Practice Question on signal modulation

Let x(t) be a signal whose Fourier transform $ {\mathcal X} (\omega) $ satisfies

$ {\mathcal X} (\omega)=0 \text{ when }|\omega| > 1,000 \pi . $

The signal x(t) is modulated with the complex exponential carrier

$ c(t)= e^{j \omega_c t }. $

a) What conditions should be put on ωc to insure that x(t) can be recovered from the modulated signal x(t)c(t)?

b) Assuming the condition you stated in a) are met, how can one recover x(t) from the modulated signal x(t)c(t)?

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Answer 1

a) ωc > 0

b) to recover x(t) from x(t)c(t), multiply x(t)c(t) by $ e^{-j \omega_c t }. $

--Cmcmican 20:56, 7 April 2011 (UTC)

Answer 2

a) wc > wm

    wc > 1000pi

b)Since y(t) = x(t) e^jwct

        So x(t) = y(t) e^-jwct

   so to demodulate multiply by e^-jwct

--Ssanthak 12:39, 19 April 2011 (UTC)

Answer 3

a) Since c(t) only produces a phase shift, there is no potential for overlap of the signal, and no conditions are needed on ωc.  Even if ωc was zero, that is fine, it just means a shift of zero (and negative would just shift it in the opposite direction.)  It has to be a real number, though, right?  Would we ever need to state that?

b) I agree with those two on the rest.

--Kellsper 22:16, 20 April 2011 (UTC)

I agree with you a) should have been, no conditions on wc. --Ssanthak 09:50, 21 April 2011 (UTC)

Back to ECE301 Spring 2011 Prof. Boutin

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