## 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 sinusoidal carrier

*c*(*t*) = cos(ω_{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 conditions 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} > ω_{m} = 1,000π must be met to insure that x(t) can be recovered.

b) To demodulate, first multiply again by cos(ω_{c}*t*). Then feed the resulting signal through a low pass filter with a gain of 2 and a cutoff frequency of ω_{c}.

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

- Instructor's comment: Good! For the test, make sure you have a clear mental picture of what is happening in the frequency domain when you do this and why it works. -pm

### Answer 2

a) w_{c} > w_{m}

w_{c} > 1000pi

b) Multiply by cos(w_{c}t) then pass it through a Low Pass Filter with a gain of 2 and a cutoff f of w_{c}

_{} H(w) = 2 [u(w+w_{c})-u(w-w_{c})]

--Ssanthak 12:49, 19 April 2011 (UTC)

- Instructor's comment: It is a bit confusing if you talk about a "cutoff f", because the variable "f" is used to denote frequency in hertz. In this course, we are using $ \omega $ for which the units are radians per time unit. -pm

### Answer 3

Write it here.