Answer Happy

(b) (8 marks) Consider a system first sampling 22(t) = 1+į cos(30007t)+sin(4000t) by a impulse train p(t) = n=-20 8(t – NT), and then using a filter with spectrum Hw) to reconstruct it, as illustrated in Fig. 2. i) (3 marks) Suppose this system uses sampling at T = 4000 s. Express the spectrum for the sampled signal xp(t), \XpW), in terms of |X2(W)], and sketch |Xp(w). ii) (5 marks) Now, suppose this system uses an advanced sampling and reconstructing tech- nique. It sets T = 2000 s for sampling and uses a band-pass filter to reconstruct. The spectrum of this band-pass filter is given in Fig. 3. The spectrum of the reconstructed signal xr(t) is given by |X,(W)= cx \H(W) Xp(W)], where c is a constant. Under this setting, sketch the spectrum Xp(w) for the sampled signal xp(t), and specify the range of wa (see Fig. 3) such that ar(t) is a perfect reconstruction.
p(t) = 38(t-nT) nu Xp (t) x2(t) H@) *r(t) Figure 2: The sampling system considered in Question (b), Problem 5.
|H(0)| 11 all — Фь -a Фа Wb=+10507 Figure 3: The spectrum |H(jw)| in Question (b)-ii), Problem 5.