Window function
Window functions are applied to avoid discontinuities at the beginning
and the end of a set of data. The smaller these discontinuities are, the
faster the side slopes drop.
The maximum order of derivative which
is zero at the ends determines the asymptotic behaviour:
- steps in the function itself: asymptotic -6 dB/oct
- continuous function, step in first derivative: -12 dB/oct
- and so on.
There is an intrinsic trade-off problem between:
and
The following windows are normalized for a MDCT on the range of [-1,+1].
- x = -1...+1
- w = (1 + x) pi = 0 ... 2 pi
Non-power-preserving analysis windows
Rectangular windows
Full size window. Actually this is a MDCT without window.
f(x) = 1 for |x| < 1 , 0 otherwise
Sometimes also written as
f(x) = sqrt(1/2) for |x| < 1 , 0 otherwise
Half size window. Actually this is a DCT Type ???
f(x) = 1 for |x| < 1/2 , 0 otherwise
How to add images ???
http://www.uni-jena.de/~pfk/wiki/*.png
Image of f(x) and spectral resolution
Triangular (aka Bartlett) window
f(x) = 1 - |x| for |x| < 1, 0 otherwise
Image of f(x) and spectral resolution
Hamming/van Hann window
f(x) = a0 - a1 * cos(w)
van Hann window: a0 = , a1 =
hamming window: a0 = , a1 =
Blackman/Blackman Harris windows
f(x) = a0 - a1 * cos(w) + a2 * cos(2w) - a3 * cos(3w)
Blackman: a0 = , a1 = , a2 = , a3 =
Blackman Harris: a0 = , a1 = , a2 = , a3 =
Blackman Nuttall: a0 = , a1 = , a2 = , a3 =
Bartlett-Hann Window
Mixture of Barlett and van Hann window:
f(x) = a0 - a1 * cos(w) - a2 * |x|
a0 = , a1 = , a2 =
Bessel window
f(x) =
Power-preserving analysis windows
Sine window
f(x) = sin(w/2)
Kaiser-Bessel-derived (KBD) window
For 0 <= x <= 1:
f(x) = Int
For x > 1:
f(x) = 0
For x < 0:
f(x) = f(-x)
(See Kaiser window.)
Other power-preserving windows
Multiple overlap windows
When using FFT or DCT for spectral analysis a sample belongs to oneanalysis window. When using windowing, samples at the boundaries are attenuated.
To reduce the effect that these samples become less important for the result,
normally windows are overlapped. So samples between two blocks are attenuated,
but they belong to two blocks: their influence is still (nearly) the same
as samples which are not attenuated. But it is possible to overlap more than
two windows. This typically makes the transition band between main slope and side slopes smaller.
Triple overlapped cosine window
The normal cosine windows do not preserve the power of the signal.
Samples which are exactly between two blocks are attenuated by 6 dB, i.e.
their power is reduced by a factor of 0.25. The overlapping reduces this to
a factor of 0.5.
Referenced By
List of mathematical topics (V-Z)
|
