Windowing is the application of a window prior to processing. That the signal s(t) is multipied by a window function w(t), before processing. To put it in a more precise context, suppose we are doing a Fourier analysis on the signal s. Then instead of computing

S(k)  =

N-1 å t=0

s(t)exp

æ ç è

-

2pikt N

ö ÷ ø

we compute

S(k)  =

N-1 å t=0

w(t)  ´  s(t)exp

æ ç è

-

2pikt N

ö ÷ ø

The reason to do this is to guarentee that the values at the endpoints of match so as to avoid leakage. For this to work the width of the support (the non-zero portion) of the window, called the windowlength option should be less than N. The window is generally centered at N/2.

Valid types of windows are Hamming, Hanning, Bartlett, Blackman, or Rectangle

Hamming Window

Hamming window is defined as

W[t]  =

ì ï í ï î

0.54  -  .46cos æ ç è 2p(t  -  c) N  -  1 ö ÷ ø if   -  r £ t  -  c  <  r 0 ow.

where c is the center of the window and 2r is the length of the window Hamming is the most frequently used type and is the default.

Hanning window is defined as

W[t]  =

ì ï í ï î

0.5  -  .5cos æ ç è 2p(t  -  c) N  -  1 ö ÷ ø if   -  r £ t  -  c  <  r 0 ow.

where c is the center of the window and 2r is the length of the window

Bartlett window is defined as

W[t]  =

ì ï í ï î

1  -   ||t  -  x|| r if  -  r£t  <  r 0   ow

The Bartlett window is triangular in shape

Rectangular window is defined as

W[t]  =

ì í î

1 if  -  r£t  <  r 0   ow

The rectangular window will give great leakage at the boundries. Leakage is considered bad.

Blackman window is defined as

W[t]  =

ì ï ï í ï ï î

0.35875  -  0.48829cos æ ç è 2p(t  -  c) N  -  1 ö ÷ ø  +  0.8cos æ ç è 2p(t  -  c) N  -  1 ö ÷ ø

for ||t - c|| < r and 0 otherwise, where 2r is the length of the window.