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ONR ltr 28 Jul 1977
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tSV.
Some Topics on Limiters and FM
Demodulators
§
>
Maung Gyi
July 1965
Technical Report No. 7050-3
Prepared under
Office of Naval Research Contract
Nonr-225(83), NR 373 360
Jointly supported by the U.S. Army Signal Corps, the
U S. Air Force and the U.S. Navy
(Office of Naval Research)
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Some Topics on Limiters and FM
Demodulators
>
Maung Gyi
July 1965
Technical Report No. 7050-3
Prepared under
Office of Naval Research Contract
Nonr-225(83), NR 373 360
Jointly supported by the U.S. Army Signal Corps, the
U S. Air Force ond the U.S. Navy
(Office of Naval Research)
SVSTEmS THEORV IRBORIITORV
SIRnFORD ELEtfROnilS IRBORRTORIES
STnnFORO UniUERSITV • STRRFORD. CRUFORRIR
SEL-65-056
SOME TOPICS ON LIMITERS AND FM DEMODULATORS
by
Maung Gyi
/
ole or in part
any purpose of
at es Government.
7050-3
Jointly s
tlv
ider
larch Contract
NR\373 360
Army Signal Corps,
\the U.S. Navy
(Office of Naval Research)
Systems Theory Laboratory
Stanford Electronics Laboratories
Stanford University
Stanford, California
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ABSTRACT
An analysis is marie of wide-deviation frequency-modulation systems
having low nominal-carrier-frequency to information-bandwidth ratios.
Since limiting plays an important role in such systems, the effects of
hard limiting of many signals in random noise are analyzed.
Expressions
are given for the output signal-to-signal ratios (SSR), the output signalto-noise ratios (SNR), and the power in any crossproduct.
The effect of
the power in each output component as a function of the input SNR is
investigated.
It is found
that for all values of input SNR's greater
than 10, the strengths of the various output components are relatively
constant.
For the case of more than two input signals, a weak signal-
boosting effect manifests itself when the input SNR's are less than 1
and the input SSR's are greater than l/lO.
The signal-suppression effect
and the signal-to-signal power sharing, together with their dependence on
input signal noise, are presented for various cases.
Expressions are presented for the autocorrelation function and the
spectral power density of hard-limited FM pulse trains, which allow the
computation of the intermodulation products in the information bandwidth
or in any other frequency interval.
The effect of the nominal-carrier
frequency on the interference ratio is exhibited graphically for various
constant deviations.
A partial interaction of the positive and negative spectra of the
modulated wave causes extraneous outputs.
For all such spurious components
to be filterable, a relation is derived between the carrier frequency, the
maximum frequency of information, and the spectral bandwidth of the modulated wave.
The analysis results in systems which have better performance
capabilities.
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111 -
SEL-65-056
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CONTENTS
Page
I.
II.
INTRODUCTION
1
A.
Background
1
B.
Outline of Presentation
2
HARD LIMITING OF
A.
SIGNALS IN RANDOM NOISE
Formulation of the Problem
1.
2.
3.
4.
5.
B.
Llmiter Input Random Process
Input Noise
Autocorrelation Function of the Input Noise .
Limiter Transfer Function
Narrowband Assumption
. . .
6
20
22
Derivation of Autocorrelation Function
Spectral Analysis and Filtering
Output Signal-to-Noise Terms
C.
Output Factors Due to Signal and Noise Interaction
D.
Smooth Limiters with Error-Function Transfer
Characteristics
E.
4
5
5
6
6
6
Limiter Output
1.
2.
3.
.
.
27
29
Asymptotic Results
1.
2.
3.
III.
p
29
34
40
Case I, Weak Signals
Case II, One Strong Signal
Case III, p Signals of Equal Strength
F.
Computed Results
49
G.
Conclusions to Chapter II
50
OUTPUT SPECTRA FOR WIDEBAND FM PULSE TRAINS GENERATED FROM
HARD-LIMITED FM SIGNALS
68
A.
Introduction
68
B.
Input Process
69
1.
2.
69
71
C.
Assumption
FM Pulse Train
Evaluation of
1.
2.
11
Expression of the O*
Spectral Zone, CQ . . .
Expression for the 2pth Spectral Zone, p ^ 0 .
SEL-65-056
IUTJ^I^\ä'\JVI
^\
74
[y(t) y(t - T)]
.
.
74
75
IV
?J\JWFW V^
v wr.. . t^
L"W
v"^ LV '-% L"V _r» ./v
\.-ü
.
k.'w _-^ .■-• _> .
• M'^'Jt "VJ« "_• '\JII\M-JirmrM-
Page
D.
E.
F.
IV.
80
1.
2.
3.
4.
80
81
82
85
Expansion of z(t)
Autocorrelation Function
Power Spectral Density
Output Voltage Vs Frequency Characteristic
....
Intermodulation Products
85
1.
2.
86
88
Interference Ratio, IR(p,n)
Pulse Shape and Interference Ratio
Results
88
1.
2.
88
91
Theoretical and Experimental Results
Advantages over Conventional Detectors
REDUCTION OF SPURIOUS OUTPUTS IN LOW-CARRIER, WIDEDEVIATION, WIDE-BANDWIDTH FM SYSTEMS
93
A.
Formulation of the Problem
93
B.
Analysis of the Problem
95
C.
Interrelationships of fc, fymm"
Filterable Spurious Outputs
D.
V.
Output Process
^^
*■ fan
^or
97
Reduction of Spurious Outputs
98
1.
2.
99
99
Filtering Technique
Phasing Technique
CONCLUDING DISCUSSION
105
APPENDIX A.
Useful Integrals of Bessel Functions
107
APPENDIX B.
Derivation of Autocorrelation Function of the
Output z(t)
110
REFERENCES
oc«.>u jrv jrtmijrtajni/ivjrMinLX **
122
SEL-65-056
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».•<«-••_'
ULT ILH
icn 'O m.
KT K.'. V.
O O O. *I '.* •." «.'■ ■Ji.'s." >
TABLES
Number
Page
1
Suppression effect for the three-signal case
2
Suppression effect for the four-signal case
3
Output signal power for
p
equal signals and no noise
61
4
Output signal power for
p
equal signals in noise
. .
62
5
Crossproducts vs
for the four-signal case
. .
63
6
Crossproducts for large
p
7
Crossproducts for large
p,
no noise (equal-signal case)
65
8
Interference ratios for various deviations and bandwidths
89
9
Comparison of computed and measured values for various
IR(p,n)
91
(S/N)
(p = 3) .
(p = 4)
57
58
in noise (equai-signal case)
64
ILLUSTRATIONS
Figure
1
Block diagram of the receiver analyzed
4
2
Contours of integration
8
3
Output signal power for the three-signal case
54
4
Output signal power for the four-signal case
55
5
Output signal power for the five-signal case
56
6
Suppression effect for
p = 3
59
7
Suppression effect for
p = 4
60
8
Crossproduct
case
9
10
S 3 p_3
^ .0
Crossproducts vs
p
vs
p
for equal-signals-in-noise
66
for equal-signals, no-noise case ...
67
Significant components of the fundamental and thirdharmonic zones of the limiter output
70
11
Block diagram of the system analyzed
71
12
Curves showing the interference ratio vs deviation for
fixed fc/fmm = 2
90
Curves showing spurious output behavior vs carrier
frequency at constant deviation
90
13
SEL-65-056
KPJOVOWATUJKM*. V:
vi
rJOKT.KJWJO">■ '>.V "> »> r> »J« >> V »-K»>"> .'■>'J- rj.'J "J> 'AVrj. •>",*■>'J-TJ« -.- v > v
■ . •/"«-.*•>"»»,. •_* •
p
Figure
14
age
Typical bandwidth of channel, using vestigial frequency
modulations
93
15
Modulated spectrum when
fh » fymm
94
16
Modulated spectrum
when
r
96
17
Modulated spectrum when
f'h =■ f ymm
„,„
fc < f
18
Lowpass filtered output
19
Block diagram and spectra of the proposed system using the
filtering technique
100
20
Demodulation process using the filtering technique
100
21
Block diagrams of the phasing technique
HXKKHKHMiKLa ,i^WJLT>LTHft*AKT<*>JUOW^ ^
(shows aliasing)
...
97
98
vii -
....
101
SEL-65-056
PRINCIPAL SYMBOLS
b.k m m
m
m
I 1 2..m
=
k 2
(N/2)
/ h,k,m m ..m L[see Eq.
(2.29)]
'
*
\
/i
1 2
f
C
n(cos ilr(t)] * 5(f + f )
c
" c
C
3(cos t(t)j * 5(f - f )
c
c
C+f
c-i+
• "l [1 + sgn (f)]
c
C+f
' i [1 - sgn (f)]
c
E(x)
complete elliptic integral of the second kind
=
„ /X
K\
\ ' 2/
•n/2
fn/*
/
ri
[1 - x
2.2 ,1/2 ^
sin <t>] '
d*
f
nominal-carrier frequency
f
modulating frequency
f
maximum frequency present in the information
0(6]
Fourier transform of
k
g(t) * g(t)
k
I (x)
modified Bessel function of the first kind of order
[o]
self-convolutions of
g(t)
(i mag i n ary argumen t)
IM(p,n)
intermodulation product at frequency
IR(p,n)
interference ratio at frequency
J m (x)
Bessel function of the first kind of order
(2pf
(2pf
+ nf )
+ nf )
m
(real
argument)
percent deviation
SEL-65-056
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L-M
i^nvj"«
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L™ UV WH VKMHU
».jt njt r M f-M
^J*
' jk^M rjirjt rM r^jw ^' ^j ^ irjir~ rfVifcr ■ ^. *■- *\- *v W4*\i iru
m
v -n.« n ■
M (w ,w ,)
two-dimensional joint characteristic function
p(y)
probability distribution function of
R [t. ,t ]
x 1 ^
E[x(t ) • x(t )] = autocorrelation function of the real
l
^
random process x(t)
R [0,T]
x
R v(T)
x
S(f)
spectral power density = Fourier transform of
= I
R(T)
e"laYl dl,
(y)
R(i)
^ = 2,^
•s—oo
S (f)
S(f) • I [1 + sgn (f)l
S_(f)
S(f) • I [1 - sgn (f)]
S+f
:Msin t(t)] x 5(f - fc)
c
x(t)
Hilbert transform of
x(t) = Cauchy's principal value of
s)
n J
t -s
ds
ß
A Af
/
k
c\
modulation index = ± — = 1+ -r-rr ' T— I
m
\
m/
6(t)
Dirac delta function = unit impulse at
G
fl
Neumann numbers = \
[2
m
if
m = 0
if
m > 0
t = 0
^ (t)
random process defined by
D(T)
MV
R(T)
—7—r
= normalized autocorrelation function
'
[J'(t) - \\f{t - r)]
R(0)
f
o(f) eiaVI df,
a- = 2Jif
•' — nil
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KH
m. ■ p
M
«-* "us n.«
-VM
nji fui
VLM
T-M rJ«
"\JI TJ*
r
SEL-65-056
-00
a(f)
Fourier trans form of
P(T)
= I
p( T) e
dx
•'-tX)
* (f)
y
characteristic function which is equal to
i
E[e
^]
=
f" e^y p(y)
dyi
^ .
2jlf
•'—uo
t(t)
information in the case of phase modulation
information in the case of frequency modulation
convolution; that is,
X00
-00
g(a) f(t - a) da = j
-00
sgn (f)
0
for
f < 0
2
for
f = 0
1
for
f > 0
f(a) g(t - a) da
J.Qo
SEL-65-056
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ACKNOWLEDGMENT
The author wishes to acknowledge the guidance of Dr. Thomas Kailath,
under whose supervision this work was done.
He also wishes to thank Mr.
P. D. Shaft of Stanford Research Institute for his suggestions regarding
the presentation of results on hard limiters.
(n»uouriuaijr**Tinwiim¥TmLkTjr*^^
xi -
SEL-65-056
V
I.
INTRODUCTION
The purpose of this research is to analyze some of the problems
peculiar to wide-deviation frequency-modulation systems having low nominalcarrier frequency to information-bandwidth ratios.
Some form of frequency
modulation is employed in many tape-recorder systems due to the fact that
the effect of random envelope variations occurring during playback can be
eliminated at the receiver by hard limiting prior to demodulation.
A
typical electromagnetic recording channel has a bandwidth from an upper
limit of a few megacycles down to a lower limit of a few hundred cycles.
If, for example, it is desired to record information down to dc, some form
of modulation must be used before the information can be so recorded.
The
parameters of a typical channel are a low nominal-carrier frequency and
the widest possible deviation ratio.
The high deviation is needed to
obtain an acceptable signal-to-noise ratio (SNR).
Since the channel band-
width is fixed, most of the modulation schemes result in reduction of the
information bandwidth that can be recorded.
In order to maximize ihe information bandwidth, a vestigial technique
is sometimes employed.
is used.
At other times conventional frequency modulation
In either case, spurious outputs arise because the bandwidth of
the channel is not wide enough to record the entire portion of the frequencymodulated spectrum that carries significant amounts of energy.
The ultimate
performance of such a system is often limited by these spurious outputs
rather than by the SNR.
on the same channel.
In other instances, many signals are multiplexed
Then it is of importance to be able to compute the
strengths of various crossproducts and the signal-to-noise ratios of each
signal.
A.
BACKGROUND
The effects of hard limiting on signals and noise have been studied by
various investigators from time to time.
by Bennett and Rice [Ref. 1]
Using the techniques developed
and by Middleton [Refs. 2,3], Davenport
studied the case of a single sine wave and additive gaussian noise when
th
passed through a v
law device and gave explicit results for the changes
- 1 -
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SEL-65-056
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in signal-to-noise ratio produced by an ideal limiter [Ref. 4].
Later
Granlund [Ref. 5] and then Baghdady [Ref. 6] studied the problem of hard
limiting of two sine waves without noise and their results included some
expressions for the evaluation of the strengths of output signals and
intermodulation products.
Davenport and Root [Ref. 7] studied the prob-
lem of bandpass limiting of two sine waves in the presence of additive
gaussian noise.
Their investigations were reinforced recently by Jones
[Ref. 8], who studied the problem in great detail in order to examine the
effects of limiters in long pulse radars and other cases where substantial
pulse overlaps occurred.
Jones studied the case of two signals in noise
and gave relations among the output signals, noise, and intermodulation
products due to bandpass hard limiting in that case only.
The present study extends the analysis to the case of bandpass hard
limiting of
p
signals in additive gaussian noise.
It is, in short, a
generalization of the problem investigated by Jones, using the nonlinear
transform technique which is convenient and mathematically rigorous.+
The results are shown to agree with those of
(2) Granlund and Baghdady, when
p = 2
(l) Jones, when
p = 2;
and when the input signal-to-noise
ratio is made arbitrarily large (no-noise case); and (3) Davenport, when
P = 1.
B.
OUTLINE OF PRESENTATION
Chapter II presents an analysis of the effect of hard limiting on
p
signals in noise.
Expressions are given for the spectral power density
and the autocorrelation of the limited output, the output SNR's, the output signal-to-signal ratios (SSR), the signal-suppression effect, and the
intensity of any crossproduct.
Some curves and numerical results are also
presented, and some asymptotic values are derived.
Chapter III presents a derivation of expressions giving the output
speocrum of wideband frequency-modulated pulse trains generated from
Prior to publication of this study, P. D. Shaft, who has also investigated
the problem, presented a paper at the IEEE Convention in New York on 24
March 1965 [Ref. 9]. Mr. Shaft was formerly with Western Development
Laboratories, Philco Corporation, and is now with the Stanford Research
Institute, Menlo Park, California.
SEL-65-056
jsjo#AKnwvtWiANLV.r^.rAr^.^v^
- 2 -
hard-limited frequency-modulated signals.
spurious outputs in the demodulated signal.
Expressions are given for the
A system of indexing the
spurious components is developed, and curves are given which show the
dependence of spurious outputs on carrier frequency, deviation, and the
information frequency.
Interference ratios are defined, *hen computed
and tabulated together with experimentally measured values.
Chapter IV presents an analysis of the "aliasing" phenomenon which is
due to the interaction of the positive and the negative spectra when the
frequency of the nominal frequency-modulated carrier has a value lower then
half the spectral bandwidth of the modulated wave.
Expressions are given
that relate the nominal-carrier frequency to the maximum frequency of
information.
Analysis and block diagrams of proposed systems are pre-
sented which are improvements over existing systems.
|IWWAWLV:S/-S.WW%^WUN^V.V-^
- 3 -
SEL-65-056
>S^'.^/v^V^V'."."rJV,0V
II.
A.
HARD LIMITING OF
p
SIGNALS IN RANDOM NOISE
FORMULATION OF THE PROBLEM
The effects of hard limiting of
p
signals in random noise are ana-
lyzed, assuming the input conditions described in the following five subsections.
The system studied is shown in the block diagram of Fig. 1, in
which it is assumed that the input signals and noise are bandlimited by
bandpass filtering centered around
f .
BANDLIMITED BY 'DEAL LIMITER
BANDPASS FILTERING
CENTER
1
FREQUENCY fc
x(t )
*"
BANDPASS FILTER
ö
1
FIG. 1.
1.
I(t)
y(t)
BLOCK DIAGRAM OF THE RECEIVER ANALYZED.
Limlter Input Random Process
The limiter input random process, defined as
p
unrelated sine waves and additive gaussian noise.
every
'.
i
in the
(l,p)
x(t),
In particular, for
interval,
!.(t) = J2S.
cos (cu.t + *.)
v
1
where the
$
consists of
1
1
1'
(2.1)
are statistically independent random variables, each having
a uniform distribution on the interval
The input
x(t)
{0,2n).
is therefore
x(t) = s^t) + s2(t) + . . . + s (t) + n(t)
= y
y/2S. cos (cr.t + $.) + n(t)
(2.2)
i=l
SEL-65-056
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* ' *
W
J»
m
j0 rjtrji'^
-_»i :■■/•_ J"-J
2.
Input Noise
The input noise
n(t)
is a sample function of zero-mean, station-
ary, narrowband gaussian noise.
It is assumed symmetrical around
fc
since the signals and noise are bandpass filtered prior to limiting.
3.
Autocorrelation Function of the Input Noise
The noise
process.
n(t)
is a sample function of a stationary random
The covariance function of
n(t)
can be readily written as
[Davenport and Root, Ref. 7, p. 169]
R(t) = R (t) cos
c
[<M
where
a.
c
CD
c
t - R
* *IM
cs
(t) sin cu t
c
1/2
cos cu t + tan
c
•i ■W"
TTTTT
(2.3)
= 2nf .
c
Since
f
c
is chosen such that
S(f)
is symmetric about it,
s+(f + fc) = s+(-f + fc;
(2.4)
s_(f - fc) = s_(-f - fc;
(2.5)
and
which gives
S K(f) = 2S (f + f )
c
'
+
c'
(2.6)
S
cs
(f) = 0
This equation implies that
3nM™r>icw™Tc>eK*y^^v*jr*jrvjr>ur<^y^3r*M^
R(t)
\ / = 2R c (t) cos a. c t
5 -
(2.7)
SEL-65-056
Adopting the usual normalization by defining
such that it contains
P(T)
unit power, one can write the covariance function of the input noise as
R (T) = NP(T) COS CD T
4.
Limiter Transfer Function
The limiter output
5.
(2.8)
y(t)
can be written in terms of its input as
j+l
for
x(t) > 0
y(t) = g[x(t)] = < 0
for
x(t) = 0
■1
for
x(t) < 0
(2.9)
Narrowband Assumption
The narrowband assumption implies that the difference between any
pair
[CU.,üü.]
■!■
J
chosen from the set
{üX. ,Cü„
X
A
, . . . ,ü) ,CO ]
p
C
is negligible
magnitudewise when compared to any member in the set; that is,
|<xi - cu | « ^k
where
(k] = {1,2,...,p,c).
for any
i,j f (k)
(2.10)
This assumption guarantees that the inter-
ference between adjacent spectral zones at the output of the limiter will
be negligible.
Hence the first spectral zone can be obtained by suitable
bandpass filtering around
B.
f .
LIMITER OUIPUT
1.
Derivation of Autocorrelation Function
The signal-to-noise ratio among the output components of the
limiter output
y(t)
can be obtained from the autocorrelation function
of the limiter output which is defined as
RV(T) = E {g[x(t)] g[x(t
SEL-65-056
TTÜUIÄ WVAJTMX/>Ll x*>Lr w«^ ,<A :<,*KT»ww>^^^^
+
T)])
(2.11)
or
R (t.f) = Ev(g[x(t)]
y
x
where
t' = t + T.
g[x(t')]}
(2.12)
Using the techniques suggested by Bennett and Rice
[Ref. 1], Middleton [Ref. 2], and Davenport and Root [Ref. 7], one can
represent the transfer function of the nonlinear device by
g(x)
=
.J_ f
f
(w)
xw
e
dw +
J_ r
f
(w) eXW dw
(2.13)
+
where
C
= the contour
w = u' + jv,
u' > u
+
C
-5
= the contour
w = u" + Jv,
-OO
/-OO
g» e-WX dx =
: (w) =
rO
_wx
(w) =
u" < u3
WX
g (x) e
1 • e
/*'
e
dx =
-wx'
-WX _,
1
dx = —
w
, .
1
dx' = w
Hence
, ,
g(x) =
The contours
C
and
if
^j I
'C
C
e
xw dw
V
+
1
f
2^ /
e
xw dw
V
+
are shown in Fig. 2.
It should be noted that both contours include the entire
w = jv
axis except the origin, which is excluded because of the presence of a
VVMTMJW^VVVMWW^^
- 7 -
SEL-65-056
/{■ •.(■/ v v.v.v.v.'y//'.
C+ CONTOUR
FIG. 2.
pole at that point.
CONTOURS OF INTEGRATION.
The integrands are identical and therefore, for
compactness, the equation of
over the contour
C
and
C
C.
C_ CONTOUR
g(x)
is represented by a single integral
For final evaluation, one can simply evaluate over
separately and add.
/ \
1
/
2nj Jn
wx dw
w
(2.14)
Therefore,
R (t,t )
yy
'
e
=-,—^
J ^r ( -^E^
^ [^(0
X
t
(2nj) ^C t -'C f
+
\Mt<)]}
t
(2.15)
By definition
,
Ex{exp [wtx(t) + wtlx(t )]}
is the two-dimensional joint
characteristic function and can be denoted by
/• dw
M (w ,w .).
x t t'
Hence
/• dw
V^'^^^/oTJtv-"^'"^
Since the input process has an independent nature, manipulation of
is particularly convenient.
The input
x(t)
■
(2 16,
M
x
is the sum of the zero-mean,
statistically independent random processes,
SEL-65-056
- 8 -
TMiTMijMtnjtr«iÄÄrmr\jif\j»^ürji'«rv»r,ji«ji.r.> -Jirjirjr njt.\**Mn*Kk'jt*±KM*ji''M?jtT 'm rji " i'r k T. w*k r « r vx«> »>-"• ."• »■>>. .'>].>.>»> v"" .> ~1«, «^.."i
x(t) = n(t) + Y s.(t)
(2.17)
i=l
and therefore
p+1
^VV)
=
11
M
x i (wfwt'
i=l
p+1
[I
ii
i^l
,..
E
x. {exp
r [w t x i (t) + w t' x.(t')]}
i
i
(2.18)
It can be shown that the Joint characteristic function for stationary
gaussian noise is [Ref. 7, Eq. (13.43), p. 289; Parzen, Ref. 10]
Mn(wt.wt,) = exp [| (w*
+
w*,)
+
Rn(T) w^,]
(2.19)
Each of the sine-wave signals
s. = J2S. cos (cü.t + 0.)
has a joint characteristic function
M
s ^W^
i
= E
fexp ^t^i
COS
e
t
+
^"t1
N/^1
COS
0 ,
t ^
(2.20a)
The exponentials are expanded using the Jacobi-Anger formula [Watson,
Ref. 11; also see Eqs. (B.lOa) - (B.lOd) on p. 115, this report] to give
00
M
(w^.w^,) = >
t t'
/_,
1
m=0
s.
00
> e c I (w^, 72S-) I (w+Iv
/2sT) E(cos mQ+ cos ne,.,)
/_, mn mtv
i
nt'
i'
t
t'
n=0
(2.20b)
- 9 -
SEL-65-056
■ ntmMtiMmjaarmjrRM juf itMiiiriununuf it^njf x^-RVTtVTL¥TiVTtVÄ ¥> \n V>M> \> vn\-f. i^J/*_WTVKi>i>M>^> yxyTn.yxVTi.ii>J.Xlf»JWT' M'«Ji"KJi">ur*_ir<'jr>Ui""J
The expectation can be simplified to
E(cos me 1 cos nG , ) = E[cos m(a).t + t.) cos n(üu.t, + <)). )'
11
L
11
E < — cos [(m + n)uj. t + nu.T + mit, + n* 1)
I ^
i
i
i
i I
+ E <- cos [(m- n)cü.t - nuj.T + (m- n)*.])
- cos
ä
(IHD.T)
i
when
n ^ m
when
m = n
(2.21)
The above simplification gives
;.(W) - 2 €m ^^t^) 1J\'/™'i)
cos
m=0
where
Im(x) = [modified Bessel function of
the first kind of order m]
r
m
1
if
m = 0
2
v.
if
m > 0
= Neumann numbers
From Eq. (2.19)
exp [RjT) wtwt,] = ^ i, [R^r) wtwtl]k
k=0
SEL-65-056
inj*Tjwuw\*AAfuy\jc\)Vv:<A)inK\MXftMAXA>uT*T
- 10
i^S) (2-22)
Therefore,
M(
n VWf)
= eXP
(I \)
eXp
exp [R (T)
(1 \)
n
Vf ]
^ ± [^(T) W^,]1
= exp (| wj) exp (| wj.)
k=0
(2.23)
which gives
R (t.f ) =
y
/- dw
/• aw
i-r
—i
—LM(w.w.)
2
W
X
(2..1)
* *
(2nj) -^C t ^C Wf
P
M)2 -^C
(2nj)
W
t •'C
W
t'
M
K «s.^t^t^
Sl
i=l
exp (| wj) exp (| w2,)
^ ^ [^(T) W^,]1
k=0
Simplifying,
00
,
Ry(t,t )
mtamsmTinananuyancMiyinv^^^^
«k /
^
V VTJ f wk-l exp /N w 2\ dw
^
1
^-p Z "kT- X t
(2 t)
t
k=0
n Ms.i(wt'wf
f k-l
/N 2 \
I wt, exp (- wt,j dwt,
i=i
[2.24)
- 11
SEL-65-056
Define:
1
^ ^r
w^
2nj J
hk(m1..m
i
c
(wVisT)
i=l
exp f- w j dw
(2.25)
i
With this definition, Eq. (2.24) can be rewritten as
Ry(t,f) = Ry(T)
00
oo
-1 1
k=0
R (T)
V
n
2
Z -TT-hk)m11..mp ( H
m =0
m =0
\i=l
P
e
m.1
COS m
AT
which gives the autocorrelation function of the limiter output as
y
y H^^i
Z_( •■■ ZJ
k;
k,m
v-o
= y
y
ZJ
k=rO
m 1 =0
[
p
+ cos (mi
A
A
1
Fi
COS
[cos (
Ii\\
11
\i=l
* m cosm.o..
ii T)
(2.26)
1
cosines can be expressed as a sum of cosines
r
(cos m.fe^ = -iy j^cos (m^,^ + m262
+
P
m =0
P
A product of
T
. .m
A
(raiCl
Vi i
. + m !; )
P P
+ m ()
ä
+
■•
+ . . . - roc) + cos (m P
Ä
m 6
pp
2 2
4
+
!
VI P-I
>(WüytWCTWV7«A'/uyrwi^</^jAWjywvy///<or^
-
)
VP
■■■2P'1
+ . . . - m
f
, + m e
p-1 p-1
p f
terms
]
)]
VP
SEL-65-056
+ m 6
11^2
(2.27)
12
where the final bracketed term represents the sum of
2
cosines by
definition.
a.
Simplification of the Contour Integral
h,
k,m, . .m
1
P
In evaluating the contour integral h,
, the first
k.m,..m
1
P
question to resolve concerns the pole at the origin.
k,m,..m
1
P
_1_ T
2]l
J Jr
k-1
1_ f
2nj Jn
k-1
11 Im. ^v^
i
P
n m. ^^
i=l
exp
exp
1
/N W 2\
dw
12
)
/N
2\
(2w;
dw
(2.28)
Note that for
w -> 0,
I (w) -> w /2 p I
(see Watson, Ref. ll), the above
integrand becomes
(k+Em.-i;
w
as
constant
w -> 0
The integrand is well behaved along the whole of the
axis except possibly at the origin.
vanish if either
1.
k
w = jv
The pole at the origin will also
or any one of the
m.
is greater than or equal to
If the dc term is eliminated from the covariance function, no pole
will occur at the origin.
All the limiter inputs have zero mean, making
the integrand analytic along the entire
w = Jv
axis.
integrals can now be reduced to a single integral.
k ,m.
2nj
rftWA/VWLVJW^A^W^JV.Vu^JV^^^
r
Hence,
P
k-1
T]
lx 1 m, (wT^s
V
1
i=l
- 13
1
The two contour
exp
(I w2) dw
SEL-65-056
.V.V_-.-.V.V.V.V.V/,i.V.V'.._V.V,".V7A'
Converting to a real integral by using the relations
(jz) = j1" J (z)
I
m
and
w = jv
i1
yields
~ -L f
k(m1..m " nj J^ J
k-1
v k-1 ex
/
N
2\
P(-2V/J
^i
rr
,
i=l
.
1
.
(k+Zm.-l)
i
7 J
j
p (-1v2) n
/■0
m
i
i=l
_
.
J
P
(k+Lm.-l)
i
i
(k+Em.)
1 -
^
i
v^1 exp (-f v2) f[ Jm (vyiiT) dv
i=l
.
1
(v/^T) dv
P
0
J
j
v^1 exp (-|v2) U Jm (v/^-) dv
J
n J
,
i
i=1
-^
(k+Zm1-l)
1
ex
^^
•/-»
,
IT V^/^) J dv
(-1)
-oo
J
p
1
2
v*- exp (-|v ) [[ Jm (v/SS") dv
i=l
i
Now
r
2
for
k +
2».
odd
0
for
k +
2».
even
(k+Im. )'
1 - (-1)
SEL-65-056
IAJU\IUTJfcaM/VJtTJUlJUAÄnJWTJW^\JWV^/v*^
14 -
L/W i«i u^ ^rv luTi'-.''-
L^
.-»
:,-V L.-«
_-» ..-% _"■-TT ^-fc .
: l-i » WW_Pt»_«_«.
and hence
'2
\k ,m, .
1
.m
P
-1) /•«
(k+lrn,-!)
1
P
_
H Jm (v/ÜT) dv
1=1
i
v""1 exp (- | v2)
I
■'0
for
/
fk
V
\
^m.J
/
+
odd
=<\
(2.29;
for
b.
k +
ra
(
I i)
Computation
of the Integral
r
b.
k.m,..m
1
P
It is convenient to define a quantity
3
k , m, . . m
1
p
\2/
(2.30)
k, m. . . m
1
P
In the autocorrelation function, this quantity appears with an index of
2, hence the sign of the term will be dropped and the quantity redefined
as
k/2
G„ /N\
r v k-l
[zj
J
o
/ N 2\
exp (- - v j
Jl
1=1
b
„
J
(v/ZS^) dv
m
If
k
^«^
+
odd
i
=<
1
P
(2.3i:
otherwise
It has been found preferable to adopt the following two forms:
1.
Strong Noise Case.
N
is greater than
S.
for all
i r (l,p).
By
an appropriate change in variable and subsequent simplification,
r i
r k-i
^ -i
k ,m, ..m
1
P
x
exp
/ x2\ r.
T
V Ti iLi Jmi
dx
if I k +
N
m
)
odd
(2.32;
*
olherwi se
- 15
rr vm
'L.-*
ur« ■!/■*
SEL-65-056
ftrw.-'Kn-u*j**M*jTnjirui r^^^.rji.TJiirj^v r-JindR./vv; i/v'il\J iTJlTJir. CUITJ r. »'• ^V (TV W^rt^ r"1* T¥* k r
L."'X
' «^
IL' ■L.'Vf
iLnicnX-IW
2.
Strong Signal Case.
Let
S
> 8.
for
1 e (2,p)
and let
S
> N.
Then by another appropriate change in variable,
1
1
p
2
/NV/
r "-1
/
N
2
j
n
(2.33)
dy
i=l
Defining
7, = ^S./S ,
7^^ = I
and
then by hypothesis
7. S 1
for
W 1
hence
fl
,
1
k
'2 -
k.m
1
k/2 yk lexp
^ f ' f^yUvv)dy
if k+
2
^
/NV/
odd
P
(2.34)
otherwise
It is easy to express these forms in terms of an infinite
series involving confluent, gaussian, or generalized hypergeometric functions.
But it is better to deal directly with Eqs. (2.3l) - (2.33) when
deriving asymptotic properties and limiting values.
Due care must be
given to convergence problems.
For numerical results, it was found to be most convenient to
carry out numerical integration on the computer directly rather than
resorting to the evaluation by summation of hypergeometric series.
See Appendix A and Refs. 12-18 for additional formulas and tables pertaining to integrals involving Bessel functions.
SEL-65-056
jijiAjtnjrruiTuu\*n*juifu^AAJvuuLvu^^
- 16
-A.v.r^'j«-.*'.* v *> r>.?> •>">•>_■'> ^>->'>">'
c.
Wideband and Narrowband Considerations
Based on the simplification and computational procedures
discussed in Sees, la and lb above, the following expressions that are
valid for any bandwidth are obtained:
oo
R
y
(x) =
oo
>
/L
k=0
k
oo
(T)
R
P
>
...
> ——,— b,
t-i
Z-,
k!
k.m. ..m
1
P
v
m =0
m =0
1
P
I
i
-
e
iX -1Y
m.
1
cos m.cjj.T
i i
(2.35a)
where
R
n
is the autocorrelation function of the input noise,
(T)
Using the expressions and notation of Eq. (2.27), one can
rewrite the above equation as
oo
oo
oo
, , = V
R (T)
V
V
m=0
1
m =0
P
y
Z
k/
\
R (T)
n
Z ••• Z -T^- k.mi..mD
k=0
. . .
P
hv2
""l
e
JP
^FT—
• cos [(m.cü, ± . . . ± m to )T]
11
P P
The Fourier transform of Eq.
(2.35b)
(2.35b) gives the power spectral
density of the limiter output,
oo
s yK(f)'
=
y
La
k=0
oo
(»6
...
£
2
y
...
y
-^—^h
k
P
JL
L,
' 2
k,m
1..m
k
m =0
1
m =0
P
k
[s
(f)
.s n-rf)iJ
[n
"
* {&[f
- (mf,
± ... ± m f )] + 5[f + v(m.f, 1 ... ± m f /JJ
)])
11
v
ll
PP
ll
pp
(2.36)
[k
and
1
S v(f) * S v(f)
implies k convolutions of S v(f) with itself
n '
n 'J
n '
S (f)
is the power spectral density of the input noise.
Kmuraxn)itA>üWüw*JTV^:</^^
17 -
SEL-65-056
^%^\^s<:^v:^,0<.T^'!:-s.lA*>N,rNi,rX,"3s.'CV*:
With the following points in mind, it is only a matter of
straightforward computation to obtain numerical results for specific
problems:
1.
As
k
increases, the
k
self-convolutions of
S (f)
may
rapidly become gaussian.
ma
y
l3e
a
2.
^LvO
etc.
reproducing density, e.g., exponential, gaussian,
3.
Convolutions with delta functions are particularly easy.
4.
In practice, one is usually interested in a certain spectral zone
and in some cases in a few terms, which simplifies matters.
5.
Use should be made of the fact that
the terms where
k + Em.
hk)m
m
exists only for
is odd and vanishes for the rest.
Hence
the terms to be computed are halved at the outset.
6.
The value of
h
k|mi..m
m
ay be so small as to be insignificant.
Up to this point no use has been made of the narrowband
restriction.
The expression for the autocorrelation of the limiter out-
put is valid for any bandwidth.
Assuming narrowband restriction and normalizing.
Rn(T) = NP(T)
and
S (f) = Na(f)
where
o(f) = I
(0) =. f
so that
a(f)
P(T)
e
dT
o(f) df = 1
is a probability distribution function.
has a density which is even about
The input noise
f ,
R (T) = Np('i) cos UJ I
SEL-65-056
- 18
i*i»»iviiVMuw\«riÄ\^w^ÄrLr^vvv^uw\jvv'*iLVL^n_v\A/--vuv,Aajv'^v'.^-> ^-'^., v" v^
^TK
^tnk^vvr. '-•. r. •;
T.'-'.''■'.>
.» .
•.•W^-FJ*?J<^>"_X'ü"
"J'.,'^,"^.V."^-1
Substituting in Eq. (2.35b),
.X)
l-W
,M = I 2-1
k=0
k!
m =0
P
m =0
cos [ (m tu ±
k,m,..m
1
P
k
k
p (T) COS (CDCT)
1
m
—jp
(2.37)
± m m )T]
P P
To simplify the above, use is made of the following expression [Ref. 7,
page 298]:
(k-l)/2
Ttl
I
(k-O: i;cos (k-2i)
c
for
k
odd
i=0
cos (CJJ T) =/
(k-2)/2
k!
k-1
2
(k-O! i!cos
(k 2l)
-
V^^TFÄ:
for k
even
(2.38)
It is readily seen that the output power spectrum is centered at the following carriers:
(k - 2i) co
± m CD
19
±
...
± m U3
(2.39)
SEL-65-056
*.nMrMrMPjtvMK*rj(Wjt\*-*j(HMnx'i>± »Jinif nj.,?^m^V>J(X>>l>^>^>^>J^Vf"JO"Ji>i>>f^>^>>J.>J.>^J.^^^>i^^^
Further
X
cc
i i ■■■ i <v.rap v^
J
*=! m =0
k=l
k odd
m =0
i
Epkw
P
(k-l)/2
i
2
(k - i)l 11
.=o
cos (i (k - 2l)a,c 1 m^j i , . . + m^nJ]T]
P P
1 1 ■■!
k=0
m =0
k even
Ry(T) = <
m
k,m, . .m
n, =0
P
cos {[(k - 21k
1
p
+ m,^, ± . . . ± m a
cos [(m1^1 t ... ± m a, )l]
P P
2.
P
,P-1
IT]
k
/
v
-p(^
(k-2)/2
V
2
1
(k-i)! il
+
If
k +
1;
if
k +
I-.)
is odd
Spectral Analysis and Filtering
Since the autocorrelation of the output exists for only the odd
integral values of
k + Lm. ,
the spectrum of the limiter output consists
of components situated in zones around the odd harmonics of the input
frequencies.
As stated before, the narrowband restriction virtually
eliminates interference of these output zones and hence any one zone can
be selected by appropriately bandpass filtering the output.
It is now assumed that the zone around the first harmonic band
around
f
is filtered.
The filtered output process is denoted by
z(t).
The general expressions for
R (l) and S (f) are cumbersome.
z
z
The analysis is not difficult but it is lengthy and requires careful manipulations.
the term
term
m
(
It is better to think that the sums of cosines constituting
cos (a
1"i
r)
are the carriers which arc shifted up or down by the
± • .. ± m o ).
ii
P P
conditions must hold:
SEL-65-056
xmiri&jrvQimu^^
To get the first-harmonic band, the following
20
,
>
'.N'.'. ."/.V'.%V.'
k +
N n^ = odd integer S 0
(2.40)
(k - 2i) ± m
±
± m
= ±1
P
which implies
|m1 ± ...
±
0,2,4,. .. ,k+l
for
k
odd
1,3,5,...,k+l
for
k
even
mp| =/
Now for each of the values of
series of terms is written.
m.
which satisfy the above equation, a
The final total of all such terms in the
first-harmonic band may in some cases be expressed in a single compact
form.
This is possible for small values of
p,
such as 0, 1, or 2.
For instance, this procedure, when applied to the case
p = 2,
gives the following result which is due to Jones [Ref. 8, Eq. (13), p.
35]:
00
R
2 b,
00
^=S 1
i=-oo
^=-oo
L
• cos [lilcü
II
C
1
k=|i| ,
-
kUI |^-|i+l||
i— P (T)
k +
|i|+2
0 +
ili + l|u)
12
2 -
Ü(üJ_
üOIT
1
where
N^/2
k,m1m2
Ml)
k,m1m2
Similar expressions can be derived easily for
- 21
p = 0
or
1
SEL-65-056
J«-«n-.«M^»i.-.«.« «n««'«B«fl.ÄM,ni.>liLnM-ni«._1»Jl>U.».'TIUM».1fc«»JTKJlKr«L1«Lr H.T^AIVaA/lSJ'JCT\i^)^A.N^^.l^\^rt.>A^^'^',_M.■W.■L^^_^i^^.\^'^ \'X'S'0-.'
It has been assumed that the input process is wide-sense stationary.
If it is not weakly stationary, the only change that would result
would be to replace
3.
T
by
(t ,t ).
Output Signal-to-Noise Terms
a.
Output Signal Power
Fortunately it is not necessary to calculate the series for
R
Z
(T)
to find the output signal-to-noise terms.
The expression for the
autocorrelation of the output consists of line spectra and continuous
noise components.
The periodic part of the output is primarily the result
of the interaction of the input signals with themselves.
The remaining
terms correspond to the random variations of the output, i.e., the output
noise.
These terms can be split into two sets, one corresponding to the
interaction of the input noise with itself and the other corresponding to
input signals interacting with the input noise.
Root's notation [Ref. 7],
R (T)
z
Using Davenport and
can be split into subsets:
p
p-1
R v/
(x) = R
(l) + V R
(T) + y R
(T)
z
s,..s
/.
s.nv/
^
s,s.nn
1
p
,i
.,ii+l
i=l
i=l
... R
nn
(T)
(2.41)
where
Rc, &„ (T) contains the output signal and cross terms [periodic
"1•• p
part of R (r)],
and R (x) represents the direct feedthrough noise
components.
One is usually interested in only a few terms belonging to
one of the
R (x)
complete series of
of
p
1.
subsets.
Hence it is not necessary to calculate the
R (i),
which is usually extremely lengthy for values
greater than 1.
The terms in
R
n(
0
are the ones that result from the interaction
i
of the signal
2.
The terms in
i
with the input noise.
RcS..i^cS.„(i)
1 11
are the ones that result from the inter-
action of both the input signals
SEL-65-056
.«»".»» .V.»«V.,V»iViV».t,v ".
- 22 -
i
and
j
with the input noise.
The very definition of these terms imposes certain obvious conditions on
the
k
and
k + Em,
m.
indices.
These conditions, together with the need that
be odd, make computing the terms involved a simple process.
The subscript
o
will be used for output and
The output signal power is given by imposing the conditions
m. = 0
for
j ^ i,
i
o
th
m
i
zone
Specifically, for the first zone,
= 2b
first
zone
for input.
k = 0
and
which give
(S.)
<Sl'o
i
2b
(2.42)
0,00..m.0..0
m. = 1
gives
0,00..100..0
-.2
8^
2
v"1 exp (- | v2) J^v/is^)
J
|!
J^v/äs^dv
(2.43)
b.
Signal and Crossproduct Components,
R
(r)
. . b
1
P
One can easily derive expressions for signal and crossproduct
S
componerts in both the wideband and narrowband cases.
Inspection of the
expressions for
must be 0.
R (i)
and
R (r)
indicates that
k
Hence,
in the wideband unfiltered case.
R
1
.s ^ P
^v .
m =0
• 1 "o.m,.1 ,m P
m =0
P
f ;
1
\i-l
r
m
cos m.ii . [
I
i
i
m,
m
>
...
>
b'
—1
: P cos [ (m
,i. -* ...im d )T]
v
^
i_^
0,m , .m
p-1
1 1
p p
1
p
2r
m,=0
m =0
1
P
TnaroTf^v.v^^'irrA-AV.T^VL'vtv^rAAM.v^'^VAKJVJ'*^
23
SEL-65-056
when
EriK
Is odd.
For the narrowband case this reduces to
V
V
\..sj^Z
P
•"•
m =0
2
n^ ' " *
b
Z
m =0
m
^l-^COS ^^l-'^Vp^
0,mii..mP
^
with the additional condition imposed on the indices that
m, + . . . ± m = ±1
1
P
For p = 1
the above reduces to
yRS(T) -
I
m1=l
m1=odd
b
o(ml tm1
cos
"VV
= 2
S
m =1
m =odd
b
cos
o.m
VlT
1
which is precisely the result given by Davenport and Root [Ref. 7].
the first zone, obviously,
ZRS(T)
For any bandwidth, for
IXi
{T)
y\s
2
1
^
=
S
m^O
For
=2b*(1 cos ay
p = 2,
00
2 ^O,™*
1 2
m2=0
C
m;m
o
1
2
COS m
i^lT
COS m
R
(T),
2V
m +m = odd
For the narrowband case (first-harmonic band),
condition imposed on the indices is
SEL-65-056
ICUltVMVKVKWVVVVmVVVVm^^
- 24 -
Z S -. So
the additional
m
± m
= ±1
Hence,
R
(x) =
z s, s„ '
>
Z-j
m =0
7 -fb.
t-> 2\0,mm
m =0
c
r
) cos [(ma, + m ^ )T]
W-, ™r,l
11
22'
m ±m =±1
i 2
Lm. =odd
i
Further simplification in this case is possible by noting that
cos [(m
+ mo0J9)T]
üJ1
contributes only when
mj = 0,
m
+ m
= 1,
that is, \
m2 = 1
and
m2 = 0,
/ 2
(b
This gives
implies
m
cos tu-T + b
= m
A
+ 1.
2
\
cos auTJ.
111^^=1
Also,
m
1
" ^
= ±1
This contribution to the first zone due to the
X
terms arising from
cos [(nUiX), - m cu
XX
£t £
)T]
is, with appropriate restric-
tions.
1 Ik.mrm II)
r
m/mJ
cos
V
L1
,
T
D^JT]
2
b
1
V
+
ir-V! - (m1
S
0,m 11
(m -1) rm 1 -1
b
0,m (m.l; rm1
cos
'^iS " "l) " ^^
COS
l[-miS " ^) " ^It)
m ^0
ll
2b
0, j. I JV1|
cos
Ifm^^ - a,) - a2]l)
+
b^10 cos ^r
+
b^01 cos
V
m1/l,0
25 -
SEL-65-056
t\j*m\rm\.m.\rm\rm u-n ^JW hn« W» L/H UKUII LTM LT» IT* L.1"»» L""* V'** ^'»X L >« k H -1» -■"» «'"X J"« _'"* WÜ-l«. ■.■"J^k.^to*-'*»«" «^ta"**» *_*• "Jü "J ""> "> "^ "Ji "J* ■"> "J* "i^ ."-AJ^W
Hence,
00
= 2
R
z s^
COS
S ^.[m^ [nyll
^1^2 " ^l^ " -2^^
m =-oo
(2.44)
This agrees with Jones' result [Ref. 8, Eq. (15), p. 36].
values of
lengthy.
p,
For higher
the computations though simple and straightforward become
Fortunately, it is not necessary to derive the above series for
the signal-to-noise computations.
Additional conditions are imposed on
the indices and these simplify matters extensively.
c.
Direct Feedthrough Noise Components,
R
These terms impose the condition that
nn
(x)
m. = 0
for every
i.
Since the autocorrelation function exists only when the sum of all the
indices is odd,
k
must be odd.
Am(T) = Z hk,oo..ok:
Hence,
P (
^
cos
V
k=l
odd
(k-l)/2
~ Y
Y
k=l
odd
i=0
-L
Z
(k-l)/2
I I
k=l
odd
For
zRnn(T),
2
k,00..0 p k,(r). k!
kl
cos p,,
[(k
2
2
-
,
" 2lKT]
0
k, .
k,00..0
K
v
(k-i): ü
'
r/,
cos [(k
„. v
"
2l
T
KT]
i=0
(k - 2i) = 1;
z R nn
that is,
1
k=l
odd
SEL-65-056
1
"kTT (k-i)! i!
i = (k - l)/2.
This gives
2
P (T)
k, 00. .0
k + 1
:~ - 1 ,
2
2
- 26 -
»,vavj>/Anjwv>wLVV«uT(vnikr*kTJiKMfjtAj(Mi^jirj«i-j«_rj ~j'Tj*.r*r^'*jfr^\*j*Tjj*:•*'■* •^JJTJT*'* 'jr*s*'* v •*'*'*•*'* ■jjiJiRjiK«"»^ w
The expression reduces to Eq. (l?) of Jones [Ref. 8] when
p = 2.
With
no restriction placed on the bandwidth, these terms for wideband cases
can be expressed as
00
«k/
-,
R
nn(T) " Z "Ti" hk(00..0
k=l
odd
V Nk
=
Z k!
k, . 2
p (T) hu
:k,00..0
k=l
odd
Also,
V) = 2lT\2,oo..ot("*°(£']
k=l
odd
C.
OUTPUT FACTORS DUE TO SIGNAL AND NOISE INTERACTION
These computations are usually lengthy though straightforward.
example, for
R„
i
n(0.
For
the conditions on the indices which must be met
are:
(i)
k +
^ m. = odd nonnegative integer
(ii)
m. =0,
(iii)
m. ^0
where
and
j e (l,p)
and
j ^ i
k ^ 0
Hence for wideband cases,
00
R
s.n
(r)
v/
= 2
GO
2 1
>
Z-«
k=l
>
sLi
m. =1
0
—r-,— h, nn
nn
n cos m.cü.T
kl
k!
k,OO..m.OO..O
ii
i
k+m. = odd
i
27 -
SEL-65-056
trpamjmjuxMcrjKmai-KxKmiL* KJUKjnutfjinjt^jrr'jcnjnijenM-n.ynjrKifKwr r».y,i>u *_k"K_k' rv^j« R_J<TU fji ••Jifj.Fj* »JI'^J« 'J* nvTJii>j^. .•_.•_!■' »»Jt"» > Ki'Kk'tyrty
For the narrow bandpass case,
^s.nv/
/L,
k=l
<L/ kl
ni.=l
k, OO..m.OO..O
pk(r)
i
^V
COS
m (ü T
i i
i
k+m. = odd
i
(k-■2)/2, k even
(k--l)/2, k odd
a)
00
1
V
k=:l
I
k
< 00. .m.O. .0
( :)
(
m.=l
(k-j): j!
j=o
k+m. = odd
i
Note that the
cos {[(k - 2j)a
- m.u>. ]TJ
cos {[(k - 2j
+ m.co, ]TJ
)üJ
terms are not permissible
owing to the condition on the indices.
For the first-harmonic band, further conditions are imposed,
k - 2j - m. = ±1;
k - 2j = (m. ± i;
k - (m. ± l)
k + (m. + 1)
Also when
k
i
k - j =
J =
is even,
m. = odd, which implies
k = m. J 1.
'
Applying
this and simplifying,
R
z s
n'^ = 2
i=-cxi
*
cos
SEL-65-056
XüVinnnrjhriM\nc\j^njvv^\^\r^f^\j¥\rä)narM^^'
k= i
I
,
2b
i +2
[\l\a-c -
k,00.. i+1 0..0
k +
i
1
i
k -
(i + l^ajl
28
M^*
it * M^.M^aiP-M* hi **• "Jt "'.to f_M - ■ - - fv> -J» r^J* ->
This result reduces to that of Jones [Ref. 8] when
p = 2.
Similarly,
expressions can be developed for the other factors of the autocorrelation
function of the filtered output.
D.
SMOOTH LIMITERS WITH ERROR-FUNCTION TRANSFER CHARACTERISTICS
The above results are applicable to the class of smooth limiters
which have a transfer function describable by an error curve
(x/a)
d|
(x) = erf (x/a) = -p: /
for
x > 0
for
x = 0
for
x < 0
J
/K
0
g(x) =/o
(-x)
by simply replacing
N
in the above analysis by
seen to be valid for any value of
smoothness of the limiter.
p.
(N + a ),
The value of
a
(2.45)
which is
controls the
For the case of one signal in noise, see
Galejs [Ref. 19], Blachman [Ref. 20], and Jones [Ref. 8].
E.
ASYMPTOTIC RESULTS
1.
Case I, Weak Signals
Since all
than
S
p
for all
signals are buried deep in noise,
i 6 (l,p).
=
3
0100..(m1=l)0..0
is greater
From Eq. (2.3)
1/2
2 f00
n j0
N
1/2
exp (-x /4) j
x
1
dx
J^1
(2.46)
29 -
SEL-65-056
m «MJtA mx MA «.* SAW.I. r. \ x rrn vx vn v R^Tttf MUMFJl rjl *JU\*L'J'J,AJ,*J-*S*-fJ W W rfVTJ <V )fV TV KVJ'VrAjrit.T VTiLF.r, »'«.TXAVfl Jiflk.1" H5 *.* "J^ LTJ-'.,».'
For further evaluation, use is made of the following arguments;
1.
The power series representing the Bessel functions
+ 2n
,(z) = n=0
1
nl(n + n
is absolutely convergent for all values of
x,
real or complex,
less than infinity.
2.
Hence the power series obtained by the multiplication of a finite
number of such series is also absolutely convergent for all finite
values of
x.
For detailed proof refer to Watson [Ref. 11, Sec. 5.41, pp. 147-148].
In
particular.
1/2
1/2
P
li
J^1
A
1/2
>©•#)
x
2
+
1/2
(where
c
,
n ^ 0
c2x
are the terms involving
This series is absolutely convergent.
+ c4x
S./N
- c6x
...)
+
and its higher powers.
Therefore the integration and sum-
mation can be interchanged, giving
SEL-65-056
u^u^TU(KJ^.^Jrnl^Tü^^icn*^.J^^uu\*^>-^*TyJVL^^
30
,
•J«-J '^V
■.'•.'> v v ■>•>"J- v -J '• '> ■>
exp
3
0(00..10..0
n 2 yNy
r T /dx
^j0
2
- c.
exp
exp (- ^p ) dx + c.
Jo
•'o
Since each integrand exists for all
dx
,-T
n g 0,
1/2
J
0 ,00 . . 1. . 0
Furthermore,
since
N
it V N
c ,
c ,
x
exp (- ^ 1
^■
/o
etc. involve
S./N
and its higher powers, and
can be arbitrarily chosen large enough such that the higher
order terms can be neglected, then, for very large
for all
dx
C2
N
such that
S^/ft
i e (l.p),
1/2
^.OO. .10. .0
Also, under these conditions, for every
power in signal
i
(2.47)
\n N
i e (l,p),
the output signal
in the first zone is
(^'o-fhf
(2.48)
Hence under strong noise conditions, input signal-to-signal ratios are
prest-ved, i.e..
S /
^ k/
o
\S /
\ k/.
' i
&WKK'T£XM^\L*)Lr±A)Ut*AWKÄ^¥-H^**J'.Kf.^A*WK*^-mJ\l^r^
for
31 -
(2.49)
j,k e (l.p;
SEL-65-056
<." O O-.1^ ^.f'■t^Vf ^
. L<'^S\.<''j:H]r*X)Cr*Jr''i.*
To compute the signal-to-noise ratio at the output, the total
output noise power in this case is given by the
noise power in
z(t),
N
o = Rnn^
+
defined as
1 Rs.nW
+
i=l
N ,
n x n
terms.
The total
is
+
lhs
JV
i i+1
••'
+ R
s 1' ' s p «»
From Eq. (2.32) and arguments similar to the ones used above,
/•00
k, m . . m
1
p
n2
As
N -» oo
as
Si/N -<• 0.
^ Jo ^
A
such that
Hence
2\ p [/y^x]1
/
exP
Si/N -► 0,
bk
00
0
(■
I ) li
i=l
then
higher
order
terms
m
bkim
for any
m. a 1 -* 0
is the only term to survive.
the noise at the output is contributed by
n x n
dx
That is,
terms only under strong
noise conditions, in which case,
x
k , 00. . 0
k-1
0
k 1
,2 - Jo
2\ P
/
exp f- ^1 ^ Jo
^
0
\ 4 // i=0
1/2
dx
/ x \ k-1 J
exp I - — 1 x
dx
exp (-t) t^/2)"1 dt
•'0
which by definition of the gamma function becomes
b k , 00.
SEL-65-056
i*XKi*Mcw^w)orvnMV\]i*^^
.0
=ir(M
n
\2/
(2.50)
32 -
•/.%•' ■•^>,'>.• ■ ''..-'v. ■> •> '.■ W_"J.r^r/r/."jv,
The output noise
N
is given by
r2/10
R
S
nn(0)
= 2
Z 3
k
n
1 , k - l"
+
k=l
odd
Substituting
k = 2J + 1,
oo
N
2
=
o
r2
V^ ^
R (0) =
z nn
^
Z
71
\
2/
JTTJ +
i):
J=0
Using the generalized factorial notation given in Ref. 13,
zRnn(0^ =J2 V^/^S
j=0
= n 2F1 (2' 2;
2;
j
(2).3 j!
J
V
Q
—
n2
In the above equation,
the second kind.
for
N -* c»
and
E(x)
denotes the complete elliptic integral of
This result could have been predicted by observing that
(-l.+l)
square wave.
the fundamental zone is
(s/n ).
Since
-» (8/n ).
When
(2.5l)
1
the limiter output is a
N
S./N-*0
N -♦ 00
S. -> 0
in such a way that
Hence the total power in
for every
(S./N).
-> 0,
i e (l,p),
the output
signal-to-noise ratio is given by
mivMÄnKNWi*iwjav<?w*w
- 33 -
SEL-65-056
for any
N
that is, a loss of 1 db approximately.
i e (l,p)
(2.52)
Hence, when noise controls the
limiter, the following general results apply:
1.
All input signal-to-signal ratios are preserved when hard bandpass
limiting occurs, i.e., signal suppression effect is absent.
2.
The signal-to-noise ratio at the output for every signal is 1 db
less than the signal-to-noise ratio at the input.
Particular cases of the above have been derived previously by Davenport
for the one-signal-in-noise case [Ref. 4] and by Jones in the twosignals-in-noise case [Ref. 8].
2.
Case II, One Strong Signal
a.
Signal and Crossproduct Power
In this case, one signal is very strong relative to the noise
and to each of the other signals.
Let the strongest signal be
S .
This
is no restriction since the numbering of the input signals is arbitrary.
Here
(S ) S V S. + N
X.
/A
I-,
\---2
1
It is also given that
i
and
> 0
P
(l.p)
for
i 1
The limiting value of the output strong signal power
evaluated under the above given conditions.
2 r
J
0,10..0
cxp (-Ny2/4S )
(S,)
is
2b
,
From Eq. (2.33),
'S.\l/2
P
dy
n j0
i=2
The integrand exists for all values oi positive indices on
y
and, by
the arguments developed previously,
SEL-65-056
- 34
• »viwMirjiruirairuirv*-. »'. F-. f ■« «^ r.-*n\.-*~* * .* i fv r. «f-. <•.*-. r. <r. «"^ ir. rf'. »rj>r. <v c . <
J
t v • . » , » • C. '.
'.'.T.'.'. f» " t' . • . • . '
(S./S ) -»0,
Jr00
Lij1(y)dy
i ^ 1,
and
for,
(S^N) - oo
Since the Integrand is 1,
b
-* 2/JT
0
and therefore,
(si)„-2(f) =
_8_
2
which is, as expected, the total energy in the first-harmonic zone.
An alternate way to arrive at the above is
r
-1
y
o
N \
exp
Ä
y
2
^(y) dy
1/2
1
/sl
-)
s/^V
Noting that
I (z)
n
e / / 2nz
ex
P
-2N,
for large
0,10..0
0 V 2N/
+
1 \ 2N,
z,
n
which is the above result.
The limiting value of any of the weak signals at the limiter
output is
^A -
2b
0,0..(m.=l)0..0
M
i
35 -
SEL-65-056
«J-l-BJ «Jl lUMd »-n «."MUMLAH-n m-IM-n ».1 »IJ\^^,IVlJV\JV\JUWUVL^r_V\JV^'V VhV^ WUVUV -> UN V> L% IT» \."N L"« LV O -> LT« L> ."» V"^ l"» '.'. ■>
where
1/2
3
0(0..(m.=l)0..0
f00 1eXP /
= 2
* Jo y
N y2\ JT
[-ls-i ) l
1/2
dy
Jo(y)
j=2
Since
(S./Sj^). -> 0
and
^0,
(N/SJ^).
it can easily be justified by
use of the above arguments that
1/2
Jrco
-1
y
J0(y) dy
1
0
This integral is the well-known Sonine and Schafheitlin integral.
0,0..(m.=l)0..0 " n
'^^//2 ■■ (j)
2r(2,r(i)
l/fi
(Sn/N). -00
1'
1
and
- 2' 2'
' Uj
1/2
as
n I S,
Theref ore, for
F
Hence,
(S./S,)
i'li
is
^~m
SEL-65-056
uini^i™ ruwvi^ yvifv^vvAn^^vv-A'A^vrJL-^JVVVLVJ^ ^
36
S./S, -* 0
1' 1
(2.54)
^0,
(2.55)
Hence, the output weak-signal to strong-signal power ratio is
for
(S/N).
1'
i -* oo
S
(S./
i). ""
i'ii
and
0
(2.56)
Therefore, if the strong signal controls the limiter, then all the other
signals are modified such that the output weak-signal to strong-signal
power ratio is one-fourth of the input power ratio.
In the case of two
signals without noise, a similar result is given [Ref. 8].
It is easy to evaluate any interfering component under the
above conditions.
It would be convenient to adopt the notation
S
„
„ „ to mean the crossproduct component at a frequency
00. .n^O. .m-O. .0
(mm - m cjj ). An example would be the evaluation of the intermodulation
J
J
component which occurs at a frequency of
start with the expression for
for
(S /N). ->
OO
and
VRS
s
(S./S^. ^0
(2^
(T)
for
-OK).
It is best to
in such cases.
i t
(2,p),
Therefore,
the ratio of the
output power of the strongest intermodulation component to the output
power of the weak signal is
S
200..iOO..0
v
When
p
i
(2.57)
/
signals are present at the input of the limiter,
such that the strongest signal
S
controls the limiter and all the
others are weak signals, the output power of the
(2^ - en.)
intermodu-
lation component approaches that of the power of the weaker signals
[Ref. 8].
b.
Noise Power
To determine which of the terms in the output power contribute
to the noise, the following analysis is adopted:
N
o
-
R (0) + R
(0) + Rc (0) + ... + R
(0)
z nnv
z s nx
z s2n
z s^.s n
3wro\Rr^;vL'^^^wnwwraA«cv<^K^^
- 37 -
SEL-65-056
Using Eq. (2.33), it can easily be shown that
m./2
/
\ ^
" ^ \V
k.m,. .m
1
p
S,/S1 - 0
if
m. ^ 0,
y
io
higher
1 - order
terms
Since
m.
k 2
;:
i=2
—
2
1
m. !
i
Jm (y) dy
i / 1,
b,
-> 0
k.m,..m
1
p
if any
m^^ / 0,
i c
(2,p;
(2.58)
The above equation Implies that the terms which contribute to the output
noise power are
n \ n
k,m 00.
Since
(S
/N).
l'i
power in
k,
->
i.e.,
x n
(S./S,). -» 0,
i'li
k = 1
n \ n
2b
s
terms and therefore
o^feff^V^
and
OO
belonging to the
terms and
are retained.
The first-order component
subset is
1,00. . 0
i: o:
the terms involving
b the lowest
/ \1/2
12
-> 2
_2_
X_'
(2.59;
n2 \Slt
The rest of the terms in the
(N/S
).
are neglected.
must be even for
SEL-65-056
ÄVKVWVV\V\MV^^\NVWW\AV^^^
n
n
subset containing higher powers of
Also, irrespective of the value of
b,
__ „
1,111,00. .0
to exist,
'
38
m ,
which
/ \1/2 r
/ \1/2
Furthermore,
iX)
R „(o)
y s1n
00
-I lh<
k=l
z R s n (o)
2
nn
n
m 00..0
COS
(X) T cos m mT
c
11
T=0
m =1
2 C0Sa) T cos
c
(I ^oo-.o)
^V
T=0
filtered
14=1,
m =2
= 2b
1,200..0
_2_ /JN_
(2.60)
.2" \S1>
Hence for
(S
/N)
. ->■ oo
and
(S./S ). -»0,
i e (2,p),
the total noise
output power is
N
2
o ^2bi(oo..o
2
+
4 / N
^l^OCO^^li^
The output signal-to-noise ratio for the strong signal is
*.-©,
(2.61)
The 3-db increase in the output SNR, which occurs due to hard
bandpass limiting, applies only to the strongest signal, which controls
SEL-65-056
- 39 -
■ Lm \Jim\jnm
VJII
\^
LTK
u-* vrm \nt i.'Oi L'-H V'H ^Ti l/H IT* J^ -m U'-» '^"n
L.'
M JUL^wW-^^'i-'PL "^.P. W
*_M
»to ■_* \ji "J« "V*! "Jt PJ« '> "JWTU* .'*J*."\Rf^Ä"bHr^.
^
J*.,
the limiter.
This result for the case of one signal was given by
Davenport [Ref. 4], and this analysis also checks with the one by Jones
[Ref. 8].
For weak signals
S.,
i ^ 1,
(2.62;
N
/o ^ * Wi
Each of the remaining
(p-l)
weak signals suffers an SNR
loss of 3 db when the limiter is controlled by the strong signal.
3.
Case III,
a.
p
Signals of Equal Strength
Signal Power
From Eq. (2.43) it is seen that the output signal power
depends solely on
b,,
m
m
.
The condition that all the signals are of
equal strength gives
i)o..o
0,00..( v
= 2 f
-J0
y
-1 ex
/
N
2\
.
Tp-1
^-^ j Vy) Jo
w
(y) dy
(2.63;
As a result of the symmetry of the integrand, it can be concluded that
all signal output powers are equal;
/o V1 - (-if^2) v^r1'
For the noiseless case and
exist [Refs. 7,8].
y) dy
p =1 or 2, closed-form solutions
For all other cases, solutions can be developed in
the form of an infinite series involving generalized hypergeometric functions .
For large
p,
the following procedure culminates in a useful
approximation.
SEL-65-056
- 40 -
mutant ■urjnkMKxnjifuiLrjtrjtrjf.rji r*n*rj'rj>.rArjT*.\Ar*J-jj-*."j,r* •\*rjßrj.rj'* ■jTjt.rs** \f -^ '.»'.* "J<r.» •j<^inj«Rji-FjiHi»_wnif»_ii"^vi«^T«_vxii*'
Define I
(P) = £ y"1 exp (- ^ yj ^(y) ^"'(y) dy
Comparing the series
Jjy) = i - ^
„2
2
j
-^
„2
„2
2 • 4
+
/
2
4
- Mi - —^
+
y
2\
2 • 4
2 • 4 • 4 • 6
M
yy
:r
with
exp
exp
and u^suming
p
=
2/
1(1/
:l
" "2
= 1
+
\2/
2T
2 • 4
large, one obtains in the region of interest,
J
0(y)
~
(ir
ex
^(y)
P
exp
(2.64a)
KD'
(2.64b)
Hence,
1
2
I(p)
Jo
2 exp
41
■«■■MranjricjtTtÄ-iuinjnjrjtjrr_»nji-RjiPJIa»11«JiMi'«.k« n^i Rji^jinji r.*fj>
RJ<tMi PJI
+
J5_
S \ 2
i \y
dy
SEL-65-056
rJiPJIrj<"VJ«rj« "J* rjini." »' . "j. • ji i").FJI• > \j«
"_»«>^JIP^
\«■"..»»<"» v"»'
By a suitable change of variable,
I(p) «
exp
/2n
2n Jo
Since the integration has been carried out to infinity, it is preferable
to replace
[p - l/2 + (N/S^]
in the denominator by
[p + (N/S )],
giving
[i(p)r
(-^T)
The normalized output signal power is by definition
(8/n2)
1.05 + 10 log10 fP + "—
in decibels
(2.65)
This equation implies that the output signal power in decibels decreases
linearly with the number of signals.
Also, if the noise power is equal
to the signal power, then the output signal power for
noise is approximately the same as for
(p+l)
p
signals with
signals without noise.
To verify the above results and also to justify the validity
of the approximation, the exact output signal power was computed by
numerical integration on a Philco 2000 computer and compared with the
results obtained from the above approximate formula.
The above formula
gives results which are within 0.1 db of the exact value for
p
greater
than or equal to 4.
p
as high
The results are tabulated for values of
as 100 (see Table 3, p. 61).
b.
Crossproduct Power
The output power of, say, so..m-0..m■..0'
interference frequency of 1 m. a).
m .co.
is given by
x i
J J
SEL-65-056
which
has
an
42 -
wu^rtRAJ*JV^A^T^nJW^J^fJvuvlJW^AA.lVvv\.'v^JV\.'v\rt^•.v i-v 'jv uv uv JV uv irtt s* i.> JV _> c« *> ..> J-H u> «> -> irh u> «-» vr .> WT W> y» k> v« »-•J.^JI -»^ »J.
0..m.O..m,..0
i
J
R
(0)
z s^-.s^
appropriate
term
T2
2 hn jr y-i j m (y)
L o
i
2
Jm (y) Jj" (y) dy
J
= s 0.,m.O..m...0
(2.66)
which by definition has an interfering frequency of
l"1^. -
m
i^A
■
Also
note that
i i
j jlm.=m -1
J i
I
i i
CD.
1
j j
j1
+ CO .
J
(m. -|)
(CD.
-CD.)
(CD.
-CD.)
and
Im.cjü. - m.coj
, = Im.ü). Iji
ijlm=m.-l
lii
CD.
CJü.
i
- m.üü.l1
ij
+ CO.
^-rJ-- (m. -g)
It can be concluded, therefore, that the output crossproduct terms situated
on either side of the arithmetic mean frequency have the same power.
Similarly, it can be shown that the enitre output spectrum in the first
zone is symmetrical about the arithmetic mean frequency.
Other results can also be derived when the inputs have equal
powers, e.g..
olm^mi^ü!Xk7;.K■n*AX^^xfA^«^^
43
SEL-65-056
(power at
power at
Output power at
I1 m
CD
m .u).
J J
i i
=
i
m .CD
j q1
m .ax
J i
m.a: .1I
1 J
i m .ui.
J i
m.iv I
1
power at
i
i ql
etc.
v.
Therefore, if some of the signals are of equal strength, the number of
components to be computed decreases very rapidly.
For the case
p - 2,
it is easy to compute the entire series
comprising the signal outputs in the first zone.
The formula (see Ref.
21)
(M - v)n
2 sin iV-
•'n
J (ax) J (ax) — M
v
x
/ 2
R(n + v) > 0,
a > 0
2,
gives
V^Vi(y;
1 2/o
dy
q=l •-
.
64
4
1
1
V
+
2
2
{
^ (2q - I)
1
~2
2
+
3
1
~2
2
64
4
I
u^
(2q - 1)'
+
5
(2.67)
SEL-65-056
44
MOWWIMAWWIWV,■MMm<WNWf:KtJMM*WWr^*JW
which is, as it should be, the total energy in the first zone.
This
special result, which states that the power contained in the intermodulation components falls off as the reciprocals of the squares of odd integers, was given by Jones [Ref. 8].
s
ii..ioo..o
is given
The output power in the crossproduct
by
q
p-q
times times
iqop-q
iqop-q
norm
2
f
-i2
1
r
p q
Jj(y) J - (y) exP
(2.68)
dy
4S,
The integrand is best evaluated by direct integration on a computer.
find an approximation to the above integral for large values of
following procedure was adopted
iqop-q
exp
JL
4S,
p,
To
the
Equations (2.64) give
y
2\ y
q-1
exp
norm
(ir
exp
(P
- <,) (ff
Sfi^p)!2
With an appropriate change of variable and subsequent simplification
2q-2
i (p)=
i
z(q-2)/2
^r
2 dz
.(q-2)/2
p
p
-2
+
^
-2+s"
z
^
45
m-M-rv m\i
WMT
w^zv^r wx.
WM W-J JTV W J
rv wv w- »-- r^ rv W-- nv «TV WJ w- « * »ru WV
(q/2)-l -z ^
'
e
dz
2
~^W
p-^3-
SEL-65-056
K
V « ■« "f - « ■ « W WV «"■
<■- <•. w- <*- «r- rv ru *r- »rw «r- *v ir» »ru irv irjn
Since the integration has been carried out to infinity, it is again preferable to replace
1/2 + N/S )
(p - q/2 + N/S )
in the denominator by
(p - q/2 +
to give
^(p) =
il)
2
+
2
+
S"
which gives
r
d)'
(2.69a)
q p q
i o -
norm
p -
= - <10 log
q
2
+
i+ N
i
s
10q log10 /p - |
10
(!)
+
i
t
i
(2.69b)
in decibels
Equation (2.69) gives the following results in decibels:
For
q = 1:
(S
For
100..0yo
1.05+10 log10 (p + |\
1
(2.70a)
7.07
(2.70b)
q = 3;
^S11100..0^o
For
« norm
+
30 1og10
p- 1+-
norm
q = 5:
^S111110..O^o
3.55 + 50 log10 lp-2
norm
+
fl/_
(2.70c)
This result agrees with Eq. (2.65).
SEL-65-056
46
iiif«vii\«uw\Ä jvv'wv»rirw\ftrv"wv%rvwi*iLSiL^iTXiJVAfVJV,A,,Vv\.vi,».-i Jti «u" >v":«.! »J-WAKX . V. <. C, >-, ,■■.■,■ ■■(■■.■.: v> . • . 'jT^rfMfJtrji'^rj' 'Jf'jß '*'*.'+'V
For
q = 7;
(vo4
For
70 log10 lp - 3 + —j- 4.41
(2.70d)
90 log10 fp - 4 + |-j - 15.29
(2.70e)
norm
q = 9:
9 p 9
i o - yo
norm
All the above expressions give results which are very close to those
obtained by numerical integration on the Philco computer for large values
of
p.
Some of the results are given in Tables 5, 6, and 7, pages 63-65.
s
The output power in the crossproduct
is
r
a n
q P
a
r
given
b
y
2 l 0
JT y"1 dy exp (- J- y2^ J^y) ^(y) ^(y)
2riq0p-q-r
norm
(2.71)
^[I2(P)]2
Proceeding on similar lines and approximating the Bessel functions in the
region of interest as indicated by Eqs. (2.64) results in
~
oo
I2(P)
I
/p.
2r+q-l
3r+q
dy exp
\
2r
+ N ,
3
s
^
q
2
4
/
y
_
By a suitable change of variable and simplification,
-r
I2(p)
„r+l /
2
p
q
2
2r
T
+
1
2
47
+
N
,(2r+q)/2
(^
T
SEL-65-056
Mnur^TRMnjcmrKMr »mi^jj-rwxi/mj-Ri-Ryrimvx^vnv'»! i.>\."H V* ^^^ 0^>j>VHj,>k>j.>vi'^>'^i>Ji>j>i^J.yi^J.>V*i>tl.'>(y>^>i>Vh:>v.\.N. ^Vi
Hence,
(^T-9)
r q p-q-r
2 lM(r M
,r+l
norm
2r
P - o
-T
3
I
+
2
_N
+
.2r+q
(2.72E
S
.r+1
= - <10 log
* 10(2r
10
_r(^)_
+ q)
loBl0 ^p - 3 _ I
+
i
+
iU
(2.72b)
in decibels
If
r = 0,
then Eq. (2.72b) reduces to Eq. (2.69)
If
r = 1,
q = 1,
S
If
21lV-2
r - 1,
13.09 + 30 log10
q = 3,
r = 1,
If
S
+
50 log10
q = 5,
2 1 p-3
2 1 0H
p - |
^
in decibels
(2.74)
p-f+l-
in decibels
(2.75)
+
then
1.61+70 1Og10
r = 2,
in decibels (2.73)
norm
S
21150P-6
* | + f"
then
9.57
21130P-4
(P
norm
S
If
then
norm
q = 1,
then
/
4
N
15.59 + 50 log10 I P " 3 + Tf
in decibels (2.76
norm
The above expressions give values which are close to those obtained by
integrating numerically on the computer.
in Tables 5 and 7 on pages 63 and 65.
SEL-65-056
nD™öwxmw*AW**xrvü**xru«)c^*^
48
Some of the results are shown
It may be added that such techniques can be used to get
approximate expressions for the case when the signal strengths are not
equal.
F.
COMPUTED RESULTS
For
p ?l 2,
Jones used the technique of expanding the integrand in
a series involving hypergeometric functions, integrating, and using the
resulting series for computational purposes [Ref. 8].
was found to be unsuitable for values of
p > 2.
This procedure
All computations were
performed on the Philco 2000 computer by carrying out the appropriate
integrations numerically.
used by Shaft [Ref. 9].
The numerical integration technique was also
Some of the results presented here follow the
same format as those in Ref. 9.
The following results are presented in tabular and graphical form:'1.
Figures 3a and 3b show the output power in the various signal
components for various
(S,/1^^
and
(Si/N^i
when
P =
3
and
when
2.
a.
One input signal is varied such that
S
b.
Two input signal« are varied such that
' (S
= S ).
{s^ = S^)
- Sg.
Figures 4a-4c show (he effect on the output power in the various
signal components when
a.
p = 4
and when
One input signal is varied keeping the other
(p - l) =3
signals equal and fixed.
b.
Two input signals,
S
c.
are varied, keeping
(S
= S )
S3
and
(S^ = S^).
■ s4.
Figures 5a-5c show the power in the output signal components for
p = 5, under the conditions specified therein.
Tables 1 and 2 give the signal-suppression effect in the case of
p = 3
5.
S2,
Three input signals are varied in such a manner that
the case of
4.
and
constant in such a way that
(s1 = s2 . s3)
3.
S
and
p = 4.
Figures 6a-6b and 7a-7c show the signal-suppression effect graphically.
Figures 3-9 and Tables 1-7 appear at the end of this chapter beginning
on p. 54.
- 49 -
->XMn>n«fi« n»j»«««_lM.niL1«jULA»rMUM».'\»jmj11«jn«/^ ItMU', ^V.WiW VWUNT-fk l^i uvirw V^
SEL-65-056
W^L^L'^L^.
U^ -N l."VL"« _% VV l"«' '. .'i .' ' < . i /• -•'. VW 'J'j'jt "> '
6.
Table 3 gives the output signal power in the case of
signals (no noise) for values of
1
p
100.
p
equal
Both computed and
approximate values using Eq. (2.65) are given.
7.
Table 4 gives the output signal power in the case of
signals in noise for
8.
3 :. p :
p
equal
100.
Table 5 gives the strength of some of the crossproducts for various
input signal-to-noise ratios in the case of four equal signals.
9.
Table 6 gives the crossproducts for
large
10.
p
equal signals in noise for
p.
Table 7 gives other crossproducts for the case of
p
equal signals
(no noise).
11.
Figures 8 and 9 display the output power in the crossproducts for
large
p
in the equal-signal case and clearly show that the values
obtained by the approximate formulas are indiscernible from the
exact values obtained through computer evaluation of the exact
expressions.
G.
CONCLUSIONS TO CHAPTER II
It has been shown that many of the limiting properties for the case
of hard limiting of
the case of
p = 2
p
inputs in random noise are the same as those for
under similar conditions.
The following conclusions
are in order:
1,
The limits on signal-to-noise ratio are:
For one signal in noise,
ii ^
(S/N)
v
'
'o
4 ~ (S/N).
For
p
2
signals in noise,
(S/N)
' o
SEL-65-056
- 50
*»W*l/WU«lAfUWWi™VkA_-JL"J«A*T^AAr>.iW> 'JtfJUJ \KM» Mf\* "^ **'> *> *> ''-»?.*>> r,J<'->»'>'rj,",,Ji7,J>'r>>j<>>">^"*.-"-j."v,j.->j."H_,""^"v-J.>>",V.%^'
For the strong-noise case, the following results which are known
for
p = 1, 2
any
p;
a.
[Ref. 8], have been shown to be applicable for
The output signal-to-signal power ratios are the same as the
input signal-to-signal power ratios for any number of inputs,
that is, there is an absence of signal suppression.
b.
For any
p
the signal-to-noise ratio at the output is 1 db
less than the signal-to-noise ratio at the input.
For one strong signal
a.
If the input
S
(S /S. )
and
(p-l)
weak signals
S.:
power ratio is very large, the behavior
approaches that of the one-signal-in-noise case as the ratio
of the input strong signal to the weak signal increases, and
S 1 » ^ S.
1
+
N
i=2
Then, for large input signal-to-noise ratios:
(1)
The range of the normalized SNR's approaches the onesignal-in-noise case.
(2)
The suppression effect manifests itself, with the amount
approaching the limiting value,
(S
v n/S.)
1'
(3)
->4(S
v 1/S.).,
1 'o
1'
1 1
that is, 6 db.
The strong signal receives an improvement in relative
SNR, which approaches a value of 2, that is, 3 db.
(4)
Each of the
(p-l)
signals suffers a loss in SNR by a
factor of 2, that is, 3 db.
b.
If the ratio
(S /S.).
for
greater than 1, the effects of limiting are the
(S /N).
is larger than approximately 20/l and
same as the equivalent two-signals-in-noise case.
c.
If the ratio
that for
(S /S.).
(S,/N)
1'
1
is greater than or equal to 20/l, and
is less than 1, then for
p > 2,
the sup-
pression effect becomes less pronounced and ultimately the weak
signals are boosted instead of being suppressed.
iri^i*rjuvinj«\iiin«««ioyt.^K^n^
- 51 -
SEL-65-056
4.
For two strong signals, say
S.,
S
and
S ,
and
(p-2)
weak signals
the behavior follows the patterns set by the two-signals-in-
noise case.
There is one exception, however, which is the boosting
of the weak signals for
(S /S ), > 20/l
and
are the very same conditions as for item 3d.
(S
/N).
< 1.
These
The strong signals
cause a beat pattern and every time the envelope of the two strong
signals falls below the noise or weak signal, the latter gets control of the limiter.
Consequently, all the weak signals get trans-
mitted without any attenuation.
The effect is similar to the one
described by Jones [Ref. 8], which is an attempt to give some
physical explanation as to the decrease in the relative signal-tonoise ratios for high input signal-to-noise ratios.
5.
In all cases the crossproducts follow the curves in Fig. 4 of Jones
[Ref. 8] very closely.
The signal power in the crossproducts
remains nearly constant for input SNR's greater than 6 and decreases
very rapidly for smaller input SNR's.
6.
From the numerical results and graphs presented, it is concluded
that the power in the output components remains relatively independent of the input SNR's for input strong-signal-to-noise ratios
greater than 10 db.
7.
For large
p,
and approximately equal signals, Eqs. (2.70) through
(2.76) are valid.
8.
For
a.
p
large, the following results were obtained:
The power in any signal or crossproduct term decreases in a
linear fashion when plotted on a log-log basis [see Eqs. (2.70) (2.76)].
b.
The sum of
the powers in all the
independent of
p.
p
signals at the output is
From Eq. (2.70),
P^lOO.Vo
-1.05 db
(2.77,
Other cases may be reduced in a similar manner to predict the effects
of limiting.
SEL-65-056
Ho&xjv*»ji&MWMJvat^^
- 52 -
that is, the power contained in the
p
signals is approximately
1.1 db less than the total power in the output first-harmonic
zone.
c.
The
p-signals-in-noise case has approximately the same behavior
due to hard limiting as the case of
[p + (N/S )]
signals with-
out noise.
d.
The power in decibels in any term at the output decreases as
10
I-.
log10 P
(2.78a)
i=l
that is,
aooiiiMJUiKKnjowTUKi^nxnivA^
Output power ~ p
- 53
ik ■')
(2.78b)
SEL-65-056
C5
All
W
w
II
i
I
w
w
w
snaaoao
DS
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g
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ro
w
O
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<N
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d
i—i
All
P—
51381030
SEL-65-056
i-mü ftjinji^-A-rjirji ^Jl'^_Ji
54
Pä PJI
rj* .•\j»r «
^JI
rj« rj r-ji
^JI
-■_<
PJ* TJI
■*
\M KM
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TABLE 2.
SUPIRESSION EFFECT FOR THE FOUR-SIGNAL CASE
(p = <
(All values In decibels)
(S1/N)1 = .
ftl
l\SlSJo) rwoi
L(vViJ
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Condition
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0
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3
3.8
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3.7
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6
7.9
1.9
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1.8
6.9
0.9
6.2
0.2
10
14.4
4.4
13,6
3.6
11.4
1.4
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0.1
15
20 3
5.3
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16.6
1.6
15.0
0.0
20
25.3
5.3
24.9
4.9
21.7
1.7
19.8
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Condition
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0
0
0
0
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0
0
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0
3
3.4
0.4
3.5
0.5
3.4
0.4
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6
6.2
0.2
6.4
0.4
6.5
0.5
6.2
0.2
10
9.5
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10.0
0.0
10.6
0.6
10.1
0.1
15
13.4
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14.4
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15.6
0.6
14.9
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20
17.5
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19.1
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20.4
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19.7
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j
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• S4
0
0
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0
0
0
0
0
0
3
3.7
0.7
3.5
0.5
3.3
0.3
3.1
0.1
6
7.1
1.1
6.9
0.9
6.4
0.4
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0.1
10
11.3
1.3
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1.1
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15
16.3
1.3
16.2
1.2
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TABLE 3.
Number of
Signals
(P)
OUTPUT SIGNAL POWER FOR
EQUAL SIGNALS AND NO NOISE
, 2
Total energy In first zone = 8/n = 0 db
(All computed values* in minus decibels)
Exact
Value
Approximate
Value
p
Number of
Signals
(p)
Exact
Value
Approximate
Value
1
0
1.05
14
12.5
12.5
2
3.9
4.1
15
12.8
12.8
3
5.6
5.8
16
13.1
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4
6.9
7.1
17
13.3
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5
7.9
8.0
18
13.6
13.6
6
8.7
8.8
19
13.8
13.8
7
9.4
9.5
20
14.1
14.1
8
10.1
10.1
30
15.9
15.8
9
10.6
10.6
40
17.1
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11.0
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11
11.5
11.5
75
19.9
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12
11.8
11.8
100
21.2
21.1
13
12.2
12.2
Values:
Exact
(S.)
i o
f
T2
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Approximate = -[1.05 + 10 log10 (p + N/S.)],
N = 0, noiseless case.
Note:
For all values of p greater than 4, the approximate formula
gives the answers to within 0.1 db. For the most part the
error is due to rounding off the values to the first place.
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TABLE 5.
CROSSPRODUCTS VS
(S/N)
FOR THE FOUR-SIGNAL CASE
S
l = S2 = S3 = S4
(All computed values in minus decibels)
2
Normalization = 8/n
Notation:
s
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1000
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6.9
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28.5
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31.9
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21.5
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SEL-65-056
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1
1
50
60
70
1—r
-{+eoand+IOdb)-(S/N)l
-70
i
10
20
FIG. 8.
30
40
NUMBER OF SIGNALS, p
CROSSPRODUCT
S „
,,
iV"3
VS
p
i
i
80 90 100
FOR
EQUAL-SIGNALS-IN-NOISE CASE.
In Figs. 8 and 9 where the approximate curve
does not coincide with the exact curve, it
is indicated by a dashed line.
SEL-65-056
OOAMKlMaVVCAMlAWOUVIJ^^
66
20
10
30
40
50
60
70
80 90 100
NUMBER OF SIGNALS, p
a
-
S
^ „-5^'
S
iV
iV"7'
S
iV-9' ViVV
-80
APPROX
w -100
I
Odb«-^-
S -120
-140
-160
-180
10
_l
J_
20
30
40
50
60
L-
70
_L
80 90 100
NUMBER OF SIGNALS, p
FIG. 9.
ifninu£RW\j<3vt7U<^it^x^rw ^K^^
b.
S
CROSSPRODUCTS VS
p
2riq0p-q-r
FOR EQUAL-SIGNALS, NO-NOISE CASE.
67
SEL-65-056
HI.
A.
OUTPUT SPECTRA FOR WIDEBAND FM PULSE TRAINS
GENERATED FROM HARD-LIMITED FM SIGNALS
INTRODUCTION
In most narrowband FM systems the nominal carrier frequency is much
larger than the maximum information bandwidths, e.g., an FM broadcast
channel bandwidth is approximately 36 kc, while the carrier is a few
megacycles.
In the process of demodulation, the received signal is sub-
jected to hard limiting.
The amount of energy in the sideband of the
third harmonic which lies in the first-harmonic band is typically less
-6
than 10
of the energy in the fundamental band. Hence the output of
the limiter, to all intents and purposes, consists of discrete spectral
zones as shown in the following sketch; which is a typical spectrum of a
limited narrowband FM signal.
When such a signal is passed through a
narrow bandpass filter, the fundamental band can be filtered out.
Since
it is essential that such a bandpass filter have an excellent time-delay
characteristic, the filtered fundamental zone does not suffer any appreciable phase distortion, nor is the spectral distribution of the first
zone disturbed.
ture.
As a result, the filtered output has no envelope struc-
Therefore this signal can be demodulated by means of a relatively
narrowband discriminator, which may be a tuned circuit or a balanced
detector of the conventional type [Ref. 22].
Such systems have been
extensively analyzed by various investigators [e.g., Middleton, Ref. 2,
and Loughlin, Ref. 23].
-fh
-(f-
-//-
Vh^-ff-
-If5fr
SEL-65-056
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68
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rv» r
As an illustration, consider a case of monotone frequency modulation
where the normalized carrier frequency is 1 and the modulating frequency
is approximately 0.5.
Assume further that the deviation is ±50 percent.
The limited spectral output is as shown in Fig. 10a.
It is clear that
in order to filter out the fundamental zone immediately prior to detection, as before, it would be preferable in this case to use a lowpass
filter of bandwidth
2(Af + f ) = 2 [Refs. 24 and 25] .
put will then be as shown in Fig. 10b.
The filtered out-
Because of the high selectivity
of the lowpass filter, excessive phase distortions in the sidebands of
the filtered output will occur.
In addition, since the filtered output
contains quite a few significant sidebands of the third-harmonic zone,
an undesirable envelope structure will be present.
The detected output
may be computed using the techniques given in Refs. 24 and 26.
In any
case, since the amplitudes of the in-band third-harmonic lower sidebands
are appreciable, the detected output will certainly have unacceptable
spurious components.
Hence, filtering the limited output is not desir-
able in low-carrier wide-deviation systems.
Since the spurious outputs usually limit the system performance, it
has been necessary to define an "interference ratio" (see Sec. E).
A
complete analysis of such a demodulation scheme is covered in this chapter.
The autocorrelation function and the power spectral density of the output
process, as well as closed-form expressions for the interference ratio,
are derived for the general case.
The closed-form expressions are used
to compute the spurious outputs which are within 2 db of the experimentally measured values.
The specifications of some of the demodulators
designed on the basis of this analysis are also presented.
B.
INPUT PROCESS
1.
Assumption
It is assumed that the limiter has a sufficiently wide bandwidth,
which would imply that the output of the limiter can be considered to be
For these systems, satisfactory performance may be obtained through
detection by means of integrating a train of arbitrary but fixed, unidirectional pulses which are generated every time the limited wave
passes through zero [Cady, Ref. 27].
- 69 -
SEL-65-056
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switching between the normalized values of -1 to +1 with negligible rise
and fall times.
2.
This assumption is usually valid in all practical systems.
FM Pulse Train
If any FM signal is an input to the limiter, then the train of
pulses generated by placing an arbitrary but fixed pulse at every crossover or at every
p
crossover (where
p
is any positive integer) of
the limited signal is termed an "FM pulse train."
An arbitrary but fixed pulse implies that the pulse shape, duration, amplitude, etc., may be arbitrarily chosen, but that once chosen,
they remain invariant.
The duration of the pulse is such that the suc-
cessive pulses do not overlap at any instant.
It may be noted that if
and
±f
at every
f
is the nominal-carrier frequency
is the peak frequency deviation, then the FM train with a pulse
p
crossover has a pulse repetition rate of
pulse deviation of
2f
and
c/P
a
±2f/p.
Conceptually, and in practice as well, one of the best ways to
generate or analyze an FM train is indicated in the block diagram of Fig.
11.
The system analyzed is described in the following sections.
WIDEBAND
x(t)
y(t)
LIMITER
T®
y(t)-y(t-T)
ZERO MEMORY
SQUARER
y(t-T)
FM PULSE TRAIN;
NOTE: T MUST BE SMALL
RECTANGULAR PULSE
OF WIDTH T AT EVERY
FIXED DELAY
T
COMPARED TO A PERIOD OF
ZERO CROSSING
UNMODULATED CARRIER
FREQUENCY.
ORIGINAL
INFORMATION
FIG. 11.
at«»nvxv-*jTUi-*J.TUI *M*M*JI rJU^Jl/^vw^^
LOWPASS
tin
FILTER
BLOCK DIAGRAM OF THE SYSTEM ANALYZED.
71
SEL-65-056
a.
Input Random Process,
x(t)
Since the results for phase modulation can be written down
directly from the results of frequency modulation, it will suffice to
analyze the FM case only.
Hence the input random process is assumed to
be any signal of the form
x(t) = cos [a^ t + $
where
x(t)
+ ^(t)]
(3.1)
is a process consisting of a sinusoidal waveform (of constant
amplitude) whose phase is the sum of the random process
random variable
$ .
If
vlf(t)
*(t)
In the FM case, the derivative
represents the original information, that is,
H/'
-Kt) ^
where
g(t)
and the
is the original information, then this
process represents phase modulation.
process
Kt)
(3.2)
g(0 d?
is the original information.
Define
1
c
c
f
r
(3.3)
g(l) d?
and
A
02 ^(t - T)
+
$c
+
kf
I
f
t-T
g(|) di
(3.4;
and therefore
x(t) = cos 6,
72
SEL-65-056
m'm.mKmM.-K.-m.Mmk.mnmiLm'rLM-r^M rtmr m r. ■
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*J*
■. * r » r. n "_w r_M ^ JT r » .■•. _h " .M
PJI
:".»'"Jk^)»,Mk f
«^.M*
b.
Square of the Limiter Output,
2/
s
y [t]
It can be shown that the output of the limiter
A
^
(-1
y(t)
is
cos me
y(t)=iy
^
(3.5)
m =1
l
odd
Since the limiter has a large bandwidth, the rise and fall times are
extremely small.
hence
y (t)
The limiter output
y(t)
switches from -1 to +1 and
has a constant value of 1 at all times.
(m
+m2-2)/2
(-1)
cos m C
^-{$1 1
m =1
i
odd
Also,
cos m2e2
m m
i 2
m =1
2
odd
(3.6)
c.
Pulse Train,
z(t)
The FM pulse train
z(t)
consists of a train of positive-
going (or negative-going) rectangular pulses of constant height and of
constant width
y(t).
T
placed at every crossover of the limiter output process
As shown in Fig. 11, the output of the limiter is delayed by a
fixed amount, say
T.
The amount of delay
T
must be small compared
with a period of the unmodulated carrier frequency.
is subtracted from
squarer.
y(t)
This delayed output
and the result passed through a zero memory
This gives the pulse train
z(t),
where
z(t) = [y(t) - y(t - T)]2
= y2(t) + y2(t - T) - 2y(t) y(t - T)
- 73 -
SEL-65-056
Using Eq. (3.6) gives
z(t) = 2[1 - y(t) y(t - T)]
Hence the expression for
pulse train
C.
z(t),
EVALUATION OF
[y(t) y(t - T)]
(3.7)
will give the expansion of the
the modifications being of a trivial nature.
[y(t) y(t - T)]
Equation (3.7) clearly shows the necessity of evaluating
[y(t) y(t - T)].
With Eqs. (3.l) - (3.5), it can be shown that
(m +m
,,
,
y(t) y(t-
^
T)
(4\
V
^
V
I
Z
m =1
m =1
odd
odd
=(-)
'
^
<L^
y
L*
V1
m =1
odd
odd
\2
cos m^ cos m2e2
^
(3.8a)
^
2
= \n
(-)/
2"2)/2
i
2
(m1+m2-2)/2
^
[cos («,(•
v
ll
l
mm
+
2
+ cos (m^ - m262)]
=C0
where
+
C2+C4
+
...
+
C2p
+
...
C
Expression for the
The
0
0
Spectral Zone,
(3.8b)
(3.8c)
represents the 2p
spectral zone.
2p
clear that the odd spectral zones do not exist.
1.
mflj
22
From Eq. (3.8c) it is
CQ
spectral zone imposes the condition on the indices that
m1 i m2 = 0
SEL-65-056
- 74
\yj*ymi\mnma\w*xm^*^i*rKMt%.mn*r\.m^njkT.%r\mnnü\w<r-iL^n^*s^*u\*fy itn
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*.*U R-UA
k 4 W %^\\>\ r\ w
T'/'.>'
v* . ? - ndn.MP_Mr.M PJi "'^M r.«
-JI
r ^-^-r^r
with
m
and in
each being positive odd integers. Hence the only
1
2
contribution to this zone from the term cos (m^ - m^) is
T1
°°
2
C0 = (I) I 1 ^^^
m
m^l
odd
\r
=5(7) 2
i ^l - e2)
l
00
i /A\2
COS m
/
cos
\
m
i(ei "
L
e
9^
<3-9)
^- mT—^
1
m =1
1
odd
From Bromwich [Ref. 28, p. 371],
00
^
cos 111.
m,X
COB
A
^_,
17-
^-i
r-1
cos m X
„—22
1 —^'"»
1 , m1 - a
m,
a-K)
1
m^l
m =1
n sin a
= lim
a->0
(i-*)
4a cos -5-
X
^(f "
)
0 < X < jt
(3.10)
Hence
C
0
2.
= 1
" f (ei " e2)
Expression for the
2pth
The contributions to the
as the total of the
0
< (ei " S) <
Spectral Zone,
2pth
cos (me, + »"g^)
n
{3 ll)
-
p ^ 0
spectral zone can be written down
te™s
and
the
cos
i^-^ " "V^
terms.
- 75 -
SEL-65-05G
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a.
The
cos (m G.^ + m-B-)
Terms
These terms can be obtained by imposing the condition on the
indices that
m
Since
m
and
1
and since
m
+ m
= 2p
or
m2 = 2p - n^
can each take on only positive odd integer values
2
p
can take all positive integers greater than zero, only
those values of
m
are acceptable which satisfy the condition
1 S n^ g (2p
1)
The application of this condition gives the subtotal
°o
2
c.
2p
=
/i\
\n/
i
2
C'
as
(m1+2p-m1-2)/2
Y
Z.
m ^l.odd
^
r
/o
m1(2p-m1)
cos [me. + (2p - m )e ]
11
12
m2=(2p-m1)
(4\
2P 1
2
"
y
i
W
2
Z.
P
/-1) - . cos [mfe.,
- ej + 2pe2]
L
v
m1(m1 - 2p)
l
1
2
2
(3.12)
m =1
odd
b.
The
cos (m101 - m^g)
Terms
These terms can be obtained by using the condition
m
i ~
Consider that
n^ - m2 = 2p,
that
m
the
m
p
and
m
2
=
that is,
±2p
m2 = n^ - 2p.
Also by noting
can have all odd integer values greater than zero and
can only have positive nonzero integer values, it can be shown
that
SEL-65-056
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- 76 -
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(2p + 1) S m1
Therefore
[m1+(mi-2p)-2]/2
2
= (!) \
2p
1
(-1)
cos [m101 - (m1 - 2p)e2]
m
m
2
l( l- p^
m =2p+l ,odd
m. =(m1-2p)
/4\2i
\nl 2
V
/L
{-i)\
m1(m1 - 2p)
cos
[
(e
lv 1
e2)
+
(3.13)
2Pe2]
m =2p+l
all odd
Now consider
(m
- m ) = -2p,
which implies
m2 = (m1 + 2p).
Imposing
the following conditions,
p ä 0,
m1 ? 1
all odd values
m2 äl
J
gives
1 S m
< oo
oo
2
pin
C
2p
= (1^
\n/
which contributes
i
2
y
Z-
C"1
as
[m1+(m1+2p)-2]/2
i^
/
0 v
m1(m1 + 2p)
cos [m101 - (m
111
+
2p)o ]
2
m =1
i
odd
-{tf\l
TmMU^TUKBMJüUVUViNVJI.rjl^^^^
/-')p , cos
^{m^ + 2p)
[m1(e1
fi
2)
2pe2]
(3.14a)
m =1
odd
77
SEL-65-056
' {■ l.C/",/?>.V.V
2
=
(i)
\n/
D
i
2
-—, w^"1js——T cos [-iWe.
- G„) - 2pe^]
L
(-m|)[(-m^) + 2p]
lv 1
2'
2
V
Z^
-mJ=l
odd
= (;) 5 1
^T1^
3 [B
cos
[m'(e,
^-2,)\ '°
i(ei -- 0e„)
2'
n
+ 2p0o]
^a'
(3.14b)
m|=-l
odd
Hence
C„ = Cl + C" + C"'
2p
2p
2p
2p
2
00
= (!) I I
vi;1^) - [^(e, - o2) . 2pe2]
(3,i5a)
odd
/1\2 I Xzli!
\J 2 2p
V
Z
fcos [m1(e1 - e2)
1
m1-2p
+
2pe2] ^ cos [m^G^e^^pe^'
"
nij^
m =-oo
odd
(3.15b)
Further
cos [(m1(e1 - o2) + 2pe2]
m
- 2p
~
ZJ
m =-oo
m =-0^
m^=-<x>
odd
odd
SEL-65-056
cos [(m1 - 2p)(e1 - e2) + 2pG1]
m
l ~
2p
- 78
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cos (nij^ - 2p)(G1 - e2)
{2p ) Y/
= cos v-pr,
h/
01^2?
m =-oo
odd
~
sin {2ph1)
sin (m1 -2p)((
^
1
-f 2)
- o
m
m
1 -2p
m =-oo
It is obvious that
^
co^_ne
= cos 0(1
_ ^
+
cos
3(, (1 _ 1)
n=-oo( odd
= 0
and, by Ref. 28,
V
Z
n=-oo
odd
sin nß
n
-i
n=l
odd
i
0 < G < n
(3.16)
Hence
V
>
A
cos
f(m1=—=
(ei - S)
m, - 2p
+ 2pe ]
2
• o2p(f
= - -" sin
K in
2
1
0 < ( < n
(3.17;
m =-oo
odd
79 -
SEL-65-056
L
-n
L
i« \j~a v* \j-m -Lnm k/w w* -Ti l_Ti U"» b-'W u *
Also, proceeding on similar lines,
cos O (e. - ej + 2pe_]
1
= - ^r sin 2pe
r o
2
2
m.
0 < 9 < n
(3.18)
odd
Substituting the results of Eqs. (3.17) and (3.18) in Eq. (3.15b) gives
C
2p
=
n
[sin
2p
2pe
2 '
Sin
2pe ]
i
0 < (ei - e2) < n
(3.19a)
=
- 1
(-i)p sm p(e, - e_) cos p(e <- e )
^
?
i
?-
p ^ 0
(3.19b)
Equations (3.8c), (3.1l), and (3.19b) give
~ (-i)p sin p(e - e ) cos p(e +e )
^
^
y(t) y(t.T) = [l-f (6,-63)] - 1 ^
p=l
r
T
9
0 < (e
D.
- e2) < n
(3.20)
OUTPUT PROCESS
1.
Expansion of
z(t)
The expansion of the FM pulse train can be rewritten by substituting the result of Eq. (3.20) in Eq. (3.7), which gives
4
z(t) = " (e! - e2)
+
"
(-i)p sm
2 - 2
P(e1
- e2) cos
P(e1
+
e2)
JT
P=I
SEL-65-056
IM
«
MM
a
M
- 80
•^RMRb.sMRVHb ^ "w « ■« w ^ »t M ir-^ wuitw ^y«%«wir'«WtaHM^y>«^w"l."k^^^«"Ä«M»,jdn*Twnw,l»fl«^k." *."' M.1 »Cl »_« k^. VI njA 1^1 f
(-l)P sin p(G1 - e2) cos p(G1 + Gg)
4
(e, - e2)
+
2 ^
4
P
sin 2pe.
sin 2pei
1
(3.21)
jt
p=l
where
= total phase of the input process
x(t)
at time
t
= total phase of the input process
x(t)
at time
(t - T)
It may be noted that
)
X
- G
A
(3.22)
= ü) T + [4f(t) - ^(t - T)]
C
G1 + G2 = 2
[ü)C
(3, 23)
(t - |) + *c] + [Ht) + ^(t - T)]
where
\|/(t) = information for phase modulation
i(t) = information for frequency modulation
It may be added that all the expressions derived above are valid for
those cases where
x(t)
is subjected to hard limiting with wideband
limiters.
;*.
Autocorrelation Function
Thr derivation of the autocoirelation function, which is quite
involved, is shown in Appendix B.
The final result is contained in Eq.
(B.19):
R (t^1) = B
z
+
^) = (!)2 {f/ + 2VT) - [vr + T) + VT -T)]
"
^
"
^
P=l
i^-00
. 2
2J^(2pß) sin^ (2pc.c
_S
^__E
nc.m) ^
E_i cos (2i^c
+
/l
na.m)vj
(B.19)
81
•xi«ir«:inafiinimmiu^vxKTty^irRv^^nyi«^7UflU(XrfK^7iJ(»i^
+
SEL-65-056
3.
Power Spectra] Density
The expressions for the power spectral density of the output
process are developed assuming rectangular pulse trains, arbitrarily
shaped pulse trains, and power spectral density with no modulation.
a.
Rectangular Pulses
In the case of rectangular pulse trains it is easily seen
that
^H±)2ty8(t).461„2f s^)
00
00
Z
Z
p=l
n=-oo
2/
\
2J (2pß)
2—
Sin
[(2^c
+ na) )
m 2}
8[f
- (2pf
+ (2pf
1
v
L
v r
"c + nf m')]J + 6[f
c + nf m')]J
4
/[-}
W<a. V*^
cT6(f)
oo
oo
+
2,
& f
f
/.R2 sm
■ 2 ^\
(2ß
-} ( - m)
>
2J (2pß) sin
2 /
I S^
p=l
b.
p
n=-oo
&[f
- (2pf
L
v
c
(2pa3
(3.24a]
+ & f + f
(
m)
\ T
+ naj ) -
2
+ nf y)]J + 5[f + (2pf + nf )'
m
c
m'
(3.24b;
Arbitrarily Shaped Pulses
For arbitrarily shaped pulse trains it is better to cast the
expression for
S (f)
into a more transparent form as follows:
SEL-65-05G
rx-wi-MMH tjvv.'v us vawit
M*
■ ma r J* na
- 82
."M
r.j» r ■ r_» f_M p.* rji
A^«
rji rjt r j» •\j> rji
^JI
rjir J r-» ■_« -•_« r_» rj -^»
-VJI
•
Sz(f) = (^) <(2fcT)2 n2 5(f)
00
U
+
00
I I
p=l
2J
n(2pß)
sin2 nfT
n=-ot
—2
p
2X2"
nfT
2(9
f
n (2pf
c
+
f
.2
"'m)
T
2
S[f - (2pfc + nfm)] + S[f + (2pfc + nfm)]
(3.25a;
2
(4)
2
<;(2fcT)
CO
00
I 1
p-l
2 2
&(f) + 2ß f
2
(T sine fl)
&(f - f ) + 5(f + f )
^
2(2pf^ + nf )2 J2(2pß)
5
2
2
(T sine fT)2
n=-oo
&[f - (2pfc + nfj] + 5[f + (2pfc + nfm)]
(3.25b:
,2
= (4r (T sine fT)^ < (2f )
00
00
1 1
p=l
2(2pf
v r
c
„ „
2„2
6(f) + 2ß f
5(f
- f y) + 5(f
+ f ;
v
v
m
m'
+ nf )2 J2v(2pß)
m'
n
n=-o<
&[f - (2pfc + nfm)] + 6[f + (2pfc + nfm)]
(3.25e)
- 83
SEL-65-056
rTMi^jB»^iv^T5^iijinj«ut-><jrrjciVK!»jfnj'^»"-»'\»i'j>f._M Vii>_»<fi-»n_Kf"jirj« nj< PL» rj rJ«fji rj» ">> r\A'J< rj* rji 'J- ".- '* 'J> '>">'> V "J« ">TJH "> \j< V« "> '
Also,
Jrllt) -> T sine fT
Hence
R (T) = (4)2 JrUt) * jTl(t) *
CW
00
*1 1
p=l
2(2pf
x
c
(2f )2 + 2ß2f2 cos a) T
c
m
m
+ nf )2 J2(2pß)
m
n
cos (2pü3
c
+ no) )T
m
n=-o
(3.26)
Let
p(t)
be any arbitrarily shaped pulse with
P(f)
as its transform,
and whose width is such that no overlapping of the successive pulses
occurs.
Then for such ," pulse train
S (f)
z
2
? )
2
S2(f) = (4)2 [P(f)]2 /(2fc)2 6(f)
00
00
1 1
p=l
2(2pf
v
^ c
+
becomes
2 2
2ß2f2
&(f
"
f )
m +
^
&(f
+
f
m'
^
+ nf )2 J2x(2pß)
m'
n
n=-o
&[f - (2pf
+ v(2pf
)]
\ »- c + nf m/)]J + &[f
L
^ c + nf m^
(3.27)
which gives the power spectral density for a train having arbitrarily but
fixed shaped pulses at every zero crossing.
c.
Output Spectra with No Modulation,
ß = 0
It can easily be shown from Eq. (3.25b) that the spectral
power density with no modulation i-
SEL-65-056
84
"
s2(f)
2
= (4)
2
(2fcT)
&(f-2pfc)
5(f) + ^
no
mod
2
sinc
(2PfcT)
+
8(f
+
2pfc)
2
p=l
(3.28)
The above result can easily be derived in a simpler way by noting that
the output with no modulation consists of a pulse train of random phase,
the repetition rate being
4.
2f
and each pulse being J TL(t).
Output Voltage Vs Frequency Characteristic
It is convenient to express the sensitivity of the system shown
in Fig. 11 in terms of the dc output voltage
for
S (f)
z
quency f
Vdc.
From the expression
it is easy to show that the rms voltage output at any freis
ßV.
dC
/i f
\
sinc f T f
m / m
(3.29)
The linearity of such a system is therefore excellent.
E.
INTERMODULATION PRODUCTS
From the power spectral density of
z(t),
it is seen that the inter-
modulation products can be tagged by two indices,
IM(p,n)
1.
The intermodulation product
p
IM(p,n)
belongs to the
The order of the intermodulation product
-«>
to
n
Hence
p
spectral
can have all integer
+oo.
The frequency of the intermodulation product
+ nf ).
cm
When the power in the wanted information is
IM(p,n)
(2pf
4.
n.
assuming all positive integer values.
values from
3.
and
would imply the following:
zone,
2.
p
then the power in the intermodulation product
2J2
(2pß) sin
n
(2puj
E
+ m) -z
E?—£
2
[2fä
is
2
sin
IM(p,n)
(a^T/2)],
is
(3.30)
2
p
_ 85 -
SEL-65-056
Since the power in
IM(p,n)
is at a frequency of
is possible for certain values of
IM(p,n)
out.
p
(2pf
and negative values of
+ nf ),
n
it
that
will lie in the wanted spectrum and hence cannot be filtered
This intermodulation product appears in the output as an inter-
ference at a frequency which is not harmonically related to the information frequency and is therefore sometimes referred to as a "spurious
output."
In such cases, it is more convenient and meaningful to define
a ratio, termed the "interference ratio," or IR(p,n).
1.
Interference Ratio,
IR(p,n)
The interference ratio
IM(p,n)
IR(p,n)
is defined as the power in
when the power in the wanted information is normalized to 1.
Hence
IR(p,n) = interference at frequency
(2pf
+ nf )
-i2
J
n
(2pß)
v
/ sin (2pu)
v
c
+ no) ) nr 2
(3.31)
üj T
pß sin
m
which can also be expressed in decibels.
Almost all existing wideband FM systems have operating parameters
such that the values of
T/2 and (2puj + no) )(T/2) are extremely
m '
cm'
small and the sines may be replaced by their arguments. This simplifiCD
cation gives
|jj2pß)
IR(p.n) =
"pß
J
= 20 log
10
|n|(2p^
Pß
db
(3.32)
86
SEL-65-056
vumA/uwuwvmuwwLmvnrLrtfuwimuvv.iiLntu'^ii.'xi.-M
+ n
L/Wuvim
wwuv tr« t/» ur« u-w v« ijn v.-^
JKU-»
«>«)•» ikf y •»•?"«« i. "»^^ "_«'.«. ',« "_»-■'_» "_» -jt'
provided
n
f
^0,
ß^O,
pe (l,oo),
has a negative integer value in
The output
z(t)
and
n ^ 0.
It may be noted that
IR(p,n).
is filtered to recover the original information.
Let the output of the final lowpass filter be
B. .
modulation components which satisfy the condition
Hence all the inter(2pf
+ nf ) > B.
are filterable and are therefore of no consequence as far as the overall
system performance is concerned.
The strongest interference usually occurs for
the highest negative value of
n
(2f
that is,
n
c
p = 1,
and for
which satisfies the condition
+ nf )
nr
^P
is the integer which satisfies
2f
c
- B0
jjp
f
m
Since the highest negative value of
(3.33)
n
will occur for the highest
in-band frequency, the worst interference will be for that
obtained when
a frequency
f
(2f
v
m
c
is replaced by
+ nf ),
m
B
n
which is
The worst interference occurs at
£p-
and has a ratio
J|n|(2ß)
IRU.n) = 20 log
+ n
10
db
(3.34)
m/
where
A
Af
ß = modulation index = ± —
m
Usually a
±k
percent system implies that
.
f
k
c
100 f
0.01 k I—
\ my
87
(3.35)
SEL-65-056
Therefoj
IR(l.n)=20 1ogl
/
Ji |[0.02 k (f li )]
I I oi
^
db
(3.36]
It may be noted further that only the first-order interference components
need be computed, since in most of the wideband FM systems the total contributions due to higher order components amount to less than 10 percent
of the first-order term,
2.
Pulse Shape and Interference Ratio
In computing the first-order effects, the interference ratio is
calculated at a point where the equality
f
+ nf ) is satisfied.
cm
Hence the interference ratio is essentially unaffected by the shape of
m
= (2pf
the pulses.
F.
RESULTS
1.
Theoretical and Experimental Results
In order to be able to select optimum system parameters, given
a set of specifications, Eqs. (3.3l) - (3.36) are used to compute:
the
interference ratios in Table 8 and the curves of Figs. 12 and 13.
1.
Table 8 shows the interference ratio in decibels for various
deviations.
The information bandwidth is assumed to be 0 to 1,
while the nominal-carrier frequency is set at 2.
ratios
IR(l,-3),
IR(l,-5),
IR(l,-7),
cause spurious outputs at frequencies
and
The interference
IR(l,-ll),
which
1, 2/3, l/2, and l/3, are
shown in the table.
2.
The curves of Fig. 12 show graphically the relationship between the
interference ratio and the deviation when the ratio of the nominalcarrier frequency to the information bandwidth is 2,
3.
The curves of Fig. 13 show the behavior of spurious output vs
carrier frequency at constant deviation
SEL-65-056
- 88
TABLE 8.
INTERFERENCE RATIOS FOR VARIOUS
DEVIATIONS AND BANDWIDTHS
Normalized information bandwidth, BW = 1
Normalized nominal carrier frequency, f /BW = 2
(All values in minus decibels)
System
Deviation
(±k«
IR(l,-ll)
f/BW = 1/3
IR(l,-7)
f/BW = 1/2
0
IRU.-B)
IR(l,-5)
f/BW = 2/3
f/BW = 1
--
—
—
10
—
—
83.6
43.6
12.5
—
108.0
—
—
20
—
—
59.9
31.8
25
—
75.1
52.4
28.2
30
—
—
46.4
25.2
37.5
—
55.4
—
—
40
—
—
37.4
20.9
42
77.1
—
—
—
50
—
42.4
30.9
17.8
Note:
Values not indicated were either negligibly small or
not computed.
The experimentally measured results, which are also shown on
Figs. 12 and 13, are in close agreement with the computed values.
For
example, spurious output measurements which were made for various deviations and for nominal-carrier and input frequencies were, in every case,
within 2 db of the theoretically computed values.
A typical example of
these experimental measurements is reproduced below from data obtained
on 6 November 1964:
Conditions
Nominal-carrier frequency:
900 kc
Deviation:
+40 percent
Test gear:
FM multivibrator modulator ]
FM pulse train demodulator
prototypes
Spectrum analyzer
89 -
SEL-65-056
SYSTEM DEVIATION tk
20
30
40
■y
T
10
T
50
-r-
A EXPERIMENTAL
O THEORETICAL
-20
g-40 -
-60 -
-80 -
100 -
-120
FIG. 12.
CURVES SHOWING THE INTERFERENCE RATIO VS DEVIATION
FOR FIXED
t It
=2.
c' mm
NOMINAL-CARRIER FREQUENCY (normaiiztd)
0
-100 |-
2
^p
A
3
*/p
flrp W/p ^/p
EXPERIMENTAL
O THEDRETICAL
-120
4
-(/-
J
L
FIG. 13. CURVES SHOWING SPURIOUS OUTPUT BEHAVIOR VS CARRIER FREQUENCY
AT CONSTANT DEVIATION,
SEL-65-056
- 90 -
In Table 9 the computed and measured values for various
IR(p,n)
are
compared.
TABLE 9.
COMPARISON OF COMPUTED AND MEASURED VALUES
FOR VARIOUS IR(p,n)
Input
(kc)
IR(p,n)
Frequency
(kc)
Computed Value
(db)
Measured Value
(db)
500
IR(l.-3)
300
-26.8
-29
IR(l,-4)
200*
-37.8
-38
lR(l,-5)
200*
-52.5
-53
550
-25.1
-27
IR(l,-6)
300
-42.4
-40
IR(l,-7)
50
-71.3
—
IR(1,-14)
400
-69.8
—
IRCl.-lS)
300
--
—_
400
250
100
The input frequency had to be rocked slightly to distinguish
between these two components.
The linearity and dc drift measurements were made on a typical
FM pulse train demodulator with a nominal-carrier frequency of 900 kc and
a deviation of ±40 percent.
The test gear used was a five-place digital
voltmeter, a frequency counter, and a PM prototype demodulator.
It was
found that the maximum deviation from best straight line through zero was
better than ±0.1 percent and that the dc drift was better than 0.01 percent per degree Centigrade.
2.
Advantages over Conventional Detectors
As far as low carriers and wide-deviation FM systems are con-
cerned, this demodulation technique affords a better way to recover
original information than the conventional FM discriminators or ratio
detectors.
The conventional detectors having a maximum deviation from
linearity of less than +1 percent to over ±50 percent of the center
carrier frequency are extremely difficult to realize [see Ref. 24, p.
494; Ref. 29, p. 1104; and Ref. 30, p. 214].
- 91 -
SEL-65-056
1.
FM pulse train demodulators represent at least an order-of-magnitude
improvement in the linearity of frequency vs output voltage.
2.
This technique results in a device having excellent dc drift
characteristics.
3.
It is cheaper and easy to implement.
4.
Since the i-f filtering before detection leaves lower sidebands of
the third harmonic of the carrier frequency (resulting in amplitudemodulation), the interference is too severe to permit satisfactory
operation of the wideband FM discriminator.
5.
The .v^'idpass filter must be quite sharp and yet must have excellent
envelope delay characteristics, which make the filtering after
limiting (prior to detection) undesirable.
These advantages have resulted in the production of FM demodulators
having th*" iollowing specifications:
Maximum
Deviation
from Linearity
Center Carrier
Frequency f
Deviation
(±^of fc)
1 to 1800 kc
going up in
octaves
40
dc to 1 Mc
depending
on f
c
0.01
7 Mc
30
dc to 5 Mc
--
1
15 Mc
30
dc to 10 Mc
—
1
SEL-65-056
Information
Bandwidth
92 -
DC Drift
(fo/°C)
(±«
0.1
IV.
A.
REDUCTION OF SPURIOUS OUTPUTS IN LOW-CARRIER,
WIDE-DEVIATION, WIDE-BANDWIDTH FM SYSTEMS
FORMULATION OF THE PROBLEM
In many systems and practically all electromagnetic recorders, the
channel has a bandwidth from a few hundred cycles to a few megacycles.
In most cases the information has a bandwidth down to dc and therefore
some form of a modulation scheme is required.
The bandwidth of the chan-
nel, the modulation scheme, the tolerable strengths of extraneous outputs,
and the signal-to-noise ratio are some of the factors which determine the
bandwidth of the information that can be recorded.
Frequently vestigial
FM recording is used in order to record information of much larger bandwidths than is possible with conventional FM.
vestigial FM recording is shown in Fig. 14.
The spectrum of a typical
A high modulation index is
needed in order to obtain an acceptable signal-to-noise ratio.
MAXIMUM EXCURSION OF
-NOMINAL-CARRIER FREQUENCY?
BANDWIDTH OF CHANNEL
FIG. 14. TYPICAL BANDWIDTH OF CHANNEL, USING VESTIGIAL
FREQUENCY MODULATIONS.
In the frequency-modulation case, some relations between the bandwidth of the modulated wave and the frequency deviation are given by Harris
[Ref. 31] and Abramson [Ref. 32].
It is assumed in the analysis by Harris
and also by Abramson that the input is a narrowband random process.
This
is shown in Fig. 15.
In the systems analyzed in this chapter, the narrowband assumption is
not valid.
p. 97.
A typical spectrum of a wideband system is shown In Fig. 17,
The spectrum centered at
f
partially overlaps the spectrum
93 -
SEL-65-056
g(t)
FREQUENCY
MODULATOR
y(t)
f
h » fm.n
Q(f)
-(f-
-fh
Sy(f)
centered at
FIG.
15.
MODULATED SPECTRUM WHEN
-f
as a result of the nominal-carrier frequency being
comparable to the information bandwidth.
f.. » f
h
ymm
Because of this partial over-
lapping, "aliasing" occurs and outputs appear in the demodulated information which cannot be attributed to a harmonic distortion of the input
information.
These so-called "suprious outputs" vanish under certain
obvious conditions which, however, are unacceptable for the reasons indicated below:
Condition
Reason for Nonacceptance
Sufficiently high nominal-carrier
frequency so that overlapping of
the spectra does not occur.
Because of transducer electromagnetic limitations.
2.
Reduced information bandwidth.
Makes the channel uneconomical.
3.
Reduced deviation.
Results in rapid deterioration
of signal-to-noise ratio.
Wang and Wade [Ref. 33] discuss this phenomenon in connection with
superregenerative parametric amplifiers.
SEL-65-056
94
The analysis of this chapter is devoted to finding acceptable methods, if
there by any, for reducing the strengths of the spurious outputs.
B.
ANALYSIS OF THE PROBLEM
It is assumed that the information
where
f
g(t)
(-fmm.*„„„)'
has a bandwidth
denotes the maximum frequency present in the information,
nun
Define
Ht) =
■/'
g(l) d|
If
f,. is the nominal-carrier frequency of a frequency-modulated wave
h
y(t), and f, » f , then after normalizing the amplitude to 1,
h
mm
y(t) = cos
It can easily be shown that if
[üJ
t + "fjj + ^(t)]
t(t)
(4.1)
is a gaussian random process with
a finite mean square value [Ref. 32],
R
where
R
(T)
density of
(T)
= i exp [-R^O)] exp
is the autocorrelation function of
y(t)
(4.2)
[R^(T)] COS üJ^
y(t).
The power spectral
is (see Fig. 15),
S (f) = \ exp [^^(0)] n{exp i^)]] * [S(f - fh) + 5(f
where the Fourier transform of
T
RJ.( )
+
fj]
(4.3)
is
Joo
exp [KJl)] exp (-icox) dx
uo
Although the information
seen from
(4.3).
g(t)
is bandlimited to
Fig. 15 that the bandwidth of
S (f)
f ■
it
can
be
is given by expression
In the following discussion it is assumed that
- 95 -
lfmml'
S (f)
exists in
SEL-65-056
a band
2f
centered at ±f. and is practically zero elsewhere,
ynun
n
Usually f„ymm
mm > f mm .
If the conditions in Fig. 15 exist, that is, fh » f „„j,. then
demodulation by any standard technique will result in the recovery of
the information
If
y(t)
ereater
than
0
g(t)
without spurious outputs.
is now translated down to another frequency, say
f^
is
f
as shown in Fig. 16, the original message can still
ynun
be recovered without any spurious outputs. This is evident from the
fact that if one reverts back to the original carrier frequency
translation, as shown in Fig. 16, one obtains
y(t)
exactly.
whole process is "reversible."
y(t)
MULTIPLIER
X
CO«
BANDPASS FILTER
BAND CENTER fj
BANDWIDTH 2f,imm
I«t)
BANDPASS FILTER
BAND CENTER
fh
BANDWIDTH 2f,ymm
ylf)
[(«V «„.)»]
-^iL. MULTIPLIER
X
C0
«hh-Wh,),l
FIG. 16.
SEL-65-056
MODULATED SPECTRUM WHEN
- 96
f'h § f ymm
fh
by
Hence th6
The case where
less than
y(t)
is translated down to a carrier frequency
f
f
is shown in Fig. 17. This condition is usually preymm
valent in wideband FM recorders. Partial overlapping of the positive and
negative spectra occurs and spurious outputs result.
The shape of the
spectrum of
S (f)
is different from S (f); and given such a z(t),
z
y
one cannot revert back to y(t) by translation. The process is therefore "irreversible."
C.
INTERRELATIONSHIPS OF
SPURIOUS OUTPUTS
If one demodulates
f ,
c
z(t)
f
and
f,,
Up
FOR FILTERABLE
and passes the output through a lowpass
filter to recover the message
y(t)
ymm
y
g(t),
MULTIPLIER
a portion of the spurious outputs
LOWPASS FILTER
z(t)
X
COS
K-«.»♦]
$,(«
■rVK'ymm
REGION OF "ALIASING"
f
£ Wc
8,(0
FIG. 17. MODULATED SPECTRUM WHEN fc < fymm
ALIASING). As a result of aliasing, y(t)
be recovered from z(t). The dotted lines
lower figure indicate the shape of S (f)
y
Sit)
for f > f
.
z
c
ymm
- 97 -
(SHOWS
cannot
in the
and all
SEL-65-056
generated due to the aliasing phenomenon is filterable (Fig. 18).
more specific,
specifi
if the lowpass filter has a bandwidth of
f,
,
To be
then from
Fig. 17 and
■£P
[f
c
minimum distortion occurs in
filterable.
(2f
f
Hence if
),
ymm'
g(t)
- (f
f )] = 2f
c'■'
c
ymm
g(t)
(4.4;
ymm
since the spurious outputs are
is bandlimited to frequencies less than
good demodulation is possible,
2(0
»EMOOULATOR
FIG. 18.
g(t)
LOWPASS FILTER fs|ffp|
LOWPASS FILTERED OUTPUT.
In the frequency-modulation case, the spectrum theoretically has an
infinite bandwidth.
In all practical systems, however,
nificant over a small frequency interval.
S (f)
All components of
is sigS (f)
out-
side this interval may be vary much below the signal-to-noise ratio of
the system, which sets a criterion in estimating f
from Eq. (4.4).
'
ymm
To illustrate the application of the result, consider a simple case of
monotone modulation.
of information
f
mm
If the modulation index
is such that onlyJ
p
ß
for the maximum frequency
lower sidebands are signifi-
cantly different from zero, then for the case of no spurious outputs in
the filtered output due to aliasing,
mm
D.
(2f
that is.
pf mm
mm
(rh)
(4.5)
REDUCTION OF SPURIOUS OUTPUTS
Since the bandwidth of the channel is less than
f
ymm
can do for minimization of spurious components is to make
Fig. 17 coincide with
S (f)
S (f)
in
throughout the entire bandwidth of the
channel, as shown by the dotted lines in Fig. 17.
This approximation
can be done by appropriate filtering or phasing techniques.
SEL-65-056
the best one
98 -
1.
Filtering Technique
It is clear that the filtering should be performed when the
conditions in Fig. 15 prevail.
The bandpass filter must have an approxi-
mately constant group delay characteristic close to the first null.
slope of the filter depends upon
ß,
f
111 III
,
f ,
K*r
f. ,
XL
The
the modulation index
the signal-to-noise ratio of the system, and the rate of cutoff of
the final lowpass filter at the output of the receiver.
Bessel function tables and a reasonably low value of
With a set of
fh,
the system
parameters can be chosen so as to result in filters that are realistic
and Inexpensive.
The complete system which implements the filtering
technique is shown in Figs. 19 and 20.
2.
Phasing Technique
The block diagram which Implements the phasing technique is shown
in Fig. 21.
The modulator of Fig. 21a and the demodulator of Fig. 21b
have both resulted from the following theoretical analysis:
y(t) = cos
[üü
t + *
+ t(t)]
= cos ((Jü t + <t> ) [cos t(t)] - sin (ou t + u^) [sin t(t)]
Taking Fourier transforms,
Y(f) =
1*
■Kt
—- 5(f
- f /) +
x
2
c
—— 5(f + f )
2
c'
* c'3{cos t(t)]
■1*
i<t>
c
^r- 5(f
- f /)
v
21
c
—— 5(f
+ f )
v
21
c'
* n(sin t(t)}
(4.6;
Define
c
f
= ,'Hcos ^(t)] * s(f + fc;
- 99
(4.7£
SEL-65-056
FREQUENCY
MODULATOR
NOMINAL
CARRIER
g(t)
J^IU BANDPASS
MULTIPLIER
LOWPASS
X
FILTER
FILTER
"«'h-'e'
RECORD
i(t)
»h-»c»
-ff-
-ff-
FIG. 19. BLOCK DIAGRAM AND SPECTRA OF THE PROPOSED SYSTEM
USING THE FILTERING TECHNIQUE.
BALANCED
2(t)
MODULATOR
FILTER
LIMITER AND
OUTPUT
DISCRIMINATOR
FILTER
INFORMATK»
cos 2irfdt
(^ ARBITRARY)
FIG. 20.
SEL-65-056
DEMODULATION PROCESS USING THE FILTERING TECHNIQUE,
- 100
FM ot f.
FREQUENCY
Jill MODULATOR -
MULTIPLIER
X
LOWPASS
FILTER
/
(OS MODULATED
vi/ OUTPUT
|iin[(a.h-«e
HILBERT
TRANSFORMER
LOWPASS
FILTER
MULTIPLIER
X
a.
Modulation
BALANCED
MODULATOR
1 cos <«i'h t
MODULATED
OUTPUT
,ln^ *
HILBERT
BALANCED
TRANS» :ORMER
MODULATOR
b.
FIG. 21.
LIMITER and
DISCRIMINATOR
LOWPASS git)
FILTER
Demodulation
BLOCK DIAGRAMS OF THE PHASING TECHNIQUE.
C
f
= ;3{cos ^(t)J * 8(f - f)
(4.7b)
= cr3{sin ^(t)] * &(f + fc)
(4.7c)
= n{sin \lf(t)] * &(f - f )
(4.7d)
c
S_f
c
S
+1
c
c
[F(f)]+ ^F(f) \ [1 + sgn f]
= F(f)
for all
f < 0
0
for all
f > 0
- 101 -
(4.8)
SEL-65-056
[F(f)]_ ^F(f) i [1 - sgn f]
= F(f)
for all
f < 0
= 0
for all
f > 0
(4.9)
Hence
i«
if
■i<t)
Y(f)=Vc+f c
ids
+± J:c
i
+£
2rIs-f c
-f c -hr^t c
(4-10)
also define
r(t) = sin
10
R(f) =
B f
(
■
[Cü
c
-i^
c
2i
* 3{cos \lf(t)J
-10
fc) +
-10
c
«
+f
(4.11)
c
c) ^T
£_!S(f .
10
+ \lr(t)]
c
f
10
«
t + $
c
2i
£_!
5(f + fc)
10
c
„
C
Defining the Hilbert transform of
-f
+
c
x(t)
* 3{sin \lf(t)j
-10
c
2
c
b
as
+f
+
-f
Joo
uo
102
c
x(t),
x(t) = Cauchy's principal value of
SEL-65-056
(4.12)
S
2
c
t
ill
ds
(4.13)
Using the relation
F(f) = [F(f)]+
and subtracting
Y(f) - R(f)
2
R(f)
from
e 'C C
2
\ +f /
\
c/+
Y(f)
+
[F(f)]_
(4.14)
gives
\
c/
(4.15)
A comparison of the above equation with Eq. (4.10) for
shows that the wanted output is precisely
Y(f) - R(f)
Y(f)
since the inter-
fering portions of the positive and negative spectra nullified each other.
The block diagram of the modulator based on this analysis is shown in
Fig. 21a.
To recover the message, the systems shown in Fig. 19 or Fig.
21b can be used.
It may be added that Hilbert transformers can be approximated to
a desired accuracy over a wide frequency interval by several methods,
e.g., an active network, a passive network, or a tapped-delay-line realization.
Detailed explanations of these methods are contained in Refs. 34
through 39.
The phasing method with modifications can be used to develop
complete systems using SSB-FM, SSB-PM or SSB-Hybrid modulation.
Some
experimental results on SSB-FM are contained in Ref. 39.
The analysis presented in this chapter has resulted in techniques
which suppress the spurious outputs that result from the interaction of
the positive and negative spectra when the nominal-carrier frequency is
- 103 -
SEL-65-056
sufficiently lowered.
Block diagrams of the proposed systems are pre-
sented in Figs. 20, 21a, and 21b.
Although the solutions are equally
effective, for reasons of economy the two systems shown in Figs. 20 and
21a would be preferable.
SEL-65-056
- 104
V.
CONCLUDING DISCUSSION
An analysis, together with experimental verification of the theoretical results, has been presented which allows the computation of the
strength in any output component and the "distortion" (i.e., the sum of
the power in all the intermodulation components in the first-harmonic
band) when many signals in noise are subjected to hard limiting.
Typical
situations in which these results have proved to be most useful are as
follows:
1.
When several modulated or unmodulated carriers are subjected to
hard limiting--a problem of common occurrence in a satellite comraunioation System ur in radar problems involving multiple echoes.
2.
When a frequency-modulated signal, after being passed through a
channel having arbitrary output vs frequency characteristics, is
subjected to hard limiting.
3.
This study may be used by various organization, such as NASA's
Inter-Range Instrumentation Group (iRIG), for drawing up specifications for various frequency-modulated instrumentation systems.
For the general case, expressions are developed which give the signalsuppression effect, the output signal power distribution, and the power
in the various crossproducts due to hard limiting.
A few results that
are well known for the cases of one signal in noise and two signals in
noise are shown to be valid for the general case.
It is also shown that the behavior on hard limiting of the manysignals-in-noise case approaches that of the one-strong-signal-in-noise
case if the strong signal power is much greater than the sum of the power
in all the remaining weak signals and noise.
It should be stressed that
the power in each of the weak signals or noise need not be equal for the
above approximation to be valid.
Further, the case of hard limiting of
many signals in noise is reducible to that of two signals in noise if the
power in each of the two signals is much greater than the sum of the power
in the remaining weak signals and noise.
The approximation is valid for
any arbitrary weak signal and noise power distribution since it is the
- 105 -
SEL-65-056
sum of the power in them that is of consequence.
Also the case of
signals in noise can be approximated as the case of
(p + l)
p
signals
without noise, as far as the behavior due to limiting is concerned.
Further, the output power in the various components of the limited signal
is relatively independent of the input strong-signal-to-noise ratios, for
ratios larger than 10 db.
For the case of
p
equal signals (with and
without noise), the results of hard limiting for a large range of
p
are
presented.
Some of the contributions of this research are listed below:
1.
A means of predicting the approximate behavior on hard limiting of
many signals and noise has been developed.
2.
Approximate expressions for the power in the various output signal
and nrossproduct components for any
given.
p,
derived analytically are
These expressions give results which are very close to
those obtained through extensive digital computer computations,
the error in most cases being negligible.
3.
The judicious selection is permitted of frequency spacings of
several modulated carriers so as to minimize the distortion over
any frequency band due to hard limiting.
4.
The strength of the interfering components present at the output
of an FM system, accomplishing demodulation by hard limiting followed by integrating an FM pulse train with arbitrary pulses at
every
p
crossover, can be easily evaluated by the results con-
tained in this report.
5.
The phenomenon of "aliasing," is analyzed, the optimum selection
of operating parameters is permitted, and systems are proposed
which minimize such distortion.
SEL-65-056
- 106
APPENDIX A.
USEFUL INTEGRALS OF BESSEL FUNCTIONS
Recently much work has been done in the evaluation of integrals
involving Bessel functions and products of Bessel functions.
Some of
the important references and pertinent equations are given below.
r
2 2
u-11
f exp (-pV) t^
JQ
(a/2p/r(^)
X
—^-+-
J V (at) dt =
2pM r (v + i)
2
/
2
V
exp (-a /4P) {*/2P) r(^)
2p^ r (v + i;
lFl(^+1;
...
^
(A.l)
with the conditions
R(fi + v) > 0,
R(p ) > 0
For tables of the confluent hypergeometric function,
jF
refer to
Slater [Ref. 12].
Ä
t
'o
| (a/2pAb/2Pr T ^+V+2K+l)
exp (-pV) J (at) J (bt) dt =
■
A+1
P
-
Z
k=0
- 107 -
r (n
+
i) r (v
(-)k(a/2P)2k(^
+
i)
V+ A+
2
^
k:((i + i),'k
SEL-65-056
. 2F1 U.-M-k; ^
+
^ \j
(A.2)
with the conditions
R(p2) > o
R(n + v + A) > -1,
where
(a)
is the generalized factorial function defined as [Ref. 13]
n
(cOn = H («+ k - i)
k=l
= a(a + 1)(a + 2) ... (a + n - 1)
nil
(A.3)
and
(a)0 =1
a ^ o
The following Weber-Schafheitlin type integrals are treated in great
detail in many references [see for example Watson, Ref. 11]
V
A 1
A+1
(b/a)
r00 .AA
^/aJ (a/2)
^i*! - iF i(^-2— )
t
J (at) J (bt) dt =
;
:
77^0
v> ' v^ >
2r (v + i) r (^- v - A+ ^
2*1 y
2
'
2
'
v +1
' ^J
(A.4a)
with the conditions
R(|i + v - A + 1) > 0,
SEL-65-056
R(A) > -1,
- 108 -
0 < b < a
(a/b)V(b/2)Ä-1r(^+
Ä
t'
J (at) J (bt) dt =
V
-A+1)
2r (v+ 1)r(^-^Ai_l)
(A.4b:
with the conditions
R(li + v - A + l) > 0,
R(A) > -1,
0 < a < b
Note that (A.4a) and (A.4b) are not continuous at
a = b
(a^"1 r(A) r (^+
t"A J (at) J (bt) dt =
2r
•'n
/v-^ + A+l\
r
v
and that
-
/v + |a + A+l\ p
Ä+
^
/M.
- v + A+ 1\
(A.5)
with the conditions
R(n + v + l) > R(A) > 0,
a > 0
For integrals involving the product of more than two Bessel functions,
it is possible to expand the other Bessel functions in powers of
to use expressions (A.4b) or (A.l).
t,
and
It was found best not to use the
above expressions for the general case as it is easier and more convenient
to carry out the integrations numerically on a digital computer.
- 109
SEL-65-056
i
APPENDIX B. DERIVATION OF AUTOCORRELATION
FUNCTION OF THE OUTPUT z(t)
I__ t
It is convenient to define
(B.l)
= [i|r(t) - ^(1 - t)]
Therefore,
output
- e^) » os^T +
z(t)
The autocorrelation function of the
m
m
is defined as
>
00
-(5) E<'S
u^cT + 6,j.(t) + ^
(sin 2p0j^^ - sin ZpO^^)
P=1
J •*
^
(sin
>
- sin
q=l
(B.2a)
where
E
denotes the expectation and
the Instant
t.
6^^^
denotes the value of
0^^
at
Hence
I
U<
. V »|v" • V' ♦ i 4'^
••i. •
V.*A
••••>
»
• *{i
v*>•Ss- • -
••{i 4^ v*> I- *s. —
(B.2b)
I
«
v.v
m
I
^
i
SEL-65-056
^
77j
- 110 -
r#:
,S^
Now,
E[lT(t)] = E[v|/(t) - tCt - T)] = 0
E[5T(t')] = 0
E{sii-> 2pe
For
<>
c
) = E/sin {2p[a) t + ilf(t)] + 2p$ ]J
unifoi-mly distributed on
E[sin 2pe
Xv
] =
[0,2n],
and
^(t)
and
t>
independent,
c
E{sin 2p[a) t + iKO]} E[cos 2pit) ]
c
c
+ E(cos 2p[cD t + i(t)]) E[sin 2pit> ]
c
c
= o
Similarly,
E(sin 2pe2t)
E(sin 2pe2t,) >
E(sin 2peitI)
Furthermore, by the independence of the random process
random variable
<J>
^(t)
and the
c
E[|T(t) sin 2peitl] = E^(t) sin ^p^f + t(t')] + 2p«c]J
- E\Kt - T) sin ^p^t' + tyit')] + 2p(t>c}
111 -
SEL-65-056
which further results in
EU^f) [sin 2peit - sin 2pe2t])
)
E{^T(t) [sin 2peit, - sin 2pe2t,]}
Therefore,
00
00
NP+Q
+ E
1 1 ^ikLp=l
(Sin
^It -
Sin 2pe )
2t
q=l
(B.3)
• (sin 2qeitI - sin 2qe2tI)
Further use may be made of the fact that
E[sin 2peit: sin 2qeit,] = ^ E[cos (2peit - 2qeit,)]
- | E[cos (2peit + 2qeitI)]
= | E cos {2pü; t + 2p<t>
- [2qoD t' + 2q$
C
C
+ 2pt(t)
+ 2q\^(t, )]]
- \ E{cos [2p(jD t + 2p4>
_}
+ 2p\lr(t;
+ 2q(jü t' + 2q$ + 2ql^f(t, )]]
c
c
0
SEL-65-056
if
- 112 -
p ^ q
Also, for all Integral value of
p
and
q £ (l,c
| E[cos (2peit + 2q0ltI)] = 0
The other terms can be evaluated in a similar manner.
R^t.f)
-ft)'
CD
2_2
2
T
E
+ R^ (t,t') +
\| ^
[cos
Hence,
2
P(eit ' ^t')
p=l
+ cos 2p(e2t - e2t,) - cos 2p(e 2t " ^t') "
COS
2p(e
it - ^f^/
(B.4)
Define
t - t' =- T,
that is,
t' = t - T
E[cos 2p(ei+ - ei4.,)] = E{cos 2p[^t + ^(t) -
CJD
t • - ^(t')]}
= E cos 2p{ü) T + [t(t) - ilf(t - T)])
E{cos 2p[ai T + 5_(t)]]
C
u
by definition of
I (t)
«
= E^Re exp {i[2p(DcT + 2p|T(t)]}J
= E^Re {exp [i2pcucT] exp [i2plT(t)]}J
Re (exp (i2pcD
T)
C
E[exp (i2pt (t)]}
"
(B.5)
By definition of the characteristic function,
*2p| (f) =
E ex
[
113
P (i^p^)]
SEL-65-056
Here is is assumed that
\lr(t -
T)
IS
^(t)
is stationary.
also stationary and depends on
Therefore
T
only.
| (t) = ilr(t)
Hence
%i (ä^^^M^"
where
Q0 ,
2plT
is the characteristic function of
2p| .
T
(B.6)
Therefore
E[cos 2p(eit - eitT)] = Rejexp ^poyr) ^{j^j
(B.7)
Similarly,
E[cos 2p(e2t - e2t,)] = E[cos {2püücT + 2p^T(t - T)J]
(B.8a)
E[cos 2p(e2t - eit,)] = E[cos {2p(iüc(T - T) + 2p5T_T(t - T)}]
(B.8b)
E[cos 2p(eit - e2t,)] = E[cos [2poüc(T + l) + 2plT+T(t)}]
(B.8c)
This technique allows computation of any specific term, given
To proceed further a specific case is assumed where
^(t)
t(t).
is a
random process:
\l/(t) = ß sin (CD t + 0> )
where
♦
m
is a random variable with
2n
P(0
m
= <>
V0
SEL-65-056
e [0,2n]
elsewhere
114 -
(B.9)
The following expansions by Jacobi and Anger are useful [Watson,
Ref. 11, p. 22]:
J
cos (z sin 9) = J0(z) + 2 y
u
n=l
sin (z sin e) = 2 Y
n=0
J
2n^
2n+l^z^
cos (z cos 9) = J0(z) + 2 >
sin
cos
^2n
(-1)
+
2ne
(B.lOa)
1 e
(B.lOb)
^
J2n(z) cos 2ne
(B.lOc)
n=l
sin (z cos 9) = 2 y (-l)"
J
2n+1(z)
cos
(2n
+
1 e
)
(B.lOd)
n=0
Also, extensive use is made of the fact that for
on
(0,2n)
and with
f(t)
and
9
9
uniformly distributed
independent [Davenport and Root, Ref. 7]
E{cos [f(t) + e]) = 0
Hence,
E[cos 2p(9lt - eit,)] = E|COS (2p[cDcT + ^T(t)]]J
= cos 2pa) T E[cos 2p£ (t)]
sin 2püo T E[sin 2p| (t)]
115 -
(B.ll)
SEL-65-056
Now,
E[cos 2p| (t)] = E{cos [2pß sin
t + «0 ) - 2pß sin
(OD
(üJ
t + $
-
CD T)]}
Efcos [2pß sin (u) t + $ )] cos [2pß sin (ou
t + * v
m
m
m
m
+ Efsin [2pß sin
(CJü
sin [2pß sin
m
OD T)]J
m
t + $ )]J
m
(CJJ
m
t + $
m
- üb
m
T)
11
Using the above expansions and taking expectations,
E[cos 2p|T(t)] = E<
Jrt(2pß) + 2 > J0 (2pß) cos 2n(üJ t + <t> )
0
Zw 2n
m
m
n=l
J0(2pß)
+
2 J J2m(2pß)
m=l
• cos 2m((x> t + it) - CD T)
m
m
m
2
+ E<
2
SEL-65-056
> J
,(2pß) sin (2n + l)(cu t + « )
^ n2n+l
m
m
n=0
> J„ , v(2pß) sin (2m +
Z-i
2m+l
m=0
116 -
1)(Cü
m
t + $
m
- Oü T,
m
(2pß) +0+0+4>-
JQ
cos 2noo T
m
2
n=l
+ 4
_ J
(2pß)
> -^r
cos (2m +
/,
2
m=0
1)(CD
T)
v
m
= jfapß) + 2 ^ jj(2pß) cos nüDmT
n=l
>
^
(B.12:
J v(2pß) cos no» T
n
m
n=-oo
Also,
Efsin 2p£ (t)l = Efsin [2pß sin (a) t + $ ) - 2pß sin (ao t + <Ji
= Efsin
[2pß
^
L h-K sin
(üD
\ mt
+ 0 /J
)] cos [2pß
rr sin
m
(üü
\ mt
-
+ <t
- Efcos [2pß sin (a) t + 4» )] sin [2pß sin (ou t + $
= E<
2
>
ZJ
J0 ,(2pß)
sin (2n +
2n+lv
1)(üü
m
t + *
CD
m
x)])
- tu
- cu
m
T)]]
' J ^
T)]
r
n=0
v
J„(2pß)
+ 2 ^
> J,,2mv(2pß)
^ ' cos 2n(cü
m t + 4> m - a)mT)'
0\ I'K/
m=l
117 -
SEL-65-056
j
- E<
J0(2pß) + 2 2^
J
2m(2Pß)
cos
2n(a)mt + tj
m=l
2
> J
(2pß) sin (2n + l)(ü) t + $ - CD T)
<^ ^n+1
m
mm
n=0
= 0
(B.13)
Substituting (B.12) and (B.13) in (B.ll) gives
E cos {2p[üü T + £ (t)]) =
c
T
> J2(2pß) cos noj T cos 2pcü T
z^
n
m
c
n=-oo
=1
J (2pß) cos (2pa> + neu )T
n
c
m
(B.14)
v
'
n=-oo
It is obvious that
E[cos 2p(e2t - e2t,)] = E{cos [2p(jucT + 2ptT(t - T)]}
= E{cos [2poü T + 2p| (t)]}
=1
J (2pß) cos (2pcü + neu )T
n
c
m
n=-oo
SEL-65-056
118
(B.15)
v
'
E[cos 2p(e2t - eit,)] = E{cos [2püüc(T - T) + 2plT_T(t - T)]}
= cos 2pcx) (T - T) E[COS 2p(; _ (t - T)]
- sin 2püüc(T - T) E[sin 2p|T_T(t - T)]
It is evident from the above that
E[cos 2p(e2t - 0lt,)] =
^
J^(2pß) cos (2poDc + nüDm)(T - l)
n=-oo
(B.ie;
Similarly, it can be shown that
E[cos 2p(eit - e2t,)] =
S
Jn(2pß) cos (2paJc + nuijil + T)
n=-oo
(B.I?;
Hence
;{cos 2p(elt - elt!) + cos 2p(e2t - e2t,) - cos 2p(e2t - elt))
cos 2p(eit - e2tI)} =
2,
Jn(2pß)[2 cos (2püJc + nujl
cos (2pa)
v
' c
= 2
2^
+ nüü )(T-T)-COS (2pa> + nw )(T+T)]
x
m
'
c
m'
J^(2pß)[l - cos (2pü3c + ncjJm)T]
n=-oo
cos (2pcjü
x
c
+ na) )T
m
119 -
SEL-65-056
= 2
jf(2pß) 2 sin2 (2^ + neu ) |
y
n=-oo
cos (2püü
+ neu )T
Also,
Rg (t,f ) =E{lT(t) 5T(f )3
'T
= E([^(t) - ^(t - T)] [t(t') - t(t, - T)])
= E[t(t) ^(f )] - E[t(t) \lr(t' - T)]
- E[t(t - T) tyit')] + E[\lr(t - T) ty{t' - T)]
2RiJr(T) - [R^(T + T) + R^T - T)]
Taking the Fourier transform gives
S^ (f) =28^(1) - [eiüjT+
iCüT
e"
] St(f)
= 2S (f) [1 - cos cuT]
=4S^(f) sin2f
(B.18)
Hence,
t t,
T
+
+
00
00
2,
v ' ) = V ) = (;)^y v^ 2 s —
T
sin
SEL-65-056
L
(2poü
c
^p=l,
^
n=-oo
„ %
2J (2pß)
P
2
+ nao ) — cos (2pa) + neu )T
nr 2J
c
m' ,
- 120 -
Since
\lf(t) = ß sin
VT)
(üü
= E[ß
t + (t> ),
Sin
it can be concluded that
(
V
+
*m)
ß
Sin
[
V
+
*m ■ V)]
= '-— COS ÜJ T
2
m
which implies
2 5(f - fj
st(f) =%-
+
5(f ^ fj
i
Hence in this case, autocorrelation of the output process,
Rz(t,t.)=Rz(T)=(^
00
2_2 + 2R^(T) - [R^T + T) + R^(T - T)]
a)V
u0
2/„\
2/„
\T
2J (2pß) sin (2püJ + nuj ) "
c
m ^
2 2^
p=l
n:
• cos (2paj
s
c
+ no)
m
)T
(B.19)
when the input random process \lf(t) = ß sin (^t + <t>m).
Eq. (B.19) is the result used extensively in Chapter III of this
report.
121
SEL-65-056
REFERENCES
1.
William R. Bennett and S. 0. Rice, "Note on Methods of Computing
Modulation Products," Philos. Mag., 18, Series 7, Sep 1934, pp.
442-424.
2.
David Middleton, "Some General Results in the Theory on Noise
through Non-Linear Devices," Quart. Appl. Math., 5, 4, Jan 1948,
pp. 445-498.
3.
David Middleton, "The Distribution of Energy in Randomly Modulated
Waves," Philos. Mag., 42, Series 7, Jul 1951, pp. 689-707.
4.
Wilbur B. Davenport, Jr., "Signal-to-Noise Ratios in Band-Pass
Limiters," J. Appl. Phys., 24, 6, Jun 1953, pp. 720-727.
5.
J. Granlund, "Interference in Frequency-Modulation Reception,"
Tech. Rep. No. 42, Res. Lab. of Electronics, Massachusetts Institute
of Technology, Cambridge, Mass., Jan 1949.
6.
Elie J. Baghdady, "Interference Rejection in FM Receivers," Tech.
Rep. No. 252, Res. Lab. of Electronics, Massachusetts Institute of
Technology, Cambridge, Mass., Sep 1956.
7.
Wilbur B. Davenport, Jr. and William L. Root, An Introduction to
the Theory of Random Signals and Noise, McGraw-Hill Book Company,
New York, 1958.
8.
John J. Jones, "Hard-Limiting of Two Signals in Random Noise," IEEE
Trans, on Information Theory, IT-9, Jan 1963.
9.
P. D. Shaft, "Hard-Limiting of Several Signals and Its Effect on
Communication System Performance" (paper presented at the IEEE
Convention, New York, Mar/Apr 1965--to be published).
10.
Emanuel Parzen, Modern Probability Theory and Its Applications,
John Wiley & Sons, 1960.
11.
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge
University Press, London, 2nd Ed., 1952.
12.
L. J. Slater, Confluent Hypergeometric Functions, Cambridge University
Press, Cambridge, 1960.
13.
Earl D. Rainville, Special Functions, Macmillan Company, New York,
1960.
14.
Yudell L. Luke, Integrals of Bessel Functions, McGraw-Hill Book
Company, New York, 1962.
SEL-65-056
- 122 -
15.
G. Eason, B. Notle, and I. N. Sneddon, "On Certain Integrals of
Lipschitz-Hankel Type Involving Products of Bessel Functions,"
Philos. Trans. Roy. Soc., 247, 1955, pp. 529-551.
16.
A. H. Van Tuyl, "The Evaluation of Some Definite Integrals Involving
Bessel Functions Which Occur in Hydrodynamics and Elasticity," Math.
Comp., 18, 87, Jul 1964, pp. 421-432.
17.
H. E. Fettis, "Note on
14, 1960, pp. 372-374.
18.
G. P. Weeg, "Numerical Integration of
f" e"X J (ilx|0 J^x^Jx'11 dx," Math. Comp. ,
J^ e
J0(TIX||) J1(X||)X
dx,"
Mathematical Tables and Other Aids to Computation, 13, 1959, pp.
312-313.
19.
Janis Galejs, "Signal-to-Noise Ratios in Smooth Limiters," IRE Trans.
on Information Theory, IT-5, Jun 1959, pp. 79-85.
20.
Nelson M. Blachman, "The Effect of a Limiter upon Signals in the
Presence of Noise," IRE Trans, on Information Theory, IT-6, Mar 1960,
p. 52.
21.
Wolfgang Grobner and Nimolaus Hofreiter, Integraltafel, Zweiter Teil,
Bestimme Integrale, Wien und Innsbruck, Springer-Verlag, 1950.
22.
S. W. Seeley, "The Ratio Detector," RCA Rev. , 8, Jun 1947, pp. 201236.
23.
B. D. Loughlin, "The Theory of Amplitude Modulation Rejection in the
Ratio Detector," Proc. IRE, 40, Mar 1952.
24.
Elie J. Baghdady, Lectures on Communication System Theory, McGrawHill Book Company, New York, 1961.
25.
Elie J. Baghdady, "Theory of Low Distortion Reproduction of FM
Signals in Linear Systems," IRE Trans, on Communications Theory,
CT-5, Sep 1958.
26.
R. Hamer, "Radio-Frequency Interference in Multi-Channel Telephony
FM Radio Systems," Proc. IEE, London, 108, Jan 1961.
27.
C. A. Cady, "A Frequency Monitor for Television Video Transmitters,"
The General Radio Experimenter, '13, 4, General Radio Company,
Cambridge, Mass., Sep 1948, pp. 1-6.
28.
T. J. I'a Bromwich, An Introduction to the Theory of Infinite Series,
Macmillan Co., Ltd., London, 1908.
29.
Langford F. Smith, Radiotron Designers Handbook, Radio Corporation of
America, Harrison, New Jersey, 1953.
123 -
SEL-65-056
30.
Harold S. Black, Modulation Theory, D. Van Nostrand Co., Inc., New
York, 1953.
31.
B. Harris, "Some Bounds on the Power Spectra of Frequency-Modulated
Sinusoidal Carriers," D.Eng.Sc. Thesis, Columbia University, 1961.
32.
Norman Abramson, "Bandwidth and Spectra of Phase-and-FrequencyModulated Waves," Report of Alto Scientific Co., Palo Alto, Calif.,
Sep 1962.
33.
C. P. Wang and G. Wade, "Noise Figure of a Superregenerative Parametric Amplifier," IRE Trans, on Communication Theory, CT-9, 4, Dec
1962.
34.
S. Darlington, "Realization of a Constant Phase Difference," Bell
Sys. Tech. J. , 29, Jan 1950, pp. 94-104.
35.
G. Fritzsche, "Practical Two-Phase Networks," Nachrichtentech. Z.,
8, Apr 1958, pp. 154-158.
36.
D. N. Vinogradov, "Determination of the Error in the Constancy of
the Phase Difference over a Frequency Band," Electrosvyaz, 27, May
1958, pp. 35-43.
37.
S. D. Bedrosian, "Design of Wide-Band 90 Degree Phase-Difference
Networks," M.E.E. Thesis, Polytechnic Institute of Brooklyn,
Brooklyn, New York, May 1951.
38.
S. D. Bedrosian, "Normalized Design of 90 Degree Phase-Difference
Networks," IRE Trans, on Circuit Theory, CT-7, Jun 1960.
39.
J. L. Dubois and J. S. Aagaard, "An Experimental SSB-FM System,"
IEEE Trans, on Communication Systems, CS-12, Jun 1964.
SEL-65-056
- 124
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Northwestern U
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Measurements Lab.
Polytechnic Institute
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Brooklyn, N.Y.
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•Central Electronics Engineering
Research Institute
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Northeastern U
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Boston 15, Mass.
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Oregon State U
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The Boeing Company
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College of Engineering
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Ohio State V
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Ball Telephone Labs., Inc.
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Whlppany Lab,
Admiral Corp.
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Chicago 47, 111.
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Airborne Instruments Lab.
Comae Road
Deer Park, Long Island, N.Y.
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Tech. Dir.
Autonetlcs
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Bell Telephone Labs.
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1
Dr. S. Darlington
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Mr. A. .1. Grossman
General Electric Co. Res. Lab.
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1
R. L. Shuey, Mgr. Info.
Studies Sec.
General Electric Co.
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GllfUlan Brothers
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Hughes Aircraft
M.illhu lioach, Calif,
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*No AF or Classified Reports.
SYSTEMS THEORY 5/65
- 3 -
Hughes Aircraft Co.
Florence at Teale St.
Culver City, Calif.
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Rm C2046
»Dir., National Physical Lab.
Hllside Road
New Delhi 12, India
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ONR/London
Sylvanla Electronics System
100 First St.
Waltham 54, Mass.
1 Attn: Librarian, Waltham Labs.
1
Mr. E.E. Hollls
Hughes Aircraft Co.
P.O. Box 278
Newport Beach, Calif,
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»Northern Electric Co., Ltd.
Research and Development Labs
Dept. 8300
Ottawa, Ontario
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Technical Research Group
Route No. 110
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IBM, Box 390 Boardman Road
Poughkeepsle, N.Y.
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IBM Poughkeepsle, N.Y.
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E. M. Davis
IBM ASD and Research Library
P.O. Box 66
Los Gatos, Calif
95031
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ITT Federal Labs.
500 Washington Ave.
Nutley 10, N.-I.
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Laboratory for Electronic^, Inc
1075 Commonwealth Ave
Boston 15, Mass.
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Northronics
Paloa Verdes Research Park
6101 Crest Roft
Palos VerdesAitates, Calif.
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Pacific Semiconductors, Inc.
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Lawndal/, Calif.
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Philia, Tech. Rep. Division
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tladlo Corp, of America
'RCA Labs., David San off Res. Cen.
Princeton, N.J.
/2 Attn: Dr. J. Sklansky
Texas Instruments, Inc.
P.O. Box 6015
Dallas 22, Texas
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Texas Instruments, Inc.
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Dallas, Texas 75222
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2
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Library
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Research
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Weitermann Electronics
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Librascope
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Melpar, Inc.
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1 Attn: Librarian /
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1
Librarian
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Palo Alto, Calif.
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Monsanto Chemical Co.
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St. Louis 66, Mo.
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Inorganic Dev.
Librarian
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Microwave and Power Tube Div.
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Raytheon Manufacturing Co.
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1
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Director of Research
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West 1 nghouse EK'ct ric Cnrp .
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Zenith Radio Corp.
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Chicago 39. 111.
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»University of Western Australia
Nedlands, West Australia
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Head of Electrical
Engineering
Sperry Gyroscope Co.
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Great Neck, N.Y.
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Engineering Library
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Sylvanla Electric Products, Inc.
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»No AF or Classified Reports.
SYSTEMS THEORY
- 4 -
UNCLASSIFIED
Security ClaMification
DOCUMENT CONTROL DATA • R&D
(tmeurtty elutltletllen ol 11»: body of mb,lt,cl mnd <nd.mn« mnotmtlon mu»r b« »nf.fd whin th, ovmrmll Mpotl I» cfM.Hfd)
2«. HE^OKT •■CUmTV C LAMIFICATrON
1. OHIOINATINO ACTIVITY (Cetpotmf author.)
UNCLASSIFIED
Stanford Electronics Laboratories
Stanford University, Stanford, California
zb
OROUP
3. RIPORT TITLE
SOME TOPICS ON LIMITERS AND FM DEMODULATORS
4. DMCHIPTIVt HOT«! (Typt ol ropor« and Inehitly dttoo)
Interim Technical Report
t- AUTHOR^ (Loot nan«, Mrai nan«, Mllal)
Maung Gyi
7«. TOTAL NO. OF PAOKJ
• . HEPO NT DATE
7b. NO. OF RCFt
39
124
July 1965
• a. ONiaiNATOR'aXsPORT NUMBKRfS;
• a. CONTRACT OR ORANT NO.
ONR Contract Nonr-
25(83)
b. PROJECT NO.
su-SEL-eö-p'se
TechnicayReport No. 7050-3
• b. OTH« REPORT NOCS; (Any othotnumbon Äalmay bm mttltnod
mim/toporO
10. AVAILAIILITY/UIMITATIOH MOTICES
Qualified requesters may obtddn copies from DDC.
dissemination by DDC limited.
11. SUPPLEMENTARY NOTES
Foreign announcement and
12. SRONSORINO MILITARY ACTIVITY
U.S. Army, Air Force and Navy (ONR)
An analysis is made of wv^e-deviation frequency-modulation systems having low
nominal-carrier-frequen/y to information-bandwidth ratios. Since limiting plays
an important role in s/ch systems, the eff^ts of hard limiting of many signals in
random noise are anal/zed. Expressions are ^iven for the output signal-to-signal
ratios (SSR), the ou/put signal-to-noise ratioss(SNR), and the power in any crossproduct. The effect of the power in each output ömnponent as a function of the
Input SNR is investigated.
It is found that for al\ values of input SNR's greater
than 10, the strengths of the various output componertts are relatively constant.
For the'case of more than two input signals, a weak signal-boosting effect manifests itself when the input SNR's are less than 1 and the input SSR's are greater
than 1-10. The signal-suppression effect and the signäl-to-signal power sharing,
together with their dependence on input signal noise, aVe presented for various
cases.
Expressions are presented for the autocorrelation function and the
spectral power density of hard-limited FM pulse trains, which allow the computation of the intermodulation products in the information bandwidth or in any other
frequency interval. The effect of the nominal-carrier frequency on the interference ratio is exhibited graphically for various constant deviations. A partial
interaction of the positive and negative spectra of the modulated wave causes
extraneous outputs. For all such spurious components to be filterable, a relation
is derived between the carrier frequency, the maximum frequency of information,
and the spectral bandwidth of the modulated wave. The analysis results in systems
which have better performance capabilities.
DD
FORM
I JAN 64
1473
IINPTASSTFTRn
Security Classification
UNCLASSIFIED
Security ClaMlfication
u.
LINK A
KEY WORDS
NOLK
LINK ■
ROLK
WT
LINKC
ROLK
COMMUNICATION THEORY
SIC-NAL-TO-NOISE RATIO
HARD LIMITING
FREQUENCY MODULATION
INSTRUCTIONS
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DD
FORM
I JAN «4
1473 (BACK)
UNCLASSIFJED
Securitv Claaalfication