Practice Question on sampling and reconstruction (related to Nyquist rate)

The signal

$ x(t)= \frac{\sin (3 \pi t)}{\pi t} $

is sampled with a sampling period T. For what values of T is it possible to reconstruct the signal from its sampling?

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Answer 1

from the Table

x(w) = u(w+3pi)-u(w-3pi)

Thus the signal is bandlimited with a w_m = 3pi

We must sample above the Nyquist Rate which is equal to 2w_m or 6pi

w_s > 6pi

T = 2pi/w_s < 2/6 = 1/3

The signal can be reconstructed for all T < 1/3.

--Ssanthak 12:09, 20 April 2011 (UTC)

Instructor's comment: Why do you say that we "must" sample above Nyquist? Is it possible that one could still be able to reconstruct when sampling below Nyquist? -pm

I guess must was a bad choice of words, we should sample above Nyquist to guarantee that we can reconstruct the signal. In this case I do not believe we can sample below the Nyquist rate because the signal is present in all frequencies from -3pi to 3pi. If the signal was asymmetric then we could sample below Nyquist provided the copies never overlap.
--Ssanthak 15:50, 20 April 2011 (UTC)

TA's comment: I think you meant that the spectrum of the signal might not always be symmetric and not the signal itself (in time domain). The signal is complex if its spectrum is not even symmetric.

Answer 2

Write it here

Answer 3

Write it here.

Back to ECE301 Spring 2011 Prof. Boutin