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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.

 

 

 

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