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Lecture 8: Discrete-Time Fourier Transform (DTFT)

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Schedule :: Transforms :: DTFT :: System Analysis :: Properties

Recall that the Z-transform is

\begin{displaymath}F(z) = \sum_{n=0}^\infty f[n]z^{-n}.\end{displaymath}

The DTFT is defined as

\begin{displaymath}F(\Omega) = \sum_{n=-\infty}^\infty f[n]e^{-j\Omega n}
\end{displaymath}

Clearly, if $f[n]$ is causal, the DTFT is the Z-transform evaluated on the unit circle. (Just like the FT is the Laplace transform evaluated on the $j\omega$ axis.) Observe that $F(\Omega)$ is $2\pi$ -periodic:

\begin{displaymath}F(\Omega) = F(\Omega+2\pi).
\end{displaymath}

Again we see this idea of the periodic repetition of the spectrum in the frequency domain and the source of aliasing.

Note that $F(\Omega)$ is a function of continuous frequency $\Omega$ -- we are not looking at samples of the spectrum (as for the DFT), and hence the function does not have to be periodic.

But, $F(\Omega)$ is periodic, and hence, $F(\Omega)$ has a Fourier series representation: its F.S. coefficient are just the samples $f[k]$ . But this representation allows us to write down an inverse DTFT : to go from $F(\Omega)$ to $f[k]$ , simply stick things in the formula for the F.S. coefficients, where the period of the function is $2\pi$

\begin{displaymath}f[k] = \frac{1}{2\pi}\int_{2\pi} F(\Omega)e^{jk\Omega}d\Omega. \end{displaymath}

(Need we point out that, yet again, in going from time domain to frequency domain the exponent has negative sign, and in going from frequency domain to time domain the exponent has a positive sign?) Notationally we will write

\begin{displaymath}f[k] \Leftrightarrow F(\Omega). \end{displaymath}

And, as for other transforms, we can talk about the amplitude and phase spectra.


\begin{example}Find the DTFT of $f[k] = a^k u[k]$.
\begin{displaymath}F(\Omega) ...
...ate. Any higher frequencies get wrapped
back around. \vspace*{5in}
\end{example}


\begin{example}An example that leads to some important insight when dealing
with...
...of like a sinc
function, except that it is periodic. \vspace*{3in}
\end{example}
(Comment on spectral leakage).


\begin{example}We can also compute an inverse transform: Let
\begin{displaymath}...
...e^{jk\Omega} d\Omega
= \frac{1}{4} \sinc(k\pi/4). \end{displaymath}\end{example}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 05). Lecture 8: Discrete-Time Fourier Transform (DTFT). Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/8_2node2.html. This work is licensed under a Creative Commons License Creative Commons License