Error Correcting Codes for Satellite Communication Channels

This paper addresses the problem of efficient forward error correction on differentially encoded, quadriphase-shift-keying (DQPSK) channels. The approach is to design codes to correct the most probable error patterns. First the probability distribution of error patterns is derived. Then a class of convolutional codes that correct any single two-bit error is described. Finally a threshold decodable code that corrects all single, and many double, two-bit errors is presented. Introduction A satellite communication system for digital data trans$i+l $i. If a 90" phase shift, or single-bit error, occurs mission must make efficient use of the available bandduring the transmission of $i, the output of the differential width and power. This can be accomplished by combindecoder will contain two single-bit errors: one in the ing forward error correction with an efficient modulation estimate of $i and one in the estimate of $i+l ${. technique. The greater efficiency of phase-shift keying Thus, the bit error rate is doubled and bit errors are cor(PSK), as opposed to frequency-shift keying, often leads related. The correlation of errors is the more serious to the choice of PSK as a modulation technique for satproblem because it severely degrades the efficiency of ellite channels [ 1 , 21. The implementation of a PSK a random-error correcting code. If, for example, a modem depends on the constraints of the system. The single-error correcting convolutional code is used for number of phase states and the type of encoding (direct forward error correction, there is no guarantee that it or differential) can be chosen to effect the desired tradeoff will correct any double-bit errors. among power, bandwidth, transmission rate, and bitForney and Bower [6] encountered this problem in error probability. In a bandwidth-limited system, quadridesigning a high speed sequential decoder. Their solution phase-shift keying (QPSK) can be used instead of binary was to perform forward error correction before difphase-shift keying to conserve bandwidth [ 1-31 , The ferential decoding [see Fig. 1 (b)]. In this scheme, the use of differential encoding to resolve the phase ambiguity error correction code (ECC) decoder need not contend at the receiver can save power that would otherwise be with double-bit errors, but it is faced with the same phase required for a residual carrier [4]. For these reasons, ambiguity that the differential encoding was used to redifferentially encoded QPSK (DQPSK) is a highly solve. Techniques to resolve this ambiguity include an efficient modulation technique for satellite communicaacquisition search at start-up and whenever the modem tion channels. Fig. 1 ( a ) shows a block diagram of a undergoes a 90" phase slippage; much of the benefit of communication system using DQPSK and forward error differential encoding is lost. correction. This paper describes a different approach o the probA disadvantage of DQPSK modulation is that each lem. Standard t-error correcting codes are designed to failure in recognizing a phase position results in a two-bit correct the error patterns most likely to appear in a bierror. A QPSK modem transmits a symbol, or bit pair, nary symmetric channel. Instead of using these codes, by phase-modulating the carrier to some phase $ chosen it seemed advisable to look for new codes that correct from the set {O0, 90", 180", 270"). In the presence of the error patterns most likely to appear in a differentially white Gaussian noise, the most likely transmission error encoded coherent QPsK system. To correct singleis a phase shift of* 90" in the estimate of $. If the symchannel errors, for example, we looked for a code that bols are encoded using a Gray code, a 90" phase error corrects the double-bit errors produced by the differential in $ is a single-bit error in the bit pair represented by decoder. Because they are highly unlikely, this code need $ [ 5 ] . When differential encoding is used, information not correct single" errors. To find such a code, it is is encoded in the difference between successive phases, first necessary to characterize the probability distribution :HEN AND R. A. RUTLEDGE IBM J. RES. DEVELOP. Dlfferential (a) Data ~ ~n Channel data decoding decoding (b) Differential Figure 1 Differentially encoded QPSK system with forward error correction and differential coding ( a ) internal to and (b) external to the error control coding of the error patterns. This is done in the following section. Next, a class of convolutional codes, which can correct any single-channel error, is presented. Then another convolutional code, which corrects all single-channel errors and 70 percent of double-channel errors, is described. Error characterization A QPSK modem transmits a pair of bits by phasemodulating the carrier to O", 90", 180", or 270". The receiving modem estimates the phase and translates it back into a bit pair. Thus, to transmit the bit pair ( I , Q ) , the modem transmits the phase 0 =AI , 0 ) 1 ( 1 ) wheref( . , .) is a I : 1 mapping of { (0, O ) , (0, I ) , ( I , O ) , ( I , I ) } onto { O", 90", 180", 270"). The receiving modem computes the maximum likelihood estimate 6 of 0 and (f, Q ) = f ' " (e^ ) . (2) The channel noise is assumed to be white and Gaussian. If the receiving modem has exact knowledge of the carrier reference phase, it can be shown [4] that the error in e ,̂ + = e & has the probability distribution p r ( 4 = 0') = ( I P r ( 4 = 9 0 " ) = Pr(4=270") = p ( l p ) ; Pr(q5= 180") =$, (4) where p = 0.5 e r f c ( V ' m ) and R is the signal-to-noise ratio. The mapping f ' ( ., .) is chosen to satisfy f " ( 0 ) Of"(0 + 180") = ( I , I ) ( 5 ) for all 0, where 0 denotes the EXCLUSIVE OR function applied element-by-element. Then it follows from (4) and ( 5 ) that the bit errors f 0 I and Q 0 Q are independent random variables with distributions P r ( f @ I = I ) = I P r ( f @ / = O ) = p , a n d P ~ ( Q O C ~ = ~ ) = ~ P ~ ( $ O C ) = O ) = ~ . (6) In other words, the channel, including the two QPSK modems, is simply a binary symmetric channel with crossover probability p . In general, however, the receiving modem does not know the carrier reference phase exactly but can estimate it to within a degrees, where a is fixed but unknown. In this case, e ̂ = 0 + a + 4 and, in general, 6 # 0 even when 4 = 0. Differential encoding is used to resolve this problem. The data sequence { (Ii, Q i ) , i = 1, 2 , . . .} is encoded into {e,}, where 0, = f ( I i , Q,) as before, but instead of transmitting Bi, the modem transmits the sequence { $i} of partial sums defined by JI" = 0; $, = qJ-] + H , , i = 1 , 2, . . .. (7) 3, = $, + a + +,, (8 ) The receiving modem estimates $ i by where the random errors 4, are independent of each other and of the input sequence $ i , and each 4, has the distribution shown in (4) . Then Oi is estimated by ^ ^ A 0, = $, $,-,, (9) (f,, 6,) =p ( 6 , ) . (10) and (I i. QJ is estimated by If the errors in the decoded sequence are denoted by ( E ~ , FJ = ( f i , Q i ) 0 (I,, ei ) , i = I , 2 , . . ., it follows from (8 1 1) that (E i . F, ) = f " ( 0 J 0 f '" (0 , + 4, + i + * ) . (E,. F i ) = ( ui, V i ) 0 ( X i , V i ) , ui, V i ) = f'" (0, +,-J 0 f'" ( 0 J , ( X , , Y i ) = J " I ( 0 , +,&l + 6,) 0 J ' " (H i +,-1) It is convenient to rewrite ( 1 2 ) as