For sigma-delta oversampled A/D converters with high-order feedback loops, stability bec...

For sigma-delta oversampled A/D converters with high-order feedback loops, stability becomes a significant consideration. An alternative approach referred to as multi-stage noise shaping (MASH) achieves high-order noise shaping with only 1st-order feedback. The structure for 2nd-order MASH noise shaping is shown in Figure P4.68-2 and analyzed in

this problem.

Figure P4.68-1 is a 1st-order sigma-delta () noise shaping system, where the effect of the quantizer is represented by the additive noise signal e[n]. The noise e[n] is explicitly shown in the diagram as a second output of the system. Assume that the input x[n] is a zero-mean wide-sense stationary random process. Assume also that e[n] is zero mean, white, wide-sense stationary, and has variance e[n] is uncorrelated with x[n].

(a) For the system in Figure P4.68-1, the output y[n] has a component yx [n] due only to x[n] and a component ye[n] due only to e[n], i.e., y[n] = yx [n] + ye[n].

(i) Determine y_x[n] in terms of x[n].

(ii) Determine P_ye(ω), the power spectral density of y_e[n].

(a) The system ofFigure P4.68 is now connected in the configuration shown inFigure P4.68, which shows the structure of the MASH system. Notice that e₁[n] and e₂[n] are the noise signals resulting from the quantizers in the sigma-delta noise shaping systems. The output of the system r[n] has a component rx [n] owing only to x[n], and a component re[n] due only to the quantization noise, i.e., r[n] = r_x[n]+r_e[n]. Assume that e₁[n] and e₂[n] are zero-mean, white, wide-sense stationary, each with variance . Also assume that e₁[n] is uncorrelated with e₂[n].

(i) Determine r_x[n] in terms of x[n].

(ii) Determine P_re(ω), the power spectral density of r_e[n].