Figure P5.76 shows the zeros of the system function H(z) for a real causal linear-phase...

Figure P5.76 shows the zeros of the system function H(z) for a real causal linear-phase FIR filter. All of the indicated zeros represent factors of the form (1−az ⁻¹). The corresponding poles at z = 0 for these factors are not shown in the figure. The filter has approximately unity gain in its passband.

(a) One of the zeros has magnitude 0.5 and angle 153 degrees. Determine the exact location of as many other zeros as you can from this information.

(b) The system function H(z) is used in the system for discrete-time processing of continuous time signals shown in Figure 4.10, with the sampling period T = 0.5 msec. Assume that the continuous-time input X_c(j Ω) is bandlimited and that the sampling rate is high enough to avoid aliasing. What is the time delay (in msec) through the entire system, assuming that both C/D and D/C conversion require negligible amounts of time?

(c) For the system in part (b), sketch the overall effective continuous-time frequency response 20 log₁₀ |H_eff (j Ω)| for 0 ≤ Ω ≤ π/T as accurately as possible using the given information. From the information in Figure P5.76 estimate the frequencies at which H_eff (j Ω) = 0, and mark them on your plot.