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Asymptotic Quantuzation for Noisy Channels

We consider the problem of asymptotic quantization in conjunction with a noisy binary symmetric channel. For a noiseless channel, Bennett's integral is a formula for the distortion of a scalar quantizer given in terms of the source density, the number of quantization points (assumed to be large), and the distribution of quantization points, or point density. In the paper we extend Bennett's integral to the case where the quantizer is used in conjunction with a noisy binary symmetric channel, assuming that channel codewords are assigned randomly. We also derive an expression for the optimum noisy channel point density.