Reduction of Quantizing Noise by Use of FeedbackReceived June 25, 1962. This paper is based in part on n dissertation by H. Austin Spung, III for the degree of Doctor of Engineering in the Yale Sehool of Enginccring. The dissertation was supported by Bell Telephone Laborntories, Murray Hill, N. J

Many information transmission systems use a discrete (digital) channel. Since most input signals are continuous, the conversion cannot be accomplished without an error which, for many cases, may be considered to have the characteristics of white noise. A method has been suggested to reduce this error by using linear feedback around the quantizer to shape the noise spectrum. Each output sample will then contain not only signal information but also information about the errors in the previous samples. Such a system is analyzed for random input signals of a rather general nature. Under assumptions allowing essentially no clipping in the quantizer and setting an upper bound on the coherence between samples of the input signal, the system can be represented by a simple model. A comparison is made of the mean-square error with and without feedback. It is shown that considerable reduction in noise power can be obtained by a slight increase in sampling rate. For example, an increase of 25 per cent in the sampling rate provides a 95 per cent decrease in error-noise power. This is equivalent to having about two additional bits per sample in the transmission channel.