Consider a signal x[n] = 0.5n(0.8)ⁿ (u[n] − u[n − 40])

(a) To compute the DFT of x[n] we pad it with zeros so as to obtain a signal with length 2^γ, larger than the length of x[n] but the closest  to it. Determine the value of γ and use the MATLAB function fft to compute the DFT X[k] of the padded-with-zeros signal. Plot its  magnitude and phase using stem. Compute then a N = 2¹⁰ FFT of  x[n], and compare its magnitude and phase DFTs by plotting using  stem.

(b) Consider x[n] a period of a periodic signal of fundamental period  N = 40. Consider 2 and 4 periods and compute their DFTs using  the fft algorithm and then plot its magnitude and phase. How do the magnitude responses compare? what do you need to make  them equal? How do the phases compare after the magnitude responses are made equal?