CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
THE EFFECTS OF PHASE AND AMPUTUDE ERRORS
ON AMPLITUDE WEJGH\ING FUNCTIONS
A Q·aduate project submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
William Lawrence Paul, Ill
May 1988
The Graduate Project of William Lawrence Paul, Ill is approved:
~--~---· ~---~-~-~~-~---
Profes . r Ray Pettit
.J
Ptf6fessor John Adams (Chair)
California State University, Northridge
ii
Acknowledgement
wish to extend my sincere gratitude to the following people:
My wife Linda, for her love and continual support.
Olyn Boyle of Hughes Aircraft Company, for encouraging me to
pursue and complete a Master of Science degree.
The Hughes Aircraft Company Fellowship program, for
providing financial support.
Professor John Adams, for his guidance.
iii
TABLE OF CONTENTS
List of Tables
vi
List of Figures
vii
List of Symbols
Xi i
Abstract
xiv
Chapter 1 I ntrod uctio n
1
Chapter 2 Spectral Analysis and Spectral Leakage with the OFT
2
2.1 The OFT Algorithm
2
2.2 Spectral Leakage
2
2.3 Measures of Performance
6
2.4 Amplitude Weighting Functions
Chapter 3 Amplitude Errors
10
15
3.1 Model
15
3.2 Theoretical Predictions
15
3.3 Simulation Results
19
Chapter 4 Phase Errors
66
4.1 Model
.66
4.2 Simulation Results
67
Chapter 5 Summary and Conclusions
94
iv
TABLE OF CONTENTS
99
Bibliography
100
Appendix
v
LIST OFTABLES
~
Table
Window parameter comparison
10
II.
Window parameter comparison with amplitude errors
95
Ill
Window parameter comparison with phase errors
97
vi
LIST OF FIGURES
Figure
1
Digital sine function
5
2a
Signal periodic over observation interval
7
2b
Signal aperiodic over observation interval
8
3
Dolph-Chebychev response
12
4
Kaiser-Bessel response
13
5
Taylor response
14
6a
Individual components of spectrum,fa=0.02 (broad view)
17
6b
Individual components of spectrum, fa=0.02 (narrow view) 1 8
6c
Individual components of spectrum, fa=0.005
20
7
Dolph-Chebychev response, fa=0.0025 cycles/sample
21
8
Dolph-Chebychev response, fa=0.005 cycles/sample
22
9
Dolph-Chebychev response, fa=0.0075 cycles/sample
23
10
Dolph-Chebychev response, fa=0.01 cycles/sample
24
11
Dolph-Chebychev response, fa=0.02 cycles/sample
25
12
Dolph-Chebychev response, fa=0.03 cycles/sample
26
13
Kaiser-Bessel response, fa=0.0025 cycles/sample
27
14
Kaiser-Bessel response, fa=0.005 cycles/sample
28
vii
LIST OF FIGURES
~
Figure
15
Kaiser-Bessel response, fa=0.0075 cycles/sample
29
16
Kaiser-Bessel response, fa=0.01
cycles/sample
30
17
Kaiser-Bessel response, fa=0.02 cycles/sample
31
18
Kaiser-Bessel response, fa=0.03 cycles/sample
32
19
Taylor response, fa=0.0025 cycles/sample
33
20
Taylor response, fa=0.005 cycles/sample
34
21
Taylor response, fa=0.0075 cycles/sample
35
22
Taylor response, fa=G.01
cycles/sample
36
23
Taylor response, fa=0.02 cycles/sample
37
24
Taylor response, fa=0.03 cycles/sample
38
25
Gain degradation, 0.0025 cycles/sample
40
26
Gain degradation, 0.005 cycles/sample
41
27
Gain degradation, 0.0075 cycles/sample
42
28
Gain degradation, 0.01 cycles/sample
43
29
Gain degradation, 0.02 cycles/sample
44
30
Gain degradation, 0.03 cycles/sample
45
31
ISLA, 0.0025 cycles/sample
46
32
ISLA, 0.005 cycles/sample
47
VIII
LIST OF FIGURES
~
Figure
33
ISLA, 0.0075 cycles/sample
48
34
ISLA, 0.01 cycles/sample
49
35
ISLA, 0.02 cycles/sample
50
36
ISLA, 0.03 cycles/sample
51
37
KEL, 0.0025 cycles/sample
53
38
KEL, O.OOS cycles/sample
54
39
KEL, 0.0075 cycles/sample
55
40
KEL, 0.01 cycles/sam-ple
56
41
KEL, 0.02 cycles/sample
57
42
KEL, 0.03 cycles/sample
58
43
Peak side lobe level, 0.0025 cycles/sample
59
44
Peak sidelobe level, 0.005 cycles/sample
60
45
Peak side lobe level, 0.0075 cycles/sample
61
46
Peak sidelobe level, 0.01 cycles/sample
62
47
Peak side lobe level, 0.02 cycles/sample
63
48
Peak sidelobe level, 0.03 cycles/sample
64
49
Quadratic phase errors
68
ix
LIST OF FIGURES
Figure
Page
50
Cubic phase errors
69
51
Fourth-order phase errors
70
52
Dolph-Chebychev response, quadratic error
71
53
Dolph-Chebychev response, cubic error
72
54
Dolph-Chebychev response, fourth-order error
73
55
Kaiser-Bessel response, quadratic error
74
56
Kaiser-Bessel response, cubic error
75
57
Kaiser-Bessel response, fourth-order error
76
58
Taylor response, quadratic error
77
59
Taylor response, cubic error
78
60
Taylor response, fourth-order error
79
61
Gain Degradation, quadratic error
80
62
Gain Degradation, cubic error
81
63
Gain Degradation, fourth-order error
82
64
ISLA, quadratic error
84
65
ISLA, cubic error
85
66
ISLA, fourth-order error
86
X
LIST OF FIGURES
Figure
£g_gg
67
KEL, quadratic error
87
68
KEL, cubic error
88
69
KEL, fourth-order error
89
70
Peak sidelobe level, quadratic error
90
71
Peak side lobe level, cubic error
91
72
Peak sidelobe level, fourth-order error
92
xi
LIST OF SYMBOLS
A
=modulation index
a(n)
=amplitude weights
ak
=normalizing constant
ea(n) =Amplitude-modulating
signal
f
=frequency
fa
=frequency of the modulating signal
N
=number of samples per OFT frame
Pk (t) =Legendre polynomial
s(n)
=sampled signal after errors and weighting
S(co)
=Fourier transform of s(n)
x(n)
=discrete-time sequence
X(k)
=output of the kth OFT filter
X(co)
=Fourier transform of x(n)
xa(n) =infinite-duration sampled sequence
Xa(co) =Fourier transform of xa ( n)
=amplitude-weighted samples=x(n)a(n)
=Fourier transform of x (n)
A
w(n)
=amplitude weighting sequence
xii
LIST OF SYMBOLS
W(ro)
=Fourier transform of w(n)
Wpeak1 (ro)=peak mainlobe gain without phase or amplitude errors
W peak2(ro)=peak main lobe gain with phase or amplitude errors
ro
=radial frequency
roa
=2nf a
<j>(t)
=phase error
a2
=variance:of the phase error
xiii
ABSTRACT
THE EFFECTS OF PHASE AND AMPLITUDE ERRORS
ON AMPLITUDE WEIGHTING FUNCTIONS
by
William Lawrence Paul,lll
Master of Science in Engineering
The effects of phase and amplitude errors on the frequency
responses of amplitude weighting functions were investigated.
Amplitude errors were modeled as single-tone amplitude
modulating signals.
Simulations were made and comparisons
made to theoretical predictions.
Quantitative comparisons were
made for different windows with similar peak sidelobe levels in
terms of the effects on main lobe gain, main lobe 3-dB
beamwidth, integrated sidelobe ratio, and peak sidelobe level.
Phase errors were modeled using Legendre polynomials and
the effects of quadratic, cubic, and fourth-order phase errors
were evaluated.
Chapter 1 introduces the subject, while Chapter 2 provides
xiv
background information on the discrete Fourier transform (OFT) and
spectral leakage, the phenomenon which causes sidelobes in OFT
filter responses.
Also included in Chapter 2 are introductions to
the weighting functions utilized in the study, and four measures of
performance which were utilized to compare the effects of errors
on each response.
Chapter 4 is a discussion of amplitude errors; how they were
modeled, theoretical predictions, and results of the simulations.
Chapter 5 examines the model for phase errors and the
corresponding simulation results.
results for both types of errors.
XV
Chapter 6 summarizes the
CHAPTER 1
INTRODUCTION
Spectral analysis of signals using the discrete Fourier
transform (OFT), or its alternate implementation the fast Fourier
transform (FFT), is common practice in many digital signal
processing applications.
imaging radar.
Particular examples include doppler and
One of the pitfalls of a time-limited (i.e. real-
--·_::; spectral analysis is the windowing effect created by the
finite time allowed to observe the signal, causing sidelobes which
can hide smaller signals.
This problem can be alleviated by the use
of non-uniform windows to amplitude-weight the signal.
A variety of windows are currently in use (see Harris [1 ]).
Selection of a particular window is dependent on the maximum
acceptable sidelobe level, mainlobe bandwidth, etc. imposed by the
system requirements.
The purpose of this project was to
investigate what effect phase and amplitude errors in the signal
have on the frequency response of three windows with -40 dB peak
sidelobes (Taylor, Dolph-Chebychev, and Kaiser-Bessel).
This was
achieved via a FORTRAN program which is in the appendix.
CHAPTER 2
SPECTRAL ANALYSIS AND SPECTRAL LEAKAGE WITH THE OFT
2.1 THE OFT ALGORITHM
The OFT algorithm
N -1
X(k)=:E x(n)e-j2nkn/N
n=O
(1)
may be used to compute samples X(k) of the Fourier transform of
the discrete-time sequence x(n), where
k=DFT filter number
N=number of time domain samples per OFT frame
2nk/N=center digital frequency of the kth filter
It can be easily shown that the OFT is periodic in the frequency
domain, i.e.
X(k+aN)=X(k)
(2)
where a is an integer constant.
2.2 SPECTRAL LEAKAGE
The OFT and FFT are commonly used for spectral analysis of
sampled signals.
Unfortunately, unlike Fourier analysis theory,
2
3
where the signal may be assumed to be of infinite duration, reallife signal analysis requires that the observation of the signal be
restricted to a finite "window" of time.
This is true for both
continuous and sampled signals, although this discussion will be
concerned with sampled signals.
The finite observation time is
mathematically equivalent to multiplying an infinite-duration
periodic sequence xa(n) by a rectangular window w(n)
x(n)=w(n)xa(n)
(3)
where
xa(n+aN)=x(n)
and
w(n)=1
=0
otherwise.
Since multiplication of two sequences in the time domain
corresponds to convolution of the Fourier transforms of those
sequences,
X(ffi)=Xa(ro)*W(ro).
( 4)
The Fourier transform W(ffi) is given by
W ( ro) = ( e -jffi ( N-1 )/2){ sin [N ffi/2]/si n [ ro/2]}
O<ffi< 2n
(5.)
The magnitude of W(ro) is the digital sine function illustrated
4
in Figure 1.
Nulls are spaced at intervals of 21t/N.
For the case of
an infinite signal composed of sinusoids at various frequencies,
the ideal spectrum consists of discrete lines at the frequencies of
those sinusoids.
However, each line in the windowed spectrum is
convolved with the above digital sine function.
In the OFT
implementation, if the frequency of a particular component of the
signal is exactly equal to the center frequency of a OFT filter, then
the remaining filters will all fall on the nulls of the sine function
and therefore exhibit no energy from that frequency component.
Of
course, the frequency typically will not precisely coincide with
the filter center, and the DFT will exhibit energy across the
spectrum.
This phenomenon is referred to as "spectral leakage",
and the result can be reduced visibility of smailer signais in the
presence of stronger ones.
Even spectral leakage due to the image
of a signal can degrade performance, depending on the location of
the signal in the digital spectrum.
Spectral leakage can be better understood by noting that
when the frequency of the signal corresponds to the filter center
frequency, 21tk/N, then the signal goes through an integer number
of cycles over the observation period NT and is periodic, exciting
the single filter.
However,
wh~n
the frequency is not equal to
5
Magnitude (dB)
0
-3
Figure 1. Digital sine function.
6
2rck/N, then the signal goes through a non-integer number of
cycles.
Thus, if the windowed signal x(n) was extended
periodically in time (with a period NT), the extended signal would
have discontinuities at NT intervals which would result in leakage
components in the frequency domain.
This is illustrated in Figures
2a and 2b.
Spectral leakage components can be reduced by use of nonrectangular window functions which weight the time-domain
samples in the middle of the observation period greatest, and those
at the ends least.
This effectively smoothes the discontinuities
and reduces sidelobes in the frequency response.
Greater sidelobe .
suppression is achieved by windows that also smooth higher-order
derivatives of the windowed function at the boundaries.
However,
the price paid for reduced sidelobes is increased main lobe
bandwidth.
2.3 MEASURES OF PERFORMANCE
The effects of phase and amplitude errors on the frequency
responses of the windows were evaluated using the following
spectral
characteristics.
7
1 . 00
0.50
0.00
-0.50
Figure 2a. Signal periodic over observation interval.
8
1 . 0 0
0.50
0. 0 0
-0. 50
Figure 2b. Signal aperiodic over observation interval.
9
Gain Degradation (GO). GO is difference between the peak mainlobe
gains with and without errors present:
GD=Wpeakl (ro)-Wpeak2(ro)
where
Wpeakl (ro)=peak mainlobe gain without phase or amplitude
errors
W peak2(ro)=peak main lobe gain with phase or amplitude
errors
Thus, a positive GO indicates that the errors caused a reduction in
peak gain, while a negative GO indicates an increase.
Integrated Sidelobe Ratio (ISLA).
ISLR is the ratio of the energy in
the sidelobes to that in the mainlobe:
oo
J
W 2 (ro)dro
-oo
-J
ro2
W 2 (ro)dro
(l)t
ISLA=
KEL. The KEL factor is defined as:
KEL=(-3 dB bandwidth)N
(7)
10
Peak Sidelobe.
Peak sidelobe level is simply the magnitude of the
highest sidelobe peak.
2.4 AMPLITUDE WEIGHTING FUNCTIONS
The windows used for this comparison were Dolph-Chebychev,
Kaiser-Bessel, and Taylor, all with -40 dB peak sidelobe levels.
Table I indicates the nominal values of the above parameters for
each window function with no errors.
Table I. Window parameter comparison.
WINDOW NAME
GAIN
DEGRADATION
ISLR
KEL
PEAK
SIDELOBE
LEVEL
DOLPH-CHEBYCHEV
0 dB
-24.2 dB
1.21
-40.00 dB
KAISER-BESSEL
0 dB
-38.6 dB
1.33
-39.98 dB
'0 dB
-32.4 dB
1.27
-40.14 dB
TAYLOR
Dolph-Chebychev Window.
This window achieves the narrowest
possible main-lobe width for a chosen peak-sidelobe level.
11
However, the equiripple sidelobe feature causes the ISLR to be
higher than the other windows surveyed.
The frequency response
of a 40-dB Dolph-Chebychev window is shown in Figure 3.
Kaiser-Bessel Window.
The zero-order pr:Jiate- spheroidal wave
function can be used to generate a window which maximizes the
energy within a given bandwidth (resulting in a low ISLR).
function is approximated by the Kaiser-Bessel window.
This
The
freauency response is of a 40-dB Kaiser-Bessel window os
illustrated in Figure 4.
The notable feature of the Kaiser-Bessel
window is that the sidelobes roll off at a 6 dB/octave rate,
compared to the equiripple Dolph-Chebychev sidelobes.
Taylor.
response.
Figure 5 shows the 40-dB Taylor weight frequency
The Taylor window achieves very low first sidelobes,
but overall has higher sidelobes than the
Kaiser-Bessel window.
12
0
- '1 0
-20
-30
-40
-50
~ ~~~~
II Ill
I
!
~~~~~ l!
-60
-70
-80+---~-~--~--~~~~~~--~~~~
.oc
.20
.40
.60
.80
Figure 3. Dolph-Chebychev response.
1 . 0 0
-
-
-- -- -- --
---
-
-- --
-
--
13
0
- 10
-20
-30
-40~
-SO
-70
- B 0 41.JJll.ll.l.~u.i.LLJ_,_lliilJ.llU
1 . l 0
.00
Figure 4. Kaiser-Bessel response.
14
0
- 10
-20
-30
-40
-50
-60
-70
-so~~~--~----~--~~~--~--~--~~~~~
.00
.20
.40
6!'
Figure 5. Taylor response.
.80
1 . 0 0
CHA.PTER 3
AMPLITUDE ERRORS
3.1 MODEL
Amplitude errors were modeled as single-tone amplitude
modulating signals of the form
ea(n)=l +Acos(2nfan/N)
(8)
where
A=amplitude of the modulating signal, normalized to the
amplitude of the c1mar (moduiation index)
fa=frequency of tl:e modulating ShJnal
N=number of samples in a single OFT frame.
This is analogous to an AM transmitted carrier signal.
The
modulation index was varied from 0 to 0.2 in increments of 0.05
for frequencies of 0.0025, 0.0050, 0.0075, 0.01, 0.02, and 0.03
cycles/sample.
3.2 THEORETICAL PREDICTIONS
The composite sampled signal after weighting and errors are
applied, is described by
15
17
Magnitude (dB)
-10
-20
-30
~:: OJooonnrt ,r~···. ··riJr·.··. '' . '01 ...· ~r~··~····· -~· ~ ·~ :
ll rrtt·l 1r·1· ;u
fl ·h·~
11 l
1
-6 0 ··~"····!\"-·"1\"···1\'···~t; ...!'l... i'\... i'i ~- ~1-ti'' r. :···, ··:·;··:-:rH·"···)\····~···7\! Y'~f "J;t: ·1\"···tt····"· ··~~· ?r -~~· -~~--1
I 1 I 1 I 1 I 1/ \ {I f £ t" f f '' ''[ ~ f•r; i i . i : i idW t 1 ' I 1 1 •i . ''I :1 r: '' f \ f I [I I \ I\ f l I I II i
1I 1 I 11
:it ii 1 i ~ f \ f I I I t 1 I I 1 I 1 I I I l
1I 1•111 tl111 II111 'I1 ~~1 I\1I {(ff : · 1\1·1:i I• ,.it 11 ,r~: ;;;i ;;,! \!:: ;~i!'l
J~ 11 • 1 ,, '''•',: ~: ti 1 1 1 e 11 11 11 'I'~
1
-7 o ,... -l,····lr···k···ll···~l···11··..11f···~~--~~~--:~!it>t·--lf···\l···li--.lli .. ~: ...,:... 1{··· li-:~--' !.. -e; .. ~t..,f1....1,f~--11---il···~f....\J ...
~
-80
.350
l~ v~~ : ~
· .. , , · ~ ·· ,, :: "'
,,
!f.
~
w
,
: ' 1t ' ~
.400
u 11 u!
fi
.450
.500
.550
.600
.650
Frequency (cycles/sample)
Figure 6a. Individual components of spectrum,fa=0.02 (broad view).
18
) . . . . . . . . T. . . . . . . .,. . . . . . . . . . . . . . . . .T. . . . ..
Magnitude (dB)
r. . . . .
.. ...........1................. ,................
T................,
............ -r ................r. . . . . . . . ,
-10 ................. ;.................;..............;...... ..
1 AMPLITUclc ':RAORS l
i FREOUENC'f 0.02 CYCLES/SAMPLE
-20
-3()
-40
-50
-60
-71l
.450
.4GO
.470
.4.30 .49r) .::'·00 .510 .5::o
Frequency (cycles/sample)
.1:'<'·0
.5'~0
.5·5·0
Figure 6b. Individual components of spectrum, fa=0.02 (narrow
view).
19
for small values of the error signal frequency fa, moving into the
sidelobes as fa increases.
An example of the interference falling
within the mainlobe can be seen in Figure 6c.
The -20 dB
difference causes the error signal main lobes to have little effect
on the overall response.
The peak amplitude of the error signal is
(A/2)2 times the peak magnitude of the primary.
For a modulation
index of 0.2, this corresponds to -20 dB.
3.3 SIMULATION RESULTS
After each time sample of the input sequence was multiplied
by the corresponding weighting and error terms, the OFT was
computed using a 1000-point OFT.
7 through 24.
Spectra are presented in Figures
Figure 11 shows the simulation result for the
theoretical case shown in Figure 6.
The following paragraphs
discuss the effects of the amplitude errors on each of the
performance parameters
cite~
above.
Simulation runs were made
for error signal frequencies which fell within the mainlobe
(0.0025, 0.005 and 0.0075 cycles/sample), in the region of
transition between mainlobe and sidelobe (0.001 and 0.002
cycles/sample), and entirely outside the mainlobe (0.003
20
-20
-.30
-40
-50
-60
-70
_
80
.450
~r
i v j ::
.
.460
.470
=--
!I-
suM!·--·--·: usa
I
1~--
Lsa:
1-M~~~~ 11 ~{!
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
1
I
I
.540
.550
Figure 6c. Individual components of spectrum, fa=O.OOS.
21
-20
-40
-50
-60
-70
-80
.450
.4(:!0
.470
.42·0 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
.550
Figure 7. Dolph-Chebychev response, fa=0.0025 cycles/sample.
22
-20
-30
-40
-50
-60
-70
-80~--_,--~~--~----~--~----~~-r----~~~--~
.450
.460
.4 70
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.5.3·0
.5+0
.550
Figure 8. Dolph-Chebychev response, fa=O.OOS cycles/sample.
23
-30
-40
\-50
-60
-70
-80~--~'r-~.----,----~--~----~~~----~L-~'~~'
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.5-30
.540
.550
Figure 9. Dolph-Chebychev response, fa=0.0075 cycles/sample.
24
-70
_
8
i 1-: lttoE:-o 2~ 1----'~JE - ~ o I~ >to ~M I
J
•
I
·
I '
I
I
.4-:::o .4F·O .470 .480 .490 .500 .5!0 .520 .530 .540 .550
Freque~' ;y (cycles/sample)
Figtre 10. Dolph-Chebychev response, fa=0.01 cycles/sample.
25
-20
-30
-40
-50
-60
-70
.450
.450
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.53·0
.540
.550
Figure 11. Dolph-Chebychev response, fa=0.02 cycles/sample.
26
-20
-30
-40
-50
-60
-70
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.5.30
.540
.550
Figure 12. Dolph-Chebychev response, fa=0.03 cycles/sample.
27
-20
-30
-40
-50
-60
-70
-80-r--~r-~.---------~--~----~~~--~~~~a_~
.45·0
A-CO
.4 70
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
Figure i 3. Kaiser-Bessel response, fa=0.0025 cycles/sample.
.55·0
28
Magnitude (dB)
0 ················:r···············r··············r···············i············ l ············r··············r··············r··············r···············i
- •j
,.,
-
L
~
~
~
~
~
~
~
~
:
:
:
:
'
~
~
:
:
:
j
E
E
:
:
:
j
E
0 ................. j................. ~ ................. f............. ··!·················j.................!·· ··············i················-~·-···············~·-··············-~
~ AMPUTUDE ERRORS '
o
1FREQUEH~'f 0.005 c;rcLES/SA~PLE
l
1
1
1
E
i
l
~
~
················-~·-···············r················r····· ·········r················r················i········ ·······r················r·············· .. r···············-~
l. . . . . . . . .!................r. . . . .l. . . . . . . . .!. . . . . . . . .!. . . . . . . !. . . . . . . . .!. . . . . . . . .!.................!
-I ! :f--1-f j -\jA _L__ ! ---1
-30 .................
-•a
-50
~
~
~
~
~
~
_i
~
-60
-70
-804----;i--~,---~----~---r----r--L~--~_J--~;~~f
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
Figure 14. Kaiser-Bessel response, fa=0.005 cycles/sample.
.550
29
-20
-30
-40
-50
-60
-70
-80~---T--~r---.----r---.----,--L,---~_L~_L~
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
.550
Figure 15. Kaiser-Bessel response, fa=0.0075 cycles/sample.
30
-10
-20
-30
-40
-50
-60
-70
-80
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
Figure 16. Kaiser-Bessel response, fa=0.01 cycles/sample.
.550
31
-10
-20
-40
-50
-60
-70
-80~---r~-.----.---~--~--~--~----~--~L_~
.450
.460
.4 70
.480
.490
.500
.510
.520
.530
.540
Frequency (cycles/sample)
Figure 17. Kaiser-Bessel response, fa=0.02 cycles/sample.
.550
32
Magnitude (dB)
0 ·················:··············· .. :-.. ··············:-················:············ . ·············:·· .. ·············:················-:-.. ············ .. :·················:
--~
~
~
~
~
~
~
}
~
~
!. . . . . . . . .!.. . . . . . . .;.. . . . . . . . .!.. . . . . . . . .;.. . . . . . . .!.. . . . . . . . .!... ...............!.. . . . . . . . .!.
-10 ..................;..................
l AMPl.!TUtiE ERRORS :
-20
;
1FREOUEN~'f O.Q3 C~CLES/SA~PLE
~
1
~
j
~
1
l
1
l
j
l
j
-30
-40
-50
-60
-7 0 ........... ··+······· ·······+················l·················l·················l················+-···············1···············+················1··· ··········1
~
-80
.450
•
•
•
•
•
0
•
1
.460
.4 70
.480
.490
.500
.510
.520
.530
~
~
j
1
.540
.550
Frequency (cycles/sample)
Figure 18. Kaiser-Bessel response, fa=0.03 cycles/sample.
33
..,Lm. ; i . r.... i ..... r . . T
Magnitude (dB)
0 ················-~·-···············~·················~·-···············~·-······:, ·. ···~-- ..........:. ......... :.......... : ........: .. . . . :
-10
;:;:~:~:r:
-
················r··············r··············r········_·:: ·r·············T···············r··:·········r··············r··············r···············l
2 o .................i................. i.................i....... ·........L................i.................i...............i.................i.................L................ ~
-3 0
-
!
40
I I i.I l I I I I I
L. I -l 1- i - -i - ! I. - 'r
I I I : l ~ l
-r-1
·················~·················~·················~··· ···········-~·-··············-~·-···············~············ ···~·················~·················~·················j
i
r .. · · · · - -·
1·1
······. .
-50
..............; ............ .i.
-60
...............
- 70
................ ................ ···············-r-··············-r-··············-r-··············-r-··············· ..................................................
········· .. ...............i................. i................. j ...............
J
-. ...............
-
........ ....... i............... i
..............:
-80~---;--~~--~----~--~----~~~--~~--~--~
.450
.460
.470
.480
.490
.500
.510
.520
.5.30
.540
Frequency (cycles/sample)
Figure 19. Taylor response, fa=0.0025 cycles/sample.
.550
34
Magnitude (dB)
a
················r·············--r··············T···············T···········
:
:
:
;
l
············r··············r··············r··············r···············1
:
:
:
!
;
:
-10 ················+···············+················!·············· ·j·················l·················i· ··············l·················!·················!·················!
j AMPUTUDE ERRORS '
'
j
j
j
j
j
j
.,
1FREQUEH~Y 0.005 C(CLES/SA~LE
~
i
1
i
j
l
-LO ................. ,................. ,................. ,............... ,................. ,................. ,................ ,................. ,................. ,.................:
-30
-40
-50
-60 .............................................. : ................ i················+·············+···············' ............................... , ............... \
:
-7 0
:
:
:
:
................ ............... ............... T ............... j.................j.................1................. ................ .. .............................. .
-80~--~~--~--~----~--~----~--~--~--~~--~
.450
.460
.4 70
.480
.490
.500
.510
.52.0
.5.30
.540
Frequency (cycles/sample)
Figure 20. Taylor response, fa=O.OOS cycles/sample.
.550
35
-20
-30
-40
-50
-60
-70
-80~----r--L-r----~--~L---~--~---L~--~~--+---~
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
Figure 21. Taylor response, fa=0.0075 cycles/sample.
.550
36
-20
-30
-40
- 50
~::
············· ·j
............ t.;~t . . ··l·············,t"···············l················t-············•( ···,~~.:.-.·t .............., ..............,
:
:.
r
:
:
::::::::::::::.1. :.
:
:.t
:
:
:
:
:
:
i :/ :; .: ::! _·: i
1
-80~---;--~,---~----~--~----r-~-r--~~--;---_,
.450
.460
.470
.480 .490 .500 .510 .520
Frequency (cycles/sample)
.530
.540
Figure 22. Taylor response, fa=0.01 cycles/sample.
.550
37
-20
-,30
-40
-50
-60
- 7 0 ---·····--···-- --·············· ···············T··············y···············(··············i················· ················ ................................
-80~----r-~-.----~----~---r----~--~r---~-L--~--~
.450
.460
.470
.4C:.O
.490
.500
.510
.520
.530
.540
Frequency (cycles/sample)
Figure 23. Taylor response, fa=0.02 cycles/sample.
.550
38
-20
-30
-40
-50
-60
-70
-80~----r-~~----~----~---r----.---~----~~--r----;
.450
.460
.470
.480 .. 490 .500 .510 .520
Frequency (cycles/sample)
.5.30
.540
Figure 24. Taylor response, fa=0.03 cycles/sample.
.550
39
cycles/sample).
Gain Degradation.
Gain degradation comparisons are shown in
Figures 25 through 30.
For very small values of fa, it can be seen
in Figure 25 that the main lobes of all three components add to
result in increased peak magnitude.
The largest degradation
(approximately 0.75 dB) occurs for fa equal to 0.0075 and 0.01
cycles/sample.
Figures 29 and 30 illustrate that as fa increases
to a value greater than the mainlobe bandwidth, the effect on gain
degradation diminishes.
Integrated Sidelobe Ratio (ISLA).
The effects of amplitude errors
on iSLR are shown in Figures 31 through 36.
In Figure 31, the error
signal sidebands are very near the filter center, thus increasing
the mainlobe gain as noted above.
This also results in a slight
reduction in ISLR as the magnitude of the error signal increases.
For larger values of fa, the error sidebands place more energy
outside the nominal (zero-error) mainlobe null-to-null bandwidth,
resulting in ISLR increasing with both frequency and magnitude of
the error signal.
This trend can be seen in Figures 32 through 36.
It may be noted that the errors have the greatest impact on the
40
Gain Deg_radation (dB)
O.OD
-0.?0
-0.40
T .r::..
L 0 F?
1<. ;>
:; E R - 8 :::: S S E L
DOLFH-CHEBI'CHEV
-0.60
AMPLITUDE ERRORS
AM FREQUENCY= 0.0025 CYClES/SAMPlE
·- 0 . 8 0
- 1 . 0 0
-1.
20~---.----.----,----~--~----~--~----~--~--~
.000
. 0 't 0
.080
.1-'(l
Modulation Index
. 1 G0
Figure 25. Gain degradation, 0.0025 cycles/sample.
.200
41
G,ain 9~gradafion (dB)
'; . o., ol .--.
-------------~
.J
--
TA.I!_-')P
0.0•1-00
0.0.350
D 0 i_ P H - C H
!=::
8 Y C H E \/
0.0300
0.0250
0. 0200
0. 0 I 50
0.0100
0. 0050
.......::
n . 0 0 0 0
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
-i""''------...,r----,---~--.---~--..--..-----r------r, - -
.OUO
.040
. 0 8 0
. 1 :~
Modulation lnd'Ox
1.1
. 1 50
Figure 26. Gain degradation, 0.005 cycles/sample.
1
. :~ o n
42
Tt\YLOR
0.80
0.70
t<A I SER-BESSEL
DC)LPH-CHEBYCHEV
0.60
0.50
0. 40
0.30
0.20
0 . 10
AMPLITUDE ERI\ORS
AM FREQUENCY= 0.0075 CYCLES{SAMPLE
0 . 0 0 --F-----r----r------r---,------.----...-----,-----.---,------.
.000
.040
.080
. 12 0
.200
Modulation Index
Figure 27. Gain degradation, 0.0075 cycles/sample.
43
Gain Dearadation (dB)
0 . 8 \J
..-----'--------------,
Tt>i'LOR
0 . 70
0.60
-···
I<:A!SER-BESSEL
D !J L F' H - C H E B Y C H E V
0 . .50
0.40
0.30
0.20
AMPLITUDE ERRORS
AM FREQUENCY" 0.01 CYCLES/SAMPLE
0 . 0 0 ~----,---...----r---.--..-----.---...,--.------.----,
.000
.040
.080
. 120
. I f3 0
.200
Modulation Index
Figure 28. Gain degradation, 0.01 cycles/sample.
44
Gain Degradation (dB)
0.0 130~
r-.~-_---------------------------.
T.A.(LOR
0.0160
·
-···
0. 0 1 I G
l<A l'3ER-BESc.F:L
D
n
L P H -· C H E B !' ( i ! F: \f
0.0120
AMPLITUDE ERRORS
AM FREQUEHCY= om. CYCLES/SAMPLE
0 . 0 10 0
0 . 0
1)
80
0.0060
.')
0
I)
.-··-·· -··
20
.. --- ..
__.., .. ---
--- .. _..,.
•• .....- •• *
0.0000
() ! I
rJ
C)
i\ iJ
() 8 (}
. I 2 I)
. 1 6 0
Modu lotion Index
Figure 29. Gain degradation, 0.02 cycles/sample.
.200
45
.
Gain Degradation (dB)
0 . 0 1 ·i 0
0
_.)I "2 0
0.0100
0.0080
-
.
-
-
.
--
T,~YLOR
-··-
I< A I SER -BE
-
DOLPH -C HE 8
/
c
r
/
r
I
\'
HE\f
-
I
/
/
_,.
-
AMPLITUDE ERRORS
AM FREQUEHCY= 0.03 CYCLES/SAMPLE
./
.
0.0060 -
./
/
/
-
/
.
0.0010 -
0.0000
_,.· .
/
-
.. /
- -------
/
------
-0.0020
I
.000
1.
.040
I
~
I
0 ·'3
I
C)
I
.
l 2 ()
I
I
I
' l
s0
Modulation Index
Figure 30. Gain degradation, 0.03 cycles/sample.
I
.:zoo
46
ISLR (dB)
-2 0-
AMPLITUDE ERRORS
AM FREQUENCY= 0.0025 CYCLES/SAMPLE
-- 2 5-
--
Ti\YLOR
-
I< A I S E ...., - 8 E . S E L
-
•••
___ _
OOLPI-1-''H>.::E (CHEV
-30~----,_
...
...................
/'
____ _
-3 5-
!--··-··-··-··-··-·· .......
~
~---
..
- 4 0 +---,--,--...-, _----.,---.--,--,--,--......~----,--,· .:..'-==:...;j..:..·-=-=-·:...:i;..:-==...:..·~·-i
.000
.040
.. 030
.1:.0
. 1
Modulation Index
Figure 31. ISLA, 0.0025 cycles/sample.
rs
0
.::'00
47
,~SLR (dB)
-L
4-r---------------------------------------------------
-2 6- :2 8-
1/ ;,, I
·~.
[ P - [3 E S S [ I_
r-------------------------------- 3
D 0 L r:-· H --
I)-
-3 2-
cHE8
'(
I~
11 E v
------------------- ..
-3 4AMPliTUDE ERRORS
AM F!IEQUEHCY= 0.005 CYCLES/SAMPLE
-3 6-3!'\-
~-·-··-··-··--··-··-··-··-··.--··-··
-40
r
000
L
040
I
I
08
I
I
I ·!·- 0
Modulation Index
I}
I
I
I
1
Figure 32. ISLR, 0.005 cycles/sample.
so
-···
l
200
48
lS~R
-Lt_
-
(dB)
-
-24
-26 -
-
-28 -30 -
--
TAYLOR
-···
KA
-
DOLPH-CHEBYCHEV
I
SER-BESSEL
------- _.. -··
_
.. -··
_.. -·· _.. -··
-32 -~-------
-·~-
-34 -
-36 -38
-_ .. -··
-40
.000
I
••
#
AMPLITUDE ERRORS
AM FREQUENCY= 0.0075 CYCLES/SAMPLE
I
.040
I
I
.080
I
I
. 12 0
Modulation Index
I
I
. 16 0
Figure 33. ISLR, 0.0075 cycles/sample.
I
I
.200
49
-
'J "")
,:..,_ L
-
- --
-24
-
-213 -
-28 -
-30 -32 -~----3 4- -
-36 -
-38 -!---" ••
-40
.000
...... .
I
Ti\YLOR
-··-
K A I SER-BESSEL
-
DOLPH-CHEBYCHEV
-------
........
,.
.......
......
..,......... ..
-;;;;;:=- ...
......-........_.,.
. _.... .. ........
. ....-·
........ ........
AMPLITUDE ERRORS
AM FREQUENCY= 0.01 CYCLES/SAMPLE
I
.040
I
I
.080
I
I
I
. 120
Modulation Index
Figure 34. ISLR, 0.01 cycles/sample.
I
. 160
I
-1
.200
50
-- --
-20
.........
-25
/.
-30
...,.............-
.........
/
~/.I
~·
./'
.
. ..
.........
~._,..·
.......- ..........................
......
--:"
.. ...-
/
TAYLOR
~ ~----------------------------;
f<AISER-BESSEL
/
DOLPH-CHEBYCHEV
~
-35
~
AMPUTUDE ERRORS
AM FREQUENCY: 0.02 CYCLES/SAMPLE
- 4 0
-+----.,-----y------r-----r-----r-----.------r------r-----r-----,
.000
.040
.080
. I 2 0
Modulation Index
. 160
Figure 35. ISLR, 0.02 cycles/sample.
.200
51
IS!o-R (dB)
- 1 :-_:.
-20
') r::
-~_)
TAYLOR
-30
f<A I SER-BESSEL
/
-.3 5
D0LPH-CHEBYCHEV
/
/
AMPUTUDE ERRORS
AM FREQUENCY-= 0.03 CYCLES/SAMPLE
-40~--~----~--~----~--~--~----~--~--~--~
.000
.040
.080
.120
· Modulation Index
Figure 36. ISLA, 0.03 cycles/sample.
. 1 60
.. 2 0 0
52
windows with lower ISLA.
KEL.
KEL factor can be seen in Figures 37 through 42 to be either
positively or negatively affected by amplitude errors depending on
the frequency of the error signal.
The largest disturbance seen
was for fa=0.02 cycles/sample (Figure 41 ), where a 4.8% increase
was seen over the cases simulated for a Kaiser-Bessel window.
In
general, the effects were very similar for all three windows.
Peak sidelobe level (PSL).
The effects on PSL are relatively minor
for low frequency error sign_als (Figures 43 and 44).
More profound
effects are seen for signals outside the mainlobe (Figures 45
through 48). !n some cases (e.g. Figure 45), PSL can be seen
increasing as the modulation index increases, then suddenly
dropping off.
This is because in these cases the first sidelobe was
close enough to the mainlobe that as the error signal increased in
magnitude, the null filled in.
recognized as a sidelobe.
Thus, the first sidelobe was no longer
Since the filter response at that
frequency has not been reduced, this does not represent a
performance improvement.
When the frequency is high enough to place the error signal
53
KEL
3
j
:)
··-··-··-··-··-··-··-··-·· -··-··-···
- ,] 2 0
3iJO
--
T/\(LOR
-··-
I< A I SER-BESSEL
-
280
DOLPH-CHEBYCHEV
---------------·
260
AMPLITUDE ERRORS
AM Fl!EQUENCY= 0.0025 CYCLES/SAMPLE
I . 2 4 \)
220
.000
.040
.oao
.1:·u
Modu lotion Index
' 1
Figure 37. KEL, 0.0025 cycles/sample.
s !)
54
KEL
340
. . - .. - ..
- ..
.--
...
-··- .. - .. -· .
.-.
. --- .. -···
Ti\YLOR
30()
-
• • - I< i\ i
~
E R - E3 E <:;.
::.
EL
0 0 L P H - C H E B '( C H E V
280
2
---------.--.--------------
f) (1
240
AMPLITUDE ERRORS
AM FREQUENCY= 0.005 CYClES(SAMPLE
2 2 0
2
-l------,r----.-·--r-----.----r-------r---...---.--...---,
.20()
0 8 !)
. 1 G0
.000
' 0 ·+ 0
(l I]
Modulation Index
Figure 38. KEL, 0.005 cycles/sample.
55
KEL
:+.:}··-
3~.)
3 :J 0
260
· · - ••
--
TAY~_OR
-···
fC6, I SER-BESSEL
- .......... - -..........
---- ----
DOLPH-CHEB (CHEV
......._
--.,--
··- ··- .....................
---- -- ---- ---
AMPLITUDE ERRORS
AM FREQUENCY= 0.007!1 CYCLES/SAMPLE
2 0 ·J
80
.000
.040
.. 0 8 0
. 12 0
·Modulation Index
. 160
Figure 39. KSL, 0.0075 cycles/sample.
.200
56
TAYLOR
KEL
1 . 3 50
··-.
1 . 300
1 . 2 50
-··--. ·- ··-.
DOLPH-CHEBYCHEV
·--- ---- --- ---- ·- ··-- ··-----~
1 . 2 0 0
I<A I SER-BESSEL
··-.
.. ...........
---- -- ---- --
-
1 . 1 50
AMPLITUDE ERRORS
AM FREQUENCY= 0.01 CYCLES/SAMPLE
1 . 1
00~--~r----r----~---r----~--~----~--~----,---~
.000
.040
.080
. 12 0
Modulation Index
. 16 0
Figure 40. KEL, 0.01 cycles/sample.
.200
57
KEL
400
380
-
360 .3 'l- 0 -
,) 2 0
--
TAYLOR
-···
f< ,6. I S E R - 8 E S S E I_
--
OOLPH-CHE81r:HEV
~---··
_ -·· -·
..
-
-
.3 0 0 -
2RO
.-··
-~--
260 2 4-0 -
- -- --
--- -·
..
..-·· -·
----
..
-
-~
-
lMPUTUDE ERRORS
AM FREQUENCY= 0.02 CYCLES{SAMPLE
220 ~
200
.000
I
I
.040
I
T
-,
I
.080
.121)
·Modulation Index
I
I
' 1
Figure 41. KEL, 0.02 cycles/sample.
I
s0
I
.:200
58
··-··-··--··--··-··-··-··-··-·· -··-···
.3 2 0
.3 0 0
--
T AY LOR
-··-
I< A I SER-B E '3 5 E L
- DO L PH-CH
----------------E B '( C HE \f
280
260
.240
AMPUTUDE ERRORS
220
AM FREQUENCY= 0.03 CYCLES/SAMPLE
200~--~----.---~---------.----.----.---.----.----,
.000
.040
.080
.1:20
Modulation Index
. I 60
Figure 42. KEL, 0.03 cycles/sample.
.200
59
Peak Sidelobe (dB)
-39.0()
AMPLITUDE ERRORS
AM FREQUENCY= 0.0025 CYCLES/SAMPLE
-39.50
__.
----
·-··-:;:::.-"
-40.00
-40.50
T A YLOR
-··-
I< A I
-
[Jr) LPH--CHEB
.040
-
.,.,...,..,.,.
.. -- .. .-- .. - .. - .. .......-··--··--···
.
~-.-
--
.000
----------
.-- __.
__. __.
SER-B E c:.
.osn
SE L
T
CHF:\1
.
1
:q
. 1 6 ()
Modulation Index
Figure 43. Peak sidelobe level, 0.0025 cycles/sample.
.200
60
Peak Sidelobe (dB)
-39.00
-39.50
.. ..-·
.........
,
..
\
-------------\----40.50
-4 1 . 0 0
\
--
TAYLOR
-···
I< A I S E R - B E S S E L
-
DOLPH-CHEBYCHEV
\
\
\
-4 1 . 50
AMPLITUDE ERRORS
AM FREQUENCY= 0.005 CYCLES/SAMPLE
.000
.040
.030
. 12 0
Modulation Index
. 160
Figure 44. Peak sidelobe level, 0.005 cycles/sample.
.200
61
PeakoSldelobe (dB)
-- ,_:..::i
-.3
i}
l
.--:----____;___-----------,
!
T ,t;, ·r L ') i<
l<A I '3EP-BES':=-EL
D 0 !_ f=' H - C H E 8 ',
~~
11 E \f
/'..
-.3 0
-38
----
-40
.000
' 0 ·1- 0
--/
' () -3 0
/
/
/
''
''
AMPLITUDE ERRORS
AM FREQUENCY= 0.007!1 CYClES/SAMPLE
' 1 :.' 0
. 1 G0
Modulation Index
Figure 45. Peak sidelobe level, 0.0075 cycles/sample.
.
'
.-, il (/
_(___
~--
-_,
62
Peak Sidelobe (dB)
-2 4
- :2 6
-~
:2 8
~~--------------------------·
- - T A (I_() R
I' 1\
i :;
D0 L P
E R - 2 E S ': E I_
~I
--
1:
H E 8 : •. -. 1-1 [ \i
-.3 0
-.3 2
-34
-38
AlllPLITUDE ERRORS
AM FREQUENCY= 0.01 CYCLES(SAMPLE
-- 4 2 -+-----,r--------.----r-----.----r------.---..----.----,.----,
(1 '~ 0
.000
. 0 e r;
. I .' U
. 1 6 0
.200
Modulation Index
Figure 46. Peak sidelobe level, 0.01 cycles/sample.
63
l
Peak~Sidelobe (dB)
- ~· n~
-25
-30
/
b'
~·~
-35
//?
pr
.
- :J'J
- 5
\
'
.
'--~. - - - - - ·
AMPLITUDE ERRORS
AM FREQUENCY= 0.0% CYCLES/SAMPLE
T .fl..
-45
''
\
/
\
LUR
\
I< A I 5 E R - 8 E S S E L
·-··--··-··
0 0 L,P H.~ C H E 8 r' t~: HE'-/
C) --1---L---,---,-----,---,-----,-1- - r - - - ' - - , - - - - - , - - - - r - - - - ,
. r> !J
o
.O.fO
. (} 0
1)
\ !_ I)
. 160
Modulation Index
Figure 47. Peak sideiobe level, 0.02 cycles/sample.
.200
64
Peak,S,iceiobe (dB)
-i)
2 ')
-25
-30
AMPLITUDE ERRORS
AM FREQUENCY= 0.03 CYCLES/SAMPlE
- ,) 5
-40
--
T,~YLOR
-···
r<A l SER-BE3SEL
DOLPH-CHEB r'CHE\1
- 4 '5 -1---.------.---,..------.---,-----,----,-----,----,----,
.000
.040
.080
. 1:2C
Modulation Index
. 16 0
Figure 48. Peak sidelobe level, 0.03 cycles/sample.
."200
65
sidebands clearly in the sidelobes (Figure 48), the three windows
behave very similarly.
similar sidelobe levels.
In the region of fa=0.03, all three have
For larger frequencies however, the
sidelobes roll off at different rates (or not at all in the DolphChebychev case).
Thus it is expected that Kaiser-Bessel window
would exhibit the most susceptibility on the average in this
region.
CHAPTER4
PHASE ERRORS
4.1 MODEL
Phase errors may be modeled as Legendre polynomials of the
form (see reference 2):
k=0,1 ,2, ...
( 11 )
In reference [2], quadratic, cubic and fourth-order terms were
evaluated at discrete samples over the interval -N/2 to N/2 to
model the respective phase errors as follows:
P 2 (n)=cr2.f5(6n2;N2-o.5)
(12)
P3(n)=v3fl(20n3;N3_:-3n/N)
(13)
P 4 ( n) =a4{9(7 on4;N4-1 5n2.tN2-3/8)
( 1 4)
where
o2==variance.of the phase error
However, for the purposes of this effort, phase errors were
evaluated in terms of the center-to-end variation in degrees:
P'2(n)=<Pm(5.036 x 1Q-3)-f5(6n2;N2-0.5)
(15)
P'3(n)==<Pm (4.558 X 1o-3)ff(2Qn3;N3-3n/N)
( 1 6)
P'4(n)=<!>m (4.072
X
1o-3)-19"(70n4;N4-1sn2;N2-3/8)
( 1 7)
where
66
67
<l>m=the maximum center-to-end phase error in degrees
The three types of phase errors used are plotted in Figures
49, 50, and 51 for <l>m=30 degrees, 60 degrees, and 90 degrees.
Equations 15, 16, and 17 may be derived from equations 12, 13 and
14 by setting
Pkmax(n)-Pkmin(n)=n/180 radians/degree
(18)
and solving for <>k·
The sinusoidal signal was multiplied by the phase error term
eJ<J>(n) and the amplitude term prior to calculation of the OFT.
4.2 SIMULATION RESULTS
Spectra for 40-dB Dolph-Chebychev, Kaiser-Bessel, and
Taylor weights are shown in Figures 52 through 60.
phase errors of 0 to 90 degrees were used.
Center-to-end
The following
paragraphs discuss the effects of phase errors on each of the
performance parameters.
Gain Degradation.
Gain degradation comparisons are shown in
Figures 61 through 63.
The results are consistent between types
of phase errors and windows, with the quadratic errors having
68
40
30
20
1 0
- 1 0
-20
-30,_---+--~r---+---~~~--~r---,_--~
-.50
-. 3 0
10
10
n/N
Figure 49. Quadratic phase errors.
___,__~
.30
.50
69
Ph·
8 9Jse .. ~rror
50
- · 3 IJ
- . 10
10
n/N
Figure 50 Cubic ph ase errors.
30
50
70
Phase Error
:: :liff-!I:jl:l::j
50
40
3 0
20
., 0
~: ~ ::r~~;~f
-.50
-.30
:1::
:1:
10
:~:~1: : !i~~~ :1
10
.30
n/N
Figure 51. Fourth-order phase errors.
.50
71
.350
.400
.450
.500
.550
Frequency (cycles/sample)
.600
Figure 52. Oolph-Chebychev response, quadratic error.
.650
72
Magnitude (dB)
0-,------···
(~
······ ,....
-1
~
CUBIC PHAS! ERiloRS
-20-.------·
-
,,
I
-40
mmm,t~ 111\•
~1
-50~
-60
/
"
r' [,\!;-~
II
~ !J
,. r
I
-70-.
....:
._,
-30
I
i
J :~
II
.,
~
f
..L.
I
1
1- M/I.X ERR=90 DEG I·····" MAX ERR=60 DEG 1--- tv1.1\X ERR=30 DEGI- NO ERRORS I
~
I
I
.350
.400
I
I
I
.550
.450
.500
Frequency (cycles/sample)
I
I
.600
.650
Figure 53. Dolph-Chebychev response, cubic error.
73
Magnitude (dB)
0
··························r······················l......................... ,........................r······················r····-···················1
.350
.400
.450
.500
.550
Frequency (cycles/sample)
.600
.650
Figure 54. Dolph-Chebychev response, fourth-order error.
74
.350
.400
.45AJ
.500
.550
Frequency (cycles/sample)
.600
Figure 55. Kaiser-Bessel response, quadratic error.
.650
75
Frequency (cycles/sample)
Figure 56. Kaiser-Bessel response, cubic error.
76
·························-:-·························-:··························:
~
i
!
i
.350
.400
.450
.500
.550
Frequency (cycles/sample)
~
~
.600
.650
Figure 57. Kaiser-Bessel response, fourth-order error.
77
Magng~ ~:..:~~.~!. ......T························r·················7T\··················T·······················r························1
-10- ··························J.························+···············I··-1--·l················-J--························l·························l
::~ ~lnn·~iili11r~ff T-:r:. A!\f~-ii-~--~~1~
~1 ::: :I
~-~-
~ ::~ ~I'::~:~~~~-1:
~
···r . -................n. n
-60- .........................................
·········l············ .....................................
-40-
: :
-50- A- .......................
1-
--I : :
:1 :
~
-70- .......................................... ···············!·············· .........................................
MAX ERR=90 DEG ·······MAX ERR=60 DEG
··
I
I
.350
.400
I
I
1--·
MAX ERR=30 DEG
I
.450
.500
.550
Frequency (cycles/sample)
1-
NO ERRORS
I
I
.600
.650
Figure 58. Taylor response, quadratic error.
I
78
.
... ... r·· ...
. 350
.400
.450
.500
.550
Frequency (cycles/sample)
Figure 59. Taylor response, cubic error.
.600
.650
79
.
·························-·························-··················
..·······...
...
...
....
...
...
.
I
.350
.400
.450
.500
.550
.600
Frequency (cycles/sample)
Figure 60. Taylor response, fourth-order error.
.650
80
Gain De_gration (dB)
0 . 6
I)
0.50
0.40
0.30
--
T,t,YLOR
-··- I<- A
-
I
SER-BESSEL
DOLPH-CHEBYCHEV
QUADRATIC PHASE ERRORS
/
/
'
0.20
0 . 10
/
/
~~...........
/
/
.·
/.·/
/.·
/
/
/
.·
/
/
/.·
.. /
..~··;....-"
~
0. 00-~~~~~~~~------~--~----~--~----~---0
10
20
30
40
50
60
70
90
80
Max Phase Error (degress)
Figure 61. Gain Degradation, quadratic error.
81
Gain Dearation (dB)
o . s cr
--
0 . 4 J
TAYLOR
-··- f<A
-
-
0.30
I SER-BESSEL
DOLPH-CHEBYCHEV
CUBIC PHASE ERRORS
0.20
0 . 10
o.
oo4-~~~~--~----~--~--~--~--~--~
0
10
20
30
40
50
60
70
Max Phase Error (degress)
Figure 62. Gain Degradation, cubic error.
80
90
82
Gain De~radation (dB)
0 . 5 ,)
0.40
0.30
--
T A
-··-
1(•.A. I
-
DO LPH-CH EBfCHEV
"(
L.OR
s ER-B E ss E L
FOURTH ORDER PHASE ERRORS
0.20
0 . 10
0
10
20
30
40
50
60
70
Max Phase Error (degress)
Figure 63. Gain Degradation, fourth-order error.
80
90
83
slightly greater impact for all three weighting functions.
The
degradation for all cases is less than approximately 0.5 dB.
Integrated Sidelobe Ratio (ISLR).
The effects of phase errors on
ISLR are shown in Figures 64 through 66.
In each case, the Taylor
and Kaiser-Bessel plots of ISLR converge for larger phase errors
(greater than 60 degrees for quadratic and cubic, 40 degrees for
fourth-order phase errors).
highest ISLR for all cases.
Dolph-Chebychev windows result in the
The impact of phase errors on ISLR was
more pronounced for higher order errors.
KEL.
KEL factor can be seen in Figures 67 through 69 The effects
of phase ermrs on KEL were relatively minor, resulting in
approximately 1Oo/o increases.
The effect was nearly uniform for
all three weighting functions.
Peak sidelobe level fPSL).
Figures 70 through 72.
The effects on PSL are illustrated in
The results of quadratic errors, as shown in
Figure 70, are somewhat misleading since this type of error
resulted in significant widening of the main beam between the 30and 40-dB points.
The result was that the main beam covered the
84
ISLR (dB)
0
T A'( L 0 R
-5
I< A I S E R - 8 E S S E '-
-10
DOLPH-CHEB I'CHEV
- 1 5
QUADRATIC PHASE ERRORS
-20
-25
---- ---- --- _.,.. ..
-30
-·
-35
0
10
20
.--·· ---. .
30
40
--··
------- -
--...::=-·~
50
60
Max Phase Errx (degn~ss)
Figure 64. ISLR, quadratic error.
70
80
90
85
-20
~__........,
-25
~~
.......-: ..
,.,.
,.,... . .
~~
:;;..-"
/
--
-30
~ ,..--
~··"
• /
/
/
-35
--
TA'(LOR
-···
I<A I SER-BESSEL
-
0
10
20
30
CUBIC PHASE ERRORS
DOLPH-CHEBYCHEV
40
50
60
Max Phase Error (degress)
Figure 65. ISLR, cubic error.
70
80
90
86
ISLR (dB)
- 10
- 15
-.:::::.-~
-20
_....._...
~~··
~. ~'--­
...,-:.~··
~
................-
.....-'::·"
-25
............. /
/.
/.·/
-30
.,...,...
_..... /
/
.
/
FOURTH ORDER PHASE ERIIOIIS
/
-35
/
0
10
20
30
--
TAYLOR
-···
KAI SER-BESSEL
-
DOLPH-CHEBYCHEV
40
50
60
Max Phase Error (degress)
Figure 66. ISLA, fourth-order error.
70
80
90
87
6 0
KEL
··-··- .. -··- ··-··--- ··--· ·- ··-··--··--··
__.- ,__,...-- ......
40
. 2 0
Jt:...-=-..:::-:..:::..:=:..:=:.:=-:::;:..::-::=-==--------
.00
TAYLOR
KAISER-BESSEL
0.80
DOLPH-CHEB (CHEV
0.60
0.40
0.20
QUADRATIC PHASE ERRORS
0
10
20
30
40
50
GO
Max Phase Error (degress)
Figure 67. KEL, quadratic error.
70
80
90
88
400
KEL
--
T AYLOR
-···
f< A I SER -8 ESS E L
--
DOLPH
-CHEBYCHEV
--- .. -- .. --- .-. -·· -· .
1 . .3 50
,...
1 . 300
-- -- ----
1 . 2 50
CUBIC PHASE ERRORS
1 .
,...,..,..
-
..
__,. .....
_,
__,.·
~/
.,..
/
/
/
2001-----r---~-----r----.----.------.----.-----~---,
0
,n
20
30
40
so
GO
Max Phase Error (degress)
Figure 68. KEL, cubic error.
70
so
90
89
+ ()
rl
KEL
--
-
'AYLr)'
• · • I< A I S E R - 8 E S S E L
D0 L P
I . 3 50
t; ··· C
H E8 YCHE V
,__., ..
-
··-··-··_ ..... ....-..1 . 30 0
-
-
.--.
--
$__...... ....
~··
. ,_- . ..
-··
_.... .-··
------------ --- ---
FOURTH ORDER PHASE ERROR$
1 . 2 50
0
10
20
30
40
50
60
70
Max Phase Error (degress)
Figure 69. KEL, fourth-order error.
80
90
90
Peak Sidelobe (dB)
QUADRATIC PHASE ERRORS
-36
-38
-404!-----~
.,· .
..- - - - - . ....__ -~ 7-=-----+ ~----.
.......... -··-··
\
-· .-'
'
-42
--
-
\
TA(LOR
.. -VA i SER-BES
-44
\
~EL
OOLPH-CHE8YCHEV
0
10
20
30
40
50
60
70
Max Phase Error (degress)
Figure 70. Peak sidelobe level, quadratic error.
80
90
91
Pe~k,S,~delobe (dB)
'
·~
-I 5
-20
-25
-30
T A r· L 0 R
-.3 5
I< A I S E R - 8 E S S E L
-40
D 0 i_ P H - C H E B Y C H E V
CUBIC PHASE ERRORS
-45
0
10
20
3~
40
50
60
70
Max Phase Error (degress)
Figure 71. Peak sidelobe level, cubic error.
80
90
92
Peak Sidelobe (dB)
0
l
'
i -
-I 0
•·-
I< ,A I S E R - 8 E S S E
I_
D 0 L P H -- C H E 8 Y C rl E V
FOURTH ORDER PHASE ERRORS
-20
-.3 0
-40
·......-··
0
10
-··
20
_.,.-·
30
. ,_.... ..
40
--·
.-··
50
_... ..
60
70
...-·
80
Max Phase Error (degress)
Figure 72. Peak sidelobe level, fourth-order error.
..
90
93
first one or more sidelobes in some cases.
This is illustrated in
Figures 52, 55, and 58.
Cubic errors had a more substantial impact on the first
sidelobes as shown in Figure 71.
A difference between weighting
functions of approximately 2 dB in peak sidelobe was affected by
phase errors larger than 15 degrees.
Fourth-order errors also had significant impact on sidelobes,
however the first sidelobes under large phase error conditions
were very close to the main lobe such that no null occurred.
This
fact, illustrated in Figures 54, 57, and 60, cause the results of
Figure 72 to be misleading.
CHAPTER 5
SUMMARY ,AND CONCLUSIONS
The frequency responses of three window functions with -40
dB sidelobes, Dolph-Chebychev, Kaiser-Bessel, and Taylor, in the
presence of amplitude and phase errors were evaluated.
The
effects of amplitude errors with frequency up to 0.03
cycles/sample and modulation index less than 0.20 were evaluated.
Quadratic, cubic, and fourth-order phase errors were considered.
Vanous trade-offs exist between the three windows which
were evaluated, relating to_ where undesired energy is located
within the spectrum.
Dolph-Chebychev windows have the
narrowest -3-dB bandwidth; however, they also exhibit uniform
sidelobes.
Kaiser-Bessel and Taylor responses both roll off (at
different rates), at the expense of slightly wider -3 dB bandwidth.
These trade-offs must be evaluated relative to the intended
application for a proper selection to be made.
An additional
consideration in selecting a particular window could be
performance in the presence of errors.
Table II summarizes the results obtained with amplitude
errors.
These are the values of gain degradation, ISLA, KEL, and
94
95
Table II. Window parameter comparison with amplitude errors.
EHROR FREQUENCY {CYCLES/SAMPLE)
U')
PARAMETER/
WINDOW
CJ
L()
1.0
0
0
0
0
""
T""
(\J
0
0
0
c)
0
0_
0
0
0
0
0
C')
GAIN DEGRADATION {dB)
i
DOLPH-CHEBYCHEV
-1 . 1
0.02
0.76
0.66
0.02
0.00
KAISER-BESSEL
-1. 1
0.02
0.85
0.80
0.00
0.01
TAYLOR
-1.1
0.04
0.83
0.74
0.00
0.00
: !SLP (dB)
-24.7 -24.0 -2 2.1 -20.9 -17.7 -1 6. 0
DOLPH-CHEBYCHEV
KAISER-BESSEL
-
-39.7 -38.0 -33.3 -29.2 -20.7 -1 7. 1
TAYLOR
-33.6 -32.2 -29.9 -28.1
DOLPH-CHEBYCHEV
1.22
1.22
1 .17
1.12
1.24
1.21
KAISER-BESSEL
1.34
1.34
1.29
1.24
1.39
1.34
TAYLOR
1.28
1.28
1.23 l1.18
1.32
1.28
-20.2 -1 6. 8
KEL
PEAK SIDELOBE {dB)
DOLPH-CHEBYCHEV
-39.2 -40.0 -29.3 -24.9
-20.5 -19.8
KAISER-BESSEL
-39.8 -41.7 -31.1 -26.5
-45.7 -20.8
TAYLOR
-39.3 -4 0.1
-39.2 -1 9. 8
'
-39.3 -27.4
96
peak sidelobe, with amplitude errors of modulation index 0.2.
The
Dolph-Chebychev window, which had the narrowest main lobe in the
baseline condition (see Table I), was least affected in the main
lobe in terms of gain degradation or increased KEL.
The Kaiser-
Bessel window maintained the lowest ISLR of the three windows
evaluated over all amplitude error conditions.
did not exhibit a consistent trend.
Peak sidelobe levels
However, it should be noted that
this parameter is not necessarily an accurate indicator of the
quality of the stopband response since, in many cases, the first
sidelobes blended in with the main lobe such that the main lobe
was severely distorted, eliminating the first null.
In these cases
the main lobe sometimes covered the first several sidelobes.
Table Ill summarizes the results for phase errors,
illustrating the parameters measured for maximum center-to-endphase errors of 90 degrees.
The gain degradation results were
somewhat different than with amplitude errors, as the DolphChebychev window was effected slightly more than the other two.
However, the largest margin was less than 0.2 dB.
Performance of
the Kaiser-Bessel and Taylor windows in terms of ISLR were
similar, and both were typically more than 3 dB below the Dolph-
97
Table Ill. Window parameter comparison with phase errors.
0
~
I
I
0
~~
~
::>
m
DOLPH-CHEBYCHEV
0.51
0.44
0.46
KAISER-BESSEL
0.36
0.26
0.41
TAYLOR
0.44
0.35
0.43
DOLPH-CHEBYCHEV
-2 i .4
-1 7. 9
-1 4. 7
KAISER-BESSEL
-26.9
-21.9 ,-1 7. 9
TAYLOR
-2 7 .1
-21 . 9
-28.6
DOLPH-CHEBYCHEV
1.31
1.29
1.24
KAISER-BESSEL
1 . 41
1.39
1 .36
TAYLOR
1.37
1 .35
1.31
DOLPH-CHEBYCHEV
-38.7
-1 5. 2
-13.6
KAISER-BESSEL
-43.3
-1 7. 5
-31 .3
TAYLOR
-40.4
-1 6. 8
-28.6
PARAMETER/
WINDOW
a
::>
0
~e5
GAIN DEGRADATION (dB)
ISLA (dB)
-
KEL
PEAK SIDELOBE (dB)
98
Chebychev window.
The Dolph-Chebychev window did again
consistently have the lowest KEL.
The Kaiser-Bessel window had
the lowest peak sidelobes; however, the above caution regarding
blending of the main lobe and first sidelobes applies in this case as
well.
In general, the frequency responses of these windows in the
presence of amplitude or phase errors exhibited the same relative
trends as in the baseline conditions, i.e. Dolph-Chebychev
maintained tr
~
narrowest main lobe in all cases, while the Kaiser-
Bessel and Taylor windows had less sidelobe energy.
BIBLIOGRAPHY
1.
F. J. Harris, "On the Use of windows for Harmonic Analysis with
the Discrete Fourier
no.
2.
Transform", Proceedings of the IEEE, vol. 66,
1, January 1978, pp 51-83.
A. M. Furukawa, "The Effects of Phase Errors on Amplitude
Weighting Functions", Graduate Project, California State
University, Northridge, May 1987.
3.
J. F. Kaiser, "Nonrecurs_ive Digital Filter Design Using the 10 -
Sinh Window Function", Proceedings of the 1974 IEEE International
Svmoosium on Circuits and Systems, April 22-25, 1974, pp 20-23.
4. J. W. Adams, Class Notes, Engineering 568A and 5688, California
State University, Northridge, Fall 1986 and Spring 1987.
APPENDIX
COMPLJfER PROGRAM
iOO
1 01
C
c
5
10
c
C
INITIALIZE
REAL W(256),MAG(11,1024),DFTMAX,TOOPI,GD(11),
CXKEL(11),XISLR(11),PKSL(11),A(256),AMIDEX(ll),FREKA(11)
REAU<8 TUPI,FREQ,SIGMA(ll) ,HALFN,STPD,DF
COMPLEX X(256),P(256),SAMP(256),DFT
TOOPI=3.142E0*2.EO
TUPI=3.1415927D0*2.DO
AMIDEX( 1)=0. EO
FREKA(l)=O.EO
IG1AX0=490
M= 1000
FL-I=FLOAT(M)
DO 5 I=l,10
SIGMA(I)=O.EO
CONTIN{JE
WRITE(5,10)
FORMAT(' HOW MANY SAMPLES? O<N<129')
READ(S,>'<)N
HALFN=DFLOAT(N)/2.DO
SELECT TYPE OF
AMPLITU~E
WEIGHTS
c
20
30
40
c
JCNT=l
WRITE(5,20)
FORMAT(' SELECT TYPE OF AMPLITUDE WEIGHTS')
WRITE(5,30)
FOR!1ATr' !=UNIFORM, 2=HAMMING, 3=DOLPH, 4=KAISER, s=TAYLOR')
READ(5,*)Iw1SLT
IF(IWTSLT.EQ.2)GO TO 200
IF(IWTSLT.EQ.3)GO TO 300
IF(IWTSLT.EQ.4)GO TO 400
IF(HlTSLT.EC.. 5)GO TO 470
C
COMPUTE UNIFORM WEIGHTS
100
DO 110 INDEX=1,N
W(INDEX)=l.EO
CONTINUE
WRITE (7, 115)
FORMAT(' UNIFORM WEIGHTS')
GO TO 500
c
110
115
c
G
COMPUTS HAHr:ING WEIGHTS
200
CONTINUE
DO 210 INDEX=l,N
W(INDEX)=.54EO-. 46EO">'<COS (TOOPI*FLOAT(INDEX) /FLOAT(N))
CONTINUE
WRITE(7,2l5)
FORMAT(' HAMMING WEIGHTS')
GO TO 500
c
210
215
c
C
c
CALL "CHEB" SUBROUTINE - COMPUTES DOLPH-CHEBYCHEV WEIGHTS
102
300
c
CALL CHEB(W,N)
GO TO 500
C
COMPUTE KAISER-BESSEL WEIGHTS
400
410
WRITE(5,410)
FORMAT(' CHOOSE ALPHA')
READ(5,*)ALPHA
PI=3.1416EO
PALPHA=P I''!\LPHA
CALL INO(PALPHA,S1)
N1=N/2
DO 430INDEX=1,N1
ARG=PALPHA>'<SQRT( 1. EO- (FLOAT(INDEX) /FI.OAT(N1) )''"*2. EO)
CALL INO(ARG,S2)
W(INDEX)=S2/S1
CONTHfCE
DO 440 JDEX=1,N1
IDUM=JDEX+N1
W(IDUH)=W(JDEX)
CONTINUE
N2=Nl+1
DO 450 KDEX=N2,N
IDUM=\1- ::CD EX+ 1
W( IDUl1) =',if (KDEX)
CONTINUE
WRITE(7,455)
FORMA'-;:·(' KAISER -BESSEL WEIGHTS')
W2ITE(7,460)ALPHA
c
430
440
450
455
460
c
C
FGRMAT('ALPHA=',E10.~)
GO TO 500
COMPUTE TAYLOR WEIGHTS
c
470
475
485
490
495
CONTINUE
WRITE(5,475)
FORMAT(' CHOOSE ALPHA')
READ(S, >'<)ALPHA
DO 485 INDEX=1,N
W(INDEX)=1. EO+O. 2EQo'<ALPH..A.*-"COS (TOOPF"FLOAT(INDEX -N/2 -1) /
CFLOAT(N))
CONTINUE
WRITE(7 ,4<10)
,
FORHAT(' TAYLOR WEIGHTS')
WRITE(7,495)ALPHA
FORMAT(' ALPHA=' ,E10.4)
GO TO 500
c
C
COMPUTE SINE WAVE
c
500
505
DF=.5DO
WRITE(7 ,505)N
FORMAT(' N=' ,13)
DO 510 INDEX=1,N
X(INDEX)=CDEXP(DCMPLX(O .D©,TUJPI*DFLOAT(INDEX)'>'<DF))
103
510
CONTINUE
c
520
CONTINUE
c
C
SELECT TYPE OF ERRORS
c
522
524
530
531
332
533
534
540
555
55 6
559
610
615
622
616
619
620
621
623
624
625
626
WRITE(5,522)
FOR1'1AT(' DO YOU WANT AMPLITUDE ERRORS?')
RSAD (5, '"") IAMSLT
IFC:Al'1SLT.EQ.O) GO TO 555
v{RITE (7, 524)
1
FGR:iAT(' AMPLITUDE ERRORS )
WRITE(5,530)
FOR11AT(' DO YOU \'IANT TO VARY (l)INDEX, OR (2)FREQUENCY? ')
READ(5,*)IVRSLT
IF (IVRSLT.EQ.2) GO TO 532
WRITE(5,531)
F0~'1AT(' CHOOSE FREQUENCY')
READ(S ,*)FREKA(l)
GO TO 534
WRITE(5,533)
FORMAT(' CHOOSE INDEX',
READ (S, '"")AMIDEX (1)
wRITE(5,54·))
FORi·!ATC' IN wrfAT INCi\EHENTs: ')
REAI: ; , *)CRHT
GO 10 559
WRITE(7,556)
FORMAT ( I NO AHPLITIJDE ERRO:\S I )
CON!INCE
WRITE(5,610)
FORMAT(' WHAT TYPE OF PK,.SE ERRG,·:S? ')
WRITE(5,615)
FORMAT(' O=NONE' 2=QUADRATIC' 3=GUBIC' 4=FOCi<TH ORDER I)
RLAD(S, *) IPHSLT
IF(IPHSLT.EQ.O)GO TO 616
WRITE(7,622)
1
1
FORMAT( P 1~ASE ERRORS )
GO TO 620
WRITE(7,619)
FORMAT(' NO PHASE ERRORS')
GO TO 625
IF(IAMSLT .EQ.1) GO TO 623
WRITE(5,621)
FORMAT(' VARY MAX PHASE ERROR IN WHAT INCREMENTS?')
READ (5 I*) s IG<1NT
GO TO 625
CONTINUE
WRITE(5,624)
FORHAT(' CHOOSE SIGMA')
READ(5, '"")SIGMA(l)
SIGHNT=O.EO
WRITE (5, 626'
FORMAT( I HOW MANY VALUES? I).
READ (5, ·~<) INUM
I 04
617
627
628
KMAX=490
IF (IPHSLT.GT.O) GO TO 628
DO 627 I=1,N
P(I)=(l.EO,O.EO)
CONTINUE
GO TO 700
CONTINUE
SIGMA(JCNT)=SIGMA(l)+SIGMNT*FLOAT(JCNT-1)
c
C
COMPUTE QUADRATIC PHASE ERROR
c
629
631
IF (IPHSLT.GT.2) GO TO 630
DO 629 I=l,N
ZIG=SIG;1A(JCNT)'>'<5 .2036E-3
P (I)=CDEXP(DCMP':.X(O. DO, ZIG:':DSQRT(5 .DO)* (6 .DO''
C::JABS ( (DFLOAT(I) HALFN) )''""'2. DO/DFLOAT(N)"''"2. DO-Q. 5DO)))
CONTINUE
WRITE(7,63l)
FORMAT(' QUADRATIC PHASE ERRORS')
GO TO 700
c
C
COHPuiE CUBIC PHASE ERRORS
c
520
639
643
IF (IPHSLT. GT. 3) GO TO 640
ZIG=SIGi,!A(JCNT)*4, 5583E-3
DO 639 I=1,N
STPD=(FLOAT(I)-HALFN)/DFLOAT(N)
P (I )=CDEXP (DCl1PLX(O .DO, ZIG,'<-DSQRT(7. DO)* (20 .DO,...
C (DABS (STPD)*'... 2 ~*STPD-3. DC'"(DFLOAT(I) -HALFN) /DFLOAT(N))))
CONTINUE
WRITE(7,643)
FORMAT(' CUBIC PHASE ERRORS')
GO TO 700
c
C
COMPUTE FOURTH ORDER PHASF. ERRORS
c
640
649
650
700
ZIG=SIGl1A (JC:-ff)'>'<4. 0724E -3
DO 649 I=l,N
P(I)=CDEXP(DC~1PLX(O. DO ,ZI8*3. JQo'<(70 .DO*DABS ( ( (DFLOAT(I)
C -HALFN) ;DFLOAT(N)) )**4. DQ,-15 .DQo'<(DABS ( (DFLOAT(I) -HALFN) /
CDFLOAT(N)))*"''2.D0+3.D0/8 TIO'~))
CONTINUE
IRITE(7,650)
FORMAT(' FOu~TH ORDER PHAS~ ERRORS')
CONTINUE
c
C
COMPUTE At1PLITUDE ERRORS
c
705
IF{IAMSLT.GT.O)GO TO 705
AMIDEX(JCNT)=O.EO
GO TO 709
IF(IVRSLT .EQ. 2)GO TO 730,
AMINCR=CRMT
FRCRMT=O.EO
GO TO 708
107
892
893
897
894
898
899
WRITE(6,893)
FORMAT('TAYLOR WEIGHTS')
WRITE(6,894)
FORM~T('NO AM INDEX=O.lO INDEX=0.20')
DO 899 K=1,M
FREK=FLOAT(K)/FM
WRITE (6, 898)FREK,MAG(l ,K) ,MAG(3 ,K) ,MAG(5 ,K) ,MAG;:7 ,K)
FORHAT(E10.4,1X,E10.4,1X,E10.4,1X,E10.4,1X,E10.4,1X,El0.4)
CONTINUE
c
c
c
900
902
901
905
910
WRITE(7,900)
FORHAT( I
I)
WRITE(7,902)
FORHAT(' SIGI1A INDEX FREQ GAIN DEGRADATION ISLR KEL PEAK SIDEWBE
CLEVEL I)
DO 905 J=l,JCXT-1
WRI:'E(7, 901)SIGM(J) ,AHIDEX(J) ,FREKA(J) ,GD(J) ,XISLR(J) ,XKEL~J)
C,P.'\SL(J)
FORHAT(EP.2,1X,E8.2,1X,E8.2,1X,E8.2,1X,E10.4,1X,El0.4,1X,El0.4)
CONTINUE
WRITE(/,910)
FORNAT('
')
END
c
c
C
COMPUTE DOLPH -CHEBYCHEV 'WEIGHTS
c
SUBROUTINE CHEB ('ll, N)
c
10
200
201
100
REAL*4 W(l28)
REAL'''8 S, BP, B, AJ, ~1\V
\v'RITE (5, 10)
FORMAT(' CHOOSE SIDELOBE LEVEL')
READ(5,*)SLL
S=10 .DO*>':(SLL/20 .DO)
BP=DEX2 (2- JO*DLOG(S+DSQRT(S**2- _.DO)) /DFLOAT Ui -1)'
B=(BP-1. DO )''"""2/ (BP+1. DO}""*.Z
M=(N+l)/2
W(l)=l.DO
W(N)=l.DO
DO 100 K=2,H
AJ=DFLOAT(N-K)*B
JJ=K-1
WW=AJ
DO 200 J=2,JJ
JP=K·J
AJ=AJ>'<DFLOAT(JP*(N-2>'<KH:+JP) )*B/DFLCAT( (K--JP-1)>'.-(K-JP))
WW=WW+AJ
IF (WW. GE. 1. D 10>"AJ) GO 11:01 ~Cll
CONTINUE
W(K)=DFLOAT(N-1)*WW/DFL()}AT{N-K)
W(N-K+l)=W(K)
CONTINUE
108
300
315
320
WW=W(M)
DO 300 I=1,N
W(I)=W(I)/WW
CONTINUE
WRITE(7 ,315)
FORMAT(' DOLPH-CHEBYCHEV WEIGHTS')
WRITE(7,320)SLL
FORMAT(' SIDELOBE LEVEL IS ',E10.4)
END
c
C---------COMPlffE BESSEL FUNCTION------------------------------C
1
2
SUBROUTINE INO(X,S)
S=l.EO
DS=l.EO
D=O.EO
D=D+2.EO
DS=DS>'<X•'<X/ (D>'<D)
S=S+DS
IF(DS-.2E-8*S)2,1,1
RETURN
END