Window function#Triangular window

Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, merging multiscale and multidimensional datasets,{{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 }}{{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}} as well as beamforming and antenna design.

File:Spectral_leakage_caused_by_%22windowing%22.svg (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.]]

{{Main|Spectral leakage}}

The Fourier transform of the function {{math|cos(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.

{{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 window method.

{{Main|kernel (statistics)}}

Window functions are sometimes used in the field of statistical 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 and curve fitting, this is often referred to as the kernel.

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.

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.

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.

{{main|Two-dimensional window design}}

Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform. They can be constructed from one-dimensional windows in either of two forms. The separable form, $W(m,n)=w(m)w(n)$ is trivial to compute. The radial form, $W(m,n)=w(r)$, which involves the radius $r=\backslash sqrt\{(m-M/2)^2+(n-N/2)^2\}$, is isotropic, independent on the orientation of the coordinate axes. Only the Gaussian function is both separable and isotropic. 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 functions vs. an Airy function, respectively.

Conventions:

• $w\_0(x)$ is a zero-phase function (symmetrical about $x=0$), continuous for $x\; \backslash in\; [-N/2,\; N/2],$ where $N$ is a positive integer (even or odd).
• The sequence $\backslash \{w[n]\; =\; w\_0(n-N/2),\backslash quad\; 0\backslash le\; n\; \backslash le\; N\backslash \}$ is symmetric, of length $N+1.$
• $\backslash \{w[n],\backslash quad\; 0\backslash le\; n\; \backslash le\; N-1\backslash \}$ is DFT-symmetric, of length $N.${{efn-ua

|Some authors limit their attention to this important subset and to even values of N. But the window coefficient formulas are still the ones presented here.}}

• The parameter B displayed on each spectral plot is the function's noise equivalent bandwidth metric, in units of DFT bins.{{rp|p.56 eq.(16)}}
• See {{Slink|spectral leakage|Discrete-time signals|Some window metrics}} and 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 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.

File:Window function and frequency response - Rectangular.svg

The rectangular window (sometimes known as the boxcar or uniform or Dirichlet window or misleadingly as "no window" in some programs{{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}}) 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:

:$w[n]\; =\; 1.$

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 window.

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

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B-spline windows can be obtained as k-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the {{slink|#Triangular window}} (k = 2) and the {{slink|#Parzen window}} (k = 4). Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows. A kth-order B-spline basis function is a piece-wise polynomial function of degree k − 1 that is obtained by k-fold self-convolution of the rectangular function.

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File:Window function and its Fourier transform – Triangular (n = 0...N).svg

Triangular windows are given by

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

where L can be N, N + 1, or N + 2. The first one is also known as Bartlett window or Fejér window. All three definitions converge at large N.

The triangular window is the 2nd-order B-spline window. The L = 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.

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File:Window function and frequency response - Parzen.svg

{{Distinguish|Kernel density estimation}}

Defining {{math|L ≜ N + 1}}, the Parzen window, also known as the de la Vallée Poussin window, is the 4th-order B-spline window given by

:$$

w_0(n) \triangleq \left\{ \begin{array}{ll}

1 - 6 \left(\frac{n}{L/2}\right)^2 \left(1 - \frac

& 0 \le |n| \le \frac{L}{4} \\

2 \left(1 - \frac

& \frac{L}{4} < |n| \le \frac{L}{2} \\

\end{array} \right\}

:$w[n]\; =\; \backslash \; w\_0\backslash left(n-\backslash tfrac\{N\}\{2\}\backslash right),\backslash \; 0\; \backslash le\; n\; \backslash le\; N$

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File:Window function and frequency response - Welch.svg

The Welch window consists of a single parabolic section:

:$w[n]=1\; -\; \backslash left(\backslash frac\{n-\backslash frac\{N\}\{2\}\}\{\backslash frac\{N\}\{2\}\}\backslash right)^2,\backslash quad\; 0\backslash le\; n\; \backslash le\; N.$

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

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File:Window function and frequency response - Cosine.svg

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

The corresponding $w\_0(n)\backslash ,$ function is a cosine without the {{pi}}/2 phase offset. So the sine window is sometimes also called cosine window. As it represents half a cycle of a sinusoidal function, it is also known variably as half-sine window or half-cosine window.

The autocorrelation of a sine window produces a function known as the Bohman window.

These window functions have the form:

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

The rectangular window ({{math|1=α = 0}}), the sine window ({{math|1=α = 1}}), and the Hann window ({{math|1=α = 2}}) are members of this family.

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

: $w[n]=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N\}\; \backslash right)+\; a\_4\; \backslash cos\; \backslash left\; (\; \backslash frac\{8\; \backslash pi\; n\}\{N\}\; \backslash right)-\; ...$

: $\backslash begin\{array\}\{l|llll\}$

\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}

This family is also known as [https://www.mathworks.com/help/signal/ug/generalized-cosine-windows.html generalized cosine windows].

{{Equation box 1

|indent =

|title=

|equation = {{NumBlk|:|$w[n]\; =\; \backslash sum\_\{k\; =\; 0\}^\{K\}\; (-1)^k\; a\_k\backslash ;\; \backslash cos\backslash left(\; \backslash frac\{2\; \backslash pi\; k\; n\}\{N\}\; \backslash right),\backslash quad\; 0\backslash le\; n\; \backslash le\; N.$|{{EquationRef|Eq.1}}}}

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

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

{{Main|Hann function}}

File:Window function and its Fourier transform – Hann (n = 0...N).svg

File:Window function and frequency response - Hamming (alpha = 0.53836, n = 0...N).svg

The customary cosine-sum windows for case K = 1 have the form

:$w[n]\; =\; a\_0\; -\; \backslash underbrace\{(1-a\_0)\}\_\{a\_1\}\backslash cdot\; \backslash cos\backslash left(\; \backslash tfrac\{2\; \backslash pi\; n\}\{N\}\; \backslash right),\backslash quad\; 0\backslash le\; n\; \backslash le\; N,$

which is easily (and often) confused with its zero-phase version:

:$$

\begin{align}

w_0(n)\ &= w\left[ n+\tfrac{N}{2}\right]\\

&= a_0 + a_1\cdot \cos \left ( \tfrac{2\pi n}{N} \right),\quad -\tfrac{N}{2} \le n \le \tfrac{N}{2}.

\end{align}

Setting $a\_0\; =\; 0.5$ produces a Hann window:

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

named after Julius von Hann, and sometimes erroneously referred to as Hanning, presumably due to its linguistic and formulaic similarities to the Hamming window. It is also known as raised cosine, because the zero-phase version, $w\_0(n),$ is one lobe of an elevated cosine function.

This function is a member of both the cosine-sum and power-of-sine families. Unlike the Hamming window, the end points of the Hann window just touch zero. The resulting side-lobes roll off at about 18 dB per octave.

Setting $a\_0$ to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by Richard W. Hamming. This choice places a zero crossing at frequency 5{{pi}}/(N − 1), which cancels the first sidelobe of the Hann window, giving it a height of about one-fifth that of the Hann window.

The Hamming window is often called the Hamming blip when used for pulse shaping.

Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes, to a nearly equiripple condition. In the equiripple sense, the optimal values for the coefficients are a₀ = 0.53836 and a₁ = 0.46164.

File:Window function and its Fourier transform – Blackman (n = 0...N).svg

Blackman windows are defined as

:$w[n]\; =\; a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N\}\; \backslash right)\; +\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N\}\; \backslash right),$

:$a\_0=\backslash frac\{1-\backslash alpha\}\{2\};\backslash quad\; a\_1=\backslash frac\{1\}\{2\};\backslash quad\; a\_2=\backslash frac\{\backslash alpha\}\{2\}.$

By common convention, the unqualified term Blackman window refers to Blackman's "not very serious proposal" of {{math|1=α = 0.16}} (a₀ = 0.42, a₁ = 0.5, a₂ = 0.08), which closely approximates the exact Blackman, with a₀ = 7938/18608 ≈ 0.42659, a₁ = 9240/18608 ≈ 0.49656, and a₂ = 1430/18608 ≈ 0.076849. These exact values place zeros at the third and fourth sidelobes, 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.

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File:Window function and frequency response - Nuttall (continuous first derivative).svg

The continuous form of the Nuttall window, $w\_0(x),$ and its first derivative are continuous everywhere, like the Hann function. That is, the function goes to 0 at {{nowrap|1=x = ±N/2,}} unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows. The Blackman window ({{math|1=α = 0.16}}) is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not.

:$w[n]=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N\}\; \backslash right)$

:$a\_0=0.355768;\backslash quad\; a\_1=0.487396;\backslash quad\; a\_2=0.144232;\backslash quad\; a\_3=0.012604.$

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File:Window function and frequency response - Blackman-Nuttall.svg

:$w[n]=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N\}\; \backslash right)$

:$a\_0=0.3635819;\; \backslash quad\; a\_1=0.4891775;\; \backslash quad\; a\_2=0.1365995;\; \backslash quad\; a\_3=0.0106411.$

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File:Window function and frequency response - Blackman-Harris.svg

A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels

:$w[n]=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N\}\; \backslash right)$

:$a\_0=0.35875;\backslash quad\; a\_1=0.48829;\backslash quad\; a\_2=0.14128;\backslash quad\; a\_3=0.01168.$

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File:Window function and frequency response - flat top.svg

A flat top window is a partially negative-valued window that has minimal scalloping loss in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components. 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, or they may be of the usual cosine-sum variety:

:$$

\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}

The [https://www.mathworks.com/help/signal/ref/flattopwin.html Matlab variant] has these coefficients:

:$a\_0=0.21557895;\backslash quad\; a\_1=0.41663158;\backslash quad\; a\_2=0.277263158;\backslash quad\; a\_3=0.083578947;\backslash quad\; a\_4=0.006947368.$

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

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Rife–Vincent windows 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): $a\_0=1;\backslash quad\; a\_1=1$ Functionally equivalent to the Hann window.

Class I, Order 2 (K = 2): $a\_0=1;\backslash quad\; a\_1=\backslash tfrac\{4\}\{3\};\backslash quad\; a\_2=\backslash tfrac\{1\}\{3\}$

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

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

Class III is a compromise for which order K = 2 resembles the {{slink|#Blackman window}}.

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File:Window function and frequency response - Gaussian (sigma = 0.4).svg

The Fourier transform of a 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.

Since the log of a Gaussian produces a parabola, this can be used for nearly exact quadratic interpolation in frequency estimation.

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

:$\backslash sigma\; \backslash le\; \backslash ;0.5\backslash ,$

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

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File:Window function and frequency response - Confined Gaussian (sigma t = 0.1).svg

The confined Gaussian window yields the smallest possible root mean square frequency width {{math|σ{{sub|ω}}}} for a given temporal width {{math|(N + 1) σ{{sub|t}}}}. 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.

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File:Window function and frequency response - Approximate confined Gaussian (sigma t = 0.1).svg

Defining {{math|L ≜ N + 1}}, a confined Gaussian window of temporal width {{math|L × σ{{sub|t}}}} is well approximated by:

:$w[n]\; =\; G(n)\; -\; \backslash frac\{G(-\backslash tfrac\{1\}\{2\})[G(n\; +\; L)\; +\; G(n\; -\; L)]\}\{G(-\backslash tfrac\{1\}\{2\}\; +\; L)\; +\; G(-\backslash tfrac\{1\}\{2\}\; -\; L)\}$

where $G$ is a Gaussian function:

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

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}}.

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

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

for any even $p$. At $p=2$, this is a Gaussian window and as $p$ approaches $\backslash infty$, this approximates to a rectangular window. The Fourier transform of this window does not exist in a closed form for a general $p$. 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 for a study on time-frequency representation of this window (or function).

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File:Window function and frequency response - Tukey (alpha = 0.5).svg

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 + 1}} observations) that is convolved with a rectangular window of width {{math|N(1 − α/2)}}.

:$$

\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\}

{{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 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 = 0}} it becomes rectangular, and at {{math|1=α = 1}} it becomes a Hann window.

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File:Window function and frequency response - Planck-taper (epsilon = 0.25).svg

The so-called "Planck-taper" window is a bump function that has been widely used in the theory of partitions of unity in manifolds. It is smooth (a $C^\backslash infty$ 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 of gravitational-wave astronomy, inspired by the Planck distribution. It is defined as a piecewise function:

:$$

\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\}

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

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The DPSS (discrete prolate spheroidal sequence) or Slepian window maximizes the energy concentration in the main lobe, 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.

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

{{Main|Kaiser window}}

The Kaiser, or Kaiser–Bessel, window is a simple approximation of the DPSS window using Bessel functions, discovered by James Kaiser.

:$w[n]=\backslash frac\{I\_0\backslash left(\backslash pi\backslash alpha\; \backslash sqrt\{1-\backslash left(\backslash frac\{2\; n\}\{N\}-1\backslash right)^2\}\backslash right)\}\{I\_0(\backslash pi\backslash alpha)\},\backslash quad\; 0\backslash le\; n\; \backslash le\; N$ {{efn-ua

|The Kaiser window is often parametrized by {{math|β}}, where {{math|1=β = {{pi}}α}}.

{{rp|p. 474}} The alternative use of just {{math|α}} facilitates comparisons to the DPSS windows.

}}{{rp|p. 73}}

:$$

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

where $I\_0$ is the 0{{Sup|th}}-order modified Bessel function of the first kind. Variable parameter $\backslash alpha$ 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 $2\backslash sqrt\{1\; +\; \backslash alpha^2\},$ in units of DFT bins, and a typical value of $\backslash alpha$ is 3.

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File:Window function and frequency response - Dolph-Chebyshev (alpha = 5).svg

Minimizes the Chebyshev norm of the side-lobes for a given main lobe width.

The zero-phase Dolph–Chebyshev window function $w\_0[n]$ is usually defined in terms of its real-valued discrete Fourier transform, $W\_0[k]$:

:$$

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.

T_n(x) is the n-th Chebyshev polynomial of the first kind evaluated in x, which can be computed using

:$T\_n(x)\; =\backslash begin\{cases\}$

\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}

and

:$\backslash beta\; =\; \backslash cosh\backslash !\backslash big(\backslash tfrac\{1\}\{N\}\; \backslash cosh^\{-1\}(10^\backslash alpha)\backslash big)$

is the unique positive real solution to $T\_N(\backslash beta)\; =\; 10^\backslash alpha$, where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.

The window function can be calculated from W₀(k) by an inverse discrete Fourier transform (DFT):

:$w\_0(n)\; =\; \backslash frac\{1\}\{N+1\}\; \backslash sum\_\{k=0\}^N\; W\_0(k)\; \backslash cdot\; e^\{i\; 2\; \backslash pi\; k\; n\; /\; (N+1)\},\backslash \; -N/2\; \backslash le\; n\; \backslash le\; N/2.$

The lagged version of the window can be obtained by:

:$w[n]\; =\; w\_0\backslash left(n-\backslash frac\{N\}\{2\}\backslash right),\backslash quad\; 0\; \backslash le\; n\; \backslash le\; N,$

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

:$\backslash begin\{align\}$

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}

which is an inverse DFT of $\backslash left(-e^\{\backslash frac\{i\backslash pi\}\{N+1\}\}\backslash right)^k\backslash cdot\; W\_0(k).$

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 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}}

File:Window function and frequency response - Ultraspherical (mu = -0.5).svg

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

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.

The window can be expressed in the time-domain as follows:

:$$

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]

where $C^\{\backslash mu\}\_\{N\}$ is the Ultraspherical polynomial of degree N, and $x\_0$ and $\backslash mu$ control the side-lobe patterns.

Certain specific values of $\backslash mu$ yield other well-known windows: $\backslash mu=0$ and $\backslash mu=1$ give the Dolph–Chebyshev and Saramäki windows respectively. See [http://octave.sourceforge.net/signal/function/ultrwin.html here] for illustration of Ultraspherical windows with varied parametrization.

{{clear}}

File:Window function and frequency response - Exponential (half window decay).svg

File:Window function and frequency response - Exponential (60dB decay).svg

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 ). It is defined by

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

where τ is the time constant of the function. The exponential function decays as e ≃ 2.71828 or approximately 8.69 dB per time constant.

This means that for a targeted decay of D dB over half of the window length, the time constant τ is given by

:$\backslash tau\; =\; \backslash frac\{N\}\{2\}\backslash frac\{8.69\}\{D\}.$

{{clear}}

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

File:Window function and frequency response - Bartlett-Hann.svg

:$w[n]=a\_0\; -\; a\_1\; \backslash left\; |\backslash frac\{n\}\{N\}-\backslash frac\{1\}\{2\}\; \backslash right|\; -\; a\_2\; \backslash cos\; \backslash left\; (\backslash frac\{2\; \backslash pi\; n\}\{N\}\backslash right\; )$

:$a\_0=0.62;\backslash quad\; a\_1=0.48;\backslash quad\; a\_2=0.38\backslash ,$

{{clear}}

File:Window function and frequency response - Planck-Bessel (epsilon = 0.1, alpha = 4.45).svg

A {{slink|#Planck-taper window}} multiplied by a Kaiser window which is defined in terms of a 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. 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}}

File:Window function and frequency response - Hann-Poisson (alpha = 2).svg

A Hann window multiplied by a Poisson window. For $\backslash alpha\; \backslash geqslant\; 2$ 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 in hill climbing algorithms like Newton's method. The Hann–Poisson window is defined by:

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

where α is a parameter that controls the slope of the exponential.

{{Clear}}

File:Window function and frequency response - GAP optimized Nuttall.svg

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

:$w\_0[n]\; =\; a\_\{0\}\; +\; \backslash sum\_\{k=1\}^\{K\}\; a\_\{2k\}\backslash left(\backslash frac\{n\}\{\backslash sigma\}\backslash right)^\{2k\},\; \backslash quad\; -\backslash frac\{N\}\{2\}\; \backslash le\; n\; \backslash le\; \backslash frac\{N\}\{2\},$

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

Additionally, starting with a set of expansion coefficients $a\_\{2k\}$ 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. 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

$$w[n]\; =\; \backslash operatorname\{sinc\}\backslash left(\backslash frac\{2n\}\{N\}\; -\; 1\backslash right)$$

• used in Lanczos resampling
• for the Lanczos window, $\backslash operatorname\{sinc\}(x)$ is defined as $\backslash sin(\backslash pi\; x)/\backslash pi\; x$
• also known as a sinc window, because: $$w\_0(n)\; =\; \backslash operatorname\{sinc\}\backslash left(\backslash frac\{2n\}\{N\}\backslash right)\backslash ,$$ is the main lobe of a normalized sinc function

{{clear}}

The $w\_0(x)$ form, according to the convention above, is symmetric around $x\; =\; 0$. However, there are window functions that are asymmetric, such as the Gamma distribution used in FIR implementations of Gammatone filters. 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

:$$

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 $\backslash alpha\; >\; 1$, and conversely weights more highly the latest samples when .{{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 }}

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

{{notelist-ua}}

{{notelist-la}}

{{reflist|1|refs=

{{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

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|title=Discrete-time signal processing

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|location=Upper Saddle River, N.J.

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|chapter=7.2

|pages=[https://archive.org/details/discretetimesign00alan/page/465 465]–478

|url-access=registration

|url=https://archive.org/details/discretetimesign00alan

}}

{{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

}}

{{cite book |title=Microsound |first=Curtis |last=Roads |publisher=MIT Press |year=2002 |isbn=978-0-262-18215-7

}}

{{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

}}

{{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.

}}

{{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

}}

{{cite web |url=http://www.cg.tuwien.ac.at/research/vis/vismed/Windows/MasteringWindows.pdf

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}}

{{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

}}

{{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

}}

{{cite book

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{{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

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{{cite journal

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}} The fundamental 1978 paper on FFT windows by Harris, which specified many windows and introduced key metrics used to compare them.

{{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

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{{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

}}

{{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

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{{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

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{{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

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{{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

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{{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

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{{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

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{{cite book

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{{cite journal

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| issue =1

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| 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.

{{Cite book

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{{cite web

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|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.

}}

{{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

}}

{{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

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{{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

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{{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

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{{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

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{{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.

}}

{{cite journal |last=Kaiser |first=James F. |date=Nov 1964 |title=A family of window functions having nearly ideal properties |journal=Unpublished Memorandum

}}

{{cite journal |last1=Kaiser |first1=James F. |last2=Schafer |first2=Ronald W. |doi=10.1109/TASSP.1980.1163349 |title=On the use of the I₀-sinh window for spectrum analysis |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=28 |pages=105–107 |year=1980

}}

{{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

}}

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}}

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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.

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{{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

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{{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

}}

{{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

}}

{{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

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

}}

{{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

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{{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

}}

{{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

}}

{{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

}}

{{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

}}

{{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

}}

{{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

}}

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

}}

{{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

}}

{{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

}}

{{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

}}

{{cite web

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{{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

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{{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

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}}

• {{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