Insignal processing andstatistics, awindow function (also known as anapodization function ortapering function^[1]) is amathematical function that is zero-valued outside of some choseninterval. 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 calledspectral 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.^[2] 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 issquare integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.^[3]

Window functions are used in spectralanalysis/modification/resynthesis,^[4] the design offinite impulse response filters, merging multiscale and multidimensional datasets,^[5][6] as well asbeamforming andantenna design.

TheFourier transform of the functioncos(ωt) is zero, except at frequency ±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.

Windows are sometimes used in the design ofdigital filters, in particular to convert an "ideal" impulse response of infinite duration, such as asinc function, to afinite impulse response (FIR) filter design. That is called thewindow method.^[7][8][9]

Window functions are sometimes used in the field ofstatistical analysis to restrict the set of data being analyzed to a range near a given point, with a weighting factor that diminishes the effect of points farther away from the portion of the curve being fit. In the field of Bayesian analysis andcurve fitting, this is often referred to as thekernel.

When analyzing a transient signal inmodal 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.^[10]

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 toFigure 2, we can observe that there is no leakage at a discrete set of harmonically-related frequencies sampled by thediscrete 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.

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. SeeWelch method of power spectral analysis and themodified discrete cosine transform.

Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform.^[11] They can be constructed from one-dimensional windows in either of two forms.^[12] The separable form,${\displaystyle W(m,n)=w(m)w(n)}$ is trivial to compute. Theradial form,${\displaystyle W(m,n)=w(r)}$, which involves the radius${\displaystyle r=\sqrt{(m-M/2{)}^2+(n-N/2{)}^2}}$, isisotropic, independent on the orientation of the coordinate axes. Only theGaussian function is both separable and isotropic.^[13] 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 ofdiffraction from rectangular vs. circular apertures, which can be visualized in terms of the product of twosinc functions vs. anAiry function, respectively.

Conventions:

• ${\displaystyle w_0(x)}$ is a zero-phase function (symmetrical about${\displaystyle x=0}$),^[14] continuous for${\displaystyle x\in [-N/2,N/2],}$ where${\displaystyle N}$ is a positive integer (even or odd).^[15]
• The sequence${\displaystyle \{w[n]=w_0(n-N/2),0\le n\le N\}}$ issymmetric, of length${\displaystyle N+1.}$
• ${\displaystyle \{w[n],0\le n\le N-1\}}$ isDFT-symmetric, of length${\displaystyle N.}$^[A]

• The parameterB displayed on each spectral plot is the function'snoise equivalent bandwidth metric, in units ofDFT bins.^{[16]: p.56 eq.(16)}
  • See spectral leakage §§ Discrete-time signals​ andSome window metrics andNormalized frequency for understanding the use of "bins" for the x-axis in these plots.

The sparse sampling of adiscrete-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.^[a] 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.

The rectangular window (sometimes known as theboxcar or uniform orDirichlet window or misleadingly as "no window" in some programs^[18]) is the simplest window, equivalent to replacing all butN consecutive values of a data sequence by zeros, making the waveform suddenly turn on and off:

${\displaystyle w[n]=1.}$

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

The rectangular window is the 1st-orderB-spline window as well as the 0th-powerpower-of-sine window.

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

B-spline windows can be obtained ask-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the§ Triangular window (k = 2) and the§ Parzen window (k = 4).^[19] Alternative definitions sample the appropriate normalizedB-splinebasis functions instead of convolving discrete-time windows. Akth-orderB-spline basis function is a piece-wise polynomial function of degreek − 1 that is obtained byk-fold self-convolution of therectangular function.

Triangular windows are given by

${\displaystyle w[n]=1-\left|\frac{n-\frac{N}{2}}{\frac{L}{2}}\right|,0\le n\le N,}$

whereL can beN,^[20]N + 1,^[16][21][22] orN + 2.^[23] The first one is also known asBartlett window orFejér window. All three definitions converge at large N.

The triangular window is the 2nd-orderB-spline window. TheL = N form can be seen as the convolution of twoN⁄2-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.

DefiningL ≜N + 1, the Parzen window, also known as thede la Vallée Poussin window,^[16] is the 4th-orderB-spline window given by

${\displaystyle w[n]=w_0\left(n-\frac{N}{2}\right),0\le n\le N}$

The Welch window consists of a singleparabolic section:

${\displaystyle w[n]=1-{\left(\frac{n-\frac{N}{2}}{\frac{N}{2}}\right)}^2,0\le n\le N.}$^[23]

Alternatively, it can be written as two factors, as in abeta distribution:

${\displaystyle w[n]=\left(1+\frac{n-\frac{N}{2}}{\frac{N}{2}}\right)\left(1-\frac{n-\frac{N}{2}}{\frac{N}{2}}\right),0\le n\le N.}$

The definingquadratic polynomial reaches a value of zero at the samples just outside the span of the window.

The Welch window is fairly close to thesine window, and just as thepower-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 symmetricbeta distributions, 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 makingasymmetric window functions.

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 theraised cosine distribution, which goes smoothly to zero at its ends.

The raised-cosine windows have the form:

${\displaystyle w[n]=a_0-(1-a_0)\cdot \cos \left(\frac{2\pi n}{N}\right),0\le n\le N,}$

or alternatively as their zero-phase versions:

Setting${\displaystyle a_0=0.5}$ produces aHann window:

${\displaystyle w[n]=0.5\left[1-\cos \left(\frac{2\pi n}{N}\right)\right]={\sin }^2\left(\frac{\pi n}{N}\right),}$^[24]

named afterJulius von Hann, and sometimes referred to asHanning, which derived from the verb "to Hann".^{[citation needed]} It is also known as theraised cosine, because of its similarity to araised-cosine distribution.

This function is a member of both thecosine-sum andpower-of-sine families. Unlike theHamming window, the end points of the Hann window just touch zero. The resultingside-lobes roll off at about 18 dB per octave.^[25]

Setting${\displaystyle a_0}$ to approximately 0.54, or more precisely 25/46, produces theHamming window, proposed byRichard W. Hamming. This choice places a zero crossing at frequency 5π/(N − 1), which cancels the first sidelobe of the Hann window, giving it a height of about one-fifth that of the Hann window.^[16][26][27]The Hamming window is often called theHamming blip when used forpulse shaping.^[28][29][30]

Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,^[16] to a nearly equiripple condition.^[27] In the equiripple sense, the optimal values for the coefficients area₀ = 0.53836 anda₁ = 0.46164.^[27][31]

This family, which generalizes theraised-cosine windows, is also known as generalized cosine windows.^[32]

${\displaystyle w[n]=\sum _{k=0}^K(-1{)}^k{a}_k\cos \left(\frac{2\pi kn}{N}\right),0\le n\le N.}$

Eq.1

In most cases, including the examples below, all coefficientsa_k ≥ 0. These windows have only 2K + 1 non-zeroN-point DFT coefficients.

Blackman windows are defined as

${\displaystyle w[n]=a_0-a_1\cos \left(\frac{2\pi n}{N}\right)+a_2\cos \left(\frac{4\pi n}{N}\right),}$

${\displaystyle a_0=\frac{1-\alpha }{2};a_1=\frac{1}{2};a_2=\frac{\alpha }{2}.}$

By common convention, the unqualified termBlackman window refers to Blackman's "not very serious proposal" ofα = 0.16 (a₀ = 0.42,a₁ = 0.5,a₂ = 0.08), which closely approximates theexact Blackman,^[33] witha₀ = 7938/18608 ≈ 0.42659,a₁ = 9240/18608 ≈ 0.49656, anda₂ = 1430/18608 ≈ 0.076849.^[34] These exact values place zeros at the third and fourth sidelobes,^[16] but result in a discontinuity at the edges and a 6 dB/oct fall-off. The truncated coefficients do not null the sidelobes as well, but have an improved 18 dB/oct fall-off.^[16][35]

The continuous form of the Nuttall window,${\displaystyle w_0(x),}$ and its firstderivative are continuous everywhere, like theHann function. That is, the function goes to 0 atx = ±N/2, unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows. The Blackman window (α = 0.16) is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not.

${\displaystyle w[n]=a_0-a_1\cos \left(\frac{2\pi n}{N}\right)+a_2\cos \left(\frac{4\pi n}{N}\right)-a_3\cos \left(\frac{6\pi n}{N}\right)}$

${\displaystyle a_0=0.355768;a_1=0.487396;a_2=0.144232;a_3=\mathrm{0.012604.}}$

${\displaystyle w[n]=a_0-a_1\cos \left(\frac{2\pi n}{N}\right)+a_2\cos \left(\frac{4\pi n}{N}\right)-a_3\cos \left(\frac{6\pi n}{N}\right)}$

${\displaystyle a_0=0.3635819;a_1=0.4891775;a_2=0.1365995;a_3=\mathrm{0.0106411.}}$

A generalization of the Hamming family, produced by adding more shifted cosine functions, meant to minimize side-lobe levels^[36][37]

${\displaystyle w[n]=a_0-a_1\cos \left(\frac{2\pi n}{N}\right)+a_2\cos \left(\frac{4\pi n}{N}\right)-a_3\cos \left(\frac{6\pi n}{N}\right)}$

${\displaystyle a_0=0.35875;a_1=0.48829;a_2=0.14128;a_3=\mathrm{0.01168.}}$

A flat top window is a partially negative-valued window that has minimalscalloping loss in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components.^[17][38] However, its broad bandwidth results in highnoise 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,^[38] or they may be of the usualcosine-sum variety:

TheMatlab variant has these coefficients:

${\displaystyle a_0=0.21557895;a_1=0.41663158;a_2=0.277263158;a_3=0.083578947;a_4=\mathrm{0.006947368.}}$

Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.^[17]

Rife–Vincent windows^[39] are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied toEq.1, reflect that custom.

Class I, Order 1 (K = 1):${\displaystyle a_0=1;a_1=1}$ Functionally equivalent to theHann window and power of sine (α = 2).

Class I, Order 2 (K = 2):${\displaystyle a_0=1;a_1=\frac{4}{3};a_2=\frac{1}{3}}$ Functionally equivalent to the power of sine (α = 4).

Class I is defined by minimizing the high-order sidelobe amplitude. Coefficients for orders up to K=4 are tabulated.^[40]

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

Class III is a compromise for which orderK = 2 resembles the§ Blackman window.^[40][41]

${\displaystyle w[n]=\sin \left(\frac{\pi n}{N}\right)=\cos \left(\frac{\pi n}{N}-\frac{\pi }{2}\right),0\le n\le N.}$

The corresponding${\displaystyle w_0(n)}$ function is a cosine without theπ/2 phase offset. So thesine window^[42] is sometimes also calledcosine window.^[16] As it represents half a cycle of a sinusoidal function, it is also known variably ashalf-sine window^[43] orhalf-cosine window.^[44]

Theautocorrelation of a sine window produces a function known as the Bohman window.^[45]

These window functions have the form:^[46]

${\displaystyle w[n]={\sin }^{\alpha }\left(\frac{\pi n}{N}\right)={\cos }^{\alpha }\left(\frac{\pi n}{N}-\frac{\pi }{2}\right),0\le n\le N.}$

Therectangular window (α = 0), thesine window (α = 1), and theHann window (α = 2) are members of this family.

For even-integer values ofα these functions can also be expressed in cosine-sum form:

${\displaystyle w[n]=a_0-a_1\cos \left(\frac{2\pi n}{N}\right)+a_2\cos \left(\frac{4\pi n}{N}\right)-a_3\cos \left(\frac{6\pi n}{N}\right)+a_4\cos \left(\frac{8\pi n}{N}\right)-...}$

The Fourier transform of aGaussian 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.^[47]

Since the log of a Gaussian produces aparabola, this can be used for nearly exact quadratic interpolation infrequency estimation.^[48][47][49]

${\displaystyle w[n]=\exp \left(-\frac{1}{2}{\left(\frac{n-N/2}{\sigma N/2}\right)}^2\right),0\le n\le N.}$

${\displaystyle \sigma \le 0.5}$

The standard deviation of the Gaussian function isσ · N/2 sampling periods.

The confined Gaussian window yields the smallest possible root mean square frequency widthσ_ω for a given temporal width(N + 1)σ_t.^[50] 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§ Sine window and the§ Gaussian window in the limiting cases of large and smallσ_t, respectively.

DefiningL ≜N + 1, aconfined Gaussian window of temporal widthL ×σ_t is well approximated by:^[50]

${\displaystyle w[n]=G(n)-\frac{G(-\frac{1}{2})[G(n+L)+G(n-L)]}{G(-\frac{1}{2}+L)+G(-\frac{1}{2}-L)}}$

where${\displaystyle G}$ is a Gaussian function:

${\displaystyle G(x)=\exp \left(-{\left(\frac{x-\frac{N}{2}}{2L{\sigma }_t}\right)}^2\right)}$

The standard deviation of the approximate window isasymptotically equal (i.e. large values ofN) toL ×σ_t forσ_t < 0.14.^[50]

A more generalized version of the Gaussian window is the generalized normal window.^[51] Retaining the notation from theGaussian window above, we can represent this window as

${\displaystyle w[n,p]=\exp \left(-{\left(\frac{n-N/2}{\sigma N/2}\right)}^p\right)}$

for any even${\displaystyle p}$. At${\displaystyle p=2}$, this is a Gaussian window and as${\displaystyle p}$ approaches${\displaystyle \mathrm{\infty }}$, this approximates to a rectangular window. TheFourier transform of this window does not exist in a closed form for a general${\displaystyle p}$. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the§ 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^[52] for a study ontime-frequency representation of this window (or function).

The Tukey window, also known as thecosine-tapered window, can be regarded as a cosine lobe of widthNα/2 (spanningNα/2 + 1 observations) that is convolved with a rectangular window of widthN(1 −α/2).

^[53][B][C]

Atα = 0 it becomes rectangular, and atα = 1 it becomes a Hann window.

The so-called "Planck-taper" window is abump function that has been widely used^[54] in the theory ofpartitions of unity inmanifolds. It issmooth (a${\displaystyle C^{\mathrm{\infty }}}$ 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 ofgravitational-wave astronomy, inspired by thePlanck distribution.^[55] It is defined as apiecewise function:

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

The DPSS (discrete prolate spheroidal sequence) orSlepian function, taper, or windowmaximizes the energy concentration in the main lobe,^[56] and is used inmultitaper 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 parameterα.^[57]

The Kaiser windows below are created by a simple approximation to the DPSS windows:

The Kaiser, or Kaiser–Bessel, window is a simple approximation of theDPSS window usingBessel functions, discovered byJames Kaiser.^[58][59]

${\displaystyle w[n]=\frac{I_0\left(\pi \alpha \sqrt{1-{\left(\frac{2n}{N}-1\right)}^2}\right)}{I_0(\pi \alpha )},0\le n\le N}$^{[D][16]: p. 73}

${\displaystyle w_0(n)=\frac{I_0\left(\pi \alpha \sqrt{1-{\left(\frac{2n}{N}\right)}^2}\right)}{I_0(\pi \alpha )},-N/2\le n\le N/2}$

where${\displaystyle I_0}$ is the 0^th-order modified Bessel function of the first kind. Variable parameter${\displaystyle \alpha }$ 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${\displaystyle 2\sqrt{1+{\alpha }^2},}$ in units of DFT bins,^[66] and a typical value of${\displaystyle \alpha }$ is 3.

Minimizes theChebyshev norm of the side-lobes for a given main lobe width.^[67]

The zero-phase Dolph–Chebyshev window function${\displaystyle w_0[n]}$ is usually defined in terms of its real-valueddiscrete Fourier transform,${\displaystyle W_0[k]}$:^[68]

${\displaystyle W_0(k)=\frac{T_N(\beta \cos \left(\frac{\pi k}{N+1}\right))}{T_N(\beta )}=\frac{T_N(\beta \cos \left(\frac{\pi k}{N+1}\right))}{{10}^{\alpha }},0\le k\le N.}$

T_n(x) is then-thChebyshev polynomial of the first kind evaluated inx, which can be computed using

and

${\displaystyle \beta =\cosh (\frac{1}{N}{\cosh }^{-1}({10}^{\alpha }))}$

is the unique positive real solution to${\displaystyle T_N(\beta )={10}^{\alpha }}$, where the parameterα sets the Chebyshev norm of the sidelobes to −20α decibels.^[67]

The window function can be calculated fromW₀(k) by an inversediscrete Fourier transform (DFT):^[67]

${\displaystyle w_0(n)=\frac{1}{N+1}\sum _{k=0}^N{W}_0(k)\cdot e^{i2\pi kn/(N+1)},-N/2\le n\le N/2.}$

Thelagged version of the window can be obtained by:

${\displaystyle w[n]=w_0\left(n-\frac{N}{2}\right),0\le n\le N,}$

which for even values ofN must be computed as follows:

${\displaystyle \begin{array}{r}{w}_0\left(n-\frac{N}{2}\right)=\frac{1}{N+1}\sum _{k=0}^N{W}_0(k)\cdot e^{\frac{i2\pi k(n-N/2)}{N+1}}=\frac{1}{N+1}\sum _{k=0}^N\left[{\left(-e^{\frac{i\pi }{N+1}}\right)}^k\cdot W_0(k)\right]e^{\frac{i2\pi kn}{N+1}},\end{array}}$

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

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 aTaylor window.
• An alternative to the inverse DFT definition is also available.[1].

The Ultraspherical window was introduced in 1984 by Roy Streit^[69] and has application in antenna array design,^[70] non-recursive filter design,^[69] and spectrum analysis.^[71]

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.^[71][72][73]

The window can be expressed in the time-domain as follows:^[71]

${\displaystyle w[n]=\frac{1}{N+1}\left[C_N^{\mu }(x_0)+\sum _{k=1}^{\frac{N}{2}}{C}_N^{\mu }\left(x_0\cos \frac{k\pi }{N+1}\right)\cos \frac{2n\pi k}{N+1}\right]}$

where${\displaystyle C_N^{\mu }}$ is theUltraspherical polynomial of degree N, and${\displaystyle x_0}$ and${\displaystyle \mu }$ control the side-lobe patterns.^[71]

Certain specific values of${\displaystyle \mu }$ yield other well-known windows:${\displaystyle \mu =0}$ and${\displaystyle \mu =1}$ give the Dolph–Chebyshev andSaramäki windows respectively.^[69] Seehere for illustration of Ultraspherical windows with varied parametrization.

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 theexponential 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^[74]). It is defined by

${\displaystyle w[n]=e^{-\left|n-\frac{N}{2}\right|\frac{1}{\tau }},}$

whereτ is the time constant of the function. The exponential function decays ase ≃ 2.71828 or approximately 8.69 dB per time constant.^[75]This means that for a targeted decay ofD dB over half of the window length, the time constantτ is given by

${\displaystyle \tau =\frac{N}{2}\frac{8.69}{D}.}$

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

${\displaystyle w[n]=a_0-a_1\left|\frac{n}{N}-\frac{1}{2}\right|-a_2\cos \left(\frac{2\pi n}{N}\right)}$

${\displaystyle a_0=0.62;a_1=0.48;a_2=0.38}$

A§ Planck-taper window multiplied by aKaiser window which is defined in terms of amodified 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.^[76] It has two tunable parameters,ε from the Planck-taper andα from the Kaiser window, so it can be adjusted to fit the requirements of a given signal.

AHann window multiplied by aPoisson window. For${\displaystyle \alpha ⩾2}$ 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 inhill climbing algorithms likeNewton's method.^[77] The Hann–Poisson window is defined by:

${\displaystyle w[n]=\frac{1}{2}\left(1-\cos \left(\frac{2\pi n}{N}\right)\right)e^{\frac{-\alpha \left|N-2n\right|}{N}}}$

whereα is a parameter that controls the slope of the exponential.

The GAP window is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order${\displaystyle K}$. 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.

${\displaystyle w_0[n]=a_0+\sum _{k=1}^K{a}_{2k}{\left(\frac{n}{\sigma }\right)}^{2k},-\frac{N}{2}\le n\le \frac{N}{2},}$^[78]

where${\displaystyle \sigma }$ is the standard deviation of the${\displaystyle \{n\}}$ sequence.

Additionally, starting with a set of expansion coefficients${\displaystyle a_{2k}}$ 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.^[79] Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.

$${\displaystyle w[n]=\mathrm{sinc}\left(\frac{2n}{N}-1\right)}$$

• used inLanczos resampling
• for the Lanczos window,${\displaystyle \mathrm{sinc}(x)}$ is defined as${\displaystyle \sin (\pi x)/\pi x}$
• also known as asinc window, because:$${\displaystyle w_0(n)=\mathrm{sinc}\left(\frac{2n}{N}\right)}$$ is the main lobe of a normalizedsinc function

The${\displaystyle w_0(x)}$ form, according to the convention above, is symmetric around${\displaystyle x=0}$. However, there are window functions that are asymmetric, such as thegamma distribution used in FIR implementations ofgammatone filters, or thebeta 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]}

Anybounded function withcompact 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

${\displaystyle x\leftarrow N{\left(\frac{x}{N}+\frac{1}{2}\right)}^{\alpha }-\frac{N}{2},}$

where the window weights more highly the earliest samples when${\displaystyle \alpha >1}$, and conversely weights more highly the latest samples when.^[80]

• Apodization
• Kolmogorov–Zurbenko filter
• Multitaper
• Short-time Fourier transform
• Spectral leakage
• Welch method
• Weight function
• Window design method

• Harris, Frederic J. (September 1976)."Windows, Harmonic Analysis, and the Discrete Fourier Transform"(PDF).apps.dtic.mil. Naval Undersea Center, San Diego.Archived(PDF) from the original on April 8, 2019. Retrieved2019-04-08.
• Albrecht, Hans-Helge (2012).Tailored minimum sidelobe and minimum sidelobe cosine-sum windows. Version 1.0. Vol. ISBN 978-3-86918-281-0 ). editor: Physikalisch-Technische Bundesanstalt. Physikalisch-Technische Bundesanstalt.doi:10.7795/110.20121022aa.ISBN 978-3-86918-281-0.
• Bergen, S.W.A.; Antoniou, A. (2005)."Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function".EURASIP Journal on Applied Signal Processing.2005 (12):1910–1922.Bibcode:2005EJASP2005...44B.doi:10.1155/ASP.2005.1910.
• Prabhu, K. M. M. (2014).Window Functions and Their Applications in Signal Processing. Boca Raton, FL: CRC Press.ISBN 978-1-4665-1583-3.
• US patent 7065150, Park, Young-Seo, "System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation", published 2003, issued 2006 

• Media related toWindow function at Wikimedia Commons
• 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
• Online Interactive FFT, Windows, Resolution, and Leakage Simulation | RITEC | Library & Tools

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