Window function: Difference between revisions

Revision as of 20:38, 16 April 2024

Revision as of 22:03, 17 April 2024

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*<math>w_0(x)</math> is a zero-phase function (symmetrical about <math>x=0</math>),<ref name=Zphase/> continuous for <math>x \in [-N/2, N/2],</math> where <math>N</math> is a positive integer (even or odd).<ref name=Rorabaugh/>

*The sequence &nbsp;<math>\{w[n] = w_0(n-N/2),\quad 0\le n \le N\}</math>&nbsp; is ''symmetric'', of length <math>N+1.</math>

*<math>\{w[n],\quad 0\le n \le N-1\}</math>&nbsp; is ''DFT-symmetric'', of length <math>N.</math>{{efn-ua

|Some authors limit their attention to this important subset and to even values of N.<ref name=Harris/><ref name=Heinzel2002/> But the window coefficient formulas are still the ones presented here.}}

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{{Distinguish|Kernel density estimation}}

Defining &nbsp;{{math|''L'' ≜ ''N'' + 1}},&nbsp; the Parzen window, also known as the '''de la Vallée Poussin window''',<ref name=Harris/> is the 4{{Sup|th}}-order ''B''-spline window given by:

:<math>

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|background colour=#F5FFFA}}

In most cases, including the examples below, all coefficients ''a''_''k''&nbsp;≥&nbsp;0.&nbsp; These windows have only 2''K''&nbsp;+&nbsp;1 non-zero ''N''-point DFT coefficients.

==== Hann and Hamming windows{{anchor|Hamming window}} ====

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</math>

Setting &nbsp;<math>a_0 = 0.5</math>&nbsp; produces a '''Hann window:'''

:<math>w[n] = 0.5\; \left[1 - \cos \left ( \frac{2 \pi n}{N} \right) \right] = \sin^2 \left ( \frac{\pi n}{N} \right),</math><ref name=MWhann/>

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This function is a member of both the [[#Cosine-sum windows|cosine-sum]] and [[#Power-of-sine/cosine_windows|power-of-sine]] families. Unlike the [[#Hann and Hamming windows|Hamming window]], the end points of the Hann window just touch zero. The resulting [[Spectral leakage|side-lobes]] roll off at about 18&nbsp;dB per octave.<ref name=JOShann/>

Setting &nbsp;<math>a_0</math>&nbsp; to approximately 0.54, or more precisely 25/46, produces the '''Hamming window''', proposed by [[Richard W. Hamming]]. That choice places a zero-crossing at frequency 5{{pi}}/(''N''&nbsp;−&nbsp;1), which cancels the first sidelobe of the Hann window, giving it a height of about one-fifth that of the Hann window.<ref name=Harris/><ref name=Enochson/><ref name=JOSHamming/>

The Hamming window is often called the '''Hamming blip''' when used for [[pulse shaping]].<ref name=sunar/><ref name=sunar2/><ref name=SRD/>

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Rife–Vincent windows<ref name=Rife/> are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied to {{EquationNote|Eq.1}}, reflect that custom.

Class I, Order 1 (''K'' = 1):&nbsp; <math>a_0=1;\quad a_1=1</math> &nbsp; &nbsp; &nbsp; Functionally equivalent to the [[#Hann and Hamming windows|Hann window]].

Class I, Order 2 (''K'' = 2):&nbsp; <math>a_0=1;\quad a_1=\tfrac{4}{3};\quad a_2=\tfrac{1}{3}</math>

Class I is defined by minimizing the high-order sidelobe amplitude. Coefficients for orders up to K=4 are tabulated.<ref name=Andria/>

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:<math>\sigma \le \;0.5\,</math>

The standard deviation of the Gaussian function is ''σ''&nbsp;&middot;·&nbsp;''N''/2 sampling periods.

{{clear}}

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==== Confined Gaussian window ====

The confined Gaussian window yields the smallest possible root mean square frequency width {{math|''σ''{{sub|''&omega;''}}}} for a given temporal width &nbsp;{{math|(''N'' + 1) ''σ''{{sub|''t''}}}}.<ref name=Starosielec2014/> These windows optimize the RMS time-frequency bandwidth products. They are computed as the minimum eigenvectors of a parameter-dependent matrix. The confined Gaussian window family contains the {{slink|#Sine window}} and the {{slink|#Gaussian window}} in the limiting cases of large and small {{math|''σ''{{sub|''t''}}}}, respectively.

{{clear}}

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==== Approximate confined Gaussian window ====

Defining &nbsp;{{math|''L'' ≜ ''N'' + 1}},&nbsp; a [[#Confined Gaussian window|confined Gaussian window]] of temporal width &nbsp;{{math|''L'' × ''σ''{{sub|''t''}}}}&nbsp; is well approximated by:<ref name=Starosielec2014/>

:<math>w[n] = G(n) - \frac{G(-\tfrac{1}{2})[G(n + L) + G(n - L)]}{G(-\tfrac{1}{2} + L) + G(-\tfrac{1}{2} - L)}</math>

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::<math>G(x) = \exp\left(- \left(\cfrac{x - \frac{N}{2}}{2 L \sigma_t}\right)^2\right)</math>

The standard deviation of the approximate window is [[asymptotically equal]] (i.e. large values of {{math|''N''}}) to &nbsp;{{math|''L'' × ''σ''{{sub|''t''}}}}&nbsp; for &nbsp;{{math|''σ{{sub|t}}'' < 0.14}}.<ref name=Starosielec2014/>

{{clear}}

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w[N-n] = w[n],\quad & 0 \le n \le \frac{N}{2}

\end{array}\right\}

</math> &nbsp;<ref name=Bloomfield/>{{efn-ua

|1=This formula can be confirmed by simplifying the cosine function at [http://www.mathworks.com/help/signal/ref/tukeywin.html MATLAB tukeywin] and substituting ''r''=''α'' and ''x''=''n''/''N''.

}}{{efn-ua

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The Kaiser, or Kaiser–Bessel, window is a simple approximation of the [[#DPSS or Slepian window|DPSS window]] using [[Bessel function]]s, discovered by [[James Kaiser]].<ref name=Kaiser1966/><ref name=Kaiser1964/>

:<math>w[n]=\frac{I_0\left(\pi\alpha \sqrt{1-\left(\frac{2 n}{N}-1\right)^2}\right)}{I_0(\pi\alpha)},\quad 0\le n \le N</math> &nbsp; &nbsp;{{efn-ua

|The Kaiser window is often parametrized by {{math|''β''}}, where {{math|1=''β'' = {{pi}}''α''}}.<ref name=Rabiner/><ref name=Crochiere/>

<ref name=Vaidyanathan/><ref name=JOSKaiser/><ref name=KaiserDPSS/><ref name=MWkaiser/><ref name=Oppenheim/>{{rp|p. 474}}&nbsp; The alternative use of just {{math|α}} facilitates comparisons to the DPSS windows.<ref name=Kaiser_Window.html/>

}}<ref name=Harris/>{{rp|p. 73}}

:<math>

w_0(n) = \frac{I_0\left(\pi\alpha \sqrt{1-\left(\frac{2 n}{N}\right)^2}\right)}{I_0(\pi\alpha)},\quad -N/2 \le n \le N/2</math>

where <math>I_0</math> is the 0{{Sup|th}}-order modified Bessel function of the first kind. Variable parameter <math>\alpha</math> determines the tradeoff between main lobe width and side lobe levels of the spectral leakage pattern. The main lobe width, in between the nulls, is given by &nbsp;<math>2\sqrt{1 + \alpha^2},</math> in units of DFT bins,<ref name=Kaiser1980/> and a typical value of <math>\alpha</math> is 3.

{{clear}}

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\end{align}</math>

which is an inverse DFT of &nbsp;<math>\left(-e^{\frac{i\pi}{N+1}}\right)^k\cdot W_0(k).</math>

Variations:

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==== Ultraspherical window ====

[[File:Window function and frequency response - Ultraspherical (mu = -0.5).svg|thumb|480px|right|The Ultraspherical window's ''µμ'' parameter determines whether its Fourier transform's side-lobe amplitudes decrease, are level, or (shown here) increase with frequency.]]

The Ultraspherical window was introduced in 1984 by Roy Streit<ref name=Kabal/> and has application in antenna array design,<ref name=Streit/> non-recursive filter design,<ref name=Kabal/> and spectrum analysis.<ref name=Deczky/>

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The GAP window is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order <math>K</math>. It is continuous with continuous derivative everywhere. With the appropriate set of expansion coefficients and expansion order, the GAP window can mimic all the known window functions, reproducing accurately their spectral properties.

:<math>w_0[n] = a_{0} + \sum_{k=1}^{K} a_{2k}\left(\frac{n}{\sigma}\right)^{2k}, \quad -\frac{N}{2} \le n \le \frac{N}{2},</math> &nbsp;<ref name=Beccaro/>

where <math>\sigma</math> is the standard deviation of the <math>\{n\}</math> sequence.

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| chapter =11

| isbn =978-1-139-50145-3

}}&nbsp; Also https://cnx.org/contents/QsVBJjB4@3.1:6R_ztzDY@4/Pulse-Shaping-and-Receive-Filtering

</ref>

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<ref name=Deczky>

{{cite book |last=Deczky |first=Andrew |chapter=Unispherical Windows |year=2001 |volume=2 |pages=85–88 |doi=10.1109/iscas.2001.921012 |isbn=978-0-7803-6685-5 |title=ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No. 01CH37196)

|s2cid=38275201 }}</ref>