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C . Division of Electromagnetic Research

(L;)RESEARCH REPORT No. EM-152 9L.

Topics in the Theory ofDiscrete Information Channels

RICHARD A. SILVERMAN and SZE-HOU CHANG

Contract No. AF 19(604)5238

A PRIL, 1960 / 0

AUG 19 %. TL &POH c Q

,/J/?PD

AFCRC-TN-60-366

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM-152

TOPICS IN THE THEORY

OF DISCRETE INFORMATION CHANNELS

Richard A. Silverman and Sze-Hou Chang

Richard A. Silverman

Sze-Hou Chang

Morris Kline Dr. Werner GerbesDirector Contract Monitor

The research reported in this document has been sponsored by theElectronics Research Directorate of the Air Force Cambridge ResearchCenter, Air Research and Development Command, under Contracts Nos.AF 19(604)5238 and AF 19(604 )3053.

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U.S. LEPARTMENT F COMMERCEOFFICE (F TECHNICAL SERVICESWASHINGTON 25, D.C.

Abstract

The present article represents the authors' contribution to the

URSI Information Theory monograph edited by J. Loeb, Vice Chairman of

Commission 6. It discusses various topics in the theory of discrete information

channels, including the general binary channel, channels with fading, cascaded

channels, and channels with memory.

Table of Contents

~Page

1. Introduction 1

2. Information sources and information rate 1

3. Channels and mutual information rate 3

4. Channel capacity 9

5. The general binary channel 17

6. Channels with fading 22

7. Cascaded channels 31

8. Channels with memory 35

References 42

1. Introduction

Many interesting aspects of information theory can be illustrated by

studying discrete channels with small input and output alphabets. In fact,

much important work has been done on the simplest of all channels, the

(memoryless) binary symmetric channel (BSC). We refer to Elias' study of

the way in which rate and probabi2ity of error in the BSC depend on the length

of the blocks in which the coding is done [1]. In the present article, we

shall be concerned exclusively with asyptotic properties of channels, i.e.

properties which are based on the assumption that coding is done in arbitrarily

long blocks. In this study, we shall sometimes consider ternary and higher

order channels as well as binary channels. In fact, the binary channel is

too simple to be representative of the general discrete channel, as can be

seen from the fact that neither input letter can be suppressed without

destroying its information rate*.

2. Information sources and information rate

The first concept of interest in information theory is that of an

information source, i.e. a device which generates a random sequence of letters

from some alphabet; the random sequence then serves as the input to a channel

(see Section 3). Let the input alphabet consist of the m letters X1, ... , xm

We shall confine our attention to independent sources, for which the probability

P 1 _< i < m, that the input letter x. is emitted at time t is statistically

independent of which letters were emitted at times prior to t and of which

*Even in the most asymmetric binary channel one does not have to use an input

symbol more often than 63 per cent (or less often than 37 per cent) of the timeto achieve capacity (see Section 5).

-2-

letters will be emitted at -times subsequent to t. Thus, the probability of

a sequence in which xi appears NI times, 1 < i < m, is Just

(p)N IyN 2 N

(Fi) 1 (p) .. (P)

According to Shannon [2], it is particularly meaningful and fruitful in

studying information sources to introduce the concept of the entropy or

information associated with a source; for a source S of the type under

consideration, this quantity is defined by

m(1) H(s) - -i P log Pi

The base to which the logarithy is taken is conventionally chosen to be 2,

in which case (1) is said to give the number of bits of information per source

letter (see remarks at the end of this section); whenever we write log we shall

mean log 2. Henceforth, it will be assumed that the source emits one letter

per second (and that the channel accepts one letter per second). With this

convention, (1) gives either the entropy of the source in bits/symbol or the

information rate of the source in bits/second. This convention allows us to use

the terms entropy (or information) and rate interchangeably; the adjustment

needed in case the source emits one letter every T seconds is obvious.

We shall not linger on the derivation of (1) from a set of properties

which it seems reasonable to expect information to have. Such derivations are

given in detail by Shannon [2], Khirchin [3] and Faddeyev [4Q. An*portant

property of H(S) is that it vanishes if the source can emit only one of the

-3-

letters xI, ..., xM, and takes its maximum value (lop m) when the source emits

all of the letters 7 ... , with equal probability. (This is in accord

with intuitive ideas of information.) Another important property of H(S)

is the fact that although S can emit mN possible sequences of length N, one of a

much smaller set of 2N H (S ) sequences is very likely to occur if N is very

large. (An exceptional case occurs if all m source letters are equally likely;

then H(S) = log m and 2NH(S) . mN.) This so-called asymptotic equipattition

property is fundamental in information theory and can be demonstrated for

much more general sburces than those studied here (in fact for any stationary

ergodic source; see McMillan [5], Khinchin [3]). Finally, we remark that the

first of Shannon's two coding theorems (the so-called noiseless coding theorem)

consists in showing that precigely H(S) binary digits (bits) per symbol are

needed to noiselessly encode the output of S into binary digits; in general

': it takes code blocks of infinite length to effect this encoding. The noiseless

coding theorem gives theoretical justification for measuring H(S) in bits/symbol

(or bits/second).

3. Channels and mutual information rate

A discrete m x n memoryless (information) channel is a probabilistic

device which accepts any of m possible input (or " transmitted") letters

xV, ..., xn and emits any of n possible output (or " received" )letters

y19 "'". yn' in accordance with the following rules:

1) For every input letter xi and output letter yj there is a

definite number Pi, 0 <p :E I 1, which represents the probability that if x,ij

is transmitted, y is received.

-4-

2) Pirery input letter gives rise to at least one output letter,

i.e.

3) The response of the device to any input letter xi is

statistically independent of its response to any Pa3t or future letter. In

Section 8 we shall abolish this requirement and consider a special kind of

discrete channel with memory.

A convenient representation of such a discrete channel is as an m *fn

channel matrix C, i.e. as the mx n rectangular array of numbers

P1I P12 Pin

(2) C P21 P22 P2n

.. "

(We shall use C to denote either a channel or the associated matrix.) By

1) and 2) each Pij lies between 0 and 1, and the sum of every row of the matrix

(2) is unity; these properties are summarized by calling C a stochastic matrix.

The study of stochastic matrices has received a great deal of attention in the

mathematical literature, especially in connection with the theory of Markov

chains (see Feller [6]). We note that the case of a noiseless channel corresponds

to the case where each row contains one 1 and(m - 1O's, while ea6h column

contains one 1 and 6 - 1)0's.

Suppose now that the input letter xi, 1 < i < m, is used with probability

Pi and define p(ij), the Joint probability that xj is emitted and yj is

received, by the relation

p(i,j) = Pi P

Then the probability that yj, 1 < j _n, is received, regardless of which xi

is transmitted, is given by

P; p(iJ)

It is convenient to represent both the numbers PiV 1 < i < m, and the numbers

Pj < j < n, as the components of corresponding column vectors P and P', i.e.

P1 FPIP P1

- P2 P' P2

p p'

P I Ptmj n

We now give a fundamental definition introduced by Shannon [2]: The

average mutual information rate (or simply the rate) of the channel C when

it is driven by a randr sequence of input letters chosen independently *

with probabilities given by the vector P (equivalently, when it is driven by

*It can be shown that for a memoryleis channel any dependence between input

letters diminishes the rate of the channel (see Feinstein [7]).

-6-

an independent source characterized by P ) is

(3) R(P ) p(i,j) log IPi Pi

bits/econd. (Recall that by our convention one inlut letter per second is

emitted by the source and accepted by the channel.)

To Justify (3), we specialize to the case where the input and output

alphabets are the same (so that in particular m - n) and use Shannon's

correction channel argurient [2], which asserts that

(4) correction rate* + mutual information rate = source rate .

The source rate is of course just

Pi log P,

The correction rate is derived as follows: Whenever the letter yj is received,

it may be correct or incorrect. Since the probability that y, originated from

2. is p (i) - p(i.,J)/P', I < i,J < m, whenever y is received we must supply

an amount of entropy

m

" i pj(i) log pj(i)

to correct it or to leave it stand uncorrected with the assurance that it is

*More precisely, the average rate of correction entropy.

-7-

correct. Since the letter y is received with probabilityP' the average

rate at which correction entropy must be supplied is

pi(i) log pj(i) =- iJ) log

J PJ

Since

m; pi =1 ,

we can also write the source rate as

m m m 1

Ppi lo P, p(iJ) log Pi

Finally, using (4), we obtain

mutual information rate 3 R(P )

m m log m m - P(ij)

P I

- ~p(i,J) logij)loP p

iiJ

which agrees with (3) for the case m = n.

L what follows we shall find it convenient to introduce the symbol

< > (angular brackets) to denote averaging with respect to the joint

probability distribution p(i,J), i.e., if f(iJ) is a function of the two

-8-

integral arguments i and J, where 1 < i <im, I< j <n, then

= p(i~)V f(i~j)

With this notation (3) becomes

R(P) -<log P(i)> + <log pj(i)> =<log , 1, >.

As an example of (3), consider the general binary channel

(5) C(aop) = I , o<a, P<1

fed bya source producing O's and l's independently, with probabilities P0 and

P1 = I - P' ' respectively. (Eq. (5) means that transmitted O's are received as

Ofs with probability a, while transmitted l's are received as O's with probability

P.) Then it is easily verified that the rate associated with (5) and the input

vector

-> P- 0

P,

is

(6) R(a ,p; P )-aPolog (a/P') + (1-a ):og((1-a)/P )+pP 1 kg(p/P + (1. )F 1o((l.+)Ppl)g,

where aP + P(1 - P.) is the probability of a received zero and 1 - Po

is0 0 0is the probability of a received 1.

-9

4. Channel capacity

Following Shannon [21, we define the capacity c(C) of the (discrete

memoryless) channel C as the largest value of the rate wrhich is achieved when

the input probabilities are varied over all possible values, ie°

Max(7) c(C) - R(P)

P

As an example consider the BSC where the channel. matrix

C - asym a

is obtained by setting 1 i - a in (5). In this case it is clear from

syrnetry (and it can be verified by direct calculaion) that capacity is

achieved for the choice P PI =1/2, ftich implies Pt = = 1/2 as well.

Substituting these values in (6) and using 1 = - a, we find

c(Csy m 1 + a log a + 0 log

for the capacity of the BSC.

The fundamental importance of the channel capacity as an information-

theoretic quantity stems from the role that it plays in Shannon's second

coding theorem (the so-called noisy coding theorem), which it is safe to say

contains most of the substantive content and technical promise of information

theory (taken together with corresponding studies of finite-length block coding

like [l]). This theorem asserts that with proper encoding it is possible to

transmit information at any rate less than capacity with arbitrarily small

- 10 -

probability of error, provided that the block length of the code is long enough,

and furthermore that regardless of the encoding scheme, errors will always be made

if one attempts to tranmit inforation at a rate greater than capacity. Much

space has been devoted in the literature to a rigorous demonstration of this

theorem for an appropriately large class of sources and channels, and the whole

subject is a difficult one which we shall not go into here. The interested

reader is referred to the papers of Khinchin [3] and the book by Feinstein [71;

the latter author played an important role in developing a rigorous proof of

the noisy coding theorem.

We turn now to the question of how the mathematical operation symbolized

by (7) is to be carried out in general. This operation involves more than a

simple maximization problem, since the vector P in question is subject to the

constraint that it be a vector with non-negative camponents which add up to

unity. We begin by writing (3) in a form which explicitly exhibits its

dependence on the input probabilities P.; this forn is

-> m n m m n

(8) R(P)m <log P >+ < o i>P 1

To incorporate the constreint*

mm P!-I ,

we addm

*For the time being we neglect the additional constraint that Pi > , < i<m ,

which will be discussed below in connection with Muroga's work.

- 11 -

to (8), where I is an undetermined Lagrange multiplier. Differentiating the

sun- of these two terms ith respect to P I < i < m , and equating the result

to zero, we obtain

(9 i 09.m! < i <.M

(Z Fpij

where i = X - (1/log e) is a new constant. Multiplying (9) by Pi and summing

over i, we find that

where c is the capacity of the channel C.

Suppose now that the channel C under consideration, with channel matrix

(2), is square (m = n) and that det(C), the deteninant of (2), is non-vanishing.

Then the inverse matrix C l , satisfying the matrix equation

(10) CC-1 . -lC 1(ii) cc c" c -I

(where I denotes the unit matrix) exists. If we denote the elements of C-1

-1by pj , I< i, J <m , then (0) reads

m i - 1 m -l <T

J-iPiJPJk PijPJk - 6ikk

where 6ik is the Kronecker delta symbol, equal to 1 when i = k and 0 when

i # k. Multiplying (9) by p1 and summing over i, we obtain

mm mpi lo1 1 < k < m

P Pjlog 1,j 9; 10Pipi II ki

- 12 -

Hence

mP ~excP~ 1 lgpj 1 <k<m,

where by exp(x) we mean 2x , the base appropriate to our choice of units.

Multiplying by -1 and summing over k, we obtain

Pkcj

(1)P - exp{c 4- +. - log ,i) 1 < i < m

Finally, following Muroga [8], we note that

(12) P; P;

so that (ii) can be simplified to

(13) P, 1 iixpm -c +j ogPij < m

Thus., the channel capacity is the number c which when substituted in (13)

gives Pi > 0 and

mPi - 1

Then with this value of c, (13) gives the corresponding rate-maximizing

values of the input probabilities Pi, 1 < i < m . Explicitly, we form the

sum

SUi

and use the relation (12) again, obtaining

- 13 -

(1 ) _og0 + e x p p l P j l o g P ii j j

whore i% will be recalled that both log and exp are taken to the base 2.

Computations are simplified by defining (after Muir~ga [83]) the auxiliary

veotor

xl-> x2

xm i

wh ,ch satisfies the matrix equation

~pi loglo P

(aS) -fo c) -ge

m

- ( p1)og p

-14 -

It follows from (15) and (16) that X -C-H , i.e. that

so that in terms of the components of X , (14) becomes simply

m

(7) c = log exp Xi

Moreover (13) becomes

m -

P. - exp(-c) P expXk, <i<m.

Since

P;i = cof(pLI,)/dtc),

where cof(pik) is the cofactor of the element Pik' we finally have

I'll .Plm

p.8) exp(- det exp(X1) exp(X)I det(C) .

where the matrix in (18) differs from the channel matrix C by having the

entries exp(X1 ), ... , exp(X ) instead of Pil' "*" pim in the i'th row.

The method just described requires modificat_ on if it leads to negative

input probabilities, and more generally if det(C) - 0 or the sizes of the input

arn1 output alphabets are different. Details of how to deal with these various

cases are given in Muroga's paper [8], to which the interested reader is

referred. There is also available a different and more easily visualized

approach to the general problem of capacity due to Shannon [9].

We conclude our present discussion by discussing a one-parameter

family of ternary channels which for suitable values of the parameter

leads to a negative probability for one of the input symbols, which must

therefore then be suppressed according to Muroga. Consider the channel

a I-a 0

1 1S= 2 T

0 1-a a

where 0 < a < 1. The inverse matrix isI I

C-I 1 a 1S 2(l-aj =2-

1 1

The row-entropy vector is

H

H 1

H

- 16 -

and the vector X is

->

-1

where

-a log a - (1-a) log (1 -a)

Using (17) and (18) we find that the snbol corresponding to the middle row of

C should be used rith the probability

1 + qexp - -

p=

+ exp (- )

if capacity is to be achieved. This symbol should be suppressed when the

numerator goes negative. This happens when a > ao, where a is the solution

of

H()- "ao0 0 log 0

1-a 0 a

or

(19) log ao a 0

The solution of the transcendental equation (19) is a o'o 0.641. When p > O

the capacity of the channel is

log { + exp bits/second

sell?

whereas when p < O, the channel matrix reduces to

0 l-a a

This is the matrix of the binary erasure channel (BEC), i.e. the channel in

which U's and l's are transmitted and O's, lts and x's are received; a is the

probability that a 0 or 1 is received correctly and 1-a the probability that

a 0 or 1 is received as an x. Since received O's and l's are always correct,

whereas a received x is always wrong and equally likely to have come from a

0 or a 1, the capacity of the BEC is obviously a bits/second. It is interesting

to note that the capacity of C is 1 bit/second both when a - 0 and a - I, but

when a - 0 the symbol corresponding to the middle row should be sent, whereas

when a - 1 it should be suppressed.

. The general binary channel

Using Muroga's method, Silverman [10] has made a detailed study of the

general binary channel C(asp) defined by (5)*. We sketch without proof some

of the results obtained in his paper:

1) The capacity c(a,P) of the general binary channel is given by the

formula

- a ) + log 1 + exp(H(a) - H(P)

The function c(a,P) has the symetries

c(a,P) = c(Pa) - c(l - a., 1 -P) - c(l - , 1 - a) ,

*ILeb [ii] has also studied some aspects of the general binary channel.

- 18 -

and defines a surface over the unit square 0 < , I <1. Lines of constant

C(a.0) are shown in Fig. I

2) Capacity is achieved if O's are transmitted with probability

PO(U,,P) - m(p.a)- .(P-)-i I + ex

Po(ao1) satisfies the relation

O.37 < (as) < 1 - 0.63 ,

which explains the footnote in Section 1, The function P0(asp) is discontinuous

at the points a - 0 and a l it has-the symmetries

Po0(al)- P 0(1.-a, i-P) i -1 P,) - 1 - Po(U-P, 1-)

Lines of constant P (asp) are shown in Fig. 2.

3) At capacity, O's are received with probability

PO(a,) 1 + exp (-)1

he function Po(a,3) has the symnetries

P Io(al, "'(p. o P 'o(1-4, 1-0 - -'U-ps 1- •

Lines of constant Po(sp) are shown in Fig. 3.

The reader interested in other properties of the general binary channel,

e.g., thfe form of the channels giving the maximally asymmetric input probability

distributions (Por-j I/e or Po --, l - l/e), the probability of error for the

0{

S1~9-

00 01 02 03 00. 0. 07 0.0.1.

09a

Fi.10 ie f osat 8,)

20 -

1.00.41

0.43

0.450.9 0.46

0.47

.480.8

0.49

0.7

O=Q49750.6

U) to Ul)N C-i0.5-- 0 0t- cli U') LO N

U U) LO c; U inC) 1 11

0 aP 0a) to F. j

04 to Ul)cj c;

PO=Q4975

0.3-

0.2

..... ..... .. 0 4 8

0.1 10.460.45

0.430.41

0.1 0.2 0.3 0A 0.5 0.6 0.7 0.8 0.9 1.0a

Fig. 2 - Lines of constant PO(ap).

21k1.0

0'

0.9 0 0

0.80

0.70

0.6

0.5-

0.4-

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Q7 0.8 0.9 1.0a

Fig. 3 -Lines of constant P0(a,fi).

- 22 -

general binary channel, etc., is referred to Silverman's paper [IO].

6. Channels with fading

The communication situation dealt with so far can be indicated schema-

tically by the diagram

T -> C -> R,

i.e., information is sent from a transmitter T to a receiver R through a channel

C, characterized by a stochastic matrix (see Section 3). (We make no distinction

between the primary information source S of Section 2 and the transmitter T,

although in general there is the problem of (noiselessly) encoding the output

of S so as to match the input of T.) We now consider the following generali-

zation of this situation Instead of one (memoryless) channel C, let there be

a family of (memoryless) channels Ca where the index a ranges from I to s,

and let the channel which is actually present at a given time of transmission

depend on the state of a random device N (N for "nature"). Schematically, we

have

N

T -- C -- ;- R,

i.e., information is sent from the transmitter T to the receiver R through the

channel C, the state of which depends on the state of nature N. We assume

that the state a, 1 < a < a . chosen by nature is statistically independent

of past and future states of N and of the transmitted sequences as wellj let

Pc be the probability that nature chooses the state a. The channel C is now

represented by the family of stochastic matrices

- 23 -

pla)pl2(a) .... Pna

P21(a) P22(a) """0 P2n(a') ~

pml(a) p62 (a) .... pa(a

Since the transition probabilities Pij(a) are now random variables, a new

element of randomness has entered into the problem. Such a model might be

used to give an abstract representation of communication in the presence of

"fading",, and with this in mind we refer to the totality of channels C.,

1 < a< s, as a channel with fading.

A natural problem in the theory of channels with fading is that of

finding the channel capacity for various conditions of knowledge of N at

the transmitter and receiver. There are four possible cases, which can be

schematically represented as follows:

N

'ICase 1. T "C -. R

N

Case 2. T -R -R

N

Case 3. T -IC -- R

N

Case 4. T -- C -- R

- 24 -

Case 1, which we have already encountered, represents the situation in which

neither the transmitter nor the receiver knows nature's state, so that N is

effectively just more noise in addition to that already included in C, Case

2 is the situation in which the receiver but not the transmitter knows nature's

state. Case 3 is the situation in which both the transmitter and the receiver

know nature's state, and Case 4 is the situation in which the transffitter but

not the receiver knows nature's state. We now give expressions for the

capacity in all four cases.

Let p(c,i,J) be the joint probability that nature chooses the state a,-that

x, is transmitted and that.7, is received. in cases 1 and 2, where the

transmitter is ignorant of nature's state, we have

(20) p(ai, J) - p, PpJ(a)

In case 3, where the transmitter's action can depend on nature's state, we have

where Pi,, is the probability of choosing xi, given that nature's state is a.

(Case 4 requires special treatmentj see below.) Generalizing the angular

bracket notation of Section 3, we write

a m n

We now derive expressions for the channel capacity in the four different cases.

Case 1. Since neither the transmitter nor the receiver knows nature's.

I

- 25 -

state, the channel has the same capacity as the average channel

5

Z P.

with transition probabilities

5

Explicitly we have, from (8)

Max PJcI - - log

P ~ Pjij

for the capacity in this case.

Case 2. Since now the receiver can use its knowledge of the pair

(c,,yj) to infer xi, we have, from (3)

max < p(asisJ)

(22) c2 - P og:I Pip(a,J)

for the capacity in this case, where p(ca,J) is the probability of the pair

(ayj)s i.e.

p(Ui) -0 p(wij)

Using (20) we can rewrite (22) as

- 26 -

(23) C C2 l:, log PjG

Let RV(P) be the rate in the channel C when using the input vector P . Then

(23) is just

Max a -(24) C2 a ;I--p E. p~a

In other words, the capacity in case 2 is the maximum average rate when

driving an the channels Ca with the same input probability vector P. Clearly,

c2 Z cl , since we are now using more detailed knowledge of the channel.

Case 3. Now the transmitter can base its choice of input probabilities

on nature's state, choosing a suitable input vector

p1 ,SPlg

Pa = P

for each state, 1 < a < a. Moreover, an in case 2, the receiver has the pair

(ap,) available to infer xi, . Therefore we have

PCpI

- 27 -

for the capacity in this case, where the maximization is over all PF,

1 < <a . Using (21), we can rewrite (25) as

Max s

%RPa

In other words, the capacity in case 3 is the maximum average rate when

driving each channel C with its own input probability vector P. * It follows

at once that

5

(26) 03 - p c(C)

i.e., the capacity is just the average capacity of the channels C. ,

1- <a < s. Clearly c3 > c2 , since now the transmitter as well as thwe

receiver is using the extra information about the channel.

Case 4 is more complicated. In this case Shannon [12] has shown that

the capacity of the channel with fading is the game as the capacity of a new

me x n (non-fading) channel C', with input letters consisting of the mS

s-tuples (xi , Xi, ... , xi )2 1 < il, 2? ... , i s. <m and output letters

consisting of the n letters y 1p 1 < J <n, where the transition probabilities

for C are defined by

a

Thus the capacity in this case is

c4 - c(C')

- 28-

and clearly c. 3 j 0 c3 . We now give two examples which illustrate the

various cases.

ExaWple 1. Suppose nature has two states 1 and 2 with probabilities

a and P, respectivelyp and suppose

1 -1 02.11 010 1 1

Then the average channel 6 is the BSC, i.e.

a. c1 + PC2 ' p0 l

so that c - 1 + a log a + P log P bits/second (see Section 4). If nature's

state is known by the receiver or by both the transmitter and the receiver,

then the capacity of the fading channel is obviously 1 bit/second, achieved

by transmitting 0's and l's with equal probability and interchanging O's and

l's at the receiver when the channel is in state 2. Eqs. (24) and (26) confirm

that c2 c 3 = 1. When only the transmitter knows nature's state, the

capacity is again 1 bit/second, achieved by transmitting O's and l's with

equal probability and interchanging O's and l's at the transmitter when the

channel is in state 2. In this instance (case 4) Sharnon's construction asserts

that 64 4 c(C'), where

0

-29-

Denoting the rows of this matrix by the vectors A1 , ... , A4 (e.g. A!,-

we have

-> -> ->

A 4 =A 2 + aA3

It follows by an argument due to Shannon [9] that the symbols corresponding

to the first and fourth rows of C have to be suppressed if capacity is to be

achieved. Dropping the first and fourth rows of Cwe get the matrix C1 ,

so that c 4 - c(Cl) 1 bit/second, as required.

Example 2. Let there be three states with equal probability, and let

101 1 1 12 2 2 2

1 1 1 s ' 2 1 1 3 0 1

Then if neither the transmitter nor the receiver knows etcirets state, the

capacity of the fading channel is that of the average channel

2 1

1 2

i.e. cl - c(U) =1 + (1/3) log (1/3) + (2/3) log (2/3) - (513) - log 3 '.082

bits/seoond. If only the receiver knows nature's state, then by (24)

max I~--c2 - [IP)+ R2(p) + R3 (P

p

-30-

By symmetry, the maximum is achieved for

Pm 2

2

and an elementary-cal.culation shows--that,-c - _ 1- (1/2) log 3 . .208

bits/second > c 1 A If both the transmitter and the receiver know nature's

state, then by (26) -

3 c(CI)

since c(C) c(C 3) and c(C 2 0. Applying Muroga's method (see Section 4),

we easily find that c(CI ) -log5 - 2 , so that c u-log 5 - -j .2153 '3 3

bits/second > c2 *

If the transmitter but not the receiver knows nature's state, then

according to Shannon's construction, c4 - c(C'), where

2 1

1 1

2 1

1 2

7 T1 2

Driving any binary channel with a source emitting O's and l's with equalprobability leads to a rate very near capacity, which explains why 03 isso near c2 (see footnote to Section 1).

- 31-

By a simple extension of the argument given in connection with the preceding

example, we have

c4 c(C') -( - c .' .082 bits/second ,

so that in this case, unlike example 1, the transmitter cannot use its

knowledge of nature's state to increase the channel capacity.

7. Cascaded channels

Let C be an x m channel matrix with output alphabet yI "' Ym ' and

let C' be an m x n channel matrix with input alphabet yY """ ym (i.e.

identical to the output alphabet of C). Then we agree to apply yi to theI

input of C whenever Yi is received at the output of C; this mode of channel

combination is called cascading. Denote by Pij and piJ the elements of theI It It

matrices C and C , and denote by Pi the elements of the matrix C obtained.

by cascading C and Co, as just described. Nhen, since obviously

M

PIj . PLkk

C is obtained from C and C' by matrix ultiplication, i.e. C" Cc ' Since

matrix multiplication is in general non-conuutative, the same is true of

channel cascading.

a 1-aThe problem of cascading identical binary channels C(adp) -l - is

especially simple, in view of the identity

a (') 1-.a(n) 1 f 1.4 C-1 la 4(27) C(ap) a (n) ,n) - 1.

P P P

- 32 -

which is easily derived from a general formula of matrix algebra (see e.g. [13]).

(We exclude the trivial cases iul, P-0 and a0, P-., corresponding to the

noiseless channels and 1 .) A number of simple consequences can

be derived from (27):

1) Since ja - pI <1, we have

(28) - n(, ) .1 C- ( o) (l. ( o )

n--Io C(,) C (a,) (M -- (0

The limiting channel C(0°)(C,P) obviously has zero capacity, in accord with

the intuitive idea that an infinite cascade of noisy channels must destroy

any information fed into it.

2) Since

(00) p(n) p(00) p.

(OY (n) (o C-DT'

all the channels Cn(ap) lie on a straight line in the (as) square passing

through the point corresponding to the limiting channel C(@°)(a,) and the

point (0,1). Thus, referring to Fig. I, we have the following simple interpre-

tation of the operation of cascading a binary channel with itself: Using (28),

draw the straight line just described; then as n increasee, Cn(a,p) approaches

0(D) (aP) along this straight line, moving alternately from one side of the

zero-capacity line.a - to the other if a - p < .

The operation of channel cascading, or equivalently of multiplying

channel matrices, is a partial ordering in the following sense: Given any two

channels C1 and C2, then C1 is said to include C2, written C1 _ C2 , if there

- 33 -

exists a channel C such that C1 C = C2 , i.e. if C2 can be obtained from

C1 by cascading. Channel inclusion has the defining properties of a partial

ordering, namely

1) C - C , for anyC ;

2) C 1 aC 2 , C2 C 3 implies C C3

On the other hand, given arbitrary channels C1 and C2 , neither of the

relations 1 :_ C2 , C2 a C1 may hold, i.e. C1 and C2 may not be comparable

In the case of binary channels, the structure of the partial ordering

is very simply displayed. Suppose we have two binary channels

- C

S -l - ' l ne' '

and assume, as we can without loss of generality, that <a, a' <a'. Then,

following a procedure suggested by Birrbaum (private comunication), we first

prove the following lemma:

A necessary and sufficient condition for C > C' is that the interval

(L.,L 2) contain the interal (I4 ., where

I , L. a'/=' , . (l..p)/(l-) , i -

(Note that in our case 0 < L, <l<,< , o <o 0 < 0.)

To show the necessity, we suppose that

J l-xC" - l-y

*If c(C) denotes the capacity of C, then c(C) > c(C') is a necessary but not

sufficient condition for C Ole

-34 -

is the channel such that C" C c'. Then, doing the matrix multiplication

explicitly, we find that- '/( ' - - y > ,/ -

Li ~ ax + (1-07 >

'since y > 0, and

L'* (1-P')/(-a') 1- U -,(-#)y < (3.-A)/(l-a) L22 -ax - (l-4)y

since x < 1.

To show the sufficiency we solve fbrmally for the parameters xy of

C", finding

a + y

Since a'- ' >0, a-p > o and z'- 'lp >0 by the hypothesis that L1 _<I ,

we see that 0 <y < x. Moreover y x < 1, since a' - l' +u1'-P <a -

is an easy consequence of the hypothesis i4 < L2 . This completes the proof

that C C' and (1,L) ' (14,) are equivalent statements.

Using the lemma, we can give a simple geometrical model of the partial

ordering of binary channels under cascading. Plot the point (a ,P) corresponding

to C in the unit square, and draw the straight lines from the point (a,p)

to the points (0,0) and (1,1), as shown in Fig. 4. Then all the points in

the region shaded with vertical lines represent channels contained in C,

i.e. channels which can be obtained from C by cascading. This follows by

applying the lemma in the part of the region lying below the line a-P, and

then noting that multiplication by 1 changes a channel into its reflection

in the line amp. Moreover, it follows from this construction that all points

Ii

-35 -

in the region shaded by horizontal lines represent channels containing C.

Similarly, points lying outside of the shaded regions represent channels

which are not comparable with C, i.e. which cannot be reached from C either

by premultiplication or postmultiplication by any binary channel.

Fig. 4 - fllustr. -ing the partial ordering of binary channels.

For further discussion of topics related to channel cascading, see

Shannon [14, Birnbaum [151 and Silverman [30].

8. Chamels with memory

We now consider a simple idealized communication system suggested by

Chang [161, which leads naturally to the study of channels with memory.

Let the channel input be a scarce emitting binary digits, and let the channel

be such that O's and l's emitted at the time t = 0 am represented by the

waveforms

fo(t) - A exp (-t/) ,

f1(t) - .a imp (-tl) ,

- 36 -

respectively, in the absence of noisee* The effect of noise will be repre-

sented by the addition of white Gaussian noise of r.m.s, voltage a.

Suppose, to consider the simplest non-trivial case, that the source produces

the digits in isolated doublets, i.e., let a pair of binary digits separated

by an amount y < V be transmitted, and then let there be a pause in

transmission long enough to allow substantially complete decay of the

exponential exp(-t/v). Finally, suppose that we use synchronous, threshold

detection at the receiver, i.e., at thi times t and t o + y corresponding to

transmission of a doublet, we interpret a positive voltage as a 0 and a

negative voltage as a 1.

We now proceed to find the capacity of this simple channel. At a time

t 0 such that the channel has recovered from the effects of previous transmitted

signals the channel is described by the matrix

1

where

a - Prob [f 0(0) + g > O]

= Prob [f 1 () + t>0] ,

2and t is a Gaussian random variable with mean 0 and variance a' 2 (Specifically,

*We have in mind, for example, a situation where C's and l's are encoded intosharp positive and negative pulses of amplitude A and then sent through achannel whose transmission characteristics resemble those of an RC filterwith time constant %, .

**By " white" noise, we mean noise with a constant power spectral density, orequivalently with a correlation function which is a delta function. Gaussianwhite noise has no memory. This is, of course, a limiting case, approachedonly when the noise band width is much greater than that of the signals.

37 -

a is the probability that a 0 transmitted at time to is interpreted as a 0,

and is the probability that a 1 transmitted at time to is interpreted as a 0.

Similarly, a62 Po and a.,, P are the corresponding probabilities at time

to + y, under the hypothesis that a 0 or a 1 was transmitted at time to,

respectively (see below) ). Clearly we have

a Prob ( > -A) 1 1 - F(-A) ,p.Prb > A) - -F(A)

where

F6x) - - eU 1/2d du*dio' -2

is the distribution function of the random variable . It follows from the

symmetry of F(x) that a + l 1, i.e. that C is a symnetric channel. At the

time to + 7 , there are two possibilities. If a 0 was transmitted at time

to y then the channel is described by the matrix

Co a 01-4PO - po

where

0- Prob [fo(O) + f(T) + t > o] - 1 - F[A(lp)]

00 =Pro" [fo (0) + fl(y) + t >] - I - Fi--A(1-p)]

and p = exp(-=/,V). Clearly we have a + po > 1 and CO is asyumtric.

Similarly, if a 1 was transmitted at to, then at time to + y the channel is

- 38 -

described by the matrix

-c1 - -a

where

Ia Prob [fr(o) + fo(y) + E > 0] - 1 - F[A(l-p)]

Pa Prob [fl(o), fl(y) + E >0] - 1 - F[A(l+p)]

This time al + Pi < 1 and again CI is asymmetric.

To find the capacity of the channel with memory, it is simplest to

proceed on a doublet basis and enlarge the channel to a new 4 x 4 memoryless

channel C', with inputs and outputs consisting of the four pairs 00, 01, 30

and 11. Thus, we consider the 4 x 4 channel matrix

ac0 (1-a)C 0

PC1 (1-P)c 1

whose elements are themselves 2 x 2 matrices. Following Chang [16], we

find the capacity of C' by using Muroga's method (see Section 4). The

row-entropy vector H' of C' is easily seen to be

H(a) + H( o ) -

0¢> H(Ga)U + NoB'aH(Pi) +H(a) 01NP

H(P) + 10)

- 39 -

where U denotes the vector and H, H, are the two-dimensional

row-entropy vectors of the channels Co0 CI, respectively. Now we must find

the auxiliary vector X' satisfying

X -- H' .

Since C' can be written as

01 C 0 0 1-0

0 C1

we have

~w1 1 1-P C-i 0C

and

- a- IH(P) U + Ho.- > -i>

-pc 1-pCL-1 0 H(+CL -P)U +H1 0 0

I -- -> --

Since CL 0 U we hav

.(iC)(H( 1)( xo ) H ()-())(H(P)U - c H,)(29) Z' =-P -> -> -> ->

-1 1 1 H( - o

-H(>)U -

40 -

where we have introdugqd the two-dimensional auxiliary vectors Xop ,

corresponding to the channels C0 , C1 , respectively, i.e., satisfying the-. > - > -> -1 -m

equations X -C00 HO and X, - -C1 H.1 . Then, if X is the auxiliary vector

corresponding to the channel C, i.e. satisfying the equation

-> xo IH(a)

Xl1 H(O)

it follows -from (29) that

-> x- cX I 1(30) XX 0

Eq. (30) expresses the four-dimensional auxiliary vector Xt of the expanded

4 < 4channel CI in terms of the two-dimensional auxiliary vectors X, Xo

and X:. of the binary channels C, CO and CI . Finally, the capacity of C in

bits/doublet is obtained from (30) by using Eq. (17). Thus, the calculation of

X and c(C ) is a simple matter, provided one has available a table of the

values of the auxiliary vector X corresponding to the general binary channel.

Such a table is given in Chang's paper [16], to which the reader is referred

for further details.

The simple example just given, where isolated doublets are transmitted,

illustrates the general approach to the problem of determining the capacity of

channels with memory of the type under consideration. More generally, one can

1-

consider the case of isolated groups of n equally spaced digits, and then the

case of a channel driven at a uniform rate with no pauses for recovery from

the effects of previously transmitted signals. (Of course, the latter case

corresponds to passing to the limit n -,w in the case of transmitting

isolated groups of n equally spaced digits.) One can also consider m x m

channels with memory, where m > 2. Such problems have been studied by Chang

and -co-workers [16], [17]. Finally, it is natural to study the case where

the noise itself has appreciable memory or even the case where there is

statistical dependence between the noise and the signals. It appears that

much remains to be done along these lines.

- 42 -

References

[I] Elias, P. "Coding for two noisy channels" , published in"Information Theory, Thfrd London Symposium",

C. Cherry, editor, Academic Press, New York, 1955.See p. 61

[2] Shannon, C. E. "The mathematical theory of communication" , BellSystem Tech. J., 27, 379-423, 623-656, 1948.

[3] Khinchin, A. Y. It Mathematical Foundations of Information Theory",

translated by R. A. Silverman and M, D. Friedman,Dover, New York, 1958.

[4] Faddeyev, D. K. "On the notion of the entropy of a finiteprobability scheme" , Uspekhi Matem. Nauk, 11,no. 1 (67), 227, 1956.

[5] McMillan, B. "Te basic theorems of information theory" ,

Ann. Math. Stat., 24, 196, 1953.

[6] Feller, W. "An Introduction to Probability Theory and itsApplications" , vol. 1, second edition, Wiley,New York, 1957. See Ch. 15.

[7] Feinsteirn A. - "Foundations of Information Theory" , McGraw-Hill,New York, 1958. See p. 29.

[8] Muroga, S. - " On the capacity of a discrete channel, I ", J.Pbys. Soc. Japan, vol. 8, no. 4, 484, 1953.

[91 Shannon, C. E. - "Geometrische Deutung einiger Ergebnisse beider Berechnung der Kanalkapazitit" , Nachrichten-technische Zeitschrift, vol. IO, no. i, i, 1957.

[10] Silverman, R.A. - "On binary channels and their cascades" , I.R.E.

Trans. Info. Th., Vol. IT-l, no. 3, 19, 1955.

[11] Loeb, J. - "Les deux types d'erreurs dues au bruit".Annales des Telecomunications, vol. 9, no. 2,1/6, 1954.

[12] Shannon, C. E. - "Channels with side information at the transmitter",

I.B.M. Journal, vol. 2, no. 4, 289, 1958.

[13] Gantmakher, F. R. - "' The Theory of Matrices", Vol. 1, translated byK. A. Hirsch, Chelsea, New York, 1959. See p. e1.

[14] Shannon, C. E. - "A note on a partial ordering for communicationchannels" , Information and Control, 1, 390, 3,8.

[15] Birnbaum, A. - "On the foundations of statistical inference" , tobe published.

[16] Chang, S. H. - " Capacity of a certain asymmetrical binary channelwith finite memory", I.R.E. Trans. Info. Th.,vol. IT-U, no. 4, 152, 1958.

[17] Chang, S. H., - "Examples of capacity calculations of channelsMcHugh, P. G. and with memory", unpublished notes.Lob, W. H.

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