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THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING
ASPECTS OF ELECTRICAL COMMUNICATION
Volume 52 May-June 1973 Number 5
Copyright © 1973, American Telephone and Telegraph Company. Printed in U.S.A.
Distortion Produced by Band Limitation
of an FM Wave
By S. 0. RICE
(Manuscript received November 21, 1972)
The bandwidth required to transmit an FM wave is related to how much
distortion is allowed in the signal. Here expressions are developed for the
distortion (interchannel interference) produced when an FDM-FM wave
passes through an ideal filter. The signal is represented by a flat (PM)
band of Gaussian noise. The formulas obtained hold only for small rms
frequency deviation, but fortunately this is an important case in micro-
wave communication systems. The theoretical expressions agree well with
Monte Carlo residts published recently by Anuff and Liou.
I. INTRODUCTION
When a frequency-modulated wave passes through a filter, distor-
tion is produced in the signal by nonlinearity in the filter phase shift
(usually the chief offender) and by the filter attenuation. Much effort
has been spent in devising methods for computing this distortion.
A related problem is "What radio frequency bandwidth is required
to transmit a given FM wave?" An approximate answer, known as
"Carson's rule," states that the required bandwidth 2f h is given by 1
2f h = 2B + 2D max , (1)
where B is the bandwidth of the baseband signal and D max is the
605
606 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
maximum amount the instantaneous frequency deviates from the
carrier frequency. Note that (1) implies a conventional FM system.
This is the only type we shall consider in this paper. We shall not be
concerned with single-sideband FM or other schemes for reducing the
rf bandwidth.
Carson's rule has been revised recently by Anuff and Liou. 2 They
make use of Monte Carlo calculations of the interchannel interference
produced when an FM wave carrying a multichannel signal passes
through an ideal filter. The ideal filter has zero attenuation and phase
shift within the passband, and infinite attenuation outside the band.
Monte Carlo calculations of interchannel interference in microwave
systems have also been made by Grierson and McGee. 3
Here we make a beginning on the analysis (in contrast to Monte
Carlo) required to calculate the interchannel interference produced
by an ideal filter.
The FM wave is cos [> £ + <p{t)~\ where u> = 2tt/ and <f(t) is a
stationary, zero-mean Gaussian process with the two-sided power
spectrum
W(f)= \ W <> \f\* B (2)
n * U) J 0, I /| > B.
In (2), Wo is a constant and B is the top baseband frequency. In order
to represent an idle channel at frequency f c , we take W v {f) = in
the narrow slots f c ^ I f\ ^ fc + A/ c , A/ c being so small that W v {f)
can be replaced, without appreciable error, by W in the integrals ap-
pearing in the analysis.
The mean-square value of fit) and the rms frequency deviation D
are given by the ensemble averages
<<p»(0> = 2IT B(rad) 2 (3)
Z) 2 = ((^'(i)/27r) 2 ) = 2W B*/Z (Hz) 2
where <p'(t) = d<f{t)/dt. This <p(t) gives a convenient approximation to
the preemphasized wave assumed by Anuff and Liou. A representative
value of Z) ma x in (1) is 4D.
The ideal filter passband extends from f — f h to /«, + //.• It is
assumed that 2f k /f « 1 and that nB < /,, < (n + 1)5 where n is
a positive integer.
Our aim is to apply results from the theory of Volterra series to ob-
tain an expression for the dominant portion of the interchannel inter-
ference when the normalized rms frequency deviation D/B becomes
small.
COMPUTING BASEBAND DISTORTION 607
For the moment, consider one-sided power spectra. Now the power
spectrum of ip(t) extends from to B and has the value 2W . The aver-
age signal power (FM) appearing in the channel (/, / + A/) when it is
busy is
S = (27r/) 2 2Tr o A/(rad/s) 2 . (4)
Let N be the average interchannel interference power which appears
in the same channel. The value of N depends upon whether the channel
is idle or busy. When the channel is idle, the interference can be heard
as crosstalk noise. In our expressions for N/S, we assume that our
particular channel is idle, that all the other channels are busy, and
that N/S is the limit obtained as A/ tends to zero.
The nature of our results is illustrated by the following expression
for N/S in the top baseband channel:
v - I \Z(n + 1 - §)T"C + 0[<2>/*)«-«l
2B
c 1 ^ (2n-2A-)!(2fc- 1)! / 1 \*
°" (2n) ! tx (k - 1) \k\*(k + 1) ! V ( n - A;) \) '
(5)
Here the integer n is determined by the filter semibandwidth f h and
the relation nB < /,, < (n + 1)5. The first three values of C on are
C o1 = 1/4, C o2 = 5/96, and C o3 = 19/10308. For large n, C on tends
to 2 2 » +1 /[n! 4 irn(n + 2)].*
Equations (5) are a special case, / = B, of (52) which gives N/S
in a channel whose frequency / satisfies f h — nB ^ / ^ B. When
£ / < /» - nB, N/S is of order (D/B)**+* and the formulas corre-
sponding to (52) do not appear to be known. However, comparison
with Monte Carlo values plotted by Grierson and McGee 3 indicates
that replacing n by n + 1 in (52) [n still given by nB < f h < (n + 1)5]
gives an expression for N/S which is not greatly in error when / is in
< / ^ f h — nB. The simplest instance of (52) holds for n — 1,
B < f h < 2B, and / in the range f h - B £ f g B:
^=l(i^) , [( 2 -W'-0-B / ) ! ]+°<™.
(6)
The explicit part of (6) decreases to zero as / decreases from B to
f h - B. For ^ / < f h - B, N/S is 0(D*/B*).
I am indebted to a reviewer for the observation that the presence of the factor
n! * in Con and the behavior of the curves in Fig. 1 strongly suggest that the formulas
give useful results subject only to D f h (instead of the more restrictive DIB) being
smnll
small.
608 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
100 -
80 -
60
<
z
(J 20
f_ = 4.17B SPECIAL
f n = 2B
f - 1.5B
■MONTE CARLO
THEORY, EQ.0.5).
0.04 0.06 0.08 0.1 0.2 0.4
NORMALIZED RMS FREQUENCY DEVIATION, D/B
0.6
0.8 1.0
Fig. 1— Signal-to-noise ratio in top channel. The dashed lines show eq. (5) for
flat baseband phase modulation. The Monte Carlo curve 4.172? is for flat baseband
PM, and the curves 1.5B, 2B, and 35 are for the typical preemphasis used by Anuff
and Liou.
It turns out that the explicit portions of (5) and (52) are obtained
by considering modulation terms of order 2n + 1 and of type
cos 2tt[(7i + 1)5 - nB~]t.
The curves labeled f h = 1.55, 25, and 35 in Fig. 1 have been
plotted to compare our eq. (5), based on the flat power spectrum (2)
for W p (f), with the Monte Carlo results given by Anuff and Liou for
a typical preemphasis curve. The solid lines and dots show Monte
Carlo values of S/N for the top baseband channel. The dashed lines
are computed from our (5). It is seen that the slopes agree well for
small D/B, but for f h = SB a separation of about 6 dB appears. For
f h = 45 (not shown) the separation increases to about 12 dB. Most
of the separation appears to be due to the difference between (2) and
the W ¥ (f) used by Anuff and Liou. This is indicated by later Monte
Carlo computations made by Anuff for the W 9 (f) of (2), and labeled
f h = 4.17 5 in Fig. 1. There is still a separation of 2 or 3 dB. This
may be due to the granularity of the Monte Carlo approximation
to W P (f) and also to the fact that the Monte Carlo filter is not quite
ideal.
COMPUTING BASEBAND DISTORTION 609
Section II contains a statement of results from the Volterra series
theory needed in our analysis. In Section III, the simplest case,
B < f h < 25, involving third-order modulation terms is discussed in
some detail. Section IV and Appendices C and D deal with the gen-
eral nB < f h < (n + l)B case. In Section V, formulas are given for
the calculation of N/S. Appendices A and B contain material which
provides some insight to the general work of Section IV. Appendix A
discusses the case <p(t) = A cos co a t, and Appendix B treats a simple
analog of the FM problem.
All of our work deals with the fiat power spectrum W v (f) defined
by (2). The chief obstacle in going to a more general JF„(/) is the
evaluation of the multiple integrals which occur in the analysis. Pos-
sibly W v (f) = A J* for | /| < B and v > -1 could be handled by the
procedure used here, but this extension has not been studied seriously.
II. RESULTS NEEDED FROM VOLTERRA SERIES THEORY
Because the carrier frequency f is at the center of the ideal filter
passband, the even-order modulation products vanish. In the notation
of Ref. 4, the Volterra series with the even terms equal to zero is
y(t) = j-j / duigi(ui)x(t - Ui)
1 /■■» r°° r*> 3
+ ;t-. / diii / du 2 / du 3 g 3 (ui, u 2 , u a ) II x ( i ~ u k ) + • • •. (7)
<J ■ J — oo J —co J — oo A-—1
When x(t) is a stationary, zero-mean Gaussian process with two-sided
power spectrum W x (f), the Mircea-Sinnreich 5 series for the two-sided
power spectrum W u (f) of y(t) becomes [eqs. (14) and (160) of Ref. 4j :
w u u) = w x (f)\Gx(f) + ifiJ^dfiW.ubGsU, /;, -/o
+ 2WI d/i '/l d fi w 4fi w 4&°M> & -/i A -/»') + • • • f
+ 5i/l d/l /I d h w *ViW*U*)W x U - h - h)
X |(?»(/i,/ a ,/-/x-/ 8 )
+ 1I2 /1 tfMttWh h> f-h- f* a - /i) + • • • r
+^r d/i /" d/z / M rf/3 / co d f* w 4M~-
O . J _oo J — 00 J —00 ./ —X
WW/-/1- ••• - /0|G.(..-)+ •••| 2 + •■•■ (8)
610 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
Here, <?«(/i. h, ■ ■ • , /») is the m-fold Fourier transform of
g m (ti, ■ • ■ ,t m ), i.e., the rath-order transfer function.
We shall need another result which can be derived from the analysis
of Section VII of Ref. 4. Let x(t) and y(t) be as in (7) and (8), and let
2/l(0 = / dui pi (ui)x(t - ui) (9)
J —00
be the linear part of y(t). Then the power spectrum of y(t) - y L (t) is
given by
W v -„ L (f) = [Series for W v (f) with X {J) replaced by 0]. (10)
This result can be established by using the series (152) of Ref. 4 for
(y(t + r)z{t)) to evaluate the four ensemble averages appearing in
the autocorrelation function of y(t) — yUO-
In problems in which cos [2tt/ * + tp(t)] enters a filter with transfer
function K(f), the normalized transfer function
r(/) =K(fo + f)/K(f ) (11)
appears. For the ideal filter of our problem, r(/) = 1 when
-fh < f < h and r(/) = when |/| > //,. Furthermore, the power
spectrum W»(f) of the output phase angle 6(f) is given by the expres-
sion obtained from (8) by replacing W x (f) by W v (f) and (?i(/i),
Ga(/i, ft, /i), • • • by [Mircea 6 and (52), (71), and (72) of Ref. 4]:
Gei(fi) = r(/i),
QMi, h, /.) = i 2 [r(/i + St + / 3 ) - r(/or(/ 2 + /,)
- r(/,)r(/i + /,) - r(/,)r(/i + /,)
+ 2r(/ 1 )r(/ 2 )r(/ 3 )],
GhUu •••,/■)- J 4 [(12345) - 1! L' (D(2345) - II £' (12)(346)
+ 2! E' d)(2)(345) + 2! £ # (1)(23)(45)
10 15 (12)
- 3! E' (D(2)(3)(45) + 4!(1)(2)(3)(4)(5)],
G^C/x, ■■■ ,U) = j-> E (-i)'-(^ - 1)'- ( E m) E'
X r(/, + • • • + fnWfn+i + • • • + /-i+J • • •
xr(/ M+ i+ ••• + /»).
The T's and /'s have been omitted and the subscripts written within
parentheses in G« 6 - In Gem the summation over t and (i>; I, m) is es-
sentially a summation over the partitions of m, I being the number of
COMPUTING BASEBAND DISTORTION 611
parts and v x , v 2i ■ • • , v ( the parts:
The summation £' extends over the N (not to be confused with the
N denoting noise power) nonidentical products that can be obtained
by permuting the subscripts on the f's. The number of terms in the
summation 2Z' is
V
N = m\/vi\i> 2 \- ■ -vtlriM- ■ -r k \ (14)
where ri is the number of equal v's in the first run of equalities in the
arrangement v x ^ v 2 ^ ■ • • ^ v t , r 2 the number in the second run, etc.
When the v's are unequal, the r's do not appear. A more complete
explanation of the notation is given in (24) to (29) of Ref. 4.
In our work, Gewn+D will be either or —1 when n ^ 1.
When ip(t) is bandlimited to \f\ ^ B and f h exceeds B, the linear
portion of 0(0 is equal to ip(t). This can be seen formally by assuming
<p(t) to have a Fourier transform F(f) which vanishes for \ f\ > B.
Then, from (9) and <?#,(/) = r(/) = 1 for | f\ < f h , it follows that
6, At) = ^ dug 9 i(u)<p(t - u)
= J" <lfG9i(f)F(f)e<>""
= j B _ B dfF(f)e i2 'f = <p(t). (15)
Most of our analysis will consist of using the combination of (8)
and (10) to obtain expressions for TI T «_ V ,(/), the power spectrum of the
difference 6{t) — ip(t) between the output and input phase angles.
in. W„- V U) when B < /,, < IB
In this section we take B < f < 2B, f h -B£ f ^ B, and assume
D/B (and consequently W B) to be small. The power spectrum of the
output phase angle is, from (8) with in place of y,
Weif) = W v (f)\v{j)
+ h \ B dfJ R df 2 W,(fi)W,(f 2 )W P V - /x - /,)
X\G e3 (f h U / - /, - f 2 ) + 0(W o B) | 2 + 0(W* o B*). (16)
612 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
From (2), W v {f[) and W v (fd, * = h 2 > can be replaced by W in
the integrals. However, W P (f - /i - h) will be retained for the
present because it serves to make the integral vanish when
|/ — /, - f 2 \ > B. For completeness, we shall carry the first line in
(16) along in the analysis even though it will vanish when we calculate
the crosstalk noise in an idle channel represented by a slot in W v at
Since the linear portion of 0(0 is equal to <p(t), the power spectrum
We-M) o f 0(0 - *(*) is s iven b y ( 16 ) with r (/) in the first line re ~
placed by zero:
W t
-,(/) = W 9 (f)\ f. fdfiW.GnV, fi, -fi)
J. » J
+ T, r d/i/"* </^ *%(/ - h - h) I G«(/n U, 1- h- h)V
3J ~ B J '" rOW). (17)
In obtaining (17), we have used the fact that the integrand in the
first line is an even function of /i.
Examination of (17) shows that the dominant terms in W 9 -M) are
0(W 3 o B 2 ) and hence correspond to third-order modulation. When /does
not lie in an idle channel (i.e., W v (f) ^ 0), some of the third-order
terms in W g (f) arise from the cross term T(f)0(W 2 o B 2 ) which requires
a knowledge of G eb for its evaluation. For this reason, we prefer to deal
with W 9 -M) [instead of We(f)l which requires only G 93 for the cal-
culation of all its third-order terms.
When ^ / ^ B and ^ fi ^ B, as in (17), all of the T's in
OM, fi, -fi) = -W) + r(/)r(0) + r(/ x ')r(/ - fi)
+ r(-/0r(/ + fi) - 2r(f)v(f' 1 )T(-f 1 ) (is)
are unity except possibly T(f + fi) which is unity if / + /i < /* and
zero if / + fi > f h . Hence, G B3 (f, fi, - fi) is zero if fi < h - f and
is - 1 if f h - f < fi It follows that
^f 'df[W Ge 3 (f,fi -fi)
\-(B- f h +f)W , f^fk-B
= J0, f^h-B- (19)
The function W v (f - /i - f 2 ) vanishes for \f - fi- ft\ > B,
and the function
G$z(fi, fi, f — f\ — f-i)
= -r(f) + r(/0r(/ - h) + r(/,)r(/ - / 2 )
+ r(/ - /, - h)YUx + / 2 ) - 2r(/x)r(/ 2 )r(/ - h - /,) (20)
vanishes in part of the square /i = ±5, f a = ±B. The result is that,
COMPUTING BASEBAND DISTORTION
613
as will be shown, the region of integration for the double integral in
(17) reduces to the shaded areas shown in Fig. 2. In Fig. 2, it is as-
sumed that fk - B £ f £ B. When ^ / £ f h - B, the double
integral in (17) is zero because G 63 is zero.
In the present case, B < f h < 2B, it is convenient to set
f» = f- h- u
(21)
so that the lines f 3 = ±B, or f x + f 2 = f ± B, mark boundaries
outside of which W 9 (f - /, - f 2 ) is zero. Equation (21) also enables
us to write the boundaries f x = / - f h and / 2 - / - f h as f 2 + ft
= f h and /i -f- fa - /*, respectively, as shown in Fig. 2.
The expression (20) for G 63 (f h f 2 , f - fi - f 2 ) is equal to -1 in
the shaded areas of Fig. 2. This follows from the fact that all of the
T's in (20) are unity except possibly r(/ - f x ), T(f - / 2 ), and
r (/i + f 2 ), which are when their arguments exceed f h . The possi-
bilities f - fi < —f h and f - f 2 < -fh are ruled out because / > 0,
and fx + /a < — /* is discarded because it makes f 3 > B. Performing
the integration over the shaded areas in Fig. 2 is equivalent to adding
f.+ f, = f + B
f J+ f 2 =f-
f,= B
Fig. 2— The three areas of integration for the double integral in eq. (17) for W e - f (f).
614 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
the areas and gives
\, f* dfif* dfrWIWM - /, - / 2 )|(? M (/i, A, / - /i - h)\ 2
" ~* ~ B = 1[(2B - /*) 2 - (5 - WWl (22)
when //, — B ^ / ^ B. As mentioned earlier, the double integral
vanishes when / ^ //, — 5.
The main result of this section is obtained by substituting the values
(19) and (22) of the integrals in the expression (17) for We- V (f):
We-,(f) = W,(f){B + / - h)*W*
+ WH(2B - f,,V - (B - /) 2 ] + 0(1F*B*) (23)
where //, and / satisfy B < f h < 2B and f h -B£f£ B, respec-
tively. When ^ / ^ //, - B, the G M 's are zero in the corresponding
ranges of integration and it follows from (16) (with Y(j) replaced by
zero) that /J% ..
We- V (f) = 0(FS5 4 ). (24)
It also appears that the third-order part of W t - P (f) is constant when
B < f< f h .
Although it may not be obvious in Fig. 2, the areas of the three
shaded regions are equal, and each contributes the same amount to
We- v U)- There is an underlying symmetry which becomes evident
when the boundaries of the three regions are written as follows:
h = B h = B h = B
/" - B f> = B h = B (25)
/, = -B h --B U = ~B
fi + f*= fh h + U = //■ /• + /!= /*■
Furthermore, the double integral in (17) can be written as
flfdfJdftfdfiWMtiWiWWMMf -/»-/■- U)
X |G#,(/i > /t,/8)| 1 (26)
where, replacing 8(x) by the limit as e -* of h(x) = 1/e for \x\ < e/2
and h(x) = for |.r| > e/2, the integration extends over three por-
tions of a three-dimensional slab bounded by the planes f\ + /•> + fa
= f ± e/2. The three portions are cut out of the slab by the planes
defined by eqs. (25). When the integration is accomplished by inte-
grating with respect to f a first (the thickness of the slab is e/3 5 and
f 3 is integrated over a length e), the areas of integration for /i and / 2
are those shown in Fig. 2.
Thus the twofold integral is equal to the sum of three equal con-
tributions where each contribution can be regarded as arising from
COMPUTING BASEBAND DISTORTION 615
a region near one of the comers of a three-dimensional cube. It turns
out that the corresponding 2/j-fold integral encountered later is equal
to the sum of (2ft + 1) !/«!(// + 1)! contributions arising from regions
near (2w + l)!/n!(« +1)! of the 2 2 " +1 corners of a (2w + l)-dimen-
sional cube. The corners arc those whose (2n + 1 ) coordinates consist
of (n + 1 ) plus B's and n minus B's.
iv. Wt- V (f) when nB < f h < (n + 1)5
For f h and / such that nB < f,, < (n + l)B, n = 1, 2, • • • , and
fh — nB ^ / ^ B, the dominant terms in W 9 - V (f) are given by
ITT/" rB , rB
Wo-Af) = w ^f)\^J g ''■^■■■J o d &
X Gec2n+i) (/, /i, — /i, • • • , /„, — /»)
l — k)\J ^ l 7 dJn-kGei2n+i)(fi, ■ ■ ■ , f-ik, f — j\
X
J2ky j\, /l) • * - , /n-A-> Jn-k)
+ 0(^" +2 B 2 » +1 ) (27)
where for A- = n it is understood that the quantity within the absolute
value signs becomes Geci„+i)(fi, ■■■ , fin, f—fx- ••• h„). No G Bm
for m < 2n + 1 appears in (27) because, from Appendix D, all such
terms vanish over the region of integration.
To aid in the evaluation of the integrals which arise in dealing with
(27) we shall use 7
/*,. ■ ./*. //(.,„) = ( M 4t)-! A' H ^- ld '
K g o m ^ L (28)
where K ^ 0, a m = X\ + .r-> + • • • + x m , and the integration on the
left extends over the region specified by .r, ^ 0, i — 1,2, • • • , m and
K ^ a m ^ L. The integrations with respect to the /,'s in our problem
extend over regions where /,■ is near -\-B or — B; and we shall use (28)
by making the change of variable /,- = B — .r, or/, = — B + .r,-.
The G«(2n+i> in the second line of (27) is different from (and, from
Appendix D, equal to —1) only if
/ + /;+■•• + /,; > f h . (29)
Setting ft = B — .<•„ i = 1,2, • • • , ft carries this inequality into
•Tl + x 2 + • • • + x n < f + nB - fh = P - Q (30)
616 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
where we have introduced the parameters
P - (n + 1)B - f h (31)
Q = B- f
and have assumed P > Q. When P < Q, the inequality (29) and its
analogues for the other terms in (27) cannot be satisfied. Consequently,
all the Gtun+W* are zero and all the modulation terms of order
(2ra + 1) vanish from (27) when P < Q.
From (28) with m = n, K = 0, L = P - Q, and H(z) = - 1, we get
nun
pi r P-Q
±mf. { ~ l)zn ~ ldz
W v (f)
- W.(f)W?(P - Q)**/n\* (32)
for the first term on the right in (27).
Now consider the Mh term in the sum in (27). For G B{2 n+\) to be — 1
instead of 0, the sum of (n + 1) of its arguments must exceed f h
(Appendix D). It can be shown that (n - A") of the arguments must
be /,', • • • , f'n-k and that the remaining (A- + 1) arguments come from
the set of 2A; + 1 elements
fh /i, •■•,/»,/- /i " /«■ (88)
There are (2k-\- l)\/(k + l)!fc! different choices of (k + 1) items
from the set (33). Let f h f 2 , ■■■ , fk+i represent the typical choice and
/!+/■+'■•+ A+l + /l' +/!+■•• + f'n-K (34)
be the typical sum of (n + 1) elements of <?« ( a«+« which exceeds j h .
Each sum is associated with a region of integration, one boundary of
which is obtained by setting (34) equal to f h . For k = 1, there are
three regions and, after the integrations with respect to the /i's have
been performed, the regions become the ones shown in Fig. 2 with f h
replaced by f h — (n — 1)B. For A: arbitrary, the regions correspond to
the corners of a (2k + 1) -dimensional cube, the corner coordinates
consisting of (k + 1) plus J5's and k minus B's. By virtue of the type
of symmetry shown by (25) and (26) for the case B < f h < 2B, each
of the (2k + 1)!/(A; + l)!k! regions contributes the same amount to
the A;th term in (27).
The first step in evaluating the kth term (A; < n) is to perform the
integrations with respect to /,', • ■ • , /»_*. Suppose that the values of
the typical choice /i, • • • , fk+i are given. Then for Gevn+i) to be equal
to —1, it is necessary that
fi + • • • + /„_.'„ > f h - U h+i. (35)
COMPUTING BASEBAND DISTORTION 617
Setting ft = B - .r„ % - 1, 2, ■ • • , n — k carries (35) into
Xi + x» + • • • + a-„_ t < (ti - k)B
- h + h + h + ■ ■ • + /* + i. (36)
Using (28) with ??* = n - k, K = 0, H(o) = -1, and L equal to the
right side of (36) shows that the quantity inside the absolute value
signs in the fcth term is equal to
(n - fc)!(n - k — 1)1 J a
= -W n - k L»- k /(n - A-)! 2 (37)
where L ^ 0.
Next, we integrate with respect to f 1} f 2 , • ■ ■ , f k+ i. The restriction
that the right side of (36) be positive gives
/i + /i + • • • + A+i > /» - (n - fc)B (38)
and the fact that the argument of TT'„(/ — fi — ■ ■ ■ — f 2 k) must ex-
ceed — B gives
h + / 2 + • • • + f k+ i < B + / - f k+2 - ■ ■ ■ - f 2k . (39)
Setting fi' B - x t for i = 1, 2, ■ • •, A; + 1 and /,• = -B + Xi for
i = k + 2, • ■ • , 2k carries (38), (39), and the L in (37) into
xi + 3 2 + • • • + asfc+i < (n + 1)5 - f h = P
.r. ■ + x 2 + • • • + x k +i > B - f + x k+ 2 + ■ • • + x %k
= Q + **+2 + • • ■ + -r 2A - (40)
L — P — xi — Xi — • • • Xk+i.
At this stage, the fcth term in (27) is, for k > 1,
W? (2ft + 1)! /■■ /■, /■, r,
X JP?-"*(P - an - • • • - x t+1 ) 2 "- 2 V(n - fc)! 4 (41)
where (2fc -f- l)!/(ft + l)!fc! is the number of equally contributing
regions of integration. For fixed x k+2 , • • • , x 2k , the integration with re-
spect to xi, • • ■ , Xk+i can be performed by using (28) with m = k -\- 1,
K = Q + x k+ 2 + • • • + x 2k , L = P, and H(z) = (P - 2 ) 2 "-". Ex-
pression (41) becomes
(ft + l)!A-!(n- k)\*J dXk+i ' ' J dxu
X t-. f (P - z) 2 "~ 2k z k dz. (42)
61S THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
The integration in (42) extends over the region defined by x\ ^ 0,
% = k + 2, • • • , 2k, and the inequality obtained by combining the
two inequalities in (40) :
•r, +2 4- - - ■ + x 2k < P - Q. (43)
Using (28) with m = k - 1, K = 0, L = P - Q, and
H(z) = t P (P - y) 2 "- 2k y k dy (44)
J Q+z
leads to a double integral which can be reduced to a single integral by
reversing the order of integration:
-^r y j P ~ Q H{z)z«-hh = {k ]_ 1} , J* y k (P - y) 2 - 2k (y - QV-'dy
_ * (2»-2fe)!(2fc-/- 1 )! Ql(p _ Q)2n _ t (45)
When (45) is used in the expression (42) for the A-th term in We- r (f),
(42) becomes
W 2n +\2n - 2k)\ * (2k- C- l)\Q ( (P - Q) 2 "~ l ,
(k + l)\k\(k - l)!(n - A;)! 4 h W - t)\{2n - l)\ ' K }
It can be verified that (46) also holds for k = 1, even though k > 1 was
assumed in the derivation. Adding the expression (32) to the sum of
(46) from k = 1 to n and interchanging the order of summation gives
the equation sought in this section :
W e - V (f) = WMW 2 o"(P - Q) 2 "n!- 4 + W 2 n+l t C (n Q l (P - Q) 2 "~ l
(*
t=a
+ 0W' +2 5 2 »+ 1 ) (47)
where n is a positive integer, nB < f h < (n + 1)B, P > Q, P and Q
are given by (31), and
_ 1 f {2k- I- l)!(2n-2fc)! f48)
Un ~ C\(2n - l)\ Jjju, (k - ()\(k - l)\k\(k + l)!(n - W { '
When P < Q, i.e., / < f h - nB, our analysis tells us only that We-*(f)
isO(W 2n+3 B 2n+2 ).
V. THE NOISE TO SIGNAL RATIO N/S
According to (4) the average signal power (FM) in the channel
(/, / + A/) when it is busy is
S = (2irf) 2 (2W )Af. (49)
COMPUTING BASEBAND DISTORTION 619
Likewise, the average interehannel interference noise power is
AT = (27r/)2[2TTV,(/)]A/ (50)
and hence
AT AS' = W„- P (f)/W . (51)
When the channel is idle, W v (f) is in (/, / + A/), and if all of
the other channels are busy, (51) and (47) give
N/S - Wl n t, Ct„Q<(P - QY»-* + 0l(W o B)*»+i]
l-o
3D*
2B 2
+ 0[(Z)/5) 4 »+ 2 ] (52)
provided nB < f h < (n + 1)5, f h - nB ^ f ^ B, and D/B is small.
In going to the second line, we have used W B = 3D 2 /(2B 2 ) from (3)
and the definitions (31) of P and Q. Equations (5) and (6) given as
examples in Section I are obtained by setting f = B and n = 1, re-
spectively, in (52).
The first few values of Ct n X 10" are listed below.
3 4
( =
1
2
II
= 1
2.5
5.0
2
5.208
19.44
2.083
3
1.832
9.375
3.906
0.193
4 0.1994 1.226 1.182 0.2122 0.0060
VI. ACKNOWLEDGMENTS
I want to express my thanks to A. A. Anuff for his suggestions and
helpful discussions regarding the comparison of the theory with the
Monte Carlo computations. Furthermore, his Monte Carlo computa-
tion of the interehannel interference for the W v (f) used in this paper
gave a very welcome confirmation of the theoretical results. I am also
indebted to Miss Judith Seery for computing values of N/S from the
rather complicated formulas.
APPENDIX A
Sinusoidal Modulation
Some idea of how the FM distortion depends upon the radio fre-
quency bandwidth when the deviation ratio, say A, is small can be ob-
tained from the case tp(t) = A cos a> a t, w a = 2irf a . The carrier fre-
620 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
quency and ideal filter are the same as in Section I, but now n is such
that n/ a < fh < (n + l)f a .
The input to the ideal filter is the real part of
exp [ju„t + j>(0] = E c m exp (ju t + jrrua a t) (53)
m=— =o
where c m = j m J m (A), J m (A) is the Bessel function of order m, and
exp (jA cos w„ = E j m Jm(A) exp (jmo) a t). (54)
Since 2/*// « 1, the filter output is nearly equal to the real part of
exp ljuot + ;e(0] = L c m exp (ju t + jmu a t). (55)
Subtracting (54) from (55) and dividing by exp [ju < + jV(0] gives
(-n-1 to \
E + E ) C »' eX P (./WOJaO (56)
-x n+1/
where the argument £ has been omitted in 6{t) and <p(<).
Replacing exp ( — j<p) on the right by its series obtained from (53)
and taking logarithms give the known first-order approximation
— p = -Im L ( E + E )clc m exp [,/(m - /)«««]
[<=-«, \m=-« m=n+l/
+ terms of order [£ (E + E)] 2 - (57)
( m m
Since A is small and c m = j m J m (A), we have for m ^
c m = c_ m = (jA/2) m /m\ + 0(A»+ 2 ). (58)
The interchannel interference in a microwave system corresponds to
the exp (jua t) and exp ( — ju) a t) components in the expression (57) for
— ip. The largest contribution to these components comes from the
values m = n + 1, I = —n, and m = — n — 1, I — n, respectively:
(A/2) 2n+1
(H ,r\ — JvtiY i-n+n+lpjuat _|_ «— n+»+l«— /Ma* - ] _2 — 1 — 1
(0 -*;-.- /»U e + ./ e J n j( n + i)i
+ 0(A 2 "+ 2 )
= _ y^'"*l cos «.« + 0(A 2 »+ 2 ). (59)
n!(n + 1)!
It follows that the average power in the cos co a t component of 6 — <p is
P(A) = 1^^-, + 0U"+>) (60)
COMPUTING BASEBAND DISTORTION 621
and dividing by the average power A 2 /2 in <p(t) = A cos «„ t gives
ave. power in ip n\ 2 (n + l)! 2
If, instead of the pure sinusoidal signal A cos o» Q t, the signal <p(t)
were a very narrow band of Gaussian noise centered at the frequency
f a , its envelope R would fluctuate slowly according to a Rayleigh proba-
bility density
p(R) = a~ 2 R exp (-R 2 /2a 2 ) (62)
where a 2 is the average power in <p(t). Replacing A in P(A) by R and
averaging with the help of (62) shows that the average of the total
power in the components of 6 — <p clustered around /„ is
A/ = r P(R)p(R)dR
_ (2n + 1 )!„«-•" 4
~ n!'(n + 1)!*2»" + ° (cr } " (63)
This expression for N can be checked by using W x (f) = a 2 8( \f\— f a )/2
in place of W x (f) = W , \f\ < B, in the analysis of Sections II to V.
Division by the average power S = a 2 in <p(t) gives
N = J2n + 1)!^-
5 n\\n+ \)\ 2 2 2 » + K h (b;
In <^(0 = A cos w a <, A is the deviation ratio and in (64) a is the rms
frequency deviation ratio. The fact that N/S varies as a* n in (64)
agrees with the case in which <p(t) has a flat spectrum. However, (64)
is larger by roughly the factor (2n + 1)!
APPENDIX B
Simple Analogue of Relation Between <p and
The relation between the FM input <p and output d is somewhat
similar to the relation between x and y given by
» " x + (STTTT- * ! " + " < 65 '
where a is real and x is a stationary, zero-mean Gaussian process with
two-sided spectrum W x (f) and autocorrelation function R T = R(t)
= (x(t + r)x(O). We are given W x (f) and want to find W„(f) and
W v . x (f).
Our aim here is to obtain some insight regarding the origin of the
various terms in the series (17) and (27) for W 9 -^(f).
622 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
From Voltcrra series theory, taking (65) to be the series, we get
Gitfi) = 1, Qtn+itfu /i, ' " ■ , /2n +1 ) = a, and G m - for all other
values of w[Ref. 4, (22), (23)]. The Mircea-Sinnreich series [Ref. 4,
(156), (160)] for Wytf) becomes
W
(/) = W x (f) ll + -4; f d/;. • • /"" d/I FT,(/0 • ■ • H^Ctfa
+ I r dfxf dj 2 W x {h)W x (h)W x U - A - A)
•J . y — CO J — GO
X | (n-!)!2-' /I dJ[ - " 7-1 df "- 1 WlU[) ' ■ ■ ^ (/ "- )a f
+ •••
X WJif- h- A A«)|a| 2 - (66)
According to (10), the power spectrum FPy_*(/) of ?/ — z is equal to
the expression obtained by replacing the 1 [i.e., G\{f)~\ by zero in the
first line of eq. (66) for W„(f).
In this particular example, W v (f) can be obtained as the Fourier
transform of the autocorrelation function (y(t)y(t + r)). Let y(t),
y(t + r), x(t), x(t + r) be denoted by y h y 2 , Xi, x 2l respectively. Then
(2/12/2) = fa*) + (2tl + 1 ) ! C(-^^" +1 ) + (4 n+l x 2 )l
+ (2^T)T 2 {xln+1 * n+1) - (6?)
From
(exp (jus, + jvx 2 )) = exp [-2-'(m 2 + v*)R - udR t ]
we have the known results
(x 1 xl n+1 ) = (xi n+1 x 2 ) = (2n + l)\RrR n o/(n\2«)
pi -x 2 j-2, ( 2 /c+ l)!(n-/c)P { }
Substituting in (67) and taking the Fourier transform :
W v (f) = f X e-to't'iyiyJdT
j — 00
;_» |_ + n!2- + n! 2
» q'fl»+'(fl,/2)*-" ] rfiq ,
+ fri(2fc + l)!(n-fc)!»J W
COMPUTING BASEBAND DISTORTION 623
The point being made in this appendix is that the terms in (69) have
[after using 1 + 2a -f a- = (1 + a) 2 ] a one-to-one correspondence
with the terms in the Mircea-Sinnreich series (66). This can be seen
with the help of
J —X
r e -i*rfr RfdT = r dfi ... r df m - 1 w x (f 1 )---w,(f m - 1 )
J -co J — co J -co
x WAf - h- ■■■ - f m -i). (70)
APPENDIX C
Inequalities for Sums of Frequencies
Let /i, f 2 , ■ ■ ■ , f->„+\ denote the (2n + 1) arguments of any one of
the Ge ( 2,.+Ws occurring in the expression (27) for W$- V (f). They satisfy
the relations
|/,-| ^ B, i= 1, 2, ■-• , 2n + 1
/. + U + • • • + /2«+l = /
where / satisfies < / ^ B and is the frequency at which W 9 - V (f) is
being evaluated.
We shall call a set of r of the /,'s an "r-tuple" and the sum of the
/,'s the "sum" of the r-tuple.
First we show that
/ - nB ^ sum of any (/; + l)-tuple ^ / + nB. (72)
Let fi+ f- 2 + • • • + /»+i represent the typical (n + l)-tuple sum.
Then (72) follows upon using |/,| ^ B on the right side of
/!+■■•+ f,H 1 = / - fn+2 ~ ■■■ ~ / tll+1 . (73)
The next inequality is
— nB ^ sum of any r-tuple ^ nB (74)
where r = 1, 2, • • • , n, n + 2, • • • , (2n +1). When r ^ n, (74) fol-
lows from |/, | ^ £, and when r ^ n -f- 2 it can be proved by using
equations similar to (73).
The number of different (n + l)-tuples is (2n + l)!/(n + l)!n!
If, for given set of values of the /,'s, any one of the (77 + l)-tuples, call
it A, has a sum greater than nB, the sum of any one of the remaining
(n -f- l)-tuples satisfies
/ — nB ^ sum of any (n + l)-tuple except A ^ nB. (75)
624 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
The left inequality follows from (72). To obtain the right inequality,
note that all (n + 1) elements (the /,'s) in A must be positive. Con-
sider any other (n + l)-tuple, say C. Then A contains k elements
1 ^ k ^ n which are not in C. Let fa, ■ ■ ■ , f„+i represent the elements
of C so that the left side of (73) gives the sum of C. Then the right side
of (73) contains k elements of A. Since the elements of A are positive,
the right side of (73) is less than / + (n - k)B, and (75) follows from
/ + (n - k)B ^ f + (n - 1)B £ nB. (76)
APPENDIX D
Values of Gj(2n+i)(/i, ■ " • » /«»+0
The notation used in this appendix is the same as that in Appendix
C.
Let G.(2»+i) stand for G H2n+ i){fi, fa, • • • , /a«+0 where the /,'s satisfy
the relations (71). Here we show that
GWh = 0, /;, > (n + 1)5 (77)
where n ^ 1 and //, is the ideal filter semibandwidth. Furthermore, for
a given set of f u fa, • • • , /2«+i, it has been shown in Appendix C that
there is at most only one (n + l)-tuple, the sum of which exceeds nB.
There may be none. When nB < f h < (n + 1)B with n ^ 1 we shall
show that
Gff(2;.+i) = — 1 if one (n + l)-tuple sum > //„ (78)
Gewn+i) = if no (n + l)-tuple sum > fh. (79)
The inequalities (72) and (74) show that all of the r's in G 9{2 n+i) are
unity (i) when f h > (n + 1)B or (m) when no (n + l)-tuple sum ex-
ceeds f h where nB < f h < (n + 1)B. Therefore, to prove (77) and
(79), it is sufficient to show that <?<> m (/i, • • • , f m ), m ^ 2, is zero when
all of the r's in its expression (12) are equal to unity.
Consider the sum
Z E'r(/i+---+/n)---r(/»-^i +••• + /-). (80)
(v;l,m) N
When all of the r's = 1, this sum is equal to the number of different
ways m different objects (fa, fa, ■ • • , f m ) can be put in t identical boxes
with no box empty (the I pairs of parentheses enclosing the arguments
of the ( r's). From combinatorial theory, this number is <S(m, I) the
Stirling number of the second kind given by the generating equation 8 ,
for n ^ 1,
*" = t S(n, k)t(t - 1). ■ ■(< - k + 1). (81)
*=i
COMPUTING BASEBAND DISTORTION 625
To illustrate (80), let m = 4 and 1 = 2. Equations (13) show that the
sum over (v; C, m) in (80) now extends over the partitions of m = 4
which have I = 2 parts. There are two such partitions: vi = 1, j/ a = 3
and v\ = 2, uo = 2. From (14), the corresponding values of N are
4!/l!3! = 4and4!/2!2!2! = 3, respectively. Hence the sum (80) is equal
to 4 4- 3 = 7. This agrees with the known value 5(4, 2) - 7. To return
to the box problem, the 7 different ways of putting 4 different objects
into 2 identical boxes with neither box empty is indicated by
(1)(234), (2)(134), (3)(124), (4)(123),
(12)(34), (13)(24), (14)(23).
The expression (12) for Ge m consists of the sum from I = 1 to I = m
of j m - l {-\) l - x {( - 1)! times the sum (80). When all the Ts are unity
this gives
Ge m = /- 1 £ (-l)'- 1 ^ - l)\S(m, ()
1, m — 1
0, ,n > 1 (82 »
where the summation is accomplished by dividing (81) by I and then
letting t — > 0. Setting m = 2n -f 1 then gives (77) and (79).
Now we turn to (78). Let /„ +1 + f n+2 + • • • + / 2 „+i be the single
(n + l)-tuple whose sum exceeds fh. Then r(/ n4 .i + • • • + ./Wi) =
and all the other T's in Gt&n+n are unity. The problem is to de-
termine the contribution of all of the terms in Ge^n+i) containing
r(/ n+ i + • • • -j- /2n+i). Subtracting this contribution from will give
the value of G^n+n.
Setting m = 2n + 1 in (12) shows that the terms in Ge^n+D con-
taining r(/, 1+ i + • • ■ + / 2n+ i) as a factor are those for which ( and the
parts Vi of the partition of (2n -f- 1) into ( parts are such that
1=2, m = n, vt = n + 1,
* = 3, n + v 2 = n, v% - n + 1, . .
( = n + \, vx 4- • • • + v n = n v n+i = n -j- 1.
Therefore, with k = I — 1, the terms are the product of
i 2 "L (-)**! E ET(/i+ ••• +/J
r(f n -, k+ i + • • ■ + /„) (84)
626 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1973
and r(/»+i + • • ■ + JWi) where now
v i + v% + • • • + vk = n
^ Vi ^ V% ^ • • ■ ^ "A;
A/ = n !/j»i!- • -vfc!ri!ri!' • ••
When all of the T's in (84) are unity, (84) becomes
ji. £ (-)"ft!fl(fi, ib) = ;*■•(- 1)- = I (85)
A = l
where the summation is performed by setting t = — 1 in the generating
equation (81). Subtracting the contribution (85) from gives the
value G«(2H+n = -1 stated in (78).
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1. Rowe, H. E., Signals and Noise in Communication Systems, Princeton, N. J.:
Van Nostrand, 1965.
2 Anuff, A., and Liou, M. L., "A Note on Necessary Bandwidth in FM Systems,
Proc. IEEE, 59 (October 1971), pp. 1522-1533.
3. Grierson, J. K., and McGee, W. F., "Microwave System Intermodulation Simula-
tion," IEEE Int. Conf. Commun- Conf. Record, Philadelphia, Pa., 4, June
12-14, 1968, pp. 403-406.
4. Bedrosian, E., and Rice, S. O., "The Output Properties of Volterra Systems
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5. Mircea, A., and Sinnreich, H., "Distortion Noise in Frequency-Dependent Non-
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6. Mircea, A., "Harmonic Distortion and Intermodulation Noise in Linear FM
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