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Window function

From Academic Kids

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.

Frequency analysis of a signal commonly involves taking a discrete, finite-length sample of the signal and performing a discrete Fourier-related transform to convert it from the time domain to the frequency domain, in order to have a frequency spectrum. However, this type of transform assumes that the signal can be reproduced by looping the sample. If the length of the sample isn't exactly a multiple of the period of the signal, then that assumption is not met, and the transform will contain errors as a result. In particular, if the end of the sample is not continuous with the beginning, the transform will be somewhat incorrect.

To eliminate any discontinuity at the edges of a sample, the sample may be multiplied by a window function before attempting any frequency analysis. This introduces some minor error of its own, but it commonly produces a sample that more closely resembles the overall signal in the frequency domain.

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 lobe,

and

  • side lobe rejection.

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

  • x = −1...+1
  • w = (1 + x) π = 0 ... 2 π
Contents

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) = √(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

The rectangular window has the smallest main lobe width, but has the worst side lobe rejection (−6dB). In other words, the rectangular window has the best frequency resolution.


Triangular (aka Bartlett) window

f(x) = 1 − |x| for |x| < 1, 0 otherwise


Hamming/von Hann window

f(x) = a + (1 − a) cos(πx)

von Hann window: a = 1/2

Hamming window: a ≈ 0.54 (precisely a = 25/46)

Blackman/Blackman Harris windows

f(x) = a0a1 cos(πx) + a2 cos(2πx) − a3 cos(3πx)

Blackman: a0 = 0.42, a1 = −0.5, a2 = 0.08, a3 = 0

Blackman Harris: a0 = 0.35875, a1 = 0.48829, a2 = 0.14128, a3 = 0.01168

Blackman Nuttall: a0 = 0.3635819, a1 = 0.4891775, a2 = 0.1365995, a3 = 0.0106411

Bartlett-Hann window

Mixture of Barlett and von Hann window:

f(x) = a0a1 cos(πx) − 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 one analysis 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.

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