| Window type | wn |
|---|---|
| Rectangular | 1 |
| Hanning | 0·5 + 0·5 cos [n /
(m+1)] |
| Hamming | 0·54 + 0·46 cos (n/
m) |
| Blackman | 0·42 + 0·5 cos (n/
m) + 0·08 cos (2n/
m) |
| Kaiser | I0( [1
- n2 / m2]1/2) / I0() |
These notes give a brief explanation of the window functions shown by the window function applet.
The fixed shape window functions (Hanning, Hamming and Blackman) are
plotted in the diagram opposite. In common with all tapered data window
functions, these are symmetrical about the centre line (n = 0)
and have a maximum value of 1, tapering towards zero on both sides
(although some window functions, such as the Hamming window, do not quite
reach zero at either end).
The following table defines the sampled values wn of each window function, for values of n between -m and +m, giving 2m + 1 sample points in total.
| Window type | wn |
|---|---|
| Rectangular | 1 |
| Hanning | 0·5 + 0·5 cos [n/
(m+1)] |
| Hamming | 0·54 + 0·46 cos (n/
m) |
| Blackman | 0·42 + 0·5 cos (n/
m) + 0·08 cos (2n/
m) |
| Kaiser | I0([1
- n2 / m2]1/2) / I0() |
The function I0 is a zero order Bessel function
of the first kind. This function can be computed by means of a power
series expansion to any required accuracy. The value of the window
parameter
determines
the amount of taper of the Kaiser window. With
= 0,
the window is a flat, rectangular shape; as
increases,
the window becomes more strongly tapered.
The versatility of the Kaiser window is evident. In the Window design
method for FIR filters, the shape of the Kaiser window can be adjusted by
choosing a suitable value for
to
give the required stopband attenuation, as shown in the table below.
| Stopband attenuation, A(dB) | Kaiser window parameter, |
|---|---|
| 50 <= A | 0·1102(A - 8·7) |
| 21 < A < 50 | 0·5842(A - 21)0·4 + 0·07886(A - 21) |
| A <= 21 | 0 |
Window functions applet:
Java 1.02 version |
Java 1.1 version
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