Discrete-time Fourier Transform (DTFT)

A slecture by ECE student Xian Zhang

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

## Outline

• Definition
• Periodicity property
• Example of computation of DTFT of a complex exponential
• Conclusion
• References

## Definition

The discrete-time Fourier transform (DTFT) of a discrete set of real or complex numbers x[n] with n=all integers, is a Fourier series, which produces a periodic function of a frequency as the variable, $\omega$. The Fourier series is:

\begin{align} \\ \mathcal{X}_1(\omega) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} \\ \end{align}

Inverse DTFT is :

\begin{align} \\ \quad x[n] & =\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{X}(\omega)e^{j\omega n}d\omega \\ \end{align}

## Periodicity property

$\mathcal{X}(\omega)$ is periodic with period $2\pi$. Because,

\begin{align} \\ \mathcal{X}(\omega+2\pi) & = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega+2\pi)n}\\ & =\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}e^{-j2\pi n}\\ & =\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\\ & =\mathcal{X}(\omega)\\ \end{align}

## Example of computation of DTFT of a complex exponential

Given: \begin{align} \\ x[n] =e^{j\omega_o n}\\ \end{align}

Proof: \begin{align} \\ \mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\ \end{align}

Firstly, let's try:

\begin{align} \\ \mathcal{X}(\omega) & = \sum_{n=-\infty}^{\infty}e^{j\omega_o n}e^{-j\omega n}\\ & = \sum_{n=-\infty}^{\infty}e^{j(\omega-\omega_o) n}\\ & = \infty, \omega = \omega_o; 0, else \\ \end{align}

How to compute something that diverges:

\begin{align} \\ e^{j\omega_o n} & =\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n}d\omega \\ \end{align}

Assume $\omega$ is between $0$ to $2\pi$.

If \begin{align} \\ \mathcal{X}(\omega) & =2\pi \delta(\omega - \omega_o).\\ \end{align} The result works for $\omega$ between $0$ to $2\pi$.

But this is not the final answer. Because Fourier transform must be periodic but the answer we got above is not periodic.

Since $\mathcal{X}(\omega)$ must be periodic with period $2\pi$.

The final answer is:

\begin{align} \\ \mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\ \end{align}

## Conclusion

Understand the basic calculation of Discrete-time Fourier Transform (DTFT) and inverse Fourier transform. Notice an important property of DTFT: the periodicity property. Master the basic DTFT computation of complex exponential.

## Reference

[1]. Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26, 2009