IEEE TRANSACTIONS ON RELIABILITY, VOL. R-35, NO. 3,1986 AUGUST

Reliability Improvement for Simplex CommunicationSystems with Multiple Channels

Giuliano BenelliUniversit; di Firenze, Firenze

Key Words-Channel coding, Hamming code, Error probability,Channel redundancy.

Reader Aids-Purpose: Compare modelsSpecial math needed for derivations: Communication channel probabilitySpecial math needed to use results: SameResults useful to: Communication engineers

Abstract-The reliability of a communication system can be im-proved by using channel coding. Complex coding and decoding are oftenrequired to achieve high performance. An alternative approach is channelredundancy in which the same information is transmitted over two ormore channels at the same time. In this way, the coding complexity re-quired to achieve a given error probability can be appreciably reduced.However, a decoder is required for each channel. This paper describes amethod which uses channel redundancy to obtain higher performancecompared to similar systems and requires only one channel decoder.

1. INTRODUCTION

The performance of a communication system can beenhanced by using a channel coding operation which addssome redundant symbols to the transmitted informationsequence in order to correct errors introduced by thetransmission channel. Many good classes of codes for er-ror correction can be found in literature [1, 2]. However,codes having high error correcting capabilities often re-quire complex encoding and decoding algorithms.

Rahman has proposed a method [3] for improving thereliability of a communication system. The information isencoded by a block code, then transmitted over severalstatistically independent channels. The output of eachchannel is connected to its own decoder; through a suitablerule, a codeword is chosen between the decoded vectors atthe channel outputs. This method often results in a higherreliability for a given cost constraint than the single chan-nel technique.

This paper analyzes a communication system in whichthe same information is transmitted over different andstatistically independent channels. The receiver combinesthe signals from each channel and performs the demodula-tion and the decoding operations on this composed signal.In this way, only a demodulator and channel decoder arerequired. A net reduction in the error probability withrespect to other similar methods can be achieved.

2. MODEL

Notation

n codeword length of a code C.k number of information symbols in a codeword.Pe(S) bit error probability at the input of the channel

decoder in the case of s parallel and statistically in-dependent channels.

Pb(s) block error probability at the decoder output inthe of s physically parallel and statistically in-dependent channels.

c (c1, c2, ..., cO) a generic codeword of a binary codeC, (ci = - 1 or + 1).

nj,i noise component introduced by communicationchannel j on transmitted symbol i.

xj,i amplitude of the signal received at the output ofthe channel j(1 j1( s) in correspondence with thetransmitted symbol i, (1 < i < n) of c.

Ai codeword number of C having a Hamming weightequal to i.

Other, standard notation is given in "Information forReaders & Authors" at rear of each issue.

Assumptions:

1. The noise introduced by each channel is white andGaussian with zero mean and power density N.

2. A signal of amplitude c,A is used to transmit sym-bol ci of a codeword; ci = - 1 or + 1.

3. COMMUNICATION SYSTEM WITHCHANNEL REDUNDANCY

The main drawback of the method proposed byRahman [3] is the need for channel decoder. These channeldecoders are not considered in the method's cost, but theycan appreciably affect the cost itself.

This section describes a communication system withchannel redundancy, which requires only one channeldecoder. Figure 1 shows the block-diagram of such asystem wherein the parallel channels use s different fre-quencies. The system can be extended to the case of spacediversity. At the same time the method of combining thesignals coming from the channels can be changed.

The information symbols are encoded through a blockcode C of type (n, k). Each codeword is transmitted over s

0018-9529/86/0800-0301$01 .00 1986 IEEE

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IEEE TRANSACTIONS ON RELIABILITY, VOL. R-35, NO. 3, 1986 AUGUST

statistically independent channels. The output is:

xi,i = cA + n1i.. (1)

CI_tIVI&EL

SOURCJ7 7IEi] L CIIAIIIU'C4IIJL IISFRJ FPiccDER OECOOEfl

[jf IVIEL

Fig. 1. Block-diagram of a communication system with s parallelchannels.

The receiver adds the outputs from each channel andforms the signal:

codes, for which the computation of the error probabilityis quite simple [l].

Figure 2 represents the bit error probability Pe(s) usings parallel channels as a function of the signal-to-noise ratiofor the Hamming code (15, 11). In figures 2-4, curve arepresents a system employing one channel, curve b aRahman system using two channels, and curve c the perfor-mance of the system described in this paper employing twoor three channels. A net reduction in the error probabilitywith respect to the Rahman method can be obtained.

10

10

lcfz

(2)Zi =1

xj i, for I , i < n103

and then it forms the binary vector r, whose components i,(1 < i < n) is:

l=, if zj >OriL °' otherwise.

10

(3) 1o

The vector r is input to the channel decoder whichthen estimates c of the transmitted codeword c.

In order to evaluate the error probability in thesystem, consider the case in which the transmitted symbol iis + 1. The signal zi is a Gaussian random signal with meansA and variance sN. The symbol ri is erroneouslydemodulated if zi < 0 and therefore the error probabilityis:

Pe(s) = gauf (A K)

.6

-1

Fig. 2. Bit (

ratios.

(4)

As an example, for s = 2, a gain of 3 dB can be obtained.The sum (2) often automatically corrects the errors in-

troduced by the channels. The symbols correctly receivedoften present higher signal amplitudes (xj,i) than the sym-bols erroneously received (41 and therefore theypredominate in (2).

The block error probability of a communicationsystem using a code (n, k) is [1]:

Pb(s) = E Ai[Pe(s)]J [1_ Pe(S)] (5)

4. RESULTS & COMPARISONS

The performance of the described techniques are com-

pared with systems using one channel and the Rahmansystem. The codes used in all these examples are Hamming

O 1 2 3 4 5E/N (dB)

error probability Pe(s) versus

6 7

the signal-to-noise

Cost

Fig. 3. Block error probability of a Hamming code (15, 11) versus

cost.

N

I

-1 vI

302

vp

BENELLI: RELIABILITY IMPROVEMENT FOR SIMPLEX COMMUNICATION SYSTEMS WITH MULTIPLE CHANNELS

Cost

Fig. 4. Block error probability of a Hamming code (31, 26) versuscost.

The different implementation costs of a system using sparallel channels with respect to the case of a single com-munication channel must be taken into account. Thefollowing cost function C(P.) is used [3]:

c(Pe) = k/Pe(l) (6)

k being a constant; in the following, it is assumed k = 1.

Figures 3 and 4 show the block error probability ob-tained for the Hamming codes (15, 11) and (531, 26)respectively versus the cost. For a given cost, the systemone channel performs better for low signal/noise ratios;for mean and high signal/noise ratios, the system proposedin this paper appreciably improves the performance withrespect both to systems using one channel and systems us-ing Rahman's decoding strategy.

REFERENCES

[1] W. W. Peterson, E. J. Weldon, Error Correcting Codes, The MITPress, 1972.

[2] S. Lin, Introduction to Error Correcting Codes, Prentice Hall, 1970.[3] M. Rahman, "Reliability of digital communication systems with

channel redundancy", IEEE Trans. Reliability, vol R-31, 1982 Oct,pp 410-413.

[41 H. Tanaka, K. Kakigahara, "Simplified correlation decoding byselecting possible codewords using erasure decoding", IEEE Trans.Information Theory, vol IT-29, 1983 Sep, pp 743-748.

AUTHOR

Giuliano Benelli; Dipartimento di Ingegneria Elettronica; Universita diFirenze; via S. Marta 3;50139 Firenze, ITALY.

Giuliano Benelli received the degree in Physics from the University ofFlorence in 1973. In 1974 he taught channel encoding in the InformationTheory course at the Scuola di Perfezionamento in Physiscs at the Univer-sity of Florence. In 1975 he joined the Electronics Institute (now Depart-ment of Electronics Engineering) of Florence, first with a ResearchFellowship and since 1981 as a Researcher. His main research interests aresource and channel encoding, digital communication, and radar systems.

Manuscript TR84-023 received 1984 March 20; revised 1985 September23; revised 1986 May 24. *** *

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(continuedfrom page 300)ment of Bayes manipulations is correct mathematically, butthe explanations of application are poor at best. One can notlegitimately switch between probability as a degree-of-belief,and as a relative frequency; nor do Bayes methods allow oneto do that.

There is some realization of the distinction betweenfunction (schematic) diagrams and logic diagrams. But, con-trary to the book, a reliability block diagram is a logicdiagram, not a function diagram. There is a very good andnotable emphasis on common cause failures throughoutmuch of the text. The subtitle of the book implies the heavy

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emphasis on risk analysis for specific types of nuclear powerplants and their equipment.

I prefer the book by Henley & Kumamoto, ReliabilityEngineering and Risk Assessment, 1981, Prentice Hall, whichcovers similar material.

If the book is read critically, much good can be gleanedfrom it. If the reader is not capable of such critical reading,then at least the reader should be skeptical.

- Ralph A. Evans, PhD, PEProduct Assurance Consultant

303