When designing discrete filters  the specifications can be given in the time domain. One can think of  converting the frequency domain specifications into the time domain.  Assume you wish to obtain a filter that approximates an ideal low-pass  filter with a cut-off frequency ω_c= Ï€/2 and that has a linear phase ˆ’N_ω.  Thus, the frequency response is

(a) Find the corresponding impulse response using the inverse DTFT  of H(e^jω).

(b) If N = 50, plot h[n] using the MATLAB function stemfor  0 ‰¤ n ‰¤ 100. Comment on the symmetry of h[n]. Assume h[n] ‰ˆ 0 for n < 0 and n > 100, plot it as well as the corresponding magnitude and phase of H(e^jω).

(c) Suppose we want a band-pass filter of center frequency ω₀ = Ï€/2,  use the above impulse response h[n] to obtain the impulse response  of the desired band-pass filter.

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