In many real applications, practical constraints do not allow long time sequences to be...

In many real applications, practical constraints do not allow long time sequences to be processed. However, significant information can be gained from a windowed section of the sequence. In this problem, you will look at computing the Fourier transform of an infinite-duration signal x[n], given only a block of 256 samples in the range 0 ≤ n ≤ 255. You decide to use a 256-point DFT to estimate the transform by defining the signal

and computing the 256-point DFT of

(a) Suppose the signal x[n] came from sampling a continuous-time signal x_c(t) with sampling frequency fs = 20 kHz; i.e.,

x[n] = x_c(nT_s),

1/T_s = 20 kHz.

Assume that x_c(t) is bandlimited to 10 kHz. If the DFT of 0, 1, . . . , 255, what are the continuous-time frequencies corresponding to the DFT indices k = 32 and k = 231? Be sure to express your answers in Hertz.

(b) Express the DTFT of in terms of the DTFT of x[n] and the DTFT of a 256-point rectangular window w_R [n]. Use the notation X (e^jω) and W_R(e^jω) to represent the DTFTs of x[n] and w_R [n], respectively

(c) Suppose you try an averaging technique to estimate the transform for k = 32:

Averaging in this manner is equivalent to multiplying the signal by a new window w_avg[n] before computing the DFT. Show that W_avg(e^jω) must satisfy

where L = 256.

(d) Show that the DTFT of this new window can be written in terms of W_R(e^jω) and two shifted versions of W_R(e^jω).

(e) Derive a simple formula for w_avg[n], and sketch the window for α = −0.5 and 0 ≤ n ≤ 255.