Windows
Welch 
Bartlett 
Hanning/Hamming 
Kaiser
Written by Paul Bourke
August 1998
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
Also known simply as the Triangular window
The Bartlett window of width N is defined as


The function is simply
The frequency response is sinc^{2} (a triangle is the convolution
of two square windows so the convolution is the product of two sinc functions).
Hanning/Hamming window
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".
The Kaiser or KaiserBessel 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 timebandwidth product. B
determines the tradeoff between the magnitude of the sidelobes 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.
