The performance of convolutional codes on the block erasure channel using various finite interleaving techniques

Consider the transmission of a finitely interleaved rate 1/n convolutionally encoded message over a channel with memory having two internal states Ξ₀ and Ξ₁ where when in state Ξ₀, the channel resembles a noiseless binary symmetric channel (BSC), whereas when in state Ξ₁, the channel is totally blocked and is well approximated by a binary-input single-output channel. Assume that the channel´s internal state is drawn at random once every h channel uses, and then remains constant for the following h channel uses. Further assume that the message is short in comparison to h, and that due to delay constraints, the message must be decoded within Nh channel uses, where N need not be large in comparison to the code´s constraint length. Such a model is appropriate for describing a convolutionally coded slow frequency hopping system with non-ideal interleaving, in which every frequency is either totally erased or else noiseless. The probability of a message error, the normalized expected number of bits in error, and the bit error rate (BER) are analytically computed for the periodic N×h bit and word interleavers, where a bit refers to a binary code symbol, and a word refers to a n-tuple of consecutive bits. An analytic expression for the BER is also given for pseudo-random word and bit interleavers and for the corresponding limiting cases of infinite interleaving, i.e., N→∞. As an example for the use of our methods, we analyze the performance of the GSM system with various interleaving depths and methods. We introduce the notion of “matched” code and interleaver pairs, and argue that this is a desirable property. Several exhaustive searches are carried out for matched codes and interleavers