Performance estimation for concatenated coding schemes

Asymptotical analysis of concatenated codes with EXIT charts [tB99] or the AMCA [HH02b] is proven to be a powerful tool for the design of power–efficient communication systems. But, usually the result of the asymptotical analysis is a binary decision, whether convergence of iterative decoding is possible at the chosen signal–to– noise ratio, or not. In this paper it is shown how to obtain the Information Processing Characteristic (IPC) introduced in [HHJF01] for concatenated coding schemes. If asymptotical analysis is performed under the assumption of infinite interleaving and infinitely many iterations, this IPC will be a lower bound. Furthermore, it also is possible to estimate the performance of realistic coding schemes by restricting the number of iterations. Finally, the IPC can be used to estimate the resulting bit error ratio for the concatenated coding scheme. As an upper and a lower bound on the bit error ratio for a given IPC exist, we are able to lower bound the performance of any concatenated coding scheme and give an achievability bound, i.e. it is possible to determine a performance that can surely be achieved if sufficiently many iterations are performed and a large interleaver is used. I. SYSTEM MODEL In the following we analyze the properties of a digital communications system consisting of a binary Bernoulli source, a channel coder, a channel, a decoder and a sink. Without loss of generality we assume, that the source emits a block of K binary information symbols U [i], i 2 f1, 2, , Kg. The encoder maps the information vector ~ U to a codeword ~ X which consists of N symbols X [n], n 2 f1, 2, , Ng. The rate of the code, which is supposed to be time–invariant, is R = K/N measured in bit per channel symbol. The codeword ~ X is transmitted over a memoryless channel that corrupts the message by substitution errors, e.g., the binary symmetric channel (BSC) or the additive white Gaussian noise channel (AWGN Channel). Modulator and demodulator are considered as being part of the channel. Additionally we introduce an (theoretically infinite) interleaver π1 before encoding that converts the end–to–end channel between ~ U and ~ V to a memoryless channel. Figure 1: System model. The corrupted received sequence ~ Y is processed by the decoder. The decoder output is the soft–output w.r.t. symbol U [i], which is the a–posteriori probability taking the received vector ~ Y and code constraints into account, i.e., V [i] def = Pr ( U [i] = 0j~ Y ) . (1) Estimated symbols Û [i] can be obtained from the vector of soft–output values ~ V . II. INFORMATION PROCESSING CHARACTERISTICS The Information Processing Characteristic [HHJF01] for symbol–by–symbol decoding and Interleaving IPCI(C) def = Ī(U ; V ) def = 1 K K ∑ i=1 I(U [i]; V [i]) (2) characterizes a coding scheme w.r.t. soft–output, i.e. IPCI is the capacity of the memoryless end–to–end channel from U to soft–output V . In [HHFJ02] we proved by information theoretic bounding that the IPC of any coding scheme can be upper bounded by: IPCI(C) min (C/R, 1) . (3) A coding scheme fulfilling (3) with equality is called ideal coding scheme. The IPCI is important for two reasons. Firstly, the characterization w.r.t. soft–output is very helpful for the analysis and comparison of coding schemes, which will be used as components of concatenated codes [HHJF01]. Secondly, the IPCI can be a result of a convergence analysis performed with EXIT charts [tB99] or the AMCA [HH02b]. In the following we will show how the IPCI can be obtained for concatenated coding schemes and a relationship between the IPCI and the bit error ratio of the hard–output Û [i] will be derived. III. INFORMATION PROCESSING CHARACTERISTIC AS RESULT OF ASYMPTOTICAL ANALYSIS To obtain the Information Processing Characteristic for symbol–by–symbol decoding and interleaving IPCI(C), firstly we have to determine the mutual information between the source symbols U and the post-decoding soft–output V of the decoder using EXIT charts or the AMCA. It is possible to obtain both, a lower bound achieved by infinite interleaving and infinitely many iterations as well as estimations of the mutual information after an arbitrary number of iterations. As long as thereby the number of iterations is restricted such that the cycles in the graph of the code do not dominate the decoding performance the result will be close to bit error performance that can be measured if the whole coding scheme is simulated. Results or intermediate results of EXIT charts and the AMCA are the mutual information between the source symbols U in parallel concatenation or the encoded symbols of the outer encoder X in serial concatenation and the respective extrinsic soft–output at the decoder side Z resp. Q. The post– decoding information V , which is the final result at the output of an iterative decoder, is created by maximum ratio combining [Bre59] of the extrinsic informations of all consituent decoders on a symbol basis. This can be modelled statistically by information combining [HH02c]. For serial concatenation we also have to assume systematic encoding of the outer code, to ensure that the post–decoding mutual information w.r.t. info bits U is the same as the post– decoding mutual information w.r.t. code bits X of the outer encoder. Exemplary, IPCI(C) for the serial concatenation of Fig. 2 will be determined in the following. Figure 2: Encoder for serial concatenation of a rate–1/2 MFD convolutional code of memory ν = 1 with a ν = 2 scrambler (Gr = 07, G = 01). As the concatenation is of extremely low complexity, it can be assumed that even in practical implementations the limit of infinite number of iterations will be closely approximated. Hence, we firstly determine the intersection point of the transfer characteristics within EXIT charts for the range of signal– to–noise ratios. Then the post–decoding mutual information is calculated using information combining. I (U ;E ) = I (X ;Y ) [b it pe r sy m bo l] ! 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 log 1 0 (E s /N 0 )=− 1.5 dB