Spectrum of Quantization Noise and Conditions of Whiteness

When the input to a quantizer is a sampled time series represented by x1, x2, x3, . . . , the quantization noise is a time series represented by ν1, ν2, ν3, . . . Suppose that the input time series is stationary and that its statistics satisfy the conditions for multivariable QT II (it would be sufficient that two-variable QT II conditions were satisfied for x1 and x2, x1 and x3, x1 and x4, and so forth, because of stationarity). As such, the quantization noise will be uncorrelated with the quantizer input, and the quantization noise will be white, i.e. uncorrelated over time. The PQN model applies. The autocorrelation function of the quantizer output will be equal to the autocorrelation function of the input plus the autocorrelation function of the quantization noise. Fig. 20.1(a) is a sketch of an autocorrelation function of a quantizer input signal. Fig. 20.1(b) shows the autocorrelation function of the quantization noise when the PQN model applies. Fig. 20.1(c) shows the corresponding autocorrelation function of the quantizer output. Corresponding to the autocorrelation functions of Fig. 20.1, the power spectrum of the quantizer output is equal to the power spectrum of the input plus the power spectrum of the quantization noise. This spectrum is flat, with a total power of q2/12. When it is known that the PQN model applies perfectly or otherwise applies to a very close approximation, one can infer the autocorrelation function and power spectrum of the quantizer input from knowledge of the autocorrelation function and power spectrum of the quantizer output, since the autocorrelation function and power spectrum of the quantization noise are known, these only need to be subtracted. Spectral analysis of quantization is very simple when the quantization noise is white and uncorrelated with the quantizer input signal. We will present methods for determining the whiteness condition based on the multivariable characteristic function of the quantizer input. Other methods for doing this exist in the literature, and it is the purpose of this chapter to to explore these methods, and in some cases, to enhance them.