2000 Conference on Information Sciences and Systems, Princeton University, March 15-17, 2000
Inner Code Rate Selection for a Concatenated Coding System on a Binary Symmetric Channel Yong Zhang and Maja Bystrom
Dept. of Electrical and Computer Engineering Drexel University Philadelphia, Pennsylvania 19104
[email protected] and
[email protected]
Abstract | Concatenated codes have long been used as a practical means of combating errors on very noisy channels. In this paper we investigate the performance of concatenated systems that employ inner convolutional codes with outer Reed-Solomon codes. We determine upper and lower bounds of the optimal inner code rate for a selected overall code rate on a binary symmetric channel. I. Introduction Shannon [1] has shown that when we transmit information at a rate R smaller than the channel capacity C and when the codeword length N is suciently large that there exists an (N; K ) block code which will make the transmission error as small as desired. on the other hand, too large an N will make the coding system complex. As a practical method for implementing codes with long block or constraint lengths, concatenated codes were rst introduced by Forney in [2] and have been successfully used in deep space and other communication systems. A concatenated coding system divides the complexity between inner and outer codes, thus reducing the total coding complexity considerably. In the application of concatenated codes, a problem that has long puzzled researchers is that with a constraint on the overall coding rate, what is the best way to select the inner code rate? Herro, et al. [3], [4] have made signi cant progress towards solving this problem by obtaining information theoretic bounds on performance using the overall channel capacity and channel cuto rate. However, these bounds are fairly loose and are not sucient to select the optimal inner code rate, because these two parameters do not re ect the impact of the outer code. In this paper, we determine bounds on the optimal inner code rate by using random coding bounds on convolutional codes. II. Concatenated Codes
Serial concatenated coding systems are typically implemented by employing two coders (an outer coder and an inner coder), as illustrated in Fig. 1. Data bitstreams from an information source are rst sent to the outer encoder, which is usually a block encoder, such as ReedSolomon (RS) coder. The output of the outer encoder forms the input to the inner encoder, which may be either a block or trellis code. Then the output of the inner encoder is sent over the physical channel. In the receiver, decoding is accomplished in reverse order. To combat burst errors, interleaving could be used between these two coders and between the inner coder and the physical channel. The combination of inner encoder, physical channel, and inner decoder which is presented to the outer coder is typically called the outer channel. Concatenated coding systems with RS codes as the outer codes and convolutional codes as the inner codes have advantages over other types of concatenated systems. They exhibit excellent performance with soft decisions in the convolutional decoder, and when powerful RS codes are employed, they can correct burst errors generated by the inner decoder. Although we consider only the binary symmetric channel (BSC) in this paper, we focus on this class of concatenated codes to form the basis of future work. III. Lower and upper bounds of convolutional codes For a number of reasons, such as performance, convolutional codes are often used as the inner codes of a concatenated coding system. Viterbi [5] has derived the lower and upper bounds on the bit error probability in decoding a convolutional code transmitted over a memoryless channel as a function of the constraint length of that code. The probability of bit error, for any (n; b; m) convolutional code ( n bits per branch, b shift registers, memory of length m) and any decoding algorithm, is lower-
Data
Outer
Symbol Interleaver
Inner
Bit Interleaver
Source
Encoder
(optional)
Encoder
(optional)
Outer Channel Physical Channel
Data
Outer
Symbol Deinterleaver
Inner
Bit Deinterleaver
Sink
Decoder
(optional)
Decoder
(optional)
Figure 1: A block diagram of a concatenated coding system. R0 is the cuto rate, which for a BSC is given by
bounded by [6]
Pb 2?Kc b[Ecsp(R)+o(Kc )]=R ; where
Ecsp (R) = E0 () 0 < < 1 ;
(1)
h
p
R0 = 1 ? log2 1 + 2 p(1 ? p)
i
:
(5)
For all but pathological channels, the bounds are asymptotically (exponentially) tight for rates above R0 .
IV. Performance of Concatenated Codes Our objective is to nd the optimal inner code rate. When ideal interleaving is used between the inner code R = E0 ()= 0 < R < C ; and the outer code, the outer channel becomes a binary and Kc is the constraint length, given by Kc = m + 1. symmetric channel. If an (N = 2k ? 1; K ) RS code is The function E0 () is the Gallager function de ned as used as the outer code, we could compute the block error XX probability as [8] E0 () = maxf?log2 [ p(x)p(y j x)1=1+ ]1+ g ; !
and
p(x)
N X
N P i (1 ? P )N ?i ; (2) (6) PB = s s i i=t+1 where X and Y are the channel input and output spaces, respectively, p(y j x) is the channel transition probabil- where ity distribution, and p(x) is the input probability distribution. For BSC channel with crossover probability p, Ps = 1 ? (1 ? Pb )k ; and t = N ?2 K : E0 () is given as We have calculated the curves in Fig. 2 using (1) and E0 () = ? (1 + )log2 p 1+1 + (1 ? p) 1+1 : (3) (4) in (6). The parameter values are m = 4, N = 31 and overall code rate Rt = 1=4. These curves are drawn The bit error probability is upper-bounded by [7] by rst selecting an inner code rate, Ri , then computing the outer code rate from the overall code rate, Rt , and ?Kc E(R)=R Pb < [12 ? 2?(R) ]2 ; (4) K . Because of the limitation of the rates of RS codes with a xed k, we can not nd an exact K corresponding where to the outer code rate. In that case, two neighboring ( K are found to compute the curves drawn with 'stars' and 'diamonds' in Fig. 2. Fig. 2(a) shows the results E (R) = ER(0;); R < 0R
