View source for Discrete-time Fourier transform Xian Zhang ECE438 slecture - Rhea

[[Category:slecture]]
[[Category:ECE438Fall2014Boutin]] 
[[Category:ECE]]
[[Category:ECE438]]
[[Category:signal processing]]
 [[Category:Discrete-time_Fourier_transform]]

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Discrete-time Fourier Transform (DTFT)
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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Xian Zhang

Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. 
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== Outline ==
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*Definition
*Periodicity property
*Example of computation of DTFT of a complex exponential
*Conclusion
*References
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== 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, <math>\omega</math>. The Fourier series is:

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

Inverse DTFT is :

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

\end{align}
</math>
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== Periodicity property ==
<math>\mathcal{X}(\omega)</math> is periodic with period <math>2\pi</math>.
Because,

<math> \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}
</math>
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== Example of computation of DTFT of a complex exponential ==
Given:
<math> \begin{align} \\
x[n] =e^{j\omega_o n}\\
\end{align}
</math>

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


Firstly, let's try:

<math> \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}
</math>

How to compute something that diverges:


<math> \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}
</math>

Assume <math>\omega</math> is between <math>0</math> to <math>2\pi</math>.

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

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

Since <math>\mathcal{X}(\omega)</math> must be periodic with period <math>2\pi</math>.

The final answer is:

<math> \begin{align} \\
\mathcal{X}(\omega) & = 2\pi rep_{2\pi}(\delta(\omega - \omega_o))\\
\end{align}
</math>
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== 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.
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== Reference ==
[1]. Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26, 2009
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