A Closed-Form Expression for the Exact Bit Error Probability for Viterbi Decoding of Convolutional Codes

In 1995, Best

published 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 Lentmaier

In 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 Best

for the two-state encoder and by Lentmaier

for 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.