## Contents

# 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*)?

You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!

### 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_{c}t

So x(t) = y(t) e^-jw_{c}t

so to demodulate multiply by e^-jw_{c}t

--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)