We consider the transmission of numerical data over a noisy channel. Two sources of error exist. The first is the quantizer where the input data is mapped into a finite set of rational numbers and the second is the channel which includes the encoder, transmitter, transmission medium, receiver, and decoder. For any given probability density on the input data and any given channel matrix, we determine the quantization values and transition levels which minimize the total mean-square error. We also determine the best quantizer structure under the constraint that quantization values and transition levels be equally spaced. For the special case of a noiseless channel both results reduce to those of Max [2]. As an example we consider the case of Gaussian input data and phase-shift keyed (PSK) transmission in additive white Gaussian noise. The transmitter is both peak and average power limited, and the system operates in real time. Both the natural and Gray codes are considered. The mean-square error, quantizer entropy, channel capacity, and information rate are computed for the system using the optimum uniform quantizer. Finally, we show that it is important to take the channel into consideration when designing the quantizer even when the system is not constrained to operate in real time. © 1969 IEEE. All rights reserved.
