On the union bound applied to convolutional codes

To evaluate the performance of a convolutional code it is common to use a union bound on either the biterror rate or the first-event error rate. Since these bounds are given as infinite sums they must be truncated to be evaluated. However, as the tail of the infinite sum is discarded, the upper bound turns into an approximation. In this paper, two methods to evaluate the union bound while taking the summation tail into account and thus maintaining the upper bound are described. The first method is based on a state matrix description of the convolutional encoder, while the second method is based on the transfer function. Results are given that compare the upper bounds and the truncated bound (approximation) to computer simulations. The standard rule-of-thumb that the union bound is tight as long as the error rates are below 10−3 is readily seen to be valid. Keywords— convolutional code, union bound, truncation of union bound, state transition matrix, transfer function bound