Effect 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.

We begin with a sampled complex sinusoid:

(6.14) |

A portion of the real part, , is plotted in Fig.5.3. 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
. *I.e.*,
, as indicated in Fig.5.4.

The windowed signal is

(6.15) |

as shown in Fig.5.5. (Note carefully the difference between and .)

The convolution theorem (§2.3.5) tells us that our multiplication in the time domain results in a convolution in the frequency domain. Hence, we will obtain the convolution of with the Fourier transform of the window . This is easy since the delta function is the identity element under convolution ( ). However, since our delta function is at frequency , the convolution shifts the window transform out to that frequency:

(6.16) |

This is shown in Fig.5.6.

From comparing Fig.5.6 with the ideal sinusoidal spectrum in Fig.5.4 (an impulse at frequency ), we can make some observations:

- Windowing in the time domain resulted in a
*``smearing''*or*``smoothing''*in the frequency domain. In particular, the infinitely thin delta function has been replaced by the ``main lobe'' of the window transform. We need to be aware of this if we are trying to resolve sinusoids which are close together in frequency. - Windowing also introduced
*side lobes*. This is important when we are trying to resolve low amplitude sinusoids in the presence of higher amplitude signals. - A sinusoid at amplitude
, frequency
, and phase
manifests (in practical spectrum analysis) as a
*window transform*shifted out to frequency , and scaled by .

As a result of the last point above, the *ideal window transform*
is an *impulse in the frequency domain*. Since this cannot be
achieved in practice, we try to find spectrum-analysis windows which
*approximate* this ideal in some optimal sense. In particular,
we want side-lobes that are as close to zero as possible, and we want
the main lobe to be as tall and narrow as possible. (Since absolute
scalings are normally arbitrary in signal processing, ``tall'' can be
defined as the ratio of main-lobe amplitude to side-lobe
amplitude--or main-lobe energy to side-lobe energy, etc.) There are
many alternative formulations for ``approximating an impulse'', and
each such formulation leads to a particular spectrum-analysis window
which is optimal in that sense. In addition to these windows, there
are many more which arise in other applications. Many commonly used
window types are summarized in Chapter 3.

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