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SaUonal Bureau of Stan...«
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APR 8 1963
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167
THE ERROR RATES IN MULTIPLE FSK SYSTEMS
AND THE SIGNAL-TO-NOISE CHARACTERISTICS
OF FM AND PCM-FS SYSTEMS
HIROSHI AKIMA
U. S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS
^ecknlcal *^ote 167
ISSUED MARCH 25, 1963
THE ERROR RATES IN MULTIPLE FSK SYSTEMS
AND THE SIGNAL-TO-NOISE CHARACTERISTICS
OF FM AND PCM-FS SYSTEMS
Hiroshi Akima
Central Radio Propagation Laboratory
National Bureau of Standards
Boulder, Colorado
NBS Technical Notes are designed to supplement the Bu-
reau's regular publications program. They provide a
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Washington 25, D.C. - Price 40 cents
II
CONTENTS
Page
1 . Introduction 1
2. The error rates in multiple FSK systenas 3
3. Signal-to-noise characteristics of FM systems 13
4. Signal-to-noise characteristics of PCM-FS systems 18
5. Comparisons between SSB, FM, and PCM-FS systems 29
5.1. General considerations 29
5.2. Absolute comparisons 29
5.3. Comparisons with the minimum-bandwidth criterion 31
5.4. Comparisons with the minimum-power criterion 33
5.5. Comparisons with the minimum- channel- capacity criterion 35
6. Conclusions •*- t 38
Acknowledgements 39
References 40
Appendix A. An approximate method of calculating the element error rate in
incoherent FSK systems when the number of frequencies in the
keying is very large 42
Appendix B. The output SNR in band-dividing FM systems 45
Appendix C. The output SNR in PCM-FS systems 47
List of Figures 54
List of Symbols ' ; 55
111
The Error Rates in Multiple FSK Systems and the Signal-to-Noise
Characteristics of FM and PCM-FS Systems
Hiroshi Akima
The element and symbol error rates in multiple FSK (frequency-shift-keying) sj'stems
and the output SNR (signal-to-noise ratio) in FM (frequency-modulation) and PCM-FS (pulse-
code-modulation-frequency-shift) systems are evaluated for wide ranges of system para-
meters, assuming that the incoming signal and noise in the demodulator are a fading-free
signal and an additive white Gaussian noise, respectively. It is shown that the required
intrinsic SNR for an assigned value of symbol error rate in multiple FSK systems can be
reduced by increasing the number of frequencies in the keying. The possibility of improving
the threshold of FM systems beyond that of conventional ones by modulating the carrier with
sampled values and demodulating the modulated wave with a band-dividing demodulator is
shown. The value of the intrinsic SNR at the threshold increases with the value of modula-
tion index in band-dividing FM systems, and with the number of quantizing levels in PCM-FS
systems when the base in the coding or the number of digits for each sample is kept constant.
The maximum output SNR in PCM-FS systems depends only on the number of quantizing levels
and not on the base, whereas the threshold decreases as the base increases. From the com-
parison of the threshold in band-dividing FM systems with that in PCM-FS systems it is
shown that the latter cannot be lower than the for^mer but can only approach the former when
the base approaches the number of quantizing levels. Brief discussions on the threshold
effects in frequency-lock and phase-lock FM demodulators suggest that the threshold of these
feedback FM demodulators cannot be improved beyond that of a band-dividing one.
1. INTRODUCTION
In radio conamunication either the amplitude, the frequency, or the phase of a sinusoidal
wave of radio frequency (carrier) is modulated by a modulating signal. A modulation of
frequency of a carrier is called frequency-modulation (FM). In general, the modulating
signal can be classified into two categories, i.e. , analog and digital, and when the modulating
signal is digital, the frequency modulation is called frequency-shift-keying (FSK or FS).
The information signal to be transmitted can also be classified into the same two
categories. The quality of the signal at the final destination can be expressed by the output
signal-to-noise ratio (SNR) and the symbol (or character) error rate corresponding to the
analog and digital information signal, respectively. The distinction between the classification
of the naodulating signal and that of the information signal must be stressed. In a pulse-code-
modulation (PCM) system, for example, the information signal is analog, although the
modulating signal is digital. The quality of the received signal in this system, therefore,
should be discussed with the output SNR.
One of the nnost prominent features of FM systems is their signal-to-noise improving
characteristic over amplitude-modulation (AM) systems, as demonstrated by Armstrong
[1936] . Many studies have been carried out to determine the signal-to-noise characteristics
of FM systems. It has been clarified that the system has noise-suppressing characteristics
I
providing the SNR at the input of the demodulator is equal to or larger than the threshold
value of approximately 10 decibels. It has also been shown that the output SNR decreases
rapidly as the input SNR decreases beyond the threshold value [Crosby, 1937; Stumpers,
1948] .^
This value of threshold, however, is valid only when we use a conventional FM
demodulator which is composed of an amplitude limiter, a frequency discriminator, and a
low-pass filter. Several demodulators have been suggested in order to improve the threshold
of FM systems. A frequency-lock demodulator, suggested by Chaffee [1939] , was used in
the Project Echo satellite communication system, and yielded a considerable amount of
threshold improvement [Ruthroff, 1961] . On the other hand, phase-lock demodulators,
suggested by Lehan and Parks [1953] as practical approaches to the optimum FM demodu-
lator, have been used in satellite tracking and telemetering systems. A phase-lock FM
demodulator was also developed for microwave telephone multiplex communication channels
[Morita and Ito, 1960] . The threshold of the frequency-lock and phase-lock demodulator
has been studied by many [Jaffe and Rechtin, 1955; Margolis, 1957; Gilchriest, 1958;
Weaver, 1959; Martin, 1960; Choate, 1960; Spilker, 1961;Enloe, 1962; etc. ], but to the
author's knowledge, the ultimate limit of improving the threshold has not yet been determined.
An important idea for analyzing the signal-to-noise characteristics of FM systems was
suggested by Lehan [1954] . He proposed the conceptual idea of dividing the receiver band-
width into many channels (the bandwidth of each channel being equal to twice the maximum
frequency of the information signal), measuring the amplitude at each channel, selecting the
channel with maximum amplitude as a signal channel from amplitude comparisons, and
assuming that the frequency is measured by the center frequency of the signal channel. In
his paper the transmitted wave is assumed to be frequency -modulated by a discrete signal,
which coincides with the original information signal at every sampling point, equally spaced
by a Nyquist interval. He also suggested a phase-lock FM demodulator as a practical
approach to his band-dividing one. This band-dividing idea was further developed, and the
signal-to-noise characteristics of ah FM system with a band-dividing demodulator were
determined [Akima, 1961, 1963; Battail, 1962] . The above studies also suggest the
similarity of FM systems to PCM-FS systems. The PCM system was invented by Reeves
[1939; 1942] , and the superiority of the system was demonstrated by Goodall [1947] , The
basic characteristics of the system were discussed by Oliver, et. al, [1948] . The essential
feature of the PCM system is sampling at every sampling point, quantizing the sampled value
into L quantizing levels, and coding the quantized value with the base in the coding N. The
system uses n elements for every sampled value, where n is equal to log L. When the
coded signal is used to frequency-modulate the carrier, the system is called PCM-FS system.
Although PCM systems were originated with the base N = 2, a better system can be
obtained if N is increased, as shown by Billings [1958] and Viterbi [1962] .
Consider that the sampled value from an analog information signal at every sampling
point, equally spaced by a Nyquist interval, is quantized into L levels, and that this quantized
signal is used to frequency-modulate the carrier. Then the modulated wave is the same as
the one in a multiple FSK system. If the number of the quantizing levels L is very large, the
modulated wave can be considered to yield approximately the same amount of information as
the continuously frequency- modulated wave by the original information signal. On the other
hand, the discrete modulation can also be considered to be a special case of PCM-FS with
the base in the coding N equal to the number of quantizing levels L. The above observations,
therefore, indicate the necessity of constructing a general theory which covers the signal-to-
noise characteristics of both FM and PCM-FS systems.
It is clear from these observations that the starting point of this theory is to analyze
the error rates in multiple FSK systems. The superiority of the multiple FSK system to
binary ones was shown by Jordan, et. al. [1955], and Robin and Murray [1958] experimentally.
Figures in brackets indicate the literature references on page 40.
Although the element error rates in the multiple FSK systemis have been studied theoretically
[Reiger, 1958; Turin, 1958; Helstrom, 1960; Viterbi, 1962] , the symbol (or character)
error rates have not yet been studied satisfactorily.*
In this paper the error studies on multiple FSK systems are extended and, based on
these studies, the output SNR in FM and PCM-FS systems are calculated for wide ranges of
system parameters. The signal-to-noise characteristics of these systems are compared
with each other.
There are several methods of comparing these characteristics of communication
systems [Jelonek, 1952; Beard and Wheeldon, 1960; Helstrom, 1960; Lieberman, 1961] .
In this paper, however, these characteristics are discussed from the standpoint of communi-
cation system engineering. For this purpose the quality of the signal at the final destination
is expressed by the symbol error rate and the output SNR corresponding to the digital and
analog information signal, respectively, and the concepts of the intrinsic bandwidth and
intrinsic SNR [de Jager and Greefkes, 1957] are effectively used in representing the
characteristics .
In order to analyze the basic characteristics of the systems it is assumed throughout
this paper that the incoming signal in the demodulator is a fading-free signal, and that the
incoming noise is an additive Gaussian noise with a flat spectrum across the bandwidth of
interest. ^
2. THE ERROR RATES IN MULTIPLE FSK SYSTEMS
The element error rate p in multiple FSK systems with coherent detectors is given by
(v-v^)^
p = \ exp( ]^1 -
\ exp; -^ )du ^dv, (1)
where N is the number of frequencies in the keying and v is the normalized amplitude of the
incoming signal voltage with the effective value of the noise voltage in each channel as a unit
[Helstrom, 1960; Lieberman, 1961] . If we define the chcinnel SNR R as the ratio of the
c
incoming signal power to the incoming noise power in each channel, the relation between v
and R is given by
R = V ^ /2. (2)
c s
In an ideal case where the bandwidth of each channel B can be equal to the reciprocal of the
unit time duration of each digit, R coincides with the alternative expression of signal energy
A digital information signal is composed of a sequence of symbols (or characters), and
each symbol (or character) is transmitted as a sequence of elements.
-4-
in each digit per noise power density. The values of p are calculated by an electronic
computer as a function of N and R , and shown in figure 1.
On the other hand, the element error rate p in multiple FSK systems with incoherent
detectors is given by
2^ 2
= , V +v
V exp' -
2
N- 1
dv
(3)
or by its equivalent
N
= n! 2. ^-'^ [ kj^^K""^^" J
^k=2
(4)
where N and v are the same as before, and the function I„(x) is a modified Bessel function
of the 1st kind of the 0th order, and ( i is the number of combinations of k out of N
[Reiger, 1958; Lieberman, 1961] . For relatively small values of N, (4) can be used. For
large values of N, however, the numerical integration of (3) is more convenient than using
(4), The values of p are calculated by an electronic computer as a function of N and R ,
and shown in figure 2.
In either case of coherent or incoherent detectors the element error rate p approaches
(N-l)/N when R tends to zero. For large value of R , on the other hand, p can be
c c e
approximated by (N-1) times that in a binary system [Helstrom, 1960] .
Some of the curves in figures 1 and 2 are compared with each other in figure 3. It
is clear from figure 3 that the coherent detection is always better than the incoherent one
but the difference between the two detections decreases as the number of frequencies N
increases. Because of the difficulties often encountered in practical implementations, we
shall continue our present study only on the incoherent detection.
Next we shall calculate the symbol (or character) error rate in multiple FSK systems.
If the information signal is an L-alphabet system, or if it consists of L symbols (or charac-
ters), and when it is transmitted over an FSK link with N frequencies, the number of elements
n for each symbol is related to L and N by
L
N
(5)
As each element error occurs independently under the conditions assumed in this paper, the
symbol error rate p can be obtained by
= 1
(1
Pe^ •
(6)
To study the system performances we shall introduce the concepts of the intrinsic
bandwidth and the intrinsic SNR. The intrinsic bandwidth is a bandwidth which is intrinsic
to the information signal. Although there is some arbitrariness in defining the intrinsic
bandwidth, it is convenient in digital systems to take a binary system as a reference system
-5-
ELEMENT ERROR RATE IN
COHERENT FSK SYSTEMS
10"
10-2
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SNR IN SIGNAL CHANNEL Rq IN DECIBELS
Figure
10"
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<
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I-
-z.
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10-6
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ELEMENT ERROR RATE IN
INCOHERENT FSK SYSTEMS
****sJ''*\*V*^Vvx
^N = 1024
^ ^512
^^256
$^^-^128
^^/64
N= 32^
8^^
^
2'
11
■
1
-5 0 5 10 15 20
SNR IN SIGNAL CHANNEL Rq IN DECIBELS
Figure 2
■7-
A COMPARISON OF ELEMENT ERROR RATE
BETWEEN COHERENT AND INCOHERENT
FSK SYSTEMS
-5 0 5 10 15 20
SNR IN SIGNAL CHANNEL Re IN DECIBELS
Figure 3
and to take the reciprocal of the duration of each element in the binary system as the
intrinsic bandwidth B.. The value of B. expressed in cycles /second is equal to the value of
the transmission rate expressed in bits /second. Then the intrinsic SNR R. in digital systems
can be defined as the ratio of the incoming signal power to the incoming noise power contained
in a band of width B..
1
When the number of frequencies N is increased in FSK systems, the duration of each
element can be increased to log N times that in binary systems in order to transmit an
equal amount of information. The channel bandwidth B can, therefore, be reduced by a
factor log N. As the intrinsic bandwidth B. is equal to the channel bandwidth B in ideal
l 1 c
binary FSK systems, we obtain the relations
B
B. /log N
1 2
(7)
and
R.
R^/log^N.
(8)
The symbol error rates for some values of L and N are calculated from the element
error rates given above, and shown in figures 4, 5, and 6 as a function of the intrinsic
SNR R.. It is clear from these figures that the required intrinsic SNR (or the required
signal power) for an assigned value of symbol error rate in multiple FSK systems can be
reduced by increasing the number of frequencies in the keying.
We shall give a brief comment on band occupancy in multiple FSK systems. As the
overall bandwidth B is equal to NB , and as the channel bandwidth B is related to the
a c c
we obtain the relation
intrinsic bandwidth B. by {!]
B /B. = N/log N.
3. 1 ^
(9)
The values of the bandwidth ratio B /B.
a 1
for some integers of N are given in Table 1.
It is clear from the table that the ratio B /B.
a 1
decreases at first, but it increases after
passing its minimum, as N increases, (The
minimum takes place at N = e = 2,7183, and
the minimum value is e/log e = 1,8842, ) It
is important to notice that the ternary system
requires a narrower bandwidth than the binary
one and the quaternary system requires the
same bandwidth as the binary one, and these
systems achieve a reduction of the required
intrinsic SNR. A greater reduction of the
required intrinsic SNR by increasing the
number of frequencies N beyond 4, however,
must be accompanied by an increase of the
bandwidth ratio B /B.,
Table 1. Bandwidth ratio
B /B. in N-ary FSK systems.
N
B /B.
a 1
2
2
3
1.893
4
2
5
2.153
6
2,321
8
2,667
16
4
32
6.4
64
10.67
128
18.29
256
32
512
56.89
1024
102.4
SYMBOL ERROR RATE IN INCOHERENT
32- SYMBOL FSK SYSTEMS
10"
10
r2
Ld 10"^
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a:
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10
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-7
10
10-8
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^
L = 32
\
N = 32
\ N = 2
\
\
\
1
\
-5 0 5 10 15 20
INTRINSIC SIGNAL-TO-NOISE RATIO Rj
IN DECIBELS
Figure 4
-10-
SYMBOL ERROR RATE IN INCOHERENT
64-SYMBOL FSK SYSTEMS
10-
10"
a.
^ 10-3
<
QC
^ 10-^
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10
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L = 64
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64 \e \a
\n = 2
\\
\\
\ \
\\
\ \
\
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1
\
-5 0 5 10 15 20
INTRINSIC SIGNAL-TO -NOISE RATIO Rj
IN DECIBELS
Figure 5
-11-
SYMBOL ERROR RATE IN INCOHERENT
256-SYMBOL FSK SYSTEMS
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\ \
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\ \
\ \
\
1
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-5 0 5 10 15 20
INTRINSIC SIGNAL-TO- NOISE RATIO Rj
IN DECIBELS
Figure 6
-12-
ASYMPTOTIC BEHAVIOR OF
INCOHERENT FSK SYSTEMS
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10-2
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10-6
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N = 2
/25
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N=280>
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INTRINSIC
5 10
SIGNAL-TO-NOISE
IN DECIBELS
15 20
RATIO R|
Figure 7
-13-
So far we have studied a system in which one carrier is modulated or keyed by n
elements for each symbol in time sequence. We can consider another system, in which n
carriers are keyed simultaneously by frequency division. In the latter system the duration
of each element can be increased to n times that in the former, and therefore, the channel
bandwidth B csin be reduced by a factor n, i. e. ,
B = B./ (n log N), (10)
C X (^
instead of (7). To keep the same error rate in these two systems the channel SNR R
should be kept constant, because the element error rate p depends only on R and N. As
the noise power in each channel is reduced by a factor n, the signal power of each carrier
in the latter can be reduced by the same factor. The total signal power in the latter, however,
should be the same as the signal power in the former, because n carriers are used in the
latter. Moreover, the overall bandwidth B in the latter is the same as in the former,
a
because we must employ n times N frequencies in the latter. It is clear from these observa-
tions that both (8) and (9) hold in the latter as well as in the former, and therefore, figures
4, 5, and 6 are valid in the latter, too.
It is interesting to observe how the multiple FSK systems behave when the number of
frequencies N is increased. This behavior has already been studied by Turin [1959] , but it
will be studied here with a higher order approximation given in Appendix A. The relations
between the element error rate p and the intrinsic SNR R. are calculated by this approxi-
mation and are shown in figure 7. In the limit of infinite N, the relation becomes a vertical
straight line at R. = log 2 (= - 1. 592 db). It is shown how slowly this critical value of R. can
be approached by increasing N. This critical value of R. corresponds to r = 1 in Turin's
paper, and therefore, it coincides with the value which is obtained by letting the channel
capacity be equal to the transmission rate in Shannon's channel-capacity theorem [Shannon,
19481 .
3. SIGNAL-TO-NOISE CHARACTERISTICS OF FM SYSTEMS
In this section we shall discuss the signal-to-noise characteristics of a communication
system in which a radio frequency carrier is frequency- modulated by an analog signal. This
system is called the frequency- modulation (FM) system in its narrower sense.
In the transmission of an analog information signal it is convenient to take a single -
sideband (SSB) system as a reference and to take the maximum frequency of the information
signal f as the intrinsic bandwidth B., i. e., B = f . The intrinsic SNR R. can be defined
•'m ii-'m 1
as the ratio of the incoming signal power to the incoming noise power contained in a band of
width B., in exactly the same manner as in the transmission of a digital information signal.
The quality of the signal at the final destination can be expressed in terms of the output
SNR R ^.
out
The minimum overall bandwidth of an FM receiver B can be expressed by the well-
known relation
B = 2(1 + m) f =2(l+m)B., (11)
a ■'m 1
where
m = i J -f = modulation index (12)
■'d •'m
-14-
and f is the maximum frequency deviation of the modulated wave.
In receiving a frequency-modulated signal there are several demodulating schemes.
One of the most popular demodulators is a conventional FM demodulator, which consists
of an amplitude limiter, a frequency discriminator, and a low-pass filter. A frequency
discriminator can be replaced by a frequency counter, which counts the nunaber of zero-
crossings in short intervals.
The output SNR R in FM systems with conventional demodulators is proportional
to the intrinsic SNR R. and is given by
R ^ = I m^R., (13)
out 2 1
if the input SNR is larger than the threshold value of approximately 10 decibels. In terms
of the intrinsic SNR the threshold can be expressed by
10 log^^R. = 13 + 10 log^^d + m) db, (14)
as is clear from (11). The relations between R. and R , below the threshold are calculated
1 out
from the data of output noise spectrum and modulation suppression ratio given by Stumpers
[1948] , and shown in figure 8. To calculate these relations the shape of IF pass-band is
assumed to be rectangular and the noise spectrum in case of no modulation is used. From
figure 8 it should be noticed that the system with the modulation index m = 2 requires less
signal power to obtain a relatively low output SNR.
We can draw an envelope of the curves in figure 8, and regard it as the improvement
limit of FM systems with conventional demodulators. As mentioned above, several tech-
niques have been suggested to improve the characteristics of FM systems. The aim of these
techniques can be considered to move the envelope toward the left. Then an important
problem is raised on the limit of moving the envelope.
In order to solve the problem we assume for a while that the modulating signal is a
discrete signal which coincides with the original information signal at every sampling point,
equally spaced in time by the Nyquist interval corresponding to the maximum frequency of
the information signal f . After Shannon's sampling theorem [Shannon, 1948] this discrete
modulating signal can be considered to yield exactly the same information as the original
information signal. Developing Lehan's band-dividing idea [ Lehan, 1954 ] we can reach a
new model of an FM demodulator, in which the incoming wave is divided into several channels.
The simultaneous measurements of amplitude and frequency are made in each channel at the
end of every Nyquist interval. The channel with a maximum amplitude is selected as the
signal channel, and the output of the frequency-measuring circuit in the signal channel is
sent to the output of the whole demodulator [Akima, 1961, 1963] . The bandwidth of each
channel B is equal to twice the maximum frequency of the original information signal f ,
i, e, ,
B = 2 f = 2 B. , (15)
c -^m 1
The relation between the channel SNR R and the intrinsic SNR R., therefore, is given by
R = R./2. (16)
c 1
15-
SIGNAL-TO-NOISE CHARACTERISTICS OF
FM SYSTEMS WITH CONVENTIONAL DEMODULATORS
(CALCULATED FROM NOISE DATA WITHOUT MODULATION)
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INTRINSIC SIGNAL-TO- NOISE RATIO R;
IN DECIBELS
Figure 8
-16-
From (11) and (15) the number of channels N is given by
N = m + 1 (17)
for an integer of modulation index m.
In our band -dividing demodulator the selection of the signal channel can make errors.
If this should happen, an additional noise power will be produced at the output of the demodu-
lator besides the output noise power due to the incoming noise in the signal channel, and the
phenomenon of "modulation suppression" or "signal suppression" will also take place. In
Appendix B it is shown that the output SNR R in the band-dividing demodulator is given by
R - 3^2 [1 - (m + l)p]^
out 2 "i 1 + m(m + 1 ) (m + 2) [ 2 - (m + 1 )p] pR. ' ^ '
where p is the probability that any noise channel is selected as the signal channel by mistake.
This probability p is related to the element error rate p in incoherent N-ary FSK systems by
p = p^/(N - 1). (19)
and it is, therefore, given as a function of the channel SNR R and the number of channels N.
From the relations given above we can determine the relations between the intrinsic SNR R.
and the output SNR R ,. ^
out
When the input SNR is so large that the probability of mis -selection of the signal channel
p is negligibly small, (18) coincides with (13). We can see, therefore, that the output SNR
is the same in both the conventional and the band -dividing FM systems if they are operating
above their threshold.
The relations between R. and R in band -dividing FM systems are shown in figure 9.
Comparing this figure with figure 8, it is clear that the band-dividing demodulator requires
less intrinsic SNR tham the conventional one in order to obtain an equal value of R ,. This
^ out
comparison apparently shows that the conventional demodulator performs better than the
baind -dividing one at low output SNR, but as is discussed below, this results from the
difference in the definition of output SNR, In both (18) and figure 9 the output noise power
under modulation is used, while figure 8 is based on noise data under no modulation. If the
output noise power under no modulation is used for band-dividing demodulators, the second
term in the denominator in (18) becomes smaller by a factor (2 - Np), and it is shown that
the envelope of the curves in figure 9 moves left a little such that it coincides with the
envelope of the curves in figure 8 at low output SNR [Akima, 1963] .
Figure 9 also shows that, in band-dividing FM systems, the value of the intrinsic SNR
at the threshold is not constant but increases as the modulation index of the system increases.
Next we shall discuss the characteristics of FM systems having a frequency- lock or
phase-lock demodulators. Since a mathematical analysis of these demodulators when they are
operating below the threshold is a difficult problem [Enloe, 1962] we can only give the
following comments.
In common in these feedback demodulators, the equivalent noise bandwidth can at best
be reduced to twice the maximum frequency of the information signal f as in the band-
dividing one. It is, therefore, only necessary to compare the mechanism of loss-of-lock in
17-
SIGNAL-TO- NOISE CHARACTERISTICS OF
FM SYSTEMS WITH BAND-DIVIDING DEMODULATORS
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INTRINSIC SIGNAL-TO- NOISE RATIO Rj
IN DECIBELS
Figure 9
the feedback demodulators with the mechanism of mis -selection of the signal channel in the
band-dividing demodulator. From the above descriptions the mechanism of mis-selection
of the signal channel in the band-dividing one is obvious. If there is a noise channel whose
band-pass filter output is larger than that of the signal channel, a mis-selection of the signal
channel will take place.
Now we take into account a noise voltage in a period which is much longer than the
Nyquist interval. The noise voltage can be analyzed into Fourier components, spacings of
which are much smaller than f . If the phases of almost all the components in a specified
band of width 2f coincide approximately with each other at some instant, the amplitude of
the composite wave of these components is much larger than the average value, and the
frequency of the composite wave is nearly equal to the center frequency of the band and does
not change rapidly around that instant. As the increase of the amplitude of the composite
wave cannot take place unless the phases of these components coincide with each other, we
can see that the frequency of the composite wave does not change rapidly whenever the
amplitude is much larger than the average value. It should be noticed that the mis -selection
of the signal channel in the band-dividing demodulator takes place under these conditions.
Since there is little difference in the waveform between the composite wave in the noise
channel and that in the signal channel when the amplitude of the former is larger than that of
the latter, it seems reasonable to consider that the feedback loop will be locked to the noise
channel in such a situation. As no instantaneous phase information of the incoming signal
itself is available at any demodulator in FM systems, we cannot expect a locked-in condition
with less probability of loss-of-lock than the element error rate in incoherent multiple FSK
systems, and therefore, we cannot expect such a condition with less probability of loss-of-
lock than the probability of mis -selection of the signal channel in the band -dividing demodu-
lator. We might, therefore, suggest that the threshold of a frequency- lock or phase -lock
demodulator cannot be improved beyond that of the band -dividing one, and that the envelope
of the curves in figure 9 provides the limit of improving the threshold in FM systems.
In this paper studes are made on the assumption that the carrier is frequency-modulated
by a discrete signal. It is clear that the band -dividing demodulator does not apply to a
continuous FM system but only to a discrete one. We might, however, suggest that the
threshold of a continuous FM system cannot be improved beyond that of a discrete system
given in figure 9, because both systems can be considered to yield exactly the same informa-
tion, by Shannon's sampling theorem [Shannon, 1948] , and no instantaneous phase information
of the signal itself is available in either case.
4. SIGNAL-TO-NOISE CHARACTERISTICS OF PCM-FS SYSTEMS
In this section we shall discuss the signal-to-noise characteristics of PCM-FS systems.
In these systems an analog information signal is sampled at every sampling point, equally
spaced by a Nyquist interval, and the sampled value is quantized into L equally spaced levels,
and the quantized signal is coded into n elements with the base in the coding N. The relations
between L, N, and n is given by
L = n". (20)
The coded signal is used to frequency- modulate the carrier. It may be sent over one N-ary
FSK link in time sequence or over n N-ary FSK links by frequency division [Oliver, et al.,
1948] . As the modulating signal is quantized, either the coherent or incoherent FSK system
can be used. At the receiver the incoming wave is demodulated by an FSK demodulator or
demodulators, and the information signal is recovered by decoding the demodulated signal.
-19-
As the Nyquist interval corresponding to the maximum frequency of the information
signal f is equal to l/(2f ) and n elements are transmitted in each interval, the bandwidth
-'m m
of each channel in the FSK demodulator B is given by
B = 2nf = 2nB (21)
c ""m i
and
B = 2f = 2B., (22)
c -^ m 1
corresponding to the transmissions in time sequence and by frequency division, respectively.
The overall bandwidth of the system B is given by
B = 2nNf = 2nNB., (23)
a ■'m 1
and the relation between the channel SNR R and the intrinsic SNR R. is given by
c J s J
R. = 2nR =v2(log,r)R , (24)
1 c N c
irrespective of the two schemes of transmission.
If errors in the FSK transmission (or mis -selections of the signal channel) take place,
an additional output noise power will be produced besides the quantizing noise, and the
phenomenon of "modulation suppression" will also taike place. In Appendix C it is shown
that the output SNR R is given by
R , = |(L-1)^ ^L^:^ , (25)
°"^ ^ 1 +(L -1) (2 - Np)Np
where p is the probability that any noise channel is selected as the signal channel by mistake,
and it is related to the element error rate p in an N-ary FSK system by
p = p /(N - 1), (26)
e
as in the band-dividing FM system. As P is given as a function of the channel SNR R and
the number of frequencies N, we can determine the relation between the intrinsic SNR R.
and the output SNR R from the relations given above.
When the input SNR is so large that the probability of mis -selection p is negligible, the
output SNR R is given by
R , = |(L- l)^ (27)
out 2
This relation shows that the maximum output SNR in this system depends only on the number
of quantizing levels L and not on the base in the coding N.
20-
SIGNAL-TO-NOISE CHARACTERISTICS OF
PCM- FS SYSTEMS (N =2)
80
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INTRINSIC SIGNAL-TO- NOISE RATIO R-,
IN DECIBELS
Figure 10
-21-
SIGNAL -TO- NOISE CHARACTERISTICS OF
PCM - FS SYSTEMS (N =3)
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INTRINSIC SIGNAL-TO- NOISE RATIO R|
IN DECIBELS
Figure 11
22-
SIGNAL-TO-NOISE CHARACTERISTICS OF
PCM - FS SYSTEMS (N =4)
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SIGNAL-TO- NOISE RATIO R;
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Figure 12
-23
SIGNAL-TO- NOISE CHARACTERISTICS OF
PCM- FS SYSTEMS (N =8)
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Figure 13
24-
SIGNAL-TO- NOISE CHARACTERISTICS OF
PCM - FS SYSTEMS (n = I)
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INTRINSIC SIGNAL-TO -NOISE RATIO Rj
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Figure 14
-25-
SIGNAL-TO-NOISE CHARACTERISTICS OF
PCM - FS SYSTEMS (n = 2)
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SIGNAL-TO -NOISE RATIO Rj
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Figure 15
26-
IGNAL-TO- NOISE CHARACTERISTICS OF
PCM - FS SYSTEMS (n =3)
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INTRINSIC SIGNAL-TO-NOISE RATIO Rj
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27-
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IGNAL-TO-NOISE CHARACTERISTICS OF
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80
70
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INTRINSIC SIGNAL-TO- NOISE RATIO Rj
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Figure 17
28-
SUMMARIZED THRESHOLD CURVES
OF PCM-FS SYSTEMS
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INTRINSIC SIGNAL -TO- NOISE RATIO Rj
IN DECIBELS
Figure 18
-29-
The signal-to-noise characteristics of PCM-FS systems with incoherent detectors are
calculated and shown in figures 10 - 17. Figures 10, 11, 12, and 13 correspond to the values
of the base N equal to 2, 3, 4, and 8, respectively, whereas figures 14, 15, 16, and 17
correspond to the number of elements n equal to 1, 2, 3, and 4, respectively. An envelope
curve can be drawn in each of these figures, and these envelope curves are summarized in
figure 18, This figure can be considered to provide the lower limit to the required intrinsic
SNR (or the required signal power) in PCM-FS systems. It is clear from the figure that the
required intrinsic SNR can be reduced by increasing the base N. This conclusion is exactly in
parallel relation with the one in multiple FSK systems.
From (20) and (23) the relation
B^/B. = (2 loggL) (N/log^N) (28)
is derived. Comparing (28) with (9) in Section 2 we can notice that the comment on the band
occupancy in multiple FSK systems caji also apply to PCM-FS systems. It must, therefore,
be stressed that a ternary or quaternary PCM-FS system can achieve the reduction of re-
quired intrinsic SNR beyond a binary one without requiring a wider overall bandwidth B for
a given number of quantizing levels L.
5. COMPARISONS BETWEEN SSB, FM, AND PCM-FS SYSTEMS
5.1. General Considerations
Next we shall compare the signal-to-noise characteristics of SSB, FM, and PCM-FS
systems from the standpoint of communication system engineering, because these systems are
the most typical systems in analog information transmission.
Consider the situation in which the same information signal is transmitted over each
channel between the same two points in each system such that the same quality, i.e., the same
output SNR in the analog case can be obtained at the common destination. We then determine
the overall bandwidth and the intrinsic SNR (or the signal power) required in each channel,
calculate the cost of each channel, and compare the cost of channels with each other. A
system which requires a less expensive channel to achieve an equal output SNR is regarded as
the better system, insofar as the signal-to-noise characteristics are concerned.
If both the required bandwidth and the required intrinsic SNR (or signal power) in the
channel in one system are smaller than those in another system, the cost of the channel in the
former system is lower than that in the latter, and therefore, the former system can be re-
garded as the absolutely better system. It is, however, not the case in general. As a rule we
must make conditional comparisons by the cost of the required channel in each system. In the
following we shall make comparisons with some simple criteria.
5.2. Absolute Comparisons
As mentioned before there exists no absolutely best system across the whole range of the
output SNR even if our attention is confined to the signal-to-noise characteristics. In design-
ing a communication system, however, the minimum required output SNR is assigned at the
beginning, and the absolute comparison can sometimes be made for an assigned value of the
output SNR, In practice, for example, we would not use a PCM system when a relatively low
output SNR, say 30 db or below, is required, because we have better systems than PCM in
such a situation.
To make an absolute comparison for an assigned value of the output SNR R , it is
-30-
ABSOLUTE COMPARISONS OF FM
AND PCM-FS SYSTEMS
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FM(b-d)
10 20
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FM(b-d)
10 20
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500
cr 20
15
FM(b-d)
10 20
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500
5 10 20 50 100 200
BANDWIDTH RATIO, Bg/B,
Figure 19
500
-31-
convenient to replot the signal-to-noise characteristics in a B /B. - R. plane, where B and
ail a
B. are the overall bandwidth of the channel and the intrinsic bandwidth of the information signal,
respectively, and R. is the intrinsic SNR. tf we can observe from the figure that both B /B.
and R. in one system are smaller than those in another, we can conclude that the former
system has an absolute superiority over the latter for the assigned value of R .
In figure 19 the signal-to-noise characteristics of FM and PCM-FS systems are re-
plotted in B /B. - R. planes for the values of R ^ equal to 30, 40, 50, and 60 decibels. For
a 1 1 '^ out ^
R , = 30 db, even if a conventional FM demodulator is used, an FM system is better than a
out ■^
PCM-FS system with the number of elements for each sample n equal to or greater than 2.
For R =40 db, however, a PCM-FS system with n = 2 or 3 can be competitive with a
conventional FM system. For R =50 db, a PCM-FS system with n up to 6 can be competi-
tive with a conventional FM system, but a band-dividing FM system can still be absolutely
better than PCM-FS systems. For R = 60 db, the situation is completely different from the
above, and a PCM-FS system can be competitive with a band-dividing FM system. It must be
noticed that the absolute superiority of a PCM-FS system to a band-dividing FM system can
never be observed for any value of the output SNR R
out
5,3. Comparisons with the Minimum -Bandwidth Criterion
Sometimes the requirement for narrow bandwidth is so definite that the minimum-band-
width criterion can apply. In this case it is convenient to replot the signal-to noise character-
istics in a B /B. - R , plane, and to compare the required bandwidth ratio B /B. to obtain
a 1 out in a 1
an equal value of the output SNR R under an equal value of the intrinsic SNR R..
In figure 20 the signal-to-noise characteristics of SSB, FM, and PCM-FS systems are
replotted in B /B. - R , planes under the values of R. equal to 20, 25, and 30 decibels,
a 1 out 1
Although an SSB system requires the narrowest bandwidth among the systems compared here,
it does not achieve any broadband gain, i.e., the same value of R. as that of R is required
in this system. Our main interest is, therefore, in the comparison between FM and PCM-FS
systems.
When R. is equal to or larger than 30 db, both the FM and PCM-FS systems operate in
the above -the -threshold region, and a PCM-FS system can be better than a band-dividing FM
system for relatively large values of R . The comparison between PCM and FM with the use
of broadband gain by Oliver, et. al. [ 1948] roughly corresponds to the comparison given here.
As stated in their paper it is important to note that, as the bandwidth ratio B /B. in our
notation is increased, R , expressed in decibels varies as log(B /B.) in an FM system, while
out a 1
it varies as B /B. in a PCM-FS system,
a 1 -^
When R. is equal to or smaller than 20 db, on the other hand, the threshold in a PCM-FS
1
system takes place before a large amount of broadband gain is obtained, and a PCM-FS system
requires a wider bandwidth than a band-dividing FM system.
32-
COMPARISONS OF SSB, FM, AND PCM-FS
SYSTEMS WITH THE
MINIMUM-BANDWIDTH CRITERION
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Figure 20
-33-
5.4, Comparisons with the Minimum- Power Criterion
On the contrary to the preceding comparison the cost of the channel is sometimes
determined essentially by the signal power. A typical case is a communication from a
satellite, where transmitter power is severely limited, while the requirement for narrow
bandwidth is not so severe. To make system comparisons with the minimum-power criterion
the representation of the signal-to-noise characteristics in an R. - R ^ plane is very con-
1 out
venient, because the intrinsic SNR R. is equivalent to the signal power. This representation
has been used throughout sections 3 and 4 of this paper. In figure 18 PCM-FS systems having
different values of system parameters were compared with each other. It is an example of
comparisons with minimum-power criterion.
In each system the minimum power for an assigned value of R ^ is obtained when the
out
system is operating on the threshold curve, i.e. , on the envelope of the curves in figures 8
to 17. In order to make system comparisons with the minimum-power criterion it is sufficient
to compare only these threshold curves. In figure 21 the threshold curves of FM systems
are compared with some of those of PCM-FS systems. In this figure the characteristic of an
SSB system is also shown as a reference. As mentioned in section 3, it should be noted that
the curve for the conventional FM system drawn with a broken line is based on noise data under
no modulation, whereas other curves are base;^d on those under modulation. It is clear from
this figure that an FM system having a band-dividing demodulator is the best system among the
systems compared here insofar as the minimum-power criterion is concerned at output SNR
greater than about 10 decibels.
From (17) and (18) the output SNR R^^^^ in a band-dividing FM system can also be written
as
out
R,=|(N-1)2 il^^^P^^ (29)
°''^ ^ 1/R. +(N - 1) (2 - Np)Np
In PCM-FS systems, on the other hand, the required intrinsic SNR can be reduced by increas-
ing the base in the coding N, as clear from figure 18. In the limit where N = L and n = 1,
equation (25) for the output SNR R in the PCM-FS system can be modified as
R =|(N-1)^ ^^^^^ (30)
"""^ ^ 1 + (N - 1) (2 - Np)Np
Comparing (29) with (30) we can see the difference, as well as the similarity, between these
systems. It is noticed that the difference between these equations is only in the first term in
the denominator and the other terms are exactly the same. The first terms in the denomin-
ators in (29) and (30) correspond to the output noise due to incoming noise into the signal chan-
nel in a band-dividing FM system and the quantizing noise in a PCM-FS system, respectively.
Although the latter term is a constant, the former term is a reciprocal of R. and is smaller
than the latter whenever R. is greater than unity, or zero decibels. As relations between p
and R. are the same in these systems, it can be concluded that, in order to obtain an equal
value of the output SNR, the required intrinsic SNR in a PCM-FS system with N = L cannot be
-34-
COMPARISONS OF SSB, FM,
AND PCM-FS SYSTEMS
WITH THE MINIMUM-POWER CRITERION
0 10 20 30 40 50
INTRINSIC SIGNAL-TO-NOISE RATIO Rj
IN DECIBELS
Figure 21
-35-
smaller than the required intrinsic SNR in a band-dividing FM system with the same value of
N whenever the intrinsic SNR is larger than zero decibels. Thus the superiority of a band-
dividing FM system to a PCM-FS system is shown theoretically, too.
It is also shown in figure 21 that, even if a conventional demodulator is used, an FM
system is better than a binary PCM-FS system unless the assigned output SNR R is larger
than approximately 55 decibels.
5.5. Comparisons with the Minimum-Channel-Capacity Criterion.
Next, we shall make comparisons from the standpoint of the channel capacity.
The channel- capacity theorem due to Shannon [ 1948] states that, if the rate of trans-
mission of information is smaller than a certain value called the channel capacity, it is
possible to send the information through the channel with an arbitrarily high reliability. The
theorem also states that, inversely, it is impossible to send the information through a channel
at a transmission rate greater than the channel capacity with an arbitrarily high reliability.
Thus the channel capacity of a channel is the maximum rate of transmission of information
through the channel, and therefore, it is one of the inherent properties of the channel.
Although there have been many discussions on the rate of transmission of communication
systems [Jelonek, 1952; etc.], we shall discuss the problem in a somewhat different manner.
Here we shall not consider the question, "Which system can transmit more information through
a channel with a given capacity?" Instead, we shall pose another question, "in order to obtain
an equal value of the output SNR, which system requires a channel with less capacity?"
The logic behind the question is as follows: As mentioned above, the channel capacity
is an inherent property of the channel like bandwidth or SNR. If a channel has the same value
of capacity C as another channel, the two channels are equivalent insofar as the potential
ability of transmitting the information is concerned. Therefore, if a system requires a channel
with larger capacity than another system, the former system is considered to be more expensive
or luxurious, and hence to be poorer theoretically. In other words we assume that the cost of
the channel to be used in a system is given as a monotonic increasing function of the channel
capacity. In our notation the channel capacity C is given by
C = B^log^d+R^) - B^log/l+ ^), (31)
where R is the overall SNR at the output of the channel or the input of the demodulator and is
defined as the ratio of the signal power to the noise power contained in a band of width B .
If we take a ratio of the required channel capacity C to the intrinsic bandwidth of the informa-
tion signal B., the ratio C/B. can be considered to be a measure of channel occupancy of
communication systems. As the ratio C/B. is determined by the intrinsic SNR R. and the
bandwidth ratio B /B., we can calculate this ratio for an assigned value of the output SNR R ^
a 1 D t- Q^^
in various systems.
In figures 22 and 23 the required channel capacities in FM and PCM-FS systems are
compared with that in an SSB system, respectively. It is clear from these figures that band-
dividing FM systems are better than PCM-FS, and that an SSB system is the best of all,
insofar as the minimum-channel-capacity criterion is concerned. It should, therefore, be
C/)
CD
O
UJ
Q
3
O
a:
<
cr.
UJ
to
o
I
_i
<
3
Q.
H
O
-36-
COMPARISONS OF THE SSB AND FM
SYSTEMS WITH THE MINIMUM- CHANNEL-
CAPACITY CRITERION.
200
RATIO OF THE REQUIRED CHANNEL
CAPACITY TO THE INTRINSIC BANDWIDTH
OF THE INFORMATION SIGNAL C/B:
Figure 22
-37-
COMPARISONS OF THE SSB AND PCM-FS
SYSTEMS WITH THE MINIMUM- CHANNEL-
CAPACITY CRITERION.
200
RATIO OF THE REQUIRED CHANNEL
CAPACITY TO THE INTRINSIC BANDWIDTH
OF THE INFORMATION SIGNAL C/B;
Figure 23
-38-
recognized that the so-called broadband systems like FM or PCM do not improve the efficiency
in the channel capacity but only achieve the interchange or trade between bandwidth and signal
power to some extent
6, CONCLUSIONS
The element and symbol (or character) error rates in multiple FSK systems are
evaluated theoretically, and based on the error studies in multiple FSK systems, the studies
are made on the essential signal-to-noise characteristics of FM and PCM-FS systems, each
of which, together with an SSB system, can be considered to be one of the typical systems for
transmitting analog information signals. As results of these studies many curves for FSK,
FM, and PCM-FS systems are given as materials for designing communication systems. SSB,
FM, and PCM-FS systems are compared with each other with the use of several criteria of
system comparison.
The main results obtained in this paper, besides the design materials, are summarized
as follows:
1. The required intrinsic SNR for an assigned value of symbol error rate in multiple
FSK systems can be reduced by increasing the number of frequencies in the keying. By
using 3 or 4 frequencies the required intrinsic SNR can be reduced beyond that in a binary
FSK system without requiring a wider overall bandwidth than in the binary system.
2. When the transmission rate is smaller than the channel capacity in multiple FSK
systems, the element error rate can be made smaller than any assigned fixed value, no
matter how small it is, by increasing the number of frequencies in the keying.
3. A conceptual FM demodulator of a new type, called the band-dividing FM demodulator,
is introduced in order to study the essential signal-to-noise characteristics of an FM
system. By frequency- modulating the carrier with sampled values from the original
information signal to be transmitted and by demodulating the modulated wave with a band-
dividing FM demodulator, it is possible to improve the threshold of an FM system
beyond the threshold of an FM system having a conventional FM demodulator consisting
of an amplitude limiter, a frequency discriminator, and a low-pass filter.
4. In band-dividing FM systems, as in conventional FM systems, the value of the
intrinsic SNR at the threshold is not constant but increases as the modulation index (or
deviation ratio) of the system increases.
5. It is suggested that the threshold in a frequency-lock or phase-lock FM demodulator
cannot be improved beyond that in a band-dividing FM demodulator.
6. In PCM-FS systems the maximum output SNR depends only on the number of quantiz-
ing levels and not on the base in the coding, whereas the values of the intrinsic SNR at
the threshold can be reduced by increasing the base when the number of quantizing levels
is kept constant. By making the base in the coding equal to 3 or 4 the threshold can be
improved beyond that in a binary PCM-FS system without requiring a wider overall
bandwidth than in the binary systena.
7. In PCM-FS systems the value of the intrinsic SNR at the threshold increases with
the number of quantizing levels when either the base in the coding or the number of
elements for each sample is kept constant.
8. When the assigned value of the output SNR is equal to or smaller than 50 decibels,
an FM system having a band-dividing demodulator can be absolutely better than a PCM-
FS system, i.e., both the required intrinsic SNR (or signal power) and the required
-39-
bandwidth in the former system are smaller than those in the latter.
9. When the intrinsic SNR of 30 decibels or larger can be used, both the FM and PCM-
FS systems operate in the above-the -threshold region, and a PCM-FS system requires
a narrower bandwidth than an FM system to obtain relatively large values of the output
SNR (55 decibels or larger). Above the threshold, as the overall bandwidth is increased,
the output SNR expressed in decibels varies as a logarithm of the bandwidth in an FM
system, while it varies as the bandwidth in a PCM-FS system.
10. When the transmitter power is required to be a minimum, an FM system having a
band-dividing demodulator is always better than a PCM-FS system. Even if a conven-
tional demodulator is used in an FM system, it is better than a binary PCM-FS system
unless the assigned output SNR is larger than approximately 55 decibels.
11. From the standpoint of the channel capacity an SSB system is always better than FM
or PCM-FS systems.
The above studies indicate that the comparison of the communication systems can give
different results according to the criterion of comparison adopted. It must be stressed,
therefore, that the selection of the system or the determination of the system parameters
depends on the condition required in each case.
The selection or the design of the communication system does not depend only on the
signal-to-noise characteristics, although it is very important. As has been shown, a PCM-
FS system is not generally superior to other systems insofar as the signal-to-noise character-
istic alone is concerned. This fact, however, does not deny the possible advantages of a
PCM-FS system, but also shows us the necessity of considering other characteristics. As is
well known, a PCM-FS system is suited for a relay system with a long chain of repeaters
because of the regeneration of the signal in each repeater [Oliver, et.al. , 1948] .
In a PCM-FS system there exists a possibility of using an error-correcting code. It is
very complicated, although not difficult in principle, to study the signal-to-noise character-
istics of a PCM-FS system using an error-correcting code, because, by adding the parity-
check digits to the information digits, we must increase the keying rate to keep the same
information transmission rate, and a higher keying rate requires a wider overall bandwidth.
These characteristics should be studied in the near future.
From the studies on the signal-to-noise characteristics of FM systems in this paper
an interesting problem is raised. This problem is to study experimentally the mechanism of
the loss-of-lock in a frequency-lock or a phase-lock FM demodulator in comparison with the
mechanism of the mis-selection of the signal channel in a band-dividing FM demodulator.
This problem is not only interesting but also important in practice.
ACKNOWLEDGEMENTS
The author wishes to acknowledge the helpful discussions with W. B. Beery, R. J.
Carpenter, R. C. Kirby, G, F. Montgomery, R. W. Plush, and many others. The author is
also grateful to J. H. Devenney for programming the computation of error rates, to
J. C. Harman and his assistants in the drafting room of the Boulder Laboratories for the
accuracy of the drawings, and to Freda L'Manian for her assistance in preparing this paper.
-40-
REFERENCES
Akima, Hiroshi, Signal-to-noise characteristics of some typical systems and their compari-
sons, Proceedings of the Navy Research and Development Clinic 1961.
Akima, Hiroshi, Theoretical studies on signal-to-noise characteristics of an FM system,
(To be published),
Armstrong, Edwin H. , A method of reducing disturbances in radio signaling by a system of
frequency modulation, Proc. IRE 24,689-740 (May 1936).
Battail, G. , Determination approximative de la position extreme du seuil de reception en
modulation de frequence, IRE Trans, on Information Theory IT-8, 108-121 (Sept. 1962).
Beard, J. V., and A . J, Wh&eldon, A comparison between alternative HF telegraph systems.
Point -to- Point Telecommunications 4, 20-48 (June 1960).
Billings, A. R., The rate of transmission of information in pulse -code -modulation systems,
Proc. lEE 105-C. 444-447 (Sept. 1958).
Chaffee, J. G.,The application of negative feedback to frequency- modulated systems, Proc.
IRE 27, 317-331 (May 1939).
Choate, Robert L. , Analysis of a phase-modulation communications system, IRE Trans, on
Communications Systems CS-8, 221-227 (Dec, 1960).
Crosby, Murray G. , Frequency modulation noise characteristics, Proc. IRE 25, 472-514
(April 1937).
de Jager, F. , and J. A, Greefkes, Frena, a system of speech transnaission at high noise
levels. Philips Tech. Rev. 19^ 73-83 (Oct. 1957),
Enloe, L, H, , Decreasing the threshold in FM by frequency feedback, Proc. IRE 50, 18-30
(January 1962).
Gilchriest, C. E. , Application of the phase -locked loop to telemetry as a discriminator or
tracking filter, IRE Trans, on Telemetry and Remote Control TRC-4, 20-35 (June 1958).
Goodall, W. M., Telephony by pulse code modulation. Bell Sys. Tech, Jour. 2^, 395-409
(July 1947),
Helstrom, C, W,, The comparison of digital communication systems, IRE Trans, on
Communications Systems CS-8, 141- 150 (Sept. I960).
Jaffe, R. , and E. Rechtin, Design and performance of phase-lock circuits capable of near-
optimum performance over a wide range of input signal and noise levels, IRE Trans,
on Information Theory IT-1, 66-76 (Mar. 1955).
Jelonek, Z, , A comparison of transmission systems. Symposium on Applications of Com-
munication Theory, London, England (Sept, 1952),
Jordan, D, B,, H. Greenberg, E, E, Eldredge, and W. Serniuk, Multiple frequency shift
teletype systems, Proc. IRE 43_, 1647-1655 (Nov. 1955).
Lehan, Frank W,, and Robert J, Parks, Optimum demodulation, 1953 IRE National Convention
Record, part 8, 101-103 (Mar, 1953),
-41-
Lehan, Frank W. , Telemetering and information theory, IRE Trans, on Telemetry and
Remote Control, PGRTRC-2, 15-19 (Nov. 1954).
Lieberman, Gilbert, Quantization in coherent and quadrature reception of orthogonal signals,
RCA Rev. 22, 461-486 (Sept. 1961).
Margolis, Stephen G. , The response of a phase-locked loop to a sinusoid plus noise, IRE
Trans, on Information Theory IT-3, 136-142 (June 1957).
Martin, Benn D. , Threshold improvement in an FM subcarrier system, IRE Trans, on
Space Electronics and Telemetry SET-6, 25-33 (Mar. 1960).
Morita, M. , and S. Ito, High sensitivity receiving system for frequency modulated wave,
1960 IRE International Convention Record, part 5, 228-237 (Mar. 1960).
National Bureau of Standards, Tables of Normal Probability Functions, U. S. Government
Printing Office, Washington, D. C. (June 1953).
Oliver, B. M. , J. R. Pierce, and C. E. Shannon, The philosophy of PCM, Proc. IRE 36,
1324-1331 (Nov. 1948).
Reeves, A. H. , U. S. Patent No. 2,272,070 (F^eb. 3, 1942); also French Patent No. 852, 183
(Oct. 23, 1939).
Reiger, Siegfried, Error rates in data transmission, Proc. IRE 46, 919-920 (May 1958).
Robin, H. K. , and T. L. Murray, Electronic multiplex 32-tone teleprinter system. National
Research Development Corporation Bulletin, England, 13, 11-17 (October 1958).
Ruthroff, Clyde L. , FM demodulators with negative feedback. Bell Sys. Tech. Jour., 40,
1149-1156 (July 1961).
Shannon, C. E., A mathematical theory of communication. Bell Sys. Tech. Jour., 27,
379-423 (July 1948), and 27^, 623-656 (Oct. 1948).
Spilker, Jr., J. J., Threshold comparison of phase-lock, frequency- lock and maximum-
likelihood types of FM discriminators, 1961 WESCON Convention Record, no. 14/2
(August 1961).
Stumpers, F.L.H.M., Theory of frequency-modulation noise, Proc. IRE, 36, 1081-1092
(Sept. 1948).
Turin, G. L. , The asymptotic behavior of ideal M-ary systems, Proc. IRE, 47, 93-94
(Jan. 1959).
Viterbi, Andrew, Classification and evaluation of coherent synchronous sampled-data tele-
metry systems, IRE Trans, on Space Electronics and Telemetry, SET-8, 13-22
(Mar. 1962).
Weaver, Charles Sinclair, A new approach to the linear design and analysis of phase-locked
loops, IRE Trans, on Space Electronics and Telemetry, SET- 5, 166-178 (Dec. 1959).
-42-
APPENDIX A . AN APPROXIMATE METHOD OF CALCULATING THE ELEMENT
ERROR RATE IN INCOHERENT MULTIPLE FSK SYSTEMS WHEN THE NUMBER
OF FREQUENCIES IN THE KEYING IS VERY LARGE.
As given in the text the element error rate p in multiple FSK systems with incoherent
detectors is given by
Pe = i f(v, v^)
1 - g (v)
dv, (A-1)
where 2 2
. V -h V
f(v, v^) = vexp(^ _^ji^(v^v) (A-2)
and
g(v) = 1 - exp '- ^ j . (A-3)
When the number of frequencies in the keying N is much larger than one, an error rate of
interest occurs at v » 1, and the function f (v, v ) can be approximated by [Turin, 1959]
f(v, V ) - expf ^-^ ) . (A-4)
N-1
The function g (v) is a monotonic increasing function of v from zero to unity as v
varies from zero to infinity. It increases very rapidly somewhere depending on the value of
N-1
N, and it increases very slowly elsewhere. We assume that g (v) can be approximated by
, I exp -—
\/2Tro- -co ^ 2 0- /
h(v) = ^ ^ exp {- ^" " ^ ^ du, (A-5)
where the two constants a and cr are so determined that, when h(v) = 1/2, h(v) and dh(v)/dv
coincide with g (v) and dg (v)/dv, respectively. From these assumptions we can obtain
2 log Tt^I (A-6)
eV^log 2 '
-43-
and
log 2
1
2
^ N- 1
TT
a log 2
(A-7)
The element error rate p is then approximated by
e
°° (v - V )^ °° 2
Pe " 2-^ X -P (- ^V^)^^ .[ -P(- ^^^^^^- <^-'^
If we transform the coordinate system (u, v) to (x, y) by the relations
and
where
^^-^ = X cos e + y sine (A-9)
v-v = - X sin e + y cos e , (A -10)
s
cot'-^cr, (A-11)
the integrating range u > v is transformed to
V - a
X > x„ = ^ , (A-12)
0
J'
1 + ,^
and the element error rate p can be given by
Pe = 2.
V I ^^p ("1-)^^ I ^^p(-t )^y = ;^ I ^^p(-^)d^-
X -oo X
0 0
(A-13)
The values of x can be determined for given values of N and v from (A -6), (A-7), and
(A-12), and the values of the integral in (A-13) can be obtained from the tables of normal
probability functions [National Bureau of Standards, 1953] .
2
In figure A-1 the relations between R = v /2 and p thus calculated approximately
are compared with those calculated from (3) in the text. It is evident that the error due to the
approximation lies within 0.2 decibels for N = 1024, and that the accuracy of the approximation
is improved as N increases.
•44-
ACCURACY OF THE APPROXIMATION
FOR EVALUATING THE ELEMENT
ERROR RATE IN INCOHERENT
FSK SYSTEMS
5.0 7.5 10.0 12.5 150 17.5
SNR IN SIGNAL CHANNEL Rq IN DECIBELS
Figure A- 1
-45-
APPENDIX B. THE OUTPUT SNR IN BAND-DIVIDING FM SYSTEMS
The output SNR R in band-dividing FM systems is given by
P
out P _ -f P „
nl n2
where P is the output signal power, and P , and P „ are the output noise power due to in-
s r- o I- nl n2
coming noise into the signal channel and the output noise power due to mis -selection of the
signal channel, respectively.
In order to calculate the output SNR the modulation index m is assumed to be an integer.
' demodulator the ovei
of channels N are given by
In our demodulator the overall bandwidth B , the bandwidth of each channel B , and the number
a c
and
B = 2(m-fl)f = 2(m-t-l)B., (B-2)
a -"m 1
B = 2f = ,2B., (B-3)
c -" m 1
N = m -I- 1, (B-4)
respectively, where f and B. are the maximum frequency and the intrinsic bandwidth of the
information signal, respectively. We also assume that the characteristics of the frequency-
measuring circuit in each channel are linear within the frequency range from {f - B /2) to
(f -t- B /2), and the output voltage of each circuit corresponding to the frequency of input
wave of (f - B /2) and (f -I- B /2) are equal to -N/2 and -l-N/2, respectively, where f is
the center frequency of the modulated wave.
As will be shown later, the value of the channel SNR R at the threshold of this demodu-
c
lator is larger than 10 decibels. Accordingly the output noise spectrum of the signal channel
can be considered to be triangular [Crosby, 1937; Stumpers, 1948] and P is given by
B /2
c
-
nl
1
R
c
0
df =
B 24R 12R.
c c 1
(B-5)
If we denote the probability that the signal channel is correctly selected as the signal
channel by q and the probability that any noise channel is selected b^ mistake as the signal
channel by p, the probabilities p and q are related to the element error rate p in incoherent
N-ary FSK systems by
(N - 1) p -H q = p + q = 1. (B-6)
e
-46-
When the ith channel (i = 0, 1, 2, . . . . , N - 1) is occupied by the signal in some Nyquist
interval, the expected value of i is given by
N-1
j - i )p + iq = i + ( ~ i ]Np.
3=0
(B-7)
From (B-7) the relation
- N-1 ,, ,, V /. N-1
1 TT— = (1 - Np)( 1 - —^
(B-8)
can be obtained. As the (N - l)/2 th channel is the center channel, i - (N - l)/2 is equal to
the average output signal voltage when the ith channel is occupied by the signal. Equation
(B-8) mieans that the modulation looks as if it is suppressed by a ratio equal to (1 - Np). This
is called the phenomenon of "modulation suppression". The output signal power for full
modulation by a sinusoidal signal P , therefore, is given by
-s - I ( ^^ » - »P>^
(B-9)
because the output voltage V corresponding to the maximum frequency deviation
f, = mf is equal to
•'d -'m
V
N
m
N - 1
s max 2 m + 1 2
Next the variance of the channel number can be calculated as follows.
(B-10)
N-1
(i - i)
.2 -2
1 - 1
Kl
.2 .2
3 - 1
.2
p + 1 q
3 = 0
(N - 1)(2N - 1)
(N - 1) i + i'
Np
/ N
1 1 N p .
(B-11)
This is the output noise power due to mis -selection of the signal channel when the ith channel
is occupied by the signal. It is clear from (B-11) that the output noise power depends on the
value of i. If an equal probability of appearance of i is assumed, the output noise power due
to mis -selection of the signal channel P is given by
N-1
n2 N
N Zj
ii-n'
(N - 1)(2 - Np)Np
12
(B-12)
i=0
^M-
-47-
From (B-1), (B-5), (B-9)., and (B-12)the output SNR R ^ is given by
out ° ''
«out = h^-'^' Y-"^ • (^-^3)
1/R. + (N - 1)(2 - Np)Np
It can also be written as
R - 3 ^2 [1 - (m+ l)p]^
out 2 i 1 + m(m+ l)(m + 2)[2 -(m + l)p]pR. • ^
(B-13) and (B-14) are given in the text as (29) and (18), respectively.
The relations between R. and R are calculated numerically, and are shown as
figure 9 in the text. We can see that the use of the triangular noise spectrum in (B-5) is
supported by this figure, because the value of R. at the threshold is larger than 13 db, which
corresponds to R = 10 db.
c
APPENDIX C. THE OUTPUT SNR IN PCM-FS SYSTEMS
The output SNR R in PCM-FS systems is given by
P
^ (C-1)
out P , + P ^ '
nl n2
where P is the output signal power, and P and P are the output noise power due to
quantizing in the transmitter and the output noise power due to errors in the FSK transmission
(or mis-selections of the signal channel), respectively.
In order to calculate the output SNR we take the spacing of the quantizing levels as the
unit of output voltage.
The quantizing noise power P is calculated as [Oliver, et al. , 1948]
1/2
1/2
^^ = \ V^dV = 1/12. (C-2)
In N - PCM systems the quantizing level number of the transmitted signal I can be
expressed by
n
k=l
-48-
where i is zero or a positive integer from 1 to N - 1. The number of the total quantizing
levels L is given by
L = n". (C-4)
We denote, again, the probability that the signal channel is selected correctly as the
signal channel by q and the probability that any noise channel is selected by mistake as the
signal channel by p. The probabilities p and q are related to the element error rate p in
N-ary FSK systems by
(N - Dp +q = p^ +q = 1, (C-5)
which is the same as (B-6).
If the errors occur in N-ary FSK transmission the quantizing level number of the
received signal i differs from that of the transmitted signal I given in (C-3). The main
problem in this Appendix is to calculate the average value and the variance of i when errors
occur in the FSK transmission.
We denote the difference between i and i, by Ai, i.e.,
r t ■^
Al ^ l - H^. (C-6)
r t
We shall start with calculating the expected value of Ai when the i, th channel
(k = 1, 2, . . . . , n) is occupied by the signal in some Nyquist interval.
If an error occurs in the kth element and i, is received instead of i, while the other
''k k
(n - 1) elements are received correctly, the difference Ai is given by
Ai = (j^ - ij^)N^'\ (C-7)
As the probability that such an error occurs is equal to pq , and as every value of j except
i has the same value of probability of being received by mistake, the expected value of Ai
when an error occurs in the kth element while the other (n - 1) elements are received correctly
is given by taking a summation of the product of pq and (C-7) over the values of j from
0 to N - 1 except i, = i, . This is calculated as
•'k k
n-1
pq
\"-^ N-l N-1
2 (^-f) + \ {^H)^ = pq'""^ ^ (Ai)
n-1 /N-1 . \^^k ,^ ..
pq ( —^ ij^ N . (C-8)
-49-
The expected value of Ai due to a single error in any element is given by taking a summation
of (C-8) over the value of k from 1 to n as
n-1 Y
pq I
N
1, ^ N
Npq-
n-1
k=l
2 ty
(C-9)
We can extend the above result to the case where m elements out of n elements are
received incorrectly. If errors occur in the specified m elements, say k th, k th, . . . .,
and k th, and if a set of the specified values of i ,1 ,...., and i is received
12, m
instead of the set of the correct values of i
, ,...., and i while the other (n - m)
k^ k
1 m
elements are received correctly, the difference Ai is given by
'k ' ^k .^ + {\ "4 .^
. k -1
+ (^k -\ 7 ■
(C-10)
As the probability that such errors occur is p q , and as every value of j except i
a a
(a = 1, 2, . . . . , m) has the same value of probability of being received by mistake, the
expected value of the difference Ai due to the m errors in the k,th, k„th, . . . . , and k th
12m
elements is given by taking a summation of the product of p q and (C-10) over the values
of 1, from 0 to N - 1 except i, = i, . This is calculated as
■'k '^ -^k k
a a a
m
P q
-l----l[{s-\>'
k,-l . . k - 1 ^
m m -^
m n-m ,,, , .m-1
P q (N-1)
L \
AT^l ^ ^ / N - 1 . \J^m
N - 1
k J
m /
m
(C-11)
where each summation is taken over the value of j from 0 to (N - 1) except i . The expected
a a
value of the difference A^ due to m errors in any combination of m elements is given by taking
a summation over any combination of m elements. In this summation every value of k appears
, ! times, because the error in the k th element is accompanied by other (m - 1) errors
m - 1 y a
in (m - 1) elements out of (n - 1) elements. The sum, therefore, is given by
-so-
il
m n-m ,^^ , ,m-l / n - 1 \ V
N - 1 . N ^,k
k=l .
m n-m ,^^
P q (N
-i)-Y^:;)Nr5^;^-yiV-^^,
2 Z 'k^
k=l '
^:n[(N-i)p]--^q<"-^^-("^-^)
Np
L - 1
(C-12) ,
When m = 1, (C-12) is identical to (C-9).
The expected value of the difference Ai due to any combination of errors is given by
taking a summation of (C-12) over m from 1 to n. From (C-5) the summation leads to
A I = Np
L - 1
J.1
(^:j)[(N-i)p]-V""'^"<"^"'^
m-l
NP( V--'t
(N - Dp + q
ln-1
Np
L - 1
■t •
(C-13)
From (C-13) the expected value of the level number of the received signal I is given by
'r - ^-^^Pi S^
^t '
(C-14)
and the relation
K-^
^ = <--p>(v^)
(C-15)
is obtained from (C-14), As the (L - l)/2 th level is the center, Jt - (L - l)/2 is equal to
the average output signal voltage when the signal voltage i - (L - l)/2 is transmitted.
Equation (C-15) means that the ratio of "modulation suppression" is equal to (1 - Np). The
output signal power for full modulation by a sinusoidal signal P , therefore, is given by
P^.i(^)a..p,^
(C-16)
-51-
2
Similarly, we can calculate the variance of i from the expected value of ( Ai) when
errors occur in N-ary FSK transmission. If errors occur in the specified m elements, say
k.th, k„th, . . . . , and k th, and if a set of the specified values of i, , i, ...... and i,
12 m '^ ■'k/ ■'k' ■'k
12 m
is received instead of the set of the correct values of i, , i, ,...., and i, while the
k k k
12 ™ 2
other (n - m) elements are received correctly, the square of the difference (Ai) is given by
(Ai)^
^1-1
N +
■1/
a
N
k -1
m
(C-17)
As the probability that such errors occur is p q , and as every value of j except i
a a
(a = 1, 2, . . . . , m) has the same value of probability of being received by mistake, the
2
expected value of the square of the difference (Ai) due to the m errors in the k^th, k th,
. . , . , and k th elements is given by taking a summation of the product of p q and
(C-17) over the values of j from 0 to N - 1 except j = i^ • This is calculated as
a a a
m
P q
"■■"I --l
\ "1
k -1
N +
• + J
N
k -1 -, 2
m ]
J
m N-1
m n-m ,^^ . .m-
p q (N-1)
'{111 0.. - '0^ ""^" " ]]
a=l ,^ -0
m-1 m N-1 N-1
„ m n-m ,^^
+ 2p q (N
-^■'{1 1 [11 (x-\)(JK-0
k +k, -2
a=l b=a+l j,^ =0 j,^ =0
a
N
\-
^k -k^
Np[
(N - Dp
ill
1 (n-1) - (m-1) r V TN^ - 1 /N-1 . \
a= 1
2 -, 2(k -1)
N ^
+ 2(Np) (N - Dp
m- 1 m
■2 (n-2) - (m-2)/ V V
I
^^-i V^i^-L ^ n'^""^""
a=l b=a+l
\yv 2 k^^
(C-18)
-52-
The expected value of the square of the difference due to m errors in any combination of m
elements is given by taking a summation of (C-18) over any combination of m elements. In
this summation every value of k appears ( . \ times, because the error in the k th
•^ a ^^ 'ym - 1 y a
element is accompanied by other (m - 1) errors in (m - 1) elements out of (n - 1) elements,
whereas every combination of k and k, appears ' „ i times, because the two errors in the
•^ a b .''^''^ ^ m - 2y
k th and k th elements are accompanied by other (m -2) errors in (m - 2) elements out of
3. D
(n - 2) elements. From (C-4)the summation leads to
Np
n- 1^
m - 1 y
J L
(N - Dp
m-1 (n-1) - (m-1)
q
12 ^ ZA 2
k=l
N
2(k-l)
^2(^P)'(^-I)[(N-^>P
m-2 (n-2) - (m-2)
q
n-1
I
N
N-1 . > ^,k+h-2 '
k=l h=k+l
(C-19)
The expected value of the square of the difference due to any combination of errors
/ v2
(AS.) is given by taking a summation of (C-19) over m from 1 to n. From (C-5) the summation
leads to
(Ai) = Np j ~^ +
L^-l
k=l
N - 1
2 k
1, 1 N
n- 1
+ 2(Np)^ I ^ ^
k=l h=k+l
V - O^-'
N-1 . \ ,,h- 1
(C-20)
The variance of the level number in the received signal SL is related to (Ai) by
(i - i )^ = (Ai)^ - (Ai)^.
r r
(C-21)
■53-
2
The last term (Ai) in (C-21) can be expressed by
{Alf = (Np)^
2 / L
-^-yi,N^-^v
k=l
(Np)'
l{^—'-\Y
-,2
k=l
n
(Np)^ { 1
/N - 1 . \^,k-l
I — ^ 1, N
k=l
-.2
n- 1 n
+ 2(Np)
L L ^\ 2 k, JLV 2 hy
k=l h=k+l
(C-22)
from (C-13). The variance of i , therefore, is given by
_- 2 __ (L - DNp
^ r r' 12
L -^ LV 2
k= 1
-2
(C-23)
This is the output noise power due to errors in the FSK transmission. It is clear from {C-23)
that the output noise power depends on the values of i . If an equal probability of appearance
of each i is assumed, the output noise power due to errors is given by taking an average over
every value of i, as
■^ k
k=l i, =0
k
^^ ;,^>^p- . (i-Np)Np| l^^^^'''
~ k=l
N-1
L ^ 2 J
i=0
(L - 1) (2 - Np) Np
12
From (C-1), (C-2), (C-16), and (C-24) the output SNR R is given by
(C-21)
R , = |(L - 1)^
out 2
(1 - Np)^
1 + (L - 1) (2 - Np) Np
(C-25)
This result is given as (25) in the text.
I
-54-
LIST OF FIGURES
Page
Figure 1. Element error rate in coherent FSK systems. 5
Figure 2. Element error rate in incoherent FSK systems. 6
Figure 3. A comparison of element error rate between coherent and
incoherent FSK systems, 7
Figure 4. Symbol error rate in incoherent 32-symbol FSK systems. 9
Figure 5. Symbol error rate in incoherent 64-symbol FSK systems. 10
Figure 6. Symbol error rate in incoherent 25 6 -symbol FSK systems. 11
Figure 7. Asymptotic behavior of incoherent FSK systems. 12
Figure 8. Signal-to-noise characteristics of FM systems with conventional
demodulators (calculated from noise data without naodulation). 15
Figure 9. Signal-to-noise characteristics of FM systems with band-dividing
de modu lat or s . 17
Figure 10. Signal-to-noise characteristics of PCM-FS systems (N = 2), 20
Figure 11. Signal-to-noise characteristics of PCM-FS systems (N = 3), 21
Figure 12. Signal-to-noise characteristics of PCM-FS systems (N = 4). 22
Figure 13. Signal-to-noise characteristics of PCM-FS systems (N = 8). 23
Figure 14. Signal-to-noise characteristics of PCM-FS systems (n = 1). 24
Figure 15. Signal-to-noise characteristics of PCM-FS systems (n = 2). 25
Figure 16. Signal-to-noise characteristics of PCM-FS systems (n = 3). 26
Figure 17. Signal-to-noise characteristics of PCM-FS systems (n = 4). 27
Figure 18. Summarized threshold curves of PCM-FS systems. 28
Figure 19. Absolute comparisons of FM and PCM-FS systems. 30
Figure 20. Comparisons of SSB, FM, and PCM-FS systems with the
minimum-bandwidth criterion. 32
Figure 21. Comparisons of SSB, FM, and PCM-FS systems with the minimum-
power criterion. 34
Figure 22. Comparisons of SSB and FM systems with the minimum- channel-
capacity criterion. 36
Figure 23. Comparisons of SSB and PCM-FS systems with the minimum-
chainnel- capacity criterion. 37
Figure A-1. Accuracy of the approximation for evaluating the element error
rate in incoherent FSK systems. 44
-55-
LIST OF SYMBOLS
(The symbols used only in Appendices are not listed here. )
B Overall bandwidth of the system.
B Channel bandwidth, which is the bandwidth of each channel in the demodulator.
c
B. Intrinsic bandwidth of the information signal. In the transmission of digital
information signal it is equal to the reciprocal of the unit duration in a binary system.
In the transmission of analog information signal it is equal to the maximum frequency
of the information signal to be transmitted.
C Channel capacity.
e Base of the natural system of logarithms = 2. 71828 . . .
j^ Maximum frequency deviation in FM systems.
f Maximum frequency of the analog information signal to be transmitted.
I (x) Modified Bessel function of the 1st kind of the 0th order.
k A positive integer.
L Number of symbols (or characters) in FSK systems; and number of quantizing levels
in PCM-FS systems.
m Modulation index (or deviation ratio) in FM systems.
n Number of elements for each symbol (or character) in FSK systems; and number of
elements for each sample in PCM-FS systems.
N Number of frequencies in the keying in FSK systems; number of channel in baind-
dividing FM demodulators; and the base in the coding in PCM-FS systems.
Number of combinations of k out of N.
p Probability that any noise channel is selected as the signal channel by mistake in
band-dividing FM and PCM-FS systems,
p Element error rate in FSK systems.
p Symbol (or character) error rate in FSK systems.
R Overall SNR, which is the ratio of the incoming signal power to the incoming noise
power contained in a band of width B .
a
R Channel SNR, which is the ratio of the incoming signal power to the incoming noise
power contained in a band of width B .
R. Intrinsic SNR, which is the ratio of the incoming signal power to the incoming noise
power contained in a band of width B..
R Output SNR, which is the ratio of the output signal power to the output noise power.
u A real variable.
V A real variable.
V Normalized amplitude of the incoming signal voltage with the effective value of the
noise voltage in each channel as a unit.
•f The ratio of the circumference of a circle to its diameter = 3. 14159. . .
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