Performance of concatenated Reed-Solomon and TCM over DMT modulation

In discrete multi-tone (DMT) modulation the information is conveyed using a set of tones. Each tone is modulated by a complex quadrature amplitude modulation (QAM) symbol and transmitted for a fixed time interval. In order to enhance the reliability of the system a concatenated coding scheme is often used: trellis coded modulation (TCM) as the inner code and Reed Solomon (RS) as the outer code. This coding scheme is used in DSL modems, particularly in ADSL. One of the basic performance measures of the system is the bit error rate (BER). Usually the designer is given the signal to noise ratio (SNR) distribution over the available set of tones, and is asked to find the BER for a given bit allocation, or to optimize bit allocation under the constraint of maximal BER. Performance analysis of the concatenated coding scheme is quite difficult and often it is necessary to use approximations, which have limited accuracy. In this work we develop a computational procedure to solve the BER estimation problem. The procedure is based on two steps. The first step is a recursive computation of the probability of the occurrence of a certain number of RS symbol (byte) errors in an RS codeword. The second step is based on a finite state machine (FSM) description of the TCM decoder output. We also discuss the extension of this scheme to a finite D-way interleaver.