As discussed in Section 4.8.2, to process sequences on a digital computer, we must quant...

As discussed in Section 4.8.2, to process sequences on a digital computer, we must quantize the amplitude of the sequence to a set of discrete levels. This quantization can be expressed in terms of passing the input sequence x[n] through a quantizer Q(x) that has an input–output relation as shown in Figure 4.54.

As discussed in Section 4.8.3, if the quantization interval Δ is small compared with changes in the level of the input sequence, we can assume that the output of the quantizer is of the form

y[n] = x[n] + e[n],

where e[n] = Q(x[n]) − x[n] and e[n] is a stationary random process with a 1^st-order probability density uniform between −Δ/2 and Δ/2, uncorrelated from sample to sample and uncorrelated with x[n], so that for all m and n.

Let x[n] be a stationary white-noise process with zero mean and variance

(a) Find the mean, variance, and autocorrelation sequence of e[n].

(b) What is the signal-to-quantizing-noise ratio

(c) The quantized signal y[n] is to be filtered by a digital filter with impulse response Determine the variance of the noise produced at the output

due to the input quantization noise, and determine the SNR at the output.

In some cases we may want to use nonlinear quantization steps, for example, logarithmically spaced quantization steps. This can be accomplished by applying uniform quantization to the logarithm of the input as depicted in Figure P4.63, where Q[·] is a uniform quantizer as specified in Figure 4.54. In this case, if we assume that Δ is small compared with changes in the sequence ln(x[n]), then we can assume that the output of the quantizer is

ln(y[n]) = ln(x[n]) + e[n].

Thus,

y[n] = x[n] · exp(e[n]).

For small e, we can approximate exp(e[n]) by (1 + e[n]), so that

y[n] ≈ x[n](1 + e[n]) = x[n] + f [n]. (P4.63-1)

This equation will be used to describe the effect of logarithmic quantization. We assume e[n] to be a stationary random process, uncorrelated from sample to sample, independent of the signal x[n], and with 1st-order probability density uniform between ±Δ/2.

(d) Determine the mean, variance, and autocorrelation sequence of the additive noise f [n] defined in Eq. (4.57).

(e) What is the signal-to-quantizing-noise ratio

( f ) The quantized signal y[n] is to be filtered by means of a digital filter with impulse response Determine the variance of the noise produced at the output due to the input quantization noise, and determine the SNR at the output.