Computing upper bounds to error probability of coded modulation schemes

To evaluate an upper bound on error probabilities of signal constellations used for transmission over the additive white Gaussian noise (AWGN) channel, enumeration of all the constellation intradistances is required. These may be infinite in number, for example, when convolutional codes are used and the constellations are lattices. Truncation of the series does not necessarily provide a bound anymore, and must be done with care. Yet the union bound is very simple, as it does not require any further knowledge about the signal constellation than the distance enumerator. In this paper, we describe some methods that can be used to evaluate error probabilities of infinite signal constellations, and that require only a finite number of terms. These methods are applicable, for example, to convolutional codes decoded with a finite-depth Viterbi algorithm and to signal constellations carved from lattices. Coded modulations based on lattices and convolutional or block codes can also be dealt with. As an example of application, we analyze a variable-rate 3-stage coded modulation encoder/decoder, which has been built and is based on a combination of convolutional codes with a single-parity-check block code.