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:

• width of main slope,

and

• side slope rejection.

The following windows are normalized for a MDCT on the range of [-1,+1].

• x = -1...+1
• w = (1 + x) pi = 0 ... 2 pi

Table of contents

1 Non-power-preserving analysis windows 1.1 Rectangular windows 1.2 Triangular (aka Bartlett) window 1.3 Hamming/van Hann window 1.4 Blackman/Blackman Harris windows 1.5 Bartlett-Hann Window 1.6 Bessel window 2 Power-preserving analysis windows 2.7 Sine window 2.8 Kaiser-Bessel-derived (KBD) window 2.9 Other power-preserving windows 3 Multiple overlap windows 3.10 Triple overlapped cosine window

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

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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.