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THE BELL SYSTEM
TECHNICAL JOURNAL
volume xliii January 1964 number 1, part 2
Demodulation of Wideband, Low-Power
FM Signals*
By SIDNEY DARLINGTON
(Manuscript received October 3, 1963)
Some theoretical aspects of the demodulation of wideband, low-power
FM signals are discussed. It is assumed that a band-limited, continuous,
analog signal is supplied to the modulator and is recovered to a fidelity
suitable for television, telephone, or carrier telephone. Much of the paper
assumes that the baseband signal is sampled and clamped before it is applied
to the frequency modulator. The combination has been called PAM-FM
and is characterized by a piecewise constant transmitted frequency.
PAM-FM can be demodulated by spectrum analysis means not suitable
for continuously varying frequencies. It is shown that a spectrum generator
can be derived from the techniques of radar pulse compression, and is equiv-
alent to an infinite set of correlators or matched filters plus means for scan-
ning their terminals.
The spectrum analysis circuit forms are compared with demodulators
using frequency detectors, with and without FM feedback, in regard to the-
oretical noise sensitivities. The theoretical sensitivities are quite similar for
spectrum analysis and FM.FB under conditions assumed. The comparisons
disclose that frequency detectors (folloived by filters) enjoy a disguised but
efficient use of a differential phase coherence which is a characteristic of
FM signals. A combination of spectrum analysis and frequency detection
is described which has some of the theoretical advantages of both.
I. INTRODUCTION
This paper discusses some theoretical aspects of the demodulation
of wideband, low-power frequency modulated signals. A wide trans-
* Parts of the material of this article were discussed by the author in lectures
at the University of California at Berkeley during May, 1963.
339
340 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
mitted bandwidth permits a saving in power. Frequency modulation
implies a constant power level, which makes peak power identical with
average power. It is advantageous, for example, when the practical
restrictions on peak power determine system power levels rather than
restrictions on average power.
More specifically, the paper is concerned with FM systems subject
to the following external requirements: A band-limited, continuous,
analog signal is supplied to the input of a coder or modulator, which
produces the transmitted signal. A demodulator reproduces the original
baseband signal to a fidelity suitable for a television channel, a telephone
channel, or a carrier system combining a number of telephone channels.
For such purposes, for example, the average errors in the output must
be more than 40 db below the baseband signal. It is assumed that a large
FM index is used, to conserve signal power. These conditions are im-
plicit in many of the conclusions. They will be referred to collectively
as "the conditions assumed here."
Several different techniques and circuit forms are compared. The
comparisons are concerned primarily, but not exclusively, with sensi-
tivities to noise. Conventional FM receivers and circuits using FM
feedback (FMFB) are included. However, more attention is paid to
techniques which are closer to (but significantly different from) so-called
frequency shift keying (FSK), a well-known method of data transmis-
sion. 1 Thus banks of correlators or matched filters appear in some of
the proposed circuits, somewhat (but not exactly) as in FSK systems.
Alternatively, the correlators or matched filters can be replaced by
circuits resembling the pulse compressors of so-called Chirp radars, 2
and one (but not the only) purpose of the paper is to note how it can
be done.
Circuits of different kinds are compared not only among themselves
but also with theoretical bounds derived from general information
theory. Thus the paper draws on four major disciplines within the gen-
eral field of communication theory and practice, namely: conventional
FM and FMFB, discrete data transmission, pulse compression radars,
and information theory.
An expert in any one of the four disciplines may find some of the
discussion quite familiar, and perhaps superfluous. However, it is un-
likely that many readers will be thoroughly familiar with the pertinent
parts of all the disciplines. Hence a somewhat tutorial approach has
been adopted. However, some of the relations between disciplines and
some of the circuit forms appear to be novel.
The purpose of the paper is to describe and compare the various
FM DEMODULATION' 341
techniques and circuit forms in simple terms. Mathematical proofs are
outside the intended scope. Except in the Appendix, only the simplest
formulas are stated explicitly, and circuits are represented only by simple
block diagrams. A complete analysis is long, tedious, and mathematically
uninteresting; a good deal of it differs only in detail from established
applications to other problems. Some of the circuit forms have not
actually been built; the block diagrams can be filled in with circuit
details in many different ways, and best ways have not all been deter-
mined. The Appendix outlines very briefly some analytical and circuit
details, which may be needed for an appreciation of some of the conclu-
sions.
II. DEMODULATION BY SPECTRUM ANALYSIS
Much of the paper concerns systems in which the analog baseband
signal is sampled, as part of the initial modulation, but is 7iot quantized.
Fig. 1 is a corresponding block diagram. Each sample is clamped during
the sample interval, and is supplied to a frequency modulator. Then
the transmitted frequency is constant over each sample interval, but
changes from interval to interval. Curve B of Fig. 2 illustrates the
variation in frequency with time. It differs from frequency shift keying
in the following way : The transmitted frequency may be anywhere in a
continuum of frequencies; it is not restricted to a finite number of dis-
crete frequencies. The distinction has important repercussions through-
out the paper.
If the sample interval is no greater than the Nyquist interval of the
baseband bandwidth, the sampling destroys no information (at least in
principle). It is assumed here that the sample interval equals the Ny-
quist interval.
Referring again to Fig. 1, the sequence of clamped samples at the
input of the frequency modulator may be called a pulse amplitude
modulation, or PAM representation of the original signal (with no gaps
between the pulses). The corresponding output of the frequency modu-
lator has been called PAM-FM. 3 It is a known means of adapting time
BASEBAND
SIGNAL
SAMPLE
PIECEWISE
CONSTANT VOLTAGE
FREQUENCY
MODULATOR
PIECEWISE
CONSTANT FREQUENCY
""*"
CLAMP
— *
CLOCK
Fig. I — Block diagram of a PAM-FM modulator.
342 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
division multiplex to frequency modulation.* (For multiplexing, signal
samples are clamped for only fractions of sample intervals and are
interleaved with the samples from other channels ahead of the frequency
modulator.) We are concerned here with a quite different feature of
PAM-FM. The piecewise constant transmitted frequency can be demod-
ulated by means of circuitry which cannot handle the continuously
varying frequency of the more usual FM signal.
It is assumed that the demodulator is synchronized to the constant
frequency intervals, as received. Some synchronization means are sug-
gested in the Appendix (Section A.9). Then either correlators or matched
filters may be used to estimate the piecewise constant frequency, sample-
by-sample. The block diagram in Fig. 3 illustrates the concept, without
filling in circuit details. A set of correlators or filters, tuned to a sequence
of closely spaced frequencies, furnishes a spectrum analysis of the signal
plus noise received over each sample interval. The signal is estimated by
finding the frequency at which the spectrum is largest.
The operation is complicated by the fact that the true frequency is
anywhere in a continuum, and must be estimated to closer than 1 per
cent of the bandwidth of the continuum. This implies something like
100 correlators or filters, or else means for interpolation which compare
the outputs of adjacent units.
2.1 A Spectrum Generator
The set of correlators or filters furnishes an analog representation of
the desired spectrum, in which positions along a sequence of output
terminals correspond to discrete values of frequency. The techniques
of radar pulse compression can be used to represent the same spectrum,
with time as the analog of frequency, at a single output terminal. Ex-
ternally, the circuit is equivalent to an infinity of correlators or filters,
with scanning means to convert the spacially distributed outputs into
a function of time.
The spectrum generation hinges on a sequence of two operations.
Fig. 4 is a block diagram. The first operation beats the received signal
with a varying-frequency local oscillator, to obtain the difference fre-
quency. Fig. 5 illustrates the frequencies of the true signal, of the local
oscillator, and of the signal at the output of the mixer. The true fre-
quency is constant over each sample interval, as before. The oscillator
frequency varies periodically, in synchronism with the signal samples.
In particular, it varies linearly over each sample interval. Thus, at the
* For example, in telemetry systems.
F.M DEMODULATION
:il:;
~^vj
a/
1
N.
tfl
3
t>"
z
111
~>
CI
w
ir
u.
J
<
Z
o
I/)
A-ORDINARY FM
B-PAM-FM
TIME.t — *■
Fig. 2 — Instantaneous signal frequencies.
output of the mixer, the variations in frequency are the same over
every sample interval, but the average varies from sample to sample.
The second operation transmits the modified signal through a pulse
compressing (dispersive) line. The nominal delay (phase slope) varies
linearly with frequency. Over any one sample interval, the instantaneous
frequency varies linearly with time. Thus the nominal delay varies
linearly with time. Fig. 6 illustrates the variations in delay with fre-
quency and time.
The variations in delay are so scaled that the tail end of the signal
sample just catches up with the head end. Then, on the basis of nominal
delays, the entire signal sample emerges from the line in a single instant
of time. Actually, of course, the nominal delay does not apply exactly
to the time- varying instantaneous frequency. Thus the signal sample
does not actually emerge from the line all at a single instant. However,
under the conditions assumed it is squeezed into a small portion of the
sample interval.
PAM-FM
CORRELATORS
OR FILTERS GIVE
SPECTRUM AS A
FUNCTION OF
TERMINAL
SELECTS TERMINAL
WITH MAXIMUM
SIGNAL AT
APPROPRIATE
SAMPLING TIMES
RECONSTRUCTS
BASEBAND SIGNAL
ON BASIS OF
TERMINALS
SELECTED
Fig. 3 — PAM-FM demodulation by correlators or filters.
344
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
FREQUENCY CONVERTER PUL SE COMPRESSOR
PAM-FM
MIXER
LOCAL
OSCILLATOR
SAWTOOTH
VARIATION IN
FREQUENCY
_ DISPERSIVE
LINE
SPECTRUM
Fig. 4 — Spectrum generation.
The compression of a signal sample into a short pulse depends only
on the variations in the instantaneous frequency, which are the same
for each sample interval. On the other hand, the time of arrival at the
output end of the line depends on the average frequency, which is the
frequency of the true signal and varies from sample to sample. The
baseband bandwidth and the FM index restrict the signal frequencies
to a utilized RF bandwidth. With a suitable choice of circuit param-
eters, the corresponding variations in arrival time cover a little less than
one sample interval. Then the true signal produces one pulse per sample
interval, whose position in a (somewhat delayed) sample interval is a
measure of the signal frequency. Fig. 7 illustrates the situation. In
other terms, beating with a swept f requency and then pulse compressing
converts PAM-FM into pulse position modulation, or PPM.*
It is now time to note specific formulas. For simplicity, let time t be
zero at the center of a typical signal sample interval. Let the true signal,
for that interval only, be
»(*) = V^PaCOS («,J + 0.);
T/2 < t < T/2. (1)
Here T is the length of the sample interval, to, and 0, are the frequency
and phase of the true signal, and P. the signal power. Let the correspond-
ing output of the mixer be
J(0 = V2P a cos (co 8 i 4- 0. - 1 qf), - T/2 <t < T/2 (2)
where q is an arbitrary constant. The instantaneous frequency is now
to, - qt, linear with respect to time. [Actual circuitry may introduce
constant changes in amplitude, carrier frequency and phase angle,
between (1) and (2), but these are trivial for present purposes.]
* In practice, the compressed pulse will have small side lobes, omitted in Fig. 7
for simplicity. See Fig. 8 below and also Section A. 9 of Appendix.
KM DEMODULATION
345
SIGNAL
MIXER
OUTPUT
V
\
/
/
/
LOCAL
OSCILLATOR
TIME — *-
Fig. 5 — Frequency conversion.
The corresponding output of the pulse compression line is approxi-
mately
5(0 = V2P. F (w. - ">*) cos M + I </f + ft - ft)
CO* = CO,. + 7/ (3)
F(X) =
smX-
The expression assumes that co c is large compared with | w, — to,. | and
I w* — w e I . For present purposes, to, and to* lie in the utilized RF hand,
and to,, is the. midband, or carrier, frequency. A derivation of (3) from
(2) is outlined in the Appendix (Hection A.l).
The processed signal S(l) may be described as a high-frequency
sinusoid multiplied by an envelope function. The frequency, co f + qt,
varies with time, but it is independent of the received signal. On the
other hand, the phase angle is ft — ft. , in which ft is a property of the
transmission line, but ft is the phase angle of the unprocessed signal
s(t). The envelope is y/2P. F(u a — «*)• It is a function of time, but
346 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
1. END OF SAMPLE
2. MIDDLE OF SAMPLE
3. START OF SAMPLE
FREQUENCY, CO-
^%,
^i^
^V,
^^,
^^. ^
^^^
SAMPLE NUMBER, j
SAMPLE NUMBER, j+1
INPUT TIME, t *■
Fig. 6 — Delay vs frequency and time.
the time is an analog representation of the frequency variable w fc . The
signal frequency u>, enters the envelope function as a parameter.
Fig. 8 is a qualitative plot of F(w s - «*). The abscissae correspond
simultaneously to time and w* . The largest F occurs at m = w, . Thus
the frequency u s may be determined by noting the time of the maxi-
mum F, and interpreting the time in terms of u>* . The envelope F may
be separated from the sinusoid by means of an envelope detector at
the output of the line. Fig. 9(a) is a block diagram.
For some purposes, it is convenient to divide S(t) into two com-
ponents, as follows :
S(t) = \/2P t F(<a, - «*) cos j8, cos (u e t + \ qt 2 - &)
(4)
- V2P. F(u, - oj fc ) sin ft sin (u c t 4- \ qt 2 - &„).
FM DEMODULATION
347
IZ^
(I
2Zi
\* ONE SIGNAL SAMPLE
TIME, t — >-
Fig. 7 — Signals at terminals of dispersive line: 1, input signal, before pulse
compression; 2, output signal, after pulse compression.
The two sinusoids are independent of the signal s(t). Physically, the
two envelope functions can be resolved by means of phase detectors.
Fig. 9(b) is a block diagram.
Consider the Fourier transform of a time function equal to s(l) in the
one sample interval, and zero elsewhere. More specifically, consider the
transform at positive frequencies u k near w e . If the same approximations
are made, as in the derivation of (3), the real and imaginary parts of
the transform are the same as the two envelope functions in (4). The
envelope function in (3) corresponds to the transform of the envelope
of the original time function.
The same remarks apply a little more generally. Suppose the ampli-
SPECTRAL FREQUENCY, 0» k »-
Fig. 8 — The function F(u, - «*)•
348 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
s(t)
s(t)
DISPERSIVE
LINE
ENVELOPE
DETECTOR
Y2P S F(0> s -0^
-*■
SWEPT
FREQUENCY
OSCILLATOR
(a)
s(t)
s(t)
DISPERSIVE
LINE
PHASE
DETECTOR
-^2P s FCOS/3 s
"*
,
»
PHASE
DETECTOR
yi" P s Fsin/3 3
SWEPT
FREQUENCY
OSCILLATOR
_ ^
| — ►
SIN
[•W]
cos[e(t)]
FIXED
FREQUENCY
OSCILLATOR
PHASE
e(t)=(y c t + ^qt 2 -/3 c
-
SP
-IT
bH
(b)
Fig. 9 — Detection of envelope and components: (a) envelope, (b) components.
tude of the received signal s(t) is modified, as well as the frequency,
before it reaches the pulse compressor. Then s(t) becomes
1(0 = V / 2P e ^(Ocos(^ + /3 2 -i^ 2 ), -T/2<t<T/2. (5)
Suppose the envelope .4 (0 is symmetrical about the center of the sample
interval. Then (3) and (4) apply except that F(X) is now the transform
of a time function equal to the new envelope during the sample interval,
and again zero elsewhere. For the analogous radar application, see Ref. 2.
The operations which convert (1) into (3) and (4) are all linear opera-
tions on the signal. If s(t) is generalized to a sum of many constant-
frequency sinusoids, the spectrum corresponding to a single sample
interval can be generated by summing the results of the operations on
the individual sinusoids. Referring again to the block diagrams, in Fig.
9(a) the output is the amplitude of the transform, and in 9(b) the two
outputs are the real and imaginary parts. We will use the collection of
sinusoids as a representation of the signal plus noise, received during
one sample interval.
Thus pulse compression techniques generate analog representations
FM DEMODULATION 349
of the transforms of signal samples. The transforms are generated as
functions of time. The constant q determines the time-vs-frequency
scale, and can be chosen so that the utilized RF band is scanned in less
than one sample interval. The width of the peak in Fig. 8 is merely the
familiar "spectrum line width" of a sinusoid of finite duration. The de-
tailed shape (in particular the tails) can be modified to some extent
through initial multiplication by an envelope function (or, alternatively,
by a shaping circuit at the output of the dispersive line).
The same remarks apply to infinite sets of correlators or matched
filters, except that the spectra are generated at specific instants as
functions of position along arrays of output terminals. One result is:
all three embodiments are equally sensitive to noise accompanying
the received signal. A choice between the three must depend on practical
compromises, limitations, etc., associated with the design of actual
circuits. (For the external equivalence between correlators and matched
filters, see, for example, Ref. 4.)
III. SENSITIVITIES TO NOISE
Demodulation by correlators, matched filters, or spectrum generators,
as described in the previous section, will be referred to collectively as
demodulation by spectrum analysis. This section compares the effects
of noise in such circuits and in conventional FM receivers and FMFB.
Between conventional FM and FMFB, some effects of noise are quite
similar and some quite different. The two circuit forms will be referred
to collectively as demodulation by frequency detection.
It is assumed that the noise is Gaussian and that it is added to the
signal before it reaches the demodulator. It may be, for example, thermal
noise associated with first stages of amplification in the receiver. In
demodulation by spectrum analysis, the noise adds random processes
to the spectra analyzed. These may be described as two independent
Gaussian processes added to the envelope functions in (4). The inde-
pendent variable in the random processes is the spectral frequency
cjA- , which is also represented by time in the pulse compression embodi-
ment. The processes are described in a little more detail in Section A.2.
It is convenient to normalize the error formulas in terms of param-
eters r, R, and T, defined as follows:
u)b — baseband bandwidth (0 frequency to cutoff)
co r = full excursion of instantaneous signal frequency (maximum —
minimum)
r = u> r /co& = bandwidth expansion ratio (6)
350 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
P s = signal power (6) (cont.)
P n = noise power in a frequency interval equal to one baseband
R 2 = PJP n = "signal power to noise density ratio" at the input of
the demodulator
T = w/u b = baseband Nyquist interval.
Under the conditions assumed here, the bandwidth expansion ratio,
r, is fairly large — order of 10 or 20. Power thresholds (denned in the
next section) set lower bounds on R, in the neighborhood of 14 to 16 db.
In practical applications, practical compromises may require a somewhat
larger R, and the bandwidth of the receiver must be a little greater than
u r (whether demodulation is by spectrum analysis or frequency de-
tection) . The power spectrum of the noise is assumed to be uniform at
the input of the demodulator, over the pertinent frequency interval.
3.1 Two Different Effects of Noise
For present purposes, one must examine two different effects of the
noise, on the recovered baseband signal at the output of the demodula-
tor. Under the conditions assumed here, the effects of the noise on the
demodulated baseband signal are quite small most of the time. These
may be called small noise errors, and their rms is one measure of circuit
performance. On the other hand, during occasional brief intervals,
peaks in the noise have a dominant effect and temporarily replace the
true signal by a random false signal. This is commonly called blocking.
It usually persists over intervals comparable with a baseband sample
interval. The average number of blockings per second is the blocking
frequency.
Fig. 10 illustrates the two effects in terms of probability densities.
It is a qualitative (not quantitative) plot of the probability density of
the error, due to noise, in the demodulated baseband signal at any one
instant. The peak near zero is substantially Gaussian and corresponds
to the small noise errors. The long tails are flat and correspond to the
probability that blocking will replace the true signal by a random
signal. The transitions between the Gaussian peak and the flat tails are
not considered further here. They are very difficult to calculate and
must be strongly dependent on design details.
The blocking frequency decreases very rapidly as the power ratio R
increases. A related parameter is the power threshold. Thresholds of FM
circuits (and also phase lock) have been defined in numerous ways for
numerous purposes. The definition which best suits our present needs
is the following: The power threshold is the signal power just sufficient
FM DF.MODULATION
351
ERROR IN RECOVERED BASEBAND SIGNAL
Fig. 10 — Qualitative form of error distribution.
to meet a specified limit on the portion of the samples which are blocked.
It can be expressed in terms of the corresponding ratio, R, in db. Under
the conditions assumed here, the specified limit on the blocking rate
may be perhaps one in a thousand or one in ten thousand.
3.2 Sin all Noise Errors
Consider first the demodulation of individual signal samples by spec-
trum analysis. Both phase coherent and phase incoherent circuit forms
are possible. More than one kind of phase coherence is of interest here.
However, it will be simplest to start with the classical kind in which
the phase of each constant frequency sample is independent of other
samples and is determined uniquely by a rule known to the demodulator.
This kind of phase coherence requires a degree of synchronization which
may be impossible in practice. However, its theoretical properties bear
on what follows.
Under the conditions assumed here, the corresponding small noise
errors are approximately as follows:
For phase coherent demodulation:
rms [small noise errors]
max [true signal]
For phase incoherent demodulation:
rms [small noise errors]
max [true signal)
2V3
7T
1
rR
4V8
TV
1
rR
(7a;
(7b)
352 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
(The maximum true signal is here one half of a full signal excursion
between equal + and - maxima.) Derivations are outlined in Section
A.3.
According to (7), the small noise errors of phase coherent spectrum
analysis are about 6 db smaller than those of phase incoherent spectrum
analysis, assuming that the phases of signal samples are determined
individually and uniquely by a suitable rule known to the demodulator.
How do these compare with the small noise errors of demodulation by
frequency detection?
Between conventional FM and FMFB, the small noise errors are
approximately the same. More exactly, they are approximately the
same functions of power level and bandwidths, which may themselves
be quite different in practical applications of the two circuit forms. An
approximate formula is
(8)
For demodulation by frequency detection:
r ms [small noise errors] = 2 1
max [true signal] v3 rR
A well-known derivation is reviewed in Section A.4.
Superficially, conventional FM and FMFB appear to be phase
incoherent. However, the (theoretical) small noise errors are almost the
same as in the phase coherent, sample-by-sample spectrum analysis.
They differ only by a voltage ratio tt/3, or 0.40 db. This makes demodu-
lation by frequency detection 5.62 db better, in regard to small noise
errors, than the phase incoherent spectrum analysis.* It suggests that
a more subtle form of phase coherence is at work, which perhaps can be
realized also by a more subtle use of spectrum analysis.
Further evidence is as follows: Consider the usual description of
noise reduction by conventional FM demodulation. (See again Section
A.4.) The frequency detector, as such, produces a demodulated baseband
signal plus a substantial amount of noise. However, when the FM
index is large, most of the noise power is at frequencies above the base-
band. Fig. 11 illustrates the usual form of the power spectrum. Then a
filter which passes only the baseband eliminates most of the noise.
To approach the noise levels of phase coherent spectrum analysis,
one must use an almost ideal baseband filter. But then the filter com-
bines past outputs of the frequency detector over a "memory time"
substantially longer than the baseband Nyquist interval. (Ideally it
* The G-db difference has been noted before, with different interpretation, by,
for example, Kotel'nikov. 6
PM DEMODULATION
353
oj
Si
14
A- AT OUTPUT OF FREQUENCY
DETECTOR
B-AFTER FILTERING
FREQUENCY, (x>
Fig. 11 — Small noise errors in frequency detection.
.should be infinite.) Fig. 12(a) is a qualitative illustration of the appro-
priate weight function, or impulse response.
What happens if the filter is constrained to have a memory no longer
than one baseband Nyquist interval? Suppose the true signal frequency
is constant over that interval. Then the best weight function, within the
constraint, is the parabola illustrated in Fig. 12(b).* The corresponding
small noise errors turn out to be exactly as in phase incoherent, spectrum
analysis.
It is not at once clear how the longer memory of the ideal, uncon-
strained filter can reduce the (small noise) errors by anything like 5 or
6 db. The original baseband signals are substantially uncorrclated over
intervals longer than one Nyquist interval. The effective correlation
time of the noise process is even shorter. However, it is the frequency
of the FM signal which has the correlation characteristics of the base-
band. The phase is further characterized by the continuity oj phase
rotations required for a constant amplitude sinusoid of varying fre-
quency. This may be regarded as a subtle kind of phase coherence
which, in fact, is used effectively by the filter in demodulation by fre-
quency detection.
The interpretation is clarified and supported by the following argu-
ment: Consider demodulation by spectrum analysis, and suppose the
transmitted signal is generated by applying a piecewise constant control
voltage to a frequency modulator. (See again Fig. 1.) Because the output
of the modulator is a continuous sinusoid, the instantaneous phase
rotation is continuous, even though its rate of change (which is the
frequency) is discontinuous. The continuity of phase rotations, from
sample to sample, has been called differential phase coherence.
* "Parabolic smoothing" is best for a finite interval, and a constant signal
plus noise power proportional to u 2 . See, for example, Ref. G.
354 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
hiri
(a)
^ |
V ^
:
M
(b)
TIME
Fig. 12 — Filter weight functions: (a) ideal band-limiting filter; (b) optimum
when constrained to one sample interval.
Fig. 13(a) illustrates the differentially coherent phase rotations. The
slope of each straight line segment is the frequency during one signal
sample interval. In contrast, if the transmitted signal is differentially
phase incoherent, the phase rotations are discontinuous between samples,
as in Fig. 13(b). This corresponds, for example, to forming a piecewise
constant frequency signal by successive selections (or keying) from a set
of phase incoherent oscillators.
Referring to Fig. 13(a), consider sample number k. The frequency
can be estimated by an incoherent spectrum analysis of signal sample
k by itself. [See again (7b) for the rms small noise errors.] Further
information can be gleaned from spectrum analyses of samples k — 1
and fc + 1. Specifically, estimates can be obtained from these samples
of the phase rotations at the beginning and end of sample interval k.
Only the difference between the two phase angles is actually needed,
and hence the absolute phase reference required for the phase coherence
of (7a) is no longer necessary.
The difference between the two estimated angles is the net phase
rotation, modulo 2ir, over sample interval fc. Dividing by the duration
FM DEMODULATION
355
Fig. 13 — Phase rotations: (a) differential phase coherence; (b) differential
phase incoherence.
T of the sample interval gives a second estimate of the frequency, but
only to modulo 2ir/T. When the noise is small, as assumed, the first
estimate is accurate enough to resolve the ambiguity. Then a weighted
sum of the two estimates gives an improved estimate of the true signal
frequency. (The small noise errors in the two estimates are substantially
uncorrected.) Small further improvements can be derived from fre-
quency and phase estimates for additional sample intervals.
An optimum combination of phase and frequency measurements of
all samples, — » to + *> , gives a 4.365-db theoretical improvement
over sample-by-sample phase incoherent spectrum analysis. (The
power ratio is 1 + \/3.) Of this, 3.979 db can be realized by using only
samples I ■ — 1, k, k + 1 to estimate the frequency of sample k. A deriva-
tion is described very briefly in Section A.5.
Why does one not realize the full 5.62 db apparent in conventional
FM demodulation? It can be interpreted as a curious effect of the sam-
pling of the original baseband signal, which is not part of the conven-
tional FM system. The interpretation is supported by what follows.
Suppose the piecewise constant frequency is applied to the frequency
356
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
detector in an (idealized) conventional FM receiver, and that the noise
level is low enough to justify the usual small noise approximations. The
output is a piecewise constant true signal, like curve A of Fig. 14, plus
noise. The noise can be reduced by sampling the output of a suitable
filter, as suggested by Fig. 1"). Can the ideal baseband filter be used, as
for an unsampled signal?
Elementary information theory includes the following: If the samples
were represented by a sequence of very short impulses, like curve B of
Fig. 14, the ideal filter would be as effective as for the unsampled signal.
However, because they are represented, in fact, by a piecewise constant
signal, like curve A, the ideal filter has two shortcomings. It produces
intersample interference. It responds to the wanted sample less effi-
ciently than to an ideal impulse.
Suppose the filter is constrained to give no intersample interference,
assuming each sample to be a constant signal over its entire sample
interval. The best filter within the constraint gives 4.365 db improve-
ment over incoherent sample-by-sample spectrum analysis, which is
TIME — *-
Fig. 14 — Filter inputs.
FM DEMODULATION
357
FREQUENCY
DETECTOR
PIECEWISE
CONSTANT SIGNAL
FILTER
Fig. 15 — Filler and sampler after frequency detector.
exactly the same as the figure for multisample spectrum analysis using
differential phase coherence. A derivation is outlined in Section A.G.
3.3 Thresholds
Consider first the thresholds of sample-by-sample spectrum analysis.
Fig. 16(a) illustrates the spectrum of the usual signal-plus-noise sample.
Fig. 16(b) illustrates the spectrum of the occasional sample which blocks.
It assumes that the frequency of the spectral maximum is used as the
estimate of the true frequency, as before. The blocking occurs when the
spectrum of the noise sample has a peak, at a random frequency, which
SPECTRAL FREQUENCY, OJfc — >-
Fig. Hi — Spectrum of a single signal-plus-noise sample: (a) the usual sample;
(b) the occasional sample which blocks.
358 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
exceeds the spectrum of signal-plus-noise at the true signal frequency.
The remarks apply to both phase coherent and phase incoherent sample-
by-sample spectrum analysis, provided the pertinent spectra are used
for each.
The corresponding blocking probabilities are approximately as follows:
For phase coherent spectrum analysis:
P = .-f-=4exp(-« 2 /4). (9a)
For phase incoherent spectrum analysis:
P = '-L^exp(-« 2 /4). (9b)
Here P is the probability that a typical sample is blocked, and blocking
of different samples is uncorrelated.
Part of the derivation is the same as for the (gross) error rates of
quantized frequency shift keying (FSK). However, there is an extra
complication. In FSK, one is interested only in the spectrum at a finite
set of discrete frequencies. The random process which is the noise
spectrum is at most weakly correlated between the pertinent frequencies.
Thus error rates have been approximated, for example, by assuming
either zero correlation 78 or a manageably simple form of correlation. 9
For our purposes, we must consider the spectra at all frequencies in
a continuum, with the certainty that correlations are high across small
frequency differences. An exact calculation would be extremely difficult.
As an approximation, one can proceed as follows: Divide the pertinent
frequency interval into, say y equal subintervals. Approximate the true
spectrum in each subinterval by a constant. Assume that the constants
for the 77 subintervals are independent random variables (over the
ensemble of noise samples) . Now one can estimate blocking probabilities
as error rates in an ij-frequency FSK system. Differences between (9)
and equations in Refs. 7 and 8 reflect further approximations, appro-
priate under the conditions assumed here. They are described briefly
in Section A.7, together with some further analytical details.
The approximation to the spectrum may be described further as
follows: The covariance of the spectrum of the noise sample is approxi-
mated by perfect correlation over each subinterval and zero correlation
between subintervals. The actual correlation across the (radian) fre-
quency difference wj — on is
FM DEMODULATION 359
. T
sin (coo — wj) —
jT (10
(w 2 — Wi) —
(see Section A.2). Equations (9) correspond to subintervals of width
2o>b , which is the | u-> — coi | at the first zeros of the true covariance
function.
We have defined the threshold as the signal power required to meet
a specified limit on the blocking frequency. The corresponding power
ratio R, used in (9), must give the single-sample blocking probability P
which corresponds to the specified blocking frequency.
Under the conditions assumed here, 1* is very small, say 0.001 or
0.0001. Then the exponentials in (9) are very small, and small per-
centage changes in R produce much larger percentage changes in P.
As a result, changes in the coefficients, multiplying the exponentials,
can be compensated by much smaller changes in R. For example, a
two-to-one change in a coefficient is offset by something like a 5-db
change in R. Two consequences are as follows: The threshold changes
only slowly with the bandwidth expansion ratio r. The threshold is
rather insensitive to the size of the frequency subintervals used in the
approximation described above.
Numerical examples of thresholds will be tabulated in Section IV,
together with small noise errors.
Slepian 10 has derived from general information theory some important
upper and lower bounds on the thresholds (as here defined) of quantized
systems, constrained to code baseband samples individually, for trans-
mission over channels wider than the baseband. It is interesting to
compare the thresholds (9) with Slepian's bounds, even though (9)
refers to unquantized systems. Since the bounds depend on the number
of quanta, one must first decide on the appropriate quantization.
Transmission and demodulation of a quantized signal, as such, in-
volve no counterpart of the small noise errors in unquantized systems.
However, when the original baseband signal is unquantized, transmis-
sion in quantized form implies quantization or round-off errors relative
to the original signal. Then, in judging system quality, one can compare
the quantization errors in a quantized system with the small noise errors
in an unquantized system. Thus it is interesting to compare thresholds
determined by (9) with Slepian's bounds for quantized systems such
that the rms quantization errors match our rms small noise errors.
Our present purposes are served by a very rough comparison, using
360
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
graphical data in Slepian's paper. Under the conditions assumed here,
the thresholds (9) are only very little above Slepian's lower bound. The
differences are very roughly \ db for sample-by-sample phase coherent
demodulation and one db for the phase incoherent form.
In principle, the thresholds can be reduced even a little further by
combining phase and frequency estimates derived from more than one
sample interval. We have seen that a second estimate of the frequency
of sample k can be derived from the phases of samples k — 1 and fc + 1.
The same is true of the phase of sample k. This permits the phase
coherent threshold to be approximated with only differential phase
coherence. More complicated operations yield a further improvement.
Referring again to Fig. 16(b), blocking occurs when a noise peak exceeds
the signal peak, in the spectrum of the signal plus noise, and is chosen
in its place. The additional phase information can be used to improve
the choice between the two peaks. However, the 2tt phase redundancy
severely limits the improvement. For the conditions assumed here, a
rough estimate is a ten-to-one reduction in the blocking frequency, or
something like a one-db reduction in the threshold at the old rate (rela-
tive to phase incoherent spectrum analysis). A few further details are
noted in Section A. 8.
The improved threshold may be slightly below Slepian's lower bound.
This is not improper, since it is obtained by violating Slepian's assump-
tion of sample-by-sample coding and decoding.
Now consider the thresholds of conventional FM demodulators and
FMFB. Fig. 17 compares simplified block diagrams of the two circuit
BAND
FREQUENCY
BASEBAND
^>
FILTER
DETECTOR
FILTER
~~*
(a)
■J MIXER
NARROW-BAND
FILTER
FREQUENCY
DETECTOR
BASEBAND
FILTER
— *-
— *"
VOLTAGE-
OSC
LL
ATOf
\
(b)
Fig. 17 — Demodulators using frequency detection: (a) conventional FM
demodulator; (b) demodulator using FM feedback.
FM DEMODULATION 361
forms. The blocking phenomenon is a well-known characteristic of
these circuits. Under the conditions assumed here, the thresholds are
significantly lower (permit lower signal power) in FMFB circuits than
in conventional FM receivers. The advantage derives from the relative
bandwidths of the filters just ahead of the frequency detectors, and
thereby depends on a fairly large bandwidth expansion ratio, r (which
is here 10 or 20). This is, of course, the reason why FMFB is of current
interest, for example for satellite communication systems. 11
Because of the nonlinear feedback loop, it is extremely difficult to
calculate for FMFB the quantitative thresholds required for specific
blocking rates. However, important parameters have been identified
and studied, for example by Enloe. 12 Good circuits have been built and
demonstrated for voice and television channels, with thresholds which
arc not far above the theoretical lower bounds. Since the quantitative
blocking rates have not been determined, the margins above the bounds
are not known exactly.
3.4 Comparisons with Other Methods
At noise levels and blocking rates appropriate for television, telephone,
and carrier telephone, FMFB and spectrum analysis of PAM-FM have
lower theoretical thresholds than binary PCM. The binary symbols
are less sensitive to noise than, say, PAM-FM samples received at the
same rate. If this were the whole story, binary phase modulation would
have the smaller threshold by a power ratio of about two.* Actually,
of course, the symbol rate must be greater than the baseband sample
rate by a factor, say p, equal to the number of binary symbols per
sample. This, in itself, raises the power threshold by factor p. Thus, if
there are more than two symbols per sample, the theoretical threshold
for binary phase modulation is larger, by a power ratio of about p/2.
The threshold ratios are about the same if one compares the binary
PCM with the following FSK system: A set of, say, 10 discrete fre-
quencies is used, spaced orthogonally in the usual signal theory sense.
One frequency from the set is transmitted during each baseband sample
interval. But this system has only 10 quantum levels. To obtain, say,
100 quantum levels one must either transmit two symbol intervals per
sample (which raises the threshold 3 db), increase the channel bandwidth
by a factor of 10, or pack the frequencies much more closely than the
orthogonal spacing. With close spacing, errors of one quantum level
* Binary phase modulation requires less power than binary frequency modula-
tion. See, for example, Sunde. 13
362 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
are more probable than larger errors, and there comes a point where they
are more like the small noise errors of the analog systems.
In principle (but not likely in practice) thresholds can be reduced
by using systems with fewer symbols or samples per second than the
baseband sample frequency. For example, two baseband samples can
be transmitted as a single analog sample provided the signal-to-noise
ratio can be doubled (>80 db instead of >40 db). Transmission at the
reduced sample rate yields a small reduction in threshold. It is paid for
by an enormous increase in the channel bandwidth, which is required
for the higher signal-to-noise ratio.
If more and more samples are combined, Shannon's fundamental
channel capacity is undoubtedly approached. Turin 14 and Golay 16
have demonstrated that two closely related systems do, in fact, approach
the theoretical capacity.*
Our formulas for demodulation by spectrum analysis assume that the
true signal is estimated by finding the maximum point in the pertinent
spectrum. The same is true of the analysis of FSK error rates in Refs.
7, 8 and 9. A well-known substitute for the determination of a maxi-
mum uses a circuit whose output is zero except when a signal-plus-noise
(in this case the spectrum) exceeds a preset threshold. The threshold is
set so that, most of the time, the peak due to the true signal and only
that peak gets through.
Under the conditions assumed here, the threshold circuit form in-
creases the theoretical power threshold by very roughly 3 db. More
exactly, the blocking probability is dominated by an exponential factor
exp (-R 2 /8) as opposed to exp (-.R 2 /4) in equations (9).
IV. CONCLUSIONS
The techniques of radar pulse compression can be used to generate
spectra of signal samples as analog functions of time. It can be done in
real time in the sense that the spectrum of each signal sample is scanned
in a time no greater than the sample interval. The spectra are the same
as would be generated by infinite sets of correlators or matched filters.
Spectrum generation of this sort may be useful for various purposes,
particularly where the parameter ranges are suitable for the sort of
hardware which has been developed for radar pulse compression.
Demodulation by frequency detection (with or without feedback)
reduces the small noise errors by a disguised but efficient use of differen-
* The increase in channel bandwidth as Shannon's limit iB approached is merely
a property of these specific modulation schemes. In principle, it is necessary only
to increase the length of the pieces of the signal which are coded as units.
FM DEMODULATION 863
tial phase coherence, which is a characteristic of FM signals. Demodula-
tion by spectrum analysis can also take advantage of the differential
phase coherence, although the pertinent operations are fairly compli-
cated. The piecewise constant signal frequency, needed for the spectrum
analysis, reduces the effectiveness by 1.24 db in the theoretical small
noise errors (which can be offset by a 15 per cent increase in the FM
index) .
Under the conditions assumed, and for thresholds as denned here, the
theoretical power thresholds of the spectrum analysis are very close to
Slepian's lower bound. The power threshold of FMFB appears to be
quite close, but just how close has not been determined.
Thus, under conditions appropriate for television, telephone, and
carrier telephone systems, the theoretical noise sensitivities are very
little different in FMFB and in PAM-FM with demodulation by spec-
trum analysis. Both techniques pose numerous practical problems,
relating to, for example, stability requirements, switching time require-
ments, synchronization to signal samples, over-all complexity, non-
linearity in response to true signal, etc. FMFB has the advantage that it
has already been used, although under somewhat special conditions.
Some theoretical thresholds and small noise errors are collected in
Tables I and II, for various blocking probabilities P and bandwidth
ratios r. They were calculated by (7) and (9) and refer to demodulation
of PAM-FM by phase coherent and incoherent, sample-by-sample
spectrum analysis. A few remarks on circuit problems are collected in
Section A.9.
The noise figures obtainable with practical circuits are of course
somewhat poorer. The degradations may be due to rather different
practical compromises in circuits using spectrum analysis and in FMFB.
Comparisons between practical noise figures may be different for differ-
ent applications.
Under some conditions, a combination of spectrum analysis and
frequency detection may be preferable to either alone. Fig. 18 is a block
diagram of one out of many possible arranagements. A spectrum ana-
lyzer furnishes a first estimate of the frequency of a PAM-FM signal,
using phase incoherent, sample -by-sample spectrum analysis. The
estimated frequency variations are generated locally by a voltage-
controlled oscillator. A mixer subtracts the oscillator frequency from
the frequency of the received signal. (The block labeled "delay" allows
for the operation time of the spectrum analysis.) Then the output of the
mixer is very low index FM, corresponding to the errors in the first
frequency estimate, plus noise.
364
THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
Table I — Thresholds and Signal-to-Noise Ratio
for Phase Coherent Spectrum Analysis
Probability of Blocking
P
Bandwidth RaLio
T = <O r /ub
Threshold
Ratio P./Pn
["Max. Demod. Signal - !*
L rms Small Errors J
(db)
(db)
0.01
10
12.1
31.3
0.005
10
12.7
31.9
0.002
10
13.4
32.6
0.001
10
13.9
33.0
0.0005
10
14.3
33.5
0.0002
10
14.8
33.9
0.0001
10
15.2
34.5
0.01
20
12.8
38.0
0.005
20
13.4
38.5
0.002
20
14.0
39.1
0.001
20
14.4
39.5
0.0005
20
14.8
39.9
0.0002
20
15.3
40.4
0.0001
20
15.6
40.7
0.01
40
13.4
44.6
0.005
40
13.9
45.0
0.002
40
14.4
45.6
0.001
40
14.8
46.0
0.0005
40
15.2
46.3
0.0002
40
15.6
46.8
0.0001
40
15.9
47.1
* At threshold signal power.
Because of the low index, it is now appropriate to use a narrow-band
filter (passing something over two baseband bandwidths) followed by a
frequency detector and a low-pass filter. The sampled output of the
filter furnishes a correction to the first frequency estimate. The theoreti-
cal threshold of the combination is the same as for phase incoherent
spectrum analysis. The theoretical small noise errors are the same as
for demodulation of PAM-FM by frequency detection. The theoretical
improvement over the small noise errors of the first frequency estimate
is 4.365 db.
If the spectrum analysis is accomplished by correlators or matched
filters, a moderate number may be sufficient even though the over-all
errors must be >40 db below the true signal. The error determination
by frequency detection can correct for a fairly coarse quantization of
the first estimate at the same time that it reduces the errors due to noise.
The over-all circuit may be described as open-loop tuning to the pass-
band of the narrow-band filter, as opposed to closed-loop tuning in
FMFB.
FM DEMODULATION
365
Table II — Thresholds and Signal-to-Noise Ratios
for Phase Incoherent Spectrum Analysis
* At threshold signal power.
Probability of Blocking
Bandwidth Ratio
Threshold
["Max. Demod. Signal - ! •
L rms Small Errors J
P
r = a> r /ub
Ratio P,/P„
(db)
(db)
0.01
10
13.3
26.5
0.005
10
13.8
27.1
0.002
10
14.4
27.6
0.001
10
14.8
27.9
0.0005
10
15.2
28.4
0.0002
10
15.7
28.8
0.0001
10
lfi.O
29.3
0.01
20
13.9
33.1
0.005
20
14.3
33.5
0.002
20
14.9
34.0
0.001
20
15.3
34.4
0.0005
20
15.6
34.7
0.0002
20
16.0
35.1
0.0001
20
10.3
35.4
0.01
40
14.4
39.6
0.005
40
14.8
40.0
0.002
40
15.3
40.5
0.001
40
15.6
40.8
0.0055
40
16.0
41.1
0.0002
40
10.3
41.5
0.0001
40
16.6
41.8
NARROW-BAND
FREQUENCY
BASEBAND
FILTER
DETECTOR
FILTER
PAM-FM
VOLTAGE -
CONTROLLED
OSCILLATOR
SAMPLER
— *•
ANALOG
BASEBAND
SIGNAL
SPECTRUM
SAMPLES
*NAI
YZbl
!
Fig. 18 — A combination of spectrum analysis and frequency detection.
366 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
V. ACKNOWLEDGMENTS
The author's work, reported here, has been strongly influenced by
conversations with numerous colleagues in the data transmission, FM
and FMFB, information theory, and radar pulse compression fields.
These included V. E. Benefi, W. R. Bennett, L. II. Enloe, E. N. Gilbert,
W. M. Goodall, C. W. Hoover, Jr., I. Jacobs, R. A. Kaenel, D. Mitchell,
S. O. Rice, 0. L. Ruthroff, and D. Slepian at Bell Telephone Labora-
tories, and T. Kailath, R. Price, and G. L. Turin while visiting the
University of California at Berkeley. Several of the references were
supplied by reviewers.
APPENDIX
A.l Spectrum Generation by Pulse Compression
For a signal sample, modified by the local oscillator, assume:
i(t) = V2Ps B{t) cos (w.« - \ qi 1 + 0.)
E{t) = outside of interval -T/2 g t ^ + T/2
E(-t) = E(t).
For the impulse response of the pulse compression line, assume:
w(t) = cos (o),7 + ) (;f - 13,.).
When | co — to ir | « co ( . , the frequency function is
YM = ( )«p[-i 2^ J'
The output of the line is s*w. Integrate only over E(l) ?± 0:
n +T/2
Sit) = V2P, E(t) cos (w.t - hqr 2 + j8.)
•'t=-T/2
•COS [udt - t) + i q(l - rf - ftj dr.
Express the integrand as a sum of cosines. Neglect the high-frequency
term. Then:
Sit) = V2Ps I \Eir) cos [ut + hqf + P* ~ 0c
J T =_T/2
+ (co, — toe — qt)r] dr.
FM DEMODULATION 367
Resolve into components per sin, cos [(a> s — cj f — qt ) t).
Recall that E{t) is even. Then E(t) sin [(u„ — co c — qt) t] is odd in t.
S(t) = y/2P. F(a. - u e - grt) cos M + £ r/ + 0, - /3 C )
A'(X) - f " ±/?(t) cos (Xr) dr.
sin X —
When E(t) = 1, -772 ^ r ^ +772, F(X) = — — - .
X
A. 2 Noise Contributions to Observed Spectrum
Following Rice, 1 but sacrificing some details of rigor to brevity, let
the noise at the demodulator input be:
n(t) = I x(<a) cos («/ + /3„) dec + / y(a>) sin (<at + /3„) do>.
The interval on to w 2 includes all signal frequencies co„ .
Phase j8„ = an arbitrary parameter in noise representation.
.r(w), i/(co) = uncorrelated, zero average random variables, with
uniform variances, and zero autocorrelations except across infinitesimal
frequency intervals.
Let Ave denote an ensemble average, or expectation.
Let uh(u) and w 2 (w) be arbitrary, except for the pertinent conditions
of integrability.
Ave \ I ;r(a>) w'i(o>) dw I x(u) w«(b>) dw} = o- 2 / «»i(o>) mj 2 (w) do
Ave < / y(u) Wiiu)) do) I y((a) W<(u) rfu>/ = cr" / U>i(co) M> 2 (o) du»
Ave < / x(u>) u>i(ft>) da) / s/(a>) w 2 (a>) da>> = 0.
/ J 6 = noise power in one base bandwidth = co^r .
Let N(oik) = the noise part of the spectrum of one signal-plus-noise
sample.
Apply Section A.l, with w s = w and co c + qt = uu- , to integrands in
u(t).
368 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
N(t) = NiM cos (<a t + \q - ft)
+ N 2 (o>k) sin (to c t + \ q — ft)
iVi(aifc) = / x{ui) P(co — cot) dco,
iVi(a)A-), Ni(uk) = independent Gaussian random processes in co* .
Appropriate choices of Wi , w 2 in the above expectation integrals give
autocovariances of Ni, N 2 .
F(co - co fc ) F(co - wy) tfco, 7=1,2.
Approximate the integration by integrating from - =o to + <x> .
Refer to Section A.l and use E{t) = 1, -T/2 ^ t ^ + T/2.
T
sin (co* — coy) —
2 ^
Ave [Ny(iOk) NyiiOj)) = TO"
Let w,- = <a k , refer to (3), and recall that R 2 = -^ = — *- , T = — .
1 ll, COhO
(cO k — (Oj)
P. = P.
i b COfcff" ' COb
Max of Signal Spectrum _ /P. _ p __ 1 „
rms Ny(cak) V Pb
A. 3 Small Noise Errors in Sample-by- Sample Spectrum Analysis
Refer to Fig. 4, S(t) of (3), and N(o> lt ) of Section A.2.
Use co* = w e -f gt and /3„ = ft .
5(0 + AT(0 = [\/2P. F(«. - w fc ) + Ni(m)]
X cos (co£ + i gi 2 + ft - ft)
+ JV 2 (co*) sin (cot + $ ^ 2 + ft - ft).
Assume (for small noise errors only) :
Ni, N 2 2 « 2P,F 2 (0), co, - co. = e, i « co b 2 .
P/uue Incoherent Spectrum Analysis. Neglecting N 2 , the envelope is
\Z2Ps F(«. - co,) + 2Vi(a>*).
Form a power series in e and solve for max with e small.
KM DEMODULATION 369
Ave t = -
(-'V
2P,
Evaluate by (3) and Section A.2 to get (7b).
Phase Coherent Sample-by -Sample Spectrum Analysis. Refer to (1).
Make the phase & a linear function of co„ :
S(t) = V2P e cos m + (w. - «,)(r/2)j.
Find the components of S(t) and N(t) in phase with a locally
generated cos |[co c + q(T/2)]t + \ qt l - p c \.
Refer to (3). Let »S C be the component of S.
S c (t) = V2P, F(co 8 - co,) cos [(«. - co,)(7V2)]
= y/W. F[2(cu, - a.,].
It can be shown that the frequency variable is also doubled between
covariances of 2Vi(w*) and its counterpart here. Hence noise is accounted
for with \ the frequency errors e.
If the frequency-dependent signal phase appears artificial, change
the time scale to I = I 4- T/2.
S(t) = \/W t cos (coi - - co t . J , ^ i ^ T.
A. 4 Small Nuisc Errors in Frequency Detection
The FM signal is now unsampled. For simplicity assume a constant
signal frequency. Resolve the noise per signal phase.
s(t) + n(l) = WW S + n.(0] cos («.< + /3.)
+ n b {l) sin (co,/ + ft)
8(0 + n(/) = p cos [co„/ + /3, + p(0], tan ^ =
^F. + w«(0 *
The unfiltered frequency error is (p. Refer to Section A.2 to get:
When n 2 « 2P S , Ave £ = ^p- = ^ P (co - co s )' 2 dco.
The ideal baseband filter passes only | co — co s | ^ co 6 .
370 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
2 e+wft 2 8 2 D 2
Ave (Filtered = ^ L 6 X rfX = "SPl " 3PT " 3^ '
A .5 Small Noise Errors in Multisample Spectrum Analysis
Refer to (1) and Fig. 13(a). Let («, , 0.) = o>„ and the midsample
phase 0. of sample a. With differential phase coherence,
0,-/3,-!= («, + «_i)(!T/2).
Let w„ ,m<r = noise contributions to observed w„ , (2/T) /3„ .
Define .r, ,y a ,z a and note the relation to errors:
x 9 = «, + n, , i/, = (2/r)j8, + to,
2, = (a, + 3,_i) - (y, - y*-i) = (n, + n,_i) - (to, - to,_i).
Let co ff + e = the following estimate of oj, :
+ »
o) ff -f- e = a:, — zl Qj z j •
J =-00
2 » 2 2 A 2
Let tr M = Ave n„ , <r m = Ave m,
Ave e 2 - [1 - 2(Q a + Q,+i) + £ (Qy + Qw)] »."
+ [Z («i - Q/+i) 2 ]
2
0"m .
Choose the Q/s for min. Ave e 2 by the calculus of variations.
Compare with Ave £ for x„ alone, which is <r„ 2 .
Min Ave £ of sum _ a m
Ave e 2 of x a alone <r„ + <r„,
Further analysis like that of Sections A. 2 and A. 3 gives
2 _ q 2
<T n — 0(T m
Min Ave £ of sum 1 . oec , u
= : — : 7= or — 4.365 db.
Ave e 2 ot x a alone 1 + Vo
A. 5 »Sma// Muse Errors in Multisample Frequency Detection of PAM-FM
Refer to Section A.4 but assume only a piecewise constant signal
frequency. Refer to Figs. 13(a) and (15).
Let 10(0 = fi^er weight factor, referred to the output sample time.
Assume w>(± °° ) =0.
FM DEMODULATION 371
+00 /•+»
/TOO ~-fX)
w(t) <p(t)dt = — w(t) <p(t)dl.
-30 J— 00
Use <p(t) = [n,b(t)/x/2P,] and a white noise approximation.
r+«>
Ave e- = ^^ jf^ [w{t)fdt.
Find ti>(0i which gives
(a) normalized response to constant frequency in sample <r,
(b) zero response to constant frequencies in samples other than a,
(c) minimum Ave e within constraints a, b.
The calculus of variations makes w(t) quadratic over each sample
interval and continuous at the boundaries. Then Ave e 2 is a quadratic
sum of the boundary values. Minimizing the boundary values is like
minimizing the coefficients Q, in Section A. 5 (with <r„ 2 = 3a m 2 ) and
gives the same result.
A. 7 Blocking Probability in Sample-by- Sample Spectrum Analysis
Refer to Sections 3.3 and A.3. Approximate ATi(o)*), N 2 (uk) by
processes piecewise constant over 77 subintervals.
Approximate s/2P 8 F(to s — w*) by s/2P a F(0) over the subinterval
s and zero elsewhere.
Let .rx , ?/\ = the components of the signal-plus-noise spectrum, scaled
(normalized) to unit variances. The probability densities are:
*-5«p[- ( *-' ) ' + *']-
Phase Coherent Sample-by- Sample Spectrum Analysis. Rotation of
the x\ , x s axes through tt/4 gives quickly
pi»>*i**«i = l[ l - Erf (71)].
Erf w - 1/! I exp (-*)*■
This is the probability of a specific X\ > x, , out of 77 — 1 X\'s, X ^ s.
Under the conditions assumed here, the probability of any one or
more is:
372 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
P m (, - l)P{xy > x a ] = ^J [l - Erf (^)]
Per Section 3.3, use
_ rwj _ r
V ~ 2^ " 2'
Pfawe Incoherent Sample-by- Sample Spectrum Analysis
P[(.rx 2 + yx 2 ) > (W + Vs 2 )} = iexp (-f ) .
Under the conditions assumed here, for one or more X's, X ^ s:
P & ^— • exp ( — — J , use n = r/2 as before.
The last approximation is here a simplification, not a necessity. For
an exact formula (given D„ , D x as above) see Ref. 7 or 8.
A. 8 Reduction of the Blocking Rate of Spectrum Analysis
Refer to Section A.5. Use x, y of A. 5. For a second estimation of w s ,
= (i/r)(/3.+i - 0.-0 - i (w.+i + «,_0
+ e - i (2fc+i - 2/«-0 " i (x.+i + *.-0 + (2w/!T)
v = unknown integer due to phase ambiguities.
Refer to Fig. 16(b). Find the integers v for the best fits to frequencies
of the two peaks in the signal-plus-noise spectrum.
With no weighting for the heights of peaks, the probability that the
closest is the correct choice is of the order of 0.9 (under the system
conditions assumed here).
The actual choice must use also the relative heights of the peaks.
Let P M (M t , M n ) = the probability density of the maxima M, , M n
at the peaks due to signal-plus-noise and noise only (respectively).
Let P t (e„ , e„) = the probability density of the observed deviations
e„ , e n of the second w s from the location of the peaks, using best v's.
Use subscripts 1, 2 for the il/'s and e's before the identification of
which peak is signal-plus-noise and which is noise only.
The best identification corresponds to the larger of
P M (Mi , M 2 ) P,(ei , e0 and P M (M 2 , Mi) P,(«a , *).
U),
FM DEMODULATION 373
P.\i gives a strong weighting except when M 2 is close to Mi .
But when M n > M s , the difference is usually small, and P n only
very rarely gives a strong weighting to a wrong choice.
Let u = M, - M n ■ Then
P(u)
P(-u)
= e
II u
A complete calculation of the probability of a correct choice would
require integration over a complicated portion of the 4-dimensional
space of M, , M „ , e a
e, ,
A .9 Some Circuit. Considerations
A few circuit considerations are described below in brief, purely
qualitative, terms.
Synchronization of Spectrum Analysis to PAM-FM Samples. Assume
the following: The spectra represent signal-plus-noise received during
intervals locally selected by a precision oscillator or clock. The length
T of the intervals is almost right, without synchronizing means. The
problem is to synchronize the start time to the start times of the true
signal samples.
Synchronizing signals might be obtained by any of several means.
One uses a very narrow band transmission channel, to send synchroniz-
ing signals from the transmitter. Others derive synchronization error
signals from the communication signal itself, which must fluctuate
sufficiently to supply the necessary information. (When the true signal
is constant from sample to sample, there is nothing to indicate the
boundaries between samples.) An error in synchronization reduces the
height of the peak in the signal spectrum (on the average). It also pro-
duces a discrepancy between values of oj s obtained from the single
sample spectrum and by the second method described in Section A.5.
In principle at least, a synchronization error signal can be derived from
either effect and can be averaged over many sample intervals to reduce
the effects of noise on the synchronization.
Shape of the Signal Sample. In (3), the tails of the function F are
neither small nor short. By Section A.l, they can be reduced by shaping
the envelope E(t) of the signal sample before forming its spectrum. A
suitable filter in the output of the spectrum generator has the same
effect. Since the best spectral maximum corresponds to the F of (3), a
practical compromise is needed. The pulse shaping problem is an old
one, but here intersample interference due to the tails is not the im-
portant problem, but rather the way the tails can increase the blocking
probability (noise-plus-tails exceeding signal-plus-noise).
:S74 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1964
Channel Bandwidth. For both ordinary FM and PAM-FM the channel
bandwidth must be a little wider than the full excursion, w r , of the
instantaneous signal frequency w„ . The so-called Carson's Rule calls
for a channel width of a> r + 2m for ordinary FM, and the appropriate
rule for PAM-FM is at least not very different. FMFB and PAM-FM
spectrum analysis can tolerate wider bands without significant changes
in thresholds and small noise errors.
Transition Intervals. In idealized models of spectrum analysis, certain
operations happen in zero time. In any actual circuits there will be
nonzero switching times. Very roughly, if a fraction a of each sample
interval is lost due to the switching times, the signal power must be
increased by factor 1/(1 - a). Thus 2 per cent lost time requires roughly
0.1 db more power. In a sense, switching times are spectrum analysis
counterparts of feedback stability problems in FMFB, although the
comparison is purely qualitative.
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13 Sunde, E. D., Ideal Binary Pulse Transmission by AM and FM, B.S.I .J., 38,
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