Document text
Intermodulation Distortion in Analog
FM Troposcatter Systems
By E. D. SUNDE
(Manuscript received May 20, 1963)
In broadband transmission over troposcatter paths, selective fading will
be encountered with resultant transmission impairments, depending on the
modulation method. An analysis has been made in a companion paper of
such selective fading, based on an idealized model of troposcatter paths. It
indicated that selective fading will be accompanied by phase nonlinearity
which in a first approximation can be regarded as quadratic over a narrow
band. A probability distribution for such quadratic phase distortion was
derived. On the premise of quadratic phase distortion, the error probability
owing to selective fading ivas determined for digital transmission by vari-
ous methods of carrier modulation.
The same idealized model and basic premise of quadratic phase distor-
tion is used here to determine intermodulation distortion in FM for a sig-
nal with the statistical properties of random noise. An approximate ex-
pression for intermodulation noise owing to specified quadratic phase
distortion has been derived, applying for any method of frequency pre-
emphasis in FM. In turn, median intermodulation noise as well as the
probability distribution of intermodulation noise Jms been determined, as
related to certain basic system parameters.
A comparison is made of predicted with measured intermodulation noise
in four troposcatter systems with lengths from 185 to 440 miles. The results
indicate that phase nonlinearity owing to selective fading can be approxi-
mated, by quadratic phase distortion, or linear delay distortion, over an
appreciable part of the transmission band ordinarily considered for tropo-
scatter systems, with a probability distribution that can be determined from
certain basic parameters of troposcatter links, such as the length and an-
tenna beam angles. However, to predict intermodulation distortion on any
system, further experimental data than are now available are required on
beam broadening by scatter.
The present random multipath FM distortion theory is shown to afford
309
400 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
a significant improvement over an equivalent single-echo theory that has
been applied on an empirical basis to troposcatter systems.
INTRODUCTION
An analysis has been made elsewhere 1 of error probabilities in high-
speed digital transmission over idealized troposcatter paths, consider-
ing both random noise and intersymbol interference owing to pulse
distortion caused by selective fading. The above analysis indicated that
a principal cause of intersymbol interference is a quadratic component
of phase distortion, or linear delay distortion. On the same basic prem-
ise an evaluation is made herein of intermodulation noise in analog
transmission by frequency modulation, as now used for transmission of
voice channels in frequency division multiplex. Expressions and curves
are given of intermodulation noise in an idealized troposcatter channel
for a signal with the properties of random noise, as related to certain
basic system parameters and comparisons are made with the results of
measurements on four troposcatter systems. 2 - 3
In random multipath transmission the received wave can be considered
the sum of a plurality of echoes, arriving over the various paths with
varying amplitudes and different delays. Although this view is con-
ceptually simple, it does not facilitate analysis of the statistical proper-
ties of the received signal and of signal distortion. In the combination
of a number of time functions, such as echoes, the analysis is greatly
facilitated by the use of Fourier transformation to determine the cor-
responding spectra. The latter can in turn be combined directly with
appropriate attention to phase relations to obtain the resultant wave.
For this reason it is preferable from the standpoint of analysis to regard
the received wave as a multiplicity of sine wave components, rather
than signal wave echoes, arriving over the plurality of transmission
paths with varying amplitudes and phases. This is the method ordi-
narily used in the analysis of the statistical properties of narrow-band
random noise, which has properties that with appropriate translation
of the basic parameters are also applicable to random multipath trans-
mission. It is the method underlying both the previous determination
of error probabilities in digital transmission owing to noise and selec-
tive fading, and the present analysis of intermodulation noise in FM.
In certain radio systems the received wave can be considered the sum
of a principal signal wave and a weaker echo, and comprehensive theo-
retical analyses have been published of intermodulation noise in FM
owing to such echo distortion, 45 ' 6 together with the results of simulative
tests. 7 For these reasons this two-path model has been adopted as a
INTERMODULATIOX DISTORTION
401
coarse simile to multipath transmission in some interpretations of the
result of measurements of intermodulation noise in troposcatter sys-
tems. 3 The limitations of this simile arc recognized in the latter publica-
tion, 3 in which it is suggested that a more refined analysis is desirable.
The idealized multipath model used in the analysis of troposcatter
digital transmission affords a significant improvement, though it has
certain predictable limitations, as shown herein.
I. TRANSMITTANCE PROPERTIES OF TROPOSCATTER LINKS
In tropospheric transmission beyond the horizon the received wave
can be considered the sum of a large number of components of varying
amplitudes resulting from a multiplicity of reflections within the com-
mon volume of the antennas. Owing to variations in the structure of
the common volume, caused largely by winds, there will be relatively
slow changes in the many reflections and thus in the amplitudes of the
component waves. When a steady-state sine wave is transmitted, the
received wave will thus exhibit random variations in its envelope and
phase, known as fading.
In addition to such transinittance variations with time at a particu-
lar frequency, there will be transinittance variations with frequency at
any given instant, as illustrated in Fig. 1. At a given instant the ampli-
tude and phase characteristics of the transmission path may be as indi-
cated in Fig. 1(a) and at a later instant as in Fig. 1(b).
Let u = u> — con represent the radian frequency relative to a reference
frequency co . When the transmission vs frequency characteristic of a
troposcatter channel varies slowly with time t, it can be represented by
(a)
ATTENUATION
/
— ^ /
PHASE - . - »«""*'
t=t,
FREQUENCY, CO-
FREQUENCY, CO-
Fig. 1 — Illustrative variations in attenuation and phase characteristics with
frequency at two instants /i and U .
402 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
T{u,t) = A(u,t)e- iv ^ l) (1)
where
A(u,t) = amplitude characteristic as a function of t for a fixed
u, or as a function of u for a fixed time t
<p(u,t) = phase characteristic.
If u = uq is fixed, both A(u , t) and <p(u , t) are random variables of
the time t, as are the time derivatives A'(uq , I), A"(u , t), <p'(u , t),
<p"(uo , 0- The probability distributions of A(uq , t) and <p(uo , t) can be
determined on the premise that they are the sum of a large number of
randomly phased components. This results in a Rayleigh probability
distribution of A(u , 1), in conformance with observations of rapid fad-
ing. To determine the probability distributions of A', A", <p' and <p",
statistical information is required regarding the rapidity of fades. This
ordinarily takes the form of the time autocorrelation functions of A(t),
or the related power spectrum of changes in transmittance amplitude.
Such power spectra can be characterized by a certain equivalent fading
bandwidth.
If the time is assumed fixed at t = t , then A(u,t ) and <p(u,t ) will
have certain random fluctuations with the frequency u that can be char-
acterized by probability distributions. This also applies to A(u,t ),
A(u,t ), <p(u,t ), and <p( w A)> where the dots indicate differentiation with
respect to frequency u. The probability distributions of A, <p, and A and
ip depend on the frequency autocorrelation functions, or the correspond-
ing power spectra of variations with frequency. The latter depend on
differences in transmission time over the various paths, and can be re-
lated to the maximum departure A from the mean transmission delay.
The amplitude and phase characteristics as a function of u at any time
*o can in general be represented by a power series as
A(u,t ) = a + a\u + a 2 u 2 + OjW + • • • (2)
<p(u,t ) = b + biu + b 2 u + btfli + ■•*. (3)
Certain basic relations have been developed by Carson and Fry and
by van der Pohl, 8 for transmission impairments in FM resulting from
attenuation and phase distortion. With the aid of these relations it can
be shown that intermodulation noise is caused principally by phase dis-
tortion rather than by amplitude distortion. Moreover, it can be shown
that the principal contributor is quadratic phase distortion represented
by b 2 u, which corresponds to linear delay distortion 2b 2 u.
[NTERMODULATION DISTORTION 403
II. PROBABILITY DISTRIBUTION OF QUADRATIC PHASE DISTORTION
From (3) it follows that
<p(u,tn) = 26 2 + (Sb,u + •••. (4)
For m = 0, i.e., at the reference or carrier frequency, the probability
distribution of b- 2 is the same as that of ip(0,t). The latter probability dis-
tribution has been determined elsewhere on the approximate premise of
a linear variation in transmission delay, with maximum departures ±A
from the mean delay. In Fig. 2 is shown the probability that ip, or 26 2 ,
exceeds A"/3 by a factor /.\ For example, there is a probability p = 0.5
that ip exceeds A"/3 by a factor k ft 1.2, and a probability p = 0.1 that
ip exceeds A/3 by a factor /oft 19.
Thus in general
ip„ = 2h(p) = fc p A 2 /3 (5)
where fcpA /3 is the value of ip, or 2/>> with a probability p of being ex-
ceeded.
Alternatively, the value of lh with a probability p of being exceeded is
/> 2 (p)=^A 2 . (6)
Thus
6,(0.5) ft^A 2 = 0.2A 2 (7)
b
6 2 (0.1) ft^A 2 = 3.2A 2 (8)
b
M0.01) ft ^A 2 = 67 A 2 . (9)
b
Thus, when A is known, together with intermodulation noise for
quadratic phase distortion, it is possible to determine the median value
of average intermodulation noise, or the value exceeded with any other
specified probability p.
III. [NTERMODULATION NOISE FROM QUADRATIC PHASE DISTORTION
In a first-order evaluation of intermodulation noise, only the quadratic
term b^u in (3) would be considered, since it will be the principal con-
tributor. The ratio of nonlinear distortion power to average signal power
at the frequency w will depend on the signal properties and on the pre-
404
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
FACTOR, k( )
10"' 2 5 1 2 5 10 2 5 10 2 2 5 IP 3
Al
:3»
10-2
N
\
i.
10-3
'>
>
ICT 4
\
V
>
V
2
10" 5
^
s ,o* 2 5 io 5 2 5 10 6 2 5 I0 7
FACTOR, k( )
Fig. 2 — Probability that lp or 26 2 exceeds A 2 /3 by a factor A;.
emphasis used in frequency modulation. It will be assumed that the orig-
inal message wave has a flat power spectrum of radian bandwidth
£2 = 2irB and the statistical properties of random noise, and furthermore
that the message wave is passed through a transmitting filter with a
power transfer characteristic
f(o>) = l+c(o,A2) 2
= l+c(//5) 2 .
At the receiving end a complementary filter is used to restore the mes-
sage wave.
As discussed in the Appendix, exact determination of intermodulation
noise from quadratic phase distortion presents formidable difficulties,
except on the premise of slight phase distortion, which is not generally
applicable to troposcatter systems. However, it is possible to obtain an
INTERMODULATION DISTORTION 405
approximate solution without the above limitation. The following rela-
tion is derived in the Appendix for the ratio p(f) of intermodulation
noise to average signal power at the frequency / = u/2tt
P(f) =^ 2 G(c,a)H(y) (11)
where c is defined by (10)
a = f/B = a/a
B = bandwidth of baseband signal = Q,/2t
D = rms frequency deviation = £2/2ir
and
y = & 2 Q 2 = (2ir)% 2 D\ (12)
The function G(c,a) depends on the pre-emphasis and is given by
expression (108) in the Appendix, which is
G(c,a) = —— " ■ — r F(c,a)
(1 + ca 2 )(3 + c)
F(c,a) - 2 - a + 2c ± cV [1 + (1 - a) 3 ] (13)
- y[l - (1 -a) 4 ] +^[1 + (1-«)1
This function is shown in Fig. 3 for pure FM and PM and for c = 10.
The particular case of c = 10 and a = 1 will be considered further in
the following, and for this case
G(16,l) = 0.192.
The function H(7) is shown in Fig. 4 and represents an approximation,
as discussed in the Appendix. It will be noted that this function departs
from proportionality with 7" for 7 ^ 0.5, reaches a certain maximum
value and then diminishes.
IV. INTERMODULATION NOISE IN TROPOSCATTER PATHS
In accordance with (0), the value of b 2 with a probability p of being
exceeded is 6 2 (p) = k p A 2 /Q. The corresponding value of 7 is given by
(12) as
7;» - -77- Utt) /;
6 (14)
= 0.GA- p (AZ)) 2 .
406 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
0.9
\
\
c = oo
'c=o
|C=16
^
0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a = f/B
Fig. 3 — Function G(c,a) for pure FM (c = 0), pure PM (c = co), and for pre-
emphasized FM with c = 16.
7o.6^8 (AD) 2
to.! « 125 ( ad;
Thus
(15)
(16)
7o.o! = 2600 (AD 2 ). (17)
The corresponding ratios p(f) at / = B with a probability p of being
exceeded
Pp {B) = 0.192 f|j ff( T ,)
o.,(B) = 0.192 ^|J tf (8A 2 D 2 )
(18)
(19)
INTERMODri.ATIOX DISTORTION- 407
p 0A (B) = 0.192 Q?Y//(125A 2 D-) (20)
Pom(B) = 0.192 (jS tf(2600AD 2 ). (21)
V. DIFFERENTIAL TRANSMISSION' DELAY A
Exact determination of the equivalent maximum departure from the
mean transmission delay requires consideration of the antenna beam
patterns as affected by scattering. On the approximate basis of equiva-
lent antenna beam angles a, it follows from the geometry indicated in
Fig. 5 that
lk±*-±1(* +"-±l) (22)
where fl £ a,v is the velocity of propagation in free space, L is the length
of the link, and
e - k - 2-4- (23)
where Ra is the radius of the earth and the factor K is ordinarily taken
as 4/3.
The equivalent antenna beam angle a from midbeam to the 3-db loss
point depends on the free-space beam angle au and on the effect of scat-
ter, which is related in a complex maimer to a and the length L, or al-
ternatively 6. Narrow-beam antennas as now used in actual systems are
loosely defined by a ^ 20/3. For these, a « a on shorter links, while on
longer links a > ao owing to beam-broadening by scatter. Analytical
determination of a for longer links appears difficult, and only limited ex-
perimental data are available at present. For broad-beam antennas,
ao y> 26/'.] and beam-broadening by scatter is in theory inappreciable.
By way of numerical example, let L = 170 miles and K = 4/3, in
which case = 0.016 radian. With a = 0.004 radian « 20 3 it is per-
missible to take a = ao . With = a = a , (22) gives A = 0.08 X 10" 6
second.
The differential delay A in general varies with time and for narrow-
beam antennas can be considered the sum of two components
Ml) = A + Ai(0 (24)
where Ao is a fixed component obtained from (22) by taking a = a ,
the free-space beam angles. The variable component A t (/) depends on
408 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
H(7)
T=b 2 n 2 = b 2 (277D) 2
Fig. 4 — Function H{y). The parameter y is the phase distortion in radians
at a frequency corresponding to the rms frequency deviation fi = 2wD radians/
second.
Fig. 5 — Definition of antenna beam angles a, take-off angle /3 and chord angle
to midbeam. With different angles at the two ends, the mean angles are used in
expressions for A.
INTERMODULATION DISTORTION 409
scatter variation with time, as does path loss, and will have a certain cor-
relation with path loss variations. Owing to the fixed component Ao , a
weaker correlation exists between A(/) and path loss variations.
Because of the dependence of A on path loss, the ratio p p of intermodu
lation noise to average signal power will depend somewhat on path loss
However, for a given path loss p p is independent of the average trans-
mitter power and thus of the average signal power at the receiver.
VI. LIMITATIONS ON FIRST-ORDER DISTORTION THEORY
The above first-order approximation applies for sufficiently narrow
signal bandwidths at the detector input such that terms in (3) of higher
order than u can be neglected. Results given by Rice for random vari-
ables (Section 3.4 of Ref. 10) indicate there is no correlation between ip
and 'ip, so that distortion owing to the term b 3 u will combine on a power
addition basis with distortion resulting from b 2 u 2 . Moreover, there is a
negative correlation factor between ip and "ip, so that on the average 64
is negative whenever bj is positive, and conversely. Hence distortion pro-
duced by bill 4 will on the average subtract directly on an amplitude basis
from that resulting from b 2 u. In the range where the function H(y)
increases linearly with 7 , intermodulation noise owing to the term 6 2 w
increases as /; 2 2 (AD) 4 . In the same range, intermodulation noise from the
term b*u will vary as b* (AD) 8 and may hence have a significant effect
for adequately large values of A/) even though 64 be much smaller than
62 • As shown later, comparisons of measured intermodulation noise with
predictions based on the above first-order theory indicate the increasing
importance of the term btU in reducing intermodulation noise as AD is
increased.
VII. TWO-PATH VS MULT1PATH DISTORTION THEORY
The above first-order distortion theory is a mathematically derived
approximation that in principle yields valid results with appropriate
limitations on signal bandwidth and frequency deviation, and which
retains the multipath feature that is essential to this end. By contrast,
the two-path or single-echo simile mentioned in the introduction has
no such basis but has been adopted principally because of the conven-
ience of available theoretical analysis. 456 A second reason is that single-
echo distortion theory yields results that in some respects are quite
similar to those obtained with multipath transmission, as shown below.
It is noteworthy that, by proper choice of echo amplitude and delay,
results similar to those for median quadratic phase distortion can be
410
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
obtained. This is illustrated in Fig. 6, which shows the median ratio
p(B) obtained from (19) as a function of D for B = 1 mc/sec with
A = 0.1 and 0.5 microsecond. In the same figures are shown the ratios
p(B) obtained on the premise that the received wave consists of a main
signal and an echo of equal amplitude delayed by 0.07 and 0.4 micro-
second. The ratio p(B) for the latter condition is obtained from a chart
given in Fig. 9 of Ref . 3, applying for FM with virtually the same pre-
10
R 15
? 20
25
30
y?
(
)>
/
V
t
%
\
1
1
0.2 0.4 0.6 0.8 1 2 4
RMS DEVIATION, D, IN MEGACYCLES PER SECOND
6 8 10
Fig. 6 — Comparison of intermodulation noise from single-echo distortion and
quadratic phase distortion at R = 1 mc/sec: (solid lines) median intermodulation
noise from quadratic phase distortion for indicated departures A from mean delay;
(dashed lines) intermodulation noise from echo of same amplitude as signal with
delays A, as indicated.
INTERMOIHLATION DISTORTION' 411
emphasis as assumed herein and given by (10). The above charts are
based on echo distortion theory applying for echoes that are much
weaker than the signal, but this premise is ignored here in extending
the theoretical results to a fictitious echo of the same amplitude as the
signal. In this connection it may be noted that simulative tests 9 indicate
that intermodulation noise is nearly proportional to echo amplitude,
even when the latter equals the signal amplitude. With both quadratic
phase distortion and single-echo distortion, intermodulation noise is
virtually proportional to the second power of signal bandwidth. Hence,
the relative comparisons in Fig. 6 could also apply for other bandwidths
than B = 1 mc/sec.
The above comparisons indicate that in applying equivalent single-
echo FM distortion theory to multipath transmission as in troposcatter
systems, with physically tenable echo delays, certain dilemmas will be
encountered. The theory could be extended beyond its validity to fic-
titious echoes of the same amplitude as the signal, to obtain virtually
the same median intermodulation noise as for quadratic phase distor-
tion. This would exclude the possibility of greater intermodulation noise
than the median value, since the greater echo is by definition the main
signal. The other procedure would be to assume an echo that is smaller
than the main signal, which is physically more acceptable and does not
violate the basic premise underlying echo distortion theory. In this case
intermodulation noise predicted on the basis of echo distortion theory
would, at least in certain cases, be much smaller than actually observed
and could not be made to conform with observations, unless the echo
amplitude is increased to the same amplitude as the signal.
Thus, if the ratio of echo amplitude to signal amplitude is r, inter-
modulation noise power based on single-echo theory will be less than
for multipath transmission by a factor r 2 . Hence it becomes necessary
to introduce a factor 1/r- to make single-echo theory applicable to multi-
path transmission. In Ref. 3, this factor has been determined empirically
from measurements to be discussed later, and is given as 9 db.
VIII. OBSERVED MEDIAN INTERMODULATION NOISE
Measurements have been made on four troposcatter links of the me-
dian value of intermodulation noise at the frequency / = B. The modu-
lating wave in these tests had a flat power spectrum, and pre-emphasis
was used that closely corresponded tor = 16 in (10).
The basic parameters of the systems on which the measurements
were made are given in Ref. 3 and are summarized in Table I. In this
table ao is the free-space antenna beam angle from midbeam to the 3-db
412 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
Table I — Basic Parameters of Troposcatter Test
Systems in Caribbean (A) and in Arctic (B,C,D)
System
A
B
C
D
Length, miles
Radio frequency, mc
Antenna/diameter, ft
a (radian)
(radian)
Ao (microsecond)
185
725
60, 60
0.0115
0.015
0.12
194
900
30,60
0.017
0.016
0.21
340
900
120, 120
0.0058
0.031
0.185
440
800
120, 120
0.0058
0.034
0.255
loss point, which may not conform with the angle a in (22) when scatter
is considered. The values of K and are taken from Ref. 3, and differs
slightly from that obtained from (23) owing to differences in antenna
elevations. The take-off angle is virtually zero and has been neglected.
The value A of A given in the table was calculated with a = ao , rather
than the actual beam angle with scatter. Systems A, B, C and D corre-
spond to paths 1, 2, 4 and 3 in Ref. 3.
In Figs. 7 and 8 are shown the ratios pj (B) expressed in db as a func-
tion of the rms frequency deviation D for different bandwidths B of the
baseband signals.
ix. comparison of theoretical with observed median values
In the same Figs. 7 and 8 are shown median values of intermodula-
tion noise obtained from (19) for each case, based on values A m of A
that afford the best average approximation to the measurements. The
latter values are somewhat greater than A , as indicated in Table II.
A ratio A„,/A or a m /a > 1 is to be expected owing to beam-broaden-
ing by scatter, and the above ratios appear reasonable in the light of
present knowledge. Thus, if the actual angles a were known so that A
could be determined, it appears plausible that satisfactory conformance
with observed intermodulation noise would be obtained.
As noted in Section V, A includes a component Ai(0 that varies with
time depending on scatter conditions and which is correlated with path
loss fluctuations. The ratio p thus depends on path loss as affected by
scatter and has a certain correlation with path loss variation, as shown
elsewhere. 3 Hence, if measurements had been made under different
path loss conditions, the derived values A„, would have been somewhat
different.
From Figs. 7 and 8 it will be noted that with the above choice of A =
A,„ it is possible to obtain better agreement between predicted and ob-
served intermodulation noise for small bandwidths B of the baseband
INTERMODULATION DISTORTION
413
< 15
?. 25
•n 50
O
z
SYSTEM A
s^Z.
*c
/
V
r :: i
s?
/
^ s
/300
</
/,
X
\s
,>/
<£/
SYSTEM B
/
^k
°<?
>°* /
•>
-'
^
V
>^'
^
-x
^
o^
1 .
1
1
400 600 1000 2000 100 200 4-00 600 1000
RMS FREQUENCY DEVIATION IN KILOCYCLES PER SECOND
Fig. 7 — Comparison of measured and calculated median intermodulation
noise: (dashed curves) measured median intermodulation noise in top channels
at indicated frequencies in kc; (solid curves) calculated median intermodulation
noise for idealized model with (he following values of the equivalent maximum
deviation A from the mean transmission delay: system A, A,„ = 0.12 microsecond
(A = 0.12); system B, A m = 0.25 microsecond (A„ = 0.12).
signal and small deviations D than for large bandwidths and frequency
deviations. This probably resides in the circumstance that the phase
distortion terms of higher order than b 2 u 2 have been neglected in the
above first-order theory, as discussed in Section VI.
The measured median ratios given in Figs. 7 and 8 are plotted in
Fig. 9 against the ratios predicted by first-order theory. It will be noted
that measured intermodulation noise is less than predicted for signal-
to-interference ratios less than about 30 db, owing to reduction in inter-
modulation noise by phase distortion of higher order than b 2 u 2 that has
been neglected in first-order theory. The results in Fig. 9 permit an
approximate empirical correction to first-order theory.
As discussed in Section VII, with single-echo distortion theory vir-
tually the same median intermodulation noise is obtained as with the
above first-order theory, provided the echo is equal in amplitude to the
414
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
5
a.
iu
0-
Z
SYSTEM C
j
!>
HI
52
(T 25
Ul
§ 30
J
111
m 35
(/)
IS 40
o
Ul
Ul
<2 50
o
z
A^
p >
if
fc
^0 -
>
rv"2'
~^' a
x*
X
/
?
^ f
,;?
</
^y
<A-
r
/
60
SYSTEM D
v^
>c i
.-o-
1 o
0"^
V*
° u -c
— o-.
"~t
^ZA
s/
*^"9
■■"''
«•"
'^pA
.
1
1
200
400 600 1000 2000 60 100 200 400 600
RMS FREQUENCY DEVIATION IN KILOCYCLES PER SECOND
Fig. 8 — Comparison of measured and calculated median intermodulation
noise: (dashed curves) measured median intermodulation noise in top channels
at indicated frequencies in kc; (solid curves) calculated median intermodulation
noise for idealized model with the following values of the equivalent maximum
deviation A from the mean transmission delay: system C, A,„ = 0.25 microsecond
(Au = 0.185); system D, A,„ = 0.55 microsecond (A» = 0.255).
mean signal. For smaller echoes, predicted intermodulation noise must
be less. This conforms with results presented in Figs. 12 and 14 of Ref.
3, which show that intermodulation noise predicted from single-echo
theory is significantly smaller than observed. To obtain a satisfactory
average relation between predictions and observations, the predicted
values must be increased by 9 db, as in Fig. 15 of Ref. 3
Table II -
- Ratio A m /A
System
A
B
c
D
Length, miles
Ao , microsecond
A m , microsecond
A„,/A
a m /ao
185
0.12
0.12
1.0
1.0
194
0.21
0.25
1.2
1.1
340
0.185
0.25
1.35
1.35
440
0.255
0.55
2.15
2.15
INTERMODILATION DISTORTION
415
60
/
_l
HI
CO
/
/
IU
Q
Z
•
A
o A3
U
2
111
£ 35
-V
5^i
4^o
r "
U.
w
h
•
A
i oy
•
A
* '>
7
o
h
J 2'
1
<
z
o
55 ?n
A «
'
/
system:
o A
• B
A C
□ D
z
<
S
HI
/
Q
IU
K
/
10
<
u
2
/
/
/
/
5 10 15 20 25 30 35 40 45 50 55 60
MEDIAN SIGNAL-TO-INTERFERENCE RATIO WITH FIRST-ORDER THEORY IN DECIBELS
Fig. 9 — Comparison of measured median signal-to-interferencc ratios with
median values based on first -order approximation with best choice of differential
transmission delay A.
X. PROBABILITY DISTRIBUTION OF INTERMODULATION NOISE
From (18) it is apparent that the probability distribution of p is di-
rectly related to that of II (y,,). This function is shown in Fig. 10 as re-
lated to (AD) 2 for p = 0.5, 0.1 and 0.01. It should be recognized that
this function as given herein is approximate, and that the errors are
likely to be greater for small values of p than for median intermodula-
tion noise as considered previously.
From the curves in Fig. 10 it is possible to obtain approximate curves
of the probability distribution of intermodulation noise, applying for
various values of A-D- as shown in Fig. 11. These curves show that the
probability distributions vary markedly with the above parameter, in
conformance with a few probability distributions derived from observa-
416 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
10-s
10
10- 3 c ' 10
(AD) 2 = ^p/8
-2 2
Fig. 10 — Function H(y p ) for various probabilities p.
tioiis. 2 Because of the approximations involved in the present first-order
distortion theory, the above probability distribution curves should be
considered illustrative and may not be accurate enough for certain
engineering applications.
XI. PREDICTION OF INTERMODULATION DISTORTION
The present first-order intermodulation theory indicates that inter-
modulation distortion depends on the delay difference A, and this would
apply also for an exact theory. For various troposcatter links with differ-
ent angles a and 0, intermodulation distortion would be the same for
equal values of A. This is exemplified by comparison of intermodulation
noise in systems B and C as shown in Figs. 7 and 8. Though these sys-
tems have different angles a and 6, intermodulation noise is virtually
the same since A is the same. Thus, if A could be determined, the above
first-order theory, in conjunction with the above experimental data,
would permit determination of intermodulation distortion for a variety
INTERMODULATION DISTORTION
417
!*, io-'
- ^
, \
^^^
\
\
\
-
\
\
1
\
\
i
d
c
b
V
-20 -15 -10 -5 5 10 15 20 25 30
DECIBELS ABOVE MEDIAN INTERMODULATION NOISE
Fig. 11 — Probability distributions of intermodulation noise for various values
of (A/)) 2 corresponding to dashed lines a, b, c and d in Fig. 10.
of conditions other than those in the tests. The above experimental
data were confined to intermodulation noise in the top channel, i.e.,
for a = 1 in Fig. 3, and for a particular pre-emphasis, c = 16. The ex-
pression for 0(c,a), or the curves in Fig. 3, permit approximate deter-
mination of intermodulation noise at other frequencies, and also for
other kinds of pre-emphasis. For example, for a = 0.3, intermodulation
noise would be greater than for a = 1 by an approximate factor 0.32/
0.19 ~ 1.7. If pure FM(c = 0) had been used in the tests, intermodula-
tion noise at a = 1 would have been increased by an approximate fac-
tor 1/0.19 « 5.2.
At present there is a principal obstacle to prediction of intermodula-
tion distortion for other values of ao than in the above experimental
systems. This is the lack of comprehensive experimental data on the
beam angle a as affected by scatter for troposcatter links of various
lengths. When and if such data become available, it will be possible to
determine A and in turn intermodulation distortion in the manner indi-
cated above for any kind of system.
418 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1904
XII. APPLICATION TO DIGITAL MULTIBAND TRANSMISSION
The distributions in Fig. 10 apply for average intermodulation noise
over brief time intervals, as determined by changes in phase distortion
with time. During each such interval the instantaneous amplitudes of
intermodulation noise will fluctuate about the average value. For a
signal with the properties of random noise, as considered here, the proba-
bility distribution of this fluctuation is approximated by the normal law.
The distribution of instantaneous amplitudes or intermodulation noise
is important in transmission by FM of a number of digital channels in
frequency division multiplex, as discussed below.
In digital transmission over troposcatter paths, the error probability
for a given signal-to-noise ratio of the receiver depends on the trans-
mission rate, as discussed elsewhere. 1 As the transmission rate is in-
creased, the error probability is ultimately determined by intcrsymbol
interference owing to selective fading, and may be excessively high.
The error probability can in this case be reduced, for a given total trans-
mitter power, by transmitting at a slower rate over each of a number
of narrower channels in frequency division multiplex. This could be
accomplished by individual transmission over each channel, which
would entail a number of independent transmitters. An alternative
method would be to use a common amplifier and to transmit the com-
bined digital signal by frequency modulation of a common carrier, as
now used for transmission of voice frequency channels in frequency
division multiplex. In the latter case, it is necessary to consider the
possibility of additional transmission impairments owing to intermodu-
lation noise.
With a sufficiently large number of digital channels in frequency divi-
sion multiplex, the combined wave will have virtually a Gaussian ampli-
tude distribution, like random noise. Hence the probability distribution
of average intermodulation noise amplitudes would be as indicated in
Fig. 1 1 for various conditions. The instantaneous amplitude will fluctu-
ate with respect to the above average values, as noted in Section X.
In binary transmission it is often assumed that the error probability
will not be excessive if the average noise power from all sources is about
12 db below the average signal power, or 18 db below the peak signal
power in on-off binary pulse transmission. From the previous curves
and expressions it appears that intermodulation noise power averaged
over short intervals will be at least 10 db below the average signal power,
with a small probability that it exceeds — 15 db. It thus appears that
intermodulation noise will not be a limiting or predominant factor even
when a large number of binary channels are combined in frequency
FNTERMODULATION DISTORTION 419
division multiplex for transmission by frequency modulation of a com-
mon carrier.
XIII. SUMMARY
In broadband transmission over troposcatter paths, selective fading
will be encountered with resultant transmission impairments, depending
on the modulation method. A previous analysis has been made of such
selective fading, based on an idealized model of a troposcatter path. It
indicated that selective fading will be accompanied by phase distortion
that in a first approximation can be regarded as quadratic, and a proba-
bility distribution curve for such quadratic phase distortion was derived.
On the premise of such quadratic phase distortion, the error probability
owing to selective fading was determined for digital transmission by
various methods of carrier modulation.
In the present study the same basic premise of quadratic phase dis-
tortion has been used in determining intermodulation distortion for a
signal with the properties of random noise, based on the same idealiza-
tion of a troposcatter path. An approximate relation for intermodulation
noise owing to quadratic phase distortion has been derived, applying
for any frequency pre-emphasis in FM. In turn, median intermodula-
tion noise as well as the probability distribution of intermodulation
noise has been determined, as related to certain basic system parame-
ters.
Median intermodulation noise predicted on basis of free-space antenna
beam angles conforms well with observations on links 18") and 194 miles
in length. For links 340 and 440 miles long it is necessary to use antenna
beam angles that are greater than the free-space angles by factors of
about 1.35 and 2.15, respectively. On long links employing narrow-beam
antennas, beam broadening is expected because of scatter. Thus if the
beam angles had been determined by independent observations or by
more elaborate theory, it is probable that predicted intermodulation
noise would conform reasonably well with observations.
The results of intermodulation noise measurements thus appear to
confirm the conclusion in a previous theoretical analysis of troposcatter
transmittance, which indicated that phase distortion owing to selective
fading could in a first approximation be represented by a component of
quadratic phase distortion, with a probability distribution that can be
determined from certain basic system parameters. This affords a simpli-
fied first-order theoretical model of selective fading in troposcatter paths
that is applicable to evaluation of resultant transmission impairments
in both analog and digital transmission.
420 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
It can be shown analytically, and it is confirmed by observations,
that the above first-order distortion theory yields intermodulation
noise that in the case of large signal bandwidths and frequency devia-
tions will be greater than observed or obtained with a more exact dis-
tortion theory. An empirical curve presented here permits determination
of the expected correction for large bandwidths and frequency devia-
tions.
It has also been demonstrated that the first-order multipath distor-
tion theory presented here affords a significant improvement over single-
echo distortion theory applied to random multipath transmission, in
that it is simpler and accounts for the probability distribution of inter-
modulation noise without certain contradictions that are inherent in
single-echo theory. Taken in conjunction with presently available data
on observed intermodulation noise on certain troposcatter links, as dis-
cussed herein, it affords a means of predicting intermodulation noise on
any system when more comprehensive experimental data become avail-
able on antenna beam broadening by scatter.
APPENDIX
Intermodulation Noise from Quadratic Phase Distortion in Pre-Emphasized
FM
General
To facilitate analysis of intermodulation noise in FM owing to attenu-
ation and phase distortion, it is customary to introduce two basic ap-
proximations. One is the use of "quasistationary theory" in conjunction
with the concept of instantaneous frequency, which is permissible when
the signal bandwidth B is negligible in comparison with the carrier fre-
quency, so that the frequency changes imperceptibly over a signal in-
terval T = 1/2/?. The other customary approximation is that distortion
a(w) + f|9(«) is sufficiently small to permit the approximation exp
[— a( w ) — #(«)] W 1 — «0) — iP(u) over the bandwidth of the
modulated carrier wave. The latter is a legitimate approximation for
most transmission systems, and greatly simplifies the analysis, but may
lead to appreciable errors in applications to tropospheric paths where
pronounced attenuation and phase distortion can be encountered. For
this reason an alternative approximate analysis is adopted herein to de-
termine intermodulation noise from quadratic phase distortion, in which
no limitation is placed on the phase distortion.
Two limiting cases are considered, from which it is possible to make an
approximate determination of intermodulation noise as related to phase
INTERMODULATION DISTORTION 421
distortion, rms frequency deviation, and bandwidth of the baseband
signal. In the first case, phase distortion is assumed adequately small,
such that the maximum phase distortion in the carrier signal band is less
than ir radians. Under this condition it is possible by use of "quasista-
tionary" theory to determine the power spectrum of intermodulation
noise without much difficulty. In the second case, no limitation is placed
on phase distortion, in which case determination of the power spectrum
becomes excessively difficult or laborious. It is possible, however, to de-
termine total intermodulation noise power at the detector output, prior
to post-detection low-pass filtering. From the manner in which total in-
termodulation noise power behaves with increasing phase distortion, it
is possible to obtain an approximate evaluation of intermodulating noise
in a narrow band, such as a voice channel.
A.l Power Spectrum of Phase Modulation
In FM the transmitted wave is of the general form
V - cos M + iKO] (25)
where the phase \f/(t) is related to the modulating wave m(t) by
iKO = k I m(l) dt (26)
Jo
where k is a constant.
The instantaneous frequency deviation is accordingly
0(0 = *'(0 = MO- (27)
If the original signal wave has a power spectrum s(«) and power pre-
emphasis p(o>) is used, the power spectrum of the modulating wave is
W m (u) = «(«)p(«). (28)
The squared rms frequency deviation yf/'(t) is
q 2 = k~ / s(u)p(u) du. (29)
Jo
In accordance with (26), \p(t) is the integral of m(t). Hence the power
spectrum of \f/(t) is given by
W+{w) = Ar 2 s(co)p(a>)/o; 2 . (30)
From (29) and (30)
,„ , s ' s(co)p(aj)/a) 2
ir^(co) = g- „ /FV — . (3i)
/ s(w)p(w) du
Jo
The power spectrum of xf/'i t) is u'W^(u).
422 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
A. 2 A utucur relation Function of Phase Modulation
The autocorrelation function of if/(t) is
fy( T ) = [ W+(<a) cos cot dw (32)
Jo
o f° s(oj)p(oj) , /„„-.
= A: / ^ COS cot du>. {66)
Jo co-
Wbeii the constant k is determined from (29), the following relation
is obtained
R
*u,
= tf
s(co)p(co)
COS cot doi
/[7%(«)p(w)d«l. (34)
'0 CO"
When the baseband power spectrum s(co) has a bandwidth ft, (34) can
be written
R*(r) = M 2
ft
o r a s(co)p(co)
cos cot da)
I s(co)p(co) duj (35)
• J
CO"
where n is the rms deviation ratio
M = C/Q = D/B. (36)
In the special case of a flat power spectrum, s(co) = s and (35) yields
B*(r) =
ft 2 / - COS COT rfco
f
/ p(co) rfco
In
With pure FM, p(w) = p = constant and (37) reduces to
■> /*' COS ftT.C
«*(t) =
M
r/.r
(37;
(38)
/0 Z*
where .r = co/fi. From (38) it follows that
^ /„x ^ / \ •> f 1 1 — cos ftr.r. ,
«*(0) - ff*(r) = m" / ^ dx
= fx 2 [Q.T Si(Qr) + cos At - 1]
2 (Or
(39)
= M
1 -
36
+
where Si is the sine integral function.
With pure PM, p(co) = co 2 and (37) yields
t) = 3/u 2 / cos Q,tx dx
RAt) = : :
(40)
= 3u 2 sin SIt/SIt
INTERMODULATION DISTORTION 423
R+(0) - R*(t) = :V'[1 -sinQr/fiT]
:nr) 2 .1 (41)
2 (flT)T (fir) 2 , I
A.3 Intermodulation from Phase Distortion
It will be assumed that the phase characteristic is of the form
<p(u) = b a + M + & 2 w 2 + b 3 « a + • • • . (42)
Phase distortion is then represented by the term
0(w) = & 2 u 2 + b 3 tt 3 + •■• (43)
where u = to — oj is the frequency relative to the carrier frequency o> .
When the transmitted wave is of the form (25), the instantaneous
frequency deviation is
u(t) = 4,'(t) (44)
and the corresponding variation in phase distortion with time is
0[u(O] = Wmf + W(t)f + • • • . (45)
In the above relation \f/'(t) is given by (27) and the power spectrum of
tf(t) by (28) multiplied by k 2 or
W r (u) = fc 2 s(w)p(o>). (40)
In determining intermodulation distortion it must be recognized that
distortion increases in the range < 0[u(t)] ^ w, diminishes in the range
7r < p[u(t)\ < 2ir, increases in the range 2ir < 0[u(t)] < 3ir, etc., as
illustrated in Fig. 12.
To determine intermodulation distortion it is thus necessary to evalu-
ate the distortion obtained when a wave with the power spectrum ( 4(i ) is
applied to a device with the output vs input characteristic illustrated
in Fig. 12. Two limiting cases will be considered below.
A.4 Intermodulation Spectrum for Small Quadratic Phase Distortion
With quadratic phase distortion only, (45) becomes
P[u(t)] = W(0f- (47)
It will be assumed that the probability that 0[u(t)} exceeds t is so small
that it is permissible to assume I3[u(t)] < x, and furthermore that u(t)
changes at a sufficiently slow rate such that fi'[u(t)] = 2lh>\p"(t) « t.
For signals with the properties of random noise, these assumptions are
424 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1904
6 a 10 12
^b 2 [u(t)] 2
77-r-
Cb)
w«»
Fig. 12 — Instantaneous phase distortion p(t) vs instantaneous frequency
deviation u(t) of signal.
permissible provided the rms phase error 7 defined by ( 12) and appearing
in Fig. 4 is much less than 1. With these assumptions, the autocorrela-
tion function of the output phase distortion is the same as for a square
law device and is given by (Ref. 10, Equation 4.10-1)
6.W(0) + 2V(r)]. (48)
The first term can be identified with a dc component that does not give
rise to noise. The power spectrum of the nonlinear output phase distor-
tion is obtained from the second component in (48) and is given by
-00
W/*\u) = 26 2 2 / R/{t) cos wt dr.
Jo
(49)
INTERMODTJLA.TION DISTORTION 425
The ratio of average intermodulation noise power at the frequency o>
to the average signal power becomes
gV%Q 2b **l ^ (T)c08wrdr (50)
P ^' " FT,( W ) fc 2 p(co)s(co)/co 2
In view of (46) the following relation applies
Rj,>(t) = k* I s(w)p(w) cos cor dw. (51)
•'0
Expression (50) can be written
2b 2 k I k R+> (r) cos wr dr
Jo
P
(«) =
k 2 p(co)s(co)/co 2
2t 2
t I k R#' 2 (t) COS ojt rfr
a Jo
r r 00
p(u>)s(a>) / s(co)p(w) dw
Jo
where
(52)
(53)
a = w /o - //£ (54)
7 = /,, M 2 o. 2 = fc 2 o. 2 = (2t)%D 2 . (55)
The following relation applies
f tf/(r) coso>r dr =\\ W*'(u)W+>(<a - u) du (56)
Jo 2 Jo
where WV(m) is the power spectrum given by (46).
In view of (56) and (46), expression (53) can be written
/ ^ «V/m 2
p(co)s(w) J p(u))s(w)da) , v
• / s(co)p(co)s(co — u)p(w — u)du.
J— co
In the special case of a flat power spectrum s(«) = 8 of bandwidth
* Ref. 10, Eq. (4C-G). In this reference the autocorrelation function is defined
differently from the definition used here and has a factor 4 in integral (51), so that
an additional factor j appears in (5G).
426 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
ft = 2irB, (57) becomes
, a 2 7 2 /M 2 1 f a , w v ,
M = ^lf° M , Q La P(M)P(W " ' i)rfW (58)
U Jo
= 1 j£ f p(.r)p(a - aOrf.r. (59)
* p(a) / p(.r)<fc '- 1
Jo
When p(.r) is of the form
p(x) = 1 + c(u/n) 2 = 1 + ex 2 (60)
relation (59) becomes
pM = inject X ^ f (1 + c * )[1 + c(a ~ x)2]dx
M -(l + ca 2 )(S + c) Ja-i
2 2 ^ '
F(c,o)
M 2 (l + ca 2 )(3 + c)
where
F(c,a) = 2 - a + 2 ° "t — [1 + (1 - a) 3 }
\ 2 ^
-^[1- (1 -a) 4 ] +f(I +(1 -«) b ]-
In the particular case of pure FM, r = and F(c,a) = 2 - a, so
that (61) yields
p(«) = ^ (2 - a)
(63)
g)\v ( 2-a;
where a = a>/S2 = //J5, D = fl/2x and 7 = b$ = b^irDf.
The above result (63) conforms with an expression derived by Rice for
this limiting case (Ref. 11, Equation 5.6).
A.5 Total Inter modulation from Quadratic Phase Distortion
The previous analysis of the power spectrum of intermodulation noise
was based on the assumption that the maximum phase distortion in the
transmission band is substantially less than 180°. Without this limita-
tion, numerical determination of the power spectrum becomes very dim-
INTBRMODULATION DISTORTION 427
cult, though a formal solution may be feasible. However, it is possible to
determine total intermodulation distortion without too much difficulty,
without limitation on the phase distortion, as shown below.
Let x designate the instantaneous amplitude of $'{t) = km(t), and
let x have a probability density
p(x) = (—, ;Ycxp (-x 2 /2*J). (64)
For large instantaneous frequency deviations yf/'{t) the derivative \j/"{t)
is on the average sufficiently small to be neglected. The total intermodu-
lation distortion in the received signal prior to post-detection low-pass
filtering is then for a nonlinear characteristic as illustrated in Fig. 12.
(65)
I = [ ' (b 2 x 2 fp(x)dx + [ ' (2tt - b&fv(x)dx
+ I (4tt - b 2 x 2 ) 2 p(x)dx +•••+( +1 (2ttw - b 2 x 1 fp(x)dx
where
L t = Ur/b 2 )K
With b 2 x" = u", 7 = bz<r x and
P(U) = \) eXP (_w2/27) ((i6)
expression (65) can be written
rh rh
I = / U*p(u)du + / (2tt - u-)-p(u)du
Jo J /,
rh
-f / (47r — u ) p(u)du + • • •
(67)
where
lj = OV)*. (68)
Writing 2nnr — u = — t, 2udu = dr, expression (67) can be trans-
formed into
/ = f Mr)dr + C h f T , p(r)dr
Jo J-x (Zir -\- rp
(69)
+ <rl * h I n 1 m pWt + -..
J-t (4tt + T) 1
428 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
where
'« = (£)'<-"*■ (70)
Total distortion / includes a mean or dc power component h that
must be subtracted from / to obtain the nonlinear component. The mean
amplitude component h Is given by
/o* = / b 2 x 2 p{x)dx + / (2tt - hx )p(x)dx
. (71)
+ / (47r — b 2 x 2 )p(x)dx + • • •
(72)
where L m and p(x) are defined as before.
With the same notation as before, (71) can be transformed into
In the above relations 7 is the phase distortion corresponding to the
rms frequency deviation as given by
y = b 2 a 2 = b 2 Q 2 = b 2i xtf. (73)
The last relations follow from (29) since a 2 is the variance of ^'(t).
The total average signal power is
S = B#(0) = m 2 [rf 1° ^ dco ]/[C P (w)dw ] = " 2/C (74)
where C is a constant depending on p(w).
The ratio of total nonlinear intermodulation noise to total average
signal power becomes
I - h „I - h
p =
(75)
= C(I - /o)
©•
A. 6 Total Intermodulation for Small Phase Distortion
For sufficiently small values of 7 = botf, such that ir/y ^> 1, only the
INTERMODULATION DISTORTION 429
first integral in (69) needs to be considered. Hence
Jo
= 3 7 2 erf (z) - 3-2V exp(-2 2 ) - 2**7* exp(z 2 ) (76)
where
£ = 7r/2 7 . (77)
With a similar approximation (72) yields
T °~l T ^ {T)dT (78)
= 7 erf (2) - 2 § exp (-z).
For 2 ^ 2,01-7 ^ tt/8:
7 ~ 37 and To 3 = 7-
Hence/ — To = 27 2 and (75) becomes
P-C21 (79)
where the constant C is defined through (74).
It will be noted that (79) is of the same basic form as (61) for the
ratio p(«) at the frequency co. In ( 61 ) the multiplier of y 2 /^ 2 is a constant,
as is the case in (79).
A. 7 Total Intermodulation for Large Phase Distortion
When 7 ^> 1, it is permissible to approximate p(r) as given by (70)
with
pM^-i-J. (80)
This approximation is valid in evaluation of the various integrals in (69)
and (72) provided that for the minimum value of t = t, exp ( — 7-/27)
« 1 . This is the case if
71-/27 « 1 or 7 » *r/2.
With (80) in (69)
430 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
Iii (81),
f ,„ T \ ,, = ?^ [(2m + l)*(32m a - 8m + 3)
J_» (2?n?r + t) 3 15
- (2m + l)*(32m 2 + 8m + 3)]
1 2 «■■ . . ,
^ — j -— tor m ^ 1 .
m- 3
(82)
(83)
For m = 1, (82) gives about 0.5 and (83) about 0.47. Hence (83) repre-
sents a good approximation of (82).
With (83) in (81)
' ^fe) ! [
, . 2V ^ e- mT/7_1
3 ,„=i m»
(84)
As a first approximation the summation can be replaced by an in-
tegral, in which case
I
mi'^c
U /■• e™* h dm
m-
(85)
With m = ?*
— for 7 » 4tt .
By a similar approximation TV as given by (72) becomes
\27r7/ _J ,«=] J-x (2m7r + t)*.
2^) [h'- + ¥ CTfc
2
MvIMM
- for 7 » 4tt .
(86)
(87)
(88)
(89)
(90)
(91)
(92)
IXTKKMOIH'LATIOX DISTORTION 431
The ratio p is obtained from (75) with / as given by (87) and In by
(91). In the limit of 7 — -> « the ratio becomes
Hi-3
p = C
(03)
c
~ 2.0 — .
A. 8 Approximation for Total I titer modulation
The general expression for the ratio p of total intermodulation noise
power to average signal power can be written in the form
P= 2 4h(y). (94)
For the limiting case of 7 — ► 0, the function h is in accordance with (79)
h = y\ (95)
For the other limiting case in which 7 — * so , the function h is in accord-
ance with (93)
h = 2.5/2 - 1.25. (96)
In Fig. 13 are shown the above two limiting cases, together with the
function // obtained from (75) as 1} = / — h , when / and I a are deter-
mined from (80) and (91 ). The approximate function h(y) is obtained
by drawing a transition curve between the above two limiting cases, as
in Fig. 13.
A.9 Approximation for Intermodulation Spectrum
The function hly) in Fig. 13 is proportional to the total intermodula-
tion noise power and can be related to the power spectrum Wi(ta) of
intermodulation noise by
h(y) = c f ir.tco) du (97;
J it
where Co is a constant. Relation ( 94) can thus be written
2c C
P =
( Wi(a>) r/co. (98)
Jo
For 7 — * 0, (98) must conform with (95), which is possible provided
the power spectrum is of the general form
432 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1904
W?{<*) = Cl7 2 ^o("/fl)
(99)
where F is any functional relation dependent only on the ratio a = to/ft.
With (99) in (98)
2
Id)
P = 1 - 2 CoCl C i [ Fo(»/0) dc
p? iJ Jo
2 ,.00
= 2- 2codC / Fo(w) dw .
/X" ■'o
This yields relation (95) provided
CffiiC I F (u) du = 1 .
Jo
(100)
(101)
/
—
/
1
\
l-
1
1
<
-~b~~
?s.
\
**N
Q
Z
<
\j
K
/
/
/
/
/
/
10" 4
/
IO a
7- = b 2 Q 2 = b 2 (2 7rD) z
Fig. 13 — Functions h(y) and #(7): 1, functions h(y) and #(7) for 7 « 1; 2,
function h(y) for 7 » 1; 3, approximate interpolated function h (7); 4, function
#(7) for 7 » 1; 5, approximate interpolated function H(y).
INTERMODULATION DISTORTION 433
From (100) it is apparent that the ratio of intermodulation noise
power to average signal power in a narrow band do> at w is
p(co) = \ 2c oCl C i Fo(«/fl) . (102)
Comparison of (102) with (61) shows that in this case
2c oCl C i F (u>/iV = J 2 a F(c,a) (103)
il (1 + ca~){6 + c)
where F(c,a) is given by (62).
In summary, for y — > the power spectrum has a fixed shape inde-
pendent of 7 and an amplitude proportional to y .
Consider next the limiting case in which 7 — > 00. In accordance with
(90) h then approaches a constant, which is possible for various power
spectra of the general form
Wi { "\a) =^F«(<o/7 n ) (104)
7"
where F^u/y") is any functional relation dependent only on the ratio
(w/7"). In this case (104) in (98) yields
2c oCl C
P =
— [ Foo(co/7 n ) da
y" Jo
2coClC f F(u) du
M' ■'0
where u = w/7".
The exponent n can be determined from consideration of the input vs
output characteristic shown in Fig. 12. If 6 2 is increased by a factor k,
the intervals between zero points are multiplied by a factor k~ , as indi-
cated in Fig. 14 for A: = 4. For a given frequency deviation, the band-
width of the power spectrum is then multiplied by a factor fc and the
amplitude of the spectrum at each frequency multiplied by a factor
AT. Hence in the case of quadratic phase distortion as considered here,
n = § in (104).
Based on the above considerations, the power spectrum at any fre-
quency w for the above two limiting cases would vary with 7 as indicated
in Fig. 13. The shape of the curves between these two limiting cases
would in a first approximation be represented by the function H(y)
shown in Fig. 13.
434 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
Fig. 14 — (a) Relation of instantaneous phase distortion (3(t) to instantaneous
frequency deviation v(l) for a given b- ; (b) relation of instantaneous phase dis-
tortion to instantaneous frequency deviation with fourfold increase in b« .
A. 10 Approximation for p(w)
The ratio p(w) of intermodulation noise power in a narrow band at a>
to average signal power in the same narrow band can be written
>(«) =
H(y).
(106)
This relation differs from (94) in that h(y) as shown in Fig. 13 is re-
placed by H(y) shown in the same figure, and C is replaced by C(u).
The constant C defined through (74) depends on the frequency pre-
emphasis p(w). The function C(w) depends both on the frequency pre-
emphasis p(w) and the frequency under consideration.
For the particular type of frequency pre-emphasis represented by
INTERMODULATION DISTORTION 435
(60), expression (106) must conform with (61). This results in the fol-
lowing approximate relation
p(«) = (§) G(c,a)H(y) (107)
where H(y) is the function shown in Fig. 13 and
GM ~ (1 + J)(3 + «) FM (m)
where F(c,a) is given by (62).
In the particular case in which c = 16 and a = f/B = 1
G(c,a) « 0.192 (109)
and (107) yields
/!(/>•) [jj xaiiBffW (no)
B
D
= 1^1 X 0.192-7* for 7«1- < 111 )
REFERENCES
1. Sunde, E. D., Digital Troposcatter Transmission and Modulation Theory,
this issue, Part 1, p. 143.
2. Clutts, C. E., Kennedy, R. N., and Trecker, J. M., Results of Bandwidth
Tests on the 185-Mile Florida-Cuba Scatter Radio Systems, IRE Trans, on
Communication Systems, 9, December, 1961, p. 434.
3. Reach, C. D., and Trecker, J. M., A Method for Predicting Interchannel Mod-
ulation Due to Multipath Propagation in FM and PM Tropospheric Radio
Systems, B.S.T.J., 42, January, 1963, p. 1.
4. Bennett, W. R., Curtis, H. E., and Rice, S. O., Interchannel Interference in
FM and PM Systems under Noise Loading Conditions, B.S.T.J., 34, May,
1955, p. 601.
5. Medhurst, R. G., and Small, G. F., Distortion in Frequency-Modulation
Systems Due to Small Sinusoidal Variations of Transmission Characteristics,
Proc. IRE, 44, November, 1956, p. 1608.
6. Medhurst, R. G., and Small, G. F., An Extended Analysis of Echo Distortion
in FM Transmission of Frequency-Division Multiplex, Proc. IEE, 103, Pt.
B, March, 1956, p. 190.
7. Carson, J. R., and Fry, T. C, Variable Frequency Electric Circuit Theory
with Applications to the Theory of Frequency Modulation, B. S.T.J. , 16,
October, 1937, p. 513.
8. van der Pohl, B., The Fundamental Principles of Frequency Modulation,
.lour. IEE, Part III, May, 1946, p. 153.
9. Albersheim, W. J., and Schafer, J. P., Echo Distortion in the FM Transmis-
sion of Frequency Division Multiplex, Proc. IRE, 40, March, 1952, p. 316.
10. Rice. S. O., Mathematical Analysis of Random Noise-I and -II, B.S.T.J., 23,
July, 1944, p. 282, and 24, January, 1945, p. 46.
11. Rice, S. O., Distortion Produced bv a Noise Modulated Signal by Nonlinear
Attenuation and Phase Shift, B.S.T.J., 36, July, 1957, p. 879.