Consider the two-stage system shown in Figure P7.37 for interpolating a sequence x[n] =...

Consider the two-stage system shown in Figure P7.37 for interpolating a sequence x[n] = x_c(nT ) to a sampling rate that is 15 times as high as the input sampling rate; i.e., we desire y[n] = x_c(nT /15).

Assume that the input sequence x[n] = x_c(nT ) was obtained by sampling a bandlimited continuous-time signal whose Fourier transform satisfies the following condition: |X_c(jΩ)| = 0 for |Ω| ≥ 2π(3600). Assume that the original sampling period was T = 1/8000.

(a) Make a sketch of the Fourier transform X_c(jΩ) of a “typical” bandlimited input signal and the corresponding discrete-time Fourier transforms X(e^jω) and X_e(e^jω).

(b) To implement the interpolation system, we must, of course, use nonideal filters. Use your plot of X_e(e^jω) obtained in part (a) to determine the passband and stopband cutoff frequencies (ω_p1 and ω_s1) required to preserve the original band of frequencies essentially unmodified while significantly attenuating the images of the baseband spectrum. (That is, we desire that w[n] ≈ x_c(nT /3).) Assuming that this can be achieved with passband approximation error δ₁ = 0.005 (for filter passband gain of 1) and stopband approximation error δ₂ = 0.01, plot the specifications for the design of the filter H₁(e^jω) for −π ≤ ω ≤ π.

(c) Assuming that w[n] = x_c(nT /3), make a sketch of W_e(e^jω) and use it to determine the passband and stopband cutoff frequencies ω_p2 and ω_s2 required for the second filter.

(d) Use the formula of Eq. (7.117) to determine the filter orders M₁ and M₂ for Parks– McClellan filters that have the passband and stopband cutoff frequencies determined in parts (b) and (c) with δ₁ = 0.005 and δ₂ = 0.01 for both filters.

(e) Determine how many multiplications are required to compute 15 samples of the output for this case.