Document text
Distortion Produced in a Noise Modulated
FM Signal by Nonlinear Attenuation
and Phase Shift
By S. O. Rice
(Manuscript received December 6, 1956)
An expression is given for the FM distortion introduced hy a transducer
whose attenuation and phase shift depend upon the frequency in an arbitrary
way. This expression appears to be difficult to evaluate, but it yields useful
approximations for the second and third order modulation terms. In all of
the work, it is assumed that the distortion is small compared to the signal,
and that the signal can be represented by a random noise having the same
power spectrum.
INTRODUCTION
A number of workers have been concerned with the problem of com-
puting the distortion introduced by a transducer when an FM wave
passes through it. Some of the earHest results were pubhshed by Carson
and Fry and by van der Pol." Several contributions to the subject have
been made recently in connection with studies of microwave radio
systems.
An excellent paper on this subject has been pubhshed recently by
R. G. Medhurst and G. F. Small. ^ Although their results differ consider-
ably in form from those given here, they are nevertheless closely related
to ours — their "sinusoidal variations of transmission characteristics"
being special cases of our "nonlinear attenuation and phase shift."
Here we treat the problem by applying a method used in a recent
paper* to study the distortion produced by an echo. Two assumptions
are made, (1) that the distortion is small compared to the signal, and (2)
that the signal can be represented by a random noise which has the
same power spectrum as the signal. In Section I, we review some known
results and put them in a form suited to our needs. Sections II and III
are devoted to the derivation of our main formulas. The principal result
is given by the triple integral (3.2) for the power spectrum of the dis-
879
880 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957
tortion. Unfortunately, the integrals are difficult to evaluate. However,
it is possible to obtain approximations for the second and third order
modulation terms. These are given in Section IV. Some miscellaneous
comments are made in Section V.
I APPROXIMATE EXPRESSION FOR THE DISTORTION 0(i)
Let the FM signal be ip'it) = dipjdi (for phase modulation the signal
would be ^(0). Then the FM wave is the real part of
y.(() = p'^'+'^<'i (1.1)
where -p = 27r/p is the carrier frequency. Let this wave pass through a
transducer having attenuation a and phase shift (3, where a. and ^ are
even and odd functions, respectively, of the frequency /. When a unit
impulse of voltage hit) is applied to the transducer input, the output is
■- g[i) = [\~"-^^-'"^'d/. (1.2)
For physical systems, g{t) is zero for negative t.
When Vi{i) is applied to the transducer input, the output is
v,{l) = r vU')g{i - t') dt'. (L3)
When Wo(0 is applied to an FM receiver, the detector output con-sists of
the original signal <p'{t) plus the distortion 6'{t) introduced by the trans-
ducer. Comparison with (1.1) shows that B{i) may be obtained by
solving
when p, tp{i), vo{t) are assumed to be known, and Vit), 6{t) unknown.
When V{t) is taken to be positive, (1.4) determines e{t) except for an
additive term of 2xn where n is an integer.
We now assume that the transducer acts like a good transmission
medium in that the output differs but little from the input. More pre-
cisely, we assume
\vo{t) -y.(0!«l. (1-5)
Since | Vi{t) | = 1, it follows that | vo(t) \ ^ 1. Transducers having ap-
preciable attenuation and delay may be regarded as two transducers in
tandem, one with constant (independent of /) values of a and (i/f which
are roughly equal to those of the original transducer, and the second
DISTORTION IN' N'OISE MODUL.4TED FM SIGNAL 881
with variable a and ^//. The first transducer produces no distortion of
the signal, and if condition (1.5) is satisfied by the second, the con-
siderations of this paper will apply.
Equation (1.4) may be written as
so that
When we write
e{i) =Imlog^. (1.6)
expand the logarithm in (1.6), and use (1.5), we obtain our approximate
expression for B{t) :
6{i) = Im[vo(0 - vMMD = Im voit)/Vi{t)
= Im [vm" £ v,(l') 9 it - i') dt' ^^ ^^
= Im / exp [ip{t' - + i<p{i') - i<p{t)]g{t - t') dt'.
J — CO
So far there is nothing essentially new in our work.^
TI AUTOCORRELATION FUNCTION OF 6{t)
In Section I, ip'(t) could be any reasonable sort of signal. In the follow-
ing work we assume that it is a Gaussian noise whose power spectrum,
w^-(J), is given to us. The power spectrum of (p(t) is
w^tf) = w^.iS)/{2^ff, (2.1)
and its autocorrelation function is
^r = I wM cos 2irfT df. (2.2)
We have written V'r instead of ^(r) or R^ir) to simplify the appearance
of the formulas which occur in our work.
Our problem is to find the power spectrum, we{f), of the distortion
6(t), given w^(f). The method of solution is much the same as that used
in Reference 4. We first find the autocorrelation function /?s(t) of 6it)
and then obtain we{f) by taking the Fourier cosine transform of Rgir).
882 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957
Let the last integral in (1.7) be F{t) so that 9(1) = Im F{t). Then
e{t)e{t + r) = ^ Re {F{t)F*{t + t) - F{t)F(t + t)] (2.3)
where F*{t + t) is the complex conjugate of F{t + r). The autocorrela-
tion function of e(t) is obtained by averaging over the ensemble of the
noise functions ¥=(0:
Reir) = av e{t)e{t + r)
= av iRe I/"" dt' f dt" exp [ipit' - + tV(i') - i<pii)\
2 [J — CO J— CO
■j;(/ - i')g{i + T - r)[exp [-tp(r - t - t) (2.4)
- tV(^") + i^H + ^)] - exp \iv{i" - t - r) + Mi")
- Ml-^ t)]].
Since g(l) is real, g*{t) = g{t). The averaging process may be carried
out by a method analogous to that used in Reference 4. The formula to
be used is
av exp [i<p{t') - i<p{t) + i(up{t") - i(up{t + t)]
= exp [-^o(l + a) + ^f-t - a^pf-f + a^r-t^r (2.5)
+ a^t-f — a^T 4- ai/'r'-i-r]
where a is either -1 or +1, and \l/r is an even function of t. When (2.5)
is used in (2.4) a double integral for Reir) is obtained. The substitutions
X = t - t',
y = t-\- T -t", ' (2.6)
convert the double integral into
The symbol R^ is chosen to agree as closely as possible with the notation
of Reference 4. There K„ was the autocorrelation of the random func-
tion, v{t), where vit + T) = <p{t) - <p{t + T), T being the echo delay.
Here, R„ is the average value of the product,
[^(0 - ip{t + y)\ [<p{t + t) - ^(i + T + x)]
which becomes the autocorrelation function of v{i) when y = x = T.
DISTORTION IN NOISE MODULATED FM SIGNAL 883
It may be verified that the expression (2.7) for i?fl(r) is an even func-
tion of T, as it should be. Expression (2.7) is the autocorrelation func-
tion we set out to find.
The distortion e{t) has an average value, 6, whose square is Rb{oo).
Since ip(t) is a noise function, its autocorrelation function \f/r goes to
zero as t approaches «= . Hence, R»( ^ ) is given by the expression ob-
tained from (2.7) by setting ff, = 0. The autocorrelation function of
e{l) ~ eis
Re-iir) = Reir) - 7?fl(co)
= i Re /"" dx r dy g{x)e-''"-'^^''^^+'''' (2.8)
■g{y)W''{e'''' - 1) - e-'"™(e-'^" - 1)].
Ill POWER SPECTRUM OF THE DISTORTION
Since d{t) has an average value which is generally not zero, its power
spectrum, We(/), has a spike of infinite height at/ = corresponding to
the power in the de component 6. When this spike is subtracted from
wsij) the remainder is the power spectrum of 6{t) — B given by
Wb- (f) = 4: j Re-eir) cos 27r/T dr. (3.1)
When we use (2.8) and note that Re-eir) is an even function of t, we
obtain
we--e(f) = [ dx [ dyg(x)g(y)e-'''^-^'+-^>' f [cos (px - py)
J-«, J-« J-„ (32)
■{e"" - I) - cos (px + py){e~''' - 1)1 cos 2x/t dr.
Reasoning similar to that given in Reference 4 shows that the inter-
channel interference spectrum, wdf), (i.e., u\(f)Af is the average amount
of distortion power received in an idle channel of mdth Af centered on
the frequency /, all other channels being busy) may be obtained from
(3.2) by replacing (e:^"' ~ 1) by (c^"' =F ff„ - l).
The power spectrum of 6(1) - 6 may be regarded as made up of
modulation products of all orders. It turns out that the contribution of
n^ order products is given by the integral of the 7?„" terms obtained
from the power series expansions of exp [±7?^]-
IV FIRST AND SECOND ORDER MODULATION TERMS
Here we shall study the first and second order modulation terms.
These arise from the first and second powers of R^ in the expansion of
£
}pr+b COS 2x/t dr =
884 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957
the quantity within the square brackets in (3.2):
2R„ cos px cos pij + J^/ sin px sin py. (4.1)
The integrations with respect to t may be performed with the help of
'M^Ree-''''\ (4.2)
V'T+J.i/'r+c cos 2x/t dr
(■4.3)
= Re- f du wJu)w^{f - u) exp | ~i2ir[bu + c(/ - w)ll
4 J-«
which follow from (2.2) and the fact that we have defined w(-/) to be
equal to w(f). In our notation the total power in a random noise function
is the integral of w(f) from / = to / = <^ -
The first order modulation term is obtained from (3.2) by replacing
the term within the square bracket by 2R„ cos px cos py. When the ex-
pression (2.6) for R„ is used, the integration with respect to t may be
performed with the help of (4.2):
r R. cos 2.fr dr = '^ Re [(e-^"'^^ - iW"^ - 1)]. (4.4)
This leads to the following expression for the first order modulation
term in (3.2)
(4.5)
,(/) I r dxg{x)e-^'^'^' cos px{e-''"^ - 1)
J— to
This is the quantity which is to be subtracted from wsJkiJ) to obtain
the interchannel interference spectrum wdj)-
The second order modulation term is handled in much the same
manner. With the help of (4.3) it may be shown that .
/ /?/ cos 2rfT dr = Re - du w^{u)w^{J - u)
.(e"'"" - 1) (e'""''-"' - 1).
From this it follows that the second order modulation term in (3.2) is
A^ [ duw^{u)w^(f- u) j dx g{x)e~'^''^*' sinp.r
2! 2 J— oo ''— "
(g-2''- _ l)(e-="«'^-"> - 1)
(4.7)
DiaTORTION IN NOISE MODULATED FM SIGNAL 885
When \po ~ ^x is so small that exp ( — ^o + ^i) may be replaced by
unity, as it is in some important practical cases, approximations may
be obtained for (4.5) and (4.7). The integral in x may be expressed as
the sum of integrals of the type
= Ga + iB, , (4.8)
r gixy"'-'""' dx = G_„ - iB...
The values of the integrals follow from (1.2) and the Fourier integral
theorem. G and B are, respectively, even and odd functions of frequency,
and Ga , Ba are their values at the frequency f = fp -\- a where fp = p/2Tr
is the carrier frequency:
G at frequency /p + a = G,, ,
B at frequency fp -\- a ^ Ba .
In this way we get the approximation
4^V.(/) [((?/ - 2Go + G-fY + (Bj - B_,Y] (4.9)
for the first order modulation term, and
-- / duw^(u)w^(f - «)[(Gu - G-.. + G/_u - G_/+u - G/
2!8J-« (4.10)
+ G.fY- + (B„ + B_„ + 5/_„ + i?_/+„ - Bf- B.f - 2Bo)']
for the second order modulation term.
Expression (4.10) is an approximation to the second order modulation
term (4.7). When most of the interchannel interference is due to second
order modulation products, (4.10) is also an approximation to Wc(/), the
interchannel interference spectrum. The following remarks may be of
some help in deciding whether (4.10) may be used.
1. For the case of phase modulation and a "flat" signal band, the
first of equations (5.3) shows that }pn and ^pT may be made as small as
we please by choosing the signal power (as measured by Po) small
enough. .Since H,, is proportional to Po , Po may be chosen small enough
to make R^ and higher order terms negligible in the expansion of the
integrand of (3.2) (unless there is some sort of sjnnmetry which cau.ses
the second order terms to vanish). In this case the interference is mostly
second order modulation and (4.7) is a good approximation to wdj).
Furthermore, as Po approaches zero, exp (— ^o + f,) approaches unity
886 THE BBLL SYSTEM TECHNICAL JOUKNAL, JULY 1957
and (4.10) becomes a good approximation to (4.7). Just how small Po
has to be depends upon the signal bandwidth, /& , and the characteristics
of the transducer.
2. For the case of FM and a flat signal band, the second of equations
(5.3) shows that even if Po is small, the difference ^o — ypr approaches co
as I T I approaches oo . To justify the use of (4.10) in this case it is neces-
sary to take into account the behavior of g(,t), the response of the trans-
ducer to the unit impulse 5{t). For example, if the duration of g{x) in
(4.7) is so brief that g{x) becomes negUgibly small before — ^o + ^^ be-
comes appreciably different from zero (which may be achieved by mak-
ing Po small enough) then (4.10) is a good approximation to (4.7).
3. When the attenuation, a, and phase shift, /3, are given for any
particular transducer, the corresponding g{t) may be obtained from (1.2) .
Once g(t) and ^o - ^r are known, the conditions under which exp (-^o +
Tp^) may be replaced by unity in (4.7) and 0(R/) terms neglected in (3.2)
may be determined by direct examination of the integrals.
As might be expected, the third order modulation results are quite
compUcated. The third order modulation term in (3.2) is
Jta r ^f f df"w,{nw,{n^,{f")
/ dx g{x)
J— no
coBVxe-^'^^Hz'' -1)(/"- 1)(/"'- 1)
where /'" = f - f - f" and z = exp (-t2xx). When i/'o is small this is
approximately
^ /" df r dnvM')^M")wM"'W + K'] (4.12)
where
H = m{f) + m(n + ™(/) + Mf -f'~ f")
- m(f - n - m{f - n - m(0) - m(/' -f /"),
(4.13)
m(/) = Gf + G^f, 7iif) = Bf ~ B.f,
and K is an expression obtained from H by replacing n by m.
V MISCELLANEOUS COMMENTS
Here we make some miscellaneous comments related to the foregoing
results.
(5.2)
(5.3)
DISTORTION IX XOISE MODULATED PM SIGNAL 887
If the transducer is perfect except for an echo, its response to a unit
impulse 5(t) is
g(t) = 8{t) + rdit - T) (5.1)
where r and T are the amplitude and the delay of the echo. The results
obtained using (5.1) agree, as they should, with the results obtained in
Reference 4. Of course, r must be assumed small compared to unity in
order that condition (1.5) may hold.
When the power spectrum of the signal is equal to a constant Pa over
the band (/„ , /&) and zero elsewhere we have for phase and frequency
modulation, respectively,
PM: w^if) = P,, U<f <U,
FM: w,{j) = Po/(27^/)^ /„ < / < A .
When fa ^ the autocorrelation functions are
PM: ^, = PoMsmv)/v,
FM: ^0 - ^pr = A[-l + cos v + vSi(v)],
V = 2wM, A = PoU{2wf,)-' = {a/f,)\
The mean square values of the signals are Pafb (radians)^ for PM and
Pofb (radians/sec)' for FM. If, for FM, o- is the rms frequency deviation
in cps (so that the "peak" deviation is, say, 4cr cps) then (2x0-)" = Pofb .
The difference ^f/o — ^t is used in the FM case to avoid difficulty at/ = 0.
It will be noticed that our formulas are such that the iZ-'s may be re-
placed by {4> — ^o)'s without altering the values of the various ex-
ponents, etc. In microwave systems the quantity A is often small in
comparison with unity.
As an example of the use of the second order modulation approxima-
tion (4.10) consider the case where the attenuation, a, is zero and the
phase shift /3 = Oiif — fpY/2 radians, 02 being small. Then, since G ?t;
1 — a, ^ i^ —ff, we have G,, ^^ 1 and
B,. ^ -[0 for / - /p -t- u] ■
= —a2u/2.
When we take the FM case of (5.2) and substitute in the approxima-
tion (4.10), the interchannel interference power spectrum is found to be
rfb D D
2! 8 J/-
[0 -I- (2a2H(/ - «))"] du
'f^U (2x1/)= (27r)H/ - U)^ t ■ V - V-' ^^ _^
= (2T)-''(a2Po/2)'=(2A-/).
888 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957
Dividing by ivM) = Po/i2irff gives the ratio of the interference power
to the signal power
{a,<rf/2)\2 - f/h) (0.6)
where the relation Po ^ i'2ir(Tf/fb has been used to eliminate 7^n . Here
ff is the rnis freciuency de-\'iation of the FM signal in cps. The expression
(5.6) agrees mth results of some earlier work done at Bell Telephone
Laboratories. In that work the second order modulation products were
summed directly.
It is interesting to apply the formulas given here to some of the
eases considered by Medhurst and Small.^ They have shown that when
(in our notation) a = -r cos 2irfT and ^ - the power spectrum of the
distortion is
we-sif) = sin' 7r/T[Wfl_e(/)]echo , (5-7)
and when a = and /3 = r sin 2irfT,
we-eif) = cos' 7r/r[wfl_fl(/)]ccUo ■ (5-8)
Here [we-li(f)Uho is the power spectrum of the distortion due to a simple
echo of amplitude r and delay T (corresponding to « - -r cos 2irfT
and (3 = r sin 2irfT). Expressions (5.7) and (5.8) may also be obtained
by setting the impulse response git) equal to
5(0+^5(^- r)±^6(i+ T)
in (3.2).
The second order modulation approximation for the a = —r cos
2TrfT, $ = case may be obtained from (4.10) and turns out to be
[ wJuHvM - fOlSr sin pT sin tJT sin iruT sin 7r(/ - u) Tf du. (5.9)
It is seen that this contains the factor siuV/T predicted by (5.7). When
(5.9) is applied to the FM case of (5.2) an integral something like (5.5)
(but more complicated) is obtained. The ratio of the second order
modulation interference power to the signal power is found to be
2[r sin pf sin TfTfia/UfUK (5.10)
where K is the quantity
K = 2a^f pjn (///2) sin (g - ^)/2T ^^ ^^^^^
K~u L y(« - y) -1
DISTORTION IX NOISE MODULATED FM SIGNAL 889
tabulated in Table 4.2 of Reference 4 and
a = 2TfT, U = 2nUr. (5.12)
The parameters a and k that appear m the table are defined by
a = f/fb and /; = 8/bT.
These formulas serve to supplement the formulas and eiu'ves given by
Medhurst and Small.
ACKNOWLEDM ENT
I wish to express my thanks to H. E. Curtis who has furnished some
of the examples given in this paper and to E. D. Smide, S. Doha, and
others for their helpful comments.
REFEHENCES
1. J. R. Carson and T. C. Fry, Variable Frequency Electric Circuit Theory With
Application to the Theory of Frequency Modulation, B.S.T.J., 16, 510-540,
Oct., 1937.
2. B. van der Pol, The Fundamental Principlea of Frequency Modulation, Jl.
I.E.F.,93. pp. 153-158, 1946.
3. R. G. Medhurst and G. F. Small, Distortion in Frequency-Modulation Systems
Due to Small Binu.soidal Variations of Transmission CharacteristicH, Proc.
I.H.K.. 44, pp. 1608-1612, Nov., 1956.
4 . W. R. Bennetl, H. E. Curti.'*, and S. O. Rice, Interchannel Interference in FM
and I'M Systems Under Noise Loading Coiidilions, B.S.T.J., 34, pp. 601-636,
May, 1955. Tlie name problem has been treated indejiendently and in much
the same way by S. V. Borodich, (In the Nordinear Distortions Ciiu.'^ed by
Variations of the Antenna Feeder in Multichannel Frequency Modulation
Systems, Radiotekhnika, Moscow, 10, pp. 3-14, 1955.
5. These results are closely associated with some given by M. K. Zinn, Transient
Response of an FM Receiver, B.S.T.J., 29, pp. 714-731, 1948.