# Quantization Noise

For years, rumors have been circulating about quantization noise: (a) The noise is additive and white and uncorrelated with the signal being quantized, and (b) The noise is uniformly distributed between plus and minus half a quanta, having zero mean and a mean square of one twelveth the square of a quanta. Yet, simple reasoning tells another story: (a) The noise is related to the signal being quantized, (b) The probability distribution of the noise depends on the probability distribution of the signal being quantized, and (c) The noise will be correlated over time if the signal being quantized is correlated over time. In spite of the simple reasoning, the rumors turn out to be true when quantizing theorems, analogous to the well known sampling theorem, are satisfied. Quantization, a nonlinear process, can be analyzed by linear sampling theory applied to the probability density distribution of the signal being quantized. In practice, the quantizing theorems are almost never perfectly satisfied (by analogy, signals are almost never perfectly band-limited so the sampling theorem is almost never perfectly satisfied). However, the rumors about quantization noise turn out to be extremely close to being true for a wide practical range of signal characteristics. When the rumors are true, signal processing, communication, and control systems containing nonlinear quantization behave like noisy linear systems and are easy to analyze. The original theory was developed by B. Widrow in his 1956 MIT doctoral thesis, applied to fixed-point (uniform) quantization. The theory was extended by Widrow and I. Kollar in the 1990's to apply to floating-point quantization. Their work on these subjects was published in a Cambridge University Press book in 2008 entitled "Quantization Noise".

Speaker: Bernard Widrow, Stanford

Room 101

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