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Example of Windowing

Let's look at a simple example of windowing to demonstrate what happens when we turn an infinite duration signal into a finite duration signal through windowing.

Complex Sinusoid:

$\displaystyle x(n) = e^{j\omega n T}, \hspace{2cm} 0 \leq \omega T < \pi
$

Notes:

This signal is infinite duration. (It doesn't die out as $ n$ increases.) In order to end up with a signal which dies out eventually (so we can use the DFT), we need to multiply our signal by a window (which does die out).

Putting all this together, we get the following:

Our original signal (unwindowed, infinite duration), is

$\displaystyle x(n) = e^{j \omega_0 n T }, \hspace{.5cm} n \in {\bf Z}
$

A portion of the real part, $ \cos(\omega_0 nT)$ , is plotted below:

\epsfig{file=eps/infDurSin.eps,width=6in}

The imaginary part, $ \sin(\omega_0 nT)$ , is of course identical but for a 90-degree phase-shift to the right.

The Fourier Transform of this infinite duration signal is a delta function at $ \omega_0 $ :

$\displaystyle X( \omega ) = 2\pi\delta( \omega - \omega_0 ) = \delta(f-f_0)
$

\epsfig{file=eps/infDurSinSpec.eps,width=5in}

The following is a diagram of a typical window function:

\epsfig{file=eps/generalWindow.eps,width=5.5in}

This may be called a ``zero-centered'' (or ``zero phase'', or ``even'') window function, which means its phase in the frequency domain is either zero or $ \pi$ , as we will see in detail later. (Recall that a real and even function has a real and even Fourier transform.) The window is also nonnegative, as is typical.

We might also require that our window be zero for negative time. Such a window is said to be `causal'. Causal windows are necessary for real-time processing:

\epsfig{file=eps/causalWindow.eps,width=5in}

By shifting the original window in time by half its length, we have turned the original non-causal window into a causal window. The Shift property of the Fourier Transform tells us that we have introduced a linear phase term.

The windowed complex sinusoid is:

$\displaystyle x_w(n) \;=\;w(n)x(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;w(n)e^{-j \omega_0 n T} \hspace{1cm} n \in {\bf Z}
$

(Note carefully the difference between $ w$ and $ \omega$ .)

\epsfig{file=eps/windowedSin.eps,width=5in}

The Convolution Theorem tells us that our multiplication in the time domain results in a convolution in the frequency domain. Hence, in our case, we will obtain the convolution of a delta function at frequency $ \omega_0 $ , and the transform of the window:

$\displaystyle X_w(\omega) \;=\;(W \ast X)(\omega) \;=\;W(\omega-\omega_0)
$

The result of convolution with a delta function is the original function, shifted to the location of the delta function. (The delta function is the identity element for convolution.)


\begin{psfrags}\psfrag{freq}{$\omega T$}\psfrag{w}{\footnotesize $\omega_0$}\begin{center}\epsfig{file=eps/windowedSinSpec.eps,width=6in} \\
\end{center} % was epsfbox
\end{psfrags}



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``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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