SaUonal Bureau of Stan...«

Library, »-«• ^^^^ APR 8 1963

^ecltnlcaL ^^ot& 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 Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by Ihr .Superintendent of Documents, U.S. Government Printing Office 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 (pulsecode-modulation-frequency-shift) systems are evaluated for wide ranges of system parameters, 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 modulation 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 comparison 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-codemodulation (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 demodulator, 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 bandwidth 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-tonoise 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 communication 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 characters), 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

a> CL

LU ' ^

10--3 q:

(T

o

cr

LjJ 10" 4

1-2

LU^ -5

LjJ 10

_)

LU

10'

10"

10'

- ~^--^^

1 1 1 1 1 1 I 1 1 1 1 1 -N = 1024

^512

/V256 /^I28 " ^

.fi4

- \N

- -

- N= 32-^

16-^

4/^

3>/

x'^^K^^X

'

-

-

2

—\u\\\\\\vi

"

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 lllllll 1 1 1 •5 5 10 15 20

SNR IN SIGNAL CHANNEL Rq IN DECIBELS

Figure
10"

10-2

CL

<

10-3

o

cr

(T 10-4

I-

-z. UJ

Ld I0"5

_l

LU

10-6

-7

10

10-8

ELEMENT ERROR RATE IN

INCOHERENT FSK SYSTEMS

****sJ''*\*V*^Vvx ^N = 1024 ^

^512

^^256

$^^-^128 ^^/64

N= 32^

8^^

2' ^

11

1

-5 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 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 intrinsic bandwidth B. by {!] we obtain the relation

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"

10r2

Ld 10"^

<

(T

2 10-4

a:

LU

10r5

10

v-6

-7

10

10-8

^

^

L = 32

\

N = 32 \ N = 2

\

\

\

1

\

-5 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-^

Ld

O

CD

>-

C/)

10-5

10-6

10,-7

10r8

^

^

L = 64

Y u

\

64 \e \a \n = 2

\\

\\ \ \

\\ \ \

\

\\ 1

\

-5 5 10 15 20

INTRINSIC SIGNAL-TO -NOISE RATIO Rj

IN DECIBELS

Figure 5
-11-

SYMBOL ERROR RATE IN INCOHERENT

256-SYMBOL FSK SYSTEMS

I0-'

,-2 10^

LjJ 10-3

<

crg 10-4

enLU

O

I

10-5

>-CO

10"

10-7

10' 8

\\^\ L = 256

V\ Y,

256\ \l6 \4 \n=2

^\\

\

\\ \ \

\\ \ \

\ \ \

1

^

\ \

-5 5 10 15 20

INTRINSIC SIGNAL-TO- NOISE RATIO Rj

IN DECIBELS

Figure 6
-12-

ASYMPTOTIC BEHAVIOR OF

INCOHERENT FSK SYSTEMS

10-

10-2

\^ 10-3

<

(T

o

cren

Ll)

LU

_l

Ld

10"

-5

10

10-6

10-^7

10

-8

—

1lu

N = 2

/25

'^ 2'0

^2^0

/^2^

N=280>

2i6a

2320

2640-

uv

^\^Jk

00\

lU

1 \\

1

-a

CM

in

1

._ ^ 1

-5-

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 observations 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 approximation 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 wellknown 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 zerocrossings 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 techniques 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)

80

if)

LlI GO

O

UJQ

O

hI _l

<

(/)

Z)

Q.

hZ)

o

70

- 60

50

O

cro

<

q:

UJ 40

CO

o

30

20

10

_l 1 1 1 1 1 1 1 1 1 1 1

/ X

/\ 11/

~ y^ -? — X X' — Jf -9-/ / -

- nf / /

— U y /~_ —

/ K J / j/~

- /j

yT -^Z / z

- /f / /^/ X :

—

1 lA / y/-

— l/K \ // V^ y/ - - /I hi Z,/^'/A

- j mi

-

— frf

-

—

1

- - MA1

-

— /%//

' -

/ ////// - i/////

— =,,17/7 1 1 1 1 1

1

1 1 1 1 1 II r

10 20 30 40 50

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 demodulator 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 banddividing 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

80

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1 1 1 ^mtlfi 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 20 30 40 50

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-oflock than the probability of mis -selection of the signal channel in the band -dividing demodulator. 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 information, 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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a:

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70

^ 60

50

40

30

20

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1 1 1 1

" N=2

MM MM MM

n= 12 MM

/ —

-

1 II r

10 -

-

9 —

8 -

-

7 -

-

-

6

5

—

- 4 —

1 -

—

^

2 -

~1 1 1 1 /§ n = 1 UTfmh 1 1 1 1

1 1 1 1 M M 1 10 20 30 40 50

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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70

60

50

40

30

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_l 1 1

1 ~ N = 31 1 1 1 Mil w \ \ 1 1 1 1

[

n = 8 — — — —

1

7 — — — —

il

6 I — 1

—

ffl

5 —

— m

—

— 1

4

3

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—

///// — — 11III — —

nil — —

Jllll 2 — — — — — — 1 —

— U '1 M n = 1 —

-| 1 1 1/ml 1 1 1 1 1 1 1 1 1 1 1 r

10 20 30 40 50

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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1 1 1 1 1 1 1 1 1 1 1 1

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10

INTRINSIC

20 30 40 50

SIGNAL-TO- NOISE RATIO R;

IN DECIBELS

Figure 12
-23

SIGNAL-TO- NOISE CHARACTERISTICS OF

PCM- FS SYSTEMS (N =8)

80

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1 1 1 1 1 1 1 1 1 1 1 1

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10 20 30 40 50

INTRINSIC SIGNAL-TO- NOISE RATIO Rj

IN DECIBELS

Figure 13
24-

SIGNAL-TO- NOISE CHARACTERISTICS OF

PCM - FS SYSTEMS (n = I)

80

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INTRINSIC SIGNAL-TO -NOISE RATIO Rj

IN DECIBELS

Figure 14
-25-

SIGNAL-TO-NOISE CHARACTERISTICS OF

PCM - FS SYSTEMS (n = 2)

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r - -

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10

INTRINSIC

20 30 40 50

SIGNAL-TO -NOISE RATIO Rj

IN DECIBELS

Figure 15
26-

IGNAL-TO- NOISE CHARACTERISTICS OF

PCM - FS SYSTEMS (n =3)

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10 20 30 40 50

INTRINSIC SIGNAL-TO-NOISE RATIO Rj

IN DECIBELS

Figure 16
27-

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IGNAL-TO-NOISE CHARACTERISTICS OF

PCM - FS SYSTEMS (n=4)

80

70

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1 1 1 1 1 1 1 1 1 1 1 1 N= 8

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INTRINSIC SIGNAL-TO- NOISE RATIO Rj

IN DECIBELS

Figure 17
28-

SUMMARIZED THRESHOLD CURVES

OF PCM-FS SYSTEMS

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80

70

60

50

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10 20 30 40 50

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 designing 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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30

25

20

15

30

25

20

15

30

25

R„„, =60 db

FM(b-d)

10 20 50 100 200 500

-I I—-^-I

<,'^ % ^ y^ FM(CONV.)

R„„, = 50 db

FM(b-d)

10 20 50 100 200 500

cr 20

15

FM(b-d)

10 20 50 100 200 500

5 10 20 50 100 200

BANDWIDTH RATIO, Bg/B,

Figure 19

500
-31-

convenient to replot the signal-to-noise characteristics in a Bail /B. - R. plane, where B and

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 replotted 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 competitive 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-bandwidth criterion can apply. In this case it is convenient to replot the signal-to noise characteristics 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

70

60

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1 Rj=30db

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40 -

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1 1 1 1

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FM

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BANDWIDTH RATIO Bq/Bj

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 con1 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 increasing 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 denominators in (29) and (30) correspond to the output noise due to incoming noise into the signal channel 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

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 banddividing 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 transmission 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 information 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 banddividing 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/)

CDO

UJQ

3

O

a:

<

cr.

UJ

too

I _i

<

3

Q.H

O

-36-

COMPARISONS OF THE SSB AND FM

SYSTEMS WITH THE MINIMUM- CHANNELCAPACITY 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- CHANNELCAPACITY 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 banddividing 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 quantizing 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 PCMFS 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 PCMFS 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 conventional 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 PCMFS system is not generally superior to other systems insofar as the signal-to-noise characteristic 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 characteristics of a PCM-FS system using an error-correcting code, because, by adding the paritycheck 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 comparisons, 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 Communication 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 maximumlikelihood 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 telemetry 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)

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 (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 A1
-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 ins

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 frequencymeasuring 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 demoduc 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

1R

c

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 =

(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 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

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

pq

n-1 Y

I

N

1, ^ N Npqn-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, 12m k„th, . . . . , and k th

elements is given by taking a summation of the product of p q and (C-10) over the values

of 1, from 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 \

N - 1 AT^l ^ ^ / N - 1 . \J^m

kmm /J

(C-11)

where each summation is taken over the value of j from 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
-soil 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, 12k„th, . . . . , and k th, and if a set of the specified values of i, , i, ...... and i, m '^ 'k/12'k' 'km

is received instead of the set of the correct values of i, , i, ,...., and i, while the

k 12k k TM 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 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 ,^^ . .mp 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 yJ 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 modulat 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 minimumpower criterion. 34

Figure 22. Comparisons of SSB and FM systems with the minimum- channelcapacity criterion. 36

Figure 23. Comparisons of SSB and PCM-FS systems with the minimumchainnel- 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 bainddividing 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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S. DEPARTMENT OF COMMERCE

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