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 x[n].

(a) Suppose the signal x[n] came from sampling a continuous-time signal x_c(t) with sampling frequency ƒ_s = 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 x[n] is written X[k], k = 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 x[n] 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:

X_avg[32] = αX[31] + X[32] + αX[33].

Averaging in this manner is equivalent to multiplying the signal x[n] 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.

Transcribed Image Text:

S x[n]. 0