Window Functions

When you multiply in the time domain, you fold in the frequency domain.

We therefore want our window function to be as narrow as possible. Ideally window should resemble the Unit Sample in the frequency domain.

Effects on Filters

Filters dampening will also depend on the window function used.

If we assume an ideal filter, the transition from amplification of $1$ to amplification of $0$ with have the same width of the main lobe of the window function.

Windows

See slides.

Rectangular Window

$$w(n) = \begin{cases} 1, &\mathrm{for}\ -M \leq n \leq M \ 0 , &\mathrm{Otherwise} \end{cases} $$

Bartlet (Triangle)

$$w(n) = \begin{cases} a - \frac{|n|}{M} , &\mathrm{for}\ -M \leq n \leq M \ 0 , &\mathrm{Otherwise} \end{cases} $$

Hanning and Hamming window

Hanning: $\alpha = 0.5$ Hamming: $\alpha = 0.54$ $$w(n) = \begin{cases} \alpha + (1 - \alpha) \cdot \cos(\frac{n\pi}{M}) , &\mathrm{for}\ -M \leq n \leq M \ 0 , &\mathrm{Otherwise} \end{cases} $$

Kaiser Window

See slides.

$\beta$ alters the side lobe amplification.