H
lA
1n
2 . 09F
2 . 09F
1n
O . Ol l 2 H
Bl
B2
B3
B5
B6
NO
F i gure A - 1 .
ECAP S c h e ma t i c D i a gram of But t e rw o r t h
Two- Po l e Fi l t e r .
B7
123
APPENDIX B
SUBROUT I NE SUBPROGRMI BESL
The f ol l ow i ng i s a c omp l e t e l is t of t h e s ubrout i n e subprogram
BESL :
100 .
101.
1 02 .
1 05 .
110.
1 15 .
1 20 .
125 .
1 30 .
1 35 .
1 40 .
1 45 .
1 50 .
160 .
165 .
170 .
1 75 .
'
'
� --
180 .
185 .
1 90 .
1 95 .
200 .
205 .
210 �
215 �
220 �
225 .
2 30 .
235 .
240 .
245 .
C:
c :
IBESLI
2-22-69
REV 4 - 2 - 2 9
J . OVNI CK
c :
SUBROUTI NE BESL [ FO , FH , FD , N , J , L , �I J
C:
GENERATI ON OF F'?-1 SPECTRUH GI VEN THE
C:
CARRI ER FRE��UENCY li'O , THE M ODULATI NG
C:
FREQUENCY �I , A ND THE PEAK CARRI ER
C:
DEVIATI ON FD
C:
c :
c :
REAL .J ( 200 ) , F ( 200 ) , PWR
C:
C OMPUTE BETA
c :
BETA= FDIHI
WRITE ( l , l ) FO , FM , FD , BETA
1 FORMAT < II5X ' I F].t SPECTRUM ' III5X ' I CARRI E R FREQU
ENCY I ' 4X ' I = I , }t� 9 . 3I5X ' I �!ODULATI NG FREQUENCY = I , F9
. 3I5X , 1 PEAK DEVI A T I ON ' , 7X , ' = ' , F9 . 3I5X , 1 DEVI ATI ON
RATI O I , 5X , ' = ' , F9 . 3III )
C:
C:
C OMPUTE BESSEL C OEFFI C I ENTS .
C:
D O 9 :M = 1 00 , 200
P=ivl- 100
A=l . O
B= A * ( P+ 1 )
I F ( P . EQ , 0 ) GO T O 3
I F ( P . EQ . 1 ) G O T O 5
DO 2 I=2 , P
2 A=A * I
GO T O 4
3 J ( M ) = 1 . 0- ( ( 0 . 5 * BETA ) * * 2 )
GO T O 6
124
"
-- --- - ---
. - ····-·
--------·-----·· ------·-----
2 50 .
252 .
255 .
260 .
265 .
270 .
275 .
280 .
285 .
2 90 .
2 95 .
300 .
305 .
310 .
315 .
320 .
. 325 .
330 .
335 .
340 .
345 .
350 .
355 .
360 .
365 .
370 .
375 .
380 .
385 .
390 .
3 92 .
3 95 .
400 .
405 .
410 .
415 .
420 .
- · - ·--··-·
-- -··-----------
----··-·-·-·-- - - - · -··-······· ··--
--- ··--·---------
4 B=A * ( P+ l )
5 J ( l-1 ) = ( ( 0 . 5 * BETA ) * * P )IA - ( ( 0 . 5 * BETA ) * * ( 2 +P ) )IB
6 D O 7 K=2 , 25
B= B * ( P+K ) * K
7 J ( f.I ) =J ( M ) + ( ( - 1 ) * * K ) * ( ( 0 . 5 * BETA ) * * ( 2 *K+P ) )IB
C:
C O�IPUTE J OF - P
C:
c :
L=H- 2 * P
J ( L )= ( ( -l ) * *P ) *J(M)
c:
C:
C OHPUTE SIDEBA!\TD FilliQUENCI ES
c :
F ( M ) = FO+ P * �1
F ( L ) = FO-P* fl1
c :
C:
sm1 TOTAL SI DEBAND POWER
C:
I F ( P . EQ . 0 ) PWR=J ( l 00 ) * * 2
I F ( P . GT . 0 ) PWR=PWR+2 * J ( M ) * * 2
TEST= l . O-PWR
I F ( TEST . LT . lE-6 ) GO TO 1 0
I F ( M . LT . 200 ) G O T O 9
WRI TE ( 1 , 8 )
8 FORJvJAT ( III5X , 1 EIWOR- S I GNI FI CANT SI DEBANDS A R
E BEI NG NEGLECTED ' )
9 CONT INUE
C:
C:
DI SPLAY DATA
C:
1 0 WRITE ( 1 , 1 1 )
1 1 FORMAT ( 5X , ' J , F ( ) I , 5X , ' P ' , 5X , ' JP ( BETA ) I , 4X , ' F
REQUENCY ' I )
D O 1 2 N=L , M
P= N- 1 00
1 2 WRITE ( l , l 3 ) N , P , J ( N ) , F ( N )
1 3 F ORHAT ( I 9 , I 8 , Fl2 . 4 , Fl 2 . 3 )
RETURN
END
The f o l l ow i n g i s a samp l e e x e cut i on of BESL for B
FM SPECTRUM
CARRI ER FHEQUENCY
f.tODULATI NG FREQUENCY
PEAK DEVI ATI ON
DEVIATI ON RATI O
=
=
=
=
1 . 000
0 . 01 0
0 . 040
4 . 000
=
4:
125
J ,F( )
91
92
93
94
95
96
97
98
99
1 00
101
1 02
103
1 04
105
1 06
1 07
1 08
109
p
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
JP ( Bl�TA )
- 0 . 000 9
0 . 0040
- 0 . 01 52
0 . 0491
- 0 . 1 32 1
0 . 28 1 1
- 0 . 4 302
0 . 3641
0 . 06 6 0
-0 . 3971
-0 . 0660
0 . 3641
0 . 4 302
0 . 28 1 1
0 . 1 32 1
0 . 0491
0 . 01 5 2
0 . 0040
0 . 0009
FREQUENCY
0 . 910
0 . 92 0
0 . 930
0 . 940
0 . 950
0 . 96 0
0 . 970
0 . 98 0
0 . 99 0
1 . 000
1 . 010
1 . 020
1 . 03 0
1 . 04 0
1 . 050
1 . 060
1 . 070
1 . 08 0
1 . 090
126
APPENDIX C
PROGRAM F'MDI ST
'
The f ol l ow i ng i s a c ompl � t e l is t of t he pr o gram F?>iDI ST :
10.
11 .
12 .
14 .
16 .
18 .
20 .
22 .
23 .
24 .
26 .
28 .
29.
30 .
31 .
32 .
33 .
34 .
36 .
38 .
39.
40 .
41 •
42 .
43 .
44 .
45 .
46 .
48 .
50.
52 .
54 .
56 .
58 .
60.
C:
C:
/FMDI ST/
REV
4-2-69
4-5-69
J . OVNI CK
c :
REAL J ( 2 00 ) , P ( 200 ) , THETA ( 200 ) , F ( 5 1 )
PI = 3 . 1 4 1 5 92 6
WRITE ( 1 , 1 )
1 FOill i AT ( ///5X , 1 FM DI STOHTI ON 1 /5X , 1 - - - - - - - - - - - - ' // ' P ( I ) , THETA ( I ) ' / )
ACCEPT Nl , N2 , ( P ( I ) , THETA ( I ) , I = Nl , N2 )
A CCEPT " S I DEBAND I NCHE:I>il<�NT = " , NS
A CCEPT 11FO= " , FO
ACCEPT " FM= " , FM
ACCEPT " FD= " , FD
FDD=FD
D O 200 NT= l , 6
CALL BESL t FO , n1 , FD , N , J , L , �1 ]
WRITE ( 1 , 40 )
4 0 F OR�tAT ( //8X , I I I , 7X , I "F' ( I ) I I )
D O 1 00 I = 1 , 5 1
T = ( I - 1 ) / ( Ft-1 * 50 )
AS=O
BS= O
AAS= O
BBS=O
DO 2 0 K=L , M
KK= 1 00+ ( K- 100 ) * NS
X = P ( KK ) * J ( K )
Y= ( K- 1 00 ) * 2 * PI * FM * T+THETA ( KK ) * PI/180
A=X * COS tY J
B=X * SI N C.Y J
Z= ( K- l00 ) * 2 * PI * f1.1
AA= - Z * B
BB= Z * A
AS=AS+A
BS= BS +B
AAS=AAS+AA
127
- -·- -·---------- -----·--- - ..... --------- - ------ --·--- ----------- ----- - - ___________ _. -·-· ·------------------·---..-·------ --- ·
62 .
64 .
65 .
68 .
67 .
70.
72 .
74 .
76
77 .
78 .
80.
•
BBS= BBS+BB
20 C ONT I NUE
FF= ( A S * BBS- BS * AA5) / ( AS * * 2+BS * * 2 )
F ( I ) = FF/ ( 2 * PI * FD )
II=I-1
WHITE ( 1 , 50 ) I I , F ( I )
50 FORMAT ( I 9 , Fl 2 . 4 )
1 00 C ONTI NU E
CALL SERI ES (F , I)
200 FD=FD+FDD
STOP
END
1
The f ol l owi ng i s a sampl � e x e c ut ion o f Fr.ID I ST f o r t h e Butt e r w o r t h two-p o l e f i l t e r :
·
FM D I STORTI ON
- - - - - - - - - - - - -
P ( I ) , T HETA ( I )
80 , 1 20
. 09647 , 230 . 07
. 1074 , 22 7 . 67
. 1 201 , 22 4 . 97
. 1 348 , 22 1 . 94
. 1 520 , 2 1 8 . 50
. 171 9 , 2 1 4 . 58
. 1 95 1 , 2 1 0 . 09
. 22 1 9 , 204 . 94
. 2525 , 1 9 9 . 02
. 2870 , 1 92 . 2 3
. 3244 , 1 8 4 . 4 9
. 36 32 , 1 75 . 80
. 4008 , 1 66 . 2 9
. 4343 , 1 56 . 1 7
. 46 09 , 1 45 . 78
. 4796 , 1 35 . 48
. 491 1 , 1 25 . 54
. 496 9 , 1 1 6 . 1 0
� 49 93 , 107 . 1 8
. 49 99 , 98 . 6 9
� 5000 , 9 0 . 50
. 5000 , 82 . 4 3
. 49 98 , 74 . 34
. 4 988 , 6 6 . 09
. 4 959 , 57 . 58
. 48 96 , 48 . 7 9 . 4786 , 3 9 . 75
1 28
. 46 1 9 , 30 . 6 1
. 43 94 , 2 1 . 56
. 41 2 1 , 1 2 . 82
. 38 1 6 , 4 . 60
. 34 98 , - 2 . 96
. 31 8 5 , - 9 . 8 0
. 2888 , - 1 5 . 91
. 26 1 5 , -2 1 . 33
. 2368 , -2 6 . 1 3
. 2147 , - 30 . 37
. 1 952 , - 3 4 . 1 2
. 1780 , - 3 7 . 45
. 1 628 , -40 . 42
. 1 4 93 , - 4 3 . 07
SIDEBAND I NCRE!vlENT
F0= 1
FM= . 005
FD= . 01 0
j
1
CARRI ER FREQUENCY
t<lODULATI NG FREQU ENCY
PEAK DEVI ATI ON
DEVIAT I ON RATI O
=
=
=
=
FM SPECTRUM
1 . 000
0 . 005
0 . 01 0
2 . 000
1 29
. - - � ..-. -·�---�--
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
0 . 8843
0 . 81 8 4
0 .· 7 3 96
0 . 6 4 94
0 . 5492
0 . 4401
0 . 3239
0 . 2026
0 . 0785
- 0 . 0466
- 0 . 1 708
- 0 . 2 924
-0 . 4094
- 0 . 5 1 97
-0 . 6215
-0. 7136
- 0 . 7 95 1
- 0 . 8646
-0 . 9204
-0 . 9612
-0 . 9866
-0 . 9 9 6 9
- 0 . 9 925
-0 . 9731
-0 . 938 1
- 0 . 8876
- 0 . 8223
- 0 . 7441
- 0 . 6542
- 0 . 5540
-0 . 4444
- 0 . 3274
- 0 . 2052
-0 . 0800
. 0 . 0462
0 . 1716
0 . 2943
0 .4 1 2 3·
0 . 5235
0 . 6260
0 . 7184
0 . 7 998
0 . 8689
0 . 9242
0 . 9642
0 . 9887
I
I
HARH ONI C ANALYSI S OF F ( X )
F ( X ) = A O + SUM ( N= 1 TO I NFI NI TY ) [ BN * S I N ( N * X ) +CN * COS ( N
*X ) )
·"·
130
FOURI ER C OEFFI CI ENTS
A O ( DC Tl!:ffii )
N
1
2
3
4
5
6
7
8
9
10
*
* *
* * *
=
0 . 000000
BN ( SI N )
CN ( C OS )
DB*
0 . 1 42 1 45
- 0 . 00208 6
0 . 0005 8 9
0 . 000028
- 0 . 000209
- 0 . 000004
- 0 . 000242
0 . 000006
- 0 . 000284
0 . 000002
0 . 985481
0 . 001 042
-0 . 001267
- 0 . 000022
- 0 . 000155
- 0 . 000033
- 0 . 000 1 22
0 . 00002 9
-0 . 000057
0 . 000004
o . oo
-52 . 6 1
- 5 7 . 06
DB BELOW FUNDAMENTAL
FUNDAMENTAL POWER = 0
GREATER THAN 60 DB DOWN
* * *
* * *
* * *
* * *
* * *
* * *
* * *
A
0 . 9 95680
0 . 0023 3 2
0 . 001 3 97
0 . 000036
0 . 000260
0 . 000033
0 . 00027 1
0 . 000029
0 . 0002 90
0 . 000005
131
APPENDIX D
SUBROUTI NE SUBPROGRAM SERI ES
The f o l l ow i ng is a c omp1 Ef t e l is t of t h e s ubrout i ne s ubprogram
SERI ES :
500 .
501 .
502 .
503 .
504 .
5 05 .
506 .
507 .
508 .
509 .
510 .
511 .
512.
513 .
514 .
515 .
516 .
517 .
518 .
519 .
C:
c :
/SERI E S/
2-22 - 6 9
REV 4-5- 6 9
J . OVNI CK
C:
SUBROUTI NE SERI ES [ F , I J
C:
c :
HARM ONI C ANALYS I S Oli' AN A RBITR4.RY PERI ODIC
c :
WAVEFOR.t'-'1 F ( X ) GI VEN 51 DATA POINTS EVENLY
C:
SPACED OVER THE FUNDAl-lENT AL PERI OD
C:
PI = J . l 4 1 5 926
C:
C OMPUTE DC T�RM
C:
C:
A=O
D O 1 I = l , 50
1 A = A + ( F ( I ) + F ( I + l ) )/100
AT=A BS CAl
IF ( AT . LT . 0 . 000001 ) A = O
WRITE ( 1 , 2 )
2 F Oill-l AT ( //5X , ' HAHMONI C A NALYSI S OF F ( X ) ' //5X , '
F ( X ) = A O + SUM ( N= l TO I NFI NI TY ) [ BN * SI N ( N * X ) +CN *
COS ( N * X ) J ' ///5X , ' FOURI ER COEFFIC IENTS ' /// )
520 . WRITE ( l , 3 ) A
521 . 3 FOR\1AT ( 5X , ' AO ( DC TERM ) = 1 , Fl 2 . 6///6X , ' N ' , 5X
' ' BN ( SI N ) I , 5X , ' CN ( COS ) I , 6X , ' DB * ' , 6X , ' A ' / )
522 . c :
523 . C :
C OMPUTE S I N AND C OS TERMS AND
5 24 . C :
RELATI VE HAR\1 0NIC U3VELS
525 . C :
526 . D O 1 2 N=l , l O
527 . B = O
528 . D O 4 I = l , 50
529 . 4 B= B+ ( ( F ( I ) + F ( I + l ) ) * ( SI N [N * ( I - l ) * PI/25 ] + SI N (N * I
* PI/2 5 ) ) ) /100
1 32
5 30 .
531 .
5 32 .
533 .
534 .
BT=ABS C BJ
I F ( BT . LT . 0 . 000001 ) B=O
C=O
D O 5 I = l , 50
5 C=C+ ( ( F ( I ) +F ( I +l ) ) * { COS CN * ( I - l ) * PI/25 l +C OS [ N * I
* PI/25 J ) )/100
5 3 5 . CT=ABS t C J
5 3 6 . I F ( CT . LT . 0 . 000001 ) C=O
537 . I F ( N . EQ . 1 ) A l = SQRT ( B * * 2 +C * * 2)
538 . I F ( A 1 . LT . 0 . 000001 ) GO T O 8
5 3 9 . A = S QRT CB * * 2 +C * * 2 J
540 . I F ( A . LT . 0 . 000001 ) G O T O 1 0
541 . · DB=20 *ALOG10 CA/A lJ
542 . I F ( DB . LT . - 60 . 0 ) GO TO 1 0
5 4 3 . I F ( DB . GT . 6 0 . 0 ) GO T O 10
5 4 3 . 5 IF ( N . EQ . 1 ) A=Al
544 . 6 WRITE ( 1 , 7 ) N , B , C , DB , A
545 . 7 FOHI\IAT ( I 7 , 2 F1 3 . 6 , F9 . 2 , Fl 0 . 6 )
546 . GO T O 1 2
547 . 8 WRI TE ( 1 , 9 ) N , B , C , A
548 . 9 FORHAT ( I 7 , 2F l 3 . 6 , 6X , 1 * * 1 , Fl l . 6 )
549 . GO TO 1 2
550 . 1 0 WHITE ( 1 , 1 1 ) N , H , C , A
551 . 1 1 FOm•tAT ( I 7 , 2F l 3 . 6 , 5X , 1 * * * 1 , F1 l . 6 )
553 . 1 2 C ONT I NU E
554 . WRITE ( 1 , 1 3 )
555 . 1 3 FORMAT ( /I/5X ' I * I ! 3X , I DB BELOW FUNDAMF..:NTAL I /5
X , I * * I ' 2X ' I FUNDA�1ENTAL POWER = 0 I /5X , I * * * I , lX ' I G
REATER THAN 6 0 DB D OWN ' // )
556 . RETURN
5 5 7 . END
The f o l l ow i n g i s a samp l e e x e c ut i on of SERIES f o r a saw- t ooth
wa ve o f un i t y peak ampl i t ud e :
HARM ONI C A NALYS I S
OF
F' ( X )
F ( X ) = A O + SUM ( �= l T O I NFINITY ) ( B N * SI N ( N *X ) �C N * C OS ( N
* X )]
FOURI ER C OEFFI C I F�NTS
A O ( DC TF�RM ) =
N
1
0 . 4 9 99 9 9
BN ( SI N )
C N ( COS )
DB*
- 0 . 3 1 8205
0 . 000000
0 . 00
133
2
3
4
5
6
7
8
9
10
*
**
***
f '
-0 . 1 5 8 945
-0 . 1 05788
-0 . 07 9158
-0 . 0 6 3 1 37
-0 . 052421
- 0 . 044737
-0 . 038947
- 0 . 034420
- 0 . 030776
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
D B BELOW FUI-."DAMENTAL
FUNDAMENTAL POWER = 0
UNDEFI NED
- 6 . 02
- 9 . 56
- 1 2 . 08
- 1 4 . 04
- 1 5 . 66
- 1 7 . 04
- 1 8 . 24
- 1 9 . 31
-20 . 28
134
A PPENDIX E
ERROR I N C OMPUTI NG li,OURIER C OE.F'FI CI ENTS
The purp o s e of one of t h e expe r im e n t a l runs of s ubrout in�
SERI ES wa s t o c ompu t e t he r e l a t i ve l e ve l s o f the harm o n i c s i n a
saw- t o o t h wa ve .
Tab l e 3 3 g i ve� t h e r e s u l t s f o r a saw- t o o t h
wa v e t ha t wa s sampl e d f i fty- one t ime s i n o n e p e r i od .
I t i s n o t e d t hat e x c e l l e n t a c c ura cy i s r e t a i n e d out t o
: at l ea s t t h e t e nt h harm o n i c cyc l e .
T h e reason f or t h i s i s t ha t
s inus o i da l f unc t i ons a r e approxima t e d a lm o s t e xa c t ly by t he
t ra pe z o i da l t e c hn i qu e of i n t e grat i o n a s t h e f o l l owing d emons t ra t i on s e rves t o i l l us t ra t e .
C ons i de r t h e i n t e gra l
b
n
2
l
=
T1
�
sin
0
Le t us a pprox i ma t e t11e func t i on
2
1
T1
=
nxdx
sin
2
l.
( E- 1 )
nx by only n i ne samp l e s
a s s hown i n F i gure E - 1 .
T o s impl i fy t h e int e gra t i on by t h e Trape z o i da l Ru l e we ca n
take a dva n t a ge o f the fo l l ow i ng i d ent i t y
l
T1
.
s 1n
2
nx
=
l
2 TT
- c os 2 nx .
( E-2 )
We may furt h e r take a dvantage o f t he fa c t t ha t t h e p e r i od of
- c os 2
nx
is
TT
s o t hat we ne e d only i nt e gra t e over a range of
135
Harmoni c
Numbe r
n
Ca l cu l a t e d
Harmon i c L e ve l
dB
A c t ua l
Harmon i c L e ve l
dB
C omput a t i on
Error
dB
1
0 . 00
o . oo
o . oo
2
- 6 . 02
-6 . 02
o . oo
3
- 9 . 56
- 9 . 54
- 0 . 02
4
- 1 2 . 08
- 1 2 . 04
- 0 . 04
5
- 1 4 . 04
- 1 3 . 98
- 0 . 06
6
- 1 5 . 66
- 1 5 . 57
- 0 . 09
7
- 1 7 . 04
- 1 6 . 91
-0 . 1 3
8
- 1 8 . 24
- 1 8 . 06
- 0 . 18
- 1 9 . 31
- 1 9 . 09
- 0 . 22
- 20 . 28
-20 . 00
- 0 . 28
9
10
Tab l e 33
Error i n C omput i ng Ha rm on i c L e ve l s for a Saw-T o o t h
Wave .
1 36
1
71
.
s 1n
2
nx
1
71
Fi gure E-1 .
Samp l ing of
1
TT
sin
2
nx .
1 37
and t h e n m ul t ip ly t h e r e s u l t by two .
TT
Thus , by t he Trape -
z o i da l Rul e we may a p prox ima t e t h e i n t e gra l b
b
n
2
=
:[
+
+
l
n
a s f o l l ows
�]
l
-- + s i n 2a + -- + s in ( 2a + 2!... )
2 TT
2 TT
2
;)
:[
l
2 TT
+
s in ( 2a +
+
l
2 TT
+ s in ( 2a + TT )
:[
l
2 TT
+
s in ( 2a + TT ) +
l
2 TT
+ s i n ( 2a
3
2
+ s i n ( 2a +
TT
) +
l
2 TT
+
3 71
)
2
�]
�]
+ s in ( 2a + 2 TT )
�]
( E- 3 )
. and s imp l i fy i n g y i e l ds
b
n
=
!!_ [
4
i_
TT
+ 2 s i n 2a + 2 s in ( 2a +
+
2 s i n ( 2a +
11
11
)
2
) + 2 s i n ( 2a + .
�
17
)
J.
( E- 4 )
Emp l oy i ng t h e i dent i t y f or s i n ( x + y ) y i e l ds t he f o l l ow i ng
result
b
n
=
1
wh i c h i s e xa c t ly t h e resul t of Equa t i on ( E- 1 ) .
( E-5 )
One can s e e
t ha t t h e t ra pe z o i da l i n t e gra t i on t e c h n i que y i e l d s exc e l l e nt
r e s ul t s , whe n appl i e d t o s i nus o ida l f unc t i ons .
Tha t t h e a b ove � e s u l t i s val id i n genera l i s expe c t e d from
t h e Uni f orm Sampl i n g Pr i n c i p l e .
S i n c e g o od a c c ura cy wa s d e s ired out t o t he t e n t h harmon i c
1 38
i n . the �1 d is t o rt i on c a l c u l a t i on� , f i fty- one sampl e s p e r
f undamental pe r i od w e r e t h us s e l e c t e d .
. 1-
139
- ··-·-·
.. ....... .....
._,..................
. . . - ··-··· ••· ·•··: ······· ·-------- --·-----·· ·- -··-·-··· --··---- --·· --C ..... ....• ••..._,.,_ _______
APPENDIX F
PROGRAM QS
' Th e f o l l ow i n g is a c ompl e t e l is t of t h e p r ogram QS :
10.
12 .
14.
16 .
18 .
20 .
22 .
24 .
26 .
28
30 .
32 .
34 .
36 .
38 .
40 .
42 .
43 .
44 .
45 .
46 .
48 .
49 .
50.
52 .
54 .
55 .
56 .
57 .
58 .
60.
62 .
64 .
66 .
•
/QS/
C:
4 - 5- 6 9
J . OVNI CK
c :
REAL F ( 5 1 ) , L
PI = 3 . 1 4 1 5 926
WRITE ( 1 , 1 )
1 FORI\IAT ( 5X , ' QUAS I - STAT I ONARY RESPONSE ' /8X , ' SI NG
LE POLE FI LTER ' // )
R= l . O
L=U . O l 5 92
C= l . 5 92
Ji'O= 1 0
FM= O . U05
FD= O . Ol O
T 02= l . O/ ( Fl\1 * 1 000 )
W0= 2 * PI * FO
W�1=2 * PI * F'M
WD= 2 * PI * FD
WDD=WD
DO 200 NT= l , 6
BETA=WD/WM
WHITE ( l , 30 ) BETA
30 FORI\lAT ( / ' BETA = ' , F6 . 3 )
WHI TE ( 1 , 40 )
40 FORMAT ( //8X , ' I ' , 7X , ' F ( I ) ' / )
D O 1 00 1 = 1 , 5 1
T= ( I - l ) / ( FN * 50 )
W l = WO+WD * COS (WM * ( T -T02 ))
W2= WO+WD * COS (WM * ( T+T02 ))
A l = ATAN2 ( R * ( l . O - ( Wl * * 2 ) * L * C ) , Wl * L1
A2=ATAN2 ( R * ( l . O- ( W2 * * 2 ) * L * C ) , W2 *L1
DA= ( A2-Al ) * 500 * Fl\1
FF=WD * COS [WH * T +DA1
F ( I ) = Fl''/WD
II=I-1
WRITE ( 1 , 50 ) I I , F ( I )
•
140
68 .
70 .
72 .
74 .
76 .
78 .
5 0 FOHMAT ( I 9 , Fl 2 . 4 )
1 00 C ONT I NUE
CALL SERI ES l F , I )
200 WD=WD+WDD
STOP
END
The f o l l ow i n g i s a sampl e exe c ut i on o f QS f or t h e s i n gl e - pol e
f i l te r :
BETA
=
1 2 . 000
I
F(I )
1 . 0000
0 . 9972
0 . 978 9
0 . 9457
0 . 8 98 5
0 . 83 8 4
0 . 76 6 6
0 . 6847
0 . 5 940
0 . 4958
0 . 3910
0 . 2797
0 . 1616
0 . 03 6 9
- 0 . 0926
-0 . 2237
- 0 . 3522
- 0 . 4744
- 0 . 58 7 3
-·0 . 68 8 9
-0 . 7778
- 0 . 8527
o 9 1 28
- 0 . 9577
- 0 . 9868
- 1 . 0000
- 0 . 9 975
-, 0 . 9795
- 0 . 9467
-0. 8999
- 0 . 8 403
- 0 . 7 6 90
-0 . 6875
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
-
-
- - ·· - - � - - -
·.
.
- --·--- - .
-- -
---
.
- -
- - �-
- -- - -
---
-
-
- ·--- --------------
.
l l.il
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
-0 . 5973
-0 . 4 994
- 0 . 3943
- 0 . 2821
-0 . 1625
- 0 . 03 6 0
0 . 09 5 1
0 . 22 7 0
0 . 3558
0 . 47 7 7
0 . 5 9 02
0 . 6913
0 . 77 96
0 . 8541
0 . 9 1 38
0 . 9583
0 . 98 7 1
1 . 0000
HARM ONI C A NALYS I S OF F ( X )
F(X ) = AO
C OS ( N *X ) l
+
Smt ( N= 1 T O I NFI NI TY ) [ BN * SI N( N * X ) +CN*
FOU RI ER C OEFFI CI ENTS
A O ( DC TERM ) =
0 . 000000
N
BN ( SI N )
C N ( COS )
DB*
1
2
3
4
5
6
7
8
9
10
0 . 077806
- 0 . 00 1 4 1 5
- 0 . 01 6497
0 . 0005 9 0
0 . 003382
- 0 . 0001 7 9
- 0 . 0006 68
0 . 0000 4 6
0 . 0001 27
-O . OOOO l l
0 . 996057
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 000000
0 . 00
-56 . 98
- 3 5 . 64
***
- 4 9 . 41
***
***
***
***
***
*
D B BELOW FUNDAMENTAL
* * FUNDAMENTAL POWl�H
0
* * * GREATER THAN 6 0 DB D OWN
=
A
0 . 99 9092
0 . 001 4 1 5
0 . 0 1 6 4 9?
0 . 0005 90
0 . 003382
0 . 000 1 7 9
0 . 000668
0 . 000046
0 . 000 1 27
O . OOOO l l