The system of Figure 7.2 is used to perform filtering of continuous-time signals with a...

The system of Figure 7.2 is used to perform filtering of continuous-time signals with a digital filter. The sampling rate of the C/D and D/C converters is fs = 1/T = 10, 000 samples/sec.

A Kaiser window w_K[n] of length M + 1 = 23 and β = 3.395 is used to design a linear-phase lowpass filter with frequency response H_1p(e^jω). When used in the system of Figure 7.1 so that H(e^jω) = H_1p(e^jω), the overall effective frequency response (from input x_a(t) to output y_a(t)) meets the following specifications:

(a) The linear phase of the FIR filter introduces a time delay t_d. Find the time delay through the system (in milliseconds).

(b) Now a highpass filter is designed with the same Kaiser window by applying it to the ideal impulse response h_d [n] whose corresponding frequency response is

That is, a linear-phase FIR highpass filter with impulse response h_hp[n] = w_K[n]h_d[n] and frequency response H_hp(e^jω) was obtained by multiplying h_d [n] by the same Kaiser window w_K[n] that was used to design the first mentioned lowpass filter. The resulting FIR highpass discrete-time filter meets a set of specifications of the following form:

Use information from the lowpass filter specifications to determine the values of ω₁, ω₂, δ₁, δ₂, and G.