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ESD-TR-66-80

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SCIENTIFIC 4 reCHNttAj INFORMATION DIVISION

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MTR-105

THE COMPUTATION OF CERTAIN COMMUNICATION CHANNEL ERROR PROBABILITIES BY AN APPLICATION OF

DIFFERENCE EQUATION METHODS

JULY 1966 S. Berkovits E. L. Cohen

Prepared for

DEPUTY FOR COMMUNICATIONS SYSTEMS

ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscom Field, Bedford, Massachusetts

Distribution of this document is unlimited.

fsNC

Project 7560

Prepared by

THE MITRE CORPORATION Bedford, Massachusetts Contract AF19(628)-5165

This document may be reproduced to satisfy official needs of U.S. Government agencies. No other repro- duction authorized except with permission of Hq.

Electronic Systems Division, ATTN: ESTI.

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Do not return this copy. Retain or destroy.

ESD-TR- 66-80 MTR-105

THE COMPUTATION OF CERTAIN COMMUNICATION CHANNEL ERROR PROBABILITIES BY AN APPLICATION OF

DIFFERENCE EQUATION METHODS

JULY 1966

S. Berkovits E. L. Cohen

Prepared for

DEPUTY FOR COMMUNICATIONS SYSTEMS

ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscom Field, Bedford, Massachusetts

Distribution of this document is unlimited.

Project 7560

Prepared by

THE MITRE CORPORATION Bedford, Massachusetts Contract AF19(628>5165

ABSTRACT

A model for a channel is given. For this model, the recursive method is presented in order to calculate the

probability of K symbol errors in a block of n m-bit symbols.

The blocks can be interleaved or not.

REVIEW AND APPROVAL

This technical report has been reviewed and is approved.

EDGAtf A. GRABHORN, Lt. Colonel, USAF Director of Communications Development Deputy for Communications Systems

111

TABLE OF CONTENTS

Page SECTION I THE MODEL 1 SECTION II RECURSIONS - PART I 3 SECTION III RECURSIONS - PART II 6 SECTION IV INPUT FOR THE PROGRAM 10 SECTION V OUTPUT FOR THE PROGRAM 11 APPENDIX PROGRAM 13 REFERENCES 18

SECTION I THE MODEL

In estimating the performance of an error-correcting device on a specific communication channel, it is necessary to find a meaning- ful, yet tractable, mathematical model for that channel. An examination of data from real channels suggests that most channels pass through three distinct phases. The first phase, which nearly any error- correcting scheme can handle successfully, is that of long periods of practically error-free transmission. The second phase, which is the antithesis of the first phase in that no scheme can expect to correct it, is that of complete loss of signal for substantial periods of time.

The third phase might be described as a generalized sputter or bursts of error bursts, and it is this phase, if it occurs frequently, for which error correctors should be designed. This third phase may

sometimes be described by means of a two-state model. The three phases suggest a Markov process with four states, but such a process

is mathematically unwiedly. However, when the three phase picture is reasonably correct and the sputter phase is accurately modeled for purposes of estimating coder performance, it is a simple matter to make corrections in such estimates for those periods when transmission

is nearly error free or when the signal is lost.

The use of such a two-state model was suggested to us by some work by Gilbert [ 1] . Gilbert describes a model for binary error distributions

in channels subject to noise bursts. Let {x.} be the error process with x. = 1 for an error in the i demodulated bit, x. = 0 for no error.

l l

Two states, designated G and B, of the channel are postulated such that at the i bit the state S. is G or B, and the state S.,n at bit i + 1

L 1+1

depends only on S.. Thus, the state sequence {S.} is a simple Markov chain described by the two transition probabilities, P: G -» B and p: B -» G. We also use Gilbert's notation Q = 1 - P and q = 1 - p for the G •* G and B -» B transitions respectively Let h,k denote respectively

the probabilities of a correct bit in B and in G Then P{x.=l] = 1 - h = h' if S. = B and Pfx.=l} = 1 - k - k1 if S. = G.

l l l

Using this model with k = 1, Gilbert obtained a good fit to certain phone line error data. The statistic fitted was the probability of occurrence of zero (= error free) runs of length at least K. (The use of k < 1 has also been considered by Elliott [5].)

At MITRE, we have found sets of values for the model parameters P, p, h, k, which yield good fits to error data for several different types of communication media. One important statistic involved is the probability of specific error densities in various length blocks.

Given that we have p, P, h,k, [2,3,4], we present the recursive method used to calculate the probability of K symbol errors in a block of n symbols where each symbol consists of m bits.

SECTION II RECURSIONS - PART I

In Appendix II of [2] (or Appendix B of [3] or [4]) we presented a brief outline of the recursive method used to calculate the probability of K symbol errors in a block of n symbols where each symbol consists of m bits. Now we present the outline in full. Since the last

documentation, the technique has been extended to permit the n-symbol blocks to be interleaved or time spread. (To interleave s blocks means to transmit sequentially the first symbol of each of s blocks followed by the second symbol of each of those blocks, etc. Thus, a given m-bit symbol is transmitted s symbol times after the symbol preceding it in its block.)

Letting T and U represent either of the states G and B, we define TOU(t) = P(xx = ••• = xt = 0 and Sfc = U|SQ = T)

TlU(t) - P(for some i S t, x. =1 and S^t - UJS^Q = T) Then

G0G(1) = Qk G0B(1) = Ph G1G(1) - Qk' G1B(1) = Ph' B0B(1) « qh B0G(1) = pk B1B(1) = qh' B1G(1) = pk' and

GOG(t) - [GOB(t-l) p + GOG(t-l) Q] k

GlG(t) = [GOB(t-l) p + GOG(t-l) Q] k' + GlB(t-l) p + GlG(t-l) Q 3

GOB(t) = [GOB(t-l) q + GOG(t-l) P] h

GlB(t) = [GOB(t-l) q + GOG(t-l) P] h' + GlB(t-l) q + GlG(t-l) P BOB(t) = [BOB(t-l) q + BOG(t-l) P] h

BlB(t) = [BOB(t-l) q + BOG(t-l) P] h' + BlB(t-l) q + BlG(t-l) P BOG(t) = [BOG(t-l) Q + BOB(t-l) p] k

BlG(t) = [BOG(t-l) Q + BOB(t-l) p] k' + BlB(t-l) p + BlG(t-l) Q

Shortly after the program was written, we discovered that GOG(m), COB(m'), GlG(m), GlB(m), BOG(ra), BOB(m), BlG(m), BlB(m) could be obtained

from a difference equation in powers of J and L (see below). Since on our computer (IBM 7030), it took under a second to compute all eight quantities, we decided not to use the difference equation. However, we work out two, and give all eight results.

GOG(t) = {GOG(t-l) Q + GOB(t-l) p] k GOB(t) = {GOB(t-l) q + GOG(t-l) P} h

The eigenvalues come from the 2nd order linear difference equation:

ft+l " ^{(Qk + qh) f}t " ^(P"^{Q) f}t-l ⁼ °- that is, 2J = Qk + qh +V(Qk + qh)² + 4hk(p-Q)

and 2L = Qk + qh-\(Qk + qh) + 4hk(p-Q) . Thus we have

GOG(t) = ax J^t + 0^ LC , GOB(t) = |8¹ ^ + j3² LL

and we get Q^ , o^ , ^ and fi^

from the initial conditions

GOG(O) = 1, G0G(1) = Qk and GOB(O) = 0, G0B(1) = Ph .

So a + OU - 1, a,J + a?L = Qk, which yields

GOG(t) = {(Qk - L)/(J - L)} JC + {(J - Qk)/(J - L)} LC

Also, ^ + 02 = 0, ^J + 02L = Ph, which yields

GOB(t) = {Ph/(J - L)} (J* - L11) All eight solutions are as follows:

GOG(m) = {(Qk - L)/(J - L)} Jm + {(J - Qk)/(J - L)} L•

GOB(m) = {Ph/(J - L)} (Jm - Lm) BOG(m) = {pk/(J - L)} (J1" - Lm)

BOB(m) = {(qh - L)/(J - L)} J• + {(J - qh)/(J - L)} L•

GlG(m) = p/(p + P) + {P/(P + P)l (Q"P)m " 1(J " qh)/(J - L>i J•

- {(qh - L)/(J - L)} Lm

P r, ,~ \in-i Ph , Tm Tm.

GlB(m) --^[1- (Q-P) ] - -jZi (J " L )

»»<«> " ife

ife «"»>" " Ti

"

*

-

"

BlG(m) - Jj [1 - (Q-P)"] - f^ (J* - L">

SECTION III RECURSIONS - PART 2

Again letting T and U represent either of the states G and B, we define

TOUI(s) = P(x. .... = x. 1W, = ••• = x = 0,S = U|S_ = T) (s-l)nri-l (s-l)m+2 sm sm '0

• P(m-bit symbol after s symbol times is correct and ends in state U (state T) TlUI(s) = P(for some 1 £ t « m, x, . . . = 1, S = uls. = T)

(s-l)mfi sm '0

= P(m-bit symbol after s symbol times has at least one bit error and ends in state u|state T)

Let GXG = GOG(m) + GlG(m), GXB = GOB(m) + GlB(m), BXG = BOG(m) + BlG(m), and BXB = BOB(m) + BlB(m)„

Then GOGI(s) = GXG • GOGI(s-l) + GXB • BOGI(s-l) and BOGI(s) = BXG GOGI(s-l) + BXB • BOGI(s-l).

(There will be similar equations in G1BI, BlBI and GOBI, BOBI, and G1BI, BlBI, but they have the same eigenvalues and will be omitted.)

TOGI(s) - [GXG + BXB] TOGI(s-l) + [GXG•BXB-GXB-BXG] T0GI(s-2) = 0 This yields the eigenvalues:

2 a = GXG + BXB + V[GXG + BXB]²-4 [GXG • BXB - GXB • BXG]

2 T = GXG + BXB - y[GXG + BXB] -4 [GXG ' BXB - GXB • BXG]

Since GXG + GXB = BXB + BXG = 1,

CJ'T = GXG • BXB - GXB • BXG = (1-GXB) (1-BXG) - GXB • BXG = 1-GXB-BXG a + T = GXG + BXB = 2 - GXB - BXG = 1 + a T .

Hence a = 1, and T = 1 - GXB - BXG = GXG + BXB - 1.

Thus we have

GOGI(s) - X • I^s + \²<r^S and BOGI(s) = u ' 1^S + u ^TS

and we get X , X , fjL and pL from the initial conditions.

G0GI(1) = GOG(m), G0GI(2) = GXG • GOG(m) + GXB • BOG(m) BOGI(l) = BOG(m), BOGI(2) = BXG ' GOG(m) + BXB • BOG(m).

Hence

\ • 1 + X* T - GOG(m), X • l² + X • T2 - GXG • GOG(m) + GXB • BOG(m).

Solving, X = [GOG(m) (1 - BXB) + BOG(m) • GXB] / (1 - T)

\2 = [GOG(m) BXB - BOG(m) GXG] / (1 - ^T) = [GOG(m) - Xj] / T - Also,

^L ' 1 + JLU • T - BOG(m), (X± • \2 + /^ • T² = BXG • GOG(m) + BXB • BOG(m).

Solving, fa = [BOG(m) (1 - GXG) + BXG ' GOG(m)] / (1 - ^T)

^ = [BOG(m) - jL^] / T

Therefore, GOGI(s) • X. * 1 + \. ' T , and BOGI(s) - jLL* 1 + M2'T >

where \. , X„ , [I., /!„ are given above. Since s is fixed for any given application, we will refer to GOGI(s) as GOGI, and to BOGl(s) as BOGI.

Consider the first i m-bit symbols of a random interleaved block.

Let GB(i,j) * P(j symbol errors in i symbols and S. - BlSn * G)' Similarly, we define GG(i,j), BB(i,j) and BG(i,j).

Then

GG(1,0) GG(1,1) BG(1,0) BG(1,1)

GOGI G1GI BOGI B1GI

GB(1 ,0) ^{m GOBI} GB(1

,U

BB(1 ,0) ⁼ BOBI BB(1

,U

Finally, for i =2, ••• , n and j = 0, "*•, i

GG(i,j) = GG(i-lJ) GG(1,0) + GB(i-l.j) BG(1,0)

+ GG(i-l.j-l) GG(1,1) + GB(i-lJ-l) BG(1,1) GB(i,j) = GG(i-lJ) GB(1,0) + GB(i-l,j) BB(1,0)

+ GG(i-l,j-l) GB(1,1) + GB(i-l.j-l) BB(1,1) BG(iJ) = BG(i-lJ) GG(1,0) + BB(i-lJ) BG(1,0)

+ BG(i-l,j-l) GG(1,1) + BB(l-l.j-l) BG(1,1) BB(i,j) = BG(t-lJ) GB(1,0) + BB(l-l.j) BB(1,0)

+ BG(i-lJ-l) GB(1,1) + BB(i-l.j) BB(1,1).

Finally,

P(random bit is in G) = a = ~^: and hence p+P

P(K symbol errors in n symbols with s blocks interleaved)

= a. [GG(n,K) + GB(n,K)] + (1 - a) [BG(n,K) + BB(n,K)]

SECTION IV

INPUT FOR THE PROGRAM

XM = No. of bits/symbol XN = No. of symbols/block

XNEST = Largest number of symbol errors to be considered (if this field is blank, XNEST = XN)

XIPER = Number of interleaved symbols (0 or 1 means 1)

IK = 0 or not equal to 0 (0 means continue with CP, SP, H, SK;

not equal to 0 means read new parameters) CP = P

SP = p H = h SK = k

10

SECTION V

OUTPUT OF THE PROGRAM

TIMEX = A8 representation of the time read from IBM 7030 Time Clock by a STRAP coded routine (one can call his routine or omit it altogether)

CP, SP, H, SK as above

ALPHA = P (random bit is in G) M (or XM) as above

N (or XN) as above

NS = No. of recursion terms to be attempted WPI = P (symbol error)

WY155 = P (no errors when s = 1)

WPMU2 = mean number of errors in a block

FCMEAN = mean number of errors given an error occurred (when s = 1) For J, L, A, B see [2,3,4]

IS = number of errors

P * P (IS symbol errors in a block)

R * P (IS symbol errors in a block|error occurred) Q • P (s IS symbol errors in a block)

QH - P (> IS symbol errors in a block)

S « mean number of bits between blocks with > IS symbol errors PBAR • contribution to mean number of errors per block made by

probabilities actually calculated (if one wants the whole mean, then NEST « 0 or N; the same applies to VAR, SVAR, and CMEAN)

11

VAR - contribution to variance about PBAR made by probabilities actually calculated by the recursion

SVAR = approximate standard deviation

CMEAN = contribution to mean number of errors given an error occurred in the block made by probabilities actually calculated by recursion

12

APPENDIX PROGRAM

C...TO CALCULATE THE RECURSION PROBABlLI 1I1S COMMON 777/ TIMEX

DIMENSION GCI515), J;B<515)t 66(515). OBI515) UIMESSIUN amtzailt ststsasj,

OIMtNSION GOG(20i>), G1G(2C 00005 404 REA01701, LK

00006 1701 FORMAT (12)

C...READ IN THE NO. OF BITS PER SYMBOL. THE NO. OF SYMBOLS,

C...REAO IN CP. SP, H, SK

00009 604 READ 702. IK, CP, SP, H, SK 0U010 702 FORMAT (11, E17.10, 3(E1B.1H) JjgOll-

OOQIi

DO 2000 K=1,LK S = XNIK) 00014

00015 00016

NEST = XNEST(K) IPERD =XIPER(K) IFINEST.EQ.O) NEST

gL=-TjHC.r

00020 300 FORMAT!1H1.A8.21X, 77H*CALCuLATIUN OF GILBERT CHANNEL ERROR PROBAB ULITIbS USING RECURSION FORMULAS*/////)

00021 MUD = M*N

=&amx = I»/(CP*SP)

PI = (SP*SKCOMP • CP*HCOMP)*GA«MA PRINT 301, CP, SP, H, SK, ALPHA FORMAT (5X, 27HCHANNEL PARAMETERS 00031

00032

00033 301 CP = .Ell.6, 5X, 5HSP ;

112X, 35HNO. OF RECURSION TERMS REQUESTED 2 E13.7, 75X,

3 S,/)

14,

15/ 7X, $P1 = $,

$ BLOCKS INTERLEAVED

13

GCC36 WN = SP - CQ 0003? WI * bK*tJ + H*SQ 0003*5 _HR = WT»«2 J^4.*H*SK*WN 000 39 WSUKT = SORT(WR)

00040 wJ * <*T • *SURT)/2.

00041 XL = (V<T - wSUKT)/2.

00042 WA = Kj "+ UN ~*{Cf»*S**tCUMP + SP*H*SKCOMP J / 1 tCP'HCOMP + iP*SKCUMP >

00G43_ WA = wA _/ WSQRT 00044 WB = WA - 1.

00u45 <IX2 = ( WA*< l.-WJ**M) /( l.-WJ) ) - ( w(i*l l.-wL**MI/l l.-WL) » 30046 HP1 = PI * WX2

00O47 WPMU2 ••• FLOAT(N) * WPf / 00046 KJ = FLOATIMUO) * ALUL,10(WJ)

Oofi*9 HL = FUJAT(MUO) * ALUGIO(WL) 00050 ^JS^TRJ. LT. -50.) F = 0.

00051 IFTRJ. GE. -50.) F = rtJ**MOQ 00052 IF(RL. LT. -50.) 0=0.

00053 IFlKL. OE. -50.) 0 = HL**MUD

00054 W Y 155 * U_ -1 (W A* ( 1 . -F ) / (1 . -hj) )-Wd*( l.-0)/l l.-WL) ) *P 1 00055 ^:^ FCMfcAtt = WPHU2 X- (1. -_rtY155)

00056 PRINT 306, W» I , WY155, WPM02 ,FC«t AN,rfJ, WL,

FORMAT (IX, 20HPRQB SYMBOL EKRUR = ,tl2.0, 1H*, 2X, 17HPRGB NO ER 1R0RS = ,fcl2.6, 1H*, 2X, aH MfcAN = ,t!2.6, IH*, 2X, SCUNOITIUNAL ME 2AN = *, tl2.6, S*S / 10X, 4HJ = , 3X, E16.10, 1H*.

315X, 4HL = , 3X,^<J.6, 1H*. 6X, 4HA = . 3X. FS.6, 1H«, 20X, 4HB =

0OC60 GGB< 1) = CP*H

00061 GlB(1) = CP - GOBI 1) JK>p 62 aoa( 1) = _S Q* H

00063 aiBii) = so - = aoetn^

00064 BOG(l) = SP*SK

aC065 ==S .ftl&(U = SP - ^ BOGUJL__

00366 IF(M.EO.l) GO TU 80 00067 00 50 J=2,M

0006B Jl = J-l

00069 TA = GOttCJl)*SP + GOG-rere^gg^

0Q§?5 606(4) • TA*SK

Q00J4 G1&1J) = TA - SOOU) */&101, 00072 TB = G0BIJ1)*S0 • G0G(J1)*LP 00073 GOB(J) = TB*H

00074 GIB(J) = TB - GOB 1J) * G1B1J1)«SU * G1G1J1)*CP 00075 TC = B0BTJ1)*S& • B0G(J1)*CP

00076 BOB(J) * TC*H

QG077 B1BIJ) = TC - BOB(J) * B1BU1)*SQ + 00078 TO = B0G(J1)*CJ + B0B(J1)*SP 00079 BOG(J) = TO*SK

00080 50 BIG(J) = TO - BOG(J) • B1G(J1)*CQ • B1BU1I*SP 00081 10 PRINT 3C3, GOGIM), GIGTH), GO

14

00083 PRINT 304, BOG(M), B1GIMI, BOB(M), BIB(M)

00084 304 FORMAT <5X, 6HB0G = , E12.5, 5X, 6HB1G = , E12.5, 5X, 6HB0B « , 1E12.5, 5X, 6HBIB = , E12.5//1

RPI1 = ALPHA*!GIStM) + &1G(H)>

WU&T 565, ftPU

HAT ll-3f_ 3i"lUI>UfiU L/AWllllfc* € 1A1 ilAf ^UrtrKUD ?^.*TtWtjn ^

00088 00089 00090

CALL TIME GGU) = GOGIHI GGC2) = GIG(M)

00094 00095 00096

BG(2) = BIG(M) BB( 1) = BOB(M) BB(2) = B1BIMI

00100 00101 00102

GOBI = GOB(M) G1BI = GIB(M) BOGI = BOG(M)

00113 00114 00115

00119 00120 00121

PRINT 9003, SIG, TAU 9003 FORMAT I8X, SSIGMA = AG = <GG(1) «GT »

$• E20.12, 10X, $TAU GXB • BG(1>) / DEN

$, E20.12/'

00125 00126 00127

GOGI = AG • CG * G BOGI - AB + CB * G

AG = (GG12) *GT + GXB * BG12>) / DfcN

15

00130 00131 00132 00133

XB_ = 18&m -i AS) / TAU GI&I = AG + CG * G BlGI = AB + CB * G

AG = (GB< 1) *GT + GXB * BB( 1) ) / DEN 0013*

00135 00136 0C137 00138 00139

CG = IG8J1} - AG) :S^zi^t8XG * OBU)

/ TAU

t Ball) * BTI / DEN

= iseui - Afl) / TAU GOBI = AG • CG * G BOBI = AS • CB * G

AG = 1GB(2) *GT * GXB * 88(2)) / DEN 001*0- =^^£^-^(66(2) - AG) / TAU

001*1 I'-j&z—m.^ 8XG * GB(2) * BB(2) *^STF^^i 00142 C& = <BBiZl - AS) / TAU

00143 00144 00145

G1BI = AG • Co * G B1BI = A6 + CB * G

PKINT 9005,G0G1t GIGI, GOBI, G1BI, BOGI, BlGI, BOBI. B1B1 fmm'L t+SX, 7MG0&I = TPi ?.^T VIr THfTTSf ^ r £12.5, 4X ,

gLJtX, 7HG18I -« , E12.5/ 5X, THBOG1 * , £12.5, 4X, TH&

, 4X, 7HaOBI = , E12.5j_*i«

1GI •

£12.5//) 00147 9002 1FINEST.EQ.1) GO TO 90

C. .CALCULATfc THE KECUKSIUN PKUBABILITIES 00148 DO 51 L1=2,N 00149

00150 00151 00152 00153 00154

= HINGlLl.NfcST)

=ftYlg*~ Li

IFtil .GT. NEST) LY ! NESfc^k IFUl .GT. NEST) GU TO 97 Nl = LI + 1

GG1N1) » GGU1)*G1GI GB(L1)*BIGI G6IN1)

00158 97 CONTINUE 00159 DU 52 L2=2.LY 00160 N2 = LY - L2

&G{t_l)*GIBl BGUl)*GlGI BGU.l)*GiBI

+ 2 i»S== N2 -

00162

00165 YAHl =GG<N2)*G0G1

&B4N2)=GB(N21*B08I GGIN2) = YAHl

+GG1N3)*G1£I +GB(N3)*81Br

+GBCH3)*61GI

*SaW»*SlBi 00165 YAH2 =BG(N2)*G0GI

00166 BB(N2)=BB(N2)*BOBI 00167 52 BG(N2) = YAH2

•BG(N3)*G1GI

•BB(N3)*818I

+B8<N2J*B0GI

•BGlN2)*G0BI

•BB<N3I*B1GI

•BG(N3)*G1BI 001«£

GGI69 0C17C

G&« 1 >*GOGI

16

HKtH>5-#-~~o3yT

00177 503 FORMAT!//////////////50X, 33H(SYMBOL UATA STARTS ON NtXT PAGE)) 00178 IFINEST.GT.35) PRINT 502

00179 502 FORMAT (1H1) 0CU80 PklHT 500, TIHEX

00181 500 FORMATttX.AB/ IX, tSYMBLU. NUHBfcSJIOX, l5Hft£CURSI0N 1 17HCQNDIT10NAL PRQBS, 15X, 13HCUM REC

2 9X, SMEAN 'THEENS)

00182 P = ALPHA*IGG(1) • GB(1)I • BETA *(BG(1I + BB(1 00183 Q = P

0018*

00185 50186 00187 00188

) )

Rl * 1. - P

SI = « * F*UD / Rl

507 FORMAT (12X,$0*t, 8X, E16.9, 1H*, 34X, E18.12, 1 E10.4, $*$, 9X, E10.4, $*i)

PBAR = 0.

1H*, 9X,

00189 0O190 00191

PVAR == O, - OQ 53 t=2, TiS

IS = 1 - I 00192

00193 00194 00195 00196 B019T 00198 00199 00200 00201 00202

00204 00205 00206 OOJQ3 00201 00209 00210

P = ALPHA*(GGU I • GB(1 )) Q = P + 0

OH = 1. - Q

• BETA *(BG(I) + BB<I))

FIST = PBAR PVAR

FIS**2

= PBAR

= PVAR

» FIS

* FIST

^K&mi^H**;

PRINT 501» 4-S. * 53 CONTINUE

F OR «*T i ^fe

1 E10.4, lh*, 9X, E10.4, $*$) 606 CONTINOE

VAR = PVAR - PBAR*PBAR

PRINT 305, PBAR, VAR, SVAR, CKEAN, TIMEX

305 FORMAT (// 25X, $PARTIAL MEAN = (, E13.6/

1 25X, SAPPROXIMATE VARIANCE = $, E13.6/

00212 00213 00214

IFIIK .tO. 0) GO TO 604 GO TO 404

END

17

REFERENCES

1. E.N. Gilbert, Capacity of a Burst-Noise Channel, Bell System Tech. J., 39, Sept. I960.

*2. S. Berkovits, E. L. Cohen, and N. Zierler, A Model Cor Digital Error Distributions, The MITRE Corp., Bedford, Mass., ESD-TR-65-146, 1965.

*3. S. Berkovits, E.L. Cohen, and N. Zierler, A Model for Digital Error Distributions, The MITRE Corp., Bedford, Mass., TM-4189, March 15, 1965.

*4. S. Berkovits, E. L. Cohen, and N. Zierler, A Model for Digital Error Distributions, First IEEE Ann. Commun.

Conv., Boulder, Colo., June 7-9, 1965.

5. E. 0. Elliott, Estimates of Error Rates for Codes on Burst-Noise Channels, Bell System Tech. J.. 42, Sept. 1963.

* References 2,3,4 are essentially the same, 2 is more accurate than 3, and 3 is more accurate than 4.

18

Security Classification

DOCUMENT CONTROL DATA - R&D

(Security classification ol title, body ol abstract and mdaxjntf annotation must bo entered whan the overall report ie classified) 1 ORIGINATIN G ACTIVITY (Corporate author)

The MITRE Corporation Bedford, Massachusetts

2a REPORT SECURITY C L»)5IFIC»TlOI

Unclassified

2 6 SROUP

3 REPORT TITLE

The Computation of Certain Communication Channel Error Probabilities by an Application of Difference Equation Methods

4 DESCRIPTIVE NOTES (Type ol report and inclusive dates)

N/A

S AUTHORS.) ILast nams. ft ret name. Initial)

Berkovits, Shimson, and Cohen, Edward L.

6 REPO RT DATE

July 1966

7» TOTAL NO. OF PACES

21

7b NO. Or REF1

8a -ONTRACT OR BRANT NO.

AF19(628)-5165

b. PROJECT NO.

7560

»a. ORIGINATOR'S REPORT NUMBER(S)

ESD-TR-66-80

8 b OTHER REPORT NOfSJ (Any other numbare that msv be ssslgnsd this report)

MTR-105

10. AVA IL ABILITY LIMITATION NOTICES

Distribution of this document is unlimited.

It SUPPLEMENTARY NOTES 12 SPONSORING MILITARY ACTIVITY Deputy for

Communications Systems, Electronic Systems Division, L.G. Hansom Field, Bedford, Mass.

13. ABSTRACT

A model for a channel is given. For this model, the recursive method is presented in order to calculate the probability of K symbol errors in a block of n m-bit symbols. The blocks can be interleaved or not.

DD ,Mft. 1473

Security Classification

Security Classif- atiw

KEY WORDS

Channel Modelling Communii/at Ions Error Control

ROLf

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Security Classification