Figure 1 shows the magnitude of the 32-point Discrete Fourier Transform (DFT) X[k], k=0,....31 of discrete time signal x[n]= e!!!! limited by window w[n]:

1 0 < n < 7

wTnl = ,

0 otherwise

(a) Sketch the Discrete Time Fourier Transform (DTFT) X(O) of x[n] (2 marks)

(b) Comment on the shape of the spectrum X[k] provided by the DFT relative to that of X(O). Why are there so many frequency components in the DFT spectrum? Explain. (8 marks)

(c) Which normalised radial frequency O do the following indices k correspond to:

• k = 16

• k = 6(4 marks)

(d) Which of the two actions described below would bring the shape of the DFT spectrum X[k] closer to that of X(O)

a) Increasing the length of the DFT to 256

b) Increasing the length of window w[n] to 32

Justify your answer. (6 marks)

Q2 (a) Determine whether each of the following statements is true or false (where u is the unit step function). Justify your answers.

i. The signal x(t) = u(t+To) - u(t-To) will not suffer from aliasing if it is sampled with a sampling period T<2T₀ .

ii. The signal x(t) with Fourier transform X(o>) = u(w+^0)- u(s-s 0) will not suffer from aliasing if it is sampled with a sampling period T < 冗.

iii. The signal x(t) with Fourier transform X(s) = u(s)- u(o)-3 0) will not suffer from aliasing if it is sampled with a sampling period T < 2n/^! . (9 marks)

(b) Consider an ideal discrete-time band-stop filter with impulse response h[n] for which the frequency response in the interval —冗 < Q < 冗 is:

(i i 兀 i i 3n\

h(Q) = LlQM/ndlQI '才

、O, elsewhere ，

Determine the frequency response of the filter whose impulse response is h\2n]. (4 marks)

(c) Consider the system of Figure 2 Gr changing the sampling rate of signal x[n] by a non-integer factor (the low-pass Hltcr of Hgurc 2 is assumed to be ideal). Determine the output x_d [n] if x\n\ = cos(3"〃/ 4), L=6 and M=7.

(4 marks)