Revision as of 17:43, 12 September 2013

Practice Problem on Discrete-time Fourier transform computation

Compute the discrete-time Fourier transform of the following signal:

$ x[n]= \sin \left( \frac{2 \pi }{100} n \right) $

(Write enough intermediate steps to fully justify your answer.)

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Answer 1

$ x[n]=\sin \left( \frac{2 \pi}{100} \right) $

$ x[n] = \frac{1}{2j} \left( e^{ \frac{j2 \pi}{100n}}-e^{- \frac{j2 \pi}{100n}} \right) $

$ X_(\omega) = \sum_{n=-\infty}^{+\infty} x[n] e^{-j\omega n} $

$ X_(\omega) = \frac{1}{2j} \left( \sum_{n=-\infty}^{+\infty} e^{ \frac{j2 \pi} {100} n} e^{-j\omega n} - \sum_{n=-\infty}^{+\infty} e^{\frac{-j2 \pi} {100} n} e^{-j\omega n} \right) $

$ X_(\omega) = \frac{\pi}{j} \left( \delta \left({\omega - \frac{2 \pi}{100}}\right) - \delta \left({\omega + \frac{2 \pi}{100}}\right) \right) by DTFT table $

Answer 2

Write it here.

Answer 3

We can set the $ x[n]=\sin \left( frac{2\pi} 100 \right) $

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