Question: Let the real discrete-time signal x[n] with Fourier transform X(e j? ) be the input to a system with the output defined by (a) Sketch

Let the real discrete-time signal x[n] with Fourier transform X(e^j?) be the input to a system with the output defined by

x[n]. if n is even, 0, y[n] = otherwise.

(a) Sketch the discrete-time signal s[n] = 1 + cos(? n) and its (generalized) Fourier transform S(e ^j?),

(b) Express Y(e^j?), the Fourier transform of the output, as a function of X(e^j?) and S(e^j?).

(c) You would like to approximate x[n] by the interpolated signal w[n] = y[n] + (1/2)(y[n + 1] + y[n ? 1]). Determine the Fourier transform W(e^j?) as a function of Y(e^j?).

(d) Sketch X(e^j?), y(e^j?), and W(e^j?) for the case when x[n] = sin(?n/a)/(?n/a) and a > 1. under what conditions is the proposed interpolated signal w[n] a good approximation for the original x[n]

x[n]. if n is even, 0, y[n] = otherwise.

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