Lecture 13 - Notes

February 3, 2016

The Discrete-Time Fourier Transform

definition: We define the Discrete-Time Fourier Transform as,

$$X\left( e^{j\omega} \right) = \sum_{k = -\infty}^{\infty} x[k]e^{-j \omega k}$$

and the inverse transform,

$$\begin{align} x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X\left(e^{j\omega}\right) e^{j \omega n} ~d\omega \end{align}$$

Example

Find the Fourier transform of,

$$\begin{align} x[n] = a^n u[n] \end{align}$$

where $|a| \lt 1$.

So,

$$\begin{align} X\left( e^{j\omega} \right) &= \sum_{k = -\infty}^{\infty} x[k]e^{-j \omega k} \\\\ &= \sum_{k = -\infty}^{\infty} a^k u[k] e^{-j \omega k} \\\\ &= \sum_{k = 0}^{\infty} a^k e^{-j \omega k} \\\\ &= \sum_{k = 0}^{\infty} \left( a e^{-j \omega} \right)^k \\\\ &= \frac{1}{1 - a e^{-j \omega}} \\\\ \end{align}$$

Example

Find the inverse Fourier transform of,

$$\begin{align} X\left(e^{j\omega}\right) = 1 + 3e^{-j\omega} + 2e^{-j2\omega} - 4e^{-j3\omega} + e^{-j10\omega} \end{align}$$

Using the properties of Fourier transforms,

$$\begin{align} x[n] = \underbrace{\delta[n]}_{1} + \underbrace{3\delta[n-1]}_{3e^{-j \omega}} + \underbrace{2\delta[n-2]}_{2e^{-j2\omega}} - \underbrace{4\delta[n-3]}_{4e^{-j3\omega}} + \underbrace{\delta[n-10]}_{e^{-j10\omega}} \end{align}$$