2. A continuous-time signal xa (t)=sin2zt+0.1 sin127t is sampled at a sampling frequency 2, 207 rad/sec. A continuous-time double differentiator system is defined as Ye (1) dt² The continuous-time signal

xe (t) is processed by a discrete-time system as shown in the following figure. x(1) C/D T x[n] Discrete-time system y[n] D/C T y,(1) (a) Compute the Fourier transforms of the continuous-time signal xe (t) and the continuous-time sampled signal x, (t). Sketch the Fourier transforms of xe (t) and x, (t). Determine all the frequencies of the Fourier transform X, (j2) within the bandwidth 2 <. (b) Determine the Fourier transform of the discrete-time sampled signal x [n] for w<. What is the discrete-time sampled signal x [n]? (c) Obtain the frequency response of the discrete-time system so that it is equivalent to the continuous-time system. Then compute the Fourier transform of the discrete-time output signal y [n]. (d) The continuous-time output signal y, (t) is reconstructed by a reconstruction ideal low-pass filter H, (j). Determine the signal yr (t). Compare y, (t) to ye (t). Are they equal? If not, state the reason why this is the case. (e) If yr (t) is not equal to ye (t), how would you design a re- construction filter H, (j) to recover the desired reconstructed sig- nal yr (t)=ye (t). Sketch the reconstruction filter H, (j). Note that H, (j) needs not be an ideal low-pass filter.