Windows

Welch

The Welch window for N points is defined as where 0 <= i < N

Centered around 0 this looks like

Commonly used as a window for power spectral estimation.
The magnitude response is

Bartlett window

Also known simply as the Triangular window The Bartlett window of width N is defined as

The function is simply

The frequency response is sinc² (a triangle is the convolution of two square windows so the convolution is the product of two sinc functions).

Also known as the raised cosine window The Hanning window for N points is defined as where -N/2 <= i < N/2

The Hamming window is

These are specific examples from a general family of curves of the form

w(i) = a + (1 - a) cos(2 pi i / N)

The magnitude response is

Shown below is the surface generated in Mathematica of the above function varying the parameter "a".

Kaiser-Bessel window

The Kaiser or Kaiser-Bessel window is an approximation to a restricted time duration function with minimum energy outside some specified band. In the discrete case it is defined as

where N is the window width, B is half the time-bandwidth product. B determines the trade-off between the magnitude of the side-lobes and the energy in the main lobe and is often specified as a half integer multiple of pi. Io is the zero order modified Bessel function of the first kind, a series expansion of which is:

The expansion to 20 decimal places is

Analytic forms of the spectrum are not available but it can be shown that the frequency spectrum in the continuous case is proportional to:

where w_b is the width of the central lobe.

Example curves for different values of B

Using Mathematics a surface plot showing the window as a function of B.