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Lecture 9: Aliasing and Leakage

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Schedule :: The DFT :: Aliasing & Leakage :: Examples :: The FFT :: Convolution

There are two effects that are introduced into the computation of the DFT.

Aliasing
Since when we compute we are necessarily dealing with a time-limited set of data, the signal cannot be bandlimited. The sampling process, with its incumbent spectral duplication, therefore introduces aliasing. This aliasing effect can be reduced by sampling faster.
Leakage
If the function $x(t)$ is not really time limited, then we truncate it in order to obtain a finite set of samples. As viewed above, the mathematics sees the sampled signal as if it were periodic in time. There are two ways of viewing what is going on. First, if we have a function $x(t)$ , we can obtain a time-truncated version of it by
\begin{displaymath}y(t) = x(t)w(t) \end{displaymath}

where $y(t)$ is the truncated version and $w(t)$ is a windowing function. In the frequency domain, the effect is to smear the spectrum out,

\begin{displaymath}Y(\omega) = \frac{1}{2\pi}X(\omega)* W(\omega) \end{displaymath}

This smearing is spectral leakage. Another way of viewing the leakage is this: if we truncate a function then make it periodic, the resulting function is going to have additional frequency components in it that were not in the original function, due to the change from end to end. The only way this does not happen is if the the signal is periodic with respect to the number of samples already.

Leakage can be reduced either by taking more samples (wider windows of data), i.e. increasing $N_0$ . It can also be reduced by choosing a different window function. However, it can never be completely eliminated for most functions.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Lecture 9: Aliasing and Leakage. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/9_2node2.html. This work is licensed under a Creative Commons License Creative Commons License