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[[Category:discrete time Fourier transform]] | [[Category:discrete time Fourier transform]] | ||
| − | = [[ | + | <center><font size= 4> |
| + | '''[[Digital_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' | ||
| + | </font size> | ||
| + | |||
| + | Topic: Discrete-time Fourier transform computation | ||
| + | |||
| + | </center> | ||
| + | ---- | ||
| + | ==Question== | ||
Compute the discrete-time Fourier transform of the following signal: | Compute the discrete-time Fourier transform of the following signal: | ||
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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! | 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=== | ===Answer 1=== | ||
| + | <math>x[n]=\sin \left( \frac{2 \pi}{100} \right)</math> | ||
| − | <math>x[n]= | + | <math>x[n] = \frac{1}{2j} \left( e^{ \frac{j2 \pi}{100n}}-e^{- \frac{j2 \pi}{100n}} \right)</math> |
| + | <math>X_(\omega) = \sum_{n=-\infty}^{+\infty} x[n] e^{-j\omega n}</math> | ||
| − | <math> | + | <math>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)</math> |
| − | |||
| − | <math>X_(\omega) = \ | + | <math>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</math> |
| + | |||
| + | :<span style="color:green"> Instructor's comment: You need to learn to find the answer without using a table. Now, I am not sure which table you used, but it must be wrong, since the anwer you obtained is not periodic with period <math>2\pi</math>.</span> | ||
| + | |||
| + | :<span style="color:red"> TA's comment: You could rather say, X(ω) equals this from -π to π and it's 2π periodic. </span> | ||
===Answer 2=== | ===Answer 2=== | ||
| − | + | ||
| + | First, write the original function as: | ||
| + | <math>x[n] = \frac{1}{2j} \left( e^{ \frac{j2 \pi}{100n}}-e^{- \frac{j2 \pi}{100n}} \right)</math> | ||
| + | |||
| + | |||
| + | Then, for w = [-pi, pi] (<span style="color:green"> Instructor's comment: You need more justification here.)</span> | ||
| + | |||
| + | <math>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)</math> | ||
| + | |||
| + | <math>X_(\omega) = \frac{100}{2j} \left( \delta \left( \frac{100}{2pi}\omega - 1 \right) + \left( \frac{100}{2pi}\omega + 1 \right) \right)</math> | ||
| + | |||
| + | which is really is: | ||
| + | |||
| + | <math>X_(\omega) = rep_2pi \frac{50}{j} \left( \delta \left( \frac{100}{2pi}\omega - 1 \right) + \left( \frac{100}{2pi}\omega + 1 \right) \right)</math> | ||
| + | |||
| + | :<span style="color:green"> Instructor's comment: You should make it clear which expressions are valid for all values of <math>\omega</math>, and which expressions are only valid for <math>\omega \in [-\pi, \pi ]</math>.</span> | ||
===Answer 3=== | ===Answer 3=== | ||
| − | + | Xiang Zhang | |
| + | We can separate <span style="color:green"> (Instructor's comment: separate? Do you mean "write"?)</span>the equation (<span style="color:green"> Instructor's comment: it's not an equation: it's a signal, or a function.)</span> to the following function | ||
| + | <math>x[n]=\frac{1}{2 j} \left( e^\frac{j 2 \pi n}{100} - e^\frac{- j 2 \pi n}{100} \right) </math> | ||
| + | |||
| + | Because based on Fourier transform equation, | ||
| + | |||
| + | <math>X_(\omega) = \sum_{n = -\infty}^{\infty} x[n] e^{-j \omega n}</math> | ||
| + | |||
| + | Substitute in x[n] | ||
| + | |||
| + | <math>X_(\omega) = \frac{1}{2 j} \left( \sum_{n = -\infty}^{\infty} e^{ \frac{j2 \pi n} {100} } e^{-j\omega n} - \sum_{n = -\infty}^{ \infty} e^{\frac{-j2 \pi n} {100} } e^{-j\omega n} \right)</math> | ||
| + | |||
| + | <span style="color:green"> (Instructor's comment: Why write the equation above if you are going to use a FT pair from a table?)</span> | ||
| + | |||
| + | From Discrete Fourier Transform pair, | ||
| + | |||
| + | <math> x[n] = e^{-j\omega_0 n} </math> DTFT to <math> X_(\omega) = 2 \pi \sum_{n = -\infty}^{ \infty} \delta \left( \omega-\omega_0 - 2\pi l \right) </math> | ||
| + | |||
| + | <span style="color:green"> (Instructor's comment: Careful above! The original signal was called x[n]; you can't reuse x[n] for a different signal.)</span> | ||
| + | |||
| + | Hence, the function <span style="color:green"> (Instructor's comment: Function? You mean "DTFT"?.)</span> will be | ||
| + | |||
| + | <math> X_(\omega) = \frac{\pi}j \left( \sum_{n = -\infty}^{ \infty} \delta \left( \omega-\omega_0 - 2\pi l \right) - \sum_{n = -\infty}^{ \infty} \delta \left( \omega+\omega_0 - 2\pi l \right) \right) </math> | ||
| + | |||
| + | <span style="color:green"> (Instructor's comment: What is <math>\omega_0</math>?)</span> | ||
| + | |||
| + | <math>x[n]=\sin \left( \frac{2\pi}{100} n \right)</math> | ||
| + | |||
| + | <span style="color:green"> (Instructor's comment: You don't need to re-write the signal.)</span> | ||
| + | ---- | ||
[[2013_Fall_ECE_438_Boutin|Back to ECE438 Fall 2013]] | [[2013_Fall_ECE_438_Boutin|Back to ECE438 Fall 2013]] | ||
Latest revision as of 11:36, 26 November 2013
Practice Question on "Digital Signal Processing"
Topic: Discrete-time Fourier transform computation
Question
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.)
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
$ 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 $
- Instructor's comment: You need to learn to find the answer without using a table. Now, I am not sure which table you used, but it must be wrong, since the anwer you obtained is not periodic with period $ 2\pi $.
- TA's comment: You could rather say, X(ω) equals this from -π to π and it's 2π periodic.
Answer 2
First, write the original function as: $ x[n] = \frac{1}{2j} \left( e^{ \frac{j2 \pi}{100n}}-e^{- \frac{j2 \pi}{100n}} \right) $
Then, for w = [-pi, pi] ( Instructor's comment: You need more justification here.)
$ 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{100}{2j} \left( \delta \left( \frac{100}{2pi}\omega - 1 \right) + \left( \frac{100}{2pi}\omega + 1 \right) \right) $
which is really is:
$ X_(\omega) = rep_2pi \frac{50}{j} \left( \delta \left( \frac{100}{2pi}\omega - 1 \right) + \left( \frac{100}{2pi}\omega + 1 \right) \right) $
- Instructor's comment: You should make it clear which expressions are valid for all values of $ \omega $, and which expressions are only valid for $ \omega \in [-\pi, \pi ] $.
Answer 3
Xiang Zhang
We can separate (Instructor's comment: separate? Do you mean "write"?)the equation ( Instructor's comment: it's not an equation: it's a signal, or a function.) to the following function
$ x[n]=\frac{1}{2 j} \left( e^\frac{j 2 \pi n}{100} - e^\frac{- j 2 \pi n}{100} \right) $
Because based on Fourier transform equation,
$ X_(\omega) = \sum_{n = -\infty}^{\infty} x[n] e^{-j \omega n} $
Substitute in x[n]
$ X_(\omega) = \frac{1}{2 j} \left( \sum_{n = -\infty}^{\infty} e^{ \frac{j2 \pi n} {100} } e^{-j\omega n} - \sum_{n = -\infty}^{ \infty} e^{\frac{-j2 \pi n} {100} } e^{-j\omega n} \right) $
(Instructor's comment: Why write the equation above if you are going to use a FT pair from a table?)
From Discrete Fourier Transform pair,
$ x[n] = e^{-j\omega_0 n} $ DTFT to $ X_(\omega) = 2 \pi \sum_{n = -\infty}^{ \infty} \delta \left( \omega-\omega_0 - 2\pi l \right) $
(Instructor's comment: Careful above! The original signal was called x[n]; you can't reuse x[n] for a different signal.)
Hence, the function (Instructor's comment: Function? You mean "DTFT"?.) will be
$ X_(\omega) = \frac{\pi}j \left( \sum_{n = -\infty}^{ \infty} \delta \left( \omega-\omega_0 - 2\pi l \right) - \sum_{n = -\infty}^{ \infty} \delta \left( \omega+\omega_0 - 2\pi l \right) \right) $
(Instructor's comment: What is $ \omega_0 $?)
$ x[n]=\sin \left( \frac{2\pi}{100} n \right) $
(Instructor's comment: You don't need to re-write the signal.)
