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:

$\displaystyle s_{\omega_0}(n) = e^{j \omega_0 n T }, \quad n \in {\bf Z}$

A portion of the real part, $\cos(\omega_0 nT)$ , is plotted in Fig.1.3. The imaginary part, $\sin(\omega_0 nT)$ , is of course identical but for a 90-degree phase-shift to the right.

**Figure 1.3:** A portion of the real part of the sinusoid $s_{\omega _0}(n)$ .
$\includegraphics[width=0.6\textwidth]{eps/infDurSin}$

The Fourier transform of this infinite-duration signal is a delta function at $\omega=\omega_0$ . I.e., $S_{\omega_0}(\omega) = 2\pi\delta(\omega - \omega_0)$ , as indicated in Fig.1.4.

**Figure 1.4:** Spectrum (DTFT) of an infinite-duration sinusoid at frequency $\omega _0$ .
$\begin{figure}\input fig/infDurSinSpec.pstex_t \end{figure}$

The windowed signal is

$\displaystyle s_R(n) = w(n)e^{j \omega_0 n T}, \quad n \in {\bf Z}$

as shown in Fig.1.5. (Note carefully the difference between

and $\omega$ .)

**Figure 1.5:** Windowed sinusoid real part.
$\includegraphics[width=4in,height=2in]{eps/windowedSin}$

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 $\delta(\omega-\omega_0)$ with the Fourier transform of the window $W(\omega)$ . This is easy since the delta function is the identity element under convolution ( $\delta \ast W = W$ ). However, since our delta function is at frequency $\omega=\omega_0$ , the convolution shifts the window transform out to that frequency:

$\displaystyle S_R(\omega) = W(\omega)\ast 2\pi\delta(\omega - \omega_0) = 2\pi W(\omega-\omega_0)$

This is shown in Fig.1.6.

**Figure:** Fourier transform of the windowed sinsuoid in Fig.1.5: Top: *Real* Fourier transform *amplitude*. Bottom: Fourier transform *magnitude* in *decibels* (dB).
$\includegraphics[width=\textwidth]{eps/windowedSinSpec}$

From comparing Fig.1.6 with the ideal sinusoidal spectrum in Fig.1.4 (an impulse at frequency $\omega _0$ ), 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 In particular, the infinitely thin delta function has been replaced by the ``main lobe'' of the window transform.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 $\omega _0$ , and phase $\phi$ manifests (in practical spectrum analysis) as a window transform shifted out to frequency $\omega _0$ , and scaled by $A e^{j\phi}$ .

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. A more comprehensive reference is the paper by Harris [88] (with some corrections appearing in [176]).

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Next: The Rectangular Window

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
Spectrum Analysis of Sinusoids
Effect of Windowing