A Closed-Form Expression for the Exact Bit Error Probability for Viterbi Decoding of Convolutional Codes
Irina E. Bocharova, Florian Hug, Rolf Johannesson, Boris D. Kudryashov · IEEE Transactions on Information Theory · 2012
In 1995, Bestpublished a formula for the exact bit error probability for Viterbi decoding of the rate$R=1/2$, memory$m=1$(two-state) convolutional encoder with generator matrix$G(D)=(1 \quad 1+D)$when used to communicate over the binary symmetric channel. Their formula was later extended to the rate$R=1/2$, memory$m=2$(four-state) convolutional encoder with generator matrix$G(D)=(1+D^{2} \quad 1+D+D^{2})$by LentmaierIn this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Bestfor the two-state encoder and by Lentmaierfor a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate$R=1/2$and$R=2/3$encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel.