Question

1. Consider the complex-valued signal r(t) with Fourier transform as shown in the figure. Keep in mind that there are no symmetry properties this signal satisfies in the fre- quency domain. In particular, the Fourier transform is zero for negative frequencies Suppose we impulse-train sample x(t) at the rate of 500 samples/second. 200 400 600 800 1000 FREQUENCY (Hz.) (a) Sketch the Fourier transform of the impulse-train sampled signal in the range of frequencies from -1000 Hz. to 1000 Hz. [12 points] (b) Is it possible to reconstruct r(t) back from its impulse-train samples? If yes, how would you do it? Give enough details to make sure that the grader will be able to implement such a system. If not, why? 13 points]

Answer 1

(a) Given that Given that we have impulse train with sampling frequency (fs) of 500 samples/sec we know that sampled signal of x(t) is given by y(f) = fs sigma^infinity_k = -infinity x(f - kfs) here r(f) is sampled signal Fourier transform We have to draw it between -1000 Hz to 1000 Hz y(f) = 500 sigma^infinity_k = -infinity x(f - 500 k) (b) Yes, we can reconstruct x(t) by passing sampled signal y(f) from band pass filter with pass band gain equal to y 500 and pass band (400 Hz, 600 Hz) \r\n