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Practice Question on Computing the Fourier Transform of a Discrete-time Signal
Compute the Fourier transform of the signal
$ x[n] = 3^n u[-n].\ $
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Answer 1
$ {\mathcal X} (\omega) = \sum_{n=-\infty}^\infty x[n]e^{-j\omega n}=\sum_{n=-\infty}^\infty 3^n u[-n]e^{-j\omega n}=\sum_{n=-\infty}^0 3^n e^{-j\omega n}=\sum_{n=-\infty}^0 \Bigg(\frac{3}{e^{j\omega}}\Bigg)^n $ Let k=-n
$ = \sum_{n=0}^\infty \Bigg(\frac{e^{j\omega}}{3}\Bigg)^k $
$ \mathcal X (\omega) = \frac{1}{1-\frac{e^{j\omega}}{3}} $
--Cmcmican 19:42, 28 February 2011 (UTC)
- TA's comments: The answer is correct. The geometric series converges because $ \color{OliveGreen}{\left|\frac{e^{j\omega}}{3}\right|=\frac{1}{3}<1} $.
Answer 2
Write it here.
Answer 3
Write it here.
