Answer Happy

All of these problems will be easier to solve if drawn approximately to scale. For all plots / sketches, label (i) your axes, and numerical values for (ii) important times / frequencies, (iii) important amplitudes / areas. Continuous-time signal x(t) is given as x(t)=cos (100лt) (a) Sketch x(t) for at least 3 periods. We will be sampling and reconstructing this continuous-time signal throughout this exercise. (b) x(t) is sampled with sampling frequency=800. Sketch the sampled signal xp(t) for the same timespan as in (a). (c) xp(t) in (b) is reconstructed using an ideal interpolator. Will the reconstructed signal differ from x(t)? If no, so state and justify your answer. If yes, sketch the reconstructed signal x (t) for the same timespan as (a).

(d) x₂(t) in (b) is reconstructed using a zero-order hold (ZOH) interpolator. Will the reconstructed signal differ from x(t) ? If no, so state and justify your answer. If yes, sketch the reconstructed signal x,o(t) for the same timespan as (a). (e) Sketch X(jw), the spectrum of x(t). This is the spectrum of the original CT signal. (f) Sketch X₂(jw), the spectrum of the sampled signal x₂(t) as given in (b). Include at least 3 replicas.

(g) Will the spectrum X(jw) of the ideally reconstructed signal x₁(t) as given in (c) differ from the spectrum X(jw) of the original signal as sketched in (e)? If no, so state and justify your answer. If yes, sketch the spectrum of the ideally reconstructed signal X(jw). (h) Will the spectrum X,o(jw) of the ZOH reconstructed signal xo(t) as given in (d) differ from the spectrum X(jw) of the original signal as sketched in (e)? If no, so state and justify your answer. If yes, sketch the spectrum of the ZOH reconstructed signal X, (jo). You should evaluate the areas for the spectral replicas. (i) You should have noticed that the reconstructed spectrum Xro(jo) in (h) has "extra" high frequencies compared to the spectrum X(jo) in (e). This is due to the non-ideal nature of the ZOH reconstruction filter. If these extra high frequencies can be seen in the frequency domain, they must manifest in the time domain. Where do you see these extra high frequencies in the time domain plot of X,o(t) in (d)?