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| − | = Practice Question on Computing the Fourier Transform of a Continuous-time Signal = | + | = [[:Category:Problem_solving|Practice Question]] on Computing the Fourier Transform of a Continuous-time Signal = |
Compute the Fourier transform of the signal | Compute the Fourier transform of the signal | ||
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=== Answer 1 === | === Answer 1 === | ||
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| + | Use answer to previous practice problem and the time shifting property. | ||
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| + | <math>\mathfrak{F}\Bigg(s(t-t_0)\Bigg)=e^{j\omega t_0}\mathfrak{F}\Bigg(x(t)\Bigg)</math> | ||
| + | |||
| + | Therefore, | ||
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| + | <math>\mathcal X (\omega)=e^{j\omega \frac{\pi}{12}}2\pi \delta(\omega-2\pi k)</math> | ||
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| + | --[[User:Cmcmican|Cmcmican]] 20:52, 21 February 2011 (UTC) | ||
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| + | :TA's comments: In the time shift property of the Fourier transform that you provided, it should be <math class="inline">e^{-j\omega t_0}</math> and not <math class="inline">e^{j\omega t_0}</math>. Another thing is that the transform of a cosine should yield only two deltas in the frequency domain. | ||
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=== Answer 2 === | === Answer 2 === | ||
| − | + | I'll try this again, using my new answer from the previous problem, and correcting my time shifting property. | |
| + | |||
| + | <math>\mathfrak{F}\Bigg(s(t-t_0)\Bigg)=e^{-j\omega t_0}\mathfrak{F}\Bigg(x(t)\Bigg)</math> | ||
| + | |||
| + | Therefore <math class="inline">\mathcal X (\omega) =e^{-j\omega \frac{\pi}{12}}\Bigg(\pi\delta(\omega-2\pi)+\pi\delta(\omega+2\pi)\Bigg)</math> | ||
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| + | --[[User:Cmcmican|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 <math>t_0</math> 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 | ||
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=== Answer 3 === | === 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 | ||
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Write it here. | Write it here. | ||
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[[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | [[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | ||
Latest revision as of 09:26, 11 November 2011
Contents
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} )\ $
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
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.
