Topic: Discrete-time Fourier transform computation

## Question

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

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

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

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

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

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

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

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.

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 $\omega \in [-\pi, \pi ]$.

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.)