Consider the system in Figure. 1. Assume that X c (j?) = 0 for |?| ? ?

Question:

Consider the system in Figure.

1. Assume that X_c(j?) = 0 for |?| ? ? / T and that denotes an ideal lowpass reconstruction filter.

2. The D/A converter has a built-in zero-order-hold circuit, so that where (we neglect quantization in the D/A converter.)?

3. The second system in Figure is a linear-phase FIR discrete-time system with frequency response H(e^j?). 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 Y_c(j?) = H_eff(j?) X_c(j?). Determine an expression for H_eff(j?) in terms of H(e^j?T) and T.

(b) If the linear-phase FIR system is such that h[n] = 0 for n 51, and T = 10^?4s, 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 stop band) within the following tolerances:?

0.99 ? |H_eff(j?)| ? 1.01,? ? ? ? ? ? |?| ? 2? (1000),

|H_eff(j?)| ? 0.01,? ? ? ? ?? 2?(2000) ? |?| ? 2?(5000).?

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^j?) that should be used. Find and sketch the weighting function W(?) that should be used. Sketch a ?typical? frequency response H(e^j?) that might result.

(d) How would you modify your results in part (c) to include magnitude compensation for a reconstruction filter H_r(j?) with zero gain above ? = 2?(5000), but with sloping passband??