Document text
Computation of FM Distortion in Linear
Networks for Bandlimited
Periodic Signals
By CLYDE L. RUTHROFF
(Manuscript received December 14, 1967)
Computations of the distortion generated in passing large-index, jre-
quency-modulated signals through symmetrical single-pole and three-pole
bandpass fillers are presented. The computation is for a bandlimited
periodic modulation signal; noise modulation is simulated by the use of
periodic noise samples in a Monte Carlo procedure.
The convergence o] the Monte Carlo procedure is illustrated for the
case oj the single-pole filter and the results are in good agreement with
measurements.
Computations oj envelope distortion are also presented. These data give
the amplitude-to-phase conversion in the receiver containing the filter to
within a constant factor, the constant being the AM/PM conversion
coefficient oj the limiter.
I. INTRODUCTION
In spite of the efforts of a large number of investigators who have
studied the problem over three decades there is no way to compute
the distortion caused by filters and other networks for arbitrary
angle modulated signals of large index or large baseband band widths.
However, by use of the Fourier method 1-4 introduced by Roder in
1937, it is possible to compute the exact responses of networks to a
frequency-modulated signal for bandlimited periodic modulation
signals.
In addition to deterministic signals of this class, noise modulation
can also be simulated and the resulting network distortion computed
by a Monte Carlo procedure. In an excellent paper, Medhurst and
Roberts 4 have described the procedure and given some results for
low index FM, pre-emphasized in accordance with CCIR standards.
1043
1044 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
Their computer program was written in Extended Mercury Autocode.
The same method, coded in Fortran ii, and extended to include the
effects of amplitude as well as phase distortion is being used to study
large index FM systems.
The results presented are for single sine wave modulation and
random noise modulation.
II. ANALYSIS
The modulating signals are restricted to those which are both
bandlimited and periodic. This class includes many signals used for
test purposes; the notable exception is the signal consisting of band-
limited Gaussian noise. More will be said of noise modulation later.
The analysis and computational procedure follows that of Med-
hurst and Roberts in Ref. 4 and is outlined briefly here. Specifically,
the signals are those which can be written as finite Fourier series.
N
n(t) = £ (a„ cosnw a t + b„ sin ruoj) radians, (1)
71 = 1
where :
u> a = 27r/ a = 2tt/T,
7 1 is the period of /*(£),
2 f 772
a n = ■= J n(t) cos nu> a t dt,
T/2
T/2
2 f
b n = Tj, / m(0 am nu a t dt.
I J -T/2
If jn(t) is the desired phase modulation, or //(0 = d/j.{t)/dt the
frequency modulation, the angle-modulated signal is
e = (2)* cos M + /*(*)] (2)
where ca e is the carrier frequency in radians per second. The FM signal
of (2) has a line spectrum with lines at w c ± Mu a , M = 1, 2, 3, • • • .
The lines always occur at these frequencies, changing only in amplitude
and phase as functions of a n ,b n . It is this feature which makes possible
a digital computer solution and, conversely, is the reason for restricting
the form of the modulating signal to that of n(t) in (1). Beginning
with (1) and (2) the major steps in the analysis are:
(i) Derive the line spectrum of (2) .
COMPUTING FM DISTORTION 1045
(m) Modify the lines in amplitude and phase in accordance with
the response of the network being studied.
(iii) Derive the envelope and phase of the modified line spectrum,
that is, determine E(t) and 0(0 where the output of the network
is written
e„ = E(t) cos [u c t + d(t)\ (3)
(w) Derive the line spectrum of E(l), 0(0, and dd/dt.
III. RANDOM MODULATION
An important measuring method in widespread use on FM systems
is the noise loading test. The importance of this method arises from
the fact that a band of thermal noise is a good approximation to a
frequency division multiplex signal which consists of a number of
voice channels. In this test a band of thermal noise in the frequency
range 0-W Hz is the baseband signal. The noise is removed by band
rejection filters in one or more narrow bands or slots ahead of the
modulator. At the receiver the power density appearing in the slots
is a measure of the intermodulation distortion in the system. The
results are usually given in the form of a signal-to-distortion ratio,
the signal being the power density at the slot frequency when the
band rejection filter is removed, that is, when the signal is present.
Computations of distortion can be made along these lines by fol-
lowing a Monte Carlo procedure with a sequence of random noise
samples generated from the periodic form of (1). A set of N sine
waves of equal amplitudes and random phases distributed uniformly
in the interval — 2v constitutes the basic signal. Figure 1 is an
example of this random noise sample for N = 10 and Fig. 2 for
N = 50. One or more amplitudes are set to zero to form the slots,
and the power in the slots as a result of network distortion is com-
puted as outlined in Section II. The process is repeated with a se-
quence of random noise samples, each sample with a set of A r inde-
pendent random phases. The distortion is averaged for the final
result. If N is large enough, if the number of sets is large enough,
and if the network transfer function is well-behaved, then the results
approach those obtained in a noise loading test.
Rice 5 has shown that such a noise representation has a normal ampli-
tude distribution as N — > <» and w„ —* 0. Bennett has computed the
amplitude distribution as a function of N. The conclusion is that with
respect to amplitude distribution the sets of random signals of the
1046 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
0.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fig 1 — A periodic random noise sample for N = 10. Peak amplitude/rms
amplitude = | -0.525 |/[1/(22V) 1/S ] = 2.34.
form (1) approximate Gaussian noise. With respect to the spectrum
the situation is otherwise; the spectrum of noise is continuous whereas
the simulation, for finite N, has a line spectrum. This means that the
results computed with the simulated noise will approximate the results
for real noise only for network responses which are smooth enough.
An example of a function which is not smooth enough is a network
response of unity at the spectral lines and zero elsewhere. In spite of
this limitation it is not expected that smoothness will be a serious prob-
lem for most cases of interest.
3.1 Modulation Index
The modulating signal p(t) can be written as follows:
n(t) = X) A n cos (nwj + a n ) radians,
(4)
where,
COMPUTING FM DISTORTION
A 2 n = a 2 n + bl
1047
a„ = —tan"
0„
The baseband is
W = Nu> a .
(5)
Using (4) to simulate noise in a phase modulation system, the ampli-
tudes A n are equal and the random phases a n are uniformly distributed
from to 2ir. If the rms phase deviation is <p radians,
A n = <p(2/N)> radians. (6)
For the FM application the amplitude terms of the frequency modula-
tion p'(t) are made equal to simulate a flat band of noise, that is,
no> a A n = A, the peak frequency deviation per sine wave. The mean
O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 2 — A periodic random noise sample for N = 50. Peak amplitude/nns
amplitude = | 0.29 |/[l/(2/V) 1/2 ] = 2.90.
1048 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
square frequency deviation is
Substituting for u„ from (5) we get
A n = (<r/W)[(2N) h ]/n. (7)
The rms phase and frequency deviations can be related to the RF
bandwidth by Carson's rule which, for noise modulation, is written
B = 2W(1 + 4a/W), (8)
where the peak frequency deviation is assumed to be 4<r. Suppose
that the line spectrum of (2) contains kN lines in addition to the
carrier, then the bandwidth of the computed spectrum is
B = kNu a . (9)
From (5), (8), and (9) we get the relation between k and a
h = 2(1 + ±<r/W). (10)
This equation is as accurate as Carson's rule and is useful for estimating
fc when a/W is given. If k is chosen too small, significant spectral com-
ponents are omitted from the spectrum; the effect is to pass the com-
plete spectrum through an ideal filter of bandwidth kNw a .
In a similar manner k and <p can be related for the phase modula-
tion case. The rms frequency deviation for the PM case is given by
1 tA^M (11)
where N is the number of tones in the baseband. Substitution of (11)
into (10) gives the desired result.
3.2 Limitations on Modulation Index
It has been shown (9), that the maximum RF spectrum bandwidth
is given by B = kNu a . From (5) the baseband bandwidth is W = Nu a .
Assuming that only negligible energy falls outside B, then B is the RF
bandwidth and the parameter fc is a bandwidth expansion factor since
fc = B/W. (12)
Now, fc and the rms frequency deviation <r are related by (10). The
product fcN is limited by the high speed storage capacity of the machine;
COMPUTING KM DISTORTION
1049
this implies a relationship between N and a/W. Let M ^ kN be the
maximum value of kN which can be accommodated in the machine.
Then,
a/W ^ 1/MM/2N - 1).
(13)
This expression is dependent upon Carson's rule and has the same
unknown precision — but it serves to demonstrate the point that if
large a/W is desired, N must be made small. In the work reported
here, M = 500 so that for N = 10, a/W ^ 6. Conversely for N = 100,
a/W g 0.375.
Because Carson's rule has an unknown precision it is necessary to
determine to reasonable accuracy the relationship between k and a/W .
With a perfect rectangular filter of bandwidth kNu a , signal-to-dis-
tortion ratios have been computed for the case N = 10. In these com-
putations, slots 1 and 10 were set to zero separately and the SDR
computed for that slot.
The results are shown in Fig. 3 as a function of a/W with the band-
width expansion ratio k as a parameter. In all cases slot 1 has the lowest
5 6 7 8
Fig. 3 — FM signal-to-distortion ratios for square filters containing kN-\-l
spectral lines and with N = 10.
1050 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
SDR. The levelling off for SDR near 124 dB is probably caused by the
computer round-off error. The negative slopes are the result of the
finite filter bandwidth of kNu a and the decreasing accuracy of the
method of harmonic interpolation in approximating the spectrum.
Increasing k improves the accuracy of the approximation.
Values of tr/W obtained from Carson's rule in the form given in
(10) are shown by the arrows in Fig. 3. Fig. 3 can be used to determine
the value of k required to compute the SDR for a given <r/W. In all
examples reported here, k and N have been chosen so that without a
filter an SDR ^ 100 dB was obtained for the values of a/W used. The
data of Fig. 3 are averages of 20 noise samples.
IV. THE SINGLE POLE FILTER
The single-pole filter is the simplest possible realizable bandpass
filter and is important for two reasons.
(i) It is widely used. For example, it is nearly optimum for use
in the IF section of a frequency feedback receiver. 7
(ii) As simple as it is, no previous method is adequate for the com-
putation of FM distortion for high frequencies and large deviations.
4.1 Single Sine Wave Modulation
A number of years ago Bodtmann 8 made extensive measurements
on a single-pole filter with both single sine wave and noise modula-
tion.* Let us compare the measured and computed results.
The transfer function of a narrow band single-pole filter is
y = — A_^ (14)
where:
/„ is the center frequency and
f e is the half bandwidth, that is, the frequencies at which the response
is down 3 dB are f ± /, .
Bodtmann's filter was centered near 70 MHz with a half bandwidth
of 1.223 MHz. The skirts fit the response of (14) to within ±0.1 dB
out to the 15 dB loss points. The measured and computed ratios of
signal-to-third harmonic distortion power are shown in Fig. 4. Notice
*It was Bodtmann's results which led to the discovery of a simple error in
existing theories. 0-11
COMPUTING FM DISTORTION
1051
0.2
0.3 0.4 0.5 0.6 0.8 1.0
RMS DEVIATION IN MHZ
2.0
Fig. 4 — Third harmonic distortion in single pole filter.
the peculiarity which occurs at a deviation of 1.2 MHz where the curves
for 360 KHz and 1 MHz modulation frequencies cross. Existing theories
do not predict this behavior which is verified here by direct computation.
4.2 Results for Random Modulation
Computations of SDR have been made for a single pole filter for
the random modulation discussed in Section III. The results, for
noise samples of 10 and 50 sine waves of equal amplitude and random
phase, are shown in Fig. 5 with Bodtmann's measured results. The
computations followed the Monte Carlo procedure described pre-
viously. The data in Fig. 5 for N = 50 is the average over two slots
at each frequency for 50 noise samples. The pairs of slots are 4 and
5, 17 and 19, and 49 and 50, corresponding to the slot frequencies
84 KHz, 360 KHz and 1 MHz, respectively. Data for all the slots
were computed in the same computer run. In the computations for
.V = 10 one slot at a time was computed, each point being the average
of 80 noise samples.
When the noise sample is simulated by 50 sine waves, the agree-
ment with the experimental data is good. The SDR's for the case of
10 sine waves per noise sample are somewhat higher reflecting the
1052 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 19G8
0.2 0.3 0.4 0.5 0.6 0.8 1.0
RMS DEVIATION IN MHZ
2.0
Fig. 5 — Bodtmann's measured results compared with noise samples.
fact that larger modulation peaks are to be found in the sample
with the larger number of sine waves.
4.3 Convergence oj the Monte Carlo Process
The SDR's of 80 individual noise samples for N = 10 are shown
in Fig. 6 in four sets of 20 each. The average SDR as a function of
8 10 12
NOISE SAMPLE NUMBER
Fig. 6 — FM SDR in a single pole filter. Ten sine waves in baseband; SDR
computed in slot 4; bandwidth expansion factor k = 10; a = 0.2 MHz; u c /W —
1.223.
COMPUTING FM DISTORTION
1053
the number of noise samples is shown in Fig. 7; the four sets of Fig.
6 are averaged in sequence. It is interesting to ask how close to the
80-sample average one would get if only 20 samples were used. As a
partial answer, the four sets of Fig. 6 were averaged separately and
the results are shown in Fig. 8. All four 20-sample averages fall
within 1 dB of the 80-sample average.
Similar data for slot 19 is presented for the case N = 50 in Figs.
9, 10, and 11. Slot 17 was also computed and the averages for both
slots are shown in Figs. 12 and 13. The results for slots 17 + 19 are
remarkably similar to those of 19 alone. The 10-sample averages
deviate from the 50 sample average by a maximum of 2.7 dB for
slot 19 and 2.3 dB for the sum of slots 17 + 19. Interestingly enough,
the 10-sample average for N = 10 deviates from the 80-sample
average by a maximum of 2.2 dB.
The behavior of the SDR of a single noise sample as a function
of <t/W is also of interest, Fig. 14 shows this behavior for each of the
first six noise samples of set 1, Fig. 6, compared with the 80-sample
60
U 58
£ 55
53
52
51
i
20 30 40 50 60
NUMBER OF NOISE SAMPLES AVERAGED
.Fig. 7 — Fluctuations in SDR of single pole filter a? a function of number- of
sets of computations. Ton sine waves in baseband; SDR computed in slot 4;
bandwidth expansion factor k = 10; a = 0.2 MHz; w e /W = 1.223.
1054 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
64
4 ^
(0 b2
DQ
2 Q
H
D
Z
O 58
5
en
§56
t-
1 Q \
\ 1
\
Pi 1
\
t
r
\
\
\
(/> 54
O
P
_i 52
<
Z
15
"> 50
r - S
r^
3 O
3
2
NUMB
8 10 12 14 16 18 20
ER OF NOISE SAMPLES AVERAGED
Fig. 8 — Fluctuations in SDR of single pole filter as a function of number of
sets of computations. Ten sine waves in baseband; SDR computer in slot 4;
bandwidth expansion factor k = 10; a = 0.2 MHz; u,/W = 1.223.
4 5 6
NOISE SAMPLE NUMBER
Fig. 9 — FM SDR in a single pole filter. 50 sine waves in baseband; SDR
computed in slot 19; bandwidth expansion factor k = 10; a = 0.2 MHz;
Ue /W = 1.223.
COMPUTING FM DISTORTION
1055
53
i/>
in 52
m
U
UJ
Q
51
50
49
48
46
rrt/H/** 00 ^
10 20 30 40
NUMBER OF SETS OF RANDOM PHASES AVERAGED
50
Fig. 10 — Fluctuations in SDR of single pole filter as a function of number
of sets of computations. 50 sine waves in baseband; SDR computed in slot 19;
bandwidth expansion factor k = 10; a = 02 MHz; u c /W = 1.223.
2 3 4 5 6 7 8
NUMBER OF NOISE SAMPLES AVERAGED
Fig. 11 — Fluctuations in SDR of single pole filter as a function of number
of sets of computations. 50 sine waves in baseband; SDR computed in slot 19;
bandwidth expansion factor A; = 10; a = 0.2 MHz; u e /W = 1.223.
1050 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1908
53
-I
O
g
z 51
o
1-
z
g
o
D -IB
/
z 4/
IS
I/)
46
J
10 20 30 40 50
NUMBER OF SETS OF RANDOM PHASES AVERAGED
Fig. 12 — Fluctuations in SDR of single pole filter as a function of number
of sets of computations. 50 sine waves in baseband; SDR computed in slots
17 + 19; bandwidth expansion factor k = 10; a = 0.2 MHz; u c /W = 1.223.
2 3 4 5 6 7
NUMBER OF NOISE SAMPLES AVERAGED
Fig. 13 — Fluctuations in SDR of single pole filter as a function of number
of sets of computations. 50 sine waves in baseband; SDR computed in slots
17 + 19; bandwidth expansion factor k = 10; a = 0.2 MHz; u c /W = 1.223.
COMPUTING FM DISTORTION
1057
m 60
u
50
30
20
SAMPLE NO.
3 Q.
5 i!k\\
4 SS^O
2 '
jVNvx
80 SAMPLE
AVERAGE '"
"N
s
0.2 0.3 0.4 0.5 0.6 0.8 1.0
RMS DEVIATION IN MHZ
Fig. 14 — Behavior of the SDR in a single noise sample as a function of
v/W. Single pole filter; slot 4; N = 10; sample set 1.
average. The same behavior has been observed for other niters. It
is clear that almost any noise sample will predict the SDR behavior
as a function of <r/W, but the actual SDR computed for the single
noise sample depends on the peakiness of the sample.
V. THE THREE-POLE MAXIMALLY FLAT AMPLITUDE FILTER
The maximally flat amplitude filter is used widely in frequency
modulation systems; it has the flattest possible amplitude response
near the midband frequency and is often used in conjunction with a
phase equalizer. The transfer function of a narrow band three-pole
bandpass filter is
1
Y =
1 - h
u
+ ;
&i -
/-/■
1r
(15)
where
/„ is the midband frequency and
] c is the filter half bandwidth; that is, the frequencies at which the
response is down 3 dB are /„ ± f e ,
6j , 6jj are both equal to 2 for an MFA filter,
1058 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
90
m
u
£80
Z
o
S 70
K
o
60
50
'40
SLOT 1
K
SLOT 4 <i
SLOT 10 a
^
x M>
•n:
CHEBYSHEV FILTER
0.1 dB RIPPLE
V
t/VJ
5 6 7 8 10
Fig. 15 — FM signal-to-distortion ratios in a three-pole MFA filter. W — 7
MHz; 3 dB filter bandwidth = 238 MHz; N = 10; A- = 50; no carrier offset.
SDR computations for an unequalized filter are presented in Fig. 15
as a function of frequency deviation. The dashed lines are 12 dB
per octave slopes placed arbitrarily to coincide with the data at
<r/W = 2. The data points are 20-sample averages. The large cross
is the SDR in slot 10 of a three pole 0.1 dB ripple Chebyshev filter
with the same skirt selectivity as the MFA filter at a frequency 256
70
Z
Qui
o: uj
OCD
[ H uj
^.
^s»v N
NN N
%V
^
v>
4 5 6 7 8 9 10
BASEBAND BANDWIDTH IN MHZ
Fig. 16 — FM signal-to-distortion ratios in a three-pole MFA filter. a/W =
3.12; 3 dB filter bandwidth = 238 MHz; jV == 10; k = 30; slot 10; no carrier
offset.
COMPUTING FM DISTORTION'
1059
MHz from the carrier. The Chebyshev filter is clearly superior to the
MFA filter in this instance. The SDR is a function of baseband W as
shown in Fig. 16 for aJW = 3.12 and slot 10. An arbitrary slope of
18 dB per octave is included. As in Fig. 15, the data points are
20-sample averages.
Fig. 17 shows the effect of a carrier frequency offset with respect
to the filter midband frequency. In the application for which this
filter was chosen, the midband frequency change over the ambient
temperature range -40°F to +140°F is about ±6 MHz.
Results for perfect phase equalization are shown in Fig. 18; arbi-
trary slopes have been added. It is clear that nearly all of the dis-
tortion in the unequalized filter is due to nonlinear phase.
VI. AMPLITUDE TO PHASE CONVERSION
In addition to the FM distortion in the filter output there is gen-
erally some envelope distortion. Since all known limiters convert
envelope modulation to phase modulation this source of distortion
must be accounted for in system design. The envelope distortion is
computed as described in Section II and it is necessary to relate
it to the AM/PM conversion of the limiter.
For good limiters the AM/PM conversion is small and can be
assumed linear, that is,
6 = Qm (16)
80
z
O (/>
I- -I
a: w
O m
S u 70
Q Q
Pz
60
SLOT 1 <
k
V
•"•o
SLOT 4 v
SLOT 10
-20
-10 10
CARRIER OFFSET IN MHZ
20
Fig. 17 — FM SDR in three-pole MFA filter as a function of carrier offset.
W = 7 MHz; 3 dB filter bandwidth = 238 MHz; N = 10; k — 50; a/W = 3.12.
10G0 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
where
m is the index of amplitude modulation for the slot of interest,
is the phase shift in radians in the same slot caused by m, and
Q is the AM/PM conversion coefficient.
The normal signal in the slot of interest is a sine wave of amplitude
A . The signal-to-AM/PM distortion ratio is given by
SDR (AM) = 20 log A/6
= 20 log A/Qvi
= 20 log A/m - 20 log Q. (17)
The first term, 20 log A/m, can be computed for the network and
the AM/PM conversion coefficient can be included separately.
The AM and FM SDR's for transitional Butterworth-Thomson
filters 12 are plotted in Fig. 19. For the Chebyshev filter the AM and
FM SDR are 72.3 and 66.3 dB, respectively. All filters were adjusted
for equal loss 256 MHz from the midband frequency. The trends are
110
m 90
80
70
60
50
40
i
\
\
\
\
LINEAF
PHAS
\ 24
dE
\
/o
CT
AV
E
k
p
\ c
\
\
>
UNEC
JUALIZE
(
*s
dE
\
3/0
CI
A\
IE
(
J
3 4
cr/W
6 7 8 10
Fig. 18 — FM SDR in a phase-equalized three-pole MFA filter. W = 7 MHz;
3 dB bandwidth = 238 MHz; N = 10; k = 50; slot 10.
COMPUTING FM DISTORTION
1061
100
U 90
o
- 80
O
5 60'
r fm
__MA
40
MFA
5 6
MFED
Fig. 19 — FM and AM SDR in three-pole transitional Butterworth-Thomson
filters. W = 7 MHz; loss 256 MHz from midband = 20 db; AT = 10; k = 30;
no carrier offset; <r/W = 3.12; slot 10.
as expected, as the filter goes from MFA to maximally flat envelope
delay (MFED) the FM distortion decreases and the AM/PM dis-
tortion increases. The effect of the limiter AM/PM conversion coef-
ficient can be included by adding —20 log Q to the curve marked AM.
The frequency responses for the filters are given by (15) ; for the
0.1 dB ripple Chebyshev filter 6 X = 1.921, 6 2 = 1.801. For the tran-
sitional Butterworth-Thomson filters the parameters are:
Filter No.
1-MFA
2
3
4
5
6-MFED
7
&<
2.0
2.103
2.201
2.294
2.383
2.466
2.547
& 2
2.0
2.092
2.182
2.268
2.352
2.433
2.510
VII. DISCUS
SION
The Fourier method for the computation of FM distortion in linear
networks has been described and some results presented for single
sine wave modulation and for random noise modulation simulated
by groups of harmonically related sine waves. The method is exact
to an accuracy determined by the round-off error in the machine.
1062 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1968
Although the computation is exact for any individual input signal,
the results for noise modulation are only approximate because the
results depend upon averaging over a finite number of periodic noise
samples. Much of the work described in this paper has been devoted
to describing the behavior of the noise computations and in the
determination of the maximum modulation index for which computa-
tions can be made with suitable accuracy.
In addition to demonstrating the nature of convergence of the noise
averaging method, a detailed comparison of this method with the
experimental results of W. F. Bodtmann provides an excellent demon-
stration of the extent to which a noise sample consisting of as few as
10 sine waves approximates a thermal noise signal. The noise simulation
with a 10 sine wave noise sample is sufficient for most applications and
accurate computations have been made for modulation indexes of
a ^ 6W where <r is the rms frequency deviation and W is the bandwidth
of the modulating signal.
It is notable that a single periodic noise sample is sufficient to
determine the shape of the curve describing the signal-to-distortion
ratio as a function of the deviation, the baseband bandwidth, or the
filter parameters. This result, illustrated in Fig. 14, can be used to
conserve computational time when optimizing the parameters of a
system.
ACKNOWLEDGMENTS
I am indebted to M. V. Schneider who guided me patiently through
the programming maze as far as I have gone, and to T. L. Osborne
who insisted from the beginning that the Chebyshev 0.1 dB ripple
filter was better than my choice of the MFA filter.
REFERENCES
1 Roder H., "Effects of Tuned Circuits Upon a Frequency Modulated Signal,"
Proc. I.R.E., 25, No. 12 (December 1937) , pp. 1617-1647.
2 Stumpers F. W. L. M., "Distortion of Frequency Modulated Signals in
Electrical Networks," Commun. News, 9, No. 3 (April 1948), pp. 82-92.
3 Panter, P. F., Modulation, Noise, and Spectral Analysis, New York: McGraw-
Hill, 1965, pp. 273-280.
4. Medhurst, R. G., and Roberts, J. H., "Evaluation of Distortion in FM
Trunk Radio Systems by a Monte Carlo Method," Proc. I.E.E., 113, No. 4
(April 1966), pp. 570-580.
5 Rice, S. 0., "Mathematical Analysis of Random Noise," B.S.T.J., 23, No. 3
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COMPUTING FM DISTORTION 1063
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