# Digital Signal Processing/Windowing

Most digital signals are infinite, or sufficiently large that the dataset cannot be manipulated as a whole. Sufficiently large signals are also difficult to analyze statistically, because statistical calculations require all points to be available for analysis. In order to avoid these problems, engineers typically analyze small subsets of the total data, through a process called windowing.

## Windowing Introduction

Windowing is the process of taking a small subset of a larger dataset, for processing and analysis. A naive approach, the rectangular window, involves simply truncating the dataset before and after the window, while not modifying the contents of the window at all. However, as we will see, this is a poor method of windowing and causes power leakage.

## Applying Windows

Application of a window to a dataset will alter the spectral properties of that dataset. In a rectangular window, for instance, all the data points outside the window are truncated and therefore assumed to be zero. The cut-off points at the ends of the sample will introduce high-frequency components.

Consider the system H(z), with input X(z) and output Y(z). We model this as:

$Y(z) = X(z) H(z)$

If we have a window with transfer function W(z), we can mathematically apply the window to our signal, X(z) as such:

$\hat{X}(z) = X(z) W(z)$

Then, we can pass our windowed signal into our system, H(z) as usual:

$\hat{Y}(z) = \hat{X}(z) H(z)$

## Leakage

If our signal is a lowpass or passband signal, the application of a window will introduce high-frequency components. Power from the original signal will be diverted from the specified frequency band into the high-frequency areas. This redistribution of power from a target band to the upper frequencies is known as leakage.

If we look at a rectangular window, we know from duality that the frequency response of that window is a sinc function, which has non-zero values all the way out to both positive and negative infinity. Convolution of the sinc function with any narrow-band signal is going to cause a very spread-out spectrum.

## Types of Windows

### Rectangular

Rectangular Window The rectangular window was discussed in Chapter 4 (§4.5). Here we summarize the results of that discussion.

Definition ( odd):

Transform:

The DTFT of a rectangular window is shown in Fig.3.1.

Figure 3.1: Rectangular window discrete-time Fourier transform.

Zero crossings at integer multiples of

Main lobe width is . As increases, the main lobe narrows (better frequency resolution).

has no effect on the height of the side lobes (same as the Gibbs phenomenon for truncated Fourier series expansions).


First side lobe only 13 dB down from the main-lobe peak. Side lobes roll off at approximately 6dB per octave. A phase term arises when we shift the window to make it causal, while the window transform is real in the zero-phase case (i.e.

### Hamming

function of hamming

0.54+0.46cos2pi n/m-1

### Blackman

0.42+.5cos(2*pi*n/m-1)+.08cos(4*pi*n/m-1)

Greatest stop band attenuation of mentioned windowing techniques at the expense of a larger transition band.