(A) Consider the bandlimited signal x(t) = sinc(4t) sinc(t). What is fs,min, the minimum sam- pling rate for ideal reconstruction? What is the energy of x? (use Parseval's identity) (B) Let x[n] denote samples of x(t) obtained by sampling at rate fs = = = 1.5fs,min (i.e., 50% faster than the minimum sampling rate). Compute analytically and sketch the DTFT Ŷ (ƒ) of the samples over 3 periods, centered around f = 0. Hint: Use the relationship between the DTFT and the Fourier transform of the impulse trained sampled waveform Σn x(nT)(t - nT). (C) For our numerical experiments, we consider a truncated version of x(t): x₁(t) = x(t)I\_To/2,To/2](t). Set To 8. What percentage of the energy of x is captured in this truncation interval? You will need to compute the energy of x₁ numerically and compare it with the energy of x computed analytically in (A). (D) Do a stem plot of samples x[n] at rate fs (of the truncated waveform in (C)). (E) Recalling that fs = 1.5fs,min, which of the following reconstruction filters can be expected to provide ideal reconstruction of x(t) (up to scale) from its samples? (1) p(t) = sinc(5t)sinc(t), (2) p(t): your answers. = = sinc(6t) sinc(t), (3) p(t) = sinc(8t)sinc(3t). Give reasons for (F) For numerical experiments, consider a truncated version of p(t): p1(t) = p(t)I[-T。/2,To/2](t). Again, set To = 8. What percentage of the energy of p is captured in this truncation interval? You will need to compute the energy of p₁ numerically and compare it with analytical computa- tion of the energy of p using Parseval's identity. (G) Using a "fast" sampling rate ffast = 16fs (or faster if needed) to emulate continuous time reconstruction, write a Matlab program that produces: (1) the upsampled sequence x [k] and a stem plot of it. (2) samples of p(t), truncated as in (f), at rate ffast, and a stem plot of the samples. Let us call these samples p[n]. (H) Implement (4), convolving x with p to obtain x, [m]. Recognizing that each increment in m corresponds to an increment in t of Tfast = 1/ffast, plot x,(t) versus t. Plot x(t) versus t in the same way. Comment on how the two waveforms are related. (I) Replot x,(t) and x(t) after delaying and scaling x,(t) to best align it with x(t) (accounting for the delay and scaling due to filtering). Eyeball the plots to comment on how good the recon- struction is.