On the spectrum of distances of a multilevel code, decoded by a multistage decoder

In this correspondence we are concerned with the error rates contributed by component codes in a multilevel code, decoded by a multistage decoder. These error rates are required for estimating the performance of the structure, but more importantly, they can serve as a design tool for achieving high coding gain. The latter objective can be reached by approaching a balance among these error rates. For evaluating the error rate contributed by a component code, it is essential to calculate the corresponding spectrum of Euclidean distances. A method for computing this spectrum is developed in this correspondence. The method is general and applicable to any signal constellation, and convolutional as well as block component codes. The error coefficient, available in the literature for some multilevel codes with an infinite size signal constellation, is shown to be significantly different from that of a practical finite constellation. Note that the complete spectrum is essential when codewords (of a component code) with Hamming distance beyond the minimum distance affect the error rate (see, e.g., Turbo codes).