You are required to design an FIR filter, h[n], with the following magnitude specificati...

You are required to design an FIR filter, h[n], with the following magnitude specifications:

• Passband edge: ω_p = π/100.

• Stopband edge: ω_s = π/50.

• Maximum stopband gain: δ_s ≤ −60 dB relative to passband.

It is suggested that you try using a Kaiser window. The Kaiser window design rules for shape parameter β and filter length M are provided in Section 7.5.3.

(a) What values of β and M are necessary to meet the required specifications? You show the resulting filter to your boss, and he is unsatisfied. He asks you to reduce the computations required for the filter. You bring in a consultant who suggests that you design the filter as a cascade of two stages: h’[n] = p[n] ∗ q[n]. To design p[n] he suggests first designing a filter, g[n], with passband edge ω’ p = 10ωp, stopband edge ω’s = 10ω_s and stopband gain δ’_s = δ_s . The filter p[n] is then obtained by expanding g[n] by a factor of 10:

(b) What values of β’and M’ are necessary to meet the required specifications for g[n]?

(c) Sketch P(e^jω) from ω = 0 to ω = π/4. You do not need to draw the exact shape of the frequency response; instead, you should show which regions of the frequency response are near 0 dB, and which regions are at or below −60 dB. Label all band edges in your sketch.

(d) What specifications should be used in designing q[n] to guarantee that h_[n] = p[n] ∗ q[n] meets or exceeds the original requirements? Specify the passband edge, ω”_p, stopband edge, ω”_s , and stopband attenuation, δ”_s , required for q[n].

(e) What values of β” and M” are necessary to meet the required specifications for q[n]? How many nonzero samples will h’[n] = q[n] ∗ p[n] have?

(f) The filter h’[n] from parts (b)–(e) is implemented by first directly convolving the input with q[n] and then directly convolving the results with p[n]. The filter h[n] from part (a) is implemented by directly convolving the input with h[n]. Which of these two implementations requires fewer multiplications? Explain. Note: you should not count multiplications by 0 as an operation.