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 ω_c 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) ω_c > 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) w_c > w_m

    w_c > 1000pi

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

        So x(t) = y(t) e^-jw_ct

   so to demodulate multiply by e^-jw_ct

--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 ω_c.  Even if ω_c 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 w_c. --Ssanthak 09:50, 21 April 2011 (UTC)

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