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.
This signal is infinite duration. (It doesn't die out as 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
A portion of the real part, , is plotted below:
The imaginary part, , 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 :
The following is a diagram of a typical window function:
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 , 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:
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:
(Note carefully the difference between and .)
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 , and the transform of the window:
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.)