Windows Package

Craig Stuart Sapp <craig@ccrma.stanford.edu>

25 Feb 1997

Getting StartedLoad the windows package by placing the pathname to the Windows package below:

<< "Windows.m"

Windows are created in two steps: First, a continuous function describing the window is defined over the domain [-1,1]. This continuous function is the name of the window followed by the letter ’F’, for example TriangleF. Next, a discrete window is sampled from this continuous function by a function using the window name and ending in the letter ’W’. A few windows have variable parameters, such as the Kaiser window. You can see if a window has any extra parameters by typing Options[windowName] (without the ’F’ or ’W’ ending).

Options @Kaiser D

8Alpha Æ 2. <

The variable $Windows contains a list of all possible window types from the windows package.

$Windows

8Blackman, BlackmanHarris, Bohman, Cauchy, Cosine, Gaussian, Hamming,Hann, HannPoisson, Kaiser, Poisson, Rectangle, Reisz, Riemann,Triangle, Turkey, VallePoussin <

To get information on a particular type of Window:

? Blackman

BlackmanW@length, options D returns the Blackman window.

There are two plotting function designed for plotting continuous windows, and their transforms: WindowPlot, and WindowTransformPlot.

Double-click to the right of the headings below to open/close sections of this notebook.

Rectangle Window

Plot @RectangleF @xD, 8x, −1.1, 1.1 <D;

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Or use the built-in function to plot a window in a standardized style:

WindowPlot @Rectangle D;

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Rectangle Window

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2 Windows.nb

WindowTransformPlot @Rectangle, PlotRange −> 8−37, 0 <D;

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Rectangle Window Transform

SeqPlot @RectangleW @9DD;

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Windows.nb 3

Triangle Window

WindowPlot @Triangle D;

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Triangle Window

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WindowTransformPlot @Triangle, PlotRange −> 8−65, 0 <D;

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Triangle Window Transform

4 Windows.nb

SeqPlot @TriangleW @9DD;

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SeqPlot @TriangleW @9, Causal −> False D, Causal −> False D;

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Cos^n Window

Options @Cosine D

8Power Æ 1<

Windows.nb 5

WindowPlot @Cosine D;

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Cosine Window

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WindowTransformPlot @Cosine, PlotRange −> 8−69, 0 <D;

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Cosine Window Transform

6 Windows.nb

SeqPlot @CosineW @9DD;

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WindowPlot @Cosine, Power −> 2, Title −> "Cosine^2 Window" D;

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Cosine^2 Window

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Windows.nb 7

WindowTransformPlot @Cosine, Power −> 2, PlotRange −> 8−105, 0 <D;

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Cosine Window Transform

SeqPlot @CosineW @9, Power −> 2DD;

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There are additional window functions called CosinLobeW which sets PowerÆ1, and RaisedCosineW function which sets PowerÆ2.

8 Windows.nb

SeqPlot @CosineLobeW @9DD;

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Hann Window

WindowPlot @Hann D;

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Hann Window

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Windows.nb 9

WindowTransformPlot @Hann, PlotRange −> 8−105, 0 <D;

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Hann Window Transform

The Hann window is the same as a Cos^2 window:

Chop@CosineW @21, Power −> 2D − HannW@21DD

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SeqPlot @HannW@9DD;

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10 Windows.nb

Hamming Window

Options @HammingD

9Alpha Æ25ÄÄÄÄÄÄÄÄ46

=

WindowPlot @Hamming, Alpha −> 25 ê 46, Title −> "Hamming Window α→0.54" D;

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Hamming Window aÆ0.54

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WindowTransformPlot @Hamming, PlotRange −> 8−55, 0 <D;

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Hamming Window Transform

The Hann window is the same as a Hamming window with a=0.5:

Windows.nb 11

SeqPlot @HammingW@9DD;

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Chop@HannW@21D − HammingW@21, Alpha −> 0.5 DD

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Blackman Window

Options @Blackman D êê N

8A0 Æ 0.426591, A1 Æ 0.496561, A2 Æ 0.0768487 <

12 Windows.nb

WindowPlot @Blackman,Title −> "Blackman Window a0 →.427 a1 →.497 a2 →.077" D;

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Blackman Window a0Æ.427 a1Æ.497 a2Æ.077

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WindowTransformPlot @Blackman, PlotRange −> 8−75, 0 <D;

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Blackman Window Transform

Windows.nb 13

SeqPlot @BlackmanW@9DD;

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Note that the Blackman window does not get to 0 at is edges:

BlackmanF @1D

0.00687876

Blackman-Harris WindowDefault parameters are for the 4-Term -96dB Blackman-Harris window:

Options @BlackmanHarris D

8A0 Æ 0.35875, A1 Æ 0.48829, A2 Æ 0.14128, A3 Æ 0.01168 <

14 Windows.nb

WindowPlot @BlackmanHarris, Title −> "Blackman −Harris Window" D;

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Blackman-Harris Window

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WindowTransformPlot @BlackmanHarris, PlotRange −> 8−125, 0 <D;

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BlackmanHarris Window Transform

Windows.nb 15

SeqPlot @BlackmanHarrisW @9DD;

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Kaiser Window

Options @Kaiser D

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WindowPlot @Kaiser, Title −> "Kaiser Window α→2.0" D;

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Kaiser Window aÆ2.0

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16 Windows.nb

WindowTransformPlot @Kaiser, PlotRange −> 8−75, 0 <D;

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Kaiser Window Transform

SeqPlot @KaiserW @9, Alpha −> 3.0 DD;

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Windows.nb 17

Less Common Windows

Riesz Window

WindowPlot @Reisz D;

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Reisz Window

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WindowTransformPlot @Reisz, PlotRange −> 8−65, 0 <D;

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Reisz Window Transform

18 Windows.nb

SeqPlot @ReiszW @9DD;

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Riemann Window

WindowPlot @Riemann D;

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Riemann Window

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Windows.nb 19

WindowTransformPlot @Riemann, PlotRange −> 8−65, 0 <D;

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Riemann Window Transform

SeqPlot @RiemannW@9DD;

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20 Windows.nb

de la Vallé-Poussin Window

WindowPlot @VallePoussin, Title −> "de la Vallé −Poussin Window" D;

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de la Vallé-Poussin Window

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WindowTransformPlot @VallePoussin, PlotRange −> 8−105, 0 <D;

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VallePoussin Window Transform

Windows.nb 21

SeqPlot @VallePoussinW @19DD;

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Turkey Window

WindowPlot @Turkey, Taper −> 0.5, Title −> "Turkey Window Taper →0.5" D;

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Turkey Window TaperÆ0.5

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22 Windows.nb

WindowTransformPlot @Turkey, Taper −> 0.5, PlotRange −> 8−85, 0 <D;

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Turkey Window Transform

WindowPlot @Turkey, Taper −> 0.75, Title −> "Turkey Window Taper →0.75" D;

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Turkey Window TaperÆ0.75

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Windows.nb 23

WindowTransformPlot @Turkey, Taper −> 0.75, PlotRange −> 8−95, 0 <D;

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Turkey Window Transform

SeqPlot @TurkeyW @19DD;

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24 Windows.nb

Bohman Window

WindowPlot @BohmanD;

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Bohman Window

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WindowTransformPlot @Bohman, PlotRange −> 8−105, 0 <D;

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Bohman Window Transform

Windows.nb 25

SeqPlot @BohmanW@19DD;

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Poisson Window

Options @Poisson D

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WindowPlot @Poisson, Title −> "Poisson Window α→2.0" D;

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Poisson Window aÆ2.0

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26 Windows.nb

WindowTransformPlot @Poisson, PlotRange −> 8−45, 0 <D;

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Poisson Window Transform

SeqPlot @PoissonW @19, Alpha −> 3.0 DD;

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Hann–Poisson Window

This window is a multiplcation of the Hann window with the Poisson window.

Windows.nb 27

WindowPlot @HannPoisson, Title −> "Hann −Poisson Window α→2.0" D;

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Hann-Poisson Window aÆ2.0

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WindowTransformPlot @HannPoisson D;

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HannPoisson Window Transform

28 Windows.nb

SeqPlot @HannPoissonW @19, Alpha −> 0.5 DD;

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Cauchy Window

Options @Cauchy D

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WindowPlot @Cauchy, Alpha −> 2.0, Title −> "Cauchy Window α→2.0" D;

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Cauchy Window aÆ2.0

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Windows.nb 29

WindowTransformPlot @Cauchy, Alpha −> 3.0, PlotRange −> 8−55, 0 <D;

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Cauchy Window Transform

SeqPlot @CauchyW@19DD;

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Gausian Window

Options @Gaussian D

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30 Windows.nb

WindowPlot @Gaussian, Title −> "Gaussian Window α→2.0" D;

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Gaussian WindowaÆ2.0

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WindowTransformPlot @Gaussian, PlotRange −> 8−55, 0 <D;

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Gaussian Window Transform

Windows.nb 31

SeqPlot @GaussianW @19, Alpha −> 3.0 DD;

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Dolph-Chebyshev Window

Note: there is no DolphChebyshevF continuous function due to the window generation algorithm.

SeqPlot @DolphChebyshevW @51, Causal −> False, Alpha −> 2.5 D,Points −> False, LineStyle −> Thickness @0.006 D, Causal −> False,Frame −> True, Axes −> False, FrameTicks −> Automatic D;

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32 Windows.nb

Barcilon–Temes Window

Not implemented

ReferencesHarris, Frederic J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,"

Proceedings of the IEEE, Vol.66, No.1 (January 1978) pp. 51–84.

Windows.nb 33