Practice Question on Computing the Fourier Transform of a Continuous-time Signal

Compute the Fourier transform of the signal

$ x(t) = \cos (2 \pi t+\frac{\pi}{12} )\ $


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Answer 1

Use answer to previous practice problem and the time shifting property.

$ \mathfrak{F}\Bigg(s(t-t_0)\Bigg)=e^{j\omega t_0}\mathfrak{F}\Bigg(x(t)\Bigg) $

Therefore,

$ \mathcal X (\omega)=e^{j\omega \frac{\pi}{12}}2\pi \delta(\omega-2\pi k) $

--Cmcmican 20:52, 21 February 2011 (UTC)

TA's comments: In the time shift property of the Fourier transform that you provided, it should be $ e^{-j\omega t_0} $ and not $ e^{j\omega t_0} $. Another thing is that the transform of a cosine should yield only two deltas in the frequency domain.

Answer 2

I'll try this again, using my new answer from the previous problem, and correcting my time shifting property.

$ \mathfrak{F}\Bigg(s(t-t_0)\Bigg)=e^{-j\omega t_0}\mathfrak{F}\Bigg(x(t)\Bigg) $

Therefore $ \mathcal X (\omega) =e^{-j\omega \frac{\pi}{12}}\Bigg(\pi\delta(\omega-2\pi)+\pi\delta(\omega+2\pi)\Bigg) $

--Cmcmican 17:43, 23 February 2011 (UTC)

TA's comments: You're almost there. You got the transform of the cosine right. However, regarding the time shift property, you still have some mistake in it. Try first to identify what is $ t_0 $ equal to in this case.
Instructor's hint: this has to do with cascading a time shift and a time scaling. Recall that the order of the operation is relevant.-pm

Answer 3

Instructor's suggestion: how about writing this as a linear combination of two complex exponentials, and then "guessing" the right transform for each exponential separately (pulling out the constants using linearity of the Fourier transform)? -pm

Write it here.


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