Let the real discrete-time signal x[n] with Fourier transformX (ejω) be the input to a s...

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

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

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

(c) Suppose that it is of interest 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^jω) as a function of Y (e^jω).

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