View source for Window function

{{Short description|Function used in signal processing}}
{{For|the term used in SQL statements|Window function (SQL)}}
[[File:Hanning.svg|thumb|A popular window function, the [[Hann function|Hann window]].  Most popular window functions are similar bell-shaped curves.]]

In [[signal processing]] and [[statistics]], a '''window function''' (also known as an '''apodization function''' or '''tapering function'''<ref name=Weisstein/>) is a [[function (mathematics)|mathematical function]] that is zero-valued outside of some chosen [[interval (mathematics)|interval]]. Typically, window functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window".  Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values.  Thus, tapering, not segmentation, is the main purpose of window functions.

The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra.  The duration of the segments is determined in each application by requirements like time and frequency resolution.  But that method also changes the frequency content of the signal by an effect called [[spectral leakage]].  Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application.  There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.

In typical applications, the window functions used are non-negative, smooth, "bell-shaped" curves.<ref name=Roads/> Rectangle, triangle, and other functions can also be used.  A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is [[square integrable]], and, more specifically, that the function goes sufficiently rapidly toward zero.<ref name=Cattani/>

== Applications ==
Window functions are used in spectral [[frequency spectrum#spectral analysis|analysis]]/modification/[[Overlap–add method#resynthesis|resynthesis]],<ref name=OLA/> the design of [[finite impulse response]] filters, merging multiscale and multidimensional datasets,<ref>{{Cite journal |last1=Ajala |first1=R. |last2=Persaud |first2=P. |title=Ground-Motion Evaluation of Hybrid Seismic Velocity Models |journal=The Seismic Record|date=2022 |volume=2 |issue=3 |pages=186–196 |doi=10.1785/0320220022 |s2cid=251504921 |doi-access=free |bibcode=2022SeisR...2..186A }}</ref><ref>{{Cite journal |last1=Ajala |first1=R. |last2=Persaud |first2=P. |title=Effect of Merging Multiscale Models on Seismic Wavefield Predictions Near the Southern San Andreas Fault |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2021JB021915 |journal=Journal of Geophysical Research: Solid Earth |date=2021 |language=en |volume=126 |issue=10 |doi=10.1029/2021JB021915 |bibcode=2021JGRB..12621915A |s2cid=239654900 |issn=2169-9313|url-access=subscription }}</ref> as well as [[beamforming]] and [[Antenna (radio)|antenna]] design.

[[File:Spectral_leakage_caused_by_%22windowing%22.svg|thumb|400px|Figure 2: Windowing a sinusoid causes spectral leakage. The same amount of leakage occurs whether there are an integer (blue) or non-integer (red) number of cycles within the window (rows 1 and 2). When the sinusoid is sampled and windowed, its [[discrete-time Fourier transform]] (DTFT) also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue DTFT, those samples are the outputs of the [[discrete Fourier transform]] (DFT). The red DTFT has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.]]

=== Spectral analysis ===
{{Main|Spectral leakage}}

The [[Fourier transform]] of the function {{math|cos(''ωt'')}} is zero, except at frequency&nbsp;±''ω''.  However, many other functions and waveforms do not have convenient closed-form transforms. Alternatively, one might be interested in their spectral content only during a certain time period.

In either case, the Fourier transform (or a similar transform) can be applied on one or more finite intervals of the waveform.  In general, the transform is applied to the product of the waveform and a window function.  Any window (including rectangular) affects the spectral estimate computed by this method.

=== Filter design ===
{{Main|Filter design}}

Windows are sometimes used in the design of [[digital filters]], in particular to convert an "ideal" impulse response of infinite duration, such as a [[sinc function]], to a [[finite impulse response]] (FIR) filter design.  That is called the [[Finite impulse response#Window design method|''window method'']].<ref name=Oppenheim/><ref name=FIRfilters/><ref name=Tuwien/>

=== Statistics and curve fitting ===
{{Main|kernel (statistics)}}

Window functions are sometimes used in the field of [[statistics|statistical analysis]] to restrict the set of data being analyzed to a range near a given point, with a [[Weighting | weighting factor]] that diminishes the effect of points farther away from the portion of the curve being fit. In the field of  Bayesian analysis and [[curve fitting]], this is often referred to as the [[kernel (statistics)|kernel]].

=== Rectangular window applications ===
==== Analysis of transients ====

When analyzing a transient signal in [[modal analysis]], such as an impulse, a shock response, a sine burst, a chirp burst, or noise burst, where the energy vs time distribution is extremely uneven, the rectangular window may be most appropriate.  For instance, when most of the energy is located at the beginning of the recording, a non-rectangular window attenuates most of the energy, degrading the signal-to-noise ratio.<ref name=HPmemory/>

==== Harmonic analysis ====
One might wish to measure the harmonic content of a musical note from a particular instrument or the harmonic distortion of an amplifier at a given frequency.  Referring again to '''Figure 2''', we can observe that there is no leakage at a discrete set of harmonically-related frequencies sampled by the [[discrete Fourier transform]] (DFT).  (The spectral nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.)  This property is unique to the rectangular window, and it must be appropriately configured for the signal frequency, as described above.

== Overlapping windows ==
When the length of a data set to be transformed is larger than necessary to provide the desired frequency resolution, a common practice is to subdivide it into smaller sets and window them individually.  To mitigate the "loss" at the edges of the window, the individual sets may overlap in time.  See [[Welch method]] of power spectral analysis and the [[modified discrete cosine transform]].

== Two-dimensional windows ==
{{main|Two-dimensional window design}}
Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform.<ref name=Hovden/> They can be constructed from one-dimensional windows in either of two forms.<ref name=Bernstein/> The separable form, <math>W(m,n)=w(m)w(n)</math> is trivial to compute. The [[Radial function|radial]] form, <math>W(m,n)=w(r)</math>, which involves the radius <math>r=\sqrt{(m-M/2)^2+(n-N/2)^2}</math>, is [[Isotropy|isotropic]], independent on the orientation of the coordinate axes. Only the [[#Gaussian_window|Gaussian]] function is both separable and isotropic.<ref name=Awad/> The separable forms of all other window functions have corners that depend on the choice of the coordinate axes. The isotropy/[[anisotropy]] of a two-dimensional window function is shared by its two-dimensional Fourier transform. The difference between the separable and radial forms is akin to the result of [[diffraction]] from rectangular vs. circular apertures, which can be visualized in terms of the product of two [[sinc function]]s vs. an [[Airy function]], respectively.

== Examples of window functions ==

Conventions''':'''

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

*The parameter '''B''' displayed on each spectral plot is the function's [[Spectral leakage#Noise bandwidth|noise equivalent bandwidth]] metric, in units of ''DFT bins''.<ref name=Harris/>{{rp|p.56 eq.(16)}}
**See {{Slink|spectral leakage|Discrete-time signals|Some window metrics}} and [[Normalized frequency (signal processing)|Normalized frequency]] for understanding the use of "bins" for the x-axis in these plots.

The sparse sampling of a [[discrete-time Fourier transform]] (DTFT) such as the DFTs in Fig 2 only reveals the leakage into the DFT bins from a sinusoid whose frequency is also an integer DFT bin. The unseen sidelobes reveal the leakage to expect from sinusoids at other frequencies.{{efn-la
|[[#Harris|Harris 1978]], p 57, fig 10.
}} Therefore, when choosing a window function, it is usually important to sample the DTFT more densely (as we do throughout this section) and choose a window that suppresses the sidelobes to an acceptable level.

=== Rectangular window ===
<!-- [[Boxcar window]] redirects here -->
[[File:Window function and frequency response - Rectangular.svg|thumb|480px|right|Rectangular window]]

The rectangular window (sometimes known as the '''[[Boxcar function|boxcar]]''' or uniform or '''[[Dirichlet kernel|Dirichlet]] window''' or misleadingly as "no window" in some programs<ref>{{Cite web |title=Understanding FFTs and Windowing |url=https://download.ni.com/evaluation/pxi/Understanding%20FFTs%20and%20Windowing.pdf |url-status=live |archive-url=https://web.archive.org/web/20240105021252/https://download.ni.com/evaluation/pxi/Understanding%20FFTs%20and%20Windowing.pdf |archive-date=2024-01-05 |access-date=2024-02-13 |website=[[National Instruments]]}}</ref>) is the simplest window, equivalent to replacing all but ''N'' consecutive values of a data sequence by zeros, making the waveform suddenly turn on and off:

:<math>w[n] = 1.</math>

Other windows are designed to moderate these sudden changes, to reduce scalloping loss and improve dynamic range (described in {{slink||Spectral analysis}}).

The rectangular window is the 1st-order ''B''-spline window as well as the 0th-power [[#Power-of-sine/cosine_windows|power-of-sine window]].

The rectangular window provides the minimum mean square error estimate of the [[Discrete-time Fourier transform]], at the cost of other issues discussed.

{{clear}}

=== ''B''-spline windows ===

''B''-spline windows can be obtained as ''k''-fold convolutions of the rectangular window. They include the rectangular window itself (''k''&nbsp;=&nbsp;1), the {{slink|#Triangular window}} (''k''&nbsp;=&nbsp;2) and the {{slink|#Parzen window}} (''k''&nbsp;=&nbsp;4).<ref name=Toraichi89/> Alternative definitions sample the appropriate normalized [[B-spline|''B''-spline]] [[basis function]]s instead of convolving discrete-time windows. A ''k''th-order ''B''-spline basis function is a piece-wise polynomial function of degree ''k''&nbsp;−&nbsp;1 that is obtained by ''k''-fold self-convolution of the [[rectangular function]].
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==== Triangular window ====
[[File:Window function and its Fourier transform – Triangular (n = 0...N).svg|thumb|480px|right|Triangular window (with ''L''&nbsp;=&nbsp;''N''&nbsp;+&nbsp;1)]]

Triangular windows are given by

:<math>w[n] = 1 - \left|\frac{n - \frac{N}{2}}{\frac{L}{2}}\right|,\quad 0\le n \le N,</math>

where ''L'' can be ''N'',<ref name=Bartlett/> ''N''&nbsp;+&nbsp;1,<ref name=Harris/><ref name=Tukey/><ref name=MWtriang/> or ''N''&nbsp;+&nbsp;2.<ref name=Welch1967/> The first one is also known as '''[[M. S. Bartlett|Bartlett]] window''' or '''[[Lipót Fejér|Fejér]] window'''. All three definitions converge at large&nbsp;''N''.

The triangular window is the 2nd-order ''B''-spline window. The ''L''&nbsp;=&nbsp;''N'' form can be seen as the convolution of two {{Fraction|N|2}}-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.
{{clear}}

==== Parzen window ====
[[File:Window function and frequency response - Parzen.svg|thumb|480px|right|Parzen window]]

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

:<math>
w_0(n) \triangleq \left\{ \begin{array}{ll}
 1 - 6 \left(\frac{n}{L/2}\right)^2 \left(1 - \frac{|n|}{L/2}\right),
 & 0 \le |n| \le \frac{L}{4} \\ 
 2 \left(1 - \frac{|n|}{L/2}\right)^3
 & \frac{L}{4} < |n| \le \frac{L}{2} \\ 
\end{array} \right\}
</math>
:<math>w[n] = \ w_0\left(n-\tfrac{N}{2}\right),\ 0 \le n \le N</math>

{{clear}}

[[File:Window function and frequency response - Welch.svg|thumb|480px|right|Welch window]]

=== Other polynomial windows ===

==== Welch window ====

The Welch window consists of a single [[parabola|parabolic]] section:

:<math>w[n]=1 - \left(\frac{n-\frac{N}{2}}{\frac{N}{2}}\right)^2,\quad 0\le n \le N.</math><ref name=Welch1967/>
Alternatively, it can be written as two factors, as in a [[beta distribution]]:
:<math>w[n]= \left(1 + \frac{n-\frac{N}{2}}{\frac{N}{2}}\right) \left(1 - \frac{n-\frac{N}{2}}{\frac{N}{2}}\right),\quad 0\le n \le N.</math>

The defining [[quadratic polynomial]] reaches a value of zero at the samples just outside the span of the window.

The Welch window is fairly close to the [[#Sine window|sine window]], and just as the [[#Power-of-sine/cosine windows|power-of-sine windows]] are a useful parameterized family, the power-of-Welch window family is similarly useful.  Powers of the Welch or parabolic window are also [[Pearson type II distribution]]s and symmetric [[beta distribution]]s, and are purely algebraic functions (if the powers are rational), as opposed to most windows that are transcendental functions.  If different exponents are used on the two factors in the Welch polynomial, the result is a general beta distribution, which is useful for making [[#Asymmetric window functions|asymmetric window functions]].

{{clear}}

=== Raised-cosine windows ===

Windows in the form of a cosine function offset by a constant, such as the popular Hamming and Hann windows, are sometimes called raised-cosine windows.  The Hann window is particularly like the [[raised cosine distribution]], which goes smoothly to zero at its ends.

The raised-cosine windows have the form:
:<math>w[n] = a_0 - (1-a_0) \cdot \cos\left( \tfrac{2 \pi n}{N} \right),\quad 0\le n \le N,</math>
or alternatively as their zero-phase versions:
:<math>
\begin{align}
w_0(n)\ &= w\left[ n+\tfrac{N}{2}\right]\\
&= a_0 + (1-a_0)\cdot \cos \left ( \tfrac{2\pi n}{N} \right),\quad -\tfrac{N}{2} \le  n \le \tfrac{N}{2}.
\end{align}
</math>

==== Hann window====
{{Main|Hann function}}
[[File:Window function and its Fourier transform – Hann (n = 0...N).svg|thumb|480px|right|Hann window]]

Setting <math>a_0 = 0.5</math> 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/>
named after [[Julius von Hann]], and sometimes referred to as ''Hanning'', which derived from the verb "to Hann".{{Citation needed|date=June 2025}}  It is also known as the  '''raised cosine''', because of its similarity to a [[raised-cosine distribution]].

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/>

==== Hamming window ====
[[File:Window function and frequency response - Hamming (alpha = 0.53836, n = 0...N).svg|thumb|480px|right|Hamming window, ''a''<sub>0</sub>&nbsp;=&nbsp;0.53836 and ''a''<sub>1</sub>&nbsp;=&nbsp;0.46164. The original Hamming window would have ''a''<sub>0</sub>&nbsp;=&nbsp;0.54 and ''a''<sub>1</sub>&nbsp;=&nbsp;0.46.]]

Setting <math>a_0</math> to approximately 0.54, or more precisely 25/46, produces the '''Hamming window''', proposed by [[Richard W.&nbsp;Hamming]].  This 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/>

Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,<ref name=Harris/> to a nearly equiripple condition.<ref name=JOSHamming/> In the equiripple sense, the optimal values for the coefficients are ''a''<sub>0</sub>&nbsp;=&nbsp;0.53836 and ''a''<sub>1</sub>&nbsp;=&nbsp;0.46164.<ref name=JOSHamming/><ref name=Nuttall/>

=== Cosine-sum windows ===
This family, which generalizes the [[#Raised-cosine windows|raised-cosine windows]], is also known as generalized cosine windows.<ref>{{cite web | url = https://www.mathworks.com/help/signal/ug/generalized-cosine-windows.html | title = xxx }}</ref>

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math>w[n] = \sum_{k = 0}^{K} (-1)^k a_k\; \cos\left( \frac{2 \pi k n}{N} \right),\quad 0\le n \le N.</math>|{{EquationRef|Eq.1}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

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

==== Blackman window ====
[[File:Window function and its Fourier transform – Blackman (n = 0...N).svg|thumb|480px|right|Blackman window; {{math|1=''α''&nbsp;=&nbsp;0.16}}]]

Blackman windows are defined as
:<math>w[n] = a_0 -  a_1 \cos \left ( \frac{2 \pi n}{N} \right) + a_2 \cos \left ( \frac{4 \pi n}{N} \right),</math>

:<math>a_0=\frac{1-\alpha}{2};\quad a_1=\frac{1}{2};\quad a_2=\frac{\alpha}{2}.</math>

By common convention, the unqualified term ''Blackman window'' refers to Blackman's "not very serious proposal" of {{math|1=''α''&nbsp;=&nbsp;0.16}} (''a''<sub>0</sub>&nbsp;=&nbsp;0.42, ''a''<sub>1</sub>&nbsp;=&nbsp;0.5, ''a''<sub>2</sub>&nbsp;=&nbsp;0.08), which closely approximates the '''exact Blackman''',<ref name=MWBlackman/> with ''a''<sub>0</sub>&nbsp;=&nbsp;7938/18608&nbsp;≈&nbsp;0.42659, ''a''<sub>1</sub>&nbsp;=&nbsp;9240/18608&nbsp;≈&nbsp;0.49656, and ''a''<sub>2</sub>&nbsp;=&nbsp;1430/18608&nbsp;≈&nbsp;0.076849.<ref name=lvanl/> These exact values place zeros at the third and fourth sidelobes,<ref name=Harris/> but result in a discontinuity at the edges and a 6&nbsp;dB/oct fall-off.  The truncated coefficients do not null the sidelobes as well, but have an improved 18&nbsp;dB/oct fall-off.<ref name=Harris/><ref name=Blackman1959/>
{{clear}}

==== Nuttall window, continuous first derivative ====
[[File:Window function and frequency response - Nuttall (continuous first derivative).svg|thumb|480px|right|Nuttall window, continuous first derivative]]

The continuous form of the Nuttall window, <math>w_0(x),</math> and its first [[derivative]] are continuous everywhere, like the [[Hann function]]. That is, the function goes to 0 at {{nowrap|1=''x''&nbsp;=&nbsp;±''N''/2,}} unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows.  The Blackman window ({{math|1=''α''&nbsp;=&nbsp;0.16}}) is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not.

:<math>w[n]=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N} \right)- a_3 \cos \left ( \frac{6 \pi n}{N} \right)</math>

:<math>a_0=0.355768;\quad a_1=0.487396;\quad a_2=0.144232;\quad a_3=0.012604.</math>
{{clear}}

==== Blackman–Nuttall window ====
[[File:Window function and frequency response - Blackman-Nuttall.svg|thumb|480px|right|Blackman–Nuttall window]]

:<math>w[n]=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N} \right)- a_3 \cos \left ( \frac{6 \pi n}{N} \right)</math>

:<math>a_0=0.3635819; \quad a_1=0.4891775; \quad a_2=0.1365995; \quad a_3=0.0106411.</math>
{{clear}}

==== Blackman–Harris window ====
[[File:Window function and frequency response - Blackman-Harris.svg|thumb|480px|right|Blackman–Harris window]]

A generalization of the Hamming family, produced by adding more shifted cosine functions, meant to minimize side-lobe levels<ref name=JOSBlack/><ref name=JOSBlack3/>

:<math>w[n]=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N} \right)- a_3 \cos \left ( \frac{6 \pi n}{N} \right)</math>

:<math>a_0=0.35875;\quad a_1=0.48829;\quad a_2=0.14128;\quad a_3=0.01168.</math>
{{clear}}

==== Flat top window ====
[[File:Window function and frequency response - flat top.svg|thumb|480px|right|Flat-top window]]

A flat top window is a partially negative-valued window that has minimal [[#Discrete-time_signals|scalloping loss]] in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components.<ref name=Heinzel2002/><ref name=SWsmith/> However, its broad bandwidth results in high [[noise bandwidth]] and wider frequency selection, which depending on the application could be a drawback.

Flat top windows can be designed using low-pass filter design methods,<ref name=SWsmith/> or they may be of the usual [[#Cosine-sum windows|cosine-sum]] variety:

:<math>
\begin{align}
w[n] = a_0 &{}- a_1 \cos \left ( \frac{2 \pi n}{N} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N} \right)\\
           &{}- a_3 \cos \left ( \frac{6 \pi n}{N} \right)+a_4 \cos \left ( \frac{8 \pi n}{N} \right).
\end{align}
</math>

The [https://www.mathworks.com/help/signal/ref/flattopwin.html Matlab variant] has these coefficients:
:<math>a_0=0.21557895;\quad a_1=0.41663158;\quad a_2=0.277263158;\quad a_3=0.083578947;\quad a_4=0.006947368.</math>

Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.<ref name=Heinzel2002/>
{{clear}}

==== Rife–Vincent windows ====

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): <math>a_0=1;\quad a_1=1</math>    Functionally equivalent to the [[#Hann and Hamming windows|Hann window]] and power of sine ({{math|1=''α''&nbsp;=&nbsp;2}}).

Class I, Order 2 (''K'' = 2): <math>a_0=1;\quad a_1=\tfrac{4}{3};\quad a_2=\tfrac{1}{3}</math>    Functionally equivalent to the power of sine ({{math|1=''α''&nbsp;=&nbsp;4}}).

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

Class II minimizes the main-lobe width for a given maximum side-lobe.

Class III is a compromise for which order ''K''&nbsp;=&nbsp;2 resembles the {{slink|#Blackman window}}.<ref name=Andria/><ref name=Schoukens/>

{{clear}}

=== Sine window ===
[[File:Window function and frequency response - Cosine.svg|thumb|480px|right|Sine window]]

:<math>w[n] = \sin\left(\frac{\pi n}{N}\right) = \cos\left(\frac{\pi n}{N} - \frac{\pi}{2}\right),\quad 0\le n \le N.</math>

The corresponding <math>w_0(n)\,</math> function is a cosine without the {{pi}}/2 phase offset.  So the ''sine window''<ref name=Bosi/> is sometimes also called ''cosine window''.<ref name=Harris/> As it represents half a cycle of a sinusoidal function, it is also known variably as ''half-sine window''<ref name=Kido/> or ''half-cosine window''.<ref name=Landisman/>

The [[autocorrelation]] of a sine window produces a function known as the Bohman window.<ref name=MWbohman/>

==== Power-of-sine/cosine windows ====
[[File:Power-of-sine windows.png|thumb|440px|Power-of-sine window functions (left) and their spectra in dB (right), for powers 0, 0.25, 0.5, 1, 2, 4]]

These window functions have the form:<ref name=PowCos/>

:<math>w[n] = \sin^\alpha\left(\frac{\pi n}{N}\right) = \cos^\alpha\left(\frac{\pi n}{N} - \frac{\pi}{2}\right),\quad 0\le n \le N.</math>

The [[#Rectangular window|rectangular window]] ({{math|1=''α''&nbsp;=&nbsp;0}}), the [[#Sine window|sine window]] ({{math|1=''α''&nbsp;=&nbsp;1}}), and the [[#Hann and Hamming windows|Hann window]] ({{math|1=''α''&nbsp;=&nbsp;2}}) are members of this family.

For even-integer values of {{mvar|α}} these functions can also be expressed in cosine-sum form:

: <math>w[n]=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N} \right)- a_3 \cos \left ( \frac{6 \pi n}{N} \right)+ a_4 \cos \left ( \frac{8 \pi n}{N} \right)- ...</math>
: <math>\begin{array}{l|llll}
\hline
\alpha & a_0 & a_1 & a_2 & a_3 & a_4 \\
\hline
 0 & 1 \\
 2 & 0.5 & 0.5 \\
 4 & 0.375 & 0.5 & 0.125 \\
 6 & 0.3125 & 0.46875 & 0.1875 & 0.03125 \\
 8 & 0.2734375 & 0.4375 & 0.21875 & 0.0625 & 7.8125\times10^{-3} \\
\hline
\end{array}</math>
{{clear}}

=== Adjustable windows ===

==== Gaussian window ====
[[File:Window function and frequency response - Gaussian (sigma = 0.4).svg|thumb|480px|right|Gaussian window, ''σ''&nbsp;=&nbsp;0.4]]
The Fourier transform of a [[Gaussian function|Gaussian]] is also a Gaussian.  Since the support of a Gaussian function extends to infinity, it must either be truncated at the ends of the window, or itself windowed with another zero-ended window.<ref name=JOSGauss/>

Since the log of a Gaussian produces a [[parabola]], this can be used for nearly exact quadratic interpolation in [[frequency estimation]].<ref name=JOSGauss2/><ref name=JOSGauss/><ref name=interpolation/>

:<math>w[n]=\exp\left(-\frac{1}{2} \left ( \frac{n-N/2}{\sigma N/2} \right)^{2}\right),\quad 0\le n \le N.</math>
:<math>\sigma \le \;0.5\,</math>

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

[[File:Window function and frequency response - Confined Gaussian (sigma t = 0.1).svg|thumb|480px|right|Confined Gaussian window, ''σ''<sub>''t''</sub>&nbsp;=&nbsp;0.1]]

==== 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 {{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}}

[[File:Window function and frequency response - Approximate confined Gaussian (sigma t = 0.1).svg|thumb|480px|right|Approximate confined Gaussian window, {{math|1=''σ''{{sub|''t''}}&nbsp;=&nbsp;0.1}}]]

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

Defining {{math|''L'' ≜ ''N'' + 1}}, a [[#Confined Gaussian window|confined Gaussian window]] of temporal width {{math|''L'' × ''σ''{{sub|''t''}}}} 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>

where <math>G</math> is a Gaussian function:

::<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 {{math|''L'' × ''σ''{{sub|''t''}}}} for {{math|''σ{{sub|t}}'' < 0.14}}.<ref name=Starosielec2014/>
{{clear}}

==== Generalized normal window ====
A more generalized version of the Gaussian window is the generalized normal window.<ref name=Chakraborty/> Retaining the notation from the [[Gaussian window]] above, we can represent this window as

:<math>w[n,p]=\exp\left(-\left ( \frac{n-N/2}{\sigma N/2} \right)^{p}\right)</math>

for any even <math>p</math>. At <math>p=2</math>, this is a Gaussian window and as <math>p</math> approaches <math>\infty</math>, this approximates to a rectangular window. The [[Fourier transform]] of this window does not exist in a closed form for a general <math>p</math>. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the {{slink|#Tukey window}}, this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window.
See also <ref name=Diethorn/> for a study on [[time-frequency representation]] of this window (or function).

{{clear}}

==== Tukey window ====
[[File:Window function and frequency response - Tukey (alpha = 0.5).svg|thumb|480px|right|Tukey window, {{math|1=''α''&nbsp;=&nbsp;0.5}}]]

The Tukey window, also known as the ''cosine-tapered window'', can be regarded as a cosine lobe of width {{math|''Nα''/2}} (spanning {{math|''Nα''/2&nbsp;+&nbsp;1}} observations) that is convolved with a rectangular window of width {{math|''N''(1 &minus; ''α''/2)}}.
:<math>
\left . 
\begin{array}{lll}
w[n] = \frac{1}{2} \left[1-\cos \left(\frac{2\pi n}{\alpha N} \right) \right],\quad & 0 \le n < \frac{\alpha N}{2}\\
w[n] = 1,\quad & \frac{\alpha N}{2} \le n \le \frac{N}{2}\\
w[N-n] = w[n],\quad & 0 \le n \le \frac{N}{2}
\end{array}\right\}
</math> <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
|1=[[#Harris|Harris 1978]] (p 67, eq 38) appears to have two errors: (1) The subtraction operator in the numerator of the cosine function should be addition. (2) The denominator contains a spurious factor of 2.  Also, Fig 30 corresponds to α=0.25 using the Wikipedia formula, but to 0.75 using the Harris formula.  Fig 32 is similarly mislabeled.
}}
At {{math|1=''α''&nbsp;=&nbsp;0}} it becomes rectangular, and at {{math|1=''α''&nbsp;=&nbsp;1}} it becomes a Hann window.
{{clear}}

==== Planck-taper window ====
[[File:Window function and frequency response - Planck-taper (epsilon = 0.25).svg|thumb|480px|Planck-taper window, ''ε''&nbsp;=&nbsp;0.25]]

The so-called "Planck-taper" window is a [[bump function]] that has been widely used<ref name=Tu/> in the theory of [[partitions of unity]] in [[manifolds]].  It is [[Smooth function|smooth]] (a <math>C^\infty</math> function) everywhere, but is exactly zero outside of a compact region, exactly one over an interval within that region, and varies smoothly and monotonically between those limits.  Its use as a window function in signal processing was first suggested in the context of [[gravitational-wave astronomy]], inspired by the [[Planck's law|Planck distribution]].<ref name=McKechan/> It is defined as a [[piecewise]] function''':'''
:<math>
\left . 
\begin{array}{lll}
w[0] =  0, \\
w[n] =  \left(1 + \exp\left(\frac{\varepsilon N}{n} - \frac{\varepsilon N}{\varepsilon N - n}\right)\right)^{-1},\quad & 1 \le n < \varepsilon N \\
w[n] =  1,\quad & \varepsilon N \le n \le \frac{N}{2} \\
w[N-n] = w[n],\quad & 0 \le n \le \frac{N}{2}
\end{array}\right\}
</math>

The amount of tapering is controlled by the parameter ''ε'', with smaller values giving sharper transitions.

{{clear}}

==== DPSS or Slepian window ====

The DPSS (discrete prolate spheroidal sequence) or [[Slepian function]], taper, or window [[Spectral concentration problem|maximizes the energy concentration in the main lobe]],<ref name=Slepian/> and is used in [[multitaper]] spectral analysis, which averages out noise in the spectrum and reduces information loss at the edges of the window.

The main lobe ends at a frequency bin given by the parameter ''α''.<ref name=KaiserDPSS/>
{|
|[[File:Window function and frequency response - DPSS (alpha = 2).svg|thumb|480px|right|DPSS window, ''α''&nbsp;=&nbsp;2]]
|[[File:Window function and frequency response - DPSS (alpha = 3).svg|thumb|480px|right|DPSS window, ''α''&nbsp;=&nbsp;3]]
|}
The Kaiser windows below are created by a simple approximation to the DPSS windows:
{|
|[[File:Window function and frequency response - Kaiser (alpha = 2).svg|thumb|480px|right|Kaiser window, ''α''&nbsp;=&nbsp;2]]
|[[File:Window function and frequency response - Kaiser (alpha = 3).svg|thumb|480px|right|Kaiser window, ''α''&nbsp;=&nbsp;3]]
|}

==== Kaiser window ====

{{Main|Kaiser window}}
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>  {{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}} 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 <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}}

==== Dolph–Chebyshev window ====
[[File:Window function and frequency response - Dolph-Chebyshev (alpha = 5).svg|thumb|480px|right|Dolph–Chebyshev window, ''α''&nbsp;=&nbsp;5]]

Minimizes the [[Uniform norm|Chebyshev norm]] of the side-lobes for a given main lobe width.<ref name=Dolph/>

The zero-phase Dolph–Chebyshev window function <math>w_0[n]</math> is usually defined in terms of its real-valued [[discrete Fourier transform]], <math>W_0[k]</math>:<ref name=DolphDef/>

:<math>
W_0(k) = \frac{T_{N}  \big(\beta \cos\left(\frac{\pi k}{N+1}\right)\big)}{T_{N}  (\beta)}
  = \frac{T_N    \big(\beta \cos\left(\frac{\pi k}{N+1}\right)\big)}{10^\alpha},\ 0 \le k \le N.
</math>

''T''<sub>''n''</sub>(''x'') is the ''n''-th [[Chebyshev polynomials|Chebyshev polynomial]] of the first kind evaluated in ''x'', which can be computed using

:<math>T_n(x) =\begin{cases}
\cos\!\big(n \cos^{-1}(x) \big) & \text{if }-1 \le x \le 1 \\
\cosh\!\big(n \cosh^{-1}(x) \big) & \text{if }x \ge 1 \\ 
(-1)^n \cosh\!\big(n \cosh^{-1}(-x) \big) & \text{if }x \le -1,
\end{cases}</math>

and

:<math>\beta = \cosh\!\big(\tfrac{1}{N} \cosh^{-1}(10^\alpha)\big)</math>

is the unique positive real solution to <math>T_N(\beta) = 10^\alpha</math>, where the parameter ''α'' sets the Chebyshev norm of the sidelobes to −20''α''&nbsp;decibels.<ref name=Dolph/>

The window function can be calculated from ''W''<sub>0</sub>(''k'') by an inverse [[discrete Fourier transform]] (DFT):<ref name=Dolph/>

:<math>w_0(n) = \frac{1}{N+1} \sum_{k=0}^N W_0(k) \cdot e^{i 2 \pi k n / (N+1)},\ -N/2 \le n \le N/2.</math>

The ''lagged'' version of the window can be obtained by:

:<math>w[n] = w_0\left(n-\frac{N}{2}\right),\quad 0 \le n \le N,</math>

which for even values of ''N'' must be computed as follows:

:<math>\begin{align}
w_0\left(n-\frac{N}{2}\right) 
= \frac{1}{N+1} \sum_{k=0}^{N} W_0(k) \cdot e^{\frac{i 2 \pi k (n-N/2)}{N+1}}
= \frac{1}{N+1} \sum_{k=0}^{N} \left[ \left(-e^{\frac{i\pi}{N+1}}\right)^k \cdot W_0(k)\right] e^{\frac{i 2 \pi k n}{N+1}},
\end{align}</math>

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

Variations:
*Due to the equiripple condition, the time-domain window has discontinuities at the edges.  An approximation that avoids them, by allowing the equiripples to drop off at the edges, is a [http://www.mathworks.com/help/signal/ref/taylorwin.html Taylor window].
*An alternative to the inverse DFT definition is also available.[http://practicalcryptography.com/miscellaneous/machine-learning/implementing-dolph-chebyshev-window/].
{{clear}}

==== Ultraspherical window ====
[[File:Window function and frequency response - Ultraspherical (mu = -0.5).svg|thumb|480px|right|The Ultraspherical window's ''μ'' 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/>

Like other adjustable windows, the Ultraspherical window has parameters that can be used to control its Fourier transform main-lobe width and relative side-lobe amplitude. Uncommon to other windows, it has an additional parameter which can be used to set the rate at which side-lobes decrease (or increase) in amplitude.<ref name=Deczky/><ref name=Bergen/><ref name=Bergen2/>

The window can be expressed in the time-domain as follows:<ref name=Deczky/>

:<math>
w[n] = \frac{1}{N+1} \left[ C^\mu_N(x_0)+\sum_{k=1}^{\frac{N}{2}}  C^\mu_N \left(x_0 \cos\frac{k\pi}{N+1}\right)\cos\frac{2n\pi k}{N+1} \right]
</math>

where <math>C^{\mu}_{N}</math> is the [[Ultraspherical polynomial]] of degree N, and <math>x_0</math> and <math>\mu</math> control the side-lobe patterns.<ref name=Deczky/>

Certain specific values of <math>\mu</math> yield other well-known windows: <math>\mu=0</math> and <math>\mu=1</math> give the Dolph–Chebyshev and [[Tapio Saramäki|Saramäki]] windows respectively.<ref name=Kabal/> See [http://octave.sourceforge.net/signal/function/ultrwin.html here] for illustration of Ultraspherical windows with varied parametrization.
{{clear}}

==== Exponential or Poisson window ====
[[File:Window function and frequency response - Exponential (half window decay).svg|thumb|480px|Exponential window, ''τ''&nbsp;=&nbsp;''N''/2]]
[[File:Window function and frequency response - Exponential (60dB decay).svg|thumb|480px|Exponential window, ''τ''&nbsp;=&nbsp;(''N''/2)/(60/8.69)]]

The Poisson window, or more generically the exponential window increases exponentially towards the center of the window and decreases exponentially in the second half. Since the [[exponential function]] never reaches zero, the values of the window at its limits are non-zero (it can be seen as the multiplication of an exponential function by a rectangular window <ref name=JOSPoisson/>). It is defined by

:<math>w[n]=e^{-\left|n-\frac{N}{2}\right|\frac{1}{\tau}},</math>

where ''τ'' is the time constant of the function. The exponential function decays as ''e''&nbsp;≃&nbsp;2.71828 or approximately 8.69&nbsp;dB per time constant.<ref name=Gade/>
This means that for a targeted decay of ''D''&nbsp;dB over half of the window length, the time constant ''τ'' is given by

:<math>\tau = \frac{N}{2}\frac{8.69}{D}.</math>
{{clear}}

=== Hybrid windows ===

Window functions have also been constructed as multiplicative or additive combinations of other windows.

[[File:Window function and frequency response - Bartlett-Hann.svg|thumb|480px|right|Bartlett–Hann window]]

==== Bartlett–Hann window ====

:<math>w[n]=a_0 - a_1 \left |\frac{n}{N}-\frac{1}{2} \right| - a_2 \cos \left (\frac{2 \pi n}{N}\right )</math>

:<math>a_0=0.62;\quad a_1=0.48;\quad a_2=0.38\,</math>
{{clear}}

==== Planck–Bessel window ====

[[File:Window function and frequency response - Planck-Bessel (epsilon = 0.1, alpha = 4.45).svg|thumb|480px|right|Planck–Bessel window, ''ε''&nbsp;=&nbsp;0.1, ''α''&nbsp;=&nbsp;4.45]]

A {{slink|#Planck-taper window}} multiplied by a [[Kaiser window]] which is defined in terms of a [[Modified Bessel function#Modified Bessel functions : I.CE.B1.2C K.CE.B1|modified Bessel function]]. This hybrid window function was introduced to decrease the peak side-lobe level of the Planck-taper window while still exploiting its good asymptotic decay.<ref name=Berry/> It has two tunable parameters, ''ε'' from the Planck-taper and ''α'' from the Kaiser window, so it can be adjusted to fit the requirements of a given signal.

{{clear}}

==== Hann–Poisson window ====
[[File:Window function and frequency response - Hann-Poisson (alpha = 2).svg|thumb|480px|Hann–Poisson window, ''α''&nbsp;=&nbsp;2]]
A [[#Hann and Hamming windows|Hann window]] multiplied by a [[#Exponential_or_Poisson_window|Poisson window]].  For <math>\alpha \geqslant 2</math> it has no side-lobes, as its Fourier transform drops off forever away from the main lobe without local minima.  It can thus be used in [[hill climbing]] algorithms like [[Newton's method]].<ref name=HannPoisson/> The Hann–Poisson window is defined by:

:<math>w[n]=\frac{1}{2}\left(1-\cos\left(\frac{2 \pi n}{N}\right)\right)e^\frac{-\alpha\left|N - 2n\right|}{N}\,</math>

where ''α'' is a parameter that controls the slope of the exponential.
{{Clear}}

=== Other windows ===

[[File:Window function and frequency response - GAP optimized Nuttall.svg|thumb|480px|GAP window (GAP optimized Nuttall window)]]

==== Generalized adaptive polynomial (GAP) window ====

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> <ref name=Beccaro/>

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

Additionally, starting with a set of expansion coefficients <math>a_{2k}</math> that mimics a certain known window function, the GAP window can be optimized by minimization procedures to get a new set of coefficients that improve one or more spectral properties, such as the main lobe width, side lobe attenuation, and side lobe falloff rate.<ref name=MWGAP/> Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.

[[File:Window function and frequency response - Lanczos.svg|thumb|480px|right|Sinc or Lanczos window]]

==== Lanczos window ====
<math display="block">w[n] = \operatorname{sinc}\left(\frac{2n}{N} - 1\right)</math>

* used in [[Lanczos resampling]]
* for the Lanczos window, <math>\operatorname{sinc}(x)</math> is defined as <math>\sin(\pi x)/\pi x</math>
* also known as a ''sinc window'', because: <math display="block">w_0(n) = \operatorname{sinc}\left(\frac{2n}{N}\right)\,</math> is the main lobe of a normalized [[sinc function]]
{{clear}}

=== Asymmetric window functions ===
The <math>w_0(x)</math> form, according to the convention above, is symmetric around <math>x = 0</math>. However, there are window functions that are asymmetric, such as the [[gamma distribution]] used in FIR implementations of [[gammatone filter]]s, or the [[beta distribution]] for a bounded-support approximation to the gamma distribution.  These asymmetries are used to reduce the delay when using large window sizes, or to emphasize the initial transient of a decaying pulse.{{citation needed|date=January 2023}}

Any [[bounded function]] with [[compact support]], including asymmetric ones, can be readily used as a window function. Additionally, there are ways to transform symmetric windows into asymmetric windows by transforming the time coordinate, such as with the below formula

:<math>
x \leftarrow N\left( \frac{x}{N}+\frac{1}{2} \right)^\alpha-\frac{N}{2}\,,
</math>

where the window weights more highly the earliest samples when <math>\alpha > 1</math>, and conversely weights more highly the latest samples when <math>\alpha < 1</math>.<ref>{{cite journal |last1=Luo |first1=Jiufel |last2=Xie |first2=Zhijiang |last3=Li |first3=Xinyi |title=Asymmetric Windows and Their Application in Frequency Estimation |journal=Chongqing University |date=2015-03-02 |volume=9 |issue=Algorithms & Computational Technology |pages=389–412 |doi=10.1260/1748-3018.9.4.389 |s2cid=124464194 |doi-access=free }}</ref>

== See also ==
* [[Apodization]]
* [[Kolmogorov–Zurbenko filter]]
* [[Multitaper]]
* [[Short-time Fourier transform]]
* [[Spectral leakage]]
* [[Welch method]]
* [[Weight function]]
* [[Window design method]]

== Notes ==
{{notelist-ua}}

== Page citations==
{{notelist-la}}

==References==
{{reflist|1|refs=
<ref name=Oppenheim>
{{Cite book
 |ref=Oppenheim
 |last1=Oppenheim
 |first1=Alan V.
 |author-link=Alan V. Oppenheim
 |last2=Schafer
 |first2=Ronald W.
 |author2-link=Ronald W. Schafer
 |last3=Buck
 |first3=John R.
 |title=Discrete-time signal processing
 |year=1999
 |publisher=Prentice Hall
 |location=Upper Saddle River, N.J.
 |isbn=0-13-754920-2
 |edition=2nd
 |chapter=7.2
 |pages=[https://archive.org/details/discretetimesign00alan/page/465 465]–478
 |url-access=registration
 |url=https://archive.org/details/discretetimesign00alan
}}
</ref>

<ref name=Weisstein>
{{cite book |title=CRC Concise Encyclopedia of Mathematics |first=Eric W. |last=Weisstein |publisher=CRC Press |year=2003 |isbn=978-1-58488-347-0 |url=https://books.google.com/books?id=aFDWuZZslUUC&q=apodization+function&pg=PA97
}}</ref>

<ref name=Roads>
{{cite book |title=Microsound |first=Curtis |last=Roads |publisher=MIT Press |year=2002 |isbn=978-0-262-18215-7
}}</ref>

<ref name=Cattani>
{{cite book |title=Wavelet and Wave Analysis As Applied to Materials With Micro Or Nanostructure |first1=Carlo |last1=Cattani |first2=Jeremiah |last2=Rushchitsky |publisher=World Scientific |year=2007 |isbn=978-981-270-784-0 |url=https://books.google.com/books?id=JuJKu_0KDycC&q=define+%22window+function%22+nonzero+interval&pg=PA53
}}</ref>

<ref name=OLA>
{{cite web|url=https://www.dsprelated.com/freebooks/sasp/Overlap_Add_OLA_STFT_Processing.html|title=Overlap-Add (OLA) STFT Processing {{!}} Spectral Audio Signal Processing |website=www.dsprelated.com |access-date=2016-08-07 |quote=The window is applied twice: once before the FFT (the "analysis window") and secondly after the inverse FFT prior to reconstruction by overlap-add (the so-called "synthesis window"). ... More generally, any positive COLA window can be split into an analysis and synthesis window pair by taking its square root.
}}</ref>
<!--
<ref name=Carlson>
{{cite book |url=https://books.google.com/books?id=V_JSAAAAMAAJ |title=Communication Systems: An Introduction to Signals and Noise in Electrical Communication |first=A. Bruce |last=Carlson |publisher=McGraw-Hill |year=1986 |isbn=978-0-07-009960-9
}}</ref>
--->
<ref name=FIRfilters>
{{cite web |url=http://www.labbookpages.co.uk/audio/firWindowing.html|title=FIR Filters by Windowing – The Lab Book Pages |website=www.labbookpages.co.uk |access-date=2016-04-13
}}</ref>

<ref name=Tuwien>
{{cite web |url=http://www.cg.tuwien.ac.at/research/vis/vismed/Windows/MasteringWindows.pdf
|title=Mastering Windows |website=www.cg.tuwien.ac.at |access-date=2020-02-12
}}</ref>

<ref name=HPmemory>
{{cite web |url=http://www.hpmemoryproject.org/an/pdf/an_243.pdf|title=The Fundamentals of Signal Analysis Application Note 243 |website=hpmemoryproject.org |access-date=10 April 2018
}}</ref>

<ref name=MWhann>
{{cite web |url=http://www.mathworks.com/help/signal/ref/hann.html |title=Hann (Hanning) window - MATLAB hann |website=www.mathworks.com |access-date=2020-02-12
}}</ref>
<!--
<ref name=MWwindow>
{{cite web |url=http://www.mathworks.com/help/dsp/ref/windowfunction.html |title=Window Function |website=www.mathworks.com |access-date=2019-04-14
}}</ref>

<ref name=Carlin>
{{cite patent
 |title=Wideband communication intercept and direction finding device using hyperchannelization
 |invent1=Carlin, Joe 
 |invent2=Collins, Terry 
 |invent3=Hays, Peter 
 |invent4=Hemmerdinger, Barry E. Kellogg, Robert L. Kettig, Robert L. Lemmon, Bradley K. Murdock, Thomas E. Tamaru, Robert S. Ware, Stuart M.
 |pubdate=1999-12-10
 |fdate=1999-12-10
 |gdate=2005-05-24
 |country=US
 |status=patent 
 |number=6898235
}} <!--template creates link to worldwide.espacenet.com-->

<ref name=Rorabaugh>
{{cite book
|first=C.Britton |last=Rorabaugh
|title=DSP Primer
|series=Primer series
|date=October 1998
|publisher=McGraw-Hill Professional
|isbn=978-0-07-054004-0
|page=196
}}</ref>

<ref name=Toraichi89>
{{cite journal |last1=Toraichi |first1=K. |last2=Kamada |first2=M. |last3=Itahashi |first3=S. |last4=Mori |first4=R. |title=Window functions represented by B-spline functions |doi=10.1109/29.17517 |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=37 |pages=145–147 |year=1989
}}</ref>

<ref name=Harris>
{{cite journal
  |ref=Harris
  |doi=10.1109/PROC.1978.10837
  |last=Harris
  |first=Fredric J.
  |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform
  |journal=Proceedings of the IEEE
  |volume=66
  |issue=1
  |pages=51–83
  |date=Jan 1978
  |url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880
  |bibcode=1978IEEEP..66...51H
|s2cid=426548
 }} ''The fundamental 1978 paper on FFT windows by Harris, which specified many windows and introduced key metrics used to compare them.''
</ref>

<ref name=Tukey>
{{cite journal |last=Tukey |first=J.W. |title=An introduction to the calculations of numerical spectrum analysis |journal=Spectral Analysis of Time Series |year=1967 |pages=25–46
}}</ref>

<ref name=MWtriang>
{{cite web |url=http://www.mathworks.com/help/signal/ref/triang.html |title=Triangular window – MATLAB triang |website=www.mathworks.com |access-date=2016-04-13
}}</ref>

<ref name=MWbohman>
{{cite web |url=https://www.mathworks.com/help/signal/ref/bohmanwin.html |title=Bohman window – R2019B |website=www.mathworks.com |access-date=2020-02-12
}}</ref>

<ref name=MWGAP>
{{cite web |url=https://www.mathworks.com/matlabcentral/fileexchange/81658-gap-generalized-adaptive-polynomial-window-function |title=Generalized Adaptive Polynomial Window Function |website=www.mathworks.com |access-date=2020-12-12
}}</ref>

<ref name=Welch1967>
{{cite journal |last1=Welch |first1=P. |title=The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms |doi=10.1109/TAU.1967.1161901 |journal=IEEE Transactions on Audio and Electroacoustics |volume=15 |issue=2 |pages=70–73 |year=1967| bibcode=1967ITAE...15...70W
|s2cid=13900622 }}</ref>

<ref name=Bosi>
{{cite book|last1=Bosi |first1=Marina |last2=Goldberg |first2=Richard E. |title=Introduction to Digital Audio Coding and Standards |chapter=Time to Frequency Mapping Part II: The MDCT |publisher=Springer US |series=The Springer International Series in Engineering and Computer Science |volume=721 |date=2003 |location=Boston, MA |page=106 |isbn=978-1-4615-0327-9 |doi=10.1007/978-1-4615-0327-9
}}</ref>

<ref name=Kido>
{{cite journal |last1=Kido |first1=Ken'iti |last2=Suzuki |first2=Hideo |last3=Ono |first3=Takahiko |last4=Fukushima |first4=Manabu |date=1998 |title=Deformation of impulse response estimates by time window in cross spectral technique |journal=Journal of the Acoustical Society of Japan E |volume=19 |issue=5 |pages=349–361 |doi=10.1250/ast.19.349
|doi-access=free
}}</ref>

<ref name=Landisman>
{{cite journal |last1=Landisman |first1=M. |last2=Dziewonski |first2=A. |last3=Satô |first3=Y. |date=1969-05-01 |title=Recent Improvements in the Analysis of Surface Wave Observations |journal=Geophysical Journal International |volume=17 |issue=4 |pages=369–403 |doi=10.1111/j.1365-246X.1969.tb00246.x |bibcode=1969GeoJ...17..369L
|doi-access=free
}}</ref>

<ref name=Enochson>
{{cite book |title=Programming and Analysis for Digital Time Series Data |first1=Loren D. |last1=Enochson |first2=Robert K. |last2=Otnes |publisher=U.S. Dept. of Defense, Shock and Vibration Info. Center |year=1968 |page=142 |url=https://books.google.com/books?id=duBQAAAAMAAJ&q=%22hamming+window%22+date:0-1970
}}</ref>

<ref name=sunar>
{{cite web |url=https://users.wpi.edu/~sunar/courses/ece3311/slides/ch16.pdf |title=A digital quadrature amplitude modulation (QAM) Radio: Building a better radio |website=users.wpi.edu |access-date=2020-02-12 |page=28
}}</ref>

<ref name=sunar2>
{{cite web |url=https://users.wpi.edu/~sunar/courses/ece3311/slides/ch08.pdf |title=Bits to Symbols to Signals and back again |website=users.wpi.edu |access-date=2020-02-12 |page=7
}}</ref>

<ref name=SRD>
{{cite book
 | last1 =Johnson
 | first1 =C.Richard Jr
 | last2 =Sethares
 | first2 =William A.
 | last3 =Klein
 | first3 =Andrew G.
 | title =Software Receiver Design
 | publisher =Cambridge University Press
 | date =2011-08-18
 | chapter =11
 | isbn =978-1-139-50145-3
}} Also https://cnx.org/contents/QsVBJjB4@3.1:6R_ztzDY@4/Pulse-Shaping-and-Receive-Filtering
</ref>

<ref name=Nuttall>
{{cite journal
 |ref=Nuttall
 | doi =10.1109/TASSP.1981.1163506
 | last =Nuttall
 | first =Albert H.
 | title =Some Windows with Very Good Sidelobe Behavior
 | journal =IEEE Transactions on Acoustics, Speech, and Signal Processing
 | volume =29
 | issue =1
 | pages =84–91
 | date =Feb 1981
 | url =https://zenodo.org/record/1280930
}} ''Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.''
</ref>

<ref name=Rabiner>
{{Cite book
| author1=Rabiner, Lawrence R.
| author2=Gold, Bernard
| title=Theory and application of digital signal processing
| year=1975
| publisher=Prentice-Hall
| location=Englewood Cliffs, N.J.
| isbn=0-13-914101-4
| chapter=3.11
| page=[https://archive.org/details/theoryapplicatio00rabi/page/94 94]
| chapter-url-access=registration
| chapter-url=https://archive.org/details/theoryapplicatio00rabi/page/94
}}</ref>

<ref name=Crochiere>
{{cite book
|last1=Crochiere |first1=R.E. |last2=Rabiner |first2=L.R. |title=Multirate Digital Signal Processing |year=1983 |chapter=4.3.1 |page=144 |publisher=Prentice-Hall |location=Englewood Cliffs, NJ |isbn=0-13-605162-6 |url=https://kupdf.net/download/multirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf
}}</ref>

<ref name=Kaiser_Window.html>
{{cite web
 |url=https://www.dsprelated.com/freebooks/sasp/Kaiser_Window.html
 |title=Kaiser Window
 |website=www.dsprelated.com
 |access-date=2020-04-08
 |quote=The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of {{math|α}} as the bin number of the edge of the critically sampled window main lobe.
}}</ref>

<ref name=MWBlackman>
{{cite web |url=http://mathworld.wolfram.com/BlackmanFunction.html |title=Blackman Function |last=Weisstein |first=Eric W. |website=mathworld.wolfram.com |language=en |access-date=2016-04-13
}}</ref>

<ref name=lvanl>
{{cite web |url=http://zone.ni.com/reference/en-XX/help/371361E-01/lvanlsconcepts/char_smoothing_windows/#Exact_Blackman |title=Characteristics of Different Smoothing Windows - NI LabVIEW 8.6 Help |website=zone.ni.com |access-date=2020-02-13
}}</ref>

<ref name=Blackman1959>
{{cite book |url=https://smile.amazon.com/Measurement-Power-Spectra-Communications-Engineering/dp/B0006AW1C4 |title=The Measurement of Power Spectra from the Point of View of Communications Engineering |last1=Blackman |first1=R.B. |author1-link=R. B. Blackman |last2=Tukey |first2=J.W. |date=1959-01-01 |publisher=Dover Publications |isbn=978-0-486-60507-4 |page=99
}}</ref>

<ref name=Heinzel2002>
{{cite tech report |first=G. |last=Heinzel |last2=Rüdiger |first2=A. |last3=Schilling |first3=R. |title=Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows |id=395068.0 |institution=Max Planck Institute (MPI) für Gravitationsphysik / Laser Interferometry & Gravitational Wave Astronomy |year=2002 |url=http://edoc.mpg.de/395068 |access-date=2013-02-10
}} Also available at https://pure.mpg.de/rest/items/item_152164_1/component/file_152163/content
</ref>

<ref name=SWsmith>
{{cite book |last=Smith |first=Steven W. |title=The Scientist and Engineer's Guide to Digital Signal Processing |url=http://www.dspguide.com/ch9/1.htm |access-date=2013-02-14 |year=2011 |publisher=California Technical Publishing |location=San Diego, California, USA
}}</ref>

<ref name=Rife>
{{citation |last1=Rife |first1=David C. |first2=G.A. |last2=Vincent |title=Use of the discrete Fourier transform in the measurement of frequencies and levels of tones |journal=Bell Syst. Tech. J. |volume=49 |issue=2 |year=1970 |pages=197–228 |doi=10.1002/j.1538-7305.1970.tb01766.x
}}</ref>

<ref name=Andria>
{{citation |last1=Andria |first1=Gregorio |first2=Mario |last2=Savino |first3=Amerigo |last3=Trotta |title=Windows and interpolation algorithms to improve electrical measurement accuracy |journal=IEEE Transactions on Instrumentation and Measurement |volume=38 |issue=4 |year=1989 |pages=856–863 |doi=10.1109/19.31004
|bibcode=1989ITIM...38..856A }}</ref>

<ref name=Schoukens>
{{citation |last1=Schoukens |first1=Joannes |first2=Rik |last2=Pintelon |first3=Hugo |last3=Van Hamme |title=The interpolated fast Fourier transform: a comparative study |journal=IEEE Transactions on Instrumentation and Measurement|volume=41 |issue=2 |year=1992 |pages=226–232 |doi=10.1109/19.137352
|bibcode=1992ITIM...41..226S }}</ref>

<ref name=Starosielec2014>
{{cite journal |last1=Starosielec |first1=S. |last2=Hägele |first2=D. |title=Discrete-time windows with minimal RMS bandwidth for given RMS temporal width |journal=Signal Processing |volume=102 |pages=240–246 |date=2014 |doi=10.1016/j.sigpro.2014.03.033
|bibcode=2014SigPr.102..240S }}</ref>

<ref name=Chakraborty>
{{cite book |doi=10.1109/ICASSP.2013.6638833 |chapter=Generalized normal window for digital signal processing |title=2013 IEEE International Conference on Acoustics, Speech and Signal Processing |pages=6083–6087 |year=2013 |last1=Chakraborty |first1=Debejyo |last2=Kovvali |first2=Narayan |isbn=978-1-4799-0356-6
|s2cid=11779529 }}</ref>

<ref name=Diethorn>
{{cite journal |doi=10.1109/78.295214 |title=The generalized exponential time-frequency distribution |journal=IEEE Transactions on Signal Processing |volume=42 |issue=5 |pages=1028–1037 |year=1994 |last1=Diethorn |first1=E.J. |bibcode=1994ITSP...42.1028D
}}</ref>

<!--
<ref name=MWtukey>
{{cite web |url=http://www.mathworks.com/help/signal/ref/tukeywin.html|title=Tukey (tapered cosine) window - MATLAB tukeywin |website=www.mathworks.com |access-date=2019-11-21
}}</ref>
-->

<ref name=Bloomfield>
{{cite book
 | last =Bloomfield
 | first =P.
 | title =Fourier Analysis of Time Series: An Introduction
 | publisher =Wiley-Interscience
 | date =2000
 | location =New York
}}</ref>

<ref name=Tu>
{{cite book |last=Tu |first=Loring W. |title=An Introduction to Manifolds |chapter=Bump Functions and Partitions of Unity |year=2008 |publisher=Springer |location=New York |isbn=978-0-387-48098-5 |pages=127–134 |doi=10.1007/978-0-387-48101-2_13 |series=Universitext
}}</ref>

<ref name=McKechan>
{{cite journal |last1=McKechan |first1=D.J.A. |last2=Robinson |first2=C. |last3=Sathyaprakash |first3=B.S. |title=A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries |journal=Classical and Quantum Gravity |date=21 April 2010 |volume=27 |issue=8 |page=084020 |doi=10.1088/0264-9381/27/8/084020 |arxiv=1003.2939 |bibcode = 2010CQGra..27h4020M
|s2cid=21488253 }}</ref>

<ref name=Kaiser1966>
{{cite book |title=System Analysis by Digital Computer |last1=Kaiser |first1=James F. |last2=Kuo |first2=Franklin F. |publisher=John Wiley and Sons |year=1966 |pages=232–235 |quote=This family of window functions was "discovered" by Kaiser in 1962 following a discussion with B. F. Logan of the Bell Telephone Laboratories. ... Another valuable property of this family ... is that they also approximate closely the prolate spheroidal wave functions of order zero.
}}</ref>

<ref name=Kaiser1964>
{{cite journal |last=Kaiser |first=James F. |date=Nov 1964 |title=A family of window functions having nearly ideal properties |journal=Unpublished Memorandum
}}</ref>

<ref name=Kaiser1980>
{{cite journal |last1=Kaiser |first1=James F. |last2=Schafer |first2=Ronald W. |doi=10.1109/TASSP.1980.1163349 |title=On the use of the I<sub>0</sub>-sinh window for spectrum analysis |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=28 |pages=105–107 |year=1980
}}</ref>

<ref name=MWkaiser>
{{cite web |url=https://www.mathworks.com/help/signal/ref/kaiser.html |website=www.mathworks.com |title=Kaiser Window, R2020a |publisher=Mathworks |access-date=9 April 2020
}}</ref>

<ref name=Kabal>
{{cite journal |last=Kabal |first=Peter |title=Time Windows for Linear Prediction of Speech |journal=Technical Report, Dept. Elec. & Comp. Eng., McGill University |year=2009 |issue=2a |page=31 |url=http://www-mmsp.ece.mcgill.ca/Documents/Reports/2009/KabalR2009b.pdf|access-date=2 February 2014
}}</ref>

<ref name=Streit>
{{cite journal |last=Streit |first=Roy |title=A two-parameter family of weights for nonrecursive digital filters and antennas |journal=  IEEE Transactions on Acoustics, Speech, and Signal Processing|year=1984 |volume=32 |pages=108–118 |doi=10.1109/tassp.1984.1164275
|url=https://zenodo.org/record/1280988 }}</ref>

<ref name=Bergen>
{{cite journal
  |last1=Bergen
  |first1=S.W.A.
  |first2=A. |last2=Antoniou
  |title=Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics
  |journal=EURASIP Journal on Applied Signal Processing
  |volume=2004
  |issue=13
  |pages=2053–2065
  |year=2004
  |doi=10.1155/S1110865704403114 |bibcode=2004EJASP2004...63B
  |doi-access=free
  }}</ref>

<ref name=Bergen2>Bergen, Stuart W. A. (2005). {{cite web | url=https://dspace.library.uvic.ca/bitstream/handle/1828/769/bergen_2005.pdf?sequence=1 |title=Design of the Ultraspherical Window Function and Its Applications}} Dissertation, University of Viktoria.</ref>

<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>

<ref name=Zphase>
{{cite web |url=https://ccrma.stanford.edu/~jos/filters/Zero_Phase_Filters_Even_Impulse.html
|title=Zero Phase Filters |website=ccrma.stanford.edu |access-date=2020-02-12
}}</ref>

<ref name=Bartlett>
{{cite web|url=https://ccrma.stanford.edu/~jos/sasp/Bartlett_Triangular_Window.html|title=Bartlett Window|website=ccrma.stanford.edu|access-date=2016-04-13
}}</ref>

<ref name=JOShann>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Hann_Hanning_Raised_Cosine.html |title=Hann or Hanning or Raised Cosine |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>
<!--
<ref name=JOShann2>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Matlab_Hann_Window.html |title=Matlab for the Hann Window |website=ccrma.stanford.edu |access-date=2020-09-01
}}</ref>
--->
<ref name=PowCos>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Power_of_Cosine_Window_Family.html |title=Power-of-Cosine Window Family |website=ccrma.stanford.edu |access-date=10 April 2018
}}</ref>

<ref name=JOSHamming>
{{Cite web |url=https://ccrma.stanford.edu/~jos/sasp/Hamming_Window.html |title=Hamming Window|website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=JOSBlack>
{{Cite web |url=https://ccrma.stanford.edu/~jos/sasp/Blackman_Harris_Window_Family.html |title=Blackman-Harris Window Family |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=JOSBlack3>
{{Cite web |url=https://ccrma.stanford.edu/~jos/sasp/Three_Term_Blackman_Harris_Window.html |title=Three-Term Blackman-Harris Window |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=JOSGauss2>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Gaussian_Window_Transform.html |title=Gaussian Window and Transform |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=JOSGauss>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Matlab_Gaussian_Window.html |title=Matlab for the Gaussian Window |website=ccrma.stanford.edu |access-date=2016-04-13 |quote=Note that, on a dB scale, Gaussians are quadratic. This means that parabolic interpolation of a sampled Gaussian transform is exact. ... quadratic interpolation of spectral peaks may be more accurate on a log-magnitude scale (e.g., dB) than on a linear magnitude scale
}}</ref>

<ref name=interpolation>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Quadratic_Interpolation_Spectral_Peaks.html |title=Quadratic Interpolation of Spectral Peaks |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=Slepian>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Slepian_DPSS_Window.html |title=Slepian or DPSS Window |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=KaiserDPSS>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html |website=ccrma.stanford.edu |title=Kaiser and DPSS Windows Compared |last=Smith |first=J.O. |date=2011 |access-date=2016-04-13
}}</ref>

<ref name=JOSKaiser>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_Window.html |quote=Sometimes the Kaiser window is parametrized by ''α'', where&nbsp;''β''&nbsp;=&nbsp;{{pi}}''α''. |website=ccrma.stanford.edu |title=Kaiser Window |last=Smith |first=J.O. |date=2011 |access-date=2019-03-20
}}</ref>

<ref name=Dolph>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window.html |title=Dolph-Chebyshev Window |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=DolphDef>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window_Definition.html |title=Dolph-Chebyshev Window Definition |website=ccrma.stanford.edu |access-date=2019-03-05
}}</ref>

<ref name=HannPoisson>
{{cite web |url=https://ccrma.stanford.edu/~jos/sasp/Hann_Poisson_Window.html |title=Hann-Poisson Window |website=ccrma.stanford.edu |access-date=2016-04-13
}}</ref>

<ref name=JOSPoisson>
{{cite web
 | url =https://ccrma.stanford.edu/~jos/sasp/Poisson_Window.html
 | title =Poisson Window
 | last =Smith
 | first =Julius O. III
 | date =2011-04-23
 | website =ccrma.stanford.edu
 | access-date =2020-02-12
}}</ref>

<ref name=Gade>
{{cite web
  | last1 =Gade
  | first1 =Svend
  | last2 =Herlufsen
  | first2 =Henrik
  | title =Technical Review No 3-1987: Windows to FFT analysis (Part I)
  | publisher =Brüel & Kjær
  | year =1987
  | url =http://www.bksv.com/doc/Bv0031.pdf
  | access-date =2011-11-22
}}</ref>

<ref name=Berry>
{{cite journal |last1=Berry |first1=C.P.L. |last2=Gair |first2=J.R. |title=Observing the Galaxy's massive black hole with gravitational wave bursts |journal=[[Monthly Notices of the Royal Astronomical Society]] |date=12 December 2012 |volume=429 |issue=1 |arxiv=1210.2778 |pages=589–612 |doi=10.1093/mnras/sts360|doi-access=free |bibcode=2013MNRAS.429..589B
|s2cid=118944979 }}</ref>

<ref name=Hovden>
{{cite journal |title=Periodic Artifact Reduction in Fourier Transforms of Full Field Atomic Resolution Images |author=R. Hovden, Y. Jiang, H. Xin, L.F. Kourkoutis |journal=Microscopy and Microanalysis |volume=21 |issue=2 |pages=436–441 |year=2015 |doi=10.1017/S1431927614014639 |pmid=25597865 |bibcode=2015MiMic..21..436H
|s2cid=22435248 |arxiv=2210.09024 }}</ref>

<ref name=Bernstein>
{{cite book
 | last1 =Bernstein
 | first1 =Matt A.
 | last2 =King
 | first2 =Kevin Franklin
 | last3 =Zhou
 | first3 =Xiaohong Joe
 | title =Handbook of MRI Pulse Sequences
 | publisher =Elsevier Academic Press
 | date =2004
 | location =London
 | pages =495–499
 | language =en
 | url =https://books.google.com/books?id=d6PLHcyejEIC&q=image%20tapering%20tukey&pg=PA496
 | isbn =0-12-092861-2
}}</ref>

<ref name=Awad>
{{cite book |last1=Awad |first1=A.I. |last2=Baba |first2=K. |chapter=An Application for Singular Point Location in Fingerprint Classification |doi=10.1007/978-3-642-22389-1_24 |title=Digital Information Processing and Communications |series=Communications in Computer and Information Science |volume=188 |page=262 |year=2011 |isbn=978-3-642-22388-4
}}</ref>

<ref name=Vaidyanathan>
{{cite journal |last1=Lin |first1=Yuan-Pei |last2=Vaidyanathan |first2=P.P. |title=A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks |journal=IEEE Signal Processing Letters |volume=5 |issue=6 | pages=132–134 |date=June 1998 |url=http://authors.library.caltech.edu/6891/1/LINieeespl98.pdf |access-date=2017-03-16| doi=10.1109/97.681427 |bibcode=1998ISPL....5..132L
|s2cid=18159105 }}</ref>
<!--
<ref name=Lyons>
{{cite web |last1=Lyons |first1=Richard |title=Windowing Functions Improve FFT Results |url=https://www.edn.com/windowing-functions-improve-fft-results-part-i/ |website=EDN |publisher=TRW |access-date=8 August 2020 |location=Sunnyvale, CA |date=1 June 1998
}}</ref>

<ref name=Fulton>
{{cite web |last1=Fulton |first1=Trevor |title=DP Numeric Transform Toolbox |url=http://herschel.esac.esa.int/hcss-doc-13.0/load/hcss_drm/ia/numeric/toolbox/xform/index.html |website=herschel.esac.esa.int |publisher=Herschel Data Processing |access-date=8 August 2020 |date=4 March 2008
}}</ref>

<ref name=Rohling>
{{cite journal |last1=Rohling |first1=H. |last2=Schuermann |first2=J. |title=Discrete time window functions with arbitrarily low sidelobe level |url=https://www.sciencedirect.com/science/article/abs/pii/0165168483900191 |journal=Signal Processing |publisher=AEG-Telefunken |access-date=8 August 2020 |location=Forschungsinstitut Ulm, Sedanstr, Germany |pages=127–138 |doi=10.1016/0165-1684(83)90019-1 |date=March 1983 |volume=5 |issue=2 |quote=It can be shown, that the DFT-even sampling technique as proposed by Harris is not the most suitable one.
}}</ref>

<ref name=Poularikas>
{{cite book |last1=Poularikas |first1=A.D. |editor1-last=Poularikas |editor1-first=Alexander D. |title=The Handbook of Formulas and Tables for Signal Processing |date=1999 |publisher=CRC Press LLC |location=Boca Raton |isbn={{Format ISBN|0849385792}} |url=http://dsp-book.narod.ru/HFTSP/8579ch07.pdf |access-date=8 August 2020 |chapter=7.3.1 |quote=Windows are even (about the origin) sequences with an odd number of points.  The right-most point of the window will be discarded.
}}</ref>

<ref name=Robertson>
{{cite web |last1=Robertson |first1=Neil |title=Evaluate Window Functions for the Discrete Fourier Transform |url=https://www.dsprelated.com/showarticle/1211.php |website=DSPRelated.com |publisher=The Related Media Group |access-date=9 August 2020 |date=18 December 2018
}} Revised 22 February 2020.
</ref>

<ref name=Puckette>
{{cite web |last1=Puckette |first1=Miller |title=Fourier analysis of non-periodic signals |url=http://msp.ucsd.edu/techniques/latest/book-html/node171.html#12499 |website=msp.ucsd.edu |publisher=UC San Diego |access-date=9 August 2020 |date=30 December 2006
}}</ref>
--->

<ref name=Beccaro>
{{citation
 | website =mathworks.com
 | title =Generalized Adaptive Polynomial Window Function
 | author =Wesley Beccaro
 | date =2020-10-31
 |access-date =2020-11-02
 | url =https://www.mathworks.com/matlabcentral/fileexchange/81658-gap-generalized-adaptive-polynomial-window-function?s_tid=LandingPageTabfx&s_tid=mwa_osa_a
}}</ref>
}}

== Further reading ==
* {{cite web
 | url =https://apps.dtic.mil/dtic/tr/fulltext/u2/a034956.pdf
 | archive-url =https://web.archive.org/web/20190408141816/https://apps.dtic.mil/dtic/tr/fulltext/u2/a034956.pdf
 | url-status =live
 | archive-date =April 8, 2019
 | title =Windows, Harmonic Analysis, and the Discrete Fourier Transform
 | last =Harris
 | first =Frederic J.
 | date =September 1976
 | website =apps.dtic.mil
 | publisher =Naval Undersea Center, San Diego
 | access-date =2019-04-08
 }}
* {{Cite book
 |doi=10.7795/110.20121022aa
 |year=2012
 |isbn=978-3-86918-281-0
 |last1=Albrecht
 |first1=Hans-Helge
 |title=Tailored minimum sidelobe and minimum sidelobe cosine-sum windows. Version 1.0
 |volume=ISBN 978-3-86918-281-0 ). editor: Physikalisch-Technische Bundesanstalt
 |publisher=Physikalisch-Technische Bundesanstalt
 }}
* {{cite journal
  |last1 =Bergen
  |first1 =S.W.A.
  |first2=A. |last2=Antoniou
  |title=Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function
  |journal=EURASIP Journal on Applied Signal Processing
  |volume=2005
  |issue=12
  |pages=1910–1922
  |year=2005
  |doi=10.1155/ASP.2005.1910 |bibcode=2005EJASP2005...44B
  |doi-access=free
  }}
* {{Cite book
 |last=Prabhu |first=K. M. M.
 |title=Window Functions and Their Applications in Signal Processing
 |year=2014
 |publisher=CRC Press
 |location=Boca Raton, FL
 |isbn=978-1-4665-1583-3
}}
* {{Cite patent
 | title = System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation
 | country-code = US
 | description = patent
 | patent-number = 7065150
 | postscript = <!--None-->
 | inventor-last =Park
 | inventor-first =Young-Seo
 | publication-date = 2003
 | issue-date = 2006
 |ref=none
}}

==External links==
*{{Commons category-inline}}
* LabView Help, Characteristics of Smoothing Filters, http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/
* Creation and properties of Cosine-sum Window functions, http://electronicsart.weebly.com/fftwindows.html
* [http://www.ritec-eg.com/Library%20&%20Tools/Windowing-Leakage-Bin-Centering-Window-Noise-Factor.html Online Interactive FFT, Windows, Resolution, and Leakage Simulation | RITEC | Library & Tools]

[[Category:Fourier analysis]]
[[Category:Signal estimation]]
[[Category:Digital signal processing]]
[[Category:Types of functions]]