In this problem, we will consider the “double integration” system for quantization with...

In this problem, we will consider the “double integration” system for quantization with noise shaping shown in Figure P4.67. In this system,

and the frequency response of the decimation filter is

The noise source e[n], which represents a quantizer, is assumed to be a zero-mean whitenoise (constant power spectrum) signal that is uniformly distributed in amplitude and has noise power

(a) Determine an equation for Y(z) in terms of X (z) and E(z). Assume for this part that E(z) exists. From the z-transform relation, show that y[n] can be expressed in the form y[n] = x[n − 1] + f [n], where f [n] is the output owing to the noise source e[n].What is the time-domain relation between f [n] and e[n]?

(b) Now assume that e[n] is a white-noise signal as described prior to part (a). Use the result from part (a) to show that the power spectrum of the noise f [n] is

What is the total noise power in the noise component of the signal y[n]? On the same set of axes, sketch the power spectra P_ee(e^jω) and P_ff(e^jω) for 0 ≤ ω ≤ π.

(c) Now assume that X (e^jω) = 0 for π/M < ω ≤ π. Argue that the output of H₃(z) is w[n] = x[n − 1] + g[n]. State in words what g[n] is.

(d) Determine an expression for the noise power at the output of the decimation filter. Assume that π/M << π, i.e., M is large, so that you can use a small-angle approximation to simplify the evaluation of the integral.

(e) After the decimator, the output is v[n] = w[Mn] = x[Mn − 1] + q[n], where q[n] = g[Mn]. Now suppose that x[n] = x_c(nT ) (i.e., x[n] was obtained by sampling a continuous-time signal). What condition must be satisfied by X_c(j Ω) so that x[n − 1] will pass through the filter unchanged? Express the “signal component” of the output v[n] in terms of x_c(t). What is the total power