Question

(a) Based on the following discrete-time signal x[n], [n] →n -2 -1 0 1 2 3 4 i. [5%] determine the Fourier transform (i.e., X(ein)) and sketch the magnitude spectrum. ii. [4%] Given the following signal Xp[n], which is the periodic version of x[n] with period 4. Derive the Fourier series coefficients of yn], i.e., {ax}. xp[n] -1 1 2 3 4 5 iii. [4%] Hence, derive the Fourier transform of ap[n], i.e., Xp(es"). iv. [5%] Based on the results in (i), (ii) and (iii), state and explain the relationship be- tween X(ein), {ak} and X (ein). v. [4%] Hence, what is the Fourier transform of the following Izp[n], i.e., Xzp(ein)? p[n] 1 1 -2 -1 0 1 2 3 4 5 6 7 8 vi. [3%] The process of forming exp[n] from 2 [n] is called zero-padding. Base on your results, explain the purpose or function of zero-padding in constructing DTFT.

Answer 1

SOLUTION TO THE GIVEN PROBLEM IS:

Given discrete - time signal is 3 Fourier transform is given by -jon X(w = Xone ne-

ro, for n= -2, -1, 3, 4 Nin] = 1 for n= 0,2 13, for mil. X (W) = I a[nje jwn narz -jw =x[0] + xa] è uta (2] ējaw X(W) = 1+ 3 e jo erro x0w) + 3 [cos co -) Si'nw] + [Cos260 – Jin 260]. Xw)= (1 +3 caso + cos20) + j (-3 sinw - Sinao). magnitude of XCW) is » 1xcw/ : 1 (1+3606 6+ cos2w27 l3Simw + Sinzw) putting values of to as o, a, 20... we plot the graph of 1X(W)/..

'tuche spectrum » 1x(w) 115,0) 1 - - - - - - 21 26