Consider the system in Figure P7.55. 3. The second system in Figure P7.55 i...

Consider the system in Figure P7.55.

3. The second system in Figure P7.55 is a linear-phase FIR discrete-time system with frequency response H(e^jω). We wish to design the FIR system using the Parks–McClellan algorithm to compensate for the effects of the zero-order-hold system.

(a) The Fourier transform of the output is . Determine an expression for in terms of

(b) If the linear-phase FIR system is such that h[n] = 0 for n < 0 and n > 51, and T = 10⁻⁴ s, what is the overall time delay (in ms) between x_c(t) and y_c(t)?

(c) Suppose that when T = 10⁻⁴ s, we want the effective frequency response to be equiripple (in both the passband and the stopband) within the following tolerances:

We want to achieve this by designing an optimum linear-phase filter (using the Parks– McClellan algorithm) that includes compensation for the zero-order hold. Give an equation for the ideal response H_d(e^jω) that should be used. Find and sketch the weighting function W(ω) that should be used. Sketch a “typical” frequency response H(e^jω) that might result.

(d) How would you modify your results in part (c) to include magnitude compensation for a reconstruction filter with zero gain above but with sloping passband?