CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
A SIGNAL MODEL TO DETECT A CW SIGNAL IN GAUSSIAN NOISE
li
WITH ALTERNATIVES FOR IMPROVED DETECTION PERFORMANCE
A graduate project submitted in partial satisfaction
of the requirement for the degree of Master of Science in
Engineering
by
Edward Milkovich
June 1979
The graduate project of Edward Milkovich is approved:
D. f'vianquen
L. Nypan
R. Pettit - Committee Chairman
California State University 1 Northridge
June 1979
'
ACKNOWLEDGMENTS
I would like to express my appreciation to Dr. R. Pettit, and
his committee members, Dr. D. Manquen and Dr. L. Nypan, for their suggestions and direction.
I especially wish to thank the following in-
dividuals from the Lockheed-California Company:
1)
Mr. Richard R. Heppe, Vice President and General Manager,
Government
Programs,
and
Mr.
Dominic
Amara,
Chief
Scientist, for their inspiration and assistance leading to
my application and acceptance to CSUN's master program.
2)
Dr. Leonard Abrams, Operations Research Scientist, for his
invaluable tutoring and analytical assistance during my
three-year quest for the Master of Science Degree.
I also wish to thank the Engineering Administration Staff at
CSUN for the opportunity to earn a Master of Science degree.
iii
'
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
iii
v
LIST OF ILLUSTRATIONS
vi
LIST OF ACRONYMS AND SYMBOLS
vii
ABSTRACT
INTRODUCTION
SECTION
1
DESIGN OF FOURIER SIGNAL MODEL
5
Introduction
5
1.1
Design and Testing
1.2
Design of FFX Generator
10
1.3
Design of Integrators
16
1.4
Threshold Detection
19
1 .5
Conclusion
23
of~!
Noise Model Generator
7
SECTION
2
KEY ELEMENT PERFORMANCE WITH
~ffiTHODS
FOR IMPROVEMENTS
24
Introduction
24
2. 1
Evaluation of RC and Linear Integration Performance
24
2.2
Evaluation of Receiver Detector Performance
36
2.3
Evaluation of Receiver Ripple Losses
48
2.4
Conclusion
51
REFERENCES
52
APPENDIX
A
COMPUTER PROGRAM TO EVALUATE NOISE GENERATOR
53
B
COMPUTER PROGRAM FOR FOURIER RECEIVER MODEL
59
iv
LIST OF ILLUSTRATIONS
Figure
1
Page
Optimum Detection for Signal of Unknown Frequency
2
and Phase
2
Block Diagram for Signal Simulator and Fourier
6
Receiver
1. 1-1
Noise Generator Distribution Comparison
11
1. 2-1
Flow Graph of Eight-Point Decimation-in-Time
12
DFT Using Butterfly Computation
1.3-1
Plot of n vs. p for One Time Constant
20
1. 4-1
Threshold Vs. Number of Samples
22
2. 1-1
RC Losses Relative to Linear Integrator for
29
Variable Signal On Time
2. 1-2
RC Vs. Linear Comparison
30
2.1-3
Signal Detectability Performance - 1 Sample
31
2. 1-4
Signal Detectability Performance - 8 Samples
32
2.1-5
Signal Detectability Performance - 16 Samples
33
2. 1-6
Signal Detectability Performance - 64 Samples
34
2.2-1
Plot of Integration Gain vs. Fourier Receiver
37
Gain as a Function of Number of Samples
2.2-2
Input Signal-to-Noise vs. Number of Samples
38
2.2-3
Signal-to-Noise Performance of Linear Detector
43
2.3-1
Amplitude Response as Function of Bin Position
50
v
LIST OF ACRONYMS AND SYMBOLS
CPS
Conversational Programming System
CW
Continuous Wave
dB
Decibels
DFT
Decimation-in-Time
FFT
Fast Fourier Transform
Pd
Probability of Detection
PFA
Probability of False Alarm
RC
Resistor-Capacitor
SIN
Signal-to-Noise Ratio
vi
ABSTRACT
A SIGNAL MODEL TO DETECT A CW SIGNAL IN GAUSSIAN NOISE
rliTH ALTERNATIVES FOR IMPROVED DETECTION PERFORHANCE
by
Edward Milkovich
Master of Science in Engineering
June 1979
The introduction of Fast Fourier Transform (FFT) and other modern digital processing techniques offer methods for achieving improved
receiving system sensitivity for detecting a continuous wave (CW) signal of length (T) imbedded in Gaussian noise.
The finite duration
signal is generally narrow band with unknown phase and frequency, the
amplitude is constant, and the starting time is unknown.
The noise is
Gaussian distributed with zero mean, and the random variables are statistically independent.
This paper addresses the performance of key
elements in a general Fourier transform receiver system designed to
detect such a signal.
vii
The performance associated with detection, post detection integration and signal position in the filter is analyzed.
Charts are
provided to show detectability performance or effects on detectability
performance.
A computer program is designed using the Conversational
Programming System (CPS) to model each element. of the system.
cise of the program provides data to confirm analyzed results.
viii
Exer-
INTRODUCTION
It is well known that the optimum detection of a phase modulated sinusoidal pulse of length (T)
in white Gaussian noise is a
filter matched to the in-phase and quadrature phase components of the
signal followed by power estimation and threshold comparison.
The
power estimated is compared to a threshold and if the threshold is
exceeded a signal detection is declared.
This threshold can be de-
fined in terms of detection and false alarm probabilities desired for
the receiver.
Such a system is shown in Figure 1.
When the signal
frequency (f) :i,s known to .exist within some range of frequencies ( W) ,
the receiver must employ multiple values of (f), thus creating a set
of parallel filters.
apart.
These filters should be spaced one bandwidth
Since the signal is of length (T), the bandwidth is approxi-
mately (1/T)
[1].
The number of filters needed is W/1/T
= TW,
and the
probability of false alarm is then approximately TW times the probability of false alarm of one filter.
The Fast Fourier Transform ( FFT) provides an efficient method
for implementing the receiver and will be used in this experiment.
In most cases it is desirable to smooth the output of the receiver to
improve the signal detectability.
plished with an integrator.
The smoothing is usually accom-
The integrator can take one of several
1
MATCHED FILTER
IN-PHASE
SIGNAL
COMPONENT
BW
= 1/T
FREQ = £ 0
V(t)=v(t)+S(t)
WHERE
v(t)=NOISE
S(t)=SIGNAL
(
-
)2
THRESHOLD
SAMPLE
@ T
+
MA :CHED FILTER
QUAD- PI-lASE
SIGNAL
COMPONENT
BW
= 1/T
FREQ = fo
(
NAL
ENT
NT
= f
o REPEAT ABOVE WITH FREQ
= f
Figure 1
ABOVE UNTIL FREQ
I
I
COMPARE
)2
o REPEAT ABOVE WITH FREQ
~ REPEAT
INTEGRATE
l
=
0
0
f0
+ 1/T
I
+ 2/T
+
W
_j
Optimum Detection for Signal of Unknown Frequency and Phase
N
3
forms, and the form it takes for physical realizations should be based
on the behavior of' the signal.
Factors to be considered are:
1)
When did the signal begin?
2)
How long will the signal be on?
3)
How much memory can be allocated?
4)
Is the frequency constant?
5)
Is the amplitude constant?
Two integrators will be evaluated
in
this experiment.
The
first is a digital equivalent of a resistor-capacitor (RC) integrator.
Each Fourier
transform is weighted
during
integrator to exponentially purge old values.
summation,
allowing
the
This simple integrator
approach is very useful in real time systems because of its ease of
implementation with the least amount of memory hardware.
The second
integrator implemented is a boxcar ("linear") integrator which treats
all the samples with the same weight; thus, complete purging occurs
after a finite time.
The linear integrator is not often used in real
time systems because of the additional memory required to match the
integrator to the signal duration.
When the integration period is complete, a thresholding operation is used to determine if a signal is present.
The threshold
value is based on the mean and standard deviation of the noise.
When
these noise statistics are obtained, the threshold is applied for a
desired false alarm rate [ 1].
at the selected threshold value.
All the integrated filters are tested
4
When the threshold is exceeded, the filter number is recorded.
Based on prior knowledge of which filter outputs contain signal and
which contain only noise, the probability of detection and false alarm
can be verified.
Subsequent sections provide the design details and performance
of key elements of the presented Fourier receiver.
Section 1 provides
design details associated with implementing a signal model for the
Fourier receiver of the type described.
Performance analysis and
methods to improve performance are provided
for
the
following
elements in Section 2:
The square root of the sum of the squares detector,
RC integration vs. linear integration, and
Signal loss due to filter ripple.
key
SECTION 1
DESIGN AND TESTING OF FOURIER SIGNAL MODEL
Introduction
A block diagram of the receiver simulated is shown in Figure 2.
Its operation is as follows.
The signal generator provides a means
for generating CW signals with the capability to control amplitude,
frequency, frequency deviation and start/stop time.
Gaussian
noise
generator
with
controllable
The output of a
standard
deviation
is
summed with the CW signal to provide a composite signal with variable
signal-to-noise
An
ratio.
FFT
computation
estimate of the simulated signal.
provides
the
spectral
The number of filter bins used for
the spectral estimate is controllable (N
= TW = number
of samples).
The real and imaginary FFT outputs are detected using the square root
of the sum of the squares detector.
The detector outputs are inte-
grated using RC and linear integration.
The presence of a signal is
determined by. obtaining the sample mean (m) and the sample standard
deviation( s)
testing
each
from
the
filter
integrated
bins which have noise only and then
filter
bin
to
see
whether
threshold related to the mean and standard deviation.
ance occurs a 'signal present' is declared.
it
exceeds
a
If an exceed-
In addition the real and
imaginary FFT outputs are combined with weighting terms (K , K ) and
1
2
integrated in a linear integrator.
The weight terms used allow for
approximating phase differences between the FFT generators and the
5
ON/OFF
AHI'LlTUDE
r-
FIU:Q
{ BIN POSITION
f
TIME
LSTANT
rl
1-
SIGNAL
GEN
SAMPLE
(N)
RC
INTEGRATOR
l AR(J)
Q!1IT!IT:
A(J)
~
FFT
B(J)
~
A2(J)+B2(J)\
~
DETECTIONS
ro1
B(J)•O .....,.
NOISE
I
POSITION
2:
!
(Sd)
o FREQ
o AMPLITUD
o BIN
LINEAl\
lNTEGRATOR,AI(J)
E
K1A(J)+K B(J)
2
Figure 2
LINEAR
INTEGRATO
JAI(J)
Block Diagram for Signal Simulator and Fourier Receiver
Cj\
7
signal.
Properly applied,
detect signals.
this
provides an alternative method
to
This method approximates a coherent detector.
An output routine prints the results of the receiver operation.
It prints the filter bin number, the amplitude of the spectrum estimate, the noise mean and standard deviation of all the frequency bins,
the probability of signal detection (Pd) and the probability of false
alarm (PFA).
1.1
Design and Testing of CW/Noise Model generator
Fundamental to any computer simulation is the need to provide a
controllable stimulus where the characteristics of the stimulus are
known.
A CW sine wave signal generator which can be sampled at the
appropriate intervals is needed in this case as a stimulus for FFT
processing.
The amplitude, frequency relative to FFT bin position,
and start/stop control must be provided to allow variation for each
test
simulation.
The sine wave for a continuous time variable is
given by:
( 1 . 1)
x ( t)
= A sin
( 2 1T ft +
Q)
A = Signal peak amplitude
where
f
= Frequency
Q
= Phase
and the bin center frequency for the Fourier filter can be derived as
follows:
The sampled Fourier transform from Reference
[2)
is:
8
N -
-j
1
~ x(kTs) e
X(r)
k
2 1T
--
kT r
s
T
=0
r = 0 , 1 , ••• , N-1
where
x(kT )
s
=
X ( 2T1Tr)
= rth
T
s
= Interval between samples of the time series
k
= Sample number
r
= Bin number
N
= Number of samples in the Fourier transform
T
= Length of the time series record
kth sample of the time series.
complex Fourier coefficient
Then using the relationship T
= NT s
and substituting into the
previous equation:
.2 1T rk
-J
N -
( 1 • 2)
X(r)
=X
1
=2:
(2 1T r)
T
X
N
(kT ) e
s
r
= 0,
1, ... , N-1
ko::-,o
The frequency of the Fourier bin in which the signal lies is
miT and x is sampled at kT •
s
Substituting these terms into 1.1 we get
a sampled form for the sine wave generator.
2 1T mkT
(
and using T
= NT s
we get
T
s + Q)
k
= 0 , 1,
••• , N-1
9
( 1. 3)
x(kT )
s
= A sin
( 2 ~mk + Q)
k
N
= 0,
1, ... , N-1
Equation 1. 3 satisfies our sine wave generator requirements.
Now we
need a noise generator with an output having a Gaussian distribution,
zero mean, and specifiable variance.
The equation for generating such
a Gaussian distribution is provided in Reference
( 1 • 4)
v = ~ -2 ln u
where
u
u ,
1
x cos ( 2 ~ u2 ) x Sd
1
= desired
sd
=
2
3 •
standard deviation
two independent random numbers generated by a
uniformly distributed number generator over (0,1)
The Gaussian distribution amplitude v by definition is a sampled function.
Now if we calculate a new v each time we calculate a
signal amplitude, x and sum x and v, we will have a signal plus noise
generator.
The complete generator algorithm is obtained by summing 1.3 and
1 .4 and providing means for offsetting the frequency in the bin and
program start/stop control.
(1.5)
A(m)
= ON
+ Sd
This leads to:
x Am x sin ( 2 ~Nk
X
,J-2
ln
U
1
I
X
(m+0.1 x binpos) +
COS
(
2
~ u2 )
Q)
10
= turn
where: ON
generator ON/OFF under program control
= amplitude of signal
= position relative to center of bin
= nominal bin number where signal will
Am
binpos
m
appear in FFT
output
Q
Sd
= phase
= noise
shift
standard deviation
A test program was written to evaluate the signal generator and
validate the noise statistics.
is given in Appendix A.
The program with computer printouts
The validation of the statistics for more
than 15,000 samples is given in Figure 1. 1-1.
The theoretical mean
( 0) and standard deviation ( 1. 0) are validated for the input conditions.
The distribution of all the samples is compared to a normal-
ized Gaussian distribution curve and excellent conformance is noted.
1.2
Design of FFT Generator
One
of
the
reasons
that
Fourier
analysis
is
of
such
wide-ranging importance in digital signal processing is because of
efficient algorithms for computing the discrete Fourier transform [4].
This algorithm is generally referred to as the FFT algorithm.
There
are numerous ways to implement this algorithm and the choice is generally related
to
hardware
considerations.
For this
experiment,
hardware considerations are not a factor; thus, the decimation-in-time
with butterfly computation was implemented.
algorithm is shown in Figure 1.2-1.
The flow graph for the
rn1lJIII'I \i !'~~~~\ill'11 ~~J~1-~1TIT!111[1 111 !111~' 1
ulml'l-1
ntlrl ·11 ~11
~I .
II!\ Il fTIT1jJII ' Ill j!li lf1i jjll!l!! HI II
1
[\ \
1
.!ll\Tii1T1Till lj. iljl!nTfjtrrmnnr
~U~ !!1_: r _''1 - · - H_f'J t- ~~~1 ~·['t'+ID'~ ~-I~r JH bi'-~-~-~rrr!~rl11r . rr~r!j Hn~_Wllu-!!_ ~tWj·:w li'~ljWI
~r ~ ~t Ill II111~~~w~lr1_~t.j~ ll~~·l 1l!Jrl'~"
''!J ' t ·tlmJ't~~~
il''
1..~
. . t~
i:. ···m·ti p, 1m~~~J~,,tirllrl1.
~--~, 1. ~-- ilr'i ~-"
_1 -1·1: il~~~~·~
!:i!l:~h ~~~~~Ju.u,'
+iuliJUu1•1 1
1 h,t~~(~')11-.ml
i
~~{_
.HW!iL
,
__
.1.
,,
~~~~I
l!iilllt
f]Iiij.iiU!JilillUi!l
illluW!ililJ:
U~
imt+H~ *ri1 Ill
_f:\
111 [
.1~' il'illl
itt +1-.. ii ~···
t+i+
...rrl_I ·:
I
L
/
_
.
. It ·t
I I
1
I
!J:.
-_·-.'i
!}'.
I
..•...
-,
1 ..... _
I
H+,,_
:ill'!
1
1
f :
,,,1111~
~ 1Ill! I~ ~ ~mwLn_ ±, .~ H~J 'tt·,,
l1 J 1rlr ill f ~ -_tr' tl1 I I 1l ~l lUI IH.IIHnnu iii: iii! uu~m'llii IIIHHJlw.
iiiit 1 x·mrfl! ~~ !
jill t tr
I ! .· .. · l'lil ll!f
!i II !ll!il
-I
-j~iljl~_~
·
nlr
j
l
J
f
.
+
_
~
_
r
f
~
U-~-~-~-~
1rJ
·
1lt"
lrtt--1
i_
t
~t
li~
i!jl
·l
J
tt11jllf
!tit
![II
rj
t++t+J+mi+++t~ti+
~~ tuJ ~~ ~t~ ~~ mn
~1111 .
n 1 1 .. I lit~1 h~~~ ,.t! 1~ m_rltlr
dfJlU 1.'_l !
· ··
1\
1
!
11 II'
r,
I
:"!
I
1.
.
,
_ .1
1
'' '
J,~ilffi~~ 'l~~li, ~
'I 11~~
I ~1~ -~
_·.
1
I.
m1Jtf
1
~_
~- -r
n1
·,
I
1
I
I
•
_1
11
_
11
"~~~~·. -1~1 '.
~
·.!~- j~ni
I
1
'
I
'
r II_
1
I
1
t. """ ....
IJI'ilj i'l
1
f
I
J Utl_~iU~~ai_._Jl1~~~-m1 1ii_l l~_'611 lliUdHjlj!ll!j'-jjW
lltllll t!Jjlrliif~~~-~,~~~urllmil !1Jilli'!JJ ii il!!lli!lJ!tiLI!Liu
1
1
111 1 J-
1
1
.
1 i 1+~q_jw!
1
i-~lf]'lH
J I! I I . I 1!: f(~y,~·JP]Hi: 1!!11 fl~ili~ ~:p )i rtltm' I: 1 li:l
MI
_ ·I-11-_lltl
I n:lll,jjfl_-1 I+L tufflrH-_tlil
!'1·[+-·q HHffi(i.'!
_ l_: ;j_.Jiif!J
!Ill~.
rrlili~ .. . 1111III I .I I , -_ ... t ......I.1 .II J.jlh~·::llli!ll~~~Jil!illilll
:lilLIIi]J !till. tJul:il 1t11WlllllJ.L.liiJ[!ii1iiijjl·
i . jt+f -~.11 I I i I
ffin . .
.. Tt+; l.f+l. . . . .
H+t f-H++t-P.-1
1
J]!i
1\'.'~~r-:-r·r
t~·ij·rl_ill L . 1-JHljri~
rdiJ_11,jj\il· ·,¥!l
lii11ltt)-!ii1 1r··p_q 1 1
+H+++t-tt_ "til
'~~II
.I_
J t 't-'-~U'
~~I\_l11~_~..I. I ·:m11 ~I ir'-111
I 111!11\i\_l~~~nilmH_':·_~·,i
I I ~iliu dlltt!;!ilifuJllJI~+J,JJ~.J
111 "r1 Jl . 1 · 1' 1 - ' 1~ · •
11 I 1I I 19 , ·-. 11
11 :I!! 11 1iu, :::1 P
il ~ l! mm LJ 1i'I 11'1'1 'I
l . ~1 _
..
. :++f ,,
-.
I
_~ :tU. I i l1li~llll 1 ,1 ~t,ll .1 !Jl! c+4-r_:l: ~]_ll 1 +~-l-h , 'J~I,
I I! I
II
.
I 'ill
.
l I II I I ITT I· I II I II l:lt :IIi Ill tiT·!·1 1 ItiT ljtp
rrr - 1+1
r1 - - - ~ ,. - - +j:
.
·
.
-~~ ll~ .J,- liilll!_d\1\!
!jt!ttti1Jll·~-I w.l~H
l':i hH Hlf i • -Jt~r~1!~r h1j
nwtJ.IIIITlHn1
. · .
:1.1
J
,
. ·
\It ::il_ill_i~j'rli~lltJii !Iilii 1jtt
.
I I I . .
'
•I
J
l
·I''
I' I! _I' II!. :! . II II i: ii ttt -n Itti-jil It!
1
.
.
.
'
l
.
.
. IJ· 11
1 1 i'ti ,, . liJ' 1 l I 11 '
1
_,J . ·.1I ,. lr ri!.·
1.
...
·r·
.
u
~·.
•
-~If!
tf.ll~lill1r'ril
JMj
'
:+··
Hlt
·
t
'
~
.
:
.'
rf
~l'
ll-llliJl!f
~+!.
·
·
·
fl
I
1 j 1 r 111
.I 1 'I j 1I I 1 11 1 • 1,1 ' f i
'
___
.__ . l ~
,,
Ill . . · _ I L - - .. _ .. rl , jl!t1i 1.' ,11 -$ t.4t lfi!uluJ. ::•t L hil!la.1
I
•
I!
I .
II li
t
L .I
II! 111 ' 'li!il!lilif'1H'i lj J II 'l ill'~fj
+i ._.
_ -l Ji , . . .
-1:
L
t 1 . . --_!- -. +1m·' !ltl!1 +
Jlrll!\j_l :Jilil!'llj 1_®
~i l' II . . I I r
. . . t ~lli ~: ]illi!mUJUI!llliiiilrlli 4J1f dv1lj!)l~l q_
. . I
I rq II I iIfi ~~il
. . 1. .
!
I
II l~ I: !1li 11ii 'I! IIi Iii! !i !!' i rrn Ii !!I iI l 'l
liiflrd . ~It Hn l
....
.
1111 H+:i) . flt lllt~j: b:J;l!iil l!i:I:!P1 ;_-_lj:_ I:I~Wn1 r_r 1i :tllfnl thfl
m. tilt I 1.t:i>
-8- T~'i !I I~ . . 1. . ~ ,, i J jfi J 11 . !Jijn.:lllj!f.1i~ ··.:t~ :: :r.! fdil!'l il ::, uu~m~
":'1
I
111
j I Ui 1 ·.
: ir. 1
· I'I l , _rI 11
Ill
• if
jr · ilt! ·! 1
••1:: 'I ::I' .:1 'II 1'' 'Ill !_" !fir 11·!,
I
I '
' I~
IlK il
I·.
' I '
It
I1
I
11
1
)1 I -j1 lj-- · - '· -1,I - t _I~ t F-b 1- 11!tilt !1· ~ !1 1 !I~ ·111
~iir~ 1111 1fi :t!ll!rllt!1 t['11l ·~ llflil·! d ·
11111 ]
i\
HI! J i-111 11 i\J' l\i 1 li!l i@j~d : l1 l!u1 fu \\1! fL 1
r · •
.
IHI
!itt IJ~llll
1 1 jlli1'1 tlil
il,' J111~I !U:!
f·ljj'JlWi;-;-I~rtrtjtfHJjiJij
HTIH 1'\!
:rtJ+HHI+l+~l'
. r~.
I .
. ifff
- lliW
L I H ..L. 1: 1iI;_
1l
.
I
t I
II
I
I!
I.
I
:
1
J
-
J
_H_-
I
0_·:J
1
11 1
t
r '-
~lltllf~_J_~~I!,
rl
mrltlr
I -.-
~-
j.
I
:
I
1
I
I
I
~w
•I
I
j
1
11
1
I
I
1
'
I
1
1
II
I
,,
!tf111JH·r1 il 1i ~
J . • .
1
'
I
I
f
j
t_
I '
-1
1
l . . ~· . _- _ . .
· -· .
Figure 1.1-1
1tu - _ litn~
1
j
rtlr
I
-li!H!fllUWDl _,· ~[i -llli Il-l 11! Iilli]
Noise Generator Distribution Comparison
1-'
1-'
12
x(O)
x(4)
X(O)
w~
X(l)
x(2)
x(6)
X(2)
~
X (3}
x(l)
x(5)
X(4)
wS
X(5)
x(3)
x(7l
Figure 1.2-1
X(6)
ws
X(7)
-1
-1
-1
Flow Graph of Eight-Point Decimation-in-Time DFT
Using the Butterfly Computation
The
design
implementation
for
this
algorithm
involves
the
following steps:
1)
Input
data must be arranged
i.e., for an 8 point DFT:
in
bit-reversed
order,
13
From
To
~~
x (000)
x (000)
x (001)
x (100)
1
x (010)
2
x (101)
x (011)
x (110)
x (100)
x (001)
x ( 101)
x ( 101)
x ( 11 0)
6
x (011)
6
x ( 111)
x ( 111)
7
0
1
3
4
5
7
0
2
3
4
5
If (n n n ) is the binary representation of the index of
2 1 0
the
sequence
x(n),
then
the
bit-reversed
sequence
is
(n0n1n2).
2)
Multiply the input data by the appropriate function
generator using the butterfly procedure.
The indexing of
the function generators and the butterfly must conform to
the following:
a)
If the function Generator is defined as W~, then
-j(2'1Tr)
N
where r takes on the values shown in Table 1.2-1 for each
W~ multiplication shown in Figure 1.2-1.
position represents each butterfly moving
right
on
the
obtained from
flow
chart.
The sequence
from left
to
The maximum value of i
is
14
= log
i log 2
N
= log
N
log 2
i
and since i is an integer
= trunc
i
Sequence
Position
X
1
0
0
x1
x4
x5
0
0
0
0
0
Repeat
~-
~
I
N.
4
_l
I
,.
0
.
.
...
~
.
.
.
I
I
I
I
t
T
Table 1.2-1
b)
.
.
.
I
I
I
I
.
.
'
I
4
t
f
A
Repeat
'
'
0
2i
I
I
I
I
r
I
1
2N
8
N.
8
XN-1
N
0
- (2~-. -J' Hi
A.
...
Repeat
I
Sequence
N)
~
N.
4
0
I
3.
10
r Values
x2
x3
Sequence
0
2
value of (3.32193 x log
I
I
J
T
I
I
I
I
I
I
I
I
I
t
I
•I
-.
I
~
T
I
I
I
r Value Determination as Function of Sample
and Sequence Position
Next, the application of the function generator to the
input data must be considered.
to apply at each series step.
crossings are interrelated.
There are N/2 butterflies
The butterfly adjacency and
The sequence of Figure 1.2-1
is repeated in parallel for N>8 and the number of adjacent
15
lovTer elements of the butterflies increased by a factor of
2 for each ith series step added.
In the design of the FFT it is desirable to have the capability to
allow the time series to be expressed as a complex variable, and the
complex function of ~
to be expressed as sin and cos functions.
This is accomplished as follows:
From 1.2 let
where Ak, Bk are real.
= cos
Rewrite W~ as:
.
( -N2 TT r )
( 2N7T r) + J. Sl.n
and substituting into 1.2
. ( Ak sl.n(
. 2 1TNrk ) + Bk cos( 2 TTNrk ) )]
J
Equation 1. 6 then represents the form the function generators
should take when implementing the FFT algorithm.
It should be pointed
out that the signal generator described in Section 1.1 does not use a
complex time function and therefore B
=0
when executing equation 1.6.
16
Finally, the real and imaginary terms out of the FFT must
i)
be combined:
for the energy spectrum
-
and for the voltage spectrum, more commonly referred to as
another 'envelope detector',
[ x(r)] =
1.3
~(real) 2
+ (Imaginary)
2
\
Design of Integrators
The use of integrators at the output of tQ.e detector improves
the
signal-to-noise detection perfor'IDance.
The performance
for a
Fourier receiver system using a linear integrator is given in Refere?ce [1 ].
The performance for an RC integrator is not given and since
it is the most commonly used it is desirable to compare these two
integrators at various signal conditions and evaluate the performance
differences.
A performance comparison of the two
made in this experiment.
( 1. 7)
X (r)
n
integrators will be
The algorithm for the RC is given by:
xn-1 ( r)
+ 2-p
X(r)
n = 1,2, ..•.
17
where:
X(r)
=
voltage amplitude in the rth frequency bin of the
nth transform
=
integrated voltage amplitude in the rth frequency
bin after n transforms
p
=
integration constant
The algorithm for the linear integration is given by:
( 1.8)
X (r)
n
=
+
X(r)
In a real time system the RC integrator can be matched to the
expected signal duration simply by selecting the value of p to give a
time constant which is optimum relative to the expected signal duration.
The integrator will purge itself with time.
On
the other hand,
the linear integrator must be partitioned in some fashion which allows
old samples to be discarded and new ones to be added.
For example, if
100 transforms are to be summed, sum the first n transforms and store
in one buffer.
Sum the next n transforms in a second buffer.
· ~inue until all 100 transforms are stored in 100/n buffers.
a signal test, sum the 100/n buffers and test for signal.
Con-
Then, for
As time
moves on and n new transforms are obtained , sum these transforms and
discard the first buffer and replace with the new sum.
signal again.
Then test for
This requirement impacts the amount of memory needed to
implement the linear integrator.
pendent on the size of N chosen.
The amount of memory needed is de-
18
To obtain the
establish a
performance of the RC integrator we must
relationship
equation 1.7.
between
the
RC
time
constant
and
p in
This is needed for the comparison in order to match the
time constant to the signal duration.
This relationship is obtained
as follows:
Let X (r)
n
=1
in equation 1.7.
For one time constant, xn
Equation
1. 7 develops in
= .63212.
the
following manner as
sample is added:
-P
x,
=2
x2
= 2-p(1-2-p)
x3
= 2-p(1-2-P)
+ 2-p
2 + 2-p ( 1-2-p) + 2-p
Substituting in the values for one time constant:
.63212
X
n
=
X
n+1
= .63212(1-2-p)
X
n+1
-
X
n
= 2-p
(1-2-p)n
and subtracting to reduce terms, we get
.63212 (1-2-p) + 2-p - .63212
= 2-P
1 - .63212
= (1-2-p)n
log .36788
=n
(1-2-p)n
log (1-2-P)
+ 2-p
each new
19
( 1. 9)
n
=
-.43429
log ( 1-2-p)
Figure 1.3-1 is a plot of equation 1.8 as a function of p and
n.
It is used in Section 2 for the performance comparison.
Since n can only have integer values, it is obvious from Figure
1. 3-1 that p cannot be an integer in most cases.
In most hardware
systems it is desirable to implement p as an integer value in order to
simplify the multiplication.
1.4
Threshold Detection
The decision threshold establishes the detection performance
and false alarm rate of the Fourier receiver.
The relationship of de-
tection performance and false alarm rate statistics as a function of
threshold and noise statistics is provided by G. H. Robertson
[ 1] .
This reference was used to establish the threshold methodology used in
the signal model of the Fourier receiver.
Frequency smoothing is per-
formed on the time-smoothed coefficients to obtain the noise mean.
The algorithm is given by:
( 1. 10)
20
I
'
I
'
i
L
I
\
I
i
I
i
I
I
I I
. l
I
1;::=::
I
I
'
I
!
1'\
I
.... I
!1-f
' I
:
'
'
'
'
'
I
I
I
'
I
'
'
I
'
I
I
'
'
I
'
'
~
!
I
i
I
I I
I
:
'
: :
i
:
'
I
I
I
I
i
..
'
i
'
'
'
I !
I I
I
I
i
I I
I
~
'
'
'I
I
I
I
i
I
I
I
I
I
'
I
I
I
I
i
I
I
I
I
I
I
I
I
'
:
I
I
I
I
I
I
I
i
I
I
i
I
I
T
I
21
where m(r,) = voltage amplitude for the rth frquency bin after
a
n transforms
= number
M
of bins in the frequency window.
A wide
window is desirable for an accurate mean.
= frequency
rd
index of the resolution element to be
examined by the threshold detector
= separation
B
desired from signal bins
Frequency smoothing using the above algorithm provides an estimate of the noise mean under the assumption that most of the frequency resolution elements in the window contain noise and relatively
few contain signal.
If this assumption is not valid, then a two-pass
algorithm could be provided to eliminate all bins which pass the
threshold the first time from the second mean computation run. Such a
two-pass algorithm was not considered in this experiment.
Next the sample noise standard deviation is determined.
(1.11)
where
Sn (rd)
= standard
deviation of voltage amplitude.
The threshold decision is then implemented by the following
equation:
22
I
I
'
I
I
I
i I
lrt'
.I
I
I
1 I
! I
I I
I
=
I
I
I
I
I
'
I
..,.
IUl
!>
I
I
I
I
i
I
I
I
' I
I
i
'
;
k-
I
fO
~
!
I
I
_l
-
'
23
X (r) - m (r )>TH x S (rd), then declare signal present,
n
n 1
n
(1.12) If
if not , no signal present ,
where
TH
= constant
depending on false alarm rate desired.
Reference [ 1 J provides charts for a continuum of false alarm vs
threshold multiplier vs number of integration samples.
Figure 1 . 4-1
was constructed from these charts for false alarm rates of 10
10-4 •
-2
and
The 10-2 values were used in the model program in order to
minimize the number of runs required to validate the false alarm rate.
The 10 -4 values are normally used on real time systems.
1.5
Conclusion
A computer program which models the Fourier receiver design
described
in the
previous section was written and is
provided in
Appendix B. The program was used to provide test data for validation
of the performance associated with the receiver power detector and the
smoothing integrators.
The resulting data closely approximated the
The performance at low signal-to-
theoretical performance expected.
noise ratios requires many runs to obtain good statistical averaging.
In
order
to
avoided.
terminated.
conserve
If general
computer
time,
agreement
was
a
large number
obtained,
the
of runs
tests
was
were
SECTION 2
KEY ELEMENT PERFORMANCE WITH METHODS FOR IMPROVEMENTS
Introduction
On attempting to implement the Fourier receiver for real time
applications, practical considerations often argue for use of elements
which are not optimum, thus resulting in loss of system performance.
This section attempts to identify the system performance for various
approaches of implementing a system design and provide experimental
results on actual implementation using the computer model supplied in
Appendix B.
2.1
Evaluation of RC and Linear Integration Performance
It is well known that good spectral density estimates can be
obtained
by averaging.
expression for
Our interest is
to
obtain a quantitative
the detectability performance associated with aver-
aging. · Using the derivations for spectral estimates we can extract
the factors for detectability improvements as follows.
For example,
given that we have 1280 time samples of the signal to be analyzed,
partitior. the samples for computation of ten separate 128... point power
spectra.
1
128
. ]2
128m-1
[ ~xk e-JW
k
= 128(m-1)
24
m
= 1,2
.•. , 10
25
and then average the ten
10
10
2:
1
s128 (w)
= iO
1
s128,m (w).
m=1
Since the individual
estimates have
identically distributed
spectra, the averaged spectral estimate will have the same mean as
that of the individual estimates.
have a smaller variance [ 2] .
However, the averaged estimate will
In fact, if K estimates are averaged,
then:
( 2. 1)
and the standard deviation will be:
(2.2)
rr
=
.J
var SK
m
\
= ..'I/1/K
· Var S ( w) \
m
The expression for signal-to-noise is signal amplitude/
noise
If the standard deviation is changing as the ~ , then the expected
improvement of the integrators i s - F times the signal-to-noise ratio
at the input to the integra tor.
(Note:
This is not to say the
expected performance of the receiver improves as
{K'
times the input
signal-to-noise ratio).
I~
is obvious
that
the
linear
integrator
performance
will
improve as ~ because all the samples are used, i.e., unweighted.
The RC integrator on the other hand weights each sample differently;
therefore, the expected improvement will be something less than~.
We can calculate the He-performance using the relationship:
26
2
O'n
Using
samples ( N
equation
= 17,
p
=
1.9
to
find
(2.3)
Var S (w),
m
and for the linear integrator
O'L~n =
f7 Var Sm(w).
The difference in performance is
2
( 2. l~)
0' RC__
2
-
..J.!!.61
2.484,
• 0588 =
O'Lin
and in dB's
(2.5)
2
aRC
---2
O'Lin
= 10
for
an
arbitrary number of
1 • 9725) and substituting the RC weighted values
into the above equation we get:
= .1461
p
log 2.484
= 3.95
dB.
27
Thus for 17 samples the linear integrator is 3. 95 dB better.
It is of interest to know if we have chosen the proper time
constant for the signal condition given.
It is obvious. that if we
made the time constant shorter, this would be analogous to throwing
signals away.
On the
other
hand,
if
we make the
time
constant
greater, the variance of the integrated noise will get smaller, but so
will the integrated signal amplitude.
We can calculate the perfor-
mance by calculating the variance ratio as before and
signal loss.
adding the
Taking advantage of the previous variance calculation
foP p = 1. 9725, we can calculate the performance as follov-IS.
previous
calculation one
time constant was 3. 4 samples.
In the
Then to
calculate the performance with a signal duration (D), where (D) is
less than 17 samples, use:
(2.6)
Perf diff
= 10
log (.1461) - 20 log (1-e-D/3. 4 )
1/D
= 10
(log .1461 +log D) - 20 log (1-e-D/3. 4 )
= -8.353
+ 10 log D - 20 log (1-e-D/3. 4 )
where
( 1-·e -D/3 · 4 ) :: integrated signal amplitude for D samples
= performance
change in equation 2 ,1-! from reference
17 samples as we decrease samples.
28
Figure 2. 1-1 was genera ted using equation 2. 6.
The minimum
loss of 0.85 dB occurs when the time constant is approximately 0.8
times the signal duration.
Thus, if the duration of the signal is
known, the RC integrator performs quite well.
The program in Appendix B was executed for various samples in
the integrator and p adjusted per Figure 1.3-1 using N/5. The spectrum
amplitude of the FFT bins with noise only was used to estimate the
mean and standard deviation of the noise.
the results obtained.
Figure 2.1-2 is a plot of
The performance for a large number of samples
shows the linear integrator performance to be plus 4 dB better than
the RC integrator.
dB.
At a few samples the performance approaches plus 5
This is probably due to the measurement error of a small number
of samples and using p equal to an integer value .
The simulation,
however, did validate the calculated performance.
Next the program was executed with signals at various
sig-
nal-to-noise ratios with the signal-to-noise ratios measured in the
narrow band FFT filter and the detection and false alarm rates measured at the output of the integrator.
2
false alarm rate of 10- .
through 2. 1-6.
All samples were run for a
The data results are shown in Figures 2.1-3
Up to 80 samples were used for Figures 2. 1-3 and
20 samples were used for Figures 2. 1-5 and 6 .
!~
and
The Robertson curves
[ 1] are plotted to show how closely the system model approaches theol"etical performance.
29
---- .
'
-~-~----·-·
'
.
.
,---------·;-:.~:---=--:T-~~----------::-r::~~~---
-i ~
.-
--
. ---
.
-----~---
.. ·--
'
·-:-_-------1·-------- ------------- -------~----_-:-----=-~-----~-~~-~~~;~-~
-----
1
- - - - : ----- ..
t~~:= :·:-~-=~{---:-::·--=·t-::--:-:-:·.+-~~--~--- ~--~ ----=---=-~-~~i~-~:J
:.::.~c.:..:.:~::...:... :.:~._:.: __--·-------·,_._._:::_i_·._.-·--·--·-·_. ..;____. _.:__...:.:._.:__.:__·__·_:__:::.......:c:.:__: __ _:_!_:..::.~....:.:... :__::.....:..:..:1....:.....:..:
__
. -----~~~.t___~_::_::_:jr:~~=-~==~-l-~::_:_::::~;
-~~=~~-~-t~~~~ ~~~L-~---~~-L~-~-f-:_:___:!:~~~---:_~_: ~-L__ :_~~-~- --- :
-= > ... ;
:::. ·:
~=;
~--~~
=-· :..:: ; ·.:.:~:
._.________-_._.__
-·
___ :·; .. :::··:~.--:!.:
;.
--. :·i: ---
~--
'
-- ..::.. 1:~:-~ ~---- -- ----=~~.=._~~=:==-~=~~-----T'-----­
.L
;-~::- ~:--
"1- . .. : -~-= ~---------- t·--·---~r-------:-~:=-~
:-:.:-::·! ::.: : :-:::_ j -- _·--; :-.:- t·
_::.:~
.
~
.-
--·-----:-,----~-------'-
•
=.::-~-------:··------!:::.c,_~_:l_ ___ ~~~ :.::c.:_~:-~~.:...: -:::;~~i~-~:- _~::::_~~l-~ji.:_
;._+s
- J.---
-L :_~ -~lc.:~,~:~~:-==:: ·~: :~=:.:.:
1
=--~~--;;;~TfEG~~~+f~~I-~}~1-~~~f.f~)tif-l~~}~j-~,t~d~:~-~~~1~~
Figure 2.1-1
RC Losses Relative to Linear Integrator
for-Variable Signal On Time
30
;;-;·1-r;·,· ~~·,~:~-~,:-~;~~~uL~
I ,.,:;11 ;.ll;l[lio'll'JiiJ''I ~~ ~~~~~~~~~~lllljlljljllll'll 11'1\ljl'\jj~;l'll"'j'!'\'illj'llJji[J\IIIll'j\li#I'I'JJ M''''~li!JI~Jil~l i!HJ'111lll(j1•'' !Jj j
. \: .. : ·~-~<~
:u. 1 ~L~~ ~i:.ll:.:i-!)1/l.:HiJljJlL dli!l.l 1~u ~~1 w~1~~~ 1,~~~~'r 1 u_l:ri,, .J!Jiil '" j ,J. 11 I ~ ,~H~l~~l2i,
;.I<,'
::,1·,,
h:,liii' ' ';,:.:::,'' J RC).()•\j~IL1'I'-~<r\E'':
E' rr'E' ,.yj'lo''·b !j:J;:I! Til:l!illllfi IJJI :':!'I!! kiWI
I Hrl:
.llj!
,!,,
•
-M,,,...(
\,1, I• .IITjl
!
! i!fl f l! 'II ilji!'!'; 1''11
1
,
I, · ,11... 1.1 :,--v:
. ·I'll
,.Jll
'"! 1:•1r' , , 'rl , 1 I: 'I1 [':;' ·'I•
11111.jl''ll
..
1.,··1,,t·
11"•·,1 I!:
1
1'. . , :' · 1I : · I' I, 1:1 i ·'I rl""
1
,1:'...·: ,. ' ..· ·!·•·1
· I''
'. ·.f
,1I.:
',!
1
i::
'1!
I
'r
1
i
I
I
i
'
I
'
I•
1
I
1
II
:
. :11 IIJ1ji.J
' ,II I'.U.I
i' 1,lr,,lrr
r I 1: 'l
::.11,' IIIII':• 1\! 1111 J:.l:; I. :tl,lt]t :l;tl\l'i!'ll,
Ill, ;:.: !J: 1
•J.l :I n':j lit! I' !ill IIii lid lLI1 :;·lil:r 'I'
~ti .::i!T::-·1
:;- ';:~~·:;\"'~f
. !:r·::-1;--·w·,,~::p:': !~~~;n; !fliiTFJ
iii 1~,:• if11!1Ji;:rrr1 ',·f!T#~F J'li!ii iJ j'l i0rJii·
rl·1·1ttj'l.,.j m!1 i+j:·i~1 rfflo/11l]
!Tj":irili!ittl
1,Wi1T['
, •. , , . , 1 II
., 1I ' ~""'.
·1'
' M1 jFJir:w·
,.1,,
,,1,, I'
. 11 I.1
11 .1:.1 r1,., ffi!jlrjl
I''I , ' ",~'II',,
' II 1 ' ..' .. 1 ., .n ., .. j I••••
'
' 1,I I, . • 1• • 11
Ill 'I ,r1'•. , 1 1I 111 ,
II I 1. ,I,.
'
1 11
1
1
1
(])
,!1I' 1,1' ;!,
,;•::,:' .i· j I ' ,,
!··.·!
:11·:··
',j.l:'li t·lj d:' .:: ,r!l: .I, t!l: I IIJ;': ::;111•11 lli1 1W' 1111 J,j! lllj'I!•IIJI: rjiH Jj 1 jiliiiH'f'l11lJ\[I'Ii I 'I 1: 11!1''
'
'"'I
'
'
.
'
~
I
I
""
'
II
"I
''
I"''
'II'
"
'
'
''
"I'
r'
'
H
I
Ill
I
'
I
I
Ill
In
I
H-'
I
I
I
1
i
I
I
l~cl"
"I
t...:>
..
:·;---ti
'··71.._
mr i... ...,._ ·r"T . . T
rr-1 m
T'I ""il
•
,::;1 :! ·;.:1•,
.,J
·,'.,
· ,:, , '1 11 :•; :r::l:,:r :lj!j':'. 1;:. 1.!: J'Ji!trr: "'t11' ·jill :i!'prl I'Jil 1: dr\ 1:!1 ·1, !11 II: 1 1 1l! I' I : 1 111 •:1 :! II
1 !' '•'''['Ill, '!Ill 11'ill''tli" ,11 11 II' ll,lj•,IJ' lllltll ''II'!' illll"l ,1:1111 ., I ,.,.1 j II ~ lffi I Iii!~ I II ''ll
"I'
II I,.,...:
'11 1"'.......
' ", .......
I I 'I!.......
'I' .,-~.
,,, ''1'
w
-~-,-,
-~- ...
--.. ~,.. ,..,_,.,'1'1 "'IW.:. ~.-,.,.T
.....,.,,. .....
·r"-''t" Hl··+h r II,,,,.J,
•.,.r,..,., -1'
I I ir· rl II'.. I'I I
1'' ..I""!,.
1 r,.,.,
1
'
I''
,,
I
'I'
"'
'·' :,:I I '1'1
1 '"1''
'' :.I I 1''1 1'1' JIJ,
' 11 I"'
I"'II'I''I I I I
·,I .. ··~::: ''.II I:J,;
1,11 II Ill 11' 1 1'1·1
,J'I· 1[11' I'
!'li I"
Ill I·ll 1 !J'i'
I'I I' I'I'
1 I
I', I'l I
.::, ·:: :I· ,' '
l'
I til ' !
.
Ul
, ... ~J ,I...
-I.
.. I"'
,.....
1 ... ·II·.
"II
.,:11'1·
I I'll!·
''JII!.I'l
'I''IJ
I" I Ill' . .J -11, ~1'\1·1~~ ,ll. 1111 .1.1~111 111\~rl ~~~ 111 .J ~··
-·I''IJJ:I 'Ill
l ...
...
,,,,
,,
,
,
.
,
'1''1'
'I
"
I'JI'I
I"
•.
,
,
'I''
:r
'I
'II
'''I
'I
'1!'11
:,
r
1~1"
,.
,
II
dl
Jl'
li
IJI
II'
lrl
1
11
1
111
1
1
1
1
111
1
,.... 1 ' 1: Z,J;I '
' ' ,, I :: ;: ' :· 1':;: <1: 1 t •lt'l J!. 1' ,,' I;:
:I I i
:I II,' 1111 1°' Ill 'I' II 1'1 I I 'I m
I It' : I I I I ·t / I
111 I~~I
~-r·-,-1-tl--·-·
··-1---' ---.---, ..... - .. ~ ........ , "-1.,.11-, --- ,.1- -~.- , ___ IT1•jH· ,1.,.1'"' ~---I'lL_ I_
I r-~
1•-r]
~T,
itII r··II I''l'I r •I jir·
+n
ttrr
[LII 1
,."I'•. 'I' I
''II",...."'
'
'
'
'
I
'
'
'I
I'
I
I
I',,
'I"
I
I'
I'
lj''
I''
I
I'
I
,,
'
I
1•
I
'I
I
'Ill
I
I
'
'""'
'
I
II"
'I
I
I
'
1'
1
"' 1:·;,::
'" ' ""I ·:'
I
'I·:,;,j
' ' ·, '
I ':t' .. I1 'l I'.t.tI ll!!'
' ' 'I ,l'
'II ' 'I l ' I 'I I" ' I ' ' 1"'Ii ; ,,' '" :II' I! I I 1,:·
I I1'; 1
I ' lrl ! t! "I
•:'
.' ;'111::
I '1;' II'
~
';']1"11"1'-1" ,'[
1
LJ,~
....,
· •J
,....
1
I
, .. J . . .
•
:. '
I.,
' t' I
r
,·
' • I
1
,,
•
I•
1 I
,
r•
•
,
.
~
'
L
I
r
~- ··j''''l"'~-·
._ l}
! '
1
I
I
'
r
rr•
j
.. j '
•
1
I
I
•
!
I I
I
•
•
1
•,
i
t
,,,
'"'
,,
'1'1"
f-
•I
--,
•"
I
I
Pl
1-'
.rP. . .
.._,
rt
rP
()
rt
(l)
•
cr
,....
!---"
~·
rt
~.,.
~
rP
ti
H-t
0
ti
El
Pl
.........
oro
'
'
f
'II'
'
tJ
,,,
"
'
Jl)
:::, 1 -~~~~.J., ' ,,·;1',.
1'1 .,I
'I' I" rc;
u
t---"l
rP
1
I
l'~'ll"l·f· 1t~,~
1•11
!ttll•jj,
I
'11
1
•
•
t
1
I
I
.,
'
I
I
r
I'
til
•
"
I
I I
•
I
1
•
I•
I
1 1I 1 I I
I I •
j•,•
,•
I'
I
I,
I
'
1111
1
i
1 1•
t
I '
f!
I
1
1 1t
I
1
t
•
''
t
11
j
~
I
tll
'I
, ••
1
I
I
l f L[III.j~··lll 1
'
'11
•II
1
t
rl
I
'T
I
1
.
If 1
I
r -t
1
•
1r I
•
•
'li
I
..
'I
I
I
I
1 II
t
'I
I '
I
I
1
1
t
r~ I
lrlj
1
h
!TT.
J
1
I
I
I
I
1
II
l
I
tlotlj
l1
I
1.
.
1
tl
••I
11 ·,l
11
It
'
1
1
,
1
,,
'I'
1'
:
1
tl
I
,j
1
1
1
111ll
••
t
r•
1
1
II'
I
I
't
•r
1
fr
1
.P
.li
I>
I
1''
'
I
1
•t
ttr!
1
l'j
til
'·,·J~~~~·1 ~.;~ "II~~)~,111'I' I',; 1 I,-~1~' ,,~.,~·~·~Jill
1 11 ,:·~~·~.
~~~~
I' I "II'"
II II t!'l: I~,1~ ~ 1 I. ~~,,
II II"II~~~,~~!]J•J 1111 !If·
'II/ l~II'1 ,l 1 1 lu~
II lu t~1 1 I: ~ 1 ~ '1'1
1 ,,I :1!1'1'1 ljll~1.~ 1 ~,~~~·:
'I
,j!j,
j,
'''I'; . : I! '•'"' '
1[' I
'i1t11i!
1
•
1
1
.j,' ' ' "I ' I
\
r
I
,,
I
1
II
1
11
1'
lr I
II ·,1 ;, I' I 11'1 I
I' I ' 'Ill "' I I I .,, I
II
I
I
I 11'1 I
1,., ''11
.... ,..-.
____I ..., ..' _,_I' ..''I ---c-··~·-··T·
,... rlrr~... ___
.,. ,I ·-n
,++-1-t
fTI'H1 ..., 11"
I I'' I'II'I 'II ,,":I
'' II ,,,
' I·11'' ]. I'''I
I'"I ·--''I".,.,.,.! ..II"I,_ ....
'I'
I r··,
'I ' ··rl
'' Itel t·' I·jt"
.I'' •''1.,.1•.' -··1
I 'I Tl_"'I.. I',·~"
..II II'11 1
1'III I' I'I III II''
I
III'' ~+n··
It· I' f+>·
''I
'
'
'
'
"
'
'I
I'
'
!
!
1 '" '·
~ l !, :
I' \ I! ' I: l ' i
! I ; , : ll :
I !
!
I
ill~
IiI I l
!1
! 1• I I : I !
! 1' ' , i ''I
" I' ~ ·:: ,
': , , .. j:' j, J, " ·:j:·:• ;"rj;·:' 1::1 .IIi "11 I"! 'l'i i'l· ·I' "i: :;:· ,, ~-ll:l"ijiHI: 1'1: .IIIII it' Lll •j'l' ~~l]lllljll[lll-: ~~~ IIJJ 1\Ji jjlrl1'~
'
' I I 'I
I '
' I ,•1 J,, l,jI' ·'·1'1' I,,
II '•II I I" q t 11'1 II I' 'I'I I I'''
lr I I I,· jl[ 'l' ii'H+
jl' II II rrl
I II ,] I
II I ..·~'
'''I''I'll ..'I ,. t'' ..,,J,,.11~II....1'1'
I •rIJ +-i+f
I •rr
'II
IT',,. .. ,I ............. r.... -X I .. ,_,,.
__ ...l ,.I.'"
,~•
~--··--···--,.I,,
,.1 ,L. .... •·'• '"'~'·"r.
..L. 'II" J11 .J1t U.
Lj,'I
: I : ~
i !
Ill
I~ I' I' Jl: )111 J!; ~
I' l\ I' 1
11 J!) I
'~l"t
I'
'
1,'; ; j I I I
I
'
I
I
I/
I:
I
I
.
I
'
I[' I I
I '
'
I
' ' ' I ' ' ' ' '" : ' I ' 1
'I I ' III, I ' 1 I' II II I, ' ' ' I ' 'I' I I II1I ' 1
I Ill'! I I I I'1 " ' II' 'I' I I I I I I I I II I I'' I I I! I '
. j ·.,I · ,\ I1. ., , . . , .. ··1:::: I.
·: 1~~· ,.,
:: 1':"!
::L],:"
'·L·:'I:
·1,11:I Si ~~u
·1 ~~~~
1'1 11'1 I I .... ,, 1'-1111·1 ~~~ ~~,, 1n1:
•
J·ll
"I·
,,.
,,,.,.~~II
IL~ 11L
l,iJ l I "·t
1
t-l..:J
(
! ...111' : j
!'
I ' I : (1 1 ~
I I
!! I
! 1 1!tl 1I ! I' '#1 n1. . . i [, ]ll
I 'I
1 \Ill
\ l !...,,...
l I jl,·1'+I
..............
-·r·-- !
- 11--.
r·•--.
{,+1..,.
. ,. .
• •j ''
••
I
!
t' ~~~ ')
1
:;::1· ,::2
:
,',. : I, '',1·1,: .: ·j;l .. ,:,lljll
t:, J.,:: :LJ·:I'tlll' ·''111::1 Ill' t'li: l'li'llltl: !\,11::!1' :l:jl•·: 1,1\lllllt'II~J, lllr II lfttj. J~~~~l~,ITI
<r' .'I
' . I I,, I
''I' ,I "'I II"
I " 'I'I ' I'·t· II'' ''I I 'II'
I I'' I''
'
'I ,'I ' I'''
''I 'I'
' 'I ,' . Irl '''[
I ' I1•, tIII' II1 ,,J I l'' I III'I
,I•' .,...,
I .. ' , . ' I ' ' .. ''1I" ""
' r· .,,
"'I' I•··
t
~·~--·r·-·
·I' ' ' ....
II'"
' '
'
'''I
;
'
I
I
1 •
'
'
,
T
'
i
,
•
"
1
•I
'
,,
'
"
t "
1
'1
'
1·~:m
I
1
,j
•
I
'
1
1
•I
"
'f
f
f
'
,l.,..fl
'
.
, '
I
I
1
'
I I
l' '
11
I
I
•
,
1 ••
t
'
I ',It
, , , ,' I
t I:
llo
1
•
I
I'
,I
,,,,
I
I
I
]'
~"tl--
I
I :
I I ' I' I t<I I
'
I
,jtl
II
I'
I
I
1
I
1
I
1'1'
\I'
1
H-i,
I
''I
'
I
II
I
I
I
rt~ ......
..., ... 1
1
II)''
1
"
,.!
1
1
1
I ' It
' II .
I
'I
1]'\{11
I
1
II :
ljl
I
,~.
1111
I
'I
11
I
I
,r·t-<-r+- -~ ....
jt'l
I
'
'II
•I
I
I
'
t
iJI I
....,.f1
't
11•
ol''j''
\
I
,,.
1
' j
!
tj('
I
t
I
I'
I
I
,j
I
II'
I
'I
'
;
I
I
I I
I I
-.-•~
n--+-•
,...,_,.. .......
ill
lj
•
I
1
I
1I
~J
1
I
'
I
1 .......-.
Jt"""";T
1.
tljl
I
I
jl~ I
I'
l
'@
I
I
I
1
I
I
t
I
'\'
i
I
1 ~:-l]llffi ~~I, 1 ~ 1-~t:f
I
It
I
!
'
1
!:1·l·~l~ ~~1~ ~ ·1 ~·~·h' 1~·:1·1::
It• '
II
I
tH~
1
,
I;
'j,
titI
1
1
1
1
I ' 'I
''
II I
•
I
llr
'''
,.r-M
1
I
1:
11 1
tl•
I
I
I!
~·
•
'
II
I
11
1
II
I'
11
'
'
1
1
~1 "'·x" ·:~·
I
"It
.. -···· . . . .
I
•t
I I I tl ItI
"
I
'
tl•t
''
'
"'
~.-~/--+~·
1.,
1
I'
,...
'
J
LJj
r-::1
I
f
•
•
TI"~. . .r
I
•I
, ] ;
',,
)!
,i
(, '
:! r;!l t,\! 1 1• '11 i , i 1 'j; ),I' .11 ! '11 '·:1 I l
1 'j!i •
I
j'i ttl
1 ,,,, ·
·
"'J
'•I'' I r1l I''' ,.I, · ,. 'IJI 1'1 I" t'' 'l'·,jll., ''JI
,1:1 \1 · ll 1 ltr 1 I 1r 1111 1'1 ' " IJII j1)1l 'I'' 1.,1
·-· .......,1.---,---.....
,v. ""'''['"' ,.•,1.,....,,.-.-1-,.
'I'T
-r:
- n ____
lll! -m "ITillll
1!-i,
rr•llli'
1
1
1' I
1
I I ' 11'1"
II'"'
'I' I I ''I''
I I 'I'
I": ' 1'1 I'I'I~ I I''I'
11 I''IHil
"I 111'1I"
I•
'I' '
'"I li1 :''II
'I ! 1I,, II I'"
I: I I' I II Ill'
I
'l I !' II jI '1' I I j II1 II'
t
t '
I l' ! '
I" i II•! IJ''
I l'I ; :•I1J I"
t!
I I I 1"1'
' I ! :" ' ll
I 1l I I
I I ! I'
.
, I"·
I .. ...,• ,.,1·::.. . . ,I~. ,1.. ..
tj··~: : :::·1
.. ... 1'1'
...
I
...
'
..
r·
...
:I,
I·· , I,.
I·
..
II
I'.!
,
r''
-I
r'
..
~~~~~~
I ·r,l
I
,,
11'
1
lr·
..
1 1\1
:,l" ·::,
·!:
!'I' I'It 1
'I" I '"
' 1''' I' I" I '
"I' :r·:
''I' 1
I lll•li!jl'
'Ill I' j'd
II 1 I1 jtj.
I I I' I
' I 'JI'IJ
I' I 1
'! 1
I 11'
I •I II'I' I:II I 1 I:1 jli
I' I IJ'I I 1 I1 I'lj 11I I,
I
I ...., ...
' .... ...
:.! ...
•.;..Jl
,v
r"l n--· 'In ''"r.
I rrl ljl r 1. . .\, l.j '''I I J
•"Tj -q+' r'!l hl
11lr !-111 fl,ll
II
: : :.t:' ' , , , : : : II . ,.: : : ,I' I' . I' : ,II ' I , ... :: , , I, I , ~~~ • , • ' i 1111111' ,. II II I I II lr I ,I I r , I
.I .
I ' '' I ',I t , : 'IIi : ~ ., :IttlI 'II : II''
I''' "' ' ' ' 'I" "'I '•t I' ' 'I I• II I I :,1 II II J,, "I 'II ' I I• I'It I
II
I' I I
I' I
'~I•'
, , "I ," 'I'" .."I,. ....
, I',,
". ''I I.. ,.I 1'"1'
"''
''II 1111 ,,J, II'
1,[11 \[•11 1'1' '!rll'jll
t•IIIIIIJ :I•,, I'jiJI I'''I''•Ill II'
Ji,j 1'1
11 f '~'I ~~~~~~~~ I II~~
'• . '"I,,,
I I I'I''
,, I ' ' 'I ,,
II I'
'I' II 11 '''I I' IIt I II' ~ l II I''
II' III ''I• ...''I
, II Ii 1.,: ' I''I 1I ''1'j d1 I"•i
I 'I ' I'I I I I' !Il ! II. !'
i'
;' '
!' l~
::: i' l
I \ ' l . i ! : : (i ~ l
: I [0!!
i ' I I I'I I l 'I
II I
1..-.,.• t;:l ...... ,
___ ! .... ,.....
-·-'!'·-· .... , ....TI ....
....
.liT ,.,.n
,+rjJ.l+·
··r
J
...
.,..!
1
1
t::''"'
:, : ~- :·
, ' ' ,. ;,.
:' I ::,, r' I'll'
't jil,'1 11 1 1.1 'j/11\ 1,:1
1
11l'jJ'II
11 '11 ' 11•,,I1t II I11 Illj I '[II'
1'1 !1'
111 '"'
11' 1111
" 1' ,"'•'j'':
"'•
I'iti' 1:•·
,: !.,1
' 11:' ·:1,1
I'I II ::.
"' I 1II'··' li!i
I
I
1
'
I
11
ill
'
''I
'
I
I
1
t
' I· ,J,, 1"1
t '
1
1
·.:·:,til "I
,, '' : ' :!I ' ;"'
:•!: '
'l'' :•·:I !:!:
111'1'1 I 'Ill
:ll! i'l' ji! II, 1!1
'I"
'
" .I'
' .1:'1,1
"1'1
'' :'::\·
' II f !, :1' ,'''\
,II1' .ifl.,·:f·l,·'
':;+:II'
I ,I ' I· 'I'11!'1''
I I jlI il·j
I I 'II'
II 11'II1 :t>JI!:·
'I" ill'I II 1:1'
I '
~~ I I j l
I
I i ! i I' I
i ::1i 'I 1I I'
i I ! I'
I : I'' ' :'I' 'l l I''It I; ; ! Il I'i ! I' I ill' 11' :I! II I"
11
1
1'_,_
........
.;:>;;.,•• L... .l
j ... J.l, llrj'·'· ""'t·]r
·'r
_.,
.1-n-·T!l
1 "T"
·' . .: , .. :
,1
: ,:r::·
." 1 ·1·· :.•J ::'II I,''il I'·t11
l'j'J'l
1•11
t·
.
:,
l't,
:1
jl':lll,,l!'i'l
1
1
:
,,
'!"
11::
:11'
II:
·
:,
:1
,;
:
·
1,
,.,
·
11
'
1
1
1
.. 1
!'
I
I,
11
I,' 'II' I I
I 1'I I 1 1'1' 1 'l 1 II I '
I•
'I" I'
' ''
'
'
' "'I I ' '"I .. 'j,''I"'
d
'It,
II"
I'"
III 'I'
I '11 ' I' ' 'I !. .' 'I''I\ t11;:
'"I I'''I IIII ,I'I I 1\Ji I' I·!Ill''I: Ill''II " I! '1'1
1•11 'I" I'll
i' 'l 'I
I
I~ -U.
1
II
II
:I:
II
I;;'
I
'
t' I I f: t:! I! : d: : i
1
111
11
1
1
I ·'
1-'
I ' ,,
' I
" I•J"•t "I r-' II ' I ' 'I' • I <ttl
' , " 1'' I
tl' "• '
Id I .,, It'
I••
I I II 1,, I;, I.I I' It jtl;'I ill
t•l
--o
•
-·
..
-1-.
.,
·--'·--·
....
],..
,,.
·'· ,,.,1 ,..
''111''"""1"1"'.
J"fl
,,
..
'
1
:
JJ.
]I!IJ..
,,
"
"I ' '
I',, '
"I''1 1 ' 'I' ' ' 'II'1 jl"I !'I"tj I 1',, 'I'II I'Ill' .. " ,, !I'
"'II' .I ' It 1 "Jll' •I' IJ~ I'iJ'I I 1''"I'
ttl
III
'''
1 ' IJ'''I
11, ! I"
111'
; ' 1: • i
'
.
1
• '
r 1 ' '
,
'
, : '
;
:
; ' l :,l
. 11 1 1 1 !! 1 ' 1 ' : ! I : 1
' I ~ I : I : I1 ! ·
1 ,
: 1! 1 : ll : \ 1 1 1 l 1• l , l 1 t :
: l1
1' 1' ) 1
....... I
J
.. I·
\....
1 ... 1 , . .
,1,,
'"'l'r[ ,i .....
'II' ... lr,! .., 1 1.,.
-J , 11 1 ..11. .1.1
,,
.1,~
.. 1
1·1•
:1-1
.1j
- 1 '['' 11,:"11' ,11.!'1•
1
1
1
1
. I
'
I ' ' 'I ' :,·I
'"
I
11'1'
,,, I
'I',' 1 'l''
II"
ll'j'
'I'I' l''I'I''
1'! 'II'
I lj'I Illjl 1'
"'jl
ill II'
I''' '' t"''' '' '' '"'
"' "I
. II I'I 1'1'
' ' :'I'',.j
I1 ll't
I ''
I I'
I I'II I'1 I'' I 'I'I 1 11'l'l lit
'II ' I'l Ii jl1' ; I,J:'
11 'I
1
:' ' 11' . " ,I
' I
' l' '
II .1:1
I ,j
I !1!1
'lli
,1 1"
:. 1 1 I I , : : ; ' I 111
l(,i
II':
: ;:
_...'I'
, ---j- .............
..
'r··j
,, I I ''' .:::I
' 'I
' I :1::::1.1-lil:il
II !''''!'''''I'
,,, "I',·:'''"
1'1 I"
'''I jl',I J,:l
'I'' 1'!,1 I1 jl<I II'I II'
I [Th"'II., .......
II 'I.
'"II'
I'
I'
' :
,1<;;,1'
!!:; I'd''111'1
I'' IId It'''
1LI1 i'II' lilti:J:
,:JJ ltll 1,J,l
.i! 1111.
I il I I ill! Ill, :I\,
:j''
1
: 1
tV-4
,, I .. · '
·-•l""m
__ ,__
I: 'Ill"
' ' ~-!: '•I
I
...
.
.I
•:1, .,·t;.J .. :
I' ' '
.I":.:I:' ' .I
"'"
'I
'"I
'
' ' ' ''
•
•
,
r'
I
'
I
l
'L.J
rTl
•
1
I I
" '
'
•
I
I
•
•
•• ,
'
1
•
1
I
'
'
•
1
I
I
I :
'
1I
~,.,
I
:'I
tl
II
'"
,
:I
I'
'Ill,,'.
I
'
'
1
I
•
•
I
I I
t
I
I '
ttl!
I
ltjl'
I'
I
•
It
'
I
I
j
11
It
I
I
I;' I
,
1'
I
I
'
'j· . .
I
lit'
II
jtl•
~·r·---~~1'
I"
I
I
"'I
I
J ., ,
··1-··r '~-- ...... ·"·~~~J. 1 ,fffrl'
'II
11
II''
I
1
'I'
1
I
11
j''
··~t rL~.
1
1•1
•
1
II
'II
j
I
I
1
I
I
'111
' "l···r. ... ~,,...,
1
I I I
I
I
II
l
I
I
I
1
'1'1
1
I•
I
,],
It
''
I
"'
I
I
•,
'1"~
I
I
11
1
It'
~---
I
·,
1<1
ol•t
I
•
•
l•t
I
•I
I
I
1
I
II
I' .
'"j't
I
'
I
f
t
I
l
I :
1
1
,
I
1
_, __
It
I
-··rln,r~
I
lid
1
1
'•
I
t
I
]I-~
I
,
I
I
I •
I
t
I
lr
t
I
I
',•
I
, •
I
I
"l' "
1<1
II
l
f
It
Ill
t
'
I
~~i
1
-rrj
I
'I
1
I
I
/
..,.,..,~- lffit'· ITt+tnt· 1
I
j
I
rf
•
I
I
1
1
I
'
•
'
t
II
I
I!
I
'I
--~~·rr•1 ,,.ll,.d
I t t'
~--···:-"• "r·t"~~T-rl . , .
I
I I
1•1•
··i•lt-i,.,. .~.,~"-t" mrl!" ~-++
t
'I
•
1
1
I
I
''
I
11
11
I
I
1
I
1 11 J
'
lit
1
I'
1
I
-~·
I
11
t
1I
I
t
I'
ll1!
I
I
I
I
I
'
1
I
I''
,1 ..."
"'
t
jll
I•
•
I
1
1
I
• I I
t
11
1
I
I' ... ,, ,
I '
jjt
II
'
• 11
I
1
1
I •
,... ,.,. '"'Jfrrl-L,,
1
I
1•1
I
11
I
1
•
1
'
II
•It
I '
':
1
I
I
I ' I '
'''
1
I
I
t
~ ··~ j•·l·
'
1"
I
;
'
I : '
''
I
•
'
'"1···· ·-·~
I
I . ·1 .... ,.. ··!·--·
1
I
: I I
I
I
111
''
It'
'I
;
1
•
'
•
I'"
'
I
'
'
,
I
' '
I
I
1!;!
''
1 II'
1
,,,,,,,
-~-----·--·--·-·
:
1'
I
•
••
I ' • I
'
I
1
'
'
•'
I
I'
1 .""~'-·n
..
'
II
••
....
'I
'
I''
I
I
:
I : I
:
I
I
I I : :
I I
•
11
'1
'
I
-·I .. . .... ... ·-··-'I
1
'I""
'
'
' I
'
11
I,'
1
~J . r···· . .
1•
')'
'I '"' .. ,.
•
.,"'. . ..... t.,·j.
~
t.,
t
'
'
1-'
I
•'
I.
1"'
tl'"
••I'
Ul
I
o'
,I
il
''
'
..
•-<
' I I '
•ftl
•I
1
,7~ ;·~;~·
OQ
·
1
"
>jl
1
'
'j'l
"I'
I
L'-'
I-'
32
,:JzlA;~~l~~r~1~~r~;s~~J=K-¥T~8.~3D/lf-~~~fu(~~"
Figure 2.1-4
Signal Detectability Performance
8 Samples
t;:nr::·;t::;
:·::1 fi '" r:::·i I::~f'rr: ~i'Tr:;l
;:nr:: P:lrl:'iW'jF'lfj'nl
i 11r\·1:I;,
I 11''1111! i! 1
11111'] 1,:I' 11\\'l'li'f
1\1'11 !lj !ltd
i! 111111
il !J':
Jj IIJU'
II ljl jl\1nII ltj~ij
rn 11 11.r
l:; 1·.' 1::~~-:·:t'j".·:·
. ''ftl:'
·:t,.'·:: ,l:f':'·lf:l::
1 ::~ 1!, .1 !l1TfH1nT:qlrllll
111: ·1!\I::Jl :111 1·1,
J.qJI:t'1,,d
1 1 II'I~~ i,t\lifl
111 ~'I'm! II11
I,;:
1 , 1111'
1
;;-;-•--;-, ;----··,·-·1:~~~-.-·r··----•r ·-;-1~:-;- T'T; ;·-;--r;t';T:T
'f'l 1 ',. ~rrt'-:f\-i lTJ1 tjttJ-;-< !'Jl 7r]'l\-i" •i't1r.
1T'+ r,•,j+l-,~
;-jl TI tHtlt > t 1 j,!llil'f
~-l 'it.
11 11 !lTjTit?T-~T1
1!t:;nllt!l
1.;,·1,;· i,•,
,: I.::' :tii" 1' J'
'·.:, 11 ·: .1,, '''i \'II1:\[·\ :lijlj il d'!DI::,/
,·11!:
'llldlj I 1~111 i lj1!1 I' l•fl lll
f'11'1·:''
~
l. • • "
,
l. 1 • : ' :II'.'·
'
1 •! •
! • I•
J!, • 11
Jl
!)
f!
1
...
, ... I .. ,.. ].... , .... ,, . .·-~ ..:..,.I:I:. PfZDJ~f\!~~I:L.
E;;EC ..roNrl'·d·l~~·l·+j·~j·IIII''''I'IJI il'tllil'l'l'l
!•.. ~~-~~~~1~ ~~~-q.l,l ~~~~~··,1·
t'llj'fll\'1'1'1
·1. ~~ • • 1 • : l .• , • • • . I
.
..
:
i •.. , •. :I. ;ru.,On.,
:.
I.
: • :, : • :. l': : ~~~ ·r·.. I I I I, -~~~ 1 I I l'jl I! •• IIi. •
• . I I·~!. . 11·1 . ~~~·"I , l I ,,. I .I ~~
.
·~ I I I
11
.......
·
...
I
..
I··
.
,,
.
....
1
..
,.
...
.
ol·
1
..
,.
"I
.,[[.,I
'
t
,I
.,11
1
,
1
II
II
~~·,·1
·1
.I
...
~~·
'I
..
,,
1
I
,
~m'
,.,,,,
.
,
JJ
I
I·
,
•
1·
... , ., .,. ....,.
, , r...... ,_____..,_._, ..,. ,, ... r·-....-. 1 . , ... l'Tt
, , • , , 'L
, ••.
, ..
H..
1 ,- .
'11']
'I' • t
r· I rr·, •
'I
r:'""
1·, 1 .•.1·1:-;-r:t • ·::·l: 1•. 1:1:1 1l:.l.t''n: •.l~~. ·1:t·.J 1] ·1 ~ [
11 ·I,;·· 1· ::·: :,n ,,,,
1
·:!11').·.:;
·
.
1
:·
~
··•
!~~~
1
ll
T,
~~~
111
rr11
~~l
11,
til
f11
11.
111
n
;~
~~~
1
1
1
1
1 •11 1 ! 1! il1! I' 1jll'I • :l .. It •'! • 1I ' \I d111 l -1 !l
1,,,
1 •' •.
I••! 1 I,!
l]
I1
l
!,1
I
1
11 1 1 1:~~~
·:!:1·:-:1·:::
::." "'!'
--i;·
,, ['I
"'
I
"''I"",
1
·'I
"
,,
I , .,tl ,,, jl•l'
,.l.j
,,,
''I
,
I'
..
"1·1
•.
,
"'
'I"
i.tl
11'
1
I,
ill!
'J!lljt•
''\'1'
,ft,,l,l,.
I'
]:!.
ol,l
IT'
/r
j,
i''
lill
1,111.1'
'
1
tl•l
,II'
I
'I'
j ''']
j\j'tl I
j!l! tmjj
jj
''1Hjtjjt \ jlll'!l
I
!r~:~}:~/-·:-,:..r~··l .. ~·. I:. 1 ,. ,lflft\1'
·;., ."1 ~~~~~. :~:· :·~ ';, :· ~:~' ,: ·: " .. 1 'r~r·;:·1~1~~f-r-rT1h ~tt!·~-t~~~ , ,· ;:..i +11 +-:1 1 Tthi :~'rr .!.+t
1
h"j
t-'·
()Q
r::
1-j
(D
.
N
!-'
I
V1
1
•
~,1_1_'
1
1
I
t 1
t-'·
I)Q
;:.1
Ill
!-'
t::l
(D
rt
(D
()
rt
Pl
cT
t-'·
!-'
1-·
rt
'-<
1-cJ
ro
t1
H-t
0
~
Ill
;:l
()
ro
"
1'1
•
r-.~~
"ll••t
• •
• " l •!
I •
,
·~-
t
I
•I'
,.
•
, •
I
jl
r • '•'
I,
t 1! •
L
)
'f\
t
It~ !:--?:·.~~-·~ :· 1 '1,4f,;,:r,,:i!lqrj;'t,·~~~~~
l. . ,,. I
1.. ,.
II
1
I
I
I
~~~
"
1
-<-t~
' I ' ' ' ,1
1;
.,..,..,.~,-·--,
• •
11
1' 1
I
II
• ·'
I
1•
,
I
1•
'I''
,,.
,,,
-<Jj ~11I'
t'
I
1
1•
•
I
I
•
•
~·
I
I
I
1
• •
..
I'
I
I
lj•
1
•
I•
'1'
I
II
Cll
Pl
.a
t-'
ro
Ul
!
I
rl-""1
H'
il•
I
t
r
1'1
1
•
n·
I
J
1'1 1
lit
I..
I
t
1
•
1
II
lr
]
1
1
I
1
I
t
t
'JII
'II
•
f
~~~llil·· ~~u,.l, ~~~~~-1 1't 1 ! 1·
Ill.
-1
I•
!1
jl
I
\I
r\ 1 ·::1~~~· ~~~~r~r~d·
~·.
, ... ,, ..... ,......
1• •
I
1
,,
1
1'
J!1
•
...........
•
...
,
• ,
•
1
t
j
II
lt-4,-,-
-
j'
1
r
'
1'
ul.i~·ll1J
lljJ
:·!1.11'
i!tl\!llt 11 1· iI 1I''\lirf
•\1; I 111::1
11!!I1• :,!
\t•! lj 1 III ' : ! 1 1:1 11;' !1·It Jlil I 11 t\11t i11:r
j,
lt•t 1 ' I
1
··I ,Jj!
''''1'"111\rl'ljl
ql, 1-1'1''11
·fl,l.i..II''III,.J.,,I
·\1'\llll!liflll II, Irt'l'II'''II \It' I I
I I I' I IWI"II I·!
i
I,..I'll'':
'I,: 'I'I Ill
I'II I II: II''',II,[,,
I I II:II' III·,.J,
\1 '1',"
II'' II.'!
tilt I;I Ii I [ I II I I'I I 'l'., I '•lj i 1'I1 Ill!'
.,,
I
r--·+1..-,··
··n
""1
+'"T' ··1'"'1""1 t•<\•nt ,...,. ·~·lr'"'j
--,lU-1~
+ 1- ,Lt.ll'1- Hft
rrti' IT' !liT
:·;''·I;:
I~""·
'I'll
'I
1
~-'
l'o-u!'li
\
:1
Iii
'1;1
:~::
I'''
ill'
''II'
:1:1
:
•.111'
ll[!fr
!:r!l'
IJIII
111 '111! i Ii • I : I''i'!I, i o-.1 ·:~I I!'
Iii Ilid!
I' I:1II I!j' i!lj iIIjl :j ljl !·lJ!1: 1 tlf !I II'
:t;\
~_; j ' r:,
, ''!
!: ;:, 71; ' l
j 1 ',
ltl; 11' I\;: 1rt; ' t!
r:\·I j i11JI 'Iiii!l !:1ft
ttl
1I Ir ljl1 IIf l ~ 1'!t1
1
I
I,,
;I: i ' ,,.
II ' ,,
I : l' :j,
I' !I! I ,,,.,
. I' 1[1111'
I: : I I' 1111.
11: 1! It:t.l ' ,,,:,; ,1t' 1,•f,;
I' I! 1
!I'1,I Iii!
:::' IIIII.. I!,, fj,
' tiII \ll•ol••<l
"!I lj':ll ,til ,,Iilli
lj
U!tl_
I.,I ':I", . ,I'1'I, ,I i I1
, t'',;,1.
l,[f.q,l,f,
,,,,,;,
iof\tl
.. 1 1,.
··~
··~·~"+-T ' ! " ' " " ·-·~l7," -....11 2~---1~ ..., ~+-.-1-f'" ~
nrr'11'1
-rh- l;'
""TI1r'11-i4T
•r·
I
--r, "'II'""'"!
I
!
""
" . II''
,,.[,,,I
"'
,,,,
'
'1'1
,,.[
'it
'I'
II"
'''11.11
I
•,
j'
HI
'I
-••
I•
1"1!\'i'
II
'I I'll .!,If II !lo
I
II
jj'
1\
Jlol
'I'
I
I
I lj'•l
~
1
''.
tl!: :' i
I:: l
It!;
I
1
~
·~· i:
i I: I l i
'I;
j
j
11 II ! l
!
... !. l""l"''
"[·I'"" ,I,' ";1,' I'"'Ji
.,Ill'''I z
.. !, 't·tl
g ,,i'h: 1'''111.,
'II''' ··IIIII'
11·';,, '•"tn I.,
I''.I I.:,·,,·;·;
I II;::·.
t-'1'111
i'l :1, 'II"''
,., ;t I'1\ 111'1
:,, I I .,.
!'II:!'' Ill' 'II \I 1'1: Ill
1 1;1: ::,"['., :,:1 ·.: ,i't'lli I
1
-~~:L!.H.
!-....
_,I.
:_.:
..
~;...!
:"
1<J1TL::I_,~:
~
l:L:
~-l:J_:
.U'
:n
·.;.. wll
t;-lj
r'l
.tr~_J!;t
tLW
i!_j_; ,!)~u. !~- '.J+ ~ill I ,, l _LJ jlU l JJ
I
l
1-l
~
!
I
I ,,,f G1
'I.
I,
I ,,I .. TT] .. , il
,,, ''I i! ' 'J,I, ,,,
II I ·1'1
" ii 1 Ji:1,, 1:[/;ll\'··1
'II n.l
]f
..·: IJit
I 1,11 I
I ,ITI ,!Ill
' II !IIt : I,~ I I !' I Ijl
'I';! ' ~
"i ', :, :, I
•'! ,:
'1
!
tl
ft
1
'I!
!1
,
i
•
,I!J
1
i
"•'It, . ..
!. :·1 ;·:1::'1
:1 r,\.;.
>fl
.. ,, :;It ·11 1 !lt!t::.l'l ~~1;11'I.,:1..i'!tjlj:·
111, :JJ'li"l i:!ll': 11 Ill' jq-:
fllljlj\1
1!11 illlljd\ I ll f.
~.,.
. I .
1·· . 1*."' ... ,
.. , •
[ ... ~~I 1
.,::!1
I I II I·jt!
11 I111
1[.,.~. I,,,, j l''rn'l.I'I I''
I ., 'II'
11..1 I 111I I, 'I, ~~~'I ., I'II'~~~
I I 1~~~~
' 1·'I' I I I I~~~~~~
I t''
I
I
"
'
1
-·--·-·•·11 ,,
"I···~----~----·~
- •.,.- ·-·- ·-~,---- ·-·• ~-1-1. "" .,.,J..j - - • "'1 '''1'14-• 111' .,, ... , 11-"j .jJ, ]r:1 i~
1
1L,_
r·l4 , •>1 1 l ,[ j,_
." '--' I
I
..
'"'
'
'
"
.
.
'
"
'I'
''
'
'
I'll
II
'I
I
I''
I'
I'
'
1
'
I'"
1
"II
'
'
I'
II
'
I
"
I
"
I'll
.
I' I'
'I'
I
11
1
1 'I jl'
" "I'' r · o I 1
•
,
•
,,,
''J'
"
I
'
'
'
'
'
'
'
'
'
"
'
'
"
'
'
'I
'''II'
i
lj'
1''
''
J
fl'l'l
'"'I
j'
I
'
I
'
'
I ::
'l
t
11 :
!
l ' 11 ! ! I j 1 d ' 1 : ' ' ~
i I !:I
~
t
!1
I I .
l
. ·I"
'I
.1 .
·jl.lll".. I, ,,,,t 'ljjfr
[rllllj,
.!, 1.. 11 1.11 lftl1 "'I 11!1
1111 I : H
! . I, . \
1t : II' ...
; . : !:....
Jl , I i , , ! 1 . : ll
·~~ . .. ,! :~,.1111.1
I i l' . I
; t , l ~ ! Il 1 I I'
il 'I ! Jjlll,.,l
, It
I' 111-11,
I! jI Lfflll!
t II
" 11, ·
·1
'
· .. '"" · "" '" · 1 ·· " ., · ·,1 II 1 1 •,1 ;lr .,,, 1< 11 , I ;It .. ,,, ... I,', ·' l1 1 1· 11 I· 1 1111 ,'•,' I 1 1 ,1' • 1
1
I ..
.. . . . . . . , .. ,l_, ____
lo ...
-r-rr· rLT .•
,1--1"'
' " I 1[1'!"" , ...
T'']''Tfr
1!1, _!,
•.•,.u
I,,.
:,
:. ,
••
:.:•,,:
• 1. ·:
...
·1·
1:'.
··~~~
.,,,
IJJI'I:·
,
1111:
,:.:1['
:
1 ::' 1 , .,1!:11'i',
1
·
,Jill
·
1
''111'
.
1!1.
mlj'
'lll\'l.jl1
1
.1I":,:1:I.,.
H
, •
,,,, ,,111
t
,
r
l
l
t
1
t
l
!1,
''II,
'I!
l't''lli
11
1
1t
I
r
1
1
,
I
1111
1 It j1 1 1
'
'
I
'''
1• 1 • " I ' I ' ' '
'
11 •• 1
I
It•!
11 • 1
lol
11 1 I ' •I ' 1
1
I!
I I 1 lj 1
1
1 I1' !
l"'tn .,.. . ; ·,,
' "I .. ;t,' ', .. ' :' ':\!/':it
Q,1 11' I .,.,
'I'' jl .. 1
''"I 'I '·j·ll·-11
'I .,,1,
"1111'1 .,,.ill·II1I ~~f,
IH,I jlI IIIll I"'
.l,l lj. II'
,.·:j.
i'l!l:
''i'l; I•'
I! I'III 11 I' j·
1
1 ' 1 :··''II
' ' ''M'
I I
., ',,': ,: ' "
"'"",,I
1'
'"
1' '' '\l.lll'jl!l'
''Lf'• 1 ' ,[1'
·Ill'
' 1 IIi'
'JI' II1'1I1 ';!!'Ill
I 1'' ' II II'11'1
'IU'•j!l1111,t1lll, l;fl
...--1
-- -··1·--- ~· ---- ---1 -·· ··---r·-.- ---- ·~"· 'i" .,j, ~· 11T' ·-~·,···
» h " ' "'.:.11•rT•r
·r -1,II'! T•.u
"""I' ~------·
I.
"' .,.1 ,,' ''!'''I'1
''" , ...11 1' "'Ill 1'!'
,,1 ''"I''"
til• ,,--~ '~n
I
11. ..,.., .. '"11''''!',,I'''
1 I'' I I '"1]
1'"
''I'" 'I'
I': 1 ,,I1 JI"
I l'I !,II1 ''II'
II111 I1 I1 I"!oI IJI I,,
jtltI i II
.::,
; J;P. ' !
':'
'r,.·,
. 1 :' j'l',: 1 '"'
,' )!,' ~~·
:• 1 1 "i'
' ; : ;1: i,r
;1: . .
!! ' J'l: I1 ' 1~1' !!!' 'IiI"
Ill I I 1 ;ii I
1 1. ,!I 1
.... 1,... ·H
"" .,,\,.,,11"11'111
I I. 'I'I lt.\J
'''illl,r
, ;
H ,·!I .
;"'.. ··: ::· ~~·~~ 1 :·t '"'I"'
.. :· .: .·;·1.:,·"\.1,
.:.; ~~~~
·1··1 ~~~· .;· I I WJ,Il.l
,. l,ljl·l:
.:1 ....., Jl'
·m' !''1,
·:; ~ ..· 1 1~ !I'IIII.IJir
11 1111 ,11.111'
'111 11 11\l ''" r,lll
~
1 , 1Ill',,_1 1· I+,
~~--: 1..... t::l.~!_,.., __ j ___ --~l:~:.:. ~u~-)~. -~~~ J;.;.1u ~2 !~L ;~.;.. :lll.Ull'·'· if hf-!1 +-l.!LL, 'IIi 11 1 ~+~ 1 11 :l:~ UJJ" 1 t!rt r1 1 .•. , J,J~ 111_1_~~ :~ \,j·
:1 !.. jt.
·
.... [I ".. ':IL:::
11)'11
:! 111 1ll 1 It'I 11' 1:111]1
I jill,,
Hitjt'
1
1 Kr1 1rt
' : I ' ':
j 1...
l
' '.L
l ; 'I:,.
' : I : T! ' ; 1,:: I IIJi
l 1: 11
' ' ;11 !::11
l
1\ : 111 '!I'll'!'
1' 1
I li'l
. TIT1
!: ~ I'!'!11
1 I 11
: i 1\''
j \ ! 1:I 1
I .
!
I
1-llr·
, ·1,
....
\ ·
.I., .··,
· 1· · --~-- ...
,.. ,, ....
- .. .... 1....
1 ·· .. ,.,. -~ ·"lj'l'l .,11· 1
.II .., 1;,,
~~11·1111
\li'\11,
,, 1I.1.I jll'iiJIII
,.II "Ill!'
1-.[ 1II',l,Ill,lflH·I
II·~~~~
;::; ;
:'II'!
'I''::!,:·
.:!
,•:·
:!.
I·
It::',,,,,
;,1
I'
''
i'lil'j'·
Ill'''
I
I
..
Ill/
,,,
,.,
'I'll
1111
I~
1
''" I
I
'
I"' I 'Ill, I, •.. 1 ' ' I ' ' ''I' I ' I
ill' [It![ [. 'I' r I I 1 fj
I I, ' '[II '',' 'I jlj 'II I 'I JJI I I I\ ol" 'I', II' I j II I
~-+·•-. . ,. . . .,~ --~~-.--~.-·-·r---- ~ ---· "'"' ..... rr l~l r'h" ._.. 'r··
II~! Tl.
"'~,
rrl ~''[ r·r l,t'l
nn ··'Tl'·l'l.. 'r' -~\1
I'[
;·'rt
::t:l
· · -.. ,._,_
t ,· ----r""l""j-: :1' ," .. '""I'''.'
, ., __
t·::l":
·".·: o~:·
·.:.· ':: ···. ,1 ::11'•ftl··1:I''l''l'l
.It'! ,.I' iI Ti'''
1·1I
,,,q ...... , ., ''I''
I I ,J'I II"
"I I'"1: I:1 ,,:,
1'1 "',1'1 I"
jl' 1
,.,.11 1':I• 'I'll
I I· 1I II''l 11'II
II
','1' . I
1,, I I 'I
I ; ' .1I . ' '
' II.
'
' I :. ' ' ; I :;: I I ' :: ' " ' ' : ~ II ' ! I I I • I" : I ' : I ' I : ' I I \I I :1: !I III I I I II I ' I:' l : II II I i I ' ' II 'I I; i ' I
lj
'
I '! :I ' ' p, I 'I' I I'll I I ' I
! :
)
1~1 ~· :.!.!
:![1,
!l!i :~:l,::), ::! !It, r:! 11· \i!r II!) :tit]!~ lilt'!
1
I
!t. :\!I lil, \ 1 li\1 lll! !1\! ,Ill tll!! \
-~-1----t·-,._~ ~ ... -· ---~ ---·---"'1-~
-..-,---- ----·-T -...., ·-~h-""1-" ']'TT1 . . ,, T'1"' f." "111-+h r "' ., •t ll ~·4ol- ...,., m·
--...,.,T r• ~·rr;- p' I "I
r·--r;=
-~-·I +
1
i ''' : i
' i ' . ' · ' ': i; I! 'I : ·,
! I;': ",!!;; i; I; i' iIi i: I' IIi i !I'!'! i i:! i I!: I! II \til i! I 1\' I ! \·I iII i\1 1111'' ii :l \IIi IiI 'iIi iII I I' I: iIi 1f I ij' j, II' I II
.,
I
'
'
I
,,
"'
"
.
I
f,
'
"'I'
''"
,,
ljl-,
I
I.,,,'
I
'II
I•
It
I"
1
11
',,[,I
II'!
.1.
IJ''
1'1'
··!·
'''\"''
'I
Il'I1 1\I1I
'I"''
"· , 1. · .I I 'I'1 • , •" " •''! ' I ' . • ' \ .1 : 1I: 1'"I""
I'! . I'! 1 \ 1 ~ ' 'I'
"1 I ; l 1 i 1I~ 1'II 1 I I''1'1. I' 1r 1 II'\
1
1 ,f l I 1 I I'"
:', .., l'
· i'
:
:
1 · ! I 1"[II
;
jl 1! , : 1 1 l , I 11
• li j
; 1 ! : 1 1''! ! 1' 1 • 1: I 11'"; 1 ,I1•
f!. .lit·,
i :
1 ,.1 1
11'1": 1 1I '"I
1I I 'I I•'• lilt
I
I
.
"I' ' "I
I
'I
,
..
'•'
'
'I
I''
I
'I
I
I
•'
'
11
11
,,,I
'I'
"I'
·II'
I
I
lol .,.1+
[,1,
-,.... ,,
----··-- ·-·t--·-· ,..... -..... ---·>·----~-,- •. , ..... , .. ~r ,.. l. ·-T ---·
.JI,
..,.. ,.;.j.,T"\
"I ""11\:. ...'II ,...~
-~· -~- -t·r
11 , ;:lt
1
',
' 'V1
'
·1'1
: :.1 '"'•, ,'j'l
,:: IIliJ· ,I1' 11 :•r,l;l:t'
IIJIIII' 11'1 i'' ·1
' j'irJii
'I' I !11
I I1 Iii' 'til1'1• II' 1.
111"1 Ioil11 ·;,,
I
"" ·...
' I!, ' ·.;·I
,, 1''"'
'''1"1'
1'1 !111I li'l
'•1! I II,!
' 1'' ;'Ill I, I ·,ol
't •11 1
111 1 :,:.
' 1'
· I ·.:·
.[I · j " l " · 111 -1 1 ·I " : ·' ·l 1:"·
I'"'''' 'ltl!'"l .. 1 11 · 1·· tl"l I',
,; .. lll...1 .J.,
1 \ .1. 1 I'' !.. ,!1'. ''" "·II·· . 11 ...
1
:: I ! l;!p I II !'ll llll 1 ll lll::llll
ljt\llji
:!:1 ...
1!1'111 11I!'··,I I'.111
!til ' 11I : ' : , IJI j:;· ::'1 1: l !Iii j:l'jl'["
'II'Ij
'
!
I
111
II!
L'
I'
I 11 I I•
,, [\tl
"'! 'I 1'1
1 I'
I
'
I•
!>
I
.... !.... ,.·--r-- ·-"·,-- --~---- ····'--·'l'l''j'··--~"
J.,
.., ··~~~ Ili '111'1'1'
,!1·1-'
...I',f.'ITll"jT'l
.l..r
1
II' II'I II' ...
'I 1 I''t I'I'1'''1
'I'1•1..1'I·"I' 'I II II"\'I
'I
"'IJ' I ''I ',,.,I
''I' " ]''111''1
I 1'11 ,,'I I,,·,
I
'I''
II
I "I' I',,
I.
' I ' ' 'I' "' I "' 'I
11
! ''I' I1I'
I
. :. ----·.k.:.. .:.t:J: .... '! ''~: :r:. '·-~·~14[ !1.: ,.1: ::!; l:i .,f ,:l!lt,d·~tll:litlilll d!i,:I.\,H il~l,!lllilllt ti,Jtli~! :11J 1\1 \1· 1
"I
l ;:'
•I
,.
•I!•'·
I
I
I"
'
I
I,
' "
': :
I''
I
I
''';'I':
... "
I>
'
,,
,ljl
1)1
I;'; I
'•
•I
Ill
I
I
'
IJ.
I
I'
"
'' ,
1
:
••
1
I
I
t
1
•
,.1
t
''
•
1
I
I '
,
•
"
'
!
1
\'
It
lltt
II
•
,.
111
1
II•J
I
I!
t
:.
I
1•1
'I
'
''
1
I'
11''1
I'
I
I!
'
' '
''It'll
'••
I
1
II
•I
I';
'
I
I '
1
, ,•.' I
,
1
11
1•
I
I
'
\
'
,
1
'
'
[
'
'
1
'
I
t '
'
I '
1I
j
~ t •
1
•
I '
1
1
'' 1
I
II
II
I
:
I'
"
I •
1
, 1
1I I I
1
1
1
I
'
1
1I
1
::
1
'
1
11I
••
1 11
1'
I
I:;;
•
I
' I
1
1
11:1
t l111
1
:'
''
11
1
I
' I : :
I
, : I I
I : I
I
,
•
•
I t
I t
I
..
t :
I ;
I : I I
•I
""
I • I
I
• , : ,
• : I
ol
t
1
''
; ' '
I
... ,,,
•I
1,
'
•
I '
: : I
: : ' :
I
1'
1
1 1'
••
•••
'
: "
:
]
1
t l 'I t
I
''
I '
'
1
1
\
,1
•
'
11
I
1
tl'oj
11
I
I'
I
I
I
I I '
'
:
I
'''
I
I
I
II
'
:,
I
1'
1ol
'
<D
1
•
"
'I
'
'•
·~~
I
''
'1
11
1
I
I
:
I
1
''
'
1
•
I
1
1
1
'''
''
'•I'
'
I ' I '
I
II
I •
'''
I
!1•
T
'
11 11
1
'
I
I
ill
1
I
••
'
'I'
I
I
I
I
II'
1
t
I!
J
I
1
ol
'II
• I
1
,II;~'
'
I
1
11
1'
I
I
II
I
t
I :
I
L.....
I
j
1,
t
(
1
I
I(
1
'
It
I
J
''I
',,,
•
··rr
I
I
1
I :
!ii]Dllt @I'
I I I
,
11
I
1
I
,,
••1
1
I
11·1
I
\
I
1'
'I
I1
I
1 I
I •
•to
I
I
I
I
til
I
1
:
I
I
• I
,I.,, .
:
'I
I
l
It
1111
1
11
I
1
I
>'1
I
>I
1l
I
•
11
: 11 ~~~~.~
I'!
I
1
1
1
1
I
•
Ill'
r'"·
I '
I
I
'
•
II I I
I
I
j
II
jll
I•
I
I
I
1
I
I
II
1
•I"
I
II
I
II
>II
1
I
I
jl
I
1
I>\
1
II
'I
!1
II!
·--·)r·" __ .,,, . .,.J,.Irl-. ·---·
1
I,
1
I
I
'I''
I
II•
1
I•
\
,
'
•
•
jl
1
t
~
1
1
\
/11
ill
I
1
I
j
I rr·" i'Itt
l
lln'
1'
I:'
1
!I
•
I
I
1
I
1. 'lr:
I
I
1
1
I
1
'
1
,n
ol
I
1
'
I
1
t
I
1
II • I I
'I
,,;>(I
1't
1
I
"I~~~~~-!~\r
1
•
I
I
'
I
1
I
I
I
I
1
1
'
r
It
'
I
1
11
1
;
~· ~Tf ~~r·l- ~,e-•lrr
I
'
1
II
mi ,.,.
•
I
I
I
tlillj 'l'l]
'I ... l,l,\1.~' 1'~1 ~ll.Jl,,
I
,J,
I
11
I
I ,
Ill
I
' •
•I
I
•
I
I
\1'
I
•
1
I
11'
I
Ill'I
II
1
1
I
1
1
I
It
,,
II'
I
I
1
I
:
I
lot
•
I
1
, ...
: ,
''
b; ·.:
I ;
111•1
1
I
1
I
I,
'to
I
u~.
' 1
I
L:'r
:
11
1
I
,t,.o
11
'''
;If
'II
I
I
I
•
HTi
'
1
1
I
1
I',
1
•
1
.Jim'l'l·
11
1
I' . "",_,. _,. ,
,--- ~ .. ,...... ,., .. ···-·-····--m:·
. . l... ~ .[...
f-S .. ,
·f: ' · : ·: ··
·:~:~:::1
~9
'·I .,..
~"'
--~')..
[r . --.. l. -r:·-~·1----···" -·· _ ,_. .,
' "'!'" .'I'"''
I"'
"i""l···ll"" j· .,,
.1':
1
I!':\' '"
T
0'
'
~
r
I
1
!-'
1
.,rli~-~Ul~),,l.
,
1
i
I
':11.':1 ·I :t.t ·!· :· 1 1• .1 .. · J.·: 11; .. : : 1!rj1d11. ' i'll,
li ;,l!i:!l:
' I
.,.,!,.,
'i'
,,
I,.
"l''ill"'lll.llt·l•l··l
!"II;
I'' I
It':;
oil,,. III ,.,,
:1' I'',,
I' i I I''''[11,1
I': I I
" ''
1,.1. ll""
···-·
o•··l--oL•I'1 ..1 -. -·--··
..,.1...I ,.!._.,.,.,...
p~
... ,..,,-.,! --~·,··r•<::'•:;;,
J. · ·.·:
·J :1:1·, :·, 11 ,. 1 .1 ':1':: ,:'1 '1,:
:! I! I I!: I-:.< '
1: '' 'l l I I! ;::I I I :'II ,[I: ;I!' I! I I:
•1
(/)
r •
<
I
1
I,
1
::; I
:I I I
;
I:'
II
I:
'I
I
II
'
'
1
'
1
I•
II
I'
'
·---·1·--.--1!'
I
I'
11
"
1
>
'.:
I
'
,J1"
I
'
I•
1
•
II
1
'
''
"
•I
I
'
'I
'
,1
'
I
•
•
'
'
tl'
'
II'
I'
•I
1 11
I
''
,'
1
I
r,
f
' '1
ol
I
'I
l'' ,,, I"'',,
II
I
'''
II
''
Il
'l
'I
)II
'I!
1
II
1
1l
1
1
1
1
I I
I
"
1
I
•
t
I
' I 'I
"I
I
J''l' ..
1<1!
Ill
I
jll
I
I
I'!'
'
"
I"
I
'Ill
l i l t !I
1
I
I>''''
•
I
I
I
1
1
I
•I'
I
t
I
1l
·r' .. ,., . . .
II
I
1
I
I'
"II
1
I
11'
1"
I ' II
!
•I
t)ll
I'll
'I'
tl
I
1
•
I
I
'
1
''!I'
t
I
)
I
'
I
I
'I
1
.,
l
lo
I
I
I•J
1
It
It
1
I I
1111
It
t
'''
•I
I
I
I
I
I"
•'
)'
)
,.1,---· ....
1
I
1
I
1
I]
•
1•,
1
•
Lh.,, ...I 11'
i'
I
1
I
I
w
w
t-rj
>--'·
r:JQ
c
ti
(i)
N
"
I--'
I
a-C/)
t-1
r:JQ.
H
r-1
"',. .
1-'
II' I r;H·!:r.
.r;' I:.·:·
:::1 ;:.i ,;. r;r
r:r:,:n111!':!!:
il,: I il~!!:,1·
lj I!ij':'::
1'1 ·: •,! i,:·:
:''•1:·:.
·::-:IT:
1.. :,
..I':.
,,.,
I
"'
>V
r1'
(tl
n
r1'
PJ
o"
~·
~~--'·-" '
n
'<
1V
o
..,t-j
0
.
Pl
H
,..,
(1
fll
-~>-
0'
Ul
\))
'""'sv
t-'
(D
(/)
,J 1 !1~
1
1'+•• ~r'
H l~t
II! i iII,! .I, II T'"
I 'I: I , II',[ ; [1
II ! I!J'I
i I II
'tl' 1-lI i,!lw-l
I': I tj-.-t IIH-.I I •'j
jl ,:I '
:,;i',
'l" 1!: ,: 1 :>11 :!:: •.;: .,,JI
It:l ',1 111
11 II[ · 1 , .,q'l 1: jttl1il l'i I.I 1::::
illf
••. !
!l•
,
,.
f
-r, ··',
.. ,.,,II
·'It'·'.,,
... ,. j ..'II,.f .i·l' ',,,8
..
r,
.,,
I .t,l.,
..I I ...1-, I,I I.
1.1,
..
itJ,,.rl
I,"
I'
,
I '
'. IIJ I' , I I I, I , 1
,
I',
'
I
.
,
II
,
I I'1
I1
I
I
I
.
'
I
I
I,,
I
I
I
I
I
I
I
II
1
I
,
"
I'
I
I
',
,
I,
1'II
1
1
1
'
'
., ... " '
''
!'1'-~.---·f·r:111'' ''''''I'
I' .,,.,~' '1'" "'
'"'"i,,I .,_,.
!'11' 1''
'I !1 II I' 1 I ' I I ' 1 'I 1 ' I I ,1_41
r
-' -~··.-----.--r---··-~~
,..,.,-.J--,-,- I·' ··w-·
t•r• ,,: • • • J •w- •• , ... 1 lw•• jl -t.. ·j•- -~1 ., •
+t·'!I rj-1 w..,' It·r·--~I ·h-jI' 'I 1
1
1,I ' :'• .:;
I:. f·r,1: 'l"!iI l I ,I. I,i:IT I, I I I, ',Ii I' i I , I• 1Ii 1I,, lj I
11I ',1I 1ill
r
ilt ,111 : n ' 1f
'I
, ' 'I'' 1
,.1 :.·'!::.
''j ,j·:;
, I 1" ' .,.,,"I. T'"!.'-'J•',
, ' ., 'Jl 1'I I j'[
,I 11·IIr1''llj ":I
I , , .. ' 1 ... ,, ' ,...
H-1'""'
1t', I1 I lj'II ll,l
I I "
f"'Nl''l'
"'P.1' 1 'I'~~
'II
i' 1[I I r, I r··
I
'1,1 l .•lt'IJ'
, I I, I I'I1I, 'I''
'
1::· :1:'
I
li' 1 I'!,:· l!l l ; -ill
il
:!\:1.:! ', ,J;
l)i
l
·~.
1
1···1
l'
I
... 1 .. 1 ,1,,- 1.. , . 1.. ,..
J.,,_ .q.i, .[.,1,1,. ,11 11.. ,. III,JII,I ,. 1 .lj,Ji:.[l,,tl .,ri:·'·· ,,l~~~~·l 11 :1111 ,,, 1 11 ![,,, 1 ,,1 11. ,L d~-1 11~ 1., 111 ! ll l l r ~~u
·:~r~-;.:-,:. . ;:-r·-.~~ ~·:~~~~~.I:· +;I::.·~·
.. · .. : .· ,:· .:: .~~ ·::·~:~.~It·: l+t--r.-~~,·1-hrd+tj ffi''1 _.tt 1~ . ~ ·n~ wu-t· ~-~+ ,t~hl+: 1~LL41-i-,#~,
1
. ! I :. I ' :'I ; .I ;,,; Il : I'. I:: I:! i! : ; I, •'I
,II'',,,I I,jlj,: I:,l'I! 1I,II' :I II I• I:'' I: I~ I . I: [I I!!:
i: Ii :ill II I ·I I, I:I, :1 I! II'•l I I! II d II 1 lj1 It[! I IIIIt tj !I ljll, I I Ill
1• tlj 't 1 Ill iif: Ii1''1q
11
1• ::I, Itlt!
1
1....
l-..
,1·1-..
....
..."'1 ,..'I .,, .• . .,.I I ",+1,I '' I' ..II11 1,.,.
:
, .. I' ~ ltnlttl
,.,1J1J,
Ij1t
r1J,, I' j"ll
..
rltl
11-I 1'1'
, , 1• "I'
'
,
'"
',
'
..
,
''
,
1"1
,
,
'
I
I ' I I,
,
I'
I
.
I
'
jl
jl
I
I
I
I
1I
II
I
,
,
I
'I
-1,:1 11 ·::;h,: ;, .::; 11 ·1 ',' 1· 1 'I:,;! iJI. lti\lil! : '11'·1, 1111 1 , 'Iii' ' I, 1l O'>J I' 1 tii~ l11 t1ll:lf ,:1 til .lj I ·1 I ·IIU 11\ 1 1j', 1' II·1 11 lll:
~,, . :-:~.,I~;,;;..')-r:~: :~;
~-7
'-r:-1
-+_]._
. lj -~,, +",IIrh t--I ~ :, mr'"
llj_jlT'
~~~~ 1ll1 y lrl,
li-:
H1I
T 1,,I " ~H'
,..:--~1,I •. :-;., , 1 ~, . I .. , ,
. , • 1~...;'
. +;~
I 1~,''!I
..:'I m~
I , 11 ,~,, 1
1
1 ,.!+t, l-i-J-i
, 11-:~ :-r,t;-;~
~ .. ,~ 1,!
111
, jtil,I :,lr,, 1
,
1:~~ !T~
,,
11,
, ,.
1
1
1
I•
!!•'·
I!!
!1I 1 11 11 '! 1 ' 1 111 • 1 I 1~ 11
til
1
11 1 1
11
1 1
1
11 11
1
1
1 1t
1
1
1
1:.'.·!::_,
·: i iili,, ·'tiiitti'
1ltljil!i
·rI'1 tl !:i·lffil
r1 irl'
lillJ:l'
l:tTI .), In !I H
,t:l
..rttl~r·
,11~~·1 ,,.,_,,[,?.i-~11
lltll
It,.,,.
~~~r 1,lli
r,
~~~,• Ilt.[_J_
1
,·v I.,.
,
, ~~~.
, . •. ' ,j ... ,l,j
•
,, It
1
• ~~~I ilL j,,
......-rt r:_... ' ·~~~~ ~~~~~~~~~'I
11
'
,
•
11 II
·•
,.
•
1
~--·r·..
'r-~1'""
·-··------1·---·
""'1'···~--r~-·-·
-,-.-1--H"~
'
,.,,,,,,.,."•-.-""~·~,,.,f~1-T-~-.-..,.l,
-4
_,
_
.
,
_
-rtlrt-r
-r"'' ...,. , H'--' ' ' , . .." . ,
''II ' II', 'I',
"I' "I I I ,,'I I'I"I','1'1,
It"'
I IV'IIIi
,'I
'II,
'1: '''I'
I'I I'll
'Ill
!I
I lj1 I 'II'
I I
I
[I I .tti---rl i'l'
1 I1 I",.,
.
'
, , 'I
I
,,
.....
,_.._..
I
I'''
'I
I,
.
'
I
I
It
1 II
1
1
1
' ! r )' 1 ,- •
j •
i \( I 1 I
1
Jt ~.....
• ~'I
I
1
1 i tl il
I
, '
l
!rt 1
,.,,....
--I
....
-·.. ' 1 'Ij:, ' II11 ,~~,jl.
,!
.. ' i lrJ>;d.
rI"l!.. IJl ],I If ,ljl !t\ 1':!I1 Il''r·I .11.
ldr
". '
Cfl 'I
.
.
,
"
I"
IJ' I' I .1.1
[tj'
It'll
I' I ,~ ,
I
, I I .
'
I
'
I.
I
I
I
Ill
I f-t IJII.IIdl,l
t
I
::·,;:. ,·I·" . ,,, .
·I· I'
::; ... ,t :I ' ,I : It I I 'II :, :IIi II I ltti :, : t N I l'l'tl·l' ,til' I ,: , , I II IIi, . :•, It'll '~I tit'' I I I
I II II I,II I
1·
..
---:-c :.:-:- .....!..:-lr'J'+ :-r-.-+
r".,
..
[H:ru_j._.lt,lt
"' "J
, ' '
' .I
I I',, . '"'I I·' I1 I'I, It ,1[, I'II"
I I tl ,!I, I 'I I ]I' I II'
I,I · I I,\! I ,I,I ::'1I II II I I,.tl II'
'I II '1'1'111
, I IIII' I I
•. .'
Z
. 1
II I ,
tl
lj I'll
I,, II I!'!,
\I I'·
. , !
~ .' '
. ·:j;'i'
, , • • I I ':ltll'
!
I l ~!I I l l I l' . 11,,,'
•
1
1 ,1.
:. , ,: ·I> · , · · , . 1 , 1 -I ... :. · .· , 1i • · , ,
1
l
·
1:
11,
·
:1
:
·
1,
tll
c:J':
,
~
1,1,
1
1:
r
1,1
I
,
1:.
1
11
1·
,
:1
:!:I
1
·
1,
·:!I
,
:
lljl
11:1
1111
:
~1~1
I
1
1
1
.1
. 1 . I' 1
1 'I' X1 , '•!
1., ''
1,' i !I·'1 I 1 11'j. , 11
...[ lj·j,j 'r.,
11., 1,1- ,11 ,II•11 IllI1 Ittj 1It•"lrlI [,'1': 'I.I I II,I1 1II,1I'1 1, . III1 j Irll[Id1'[ 1
. 1
1 ,1.. ' I1,_..
.I''' I , I, !11Jljlj
1
·--··--....
r;a ;.LI.,.,..,f-''r
..•.
,·1,,,1 "f'-f'l"
·t-··j-r+
r···
...1_,_rft
,
: . 1.·, ,._, 1
, ....
. :, -- ,II____:,, :___· · :. ,: : ·-·"'':. 11, 1,_...,_
1: ., I...
:" 1, ·1 • ~~~ :, '~T11·--1,
1i
1'
:
'i;
1,
:
1
,
111
1
,
1
:
:
.
,
1
1
,
·
1
,
1
:
:
1
:
1
j
1
,
.
1
:
1
.
,
1
1
1
:
:
,
1
:
·
,
,
1
1 1! I I ! l : !! I 1 1!! ! II 1: j 1! I I! [ '1 ~~
I; :I I \ 0
I
I'
'!
! 1' : ; l
I ' : ll : I
d1! lll ~ 11 d
I l; I : l !1
tI
t I 11! 1 I I jl l,1I
1
1
1
1.
1
,
....
!.1
I
.
_t:'\
.
I""
"IX
"1''''
·1-l·-..
1
..
,,
,I,
....
'"lf'"t
I,
11
'''jl'li
l
..
,l.'·t'
it·l~:
11lli1·
.. - lr\· I' 'W,II. l'l''lfll
·
~- :
\:1
''! ,,,: •,;.r rili [', l't' '~· •::,
1, ''1 1!:1 :t 1''1 ij, lli'j',., ti r' ',"
11' 1 ·tf
·, 1 'r·~J, ! ~,tht
11 1
· 1!1'I ··I
II jl1 I j
l ·,.:j'
,, . :z
"
· 1,. ·-~·
· ;:,., !:""' ,;,il'l'
,, t' · 11id'J.'i' ''""t' ,, '"dl:j'l · 1'1' 11" tiU I'JI 'II .. , · ,, •I 1 tr·: 1 · t11, II l:t !111 ,
·, I11
1 t1
~:-:~,·. :o·~.
-~ .. ~. r~-:- ~.-.-!-:-H -h~-r~~~ tf:-+~: ~:~+t-r. l~~TT'~. 1: !b: :~+: .lr., tt~
i-f-t~r-Hi: i.. TJ r, t-t-r+i ::t -r~+1 ~--- ..-rr. Lr-~; ~tir[~+1 q-r-!.. ·~r H+~r-.,-n
. , ' , ,1 •H ,
' ,
' Iz
,
.
,
I
I
'
,
r , II , I. I, 1I t'' . ft Il 'I t, ,, II', ' , jill " ,II' r I ' II II ,, ,!, I I,tl I II I jl: ,I I dI ljll II1 '1 ill ' I' I .1 I ' iljl ' I)! f:.1 I I 1I II I
• : ,J
;,': :
l
,
·
I
l
t:l ,1: 'J,, il: t!
.\,,:.
'tit 1111 11:1 i1 1! :r 1
l i;
I 11 lll
ill 111
l :11 ltll
,1 t.
It! 1 1
:·'r
1
1
::::!
111: 1,'1'1 111 ' !,11jjli:
'II! riiiJI:·
,, ,. I '
' I' 1,:
I,
"! .. ·:::l'::x:::li1:t
I, I, I"!, ...!tl:l:lti
,.II, 11Ji!:.
II I tl•, ·::'!'Pi
,1,,1 t-1' I'll !jlI '1' 11,1t:jl',I
ltlf 'lj,l,! li!li:ll
It I1!11111
, I Ill
I ill' I Itilr-. In')'"!'" ..
·--+·
'd
.. :•lj-.1
. , "'t-1 t·
d ...
'1i' It -'·· -·· .,_,. _,_ .... II'I ,,,[ rl -'-lrr
'II .. ,_,.. _ '1" i rl·rt-tli.Jt
1
I 'I,. ~ I .
I """' ,I . ..:I I''I ; :' ,; I I I',1 If,ltI I'rJ:I ,: lj",
'I I' I'['l· 1'1 1II'tl 1I Illr · Ill 11"Ii, ,I', j I j I j; i!' ['11'11 ,.! ,I I Ill I 1
I'11 • I I, 1Ill '!'
' "1 1 t,! t[il
'I·jl
"':
'• 1 IIIr i:
1 !
,
,,
.!r
••I ,,r "'I~
•
•r~''
I •rl
I '
•
I '
1 1
I " ' ' II•
· , ' ,
I I 1
I, I I1 1 . 11 1.1
I 1
f
I ..!,11
[
....
, .....•!-] I, .
. 1 .
f.\
...
1
,
..
,
..
I,!.
"
I , , ..
.tl1.:.rt
't'
~·~
1··
,.1
, , ) . , ' lr
ll·'j:-~~
,1!
11:1,
I.L
,
.., 1 1
1
1
1
1
11
•••
'H
,
,
.'
..
:i\~J":
·I:
·::·
1.,1
::.,
1:1·
:
1,111
I
.,
:1
1
,
,
:\.
,
:
11
:
1.\',1'
I::
'·
~~
,
~
:1
~
:1
I'
•
1111
I
~:~,
1~[
'11,
,
:
"· 1. I .j,
., . 1. ,.1
,. I". 1':1 1 '1'1 .. ,.. , 1I , I 1 1'• ,. 1, 1,1I" 1' 1, I I1 ,111 '1 11 I,,' I 1' I,11 1 1 1. 11 I'1 'I til 11'1 .,It! I I , 1 II I 1 .. ~u~
I
.
..
"
-o
......
..I ~- ~----~ .. -~ :;:;1 --n-.. ~·- ~--~ ~ ·- --- ~ ..,_t·-,., ,r.:t . ··- ,~. .
~- 1,11nI I!"I
...,--1-H"--'
-W. u.
-.-~I! I -rnII 1''1
1I ,'I I I',I _,_,_,_11
..
I , ,.
. . ,".':'I
. "' 'I ir-.- '::I
"'I"''
'l'jI ,.,,
I'I "I:':
I' i.,
' , ·'·
~' 1'1'
. 'I 'II :. , Ill'I I'"I
I'"
'!'' ..1"1
'III
jllI 14' -1 r.~I +I"I!
I '
'I
'II
!
i"
I
I,
'I
I
i
.::::-'I'
I';
II.
i;
!1
'I
I'
I
tl
'I
I
•
'..
, . . . . , ,lj, .1: ,,1. 1-1:.,1. z' ' . , '
I''
I
,. ,J,I
"'I I.,!. , j,li
I I•
'II,.,
I I
•
1
I
II'I I'
i':·[iil'
!. :j )[ii1!
. .rl
1[1ill''jl. I', I ',,I 1..,1·11111•1'
!1;1 ''''IlL:
:!It 1111'•
11 1
11 1, ",j rll;
1
, . 110 ' ' ... I,. II ,'1' ..;:.b:
''I I,~ i ' .::.111!1
'' ~, '
, I, .,, , I' II I"' ,,II Ill, I, I {I!
:
l i i I !L !
tl
!
I
I, ! ' I1llijl
I ~ IItl ,I,I,I I l II Ijlit~
:
1,1,I 1 I'I I'I" I, ,! I 1I
i It II , w
1
~·~·
---,tJJ- ---.... - ----·-- ....
,-~
_,.,-.-,-L -+~·r--,- r,.,.,, .. ·rr· ·~· ..w.ffi~-- ·-·-- .. ''If . - ... .tnj ''lTir·t ~1- -,_r,' r,1'•.1 [~",_It
ll·r
I ~
' '
I
"
I'
I.
''I
,
.
"
'I''
I
Ill
I,'
"I
I'
I
'I
I
,
'II'
I
''I,
,
I
,
II
I1 I,II I I 'I'I
II
I,
I,
I
I
'I ,.. I"
.
.
'•
.,
,
'
I
I
,,
.
I
1'
1
I
jl
II'
I
'
,
I
I I
I
i'
I'
I .
,,,
Iii
,,
'I I ' 'I II •, I'
''II
1'
,.,, , I
,
'
'
.
'
'
I
1
!'
(I)
I l' •' •' ·I
i: l ~I
I 1 ! j : i t
:I : i j ! •
i I
I i
.
.
.
. .
. I ..
. ,.,."''"I
''I'"
I.,,I• I!'i'ld ''!'"I'
·I. I' I.l I "tl 1"1i l'Ii l I 1,11 :!'•i I'IJ,,,
,I I ''ilI ,,.
,I,.I I'
I I, !,, lI..I 1t 11I , I , I L
:,,[1_
'I
:,.·.I
:,J.:::
1:):,1;
•·'i'il,
I';:
It:.:
,II.:
.ill
ttl'
1:
.II'!
·1'
1
'
,::ljllr'l,lill:ttj'illl,i
!\Iii
·'ijlt:,,l:
II
I' • l
f~i: --:;'! --·I • - --·-r··7T
•! I; l 1 I i: • !'
~ .....
:! ' i • :I i 1' I ~I j) 1'1: ! ,
f
! l1 I' { i I
;
i II ;
.... , ---1 --1·....
,.,, · , __, .,,_.1,
, "II ltt-1f;'.
•··, ., ·, -j 1i j 1j !·1 ,,J
-.... rw ·
,_ u[t
....
:1JIT!I- 1
.I 1IJ'llj
1' I " ·,,_,,11[,,_,1
''!'"
"
I'I ill·
'"'1'1.
'"''''I'
...
'I"
I
I'
,,
I
''
I'
''!
'I'
'II'
!l,jl'
I
'
"
•
'
.. '
' ,.
'"
",, "
I'
'I,.
'
"I
'I'!
II'~"
"
,,
I
I'
I'"
I
I,
I
I
I
,
I'
I ' I 11) I[,I Ill , II I ' 1'I1
.
I
"
jl
,
I
I
! :
I.' :
i I r ! :: ! ! : ~
I I I : i I ;I ; l I I i i I ! ! II \ r 1 : ' I; l
I r
I ' 1
I I I 1! : I i I jl !' I I 1
1 I'
.,1I ,I ,
!. " ......
,,,
,
..
,.
1·1.
""
,,,,
,.1
"i'
·.1
'It·
,.
I
.I'
l,jl
,,.jltilj''ll
,(JI .I, ,,,,
1
1 ''Ill'" ,.,ljl·, •"
I :'
j I ; I i I;:
) :; 1 : I' i' I: 1
II I :I:' l 1 II I 1
:I' i: 11'l'!1 I:!"'Ill 1111•1
~I II II 'tl I '1 I1'I I.. , 'I III I·:
IiI II I I ' ' l I 11 I
1
,.' , I
.' ""
.,, 'I .,I,,.,,
II ·I' 11
I it'' ''Iii
1'I !,.,I. I I' I:I :! 'I,,,,
,I,
,.I •tl "I I I jl:l III It
' "I
"
'•I.(
' '
' I' 1"1
'
'
I
tl I 1•11
f·,'!.'
..,I... .,"I''',"Q "'.
•.
"T"" ...,.,-.-,-, _,_1_,.,_
u.
I r!- 1.. -W -+ [1,
'"'I
,J.
1.. 1, f·- ,J
"l"
rl
ri
+
·
',''
.L ll· I' '1"
1
1
•: r
-1.' I'
,'! ''I'II'1,'1;,:,
.!1' ' ' ! 1:·1
i'l: ill',
l'i',
I' 1 [ ''II ,[!,11Ill1 I'li',I j'i
II I'Ill'l [I'
'II; I111I ill
"!
!
I
II .Ill
1'1 •,III' ,;1: [[:ljll:
l
'I'1 'I ' 1 I II ,I
I 1I l'jl II Ill
' l'tllII ,[,, 1I'ij
1
1
' ' ·I·
I
I "
" I" ... , ,., " 1 ., , I "I I"
I"
I, 11 t 1.1, II,
'II 'II .,j: :·1 "I' ill-,,
II . !', ·til I,, I II ,j I 'I' 'I' I Jill
I ! ·I ,' .
I '
'' I
't' ' '
I + I ! ' i •I•~ I 1,'1'1
; : I"'I I "•jII'Ill
1
J• 1
lid IJI i I l1'1!1l I ltjl
11'I I·i1i l .,,'I I I I!I+ I;j :I I ! I l II,1 1,1II II "I .,,! II•II Ij I I1'~1I 111 II"li 1 I I, J,,II
• •
,,
•
•
'
I
It
--·-·-,......... J- i
--··1-H-'·
q ...... , .....
,,lltrl J_,_f, +--·r -r;-_,_.,1 ,_..1
,_,..,_]·, .• ,.-lr!·"-- '""
Ii · I 'i.. : · ....I ,' ·. I'['I ·, ,' · 1II,.:'r !, III:, I:I!:
i ' ! I I! ' If II Ii ! I I I I ~ i I Ii:I , II II' .i i ,1
' .II 1:' :[it'll
jl , i i I ! Ii I I: I ! , I lj j ,,, I f I i IJ I Ill III ' I j'
,i I .' i,
I
:11'! 1,1, I j 11j
jl~~lj ,1111I: i,~~~,I I jl-, 1
" 1l 1' 1Ir j,t-ill 1'1t1r, I 1' ,,1,It,i"tI t' 1I Irt t41
111 i11fIII, 11i,1 1!l11II:1!• J., ,!1I' I,Ill1)I~r•I
1·1,
~ '~• ..
• .. - 1~
-·- ·-'·••
..........
••
•l
r''
•
tit H
I
, •.•••
I
~
I
':,
1 ,,
:
,
11
t
••
\
....
ljt
.. ,
.....
I'
1
• ' I ',
1
,.
1 111
•
:~::~:':;
1
1
1
1
'
I
'''It
'
'
:
1
, I • •
•
•
1
,,
1
1 •
'I
; ;
•
11
I '
:
:.
••••
I
1 1 1 1'
''
1
"
:
"'
' 1
11 1
o
'I''
t·~ ,J.~~t' -~~~~~~~~!I~~~ Ji~,J-~.1~ ,J,r[l,,llt·fllil~' l!l'li.ll~lr''lli' U:i'llf~~~~lll
,1
'
1.,., .. ,..
1
t
]:j ·::
ltlt
I
•
I : :
•
r
I
-~1·
1
",
1
J
r
1
I
•
:
I
I'
,,
I'
•
I.
)o
1 1~1@·
•
t
,,
I
I
'
I
t
I''
I
j 1
1 1
'
1
'
til,
j
1
,I
I
I.,
j
•IJ
1
1
I
~~
1,
1~~~~:~ilt. 1~;~:1i lr1
1
•
I
I
!1•
1
tl
·'t·~'li
,,
j
I
1
''I
1 I
I
~!1 1 jll 1 i 1 1I,IIT'jl, 1 111,ijJ~~~~~I·ll~~
fl
t
••
1
~'j'i 1:l 1 ~tlfl~l~l w, m
ill!Jil'.il· i~Jr .l;iI
.t
I
I
I
I
I
I
tl
1
-~~·
~~II~ ·lii,.,IJ,~-\.,., ,r~,~ 11~~·t~lllllll'11
~~~~ ~ ~-~
h:ll_,_~_u
I ;,:"L~~ ··r·.lt:Li:-t1\--~-: ~'"':!1;~: '~TI +,LWt'~I' Lr'l..;jJ-~ 1'-'-Ill~ ~ - tr'LII"..l1,~:--c~~
..
_ .~- ·- ·
•~· _\_~'-I' .:ll. 1 +l·~""n\_1, •rdl·r[l -·~ji
]'t
11 1
11 1 f
1
• •
1
'II"
j
t
I
i i'li
1'
It
I
11•
m' r '
1• ,
1
I II'' .
1
)i\l,
-f~:. \!. l·
I,
,I
t-,1 1,~~L
i'il~i~·
- - ,~.1
:
lljl
••
1
r
•
1
I
~---~-- ,-...;,~--~~
1
1
''
,.1 ..
---!.::
!
•
I'
''
1
••
1
1
1 1
I'
'[Ill'
•
·-~:----~•--L-':
• ,
•-<f'IT
''
!,til,
.~,
---~~~~---1:=:::--j__:__
1
lt•l•
. ',,:
,.,,.~ ~
,
HII
,
1
1
'·':::· 1 ~)::!·
rl
,,.,.I
1
I:
·•·rt
~n.: :~m1 ..'·' ' ~' ' t~ ~,·TiltJ ~n1 I''~ ),;~ iIi ~ ~ ~ ~ 1 , 1: i!j 11 , 1 ,~ !1: ~ ~.:1f''''Ili' ~ ~ ·1 1 Jt~ ~1 1-~ 1 1 1ij'l r~ 'I fl~ ~ ~ !. )+·1 :1 ~ ~ 1
"I
....'l I,' ...
, , ' ..
F::
l l1j
'
j
' '
''
•
,_,_;
,,,
+r
']'
1
rAJ• '
1
,
1
1
H1"•
I
1
I I
•
'' I ' '
•
r
1
I
<I
1
I
r
1
I
li't
1
I I
11 ,
{
l
I I :
I
, 1
1
I
11
'
1
I
1
1
•I
I
,
I
I
'
'I
'
1
1
I
•
1
1
1
I
I
,
1
It
11
I
I
'
1
1
I
1
I
1
·A~~ : · ·
~ ~""''""'I~
~
.H1•
1-t:J
1 1
'',1 1 :,~.,,
,,.,.,,_.,
,_.. __ ,, •..-...,,j,,.--1-~-:·~··
'' I! .. : I'__ :,."._,.!.-·
'i' II ; ' :I,,, ,. :I .
I' ····t·r''~···:··"
' ,'·,: :::t!.i'j,'::::,:.
:.;11 ::P' URAL1LtT
;. li'; 1 ' 1.~tY.,
i'' 1'11' T1'
,! .•.•
--~~,,.L.
1 1
t:1
I' !I:!! tllli 't!1I1jll11,1',~:-··-Ill,,it' IIII'I ''I"'[H'I"''"-ll.,,_j,.
!Ilijlll'lll
11~1111!' Ii' !l[i· tl"['ll~!
I! 1''1 'tilIIlt!JIII!Iljl
1
:1 1111 ::,1[ljll'lli'l
11! 1' 'rlltl[jlllll41
111:, It:,,, :i ![I1'IIIII!
,1[ 'H':1I,
fl·jl 'l':1H1!i wlll[lt
:t 1 I'I I II::: '1 11!'
rr
• •
1
[~"-
-.!.:
•
I,_..
--
'
: •'
'
+
'
!
I
:
1,
11
: I
I1
'
I
I
•j
I:
•
•
I
1
1
,
I
I
•
I
'
I
I
:
-~-- ·T·j~-'I
•
•
';
:
1
I
' ' '
1!
1
l"
t---·
I
',
1
1
I :
I
,.
--"·
I
I
11
' I I
--1---. .- --·-I·-·~·
·I·,
I
I
'I
I :
:
' : I
I
""j
!I I
I J
t
II•'
I
t
t
1"
,.
I
:
1;
+
1
1
I
I
•
'
•
;
I. I:
I I'
t
•
I
lo
'
1
'
•
I I
' : I
'
•
•
1
I
1 11
"
11
I
•
·~
I
1
1
:
I
I )
1
1
;,~~ ,lllilJi~
I
I I I
I
I
~~It.!, .. ,_
I I
I
;
'
I
•
•
I
I :
•
I
1
tllo
•
'II,,,
I I I I'
'
l1
• I'!
,,
,
I
'
'I'
11
,,
1
I
I
I
I
I
'
1 t
1
I ' :
11
1•
I
t
1111
I'
1
1
I
:
I
I
'I '.
I ' : I
11
,j1,
1
I I
I I
I
,
·-"!'"' ,. .
1
'•
•
t
II
'•I
1
1
•
fr
1
1
11
I
.
t
rl,
·i,-1 -[,.,
I
, ~,II''
r
I
I
I
I
I
11 n'l n
1
ii"+
I I
r~- tli~
:
I
II
:'I
1
I
I
J
I
1
•
IL...
'
.~
I
:I
11 1 1
,, •
I
I
I
,
I '
1
1
1 1
11
1
1
l•1
I
11
1
•
.,
I
1
,
1
I
I
I
'
1
i
It\
It
11
'II
'
1
]1
I
I
"'
1
I
I
I
-~· I' II '
1
1\
~~~l~l,,fl, tl~j tJtl
'··!HI· L lt'l' ltl
I
I
lj
I
11
t'
I
'
I
j
1!
t
1 ~llll 1 r
'11ll\li1['1
,.,Ill ·nj
I i I, '
l, L
,til "'I I"
~ ~t-tl' 'I'~ ~·rt'JI
1
11
l
.Jl rt
I
T'-
'
,.,.1
1
•I!
I
'
I I
:
:
"lli,11
l
ilfl\ii'
•
1
1
11
I
I
IJI
, •
1
:
I
1
I
Tl
I
1I
,,,I
I
II
1
··-~l'" .I'
I
• •
'
'I
11•
•
'
ll;l~:ll
1
:,I,:,·~,
'
I :
11
I
11
I
I' I I
rl:~·- .-d~~~~
,;
1"
h
I I
1
"',IJI" ,, ..
I
\
I
,
11
I'
I' I
I
'"I
j
1
II
•
I
1
1
"'I
I
'
;
1
;,!'
m'I
J
':
I
,.
,:'t!J~
•
-x ,.
:
1
1!
I '
1
I•
,.,.
I
1
1
:
I
1
1
''
·ffi· .
I •'
1
--~--- , ".-.., 1~-''"I'
'
1
II
-~-~
,----!··~· .. ·--~,---- I---]----j--J.··-l·"'
J'""I-.. - ~- 1,.
'
I
I I
,
f--
1
,
I
I
1
'
'
I
1
..t
I
J.
' '".I
1
<f).·:--:
1
I
It•
,
' ;
•ill
1
1
j
11 1
1
t
:
1
w
.::-
35
Interpreting
these
figures
it
appears
that
the
linear
integrator signal detectability is only 2 dB better than with the RC
integrator.
This fact
used
is not startling and
in
combining
the
Fourier
is attributed
detector
type
real
outputs.
The next section explores this phenomenon.
and
to
the
imaginary
Next the program was executed 20 times with a signal on time of
1.25 times the time constant.
This was accomplished by setting p =
·1. 9725, integrating 17 samples with the signal turned on for the last
five samples.
F'or· the linear integrator 5 samples were integrated.
The results are shown in Table 2.1-1 in the Pd column
RC Integrator
pd
Linear
Integrator
pd
Equivalent
Signal Level
Difference
Bin 27
.20
.40
>+2 dB
Bin 32
.45
.45
0 dB
Bin 37
.65
.6
Table 2.1-1
-.25 dB
Signal Detection Comparison
Using Figure 2. 1. 4 to translate from Pd to dB's of signal level
difference change, we get the results shown in thEl right hand column
of Table 2. 1-1 .
Even with the fluctuation, the test results show a
marked improvement in the RC integra tor over the performance obtained
using a time constant which was 1 /5 of the signal on time.
Due to the
36
large fluctuation in the test results we cannot confirm the analysis
exactly,
but
the results are close enough to draw the conclusion
indicated by the analysis.
2.2
£valuation of Receiver Detector
Performa~
Equation 2.2 predicts that the standard deviation will decrease
at a rate of
..JK' due
to integration, resulting in a 3 dB improvement
in signal-to-noise ratio for each doubling of K.
The measurements on
the Fouri.er receiver realized only half of this gain.
confirmed by Reference 1 and is shown in Figure 2. 2-1.
This fact is
In fact, for a
lar-ge number of samples. the system gain improves only at 1 • 5 dB per
doubling of samples.
The only logical way this can occur is that the
signal-to-noise ratio in the Fourier receiver is being changed by the
detector.
This change approaches
2
(SIN) out - (SIN).~n
for weak signal-to-noise ratios.
larg~:
This explains the fact that for
samples the 4 dB integration gain in the comparison of linear
vs. RC integration was only a 2 dB system gain.
Figure 2. 2-2 is a
plot of' the input signal-to-noise ratio as a function of the number of
samples.
Using Figure 2.2-2 to obtain the number of samples (N) as a
function of (S/N) ratio and then entering Figure 2.2-1 at N we confirm
that the 1. 5 dB improvement occurs at SIN ratios of less than unity.
The analysis for an envelope detector is quite difficult.
This
arises in part because the cumulative probability distribution of the
'.
1
~,.J., ,,.__J.~t·-r•_J-_l_,.t:-1;~
,gf..r-t,
l
; ~ ~m71 j•iii~Jt'-:_~:·
t-r·o TTi"'J' rz•
"lilfj·~". ~---~ _ .:_ ···:nI!":~::r:··-~
L.
·--- 1....-f:-J-=-En=
.,.... ·· ·--- -r;-·jfiTE'
+f-+-,·lr ... ~
· .:. tt\
-~ 11
1-H......- ·I ~lh
~,_
:r ··· r;.-. . .,.,. · ·-·~·--·
--+tt;TJCTI~tttftr'".l.,-tt!
· r ·•·· ·-"· ··-· ,; . .f~~~_:. . ... -~:· -~~-r"rttrn~~·ri
c·i·h·-·+'t
r---...
--llf~·_
m·
·1
'h.~·~·
'
i
,,.J
.
.
.
:r;
..
y
:_--...
f
1
lj~,_
~~_-i·:;.
_,i!·:·,·--+··.
_
..
~
_
i
(
"
!H:
;tl.
:1;.
......
·~
r:
...
1~-1 •ry1 :!·· •r~ ·-·-- -· •:•· h· --J.
'-1+
11 r·:- •- -·j· ·- .J h: ... ..
++ +~ ; tW •rr• .... ... .. n;r -· ...... ·, jf·• 1;:· :r•· ... ·· • ::I· ........ ·• ..
·
iT +it-::.. -~~f q:: .....:: ~f · -+f f :T'¢:1..;..., . . ·• ~ ;~ ;. - · ~ ± H·1~-"\LH1tt:t ~ ri:V:: . ; ~ -- · [Jl' r!-i1 iiT ;. ... ·· - p:· ,:, ···
..
1+-r•·T
t-n+-+ .t71
. . . 1 ~~ ~--~·t···
·-·· -~-
ff.
·t··
•••· ---
t -
"T-·-
. -·-
...
'1_1_'
Hi ..... ,...
.......
. ·
m
1
iJ
·r·· ···· --
r-- ....
• · r • ...
t
:~: :: ~- . · : · ~ · .•· i~i :.is 'iii . ::- i~u ·;;; : : : .,.
fH! r:: :;~ .: · !iii H~ ; -~-: - - -1 t q+, ;}l :ii '~::. :- ~;: p;: :~:: ·
•~· ::::::::,
-~· ........ -- . . ~ ::!:.,............
1 .f.r -J~
--:P ..... ,... . . . . .... ttt~J- ilF
::-rl :n- .ri:j:.... - ·rr· .,,. ..... ---~! T,J ..• ,, ,,.1 ;:- ,....... ,., ··: .... ~
,+q;.+lF~~~-:::j:::: :::Hnt : ;: : ?: . .-: i=l i~ m:~~ :~: : : :~ ji~: ~" ·= .: : .. :ftt fW. tn: ;~: : ; ~ :: :;: ;:;1 ~~- :t trt ~tt; :t;· ~:: .;r: :: .: .. ~j: ¥:: :: .
n,- w- ....
..:fl_!:·ll . -J
":£ j, .... +(ID!1I "I,,,. J.... '" ·v lil t,, "' ,, . . ' j.:·· mrr
~~i~r+Jf:
:!;: ::;·:::· ;.•;p~ ±}~ _-l{lt_ rf: ;t~ ~ r~- }_-~- =P~~~ ~=· -~ -~ j +lrf tnn~ !~~r·.:·. ~J:t~;~ ~ ~t; f1; ~~~- ~~;:ff1
·r·;-t, lr;c
.... .... -f
]1t .J., . ,. J.......['1'- ......... -tftf
. .
,L,, . , ,.. ·
• +-+"" ~it!
,,.,
Ji~
lrr" ~
r:·· ., ..
·~-: ~: -~·
·+
:~ ·
+
t'N ., , , .,
fl~i .1· ~~·-;;._~·~~ct.,
~
t± L[l1 f! !,,, .....)..- .... r-H
t-tfr-_..,.~t;-;_·~;!r, ........... ·_j..~- ·_ill··· ,... ·r· H II' r~ [q; ., .. ,.•_......... !'r' ,;_~ ·-..... - !t+t- pt_ t ~ij: ~~
:r_· .::- ;, ..... -j~, !H·H;;r ni·t:;;J-- .. "~f":·
Tf:t:rtr~r~::;. :r:: :~: ::. : : :;; ·::. :: : :. -R+t ;-;:~: :p: 1:;:: ::: :: : : - :!·; ;r;: . ~: :. r:r..r . +I i .,·n:: !;if.,:. .. : It ;;; :;:: :~ : --~ ::r l t wn J];.; ·:: [~P ·. 'ir:; ;~ ::
t·r+ d·-- "'·!:--· · .......... ·"' ... ······ d-!r .. ,.., ............. - d· ,...C..··- +-j:_j: 1-!-J ,, · 4;::!~!· ..... · :!1·'" ·:· · · · I -dill::-;:.; ,:lr ....... rl- ....
fi:t-:1 :r;. ;::~::f::::t::- l- -~ :.. ~t - H-rp ~f,1 ::: ;:_ -::_t::- . S: ::. ... . I r:T TT'' ~~~t.: ......... !If, :;: :; .. . _:. , I • ,; : I• ,j[ ;::· ......;,. "'
I' ---.f:.. . u·..5· .... "·[·· ... Ltr
:!:.,...............--r-- :ii"""""'"
,_._.t. ' . . '"ll"';.<~!.ln~·!r.·+···--·1m-.;'8_:'1 '_r,. !:;... :..... 1;"
,...... :;f1.,...,
!tl; ilL+:·1 j'"''""""''l
- ::_,. It•
-q,l d
,,.
.. , .. ,....,.. ,.................
t·t1i rr•·:' "'
., ...
fl[''•",·:;;
. ·r· riH
11'''t:~l·.r
c:· ...........
f+'-:; ·H- ...,................... E;l ...............
·ro 1~.
w··:·r""l'" ..
. t•! : ., .. ,..~·--:· ... ,, •. ,...... ·-
·1·:--· ........ T.
;:t.
~-dH
--~
-+-~ -,.,.~- ........ -~-- ···-\-·· ··· · ·.,.. ··· --- ·- ··
fli~
··rm·~
"'~·1'
··· ··· ·· :r.· ,....... ··· · ·
-r
, , -. ..
I!' .., ·· ··· ... tr ···~-
'"Hfff
·
1 ttl· t7r• 1'''
~"
r··· . . . -
-r-•-- .... --·J-- ::y . . :l:j:; ::- !12' ~-,,i ·.. , '1' !.... +· r· ·:r ... - U.r;r' rrf· :rt· ·r.. ··[F:SvC.::P. -; r+stt.~±~~;c: 'l? .. :)J :~ :~:~:·
~j=;:: ~~ :~::k:1::::i~~ :EL:Sr;, :;~I~£ ~:::z: •:: ::r•:::1~- ;:::~:::::::t(tfr.d !I ;;~~:~:~)1r ~: §· 1TI..!' -mo:~~=4 · · . :.:~. :.
t-:'-- ,... ·--+ ::t: +rT•f. .......1 i'-- ...... v-+-h' h··v~~-"" -rr ........... -1 8. 'T ,,,. ,,........ .
l=t::: ~::: :~:r.::~l:~~n ~ ~-: .I j-;_;:U_t: :~:t::.:: ::::; ~1: :;:::::_v.~- ;W~k +: :::: tf ::·::: :. :r~,Jr;,: :::r ::::::::: :·: ::: ~1
-t--r---'~--=p--H!..t::tf:§.h 1· WL: ·· ptv ·
li'' '".,,.1
r,..+·t
'' :· ····· ····
{[TfS:f·--· .
1)1
i
1t 1
*'
''1
:-r· :,·
rnl!'r' '"· .., · ··
~'~ ~~ : ·:41 u-~~· :{!·-[T'elf~,;~
Ef):+
~ _& ·•· · 1-rrl-i:i·:J·~···,
hl; :_8:}: ::·,:· J~:-I~
-~-~[::I~ i~ . :· :.: ~~L~.
c-I: ..........
[J<PT~::..t...
1 ~ ; :rf~~~~~.- --: +~ . ~> ~/ ;._: ;: . -~~-~~~~:~ ~~~~ ;;;~\;_. :::·_/: '.: ..: ···r, ·:~~~-~ ~:~: :~::;~~ ~+~r
:.::: :::: :·l: -J 1·--:: · ~:d!~!:<~ 1- ·j t::rr·~:: .... .. . 1: . ~~t: ·::: -. · I
·'-"''--[[--.
J
..
·: •.•
f',:··-- .... ,._ .....
1
::,
I
I
1
1 ··
-:-:-:-· ·:::-·::-t-::·=J:· T ·::- ---'i!}H:±:;·-:~ ·-.~~, .. -r ·-:
["""' E--1 .. I t .. .
f·l .' .... r:.t.Jl 11..
'"l;i·:;.~-~-~r·- '_~::~::,::::~::::~t
....... I
·ri·f nm-........
-:-·~-- t·~-: ::-·- - . ., li77= ,:~- -.-rl-rt·
...
rn
·_ I'! 1
I.
.. ,
, ..
Il
I
I
•.
..
·I
·:.
·t.' ~_:
...
'1"'
...
Figure 2.2-1
· 'l
__
t··· ....
!- ~... . .
I , .I
1
~-;--;--:-::--
--1-:- --- - · ~+:- -;-,-: ·:·... t--T:r;l·-- ' r 1]'
t
":1'•.""1'
'+
': :·_ .· -~;:~:::::>1:::
,--~r ..
-;i-::-
I.
".
..
1
Tf'" "
::·:.::---
--r-
1
:·;::~.;::.·.:
••
:
III
.... ·
••
lUJER'}P·IIIH:f•f:D~!Hf!c MLLS.
"[''·::[ .......
..
r::;J:;:- -. - . --, : -:!f·
:-_:-:+-:-:-:-:
:-::T::· :: . -::-T..-Ut
f'"' ;:-- '"'!','"
' . ......
- :::-~~~ ~:~: ;_:::1:·::1·--~·
' " ' .....
ILjll
·: ,.,:
'"' .... ' I.....
' . . .
. ; . . . . . . . . . , . . . I I'....
t; 'I
.. j. ,, .. ::·
I
: ..... , . . . . . . . . . . '
..
..
Plot ot integration G.o!n vs. Fourier R~cet'-,rer Gain As
A Function of Number of SHmplea
w
-....j
1
i~:g~it \:n:;ii
1
;:::
::~ ·~· :L :::: ::: : -
.
rvr: W: ;;T: r:: ih. :: - ii!·l:::- :
fmi' r+ m: .. F~ -n+r
~-; ..
t~f
lili·-W ·"
·~; ltr·i::•·----------·f··"----· -- I~N\ittr•l!·:r:·•··· -.;;\....
f+.r tt:r····-----------·-- rl----- ··· m·'-<;_ ·t't r:t.r. ......... ;-tt--------
---.-1-t+}F 1~~ _i:f! ~r·· ·~- ...
---;·r
"." ft,, ..,........
l·T"'''':-''1'"''''"
....... ~-- ... ... .
---..
~----, ......----·--·
rn~ ~-- ...+ ,~ .. 1····· -·· ·1····~·· ··· ·:·-
It
t
t
r t'l?
-·,r··-•,••·..,..--01r.-.: ..1-liil--n'··
t ... !tt. l7 ·-· · '"1-iit·.,, ·
ttr,
''1T' ~·
ip.
tt.
- ....
-m
-
•'t
t
1!' ·-- - -1--1r
tH,·t•'\'":--··-~r':·-------···-····
\L ..... ;...... "-- 1., .......... --:.....
17"' - ..•• ··:r.·
:lr'\
:_r ... --· -
•p:::_ · ·. ::::;::_.:;_::,:._:· _ ,,
r
"I' ... ---
I
I
r
rir
R! 1-i
,l'h•·;;:::~----f#::''-------
;flr :Hq::.~- ·t ::::.:--- iT} ;tp!lll: Tr.R':: ~n::::
r:-:; ~~ i;: ::: : r .: _ ~ : . :n.; -1-P' 11li: ·:~ :~::i\ : - r;t! :::· .:
81 .~~ . , -::r -- 1· t::t• r'-t· rr\, 1~
:rtF I'!' j·i• ;;, ,: ..... - ' H-- ··-· ---l~t 1'}: ;;t: :~:: :~ ~,_:.IJ ~::~-rr·l ffn ~u; !F ~: ::::::. :§; •:: ~· =.. fft·[lE[ J; E[·: _- :LL .:.
h +-+r h ,. 't'· ·:;---r j- :----'j hi i .-;• ,;-; ..... ----- ~I= ....h\ - ' ·r-t+'
111
}+ ;;: ' ,:t4E~ ~;;i f ; :~, f;~; ~Bt
!l
tt;::::•·--
11 ' ···- ... -rt1;
" ..•
HI._;,....
pi· ·•· ...
--~-
!1'" t•
llt;JE·----
. -4:t-f t t•':
Hr· ~-.
,.,, ·'i· ,....... ..
:·i::t
:ffii
lth J+: fui 2: ~~-:: ::; : :~ ··~~:.t:-1
. ;:' ;: f ': , . :t :' • • •• •. · '.
*
·
;
~;;·1-=i~.r.Tf!l·l
irrt·;rllt~~~·~;J: _; •· h;~~·· ;Jl]'~- _FL22
q. ~ ·- · ·•· · ·
1
!
I ... , : ' ., .. :-- . ":' ,, ..... --
·J·
-,- ... "" ----1
t·-·· :·-- .. ,....... ·
+. . ' - ~ ~t
·•r· " ...
.
'r:.
~;:: :~,·
1.. -- 'I" j·-- ... -
;rr
, ___ ....
1
If·-., . . . . . . . .
]]
~::- ;; ~ •.•
! l · · - -··h·· rc:--··---- -----
...
~
-~-.: ~j w~ ·::. :::: •:• · J-t:--·......:. -: : 11f; r;r•··--·~-::. ~. . ' rl·-- 1, .......
: : : !l;.L.
;::... ...: .-. '4::j- .tt:r·r
:~::V·ln(~_!-~ - ~ t~~t>t-·~- .r(~ ntttt~:t~:~:-r$~t·t-- _1·;~:: ~;:- ·::_~~:~-~11--l:;::-:-.- -i- - -'- :;~:+-_-; +;:~-f- .:- -~- ; : - r~·H-r-r·•··-------- -~~~~
;;~ .::J-;::·-- _.--~-1.,.........
.:r._i-~t;_ ;;;; ~: . :;. ;.~- j ~-:_-_' ~- •. -i· ~,:
:;:_~~: :!;. -_ ~ tliE+
........ ,...... - • -··
..... -+-rr-jr+.,:··•-------jrl•r•·,··--j±;:.·f-- .......
:~:. :::: :>::: -·~t · : :::- -:---• ~Ptt~: F:::::: : .- +:i:: :.• := t:-¢t -::::T:~: ~:: ~:: •. ~ :;:=::::
~:'-::·1-1_-1--~Hr~'l-- ·--- '·- ··:· :: .. ----l- : L-r 1·
-+- L.J...... -· · f ~[- --- J~iir l•r· :., .. :. ··-- - ·1·· -:-. - f.+:!.~:~::
·.~.
J r"
-J
T:l--- ; Li
..
t\:11:··------j
t+li,rr·:--··-·-·i
ttif
--- - - .,,. ---" -
·t·
n-' ..
O+
;;;: ... --- .;, .;,. ,;_ -- - t- ,,
1
j
'"I11WI'] l;t'•li•
'I~!'!:· ... ,.... !·!· tl·· .• ~ .
- ·h !'lr! lr~' i1il .;.....
11 lr
!•:· ·'· ···-- tj•• .,. ---- · · - •-; r . :I'''
"l!ir 111'hr 1~', .... ·
r
r r.;r·:h
::: ti:f' lftg 1fr!m:;:~:::: : .: r;r:!;:- ::::~ :Til 1; illi!'li:ni::::--,: 1 ~l;f):ic· : •tr . Tt\! Hh;:T :~ : H!: ::: : ·:· · ~ tH- ttt.: HP ,r:. :;~ :·::: ;~' ~::::: :f~
rrr.., "[: - - ... =rj_ ....-~- ..' " -HHt'tr_ -~·· ;~;'
-r-~
1+-.~ 'I!' ,.,. ·- ·:· ...
r ll
· . m: lh-I<T
t-1
,mw,!:l;I;,;J::--1.H: -i>:
"'. ·; I- "',·+·+=·~l!(::,.:~l::~~~
i;;: ::::::: . t:
-f-!fllit"::····----•-•
, H-i·•il·--···· ·1 11• ··-----,,
I~' l! lHt n~t =- -_-+:- ~fi7. ~
\ii·d;-•""""['·•!"',""
·~······ ·•
+ W.·- t'tjlli':7f'"'··
+ l~l-tj.H 'lf1'1""'"'
·t·,rrn:d:t~·
1•·t· ·· ~
tt• ·• 1• ·· • •
· t~ Hhtit +r:. . ·
+Ht
t---------:
...
,, .. ' ' ' " + I ... ·
[
':._:_:...:._~
1
•-··· .. •-t•··--·1···· .. ···~1···,-·..
1,-..
···:·:~:·
·- -.- ...~·r~j·!·
' I..
II --~~:·:·········--~!:•=····
.... :· .... . r· . ;I ..... ....
.... , ... ·I· I
_·•·j·
"""~
r·tt1
rr~········
·····
--r-h-:
~~~1 ~~~:[1~::~ J_··_ 1J_.:~r-- -r -Et~ i~~ :~_: ~: ~-~-~l ;~: :
1
r+-:----·-1 ........
t·l·--• "''
·
=
=
~· .. ··~:~·:.::._!'
--!'-,···
rrt·
..............
Fl&ure 2.2-2
'
[nput
· ··!
···
1'-< • 1..
~.-.....~.
··
--. ·:·
·· ··
··
t
...
•
t·
. N~~.h_~:~~ :_ ;;~ ~ ;!~[T~,~- _:;T;, 'r'r~·-·~ :~~~:J .J l : : •: :;-::;:::~::·:···~· : .. i
.·=··rn··:_:··
' .. ,. . ... II~---_,
. Ill
:·--L··---:---:..J;
:: r: , ... ,, __ ---·
Sl~nul-ro-Noise
v~.
'I
'
,
• . ,.
Number uf
.
. ...
1
'··,·'
, .....
II
Sample~:~
w
co
~nm
of
Ravleigh or Rician variables is not !mown in closed form [ 5] .
However,
a
simplistic
analysis of the S/N ratio behavior can be
rationalized as follows.
The probability density function for the envelope of a sinusoid
in additive white Gaussian noise is [
p(V)
=v
e -1/2V
2
6].
-~)
e -r I 0 (VV2r
vihere
V
= the
envelope voltage in units of RNS noise voltage
r -· the signal-to-noise ratio
Rice derived the limiting forms of this distribution for very
small and very large values of signal-to-noise ratio.
When r is small
the distribution is approximately Rayleigh with
r, 2. 't' J'
(2.8)
m
= Output
mean =
-11i72'
a input noise
"V
.. , ~
.. Output" Standard Deviation =
42
1Tf
2
' crinput
noise
The signal-to-noise ratio at the input is S/ a input noise and
the output is
(SIN)
0
=.,j m2
\
+ s2
cr output
-m
40
and substituting using input parameters
_,
(2.9)
SIN
=
1~ 12 ~input
The above equations
2
2'
+ S -
~ 12 ~input
~input
ignore
the
cross
product terms on the
premise that we are looking for the average DC signal.
Equation 2.9 is plotted in Figure 2.2-3.
Inspection of Figure
2.2-3 confirms that at (SIN) ratios less than unity the output is
related to the input in a squaring relationship plus an additional
loss of 4 dB.
Then at high (SIN) the output is directly proportional
to the input plus an additional gain of 3 dB.
The loss at low signal-to-noise ratio is caused by the noise
capture of the detection process which occurs when taking the square
root of the sum of the squares of the real and imaginary FFT outputs.
The large noise term dominates the signal such that the resulting
output does not change in proportion to the input signal.
Since this analysis is based on a simplistic model and ignores
the AC cross product terms that occur, it is desirable to confirm this
model.
The program in Appendix B was used to obtain the output signal
level as a function of input signal level.
noise.
The following data were obtained.
This was done without
41
Input
Bin 27
Bin 32
Output Signal Level (Amp)
Bin 37
Bin 27
Bin 32
Bin 37
39.83
44.8
50.26
28.45
32
35.9
.3
11.07
19.2
21.54
• 15
8.53
9.6
10.77
.7
-1 dB
+ 1 dB
.5
Table 2.2-1
Input Signal Level vs. Output Signal Level
Next the program was used with signal amplitude established at
the level that generated Table 2.2-1 values but noise was added for a
variety of (SIN) ratios to obtain the results delineated in Table
2.2-2.
Input (SIN) dB
Output Signal + Noise (Amp)
Bin 27
Bin 32
Bin 37
Bin 27
Bin 32
Bin 37
+14
+15
+16
41.36
45.62
51.04
+11
+12
+13
28.54
31.7
32.51
+ 6.6
+ 7.6
+ 8.6
17.96
22.47
24.85
+ 0.6
+ 1. 6
+ 2.6
12.04
13.31
13.74
Table 2.2-2
Noise Level
Input Signal-to-Noise Ratio vs. Output Signal-Plus
Using the program supplied mean value of the noise, the signal
level was computed based on the relationship
(Sig + Noise)
2
= Sig2
+ Mean
2
42
(2. 10) Sig =
~ (Sig
+ Noise)
2
- Mean
2
'
This provided the following values for the signal:
Calculated Output Signal Level
Bin 27
Bin 32
Bin 37
40.13
44.5
50.05
26.73
30.08
30.93
14.91
20.12
22.74
6.7
8.78
9.4
Table 2.2-3
Calculated Output Signal Level Using
Output Noise Mean of 10.00
Comparing Table 2.2-3 and 2.2-1 we observe good conformance of
the
signal level.
accuracy of the test.
The difference is well within the measurement
This validates the analysis and is a sufficient
basis to draw the coricl us ion that to improve the Fourier system performance we must devise an alternative to the linear detector which
avoids squaring and square root operations.
Since the signal is unknown the question to be answered is,
where does the signal appear for each transform computed?
We can an-
ticipate what happens if the signal is in the center of the FFT bin
because phase is the only unknown.
This unknown phase will result in
the signal appearing in a predictable manner in the real and the imaginary bi.ns as follows:
Let
43
Figure 2.2-3
Signal-To-Noise Performance of Linear Detector
44
(2.11) Signal= A sin (27Tkm+ Q)
k = 0,1,
••• , N-1
N
and the Fourier transformed signal is:
N-1
( 2. 12)
Real =
L
Signal x cos ( 2 7T km)
N
k=o
Imaginary =
N-1
}:
k=o
Signal x sin ( 2 7T km)
N
Then substituting for the sum in 2.11 and multiplying this with
2.12weget:
N-1
Real
=
:E
Ax (sin(
2
2
7TNkm) cos (Q) +cos ( ;km) sin (Q))
k=o
x cos 2 7Tkm
N
and the summation of the sin ( 2 7T km) x cos ( 2 7T km) is equal to zero.
N
Then
N-1
( 2. 13) Real
=~
A x(cos(
k=o
and the imaginary is:
N-1
(2.14) Imaginary =
L
k=o
2
27Tkm
N )) x sin Q
N
45
This establishes a basis for predicting how the signal will
distribute itself in the FFT real and imaginary outputs.
From equation 2.2 we know that combining two independent samples improves the standard deviation by 3 dB.
Using the case where
the phase shift is 45°, equal amounts of signal would appear in the
real and imaginary outputs and the signal will be down 3 dB from the
case where
the
phase
is
zero.
imaginary terms directly,
standard
deviation
and
real
and
imaginary
terms,
if we
added
the
real
and
we would get a 3 dB improvement in the
with
. performance is unchanged.
Then,
the
3
dB
signal
loss,
the
overall
It appears logical that if we combine the
weighted
by
the
appropriate
amplitude
relationships, we should be able to account for all phase variations
between 0 and 360°.
This is accomplished by multiplying equation 2.13
by sin Q and 2.14 by the cos Q.
N-1
(2.15) Real=
~Ax (cos( 2 ~km)) 2
x (sinQ)
2
k:O
N-1
(2.16)
Imaginary=~
Ax (Sin(
2
~km)) 2
x (CosQ)
k=o
and then summing to get the desired result
Output
= Real + Imaginary
2
46
This scheme will take care of all cases where the signal is in
the center of the bin.
There is a logical way of quantizing the phase
(e.g., in 30° increments) to avoid numerous computations.
In a real
time system the actual quantized values would have to be determined by
tradeoff analysis of the degree of performance desired vs. cost to implement.
Another condition that must be accounted for is when the signal
is
not
at
the
bin
center.
It is
intuitively obvious
that
the
difference frequency (frequency of Fourier generators - frequency of
signal generator) will appear as the weighting function because the
difference
frequency replaces the
phase
term.
This adds another
dimension of hypothesis testing that must be accounted for.
Any
attempt to account for this dimension is beyond the scope of this
paper; however, it is important to know that we can solve this problem
without resorting to Fourier computations.
It is desirable to validate the concept using the Appendix B
model.
This was accomplished for three test cases.
0
0
One transform at 0
0
Ten transforms linearly integrated at 0
0
Ten transforms linearly integrated at 30 0 phase
phase
0
phase
The model was run sixty times for each test case with the following results.
47
Input Signals
Level
.314
One Transform
0
0 Phase
Linear Weighted
pd
pd
Ten Transforms
0
0 Phase
Linear Weighted
pd
pd
Ten Transforms
i0° Phase
Linear Weighted
pd
pd.
0 dB
.28
.43
.28
-1 dB
.23
.3
.249
-2 dB
.216
.2
.099
-9 dB
.1
.56
.02
.53
.089
-10 dB
.05
.45
.12
.35
.079
-"11 dB
.05
.33
.03
.33
.007
.005
.006
.005
PFA
.006
Table 2.2-4
.0026
Linear vs. Weighted Detector Comparisons
Inspection of the test results shows that:
1)
Using Figure 2.1-3 the detection performance for one sample
is about
2 dB better for the weighted detector. Consider-
ing that the false alarm probability is three times less
and using reference [ 1] we see that this is equivalent to
adding another 1.2 dB to the detection performance for an
overall 3.2 dB increase.
Taking into account
th~
accuracy
of the test, it is safe to say that a 3 dB improvement is
possible,
which is the gain expected for a coherent
detector.
2)
In going from one sample to ten samples we gain more than 9
dB.
rle expected 10 dB.
Again within the accuracy of the
test we conclude that 10 dB is possible and that this is 5
dB better than the conventional linear detector.
48
Since there was no performance difference between 0
3)
0
and
30° phase, the weight concept works for all phase angles.
The
concept
of using weighted
FFT bins to
collect all
the
signal. energy spilled into real and imaginary bins is a viable one and
can
be
extended
to
collect
the
energy
in
adjacent
bins.
This
extension is described in the next section.
2.3
Evaluation of Receiver Ripole Loss
The time-truncated sinusoid of the unweighted Fourier window
has the familiar (
Sin
X
X
2
) power spectrum where x is the bin position
within the bin in radians and x
=
rr/2 is the bin edge.
This response
results in signal loss as the signal moves from the bin center to the
edge of the bin, i.e., ripple.
The ripple depth at the bin edge is
obtajned from
(2. 17) R
= 10
log
[sin
rr/2 ]
rr 12
2
The ripple loss, assuming on the average the signal can occur anywhere
in the bin, is given by
rr/2
(2.18) Loss ::: -10 log
t- !·
dx
- Tr/2
=
1. 11 dB
Thus we can expect to lose 1.11 dB on the average due to ripple.
This
loss of energy is due to signals falling in adjacent bins, and the
amount of signal is dependent on where the signal is located in the
49
bin.
Figure 2. 3-1
illustrates this response,
and
determine the amplitude response in adjacent bins.
from
it
we can
This is done by
moving one bin position, then two bin positions on each side of the
signal and reading the amplitude from Figure 2.3-1.
Thus for a signal
at the bin edge (between N and N+ 1) we obtain the following amplitude
response:
Bin Number
Table 2.3-1
Sig Amp
N-2
. 11
N-1
.21
N
.63
N+1
.63
N+2
.21
N+3
• 11
Amplitude Response to Signal Between Two FFT Bins
It seems logical to take the previously described concept of
vJeighting functions and apply it here to adjacent bins.
The weights
to be applied for a bin edge signal would be those of Table 2. 3-1 .
If
this weighting is applied to the output of a linear detector we would
expect to reduce the ripple to 2 dB.
The program in Appendix B was used with the weights of Table
2.3-1 applied and the signal at the bin edge.
confirmed our expectations.
The results obtained
The Probability of Detection, Pd, with a
50
__
!~----·--~~=--~~---
-~-
~~ ~-::-~-~ 1:~~~~ :~~~L-~~;i~~~u~~~~-~; ;J_~-~-~~~-~-~=-1 :~~ ~~ :~ ~n~~ ~~ ~- ~ :~r :~ ~;
~l:J~!~~~~=~-;J'=-: jE~J-~!i2~j=J0--1]-_r~c~f::
--.- -----· ------------- :~~~~"--~i---~~~:~-~=--1=~- --~~; :~~----_
=.::._;___·----~~~------------~}:·.::::.:.:
.: __: i-
·.:..:: __ t::
__ ,
j_·- . ·:
·.--1·:··
·---- t·
___ ,
~-d~~2t~:g~j~~~T'~~~~~~L~~- =: ;~-:~~~
:
- :
j·-.:
·-----L
··•·!
-=i:--:·-- ---;-
--~~
.. ,_ --.i:~_::
~f:~~-::: . . :~1 ~-::~-~-:
-----------
~-=:r~~-~=:=~-3=~-==
iliiE~~t~~f[~:;~§EIIt!]
Figure 2.3-1
Amplitude.Response as Function of Bin Position
51
2 dB increase in level had the same probability of detect as a signal
in the mid bin position.
2.4
Conclusion
The analysis of the three key elements of the Fourier receiver
revealed their behavior in the system and, more importantly, how their
behavior can be changed .
Fundamental to each element is the notion
that if one is willing to pay the price to process the signal, taking
into account what the signal is doing in relation to time, then a considel"'able amount of system gain is possible.
be consider•ed in two domains, i.e.,
Signal variations must
frequency vs. time and signal
amplitude vs. time.
In real world system design these two domains are changing a.."ld
the concept of hypothesizing where the signal is located fits in with
the system needs.
That is, we obtain the gains identified in Section
2 and obtain further information about the signal that could have
other . important implications.
The concept of using the real and
imaginary terms with weighting functions is a simplistic approach that
allows con&iderable flexibility in the hypothesis testing to look for
and track the signal.
of large
capacity
Modern digital techniques and the availability
digital
hardware
systems make
it
possible
to
consider such a concept in the system design of a real time digital
Fourier receiver .
REFERENCES
1.
Robertson, G. H., "Operating Characteristics for a Linear Detector
of CW Signals in Narrow-Band
Gaussian Noise , "
The
Bell System
Technical Journal, April 1967.
;'2.
Schwartz,
)1Qaly~is,
3.
Texas
M.,
and
L.
Shaw,
Signal Processing Discrete Spectral
Detection, and Estimation, McGraw-Hill, 1975.
Instruments,
Inc.,
T .I. Programmable 58/59 Applied Statis-
tics, Texas Instruments, Inc., 1977.
4.
Oppenheim, A.
V. ,
and R.
W.
Schafer , Digital Signal Processing,
Prentice-Hall, Inc., New Jersey, 1975.
5.
WhaJ.en, A. D., Detection of Signals in Noise, Academic Press, New
York, 1971.
6.
Rice, S. 0., "Mathematical Analysis of Random Noise," Bell System
Tech. Jour., Vol. 23, July 1944; Vol 24, Jan 1945.
52
APPENDIX A
COMPUTER PROGRAM TO EVALUATE NOISE GENERATOR
A computer program to test the
CW/Noise Model Generator de-
scribed in section 1. 1 was written using CPS.
in Table A-1.
The program is listed
The executive is set up to execute N noise samples.
On
each run the N noise samples amplitudes are obtained from SIGEN subroutine.
The noise generator random number is provided by the system
library through the routine called random.
When the N sample ampli-
tudes are obtained, subroutine Stats is called to determine the mean
and standard deviation.
deviation per Table A-2.
Stats outputs the computed mean and standard
The program then calculates the noise sam-
ples amplitude distribution for + 4
sta~dard
deviations from the mean.
When 30 runs are completed the distribution is outputted per Table
A-3.
The CPS language used is a combination of CPS PL/I language.
Most of the statements used in this paper are PLII and are quite similar to F'O.RTRAN.
For further details see ( 1).
To assist in under-
st.anding the code used in this paper, the following interpretations
are provided.
1)
International Business Machine Corporation, Conversational
Programming System (CPS) Terminal Users Manual, IBM Corporation,
1133 Westchester Ave., White Plains, N.Y. 10604
53
GET LIST (Variable
, Variable
.. )
Data is input into the computer at execution time by means of
the GET LIST statement .
DECLARE & ALLOCATE
Statements to allocate array buffers for program use
DO-END Pair
The DO-END pair provides convenient way of repeatedly executing
a group of statements
PUT LIST & PUT EDIT
Output routines,
outputing data called for in variables
following the statement
LABEL:
PROCEDURE (identifier, ..• )
A procedure is a sequence of statements, the first of which is
a
PROCEDURE
statement
and
the
last
an
END
statement.
A
procedure must have a label and that label is the name of the
procedure.
An
optional
parameter
list
may
follow.
subroutine procedures are invoked through the use of CALL
statements.
The
55
PROGRAM CODE
EXECUTIVE
GET LIST(N,Am,Sd);
DECLARE F(-20:20) DEC(4) CONTROLLED;
ALLOCATE F,AR,A,B;
DO KK=O TO 30;
CALL sigen;
AR=A;
CALL Stats;
DO K=-20 TO 20;
DO J=O TO N-1;
IF m+s/5*(K-.5)<AR(J)&AR(J)<m+s/5*(K+.5) THEN F(K)=F(K)+1;
END;
END;
END;
PUTLIST(' J
Pd
');
00 J--20 TO 20;
PUT EDIT(J,F(J))(F(3,0),X(2),F(4));
END;
FREE F,AR,A,B;
STOP;
SIGNAL GENERATOR
sigen: PROCEDURE
kk:O;
kkk:kkk+ 1;
DO J:O TO N-1;
freqa=on*Am+1.122*sin(2*pi*(N/4+5+.1*binpos)*kk/N+pha);
freqb=on*Am*.891*sin(2*pi/N*(N/4-5+.1*binpos)*kk+pha);
freq=on*Am*sin(2*pi*(N/4+.1*binpos)*kk+pha);
A(J)=Sd*cos(2*pi*random)*sqrt(-2*log(random))+freqa+freqb+freq;
kk=kk+1;
END;
B=O;
END s.igen;
Table A-1
Noise Generator Test Program
56
STATISTICS COHPUTATION
Stats: PROCEDURE (Q);
m, s=O;
DO J=O TO N/4-7;
M=Q(J)+M;
END;
DO J:N/4+7 TO N/2-1;
M=Q(J)+M;
END ;
rn=m/(N/2-13);
00 J:O TO N/4-7;
s=(Q(J)-m)**2+s;
END ;
DO J=N/4v7 TO N/2~1;
s=(Q(J)-m)**2+s;
END ;
s-sqrt(s/(N/2-13));
END Stats;
PUT LIST ('MEAN .•.. STD DEV 1)
PUT LIST (m,s)
Table A-1
Noise Generator Test Program
57
MEAN
0.0667
MEAN
0.0142
MEAN
0.0343
MEAN
-0.0191
MEAN
0.0664
MEAN
-0.0247
MEAN
-0.1365
MEAN
-0.0951
MEAN
0.0804
MEAN
0.0112
MEAN
0.1143
MEAN
-0.0427
MEAN
o-.0510
HEAN
-0.0150
MEAN
-0.1205
MEAN
0.0379
lviEAN
-0.0007
MEAN
-0.0784
MEAN
-0.0469
MEAN
0.0385
MEAN
-0.0046
MEAN
-0.0016
MEAN
-0.0442
MEAN
0.0638
Table A-2
STD DEV.
1.0446
STD DEV.
1.0168
STD DEV.
0.9452
STD DEV.
1.0343
STD DEV.
0.9620
STD DEV.
1. 0092
STD DEV.
1.0740
STD DEV.
0.9634
STD DEV.
0.9892
STD DEV.
0.9755
STD DEV.
0.9543
STD DEV.
1.0141
STD DEV.
1.0287
STD DEV.
1.0484
STD DEV.
0.9926
STD DEV.
0.9932
STD DEV.
1.0754
STD DEV.
1. 0076
STD DEV.
0.9718
STD DEV.
0.9811
STD DEV.
0.9884
STD DEV.
0. 9777
STD DEV.
1. 0831
STD DEV.
0.9898
Sample Mean and Standard Dev. Outputs
58
N
.5.11
Am
Q
Sd
1.
J
Pd
-20
2
-19
0
2
-18
-17
-16
-15
-14
-13
-12
-11
-10
22
26
47
72
77
-9
180
238
-8
-7
-6
-5
337
466
565
716
-4
886
-3
-2
1221
-1
1163
1267
0
1
2
1297
1263
1137
3
4
5
6
1069
7
8
9
10
11
12
Table A-3
5
12
921
764
637
455
369
261
169
117
13
14
73
39
28
15
16
16
8
17
7
18
5
19
2
20
0
Ampli.tude Distribution
APPENDIX B
COMPUTER PROGRAM FOR FOURIER RECEIVER MODEL
A computer program for the Fourier Receiver given in Figure 2
and described in Section 1 was written using CPS.
made to optimize the program.
No attempts were
In general, operation is as follows.
The executive section collects the input condition (sample shown in
Table B-1) .
It executes the program using the PROCEDURE subroutines
as necessary and then outputs all parameters as indicated by the input
control units and out put procedure in the executive.
A sample output is given in Table B-2.
as described in the text.
All equations used are
The variable names have been changed to be
compatible with the computer terminal input keyboard.
A design constraint of CPS is the amount of memory allocated to
one user.
This design constraint limited the number and size of the
arrays that could be used in the'program.
Thus the largest that N can
be is 128 and the largest that MM can be is 20.
The sine wave signal
generator was set up to provide three sine wave signals, one at input
amplitude Jl.m and one -5 FFT bins away at -1 dB relative to Am and the
other +5 FFT bins away at + 1 dB relative to Am.
generator alwa.ys appears
convenience
of
in FFT bin N/4.
interpreting
the
computer listing of the program:
59
data
runs.
The center sine wave
This was done for the
The
following
is
a
60
I*
I* EXECUTIVE
1 ..
2•
~PS
PnOGRAM COOE
*li
*I;
3.
4.
GET U STO!,Am,Sd,hlnr>os,P, TH,~D;
GET LIST(Amon,MM,pha,rcon,llnon,coron);
DE r LAP. E AI { 0: 12 8) DEC ( 6) ;
6"
i.
f'ECLARE A(0:128) DEC(14);
DECLARE AR(0:128) DEC(6);
DECLARE B(0:128) OEC(14);
DECLARE C(0:128) OEC(6);
DECLARE 0(0:128) DEC(6);
DECLARE E(0:41) DEC(6);
DECLARE G(0:128) DF.C(3);
0 EC LA RE H ( 0 : 12 8 ) DEC ( 3 ) ;
OECLARE J (0: 128) DEC(3);
DECLARE ARS(0:128) DEC(6);
kkk,AR,AI,ARS,D,G,H,I,C•O;
s.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18,
19.
20.
21 ..
22.
2 3.
2 4.
2 s.
26.
2 7.
28.
2 9.
30.
31.
32.
33.
34.
35.
36.
3 7.
38.
39.
E=O;
DO X=l TO
DO JJ=l TO P-1;
IF JJ<Amon THEN GO TO jmp;
on=l;
jmp:
so ..
CALL s i ~en;
CALL f ft~en;
CALL RC;
CALL UN;
END ;
IF rcon=O THE~! GO TO ned;
CAll Stats(;\R);
PUT LIST( 1 RC RUN I =',X);
CAll OET(G);
C(X) =m;
r. (X+ 2 0) =s;
ned:
IF linon=O THE~·! GO TO edn;
AR=AIIM;
CALL Stats(AR);
ARS=AR+ARS;
PUT ll ST( 1 L It! Rml I
CALL DET(H);
=',X);
E(X)=m;
40.
41.
42.
43.
44.
45.
46.
4 7.
48.
49.
~U·1i
on=O;
edn:
E CX+20) =s;
IF cor on=O THEN GO TO den;
DO J=O TO Nl2-l;
AR(J)=AR(J+t-!12);
END ;
CALL Stats(M);
PUT lf ST{ 1CORR Rut' #=I, X);
CAL l DE T ( t ) ;
D(X)=m;
D CX+2 0) =s;
61
51.
den:
52.,
53.
54.
55.
56.
57.
58.
1 i no:
59.
co r o:
60.
61.
62.
63.
ski :
64.
65.
66.
6 7.
68 e
69.
70.
nedo:
71.
72.
7 3.
edna:
76.
77.
78.
iks:
749
75.
79.
80.
81.
82.,
83.
84.
h5.
86.
8 7.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
AR, A I =0;
END ;
IF rcon=O 1H EN GO TO 1 i no;
CA LL s ta t ( C ) ; .
PUT u ST < 'r.c f.1EAN =', m);
IF 1 i non=O THEN GO TO core;
CALL stat(E);
PUT ll ST( 1 L H! f·1EAN = 1 ,m);
I F 1i non =0 THEN G0 TO s k l ;
CALL s ta t ( !") ;
PUT ll ST( 1CORR t-1EAt!= 1 ,m);
DO J= 1 TO 2 0;
C(J)=C(J+20);
E ( J ) =E ( J+ 2 0 ) ;
[l(J)=D(J+20};
Etm ;
IF rcon=O 1l!Et! GO TO nedo;
r. ALL s tat ( C } ;
PUT ll ST( 1 P.C STD DEV = 1 ,m);
IF linon=O THEN GO TO edno;
CALL s ta t(E);
PUT llST('LIN STn nEV = 1 ,m);
IF coron=O THE~! GO TO iY.s;
CALL stat(D);
PUT ll ST( 1CORR STD DEV= 1 ,m);
G(l),H(l),l (1)=0;
DO J=2 TO NI4-7,N/4+7 TO N/2-1;
G(1)=G(l)+G(J);
!--' ( 1) ·=H ( 1 ) + H ( J ) ;
I (1)=1 (1)+1 (J);
END ;
pf =Mf,1* CN I 2 -15) ;
pf r =G C1 ) I pf ;
pf 1 =H C1) I pf ;
. pfc=l ( 1) lpf;
derr:
I F rcon =0 THEN GO TO de r r;
PUT tiSTC 1 RC OET 1 ) ;
PUT USTC'PFA= 1 ,pfr};
CALL outpt(G);
IF linon=O THEN GO TO core;
PUT U ST ( 1 lt N OE TS ' ) ;
PUT LJST( 1 PFA=',p~1);
CALL outpt (It);
CALL Stats(ARS);
m=m/HM;
s =s/r.1M;
PUT U ST('
run mean = ',m};
PUT U ST C ' r un s t d re v =' , s ) ;
PUT U ST( 'AVG SIG AMP IS 1 ) ;
CALL outpt(ARS);
62
101.
102.
105.
IF coron=O THEr-! GO TO ret;
PUT LJST( 1 COP.R DET ');
PUT LISTC 1 PFA= 1 ,pfc);
CALl out p t ( f ) ;
D0 J = 0 TO NI 2 -1 ;
106.
10 7.
END ;
core:
103.
104.
ARS(J)=ARS(J+N/2);
r.AL L S tats ( AR S);
m=m/t1t,1;
s =s /~U~;
PUT liST( 1run Mf'i=ln =',M);
PUT ll S T ( 1 r u n s t rf rle v
s};
108.
109.
110.
111.
=' ,
112.
113.
114.
115.
PUT t I ST ( I AVfi sf G M1P Is
ret:
CALL outpt(ftRS);
STOP ;
I};
r ,
63
I*
116~
117.
118.
119.
120 ..
121.
122.
12 3.
12 4.
125.
12 6.
127.
12 8.
12 9.
130.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
145.
146.
147.
148 •
14 9 •
150.
151..
152.
153.
154.
155.
156.
157.
158.
159.
160.
161.
162.
16 3.
164.
16 5 •
16 6 •
167.
,.
fftgen:
Ft~T
S PE CTP. Vt't
conr UTA T I Otl
PROrEDURE ;
u=3. 32193*1or-:100!);
j =0;
rlrev:
DO HHILE(j<N);
x=O;
1 =j;
dc=TP.l!t'C Ol/2+.1);
ac=1;
arev:
00 \'1HILE(ac<=TP.UNC0 1 /2+.1));
IF l<dc THEN GO TO incr;
.x=x+nc;
1 =i-de;
incr:
ac=2*nc;
rlc=trunc(dc/2+.1);
ENO arev;
IF .x> j THEN GO TO sk 1p;
y=A(x);
A (X) =A ( j);
A (J) =y i
y=B(.x);
B ( x) =R (J);
B(")=y·
. J
,
skip:
j=j+1;
END d rev;
u = t r u nc ( u + • 1) ;
r=l;
bflyft: DO v=1 TO u;
j=O;
q=TRUfJC0!/2/r+.l);
bflyps: DO p=1 TO q;
m=O;
b f 1 y p; p : D0 s =1 TO r ;
wr =cos ( 2 *Pi *m/ N);
wt=s inC2•pi *m/r!);
k=j+r;
asr=A(j)+A(k)•wr-B(k)•wt;
nsi =B(j)+B(k)*wr+A(k)•wJ;
udr=A(j )-A(k)*\·lr+B (k)*\'d;
ndi=B(j)-B(k)*wr-A(k)*wi;
A(j)=asr;
A(k)=arlr;
R(j)=asl;
B(k)=adl;
m=m+q;
j=j+1;
END b f 1 yg p;
j =k +1;
E NO b f 1 yp s ;
r =TR UNC ( 2 * r) ;
END bflyft;
*I;
"
'
64
167.1
167.2
168.
16 9.
170.
171.
172.
173.
174.
175.
176.
176.1
176.2
177.
rORRELATIOfJ OF.TECTOP.
;
00 j=O· TO 1'12-1;
I*
*I;
A(j+NI2)=B(j)*cos(pha)+/\(j)*sin(pha);
Etm ;
no
j
=o To
~!12-1;
a=A(j);
b =B ( j);
A Cj ) =s q r t ( a * * 2 +b * * 2 ) ;
ENf'! ;
END fftp;en;
I*
,.
s i p; en:
178.
179.
180.
181.
182.:
183 •.
184 •·
SIGfl/l.l DF.TFCTOn
*li
PROC F.DUP.E ;
kk=O;
kkk=kkk+ 1;
00 &J=O TO ~!-1;
-F r ~q a =on* Arn * s i n ( 2 * r i * ( N I 4 + 5 + • *b i n po s ) * k k It 1) ;
f re qb =on */\Pl*S i n ( 2 *Pi It!* 0"/4- 5+. 1 *b i n po s) * kk)
freq=on*Am*si n(2*pi /t'*0!14+.l*bi noos)*kk+nha
A(J)=Sd*cos(2*pi*random)*sqrt(-2*lo~Crandom)
184.1
A(J)=/\(J)+frP.q+.89*frro?;
A(J)=A(J)+l.12*freqb;
kk=kk+1;
184.2
185.
186.
187.
B=O;
188.
188.1
EtJD si,;en;
i*
HiTEGTIATOR
188. 2
189.
190.
;
PROCEOURE ;
DO J=O TO Hl2;
AR(J)=(1-2**-P)*AR(J)+2**-P*A(J);
END ;
RC:
191.
END ;
192.
193.
194.
195.
196.
19 7.
19 8.
END RC;
LIN:
PROCEDURE ;
DO J=O TO t!-1;
AI (J)=/\1 (d)+A(J);
END ;
END ll tl;
*i;
65
I* STATtSTJr.S r.ot1PUTJ\Tinn
198.1
198.2
199.
200.
2 01.
202.
203.
204.
2 05.
206.
20 7.
208.
209.
210.
211.
S tats:
EN!"' ;
f'O ,J=N14+7 TO t!l2-1;
m=Q ( J )+m;
Et!D ;
m=ml U! 12-15);
DO d=2 TO f!l4-7;
s =( 0. ( J) -m) * * 2 +s ;
ENI"' ;
no J=t114+7 To ~!/2-1;
s=(Q(J)-m)**2+s;
E~m
;
s=sqrt(sl(~l2-15));
Etm Stats;
I*
StrrNAL DETECTIOn
,.
I"'ET:
219.
220.
2 21.
222.
P P.OCE NJRE ( f");
m, s=O;
no J=2 To tJI4-7;
rr~=Q(J)+m;
212.
213.
214.
215.
215.1
215.2
216.
217.
218.
*I;
,.
by11as:
*I;
PROCED UP. F. ( Z);
r 0 d = 2 TO ~ !/ 2 -1 ;
IF AR(J)-m<s*Tii THEN GO TO by11as;
Z(d)=Z(J)+1;
PUT ltST(J);
nm ;
2 23~
PUT ED I T(Af1(t!l4-5) ,AP.(fll4) ,Ar. CN/4+5))
I F b i n po s =0 TI! F. t! G0 TO end t ;
224.
225.
226.
no
PUT LIST ( 'll ST OF B I t!S I flTE RPT');
J=2 TO Nl2-1;
B(J)=.3*AR(J-1)+.3*AR(J+2)+AR(J)+AP.(J+l);
ENIJ ;
CALl S tats (E);
no J=2 To t!/2-1;
IF B(J)-m<s*Til THEf! GO TO byps;
Z CJ+ NI 2 ) Z Cd+ r! I 2 >+ 1 ;
227.
228.
229.
230.
231.
2 32 ~
2 33.
234.
235.
235.1
235.2
236.
2 37.
238.
239.
240.
241.
242.
=
PUT tl ST ( J);
byps:
ENn ;
endt:
ENO ;
PUT ED t T ( ~ Ol I 4- 5), B (tl I 4) , B ( N I 4 + 5))
I*
,.
stat:
A VEP..AG !ttl ROUT H!E
PROr.ED URE ( P.);
m=O;
no t1=1 TO t1M;
m=R ( J )+m;
ENf' ;
m=m/n~~;
END stat;
*I;
66
242.1
242.2
243.
244.
245.
246.
247.
248.
249.
250.
I*
.
,
outpt:
OUTPUT P.OtiT I t!E
*/;
PROCEDURE ( RR);
no J=N/4-S,t!/4,t!/4+5;
o =R R ( J-1) /m1;
b =n nC.!> nu1;
r. =P.P. ( J+ 1) /Mt~;
PUT EDIT(a,h,c)(r-(10,3),X(2));
END ;
E Nn outpt;
67
INPUTS:
Number of FFT samples
Sd
=
=
=
binpos
=
Numeric control to determine where signal
N
Am
Sine wave signal amplitude
Noise standard deviation
sine
wave
center.
appears
relative
to
FFT
bin
One unit equals one-tenth of a
bin position .
Amon
=
Numeric control for when sine wave signal
comes on relative to first transform computed.
Zero means signal is on for first
transform.
pha
= Signal phase (in radians) relative to FFT
signal generator
1.'1
= Number of integrator samples
P
=
=
=
TH
rcon
RC integrator time constant control
Threshold control
Output
control
for
RC
integrator.
One
equals "ON" and zero equals "OFF
Linon
= Output control for Lin integrator.
One
equals "ON" and zero equals "OFF"
Coron
=
Output control for synthesized correlation
detector.
One equals "ON" and zero equals
"OFF"
Table B-1
Signal Model to Detect CW Signal in Gaussian Noise
68
li'
OUTPUTS:
1)
(Refer to Table B-2)
For each comolete integration period the bin numbers that
have been detected and the amplitudes for the signal bins
are printed for all integrators that are turned on.
2)
After MM runs the means and standard deviations for all
integrators are printed.
3)
For each integrator the detection statistics PFA for all
nine bins and the Pd for all signal bins and bins on each
side of the signal bins are printed.
The average signal
amplitude (average of MM runs) for the same bins the Pd's
are outputed on are also printed.
4)
The mean and standard deviation for the amplitude of the
noise bins for MM runs are outputed.
Table B-1
Signal Model to Detect CW Signal in Gaussian Noise ( cont' d)
'
69
INPUT CONDITIONS
xeq
N
128
Am
..L28
Sd
.l
b i npos
.9..
p
.Q.
TH
2:_.7
M
1
Amon
.Q.
,,,M
2:..0
ph a
.Q.
rcon
.Q.
li non
1
co ron
1
Table B-2
Sample of Input Conditions
70
.INTEGRATION PERIOD DETECTIONS AND SIGNAL AMPLITUDES
liN RUN I = 1
52
12.019
18.061
CORR RUN #= 1
11.788
17.507
ll N RUN I = 2
49
14.951
9.541.
CORR RUN I= 2
37
13.864
9.353
LIN RUN I
28
32
37
21.594
=3
24.892
2 3. 5 39
23.917
21.810
34.029
34.037
26.773
34.035
14.637
19. 424
13.411
12. 38 4
16.383
16.646
15 .. 722
15.502
23.195
23.224
CORR RUN I= 3
7
27
28
32
37
21,.010
ll N RUN I
27
26.595
CORR RUN
27
23.796
Ll tl RUN I
=4
4
#=
=5
27
46
25.852
CORR RUN
22.204
UN RUN I
9
22.630
5
#=
=6
.
CORR RUN I= 6
32
37
21.006
22.016
Table B-3
22.937
Sample of Outputs
71
DETECTION STATISTICS
LIN MEAN
= 10.363420724869
CORR MEAN= -.42964247763157
LIN STD DEV =
5. 4170293331146
CORR STD DEV= 8.1705135822296
LIN DETS
PFA= .0071428571428571
o.ooo
o.ooo
o.ooo
0.250
0. 050
0.150
0. 000
0.300
0. 000
run mean a 10.363415075808
run std dev = 1.4000601958317
A VG S I G AH P t S
11.274
10.311
18.111
10. 721
19.499
10.440
10.264
24.154
10.614
CORR DET
PFA= .0020408163265306
run
run
0. 050
o.ooo
0.250
o.ooo
0.200
o.ooo
o.ooo
0.550
0. 000
mean = -.42964188420043
std dev = 1.4738285135561
A VG S I G Ar·1 P 1 S
**
3. 02 9
15. 870
-1.852
17~291
-0.989
21.521
115. XEQ "STOP".
Table B-3
1.896
1.196
1.973
Sample of Outputs (Cont'd)