An engineer is asked to evaluate the signal processing system shown in Figure P7.21-1 an...

An engineer is asked to evaluate the signal processing system shown in Figure P7.21-1 and improve it if necessary. The input x[n] is obtained by sampling a continuous-time signal at a sampling rate of 1/T = 100 Hz.

The goal is for H(e^jω) to be a linear-phase FIR filter, and ideally it should have the following amplitude response (so it can function as a bandlimited differentiator):

For one implementation of H(e^jω), referred to as H₁(e^jω), the designer, motivated by the definition

chooses the system impulse response h1[n] so that the input–output relationship is

Plot the amplitude response of H₁(e^jω) and discuss how well it matches the ideal response. You may find the following expansions helpful:

We want to cascade H₁(e^jω) with another linear-phase FIR filter G(e^jω), to ensure that for the combination of the two filters, the group delay is an integer number of samples. Should the length of the impulse response g[n] be an even or an odd integer?

Explain.

(c) Another method for designing the discrete-time H filter is the method of impulse invariance. In this method, the ideal bandlimited continuous-time impulse response, as given in Eq. (P7.21-1), is sampled.

(In a typical application, < π/T.) Based on this impulse response, we would have to create a new filter H₂ which is also FIR and linear phase. Therefore, the impulse response, h₂[n], should preserve the odd symmetry of h(t) about t = 0. Using the plot in Figure P7.21-2, indicate the location of samples that result if the impulse response is sampled at 100 Hz, and an impulse response of length 9 is obtained using a rectangular window.

(d) Again using the plot in Figure P7.21-2, indicate the location of samples if the impulse response h₂[n] is designed to have length 8, again preserving the odd symmetry of h(t) about t = 0.

(e) Since the desired magnitude response of H(e^jω) is large near ω = π, you do not want H₂ to have a zero at ω = π. Would you use an impulse response with an even or an odd number of samples? Explain.