Rate Allocation for Serial Concatenated Block Codes - Information ...

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Rate Allocation for Serial Concatenated Block Codes Maja Bystrom and Robert A. Coury

Abstract While serial concatenated codes were designed to provide good overall performance with reasonable system complexity, they may arise naturally in certain cases, such as the interface between two networks. In this work we consider the problem of constrained rate allocation between nonsystematic block codes in a serial concatenated coding system with either ideal or no interleaving between the codes. Given constraints on system parameters, such as a limit on the overall rate, analytic guidelines for the selection of good inner code rates are found by using an upper bound on the average system block error rate.

Keywords: channel coding, concatenated codes

I. I NTRODUCTION Serial concatenated block codes were developed to achieve a low average system error rate, without the decoding expense of applying a single long-block-length code [1]. However, a serial concatenated coding system may naturally occur in a heterogeneous network at the interface of two subnetworks if no transcoding is applied. In practice this may be realized when data either are coded separately from a transmission protocol or are transmitted between networks employing different protocols, and are not decoded and then re-encoded at the network interface. In these cases there may be constraints either on the total available rate or the set of available codes, and it may be useful to adapt code rate in the face of varying source rates, bandwidth, and/or channel conditions. This work was supported in part by NSF award #9870418, and was presented in part at the 2002 International Symposium on Information Theory. M. Bystrom is with the ECE Department, Boston University.

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This work addresses this possibility in one manner, by developing an analytical approximation to appropriate channel code rates, given selected system constraints. Consider the serial concatenated block coding scheme illustrated in Fig. 1. This serial coder consists of two nonsystematic block codes with a binary symmetric channel (BSC) as the inner (physical layer) channel. The inner coder together with the BSC form the outer channel which is presented to the outer coder. Between the two coders is an optional interleaver/deinterleaver. When an interleaver with sufficient depth is employed, the outer channel appears to be a BSC with crossover probability equal to the decoded bit error rate out of the inner decoder. If the code concatenation represents a system such as the heterogeneous network described previously, possible constraints on a concatenated coding scheme could be the overall code rate and the block lengths of the constituent codes. Typically, appropriate code rates for the constituent codes in the concatenated system have been found through heuristic methods and simulations. In 1988 Herro, et al. [2] first employed information-theoretic techniques to conclude that there is an optimal rate for the inner code of a concatenated block coding system, and that this optimum depends on the overall system capacity or cutoff rate. Furthermore, they concluded that given a fixed overall rate there exists an optimum rate distribution which minimizes the overall decoded bit error rate, and that this distribution is a function of the inner code block length. However, they did not present a method for finding the optimal rates, nor did they consider the effects of rate constraints. Similar results for concatenated coding systems using inner convolutional codes can be found in an earlier paper [3]. The rate tradeoff is further illustrated in Lin and Costello [4] with an example based on the work of Odenwalder [5]. In more recent work on turbo codes by Schlegel and Perez [6] it can be seen that performance varies by adjusting the constituent codes and interleaver. In this work we expand upon the results presented in [2] by assuming selected constraints on system parameters, and develop a tractable method for determining appropriate code rates as functions

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of these constraints. We consider the effect of both the inner and outer coders, as well as the channel SNR, on the overall block error rate and develop an analytic guideline for selection of appropriate inner code rates, extending the work in [2] to supply a guideline for appropriate rate selection that can readily be used in an adaptive system or can serve as the starting point for a heuristic or exhaustive search for optimal allocations. Given the computational complexity that would be involved in determining optimal operating points for wide ranges of physical channel qualities and system parameters, it would be useful to have an initial point in order to constrain searches. In Section II we further discuss the concatenated block coding system and the ensemble performance bound employed. This upper bound on overall block error rate is used to determine a guideline for selection of the rates of the constituent binary codes as a function of the selected constraints. The analytic guidelines are given in Section III and are compared to a concatenated block coding scheme with BCH constituent codes in Section IV.

II. S ERIAL C ONCATENATED B LOCK C ODES We denote the inner and outer code rates of the serial concatenated block coding scheme of Fig. 1 as






and

    , respectively, and assume that the block lengths, 


and

 , of the

codes are predetermined, as might occur in a practical coding system, while the information block

 , are permitted to vary. The overall system code rate, , is the product of the 
 . A constraint on the overall system rate,  , will rates of the two constituent codes, i.e.,  must be satisfied. In general, the subscript  will denote a parameter be imposed, so that lengths,




and

which is associated with the inner channel, the physical channel, while the subscript  indicates an association with the outer channel. Thus, the inner channel, respectively, while

"!

"!#


and

and

$&% !

$&% !#


denote the cutoff rate and bit error rate of

denote the cutoff rate and bit error rate of the

outer channel. An exception to this convention is that the block error rate of the outer channel is denoted by

$' !#
, while the overall block error rate of the system, that is, the decoded block error

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rate after the outer decoder is then denoted as $('

! .

In the system shown in Fig. 1, the inner (physical) channel is a BSC with crossover probability

$&% !#
, thus the cutoff rate )"!#


is given by

"! * ,+.-0/21 , 3 5 4687 $ % ! *:9 ,+;$ % ! *= for

@BADC 2EFHG

(1)

@  . The channel cutoff rate can be viewed to be the largest practical rate at which arbitrarily

small probability of error can be assumed for many codes. Assuming ideal interleaving is employed, the outer channel is a BSC as well, and thus

"!

is also given by (1) with

$I% !

being the decoded

bit error rate seen by the outer decoder. In [7] the average block error probability for a block code on a discrete memoryless channel is shown to satisfy

(2) $&' ! *&J L KLMONQPSRUT8NWV @BADC EFHG

X* 9Z*=< is the code-rate-dependent random coding exponent. where is the code block length and Y 9Z*=