The quantizer Q(·) in the system S1 (Figure P4.45-1) can be modeled with an additive noi...

The quantizer Q(·) in the system S1 (Figure P4.45-1) can be modeled with an additive noise. Figure P4.45-2 shows system S₂, which is a model for system S₁

The input x[n] is a zero-mean, wide-sense stationary random process with power spectral density The additive noise e[n] is wide-sense stationary white noise with zero mean and variance σ²_e . Input and additive noise are uncorrelated. The frequency response of the low-pass filter in all the diagrams has a unit gain.

(a) For system S₂ find the signal to noise ratio: Note that y_x [n] is the output due to x[n] alone and ye[n] is the output due to e[n] alone.

(b) To improve the SNR owing to quantization, the system of Figure P4.45-3 is proposed:

Where N > 0 is an integer such that πN << M. Replace the quantizer with the additive model, as in Figure P4.45-4. Express y_1x[n] in terms of x[n] and y_1e[n] in terms of e[n].

(c) Assume that e[n] is a zero mean wide-sense stationary white noise that is uncorrelated with input x[n]. Is y_1e[n] a wide-sense stationary signal? How about y₁[n]? Explain.

(d) Is the proposed method in part (b) improving the SNR? For which value of N is the SNR of the system in part (b) maximized?