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Window function

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</style><div role="note" class="hatnote navigation-not-searchable">For the term used in SQL statements, see <a href="Window_function_(SQL)" title="Window function (SQL)">Window function (SQL)</a>.</div>

In <a href="Signal_processing" title="Signal processing">signal processing</a> and <a href="Statistics" title="Statistics">statistics</a>, a window function (also known as an apodization function or tapering function<a href="#cite_note-Weisstein-1">[1]</a>) is a <a href="Function_(mathematics)" title="Function (mathematics)">mathematical function</a> that is zero-valued outside of some chosen <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a>. 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 <a href="Spectral_leakage" title="Spectral leakage">spectral leakage</a>. 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.<a href="#cite_note-Roads-2">[2]</a> 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 <a href="Square_integrable" class="mw-redirect" title="Square integrable">square integrable</a>, and, more specifically, that the function goes sufficiently rapidly toward zero.<a href="#cite_note-Cattani-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Window functions are used in spectral <a href="Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">analysis</a>/modification/<a href="Overlap%E2%80%93add_method#resynthesis" title="Overlap–add method">resynthesis</a>,<a href="#cite_note-OLA-4">[4]</a> the design of <a href="Finite_impulse_response" title="Finite impulse response">finite impulse response</a> filters, merging multiscale and multidimensional datasets,<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> as well as <a href="Beamforming" title="Beamforming">beamforming</a> and <a href="Antenna_(radio)" title="Antenna (radio)">antenna</a> design.

<div class="mw-heading mw-heading3"><h3 id="Spectral_analysis">Spectral analysis</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spectral_leakage" title="Spectral leakage">Spectral leakage</a></div>

The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the function 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.

<div class="mw-heading mw-heading3"><h3 id="Filter_design">Filter design</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Filter_design" title="Filter design">Filter design</a></div>

Windows are sometimes used in the design of <a href="Digital_filters" class="mw-redirect" title="Digital filters">digital filters</a>, in particular to convert an "ideal" impulse response of infinite duration, such as a <a href="Sinc_function" title="Sinc function">sinc function</a>, to a <a href="Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR) filter design. That is called the <a href="Finite_impulse_response#Window_design_method" title="Finite impulse response">window method</a>.<a href="#cite_note-Oppenheim-7">[7]</a><a href="#cite_note-FIRfilters-8">[8]</a><a href="#cite_note-Tuwien-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Statistics_and_curve_fitting">Statistics and curve fitting</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kernel_(statistics)" title="Kernel (statistics)">kernel (statistics)</a></div>

Window functions are sometimes used in the field of <a href="Statistics" title="Statistics">statistical analysis</a> to restrict the set of data being analyzed to a range near a given point, with a <a href="Weighting" class="mw-redirect" title="Weighting"> weighting factor</a> that diminishes the effect of points farther away from the portion of the curve being fit. In the field of Bayesian analysis and <a href="Curve_fitting" title="Curve fitting">curve fitting</a>, this is often referred to as the <a href="Kernel_(statistics)" title="Kernel (statistics)">kernel</a>.

<div class="mw-heading mw-heading3"><h3 id="Rectangular_window_applications">Rectangular window applications</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Analysis_of_transients">Analysis of transients</h4></div>

When analyzing a transient signal in <a href="Modal_analysis" title="Modal analysis">modal analysis</a>, 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.<a href="#cite_note-HPmemory-10">[10]</a>

<div class="mw-heading mw-heading4"><h4 id="Harmonic_analysis">Harmonic analysis</h4></div>

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 <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (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.

<div class="mw-heading mw-heading2"><h2 id="Overlapping_windows">Overlapping windows</h2></div>

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 <a href="Welch_method" class="mw-redirect" title="Welch method">Welch method</a> of power spectral analysis and the <a href="Modified_discrete_cosine_transform" title="Modified discrete cosine transform">modified discrete cosine transform</a>.

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional_windows">Two-dimensional windows</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Two-dimensional_window_design" title="Two-dimensional window design">Two-dimensional window design</a></div>

Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform.<a href="#cite_note-Hovden-11">[11]</a> They can be constructed from one-dimensional windows in either of two forms.<a href="#cite_note-Bernstein-12">[12]</a> The separable form, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(m,n)=w(m)w(n)}">

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<annotation encoding="application/x-tex">{\displaystyle W(m,n)=w(m)w(n)}</annotation>

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</math><img src="./e973fc304055bf440d9bec5531a928254ab4c263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.194ex; height:2.843ex;" alt="{\displaystyle W(m,n)=w(m)w(n)}" loading="lazy"> is trivial to compute. The <a href="Radial_function" title="Radial function">radial</a> form, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(m,n)=w(r)}">

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<annotation encoding="application/x-tex">{\displaystyle W(m,n)=w(r)}</annotation>

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</math><img src="./953fcedff1b431c30c56fa1c58b385548c6c8402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.334ex; height:2.843ex;" alt="{\displaystyle W(m,n)=w(r)}" loading="lazy">, which involves the radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {(m-M/2)^{2}+(n-N/2)^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle r={\sqrt {(m-M/2)^{2}+(n-N/2)^{2}}}}</annotation>

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</math><img src="./f0bdc7c58ee00655f261bdaa21c589ae5c1876e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:33.31ex; height:4.843ex;" alt="{\displaystyle r={\sqrt {(m-M/2)^{2}+(n-N/2)^{2}}}}" loading="lazy">, is <a href="Isotropy" title="Isotropy">isotropic</a>, independent on the orientation of the coordinate axes. Only the <a href="#Gaussian_window">Gaussian</a> function is both separable and isotropic.<a href="#cite_note-Awad-13">[13]</a> The separable forms of all other window functions have corners that depend on the choice of the coordinate axes. The isotropy/<a href="Anisotropy" title="Anisotropy">anisotropy</a> 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 <a href="Diffraction" title="Diffraction">diffraction</a> from rectangular vs. circular apertures, which can be visualized in terms of the product of two <a href="Sinc_function" title="Sinc function">sinc functions</a> vs. an <a href="Airy_function" title="Airy function">Airy function</a>, respectively.

<div class="mw-heading mw-heading2"><h2 id="Examples_of_window_functions">Examples of window functions</h2></div>

Conventions:

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(x)}</annotation>

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</math><img src="./6e2e240f4a8bf7796a983ab6c130d39e64c6d73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.857ex; height:2.843ex;" alt="{\displaystyle w_{0}(x)}" loading="lazy"> is a zero-phase function (symmetrical about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">),<a href="#cite_note-Zphase-14">[14]</a> continuous for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [-N/2,N/2],}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [-N/2,N/2],}</annotation>

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</math><img src="./0377d29e5715ebdd13819c1cc07c4638a0156afd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.73ex; height:2.843ex;" alt="{\displaystyle x\in [-N/2,N/2],}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is a positive integer (even or odd).<a href="#cite_note-Rorabaugh-15">[15]</a></li>

<li>The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{w[n]=w_{0}(n-N/2),\quad 0\leq n\leq N\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{w[n]=w_{0}(n-N/2),\quad 0\leq n\leq N\}}</annotation>

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</math><img src="./5f28eb10c2426fe0deb507ce933eb6b5df2c020e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.101ex; height:2.843ex;" alt="{\displaystyle \{w[n]=w_{0}(n-N/2),\quad 0\leq n\leq N\}}" loading="lazy"> is symmetric, of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1.}">

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<annotation encoding="application/x-tex">{\displaystyle N+1.}</annotation>

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</math><img src="./b17ba5b28fc1d2e352a346d0f1ed4807e68409f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.713ex; height:2.343ex;" alt="{\displaystyle N+1.}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \{w[n],\quad 0\leq n\leq N-1\}}</annotation>

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</math><img src="./f506d61ded38a4f9d8ecd59004fbf34e793d81f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.855ex; height:2.843ex;" alt="{\displaystyle \{w[n],\quad 0\leq n\leq N-1\}}" loading="lazy"> is DFT-symmetric, of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N.}">

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<annotation encoding="application/x-tex">{\displaystyle N.}</annotation>

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</math><img src="./356b8b60a047de347b447f2bdafaaccf47502031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle N.}" loading="lazy"><a href="#cite_note-18">[A]</a></li></ul>

<ul><li>The parameter B displayed on each spectral plot is the function's <a href="Spectral_leakage#Noise_bandwidth" title="Spectral leakage">noise equivalent bandwidth</a> metric, in units of DFT bins.<a href="#cite_note-Harris-16">[16]</a>: p.56 eq.(16)

<ul><li>See spectral leakage §§ <a href="Spectral_leakage#Discrete-time_signals" title="Spectral leakage">Discrete-time signals</a> and <a href="Spectral_leakage#Some_window_metrics" title="Spectral leakage">Some window metrics</a> and <a href="Normalized_frequency_(signal_processing)" title="Normalized frequency (signal processing)">Normalized frequency</a> for understanding the use of "bins" for the x-axis in these plots.</li></ul></li></ul>

The sparse sampling of a <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (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 href="#cite_note-19">[a]</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.

<div class="mw-heading mw-heading3"><h3 id="Rectangular_window">Rectangular window</h3></div>

The rectangular window (sometimes known as the <a href="Boxcar_function" title="Boxcar function">boxcar</a> or uniform or <a href="Dirichlet_kernel" title="Dirichlet kernel">Dirichlet</a> window or misleadingly as "no window" in some programs<a href="#cite_note-20">[18]</a>) 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:

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<annotation encoding="application/x-tex">{\displaystyle w[n]=1.}</annotation>

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</math><img src="./f071fdd6a0a02d6d9dc3c1325d88a6dda6a92781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.26ex; height:2.843ex;" alt="{\displaystyle w[n]=1.}" loading="lazy"></dd></dl>

Other windows are designed to moderate these sudden changes, to reduce scalloping loss and improve dynamic range (described in <a href="#Spectral_analysis">§ Spectral analysis</a>).

The rectangular window is the 1st-order B-spline window as well as the 0th-power <a href="#Power-of-sine/cosine_windows">power-of-sine window</a>.

The rectangular window provides the minimum mean square error estimate of the <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a>, at the cost of other issues discussed.

<div class="mw-heading mw-heading3"><h3 id="B-spline_windows">B-spline windows</h3></div>

B-spline windows can be obtained as k-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the <a href="#Triangular_window">§ Triangular window</a> (k = 2) and the <a href="#Parzen_window">§ Parzen window</a> (k = 4).<a href="#cite_note-Toraichi89-21">[19]</a> Alternative definitions sample the appropriate normalized <a href="B-spline" title="B-spline">B-spline</a> <a href="Basis_function" title="Basis function">basis functions</a> 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 <a href="Rectangular_function" title="Rectangular function">rectangular function</a>.

<div class="mw-heading mw-heading4"><h4 id="Triangular_window">Triangular window</h4></div>

Triangular windows are given by

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<annotation encoding="application/x-tex">{\displaystyle w[n]=1-\left|{\frac {n-{\frac {N}{2}}}{\frac {L}{2}}}\right|,\quad 0\leq n\leq N,}</annotation>

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</math><img src="./2da35f9130dc00bdd045a305c261da1c16faecb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.322ex; height:8.176ex;" alt="{\displaystyle w[n]=1-\left|{\frac {n-{\frac {N}{2}}}{\frac {L}{2}}}\right|,\quad 0\leq n\leq N,}" loading="lazy"></dd></dl>

where L can be N,<a href="#cite_note-Bartlett-22">[20]</a> N + 1,<a href="#cite_note-Harris-16">[16]</a><a href="#cite_note-Tukey-23">[21]</a><a href="#cite_note-MWtriang-24">[22]</a> or N + 2.<a href="#cite_note-Welch1967-25">[23]</a> The first one is also known as <a href="M._S._Bartlett" title="M. S. Bartlett">Bartlett</a> window or <a href="Lip%C3%B3t_Fej%C3%A9r" title="Lipót Fejér">Fejér</a> 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 <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>N⁄2-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.

<div class="mw-heading mw-heading4"><h4 id="Parzen_window">Parzen window</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Kernel_density_estimation" title="Kernel density estimation">Kernel density estimation</a>.</div>

Defining L ≜ N + 1, the Parzen window, also known as the de la Vallée Poussin window,<a href="#cite_note-Harris-16">[16]</a> is the 4th-order B-spline window given by

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<annotation encoding="application/x-tex">{\displaystyle 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\leq |n|\leq {\frac {L}{4}}\\2\left(1-{\frac {|n|}{L/2}}\right)^{3}&{\frac {L}{4}}<|n|\leq {\frac {L}{2}}\\\end{array}}\right\}}</annotation>

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</math><img src="./bc9ee81dfd7d111f1881e1a2783ac20db15c4cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:53.416ex; height:10.843ex;" alt="{\displaystyle 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\leq |n|\leq {\frac {L}{4}}\\2\left(1-{\frac {|n|}{L/2}}\right)^{3}&{\frac {L}{4}}<|n|\leq {\frac {L}{2}}\\\end{array}}\right\}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle w[n]=\ w_{0}\left(n-{\tfrac {N}{2}}\right),\ 0\leq n\leq N}</annotation>

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</math><img src="./cfa5797bd28cc542de9f224e4b9c7c0c37bff91a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.262ex; height:4.843ex;" alt="{\displaystyle w[n]=\ w_{0}\left(n-{\tfrac {N}{2}}\right),\ 0\leq n\leq N}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Other_polynomial_windows">Other polynomial windows</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Welch_window">Welch window</h4></div>

The Welch window consists of a single <a href="Parabola" title="Parabola">parabolic</a> section:

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<annotation encoding="application/x-tex">{\displaystyle w[n]=1-\left({\frac {n-{\frac {N}{2}}}{\frac {N}{2}}}\right)^{2},\quad 0\leq n\leq N.}</annotation>

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</math><img src="./fa3a944c6a5c2051217a53dd60a9693dd76c98a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.377ex; height:8.509ex;" alt="{\displaystyle w[n]=1-\left({\frac {n-{\frac {N}{2}}}{\frac {N}{2}}}\right)^{2},\quad 0\leq n\leq N.}" loading="lazy"><a href="#cite_note-Welch1967-25">[23]</a></dd></dl>

Alternatively, it can be written as two factors, as in a <a href="Beta_distribution" title="Beta distribution">beta distribution</a>:

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<annotation encoding="application/x-tex">{\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),\quad 0\leq n\leq N.}</annotation>

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</math><img src="./cc47bb1747abbd9050569f9ea82f78acdb5fa494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:53.148ex; height:8.009ex;" alt="{\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),\quad 0\leq n\leq N.}" loading="lazy"></dd></dl>

The defining <a href="Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a> reaches a value of zero at the samples just outside the span of the window.

The Welch window is fairly close to the <a href="#Sine_window">sine window</a>, and just as the <a href="#Power-of-sine/cosine_windows">power-of-sine windows</a> are a useful parameterized family, the power-of-Welch window family is similarly useful. Powers of the Welch or parabolic window are also <a href="Pearson_type_II_distribution" class="mw-redirect" title="Pearson type II distribution">Pearson type II distributions</a> and symmetric <a href="Beta_distribution" title="Beta distribution">beta distributions</a>, 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 <a href="#Asymmetric_window_functions">asymmetric window functions</a>.

<div class="mw-heading mw-heading3"><h3 id="Raised-cosine_windows">Raised-cosine windows</h3></div>

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 <a href="Raised_cosine_distribution" title="Raised cosine distribution">raised cosine distribution</a>, which goes smoothly to zero at its ends.

The raised-cosine windows have the form:

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<annotation encoding="application/x-tex">{\displaystyle w[n]=a_{0}-(1-a_{0})\cdot \cos \left({\tfrac {2\pi n}{N}}\right),\quad 0\leq n\leq N,}</annotation>

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</math><img src="./5eba7ee5f69aa467ea1721de8cfbade35da8996b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.645ex; height:4.843ex;" alt="{\displaystyle w[n]=a_{0}-(1-a_{0})\cdot \cos \left({\tfrac {2\pi n}{N}}\right),\quad 0\leq n\leq N,}" loading="lazy"></dd></dl>

or alternatively as their zero-phase versions:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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}}\leq n\leq {\tfrac {N}{2}}.\end{aligned}}}</annotation>

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</math><img src="./f3c63c956c651adceb75338138a2e382f19b1805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:52.719ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}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}}\leq n\leq {\tfrac {N}{2}}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Hann_window">Hann window</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Hann_function" title="Hann function">Hann function</a></div>

Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=0.5}">

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<annotation encoding="application/x-tex">{\displaystyle w[n]=0.5\;\left[1-\cos \left({\frac {2\pi n}{N}}\right)\right]=\sin ^{2}\left({\frac {\pi n}{N}}\right),}</annotation>

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</math><img src="./cd8d3e844a1443d7bedd57240588cf19fcd28af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.164ex; height:6.176ex;" alt="{\displaystyle w[n]=0.5\;\left[1-\cos \left({\frac {2\pi n}{N}}\right)\right]=\sin ^{2}\left({\frac {\pi n}{N}}\right),}" loading="lazy"><a href="#cite_note-MWhann-26">[24]</a></dd></dl>

named after <a href="Julius_von_Hann" title="Julius von Hann">Julius von Hann</a>, and sometimes referred to as Hanning, which derived from the verb "to Hann". It is also known as the raised cosine, because of its similarity to a <a href="Raised-cosine_distribution" class="mw-redirect" title="Raised-cosine distribution">raised-cosine distribution</a>.

This function is a member of both the <a href="#Cosine-sum_windows">cosine-sum</a> and <a href="#Power-of-sine/cosine_windows">power-of-sine</a> families. Unlike the <a href="#Hann_and_Hamming_windows">Hamming window</a>, the end points of the Hann window just touch zero. The resulting <a href="Spectral_leakage" title="Spectral leakage">side-lobes</a> roll off at about 18 dB per octave.<a href="#cite_note-JOShann-27">[25]</a>

<div class="mw-heading mw-heading4"><h4 id="Hamming_window">Hamming window</h4></div>

Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}">

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</math><img src="./693ad9f934775838bd72406b41ada4a59785d7ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" loading="lazy"> to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by <a href="Richard_W._Hamming" class="mw-redirect" title="Richard W. Hamming">Richard W. Hamming</a>. 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.<a href="#cite_note-Harris-16">[16]</a><a href="#cite_note-Enochson-28">[26]</a><a href="#cite_note-JOSHamming-29">[27]</a>

The Hamming window is often called the Hamming blip when used for <a href="Pulse_shaping" title="Pulse shaping">pulse shaping</a>.<a href="#cite_note-sunar-30">[28]</a><a href="#cite_note-sunar2-31">[29]</a><a href="#cite_note-SRD-32">[30]</a>

Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,<a href="#cite_note-Harris-16">[16]</a> to a nearly equiripple condition.<a href="#cite_note-JOSHamming-29">[27]</a> In the equiripple sense, the optimal values for the coefficients are a0 = 0.53836 and a1 = 0.46164.<a href="#cite_note-JOSHamming-29">[27]</a><a href="#cite_note-Nuttall-33">[31]</a>

<div class="mw-heading mw-heading3"><h3 id="Cosine-sum_windows">Cosine-sum windows</h3></div>

This family, which generalizes the <a href="#Raised-cosine_windows">raised-cosine windows</a>, is also known as generalized cosine windows.<a href="#cite_note-34">[32]</a>

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w[n]=\sum _{k=0}^{K}(-1)^{k}a_{k}\;\cos \left({\frac {2\pi kn}{N}}\right),\quad 0\leq n\leq N.}">

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</div>

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

<div class="mw-heading mw-heading4"><h4 id="Blackman_window">Blackman window</h4></div>

Blackman windows are defined as

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<annotation encoding="application/x-tex">{\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),}</annotation>

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</math><img src="./ecaab90e2a8c2c1f6b47b17ce675ea00cfcbfe73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.92ex; height:6.176ex;" alt="{\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),}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1-\alpha }{2}};\quad a_{1}={\frac {1}{2}};\quad a_{2}={\frac {\alpha }{2}}.}</annotation>

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</math><img src="./16d9ad0b98e8757a9bbf42478b11390e2e3be513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.156ex; height:5.176ex;" alt="{\displaystyle a_{0}={\frac {1-\alpha }{2}};\quad a_{1}={\frac {1}{2}};\quad a_{2}={\frac {\alpha }{2}}.}" loading="lazy"></dd></dl>

By common convention, the unqualified term Blackman window refers to Blackman's "not very serious proposal" of α = 0.16 (a0 = 0.42, a1 = 0.5, a2 = 0.08), which closely approximates the exact Blackman,<a href="#cite_note-MWBlackman-35">[33]</a> with a0 = 7938/18608 ≈ 0.42659, a1 = 9240/18608 ≈ 0.49656, and a2 = 1430/18608 ≈ 0.076849.<a href="#cite_note-lvanl-36">[34]</a> These exact values place zeros at the third and fourth sidelobes,<a href="#cite_note-Harris-16">[16]</a> 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.<a href="#cite_note-Harris-16">[16]</a><a href="#cite_note-Blackman1959-37">[35]</a>

<div class="mw-heading mw-heading4"><h4 id="Nuttall_window,_continuous_first_derivative">Nuttall window, continuous first derivative</h4></div>

The continuous form of the Nuttall window, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x),}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=0.355768;\quad a_{1}=0.487396;\quad a_{2}=0.144232;\quad a_{3}=0.012604.}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Blackman–Nuttall_window">Blackman–Nuttall window</h4></div>

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=0.3635819;\quad a_{1}=0.4891775;\quad a_{2}=0.1365995;\quad a_{3}=0.0106411.}</annotation>

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</math><img src="./3d65d7b5704532d7b3c8da4fee1f895425fb03fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:72.032ex; height:2.509ex;" alt="{\displaystyle a_{0}=0.3635819;\quad a_{1}=0.4891775;\quad a_{2}=0.1365995;\quad a_{3}=0.0106411.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Blackman–Harris_window">Blackman–Harris window</h4></div>

A generalization of the Hamming family, produced by adding more shifted cosine functions, meant to minimize side-lobe levels<a href="#cite_note-JOSBlack-38">[36]</a><a href="#cite_note-JOSBlack3-39">[37]</a>

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<annotation encoding="application/x-tex">{\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)}</annotation>

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</math><img src="./55aa618bd5010d83e5df4ae06fbb3abd509f12fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.042ex; height:6.176ex;" alt="{\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)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=0.35875;\quad a_{1}=0.48829;\quad a_{2}=0.14128;\quad a_{3}=0.01168.}</annotation>

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</math><img src="./cda1939e3e4e4cf95e0a7d500dd0be12f09d0fc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:62.733ex; height:2.509ex;" alt="{\displaystyle a_{0}=0.35875;\quad a_{1}=0.48829;\quad a_{2}=0.14128;\quad a_{3}=0.01168.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Flat_top_window">Flat top window</h4></div>

A flat top window is a partially negative-valued window that has minimal <a href="#Discrete-time_signals">scalloping loss</a> in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components.<a href="#cite_note-Heinzel2002-17">[17]</a><a href="#cite_note-SWsmith-40">[38]</a> However, its broad bandwidth results in high <a href="Noise_bandwidth" class="mw-redirect" title="Noise bandwidth">noise bandwidth</a> 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,<a href="#cite_note-SWsmith-40">[38]</a> or they may be of the usual <a href="#Cosine-sum_windows">cosine-sum</a> variety:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

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</math><img src="./2e117d087bf8d6345fb6ea8ffc7b5aa7f6faae4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:44.672ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></dd></dl>

The <a rel="nofollow" class="external text" href="https://www.mathworks.com/help/signal/ref/flattopwin.html">Matlab variant</a> has these coefficients:

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=0.21557895;\quad a_{1}=0.41663158;\quad a_{2}=0.277263158;\quad a_{3}=0.083578947;\quad a_{4}=0.006947368.}</annotation>

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</math><img src="./2c623ba11533abb19bb67a3b7d192c6fb37e6d9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; max-width: 600; width:100.018ex; height:2.509ex;" alt="{\displaystyle a_{0}=0.21557895;\quad a_{1}=0.41663158;\quad a_{2}=0.277263158;\quad a_{3}=0.083578947;\quad a_{4}=0.006947368.}" loading="lazy"></dd></dl>

Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.<a href="#cite_note-Heinzel2002-17">[17]</a>

<div class="mw-heading mw-heading4"><h4 id="Rife–Vincent_windows">Rife–Vincent windows</h4></div>

Rife–Vincent windows<a href="#cite_note-Rife-41">[39]</a> are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied to <a href="#math_Eq.1">Eq.1</a>, reflect that custom.

Class I, Order 1 (K = 1): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=1;\quad a_{1}=1}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=1;\quad a_{1}=1}</annotation>

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</math><img src="./ae26a27f3a5175b7e7506a7922b1d29b3129eb9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.446ex; height:2.509ex;" alt="{\displaystyle a_{0}=1;\quad a_{1}=1}" loading="lazy"> Functionally equivalent to the <a href="#Hann_and_Hamming_windows">Hann window</a> and power of sine (α = 2).

Class I, Order 2 (K = 2): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=1;\quad a_{1}={\tfrac {4}{3}};\quad a_{2}={\tfrac {1}{3}}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=1;\quad a_{1}={\tfrac {4}{3}};\quad a_{2}={\tfrac {1}{3}}}</annotation>

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</math><img src="./9af5db1bb38c7dc4b8fe7ae5fd3960f97503cbbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.339ex; height:3.676ex;" alt="{\displaystyle a_{0}=1;\quad a_{1}={\tfrac {4}{3}};\quad a_{2}={\tfrac {1}{3}}}" loading="lazy"> 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.<a href="#cite_note-Andria-42">[40]</a>

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 <a href="#Blackman_window">§ Blackman window</a>.<a href="#cite_note-Andria-42">[40]</a><a href="#cite_note-Schoukens-43">[41]</a>

<div class="mw-heading mw-heading3"><h3 id="Sine_window">Sine window</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle w[n]=\sin \left({\frac {\pi n}{N}}\right)=\cos \left({\frac {\pi n}{N}}-{\frac {\pi }{2}}\right),\quad 0\leq n\leq N.}</annotation>

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</math><img src="./bc2af9f7febc105f1184bd114a1484bf16cf7325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.022ex; height:4.843ex;" alt="{\displaystyle w[n]=\sin \left({\frac {\pi n}{N}}\right)=\cos \left({\frac {\pi n}{N}}-{\frac {\pi }{2}}\right),\quad 0\leq n\leq N.}" loading="lazy"></dd></dl>

The corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(n)\,}">

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(n)\,}</annotation>

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</math><img src="./50dec0b51baef5071ead565d832977a5b9bb1ef0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.309ex; height:2.843ex;" alt="{\displaystyle w_{0}(n)\,}" loading="lazy"> function is a cosine without the π/2 phase offset. So the sine window<a href="#cite_note-Bosi-44">[42]</a> is sometimes also called cosine window.<a href="#cite_note-Harris-16">[16]</a> As it represents half a cycle of a sinusoidal function, it is also known variably as half-sine window<a href="#cite_note-Kido-45">[43]</a> or half-cosine window.<a href="#cite_note-Landisman-46">[44]</a>

The <a href="Autocorrelation" title="Autocorrelation">autocorrelation</a> of a sine window produces a function known as the Bohman window.<a href="#cite_note-MWbohman-47">[45]</a>

<div class="mw-heading mw-heading4"><h4 id="Power-of-sine/cosine_windows">Power-of-sine/cosine windows</h4></div>

These window functions have the form:<a href="#cite_note-PowCos-48">[46]</a>

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<annotation encoding="application/x-tex">{\displaystyle w[n]=\sin ^{\alpha }\left({\frac {\pi n}{N}}\right)=\cos ^{\alpha }\left({\frac {\pi n}{N}}-{\frac {\pi }{2}}\right),\quad 0\leq n\leq N.}</annotation>

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</math><img src="./8b81af4c85e4b5e6909b3dfb7fef170a42a02f7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.591ex; height:4.843ex;" alt="{\displaystyle w[n]=\sin ^{\alpha }\left({\frac {\pi n}{N}}\right)=\cos ^{\alpha }\left({\frac {\pi n}{N}}-{\frac {\pi }{2}}\right),\quad 0\leq n\leq N.}" loading="lazy"></dd></dl>

The <a href="#Rectangular_window">rectangular window</a> (α = 0), the <a href="#Sine_window">sine window</a> (α = 1), and the <a href="#Hann_and_Hamming_windows">Hann window</a> (α = 2) are members of this family.

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

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<annotation encoding="application/x-tex">{\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)-...}</annotation>

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</math><img src="./3449e818cb0c7c261fe536655c3b5145fd4eda9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.334ex; height:6.176ex;" alt="{\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)-...}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\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\times 10^{-3}\\\hline \end{array}}}</annotation>

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</math><img src="./0f070a84830409458ffb459a636cc21f99eb1ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:62.051ex; height:20.509ex;" alt="{\displaystyle {\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\times 10^{-3}\\\hline \end{array}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Adjustable_windows">Adjustable windows</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Gaussian_window">Gaussian window</h4></div>

The Fourier transform of a <a href="Gaussian_function" title="Gaussian function">Gaussian</a> 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.<a href="#cite_note-JOSGauss-49">[47]</a>

Since the log of a Gaussian produces a <a href="Parabola" title="Parabola">parabola</a>, this can be used for nearly exact quadratic interpolation in <a href="Frequency_estimation" class="mw-redirect" title="Frequency estimation">frequency estimation</a>.<a href="#cite_note-JOSGauss2-50">[48]</a><a href="#cite_note-JOSGauss-49">[47]</a><a href="#cite_note-interpolation-51">[49]</a>

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<annotation encoding="application/x-tex">{\displaystyle w[n]=\exp \left(-{\frac {1}{2}}\left({\frac {n-N/2}{\sigma N/2}}\right)^{2}\right),\quad 0\leq n\leq N.}</annotation>

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</math><img src="./e201c829f6aabf4339fc0a1929367d5613dfe290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.248ex; height:7.509ex;" alt="{\displaystyle w[n]=\exp \left(-{\frac {1}{2}}\left({\frac {n-N/2}{\sigma N/2}}\right)^{2}\right),\quad 0\leq n\leq N.}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \sigma \leq \;0.5\,}</annotation>

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</math><img src="./3e43f6fb554bba3e615f0c3cdbb425c8f0149c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.432ex; height:2.343ex;" alt="{\displaystyle \sigma \leq \;0.5\,}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading4"><h4 id="Confined_Gaussian_window">Confined Gaussian window</h4></div>

The confined Gaussian window yields the smallest possible root mean square frequency width σω for a given temporal width (N + 1) σt.<a href="#cite_note-Starosielec2014-52">[50]</a> 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 <a href="#Sine_window">§ Sine window</a> and the <a href="#Gaussian_window">§ Gaussian window</a> in the limiting cases of large and small σt, respectively.

<div class="mw-heading mw-heading4"><h4 id="Approximate_confined_Gaussian_window">Approximate confined Gaussian window</h4></div>

Defining L ≜ N + 1, a <a href="#Confined_Gaussian_window">confined Gaussian window</a> of temporal width L × σt is well approximated by:<a href="#cite_note-Starosielec2014-52">[50]</a>

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<annotation encoding="application/x-tex">{\displaystyle 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)}}}</annotation>

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</math><img src="./3094fa9469d1e65bfdb6a4e31cdafd037a8822ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.303ex; height:7.843ex;" alt="{\displaystyle 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)}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a Gaussian function:

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<annotation encoding="application/x-tex">{\displaystyle G(x)=\exp \left(-\left({\cfrac {x-{\frac {N}{2}}}{2L\sigma _{t}}}\right)^{2}\right)}</annotation>

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</math><img src="./7f38de68d5e2a1107a41cd84a3d87e04a891c75d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:29.529ex; height:8.843ex;" alt="{\displaystyle G(x)=\exp \left(-\left({\cfrac {x-{\frac {N}{2}}}{2L\sigma _{t}}}\right)^{2}\right)}" loading="lazy"></dd></dl></dd></dl>

The standard deviation of the approximate window is <a href="Asymptotically_equal" class="mw-redirect" title="Asymptotically equal">asymptotically equal</a> (i.e. large values of N) to L × σt for σt < 0.14.<a href="#cite_note-Starosielec2014-52">[50]</a>

<div class="mw-heading mw-heading4"><h4 id="Generalized_normal_window">Generalized normal window</h4></div>

A more generalized version of the Gaussian window is the generalized normal window.<a href="#cite_note-Chakraborty-53">[51]</a> Retaining the notation from the <a href="Gaussian_window" class="mw-redirect" title="Gaussian window">Gaussian window</a> above, we can represent this window as

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<annotation encoding="application/x-tex">{\displaystyle w[n,p]=\exp \left(-\left({\frac {n-N/2}{\sigma N/2}}\right)^{p}\right)}</annotation>

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for any even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. At <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}">

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</math><img src="./d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy">, this is a Gaussian window and as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> approaches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

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</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">, this approximates to a rectangular window. The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of this window does not exist in a closed form for a general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the <a href="#Tukey_window">§ Tukey window</a>, 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 <a href="#cite_note-Diethorn-54">[52]</a> for a study on <a href="Time-frequency_representation" class="mw-redirect" title="Time-frequency representation">time-frequency representation</a> of this window (or function).

<div class="mw-heading mw-heading4"><h4 id="Tukey_window">Tukey window</h4></div>

The Tukey window, also known as the cosine-tapered window, can be regarded as a cosine lobe of width Nα/2 (spanning Nα/2 + 1 observations) that is convolved with a rectangular window of width N(1 − α/2).

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<annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{lll}w[n]={\frac {1}{2}}\left[1-\cos \left({\frac {2\pi n}{\alpha N}}\right)\right],\quad &0\leq n<{\frac {\alpha N}{2}}\\w[n]=1,\quad &{\frac {\alpha N}{2}}\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}}</annotation>

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</math><img src="./ab34271ac62f34c531c4d6cc0b8ffd2e9793623a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.575ex; margin-bottom: -0.263ex; width:46.897ex; height:12.843ex;" alt="{\displaystyle \left.{\begin{array}{lll}w[n]={\frac {1}{2}}\left[1-\cos \left({\frac {2\pi n}{\alpha N}}\right)\right],\quad &0\leq n<{\frac {\alpha N}{2}}\\w[n]=1,\quad &{\frac {\alpha N}{2}}\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}}" loading="lazy"> <a href="#cite_note-Bloomfield-55">[53]</a><a href="#cite_note-56">[B]</a><a href="#cite_note-57">[C]</a></dd></dl>

At α = 0 it becomes rectangular, and at α = 1 it becomes a Hann window.

<div class="mw-heading mw-heading4"><h4 id="Planck-taper_window">Planck-taper window</h4></div>

The so-called "Planck-taper" window is a <a href="Bump_function" title="Bump function">bump function</a> that has been widely used<a href="#cite_note-Tu-58">[54]</a> in the theory of <a href="Partitions_of_unity" class="mw-redirect" title="Partitions of unity">partitions of unity</a> in <a href="Manifolds" class="mw-redirect" title="Manifolds">manifolds</a>. It is <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> (a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"> 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 <a href="Gravitational-wave_astronomy" title="Gravitational-wave astronomy">gravitational-wave astronomy</a>, inspired by the <a href="Planck's_law" title="Planck's law">Planck distribution</a>.<a href="#cite_note-McKechan-59">[55]</a> It is defined as a <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> function:

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<annotation encoding="application/x-tex">{\displaystyle \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\leq n<\varepsilon N\\w[n]=1,\quad &\varepsilon N\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}}</annotation>

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</math><img src="./f50bb4690593ab87a47ed71fec1945a3f95b9d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:55.262ex; height:16.509ex;" alt="{\displaystyle \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\leq n<\varepsilon N\\w[n]=1,\quad &\varepsilon N\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading4"><h4 id="DPSS_or_Slepian_window">DPSS or Slepian window</h4></div>

The DPSS (discrete prolate spheroidal sequence) or <a href="Slepian_function" title="Slepian function">Slepian function</a>, taper, or window <a href="Spectral_concentration_problem" title="Spectral concentration problem">maximizes the energy concentration in the main lobe</a>,<a href="#cite_note-Slepian-60">[56]</a> and is used in <a href="Multitaper" title="Multitaper">multitaper</a> 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.<a href="#cite_note-KaiserDPSS-61">[57]</a>

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

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<div class="mw-heading mw-heading4"><h4 id="Kaiser_window">Kaiser window</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kaiser_window" title="Kaiser window">Kaiser window</a></div>

The Kaiser, or Kaiser–Bessel, window is a simple approximation of the <a href="#DPSS_or_Slepian_window">DPSS window</a> using <a href="Bessel_function" title="Bessel function">Bessel functions</a>, discovered by <a href="James_Kaiser" title="James Kaiser">James Kaiser</a>.<a href="#cite_note-Kaiser1966-62">[58]</a><a href="#cite_note-Kaiser1964-63">[59]</a>

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<annotation encoding="application/x-tex">{\displaystyle w[n]={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}-1\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad 0\leq n\leq N}</annotation>

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</math><img src="./337ca65bf1465649348a8eb22356662c4188b092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.23ex; height:11.176ex;" alt="{\displaystyle w[n]={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}-1\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad 0\leq n\leq N}" loading="lazy"> <a href="#cite_note-70">[D]</a><a href="#cite_note-Harris-16">[16]</a>: p. 73 </dd>

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(n)={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad -N/2\leq n\leq N/2}</annotation>

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</math><img src="./45ee1fd440b21d733ece89fc46bea52d5e0026d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.156ex; height:11.176ex;" alt="{\displaystyle w_{0}(n)={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad -N/2\leq n\leq N/2}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation>

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</math><img src="./893d08e90ea73781dc133414d661529d0651ca80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}" loading="lazy"> is the 0th-order modified Bessel function of the first kind. Variable parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {1+\alpha ^{2}}},}">

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<annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {1+\alpha ^{2}}},}</annotation>

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</math><img src="./399e26ebc6c758fb6163587947a58214c832463a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.678ex; height:3.509ex;" alt="{\displaystyle 2{\sqrt {1+\alpha ^{2}}},}" loading="lazy"> in units of DFT bins,<a href="#cite_note-Kaiser1980-71">[66]</a> and a typical value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is 3.

<div class="mw-heading mw-heading4"><h4 id="Dolph–Chebyshev_window">Dolph–Chebyshev window</h4></div>

Minimizes the <a href="Uniform_norm" title="Uniform norm">Chebyshev norm</a> of the side-lobes for a given main lobe width.<a href="#cite_note-Dolph-72">[67]</a>

The zero-phase Dolph–Chebyshev window function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}[n]}">

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</math><img src="./e0d68356ad0cc45bf520fbe5be3266aa7c431cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.407ex; height:2.843ex;" alt="{\displaystyle w_{0}[n]}" loading="lazy"> is usually defined in terms of its real-valued <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}[k]}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}[k]}</annotation>

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</math><img src="./af1b9baa4526ac8ec5f39c3cd63782248015037f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.753ex; height:2.843ex;" alt="{\displaystyle W_{0}[k]}" loading="lazy">:<a href="#cite_note-DolphDef-73">[68]</a>

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<annotation encoding="application/x-tex">{\displaystyle 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\leq k\leq N.}</annotation>

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</math><img src="./7a0b570b5db14693c1fb9e16f6e392c22e18ca01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.394ex; height:8.343ex;" alt="{\displaystyle 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\leq k\leq N.}" loading="lazy"></dd></dl>

Tn(x) is the n-th <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomial</a> of the first kind evaluated in x, which can be computed using

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<annotation encoding="application/x-tex">{\displaystyle T_{n}(x)={\begin{cases}\cos \!{\big (}n\cos ^{-1}(x){\big )}&{\text{if }}-1\leq x\leq 1\\\cosh \!{\big (}n\cosh ^{-1}(x){\big )}&{\text{if }}x\geq 1\\(-1)^{n}\cosh \!{\big (}n\cosh ^{-1}(-x){\big )}&{\text{if }}x\leq -1,\end{cases}}}</annotation>

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</math><img src="./d68b2615d261dae9351bb3e5bad374b14d58c842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:54.969ex; height:10.009ex;" alt="{\displaystyle T_{n}(x)={\begin{cases}\cos \!{\big (}n\cos ^{-1}(x){\big )}&{\text{if }}-1\leq x\leq 1\\\cosh \!{\big (}n\cosh ^{-1}(x){\big )}&{\text{if }}x\geq 1\\(-1)^{n}\cosh \!{\big (}n\cosh ^{-1}(-x){\big )}&{\text{if }}x\leq -1,\end{cases}}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \beta =\cosh \!{\big (}{\tfrac {1}{N}}\cosh ^{-1}(10^{\alpha }){\big )}}</annotation>

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</math><img src="./48e2e4b35928506ba50a15f9e8450111e044c479.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.801ex; height:3.509ex;" alt="{\displaystyle \beta =\cosh \!{\big (}{\tfrac {1}{N}}\cosh ^{-1}(10^{\alpha }){\big )}}" loading="lazy"></dd></dl>

is the unique positive real solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{N}(\beta )=10^{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle T_{N}(\beta )=10^{\alpha }}</annotation>

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</math><img src="./d1f4087dc3fd71a4c134683b129c0a9b36d73476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.898ex; height:2.843ex;" alt="{\displaystyle T_{N}(\beta )=10^{\alpha }}" loading="lazy">, where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.<a href="#cite_note-Dolph-72">[67]</a>

The window function can be calculated from W0(k) by an inverse <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT):<a href="#cite_note-Dolph-72">[67]</a>

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</munderover>

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(n)={\frac {1}{N+1}}\sum _{k=0}^{N}W_{0}(k)\cdot e^{i2\pi kn/(N+1)},\ -N/2\leq n\leq N/2.}</annotation>

</semantics>

</math><img src="./0a8a2a14eb821e70f80ae2229ff8390530de4a2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.621ex; height:7.509ex;" alt="{\displaystyle w_{0}(n)={\frac {1}{N+1}}\sum _{k=0}^{N}W_{0}(k)\cdot e^{i2\pi kn/(N+1)},\ -N/2\leq n\leq N/2.}" loading="lazy"></dd></dl>

The lagged version of the window can be obtained by:

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<annotation encoding="application/x-tex">{\displaystyle w[n]=w_{0}\left(n-{\frac {N}{2}}\right),\quad 0\leq n\leq N,}</annotation>

</semantics>

</math><img src="./3af173b60d19716e14882dabdec080868fa3e0ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.321ex; height:6.176ex;" alt="{\displaystyle w[n]=w_{0}\left(n-{\frac {N}{2}}\right),\quad 0\leq n\leq N,}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math><img src="./24f3590133d3da32b13aa211153542e96d14a46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:89.35ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></dd></dl>

which is an inverse DFT of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-e^{\frac {i\pi }{N+1}}\right)^{k}\cdot W_{0}(k).}">

<msup>

<mrow>

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<msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \left(-e^{\frac {i\pi }{N+1}}\right)^{k}\cdot W_{0}(k).}</annotation>

</semantics>

</math><img src="./f64d7bb777afa30cbd6c2da16f1be79421afd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.954ex; height:6.676ex;" alt="{\displaystyle \left(-e^{\frac {i\pi }{N+1}}\right)^{k}\cdot W_{0}(k).}" loading="lazy">

Variations:

<ul><li>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 <a rel="nofollow" class="external text" href="http://www.mathworks.com/help/signal/ref/taylorwin.html">Taylor window</a>.</li>

<li>An alternative to the inverse DFT definition is also available.<a rel="nofollow" class="external autonumber" href="http://practicalcryptography.com/miscellaneous/machine-learning/implementing-dolph-chebyshev-window/">[1]</a>.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Ultraspherical_window">Ultraspherical window</h4></div>

The Ultraspherical window was introduced in 1984 by Roy Streit<a href="#cite_note-Kabal-74">[69]</a> and has application in antenna array design,<a href="#cite_note-Streit-75">[70]</a> non-recursive filter design,<a href="#cite_note-Kabal-74">[69]</a> and spectrum analysis.<a href="#cite_note-Deczky-76">[71]</a>

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.<a href="#cite_note-Deczky-76">[71]</a><a href="#cite_note-Bergen-77">[72]</a><a href="#cite_note-Bergen2-78">[73]</a>

The window can be expressed in the time-domain as follows:<a href="#cite_note-Deczky-76">[71]</a>

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<annotation encoding="application/x-tex">{\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]}</annotation>

</semantics>

</math><img src="./fe5e8dc43d2c5edbe601e3abf542a177689f4276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:63.092ex; height:10.176ex;" alt="{\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]}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{N}^{\mu }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{N}^{\mu }}</annotation>

</semantics>

</math><img src="./57dc081f4410674cee94ed6228ca7579e631f68e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.353ex; height:3.176ex;" alt="{\displaystyle C_{N}^{\mu }}" loading="lazy"> is the <a href="Ultraspherical_polynomial" class="mw-redirect" title="Ultraspherical polynomial">Ultraspherical polynomial</a> of degree N, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> control the side-lobe patterns.<a href="#cite_note-Deczky-76">[71]</a>

Certain specific values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> yield other well-known windows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>

</semantics>

</math><img src="./3753282c0ad2ea1e7d63f39425efd13c37da3169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =1}</annotation>

</semantics>

</math><img src="./820c85551af65c6bedaf1b895fbd99bd9e23ec4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =1}" loading="lazy"> give the Dolph–Chebyshev and <a href="Tapio_Saram%C3%A4ki" title="Tapio Saramäki">Saramäki</a> windows respectively.<a href="#cite_note-Kabal-74">[69]</a> See <a rel="nofollow" class="external text" href="http://octave.sourceforge.net/signal/function/ultrwin.html">here</a> for illustration of Ultraspherical windows with varied parametrization.

<div class="mw-heading mw-heading4"><h4 id="Exponential_or_Poisson_window">Exponential or Poisson window</h4></div>

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 <a href="Exponential_function" title="Exponential function">exponential function</a> 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 <a href="#cite_note-JOSPoisson-79">[74]</a>). It is defined by

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<annotation encoding="application/x-tex">{\displaystyle w[n]=e^{-\left|n-{\frac {N}{2}}\right|{\frac {1}{\tau }}},}</annotation>

</semantics>

</math><img src="./141d1630d759c1ebf5120671e36f86a6117f259c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.419ex; height:4.009ex;" alt="{\displaystyle w[n]=e^{-\left|n-{\frac {N}{2}}\right|{\frac {1}{\tau }}},}" loading="lazy"></dd></dl>

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.<a href="#cite_note-Gade-80">[75]</a>

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

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {N}{2}}{\frac {8.69}{D}}.}</annotation>

</semantics>

</math><img src="./b52d2fb0660c1425b665744694d87d3dc9b77c12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.817ex; height:5.176ex;" alt="{\displaystyle \tau ={\frac {N}{2}}{\frac {8.69}{D}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Hybrid_windows">Hybrid windows</h3></div>

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

<div class="mw-heading mw-heading4"><h4 id="Bartlett–Hann_window">Bartlett–Hann window</h4></div>

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<annotation encoding="application/x-tex">{\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)}</annotation>

</semantics>

</math><img src="./e64d33f0c717a2b40af2ed16c9fff5f58d95b937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.048ex; height:6.176ex;" alt="{\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)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=0.62;\quad a_{1}=0.48;\quad a_{2}=0.38\,}</annotation>

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</math><img src="./8ee6247f1eb891561e1e60413c940bc17a5b70c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.65ex; height:2.509ex;" alt="{\displaystyle a_{0}=0.62;\quad a_{1}=0.48;\quad a_{2}=0.38\,}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Planck–Bessel_window">Planck–Bessel window</h4></div>

A <a href="#Planck-taper_window">§ Planck-taper window</a> multiplied by a <a href="Kaiser_window" title="Kaiser window">Kaiser window</a> which is defined in terms of a <a href="Modified_Bessel_function" class="mw-redirect" title="Modified Bessel function">modified Bessel function</a>. 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.<a href="#cite_note-Berry-81">[76]</a> 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.

<div class="mw-heading mw-heading4"><h4 id="Hann–Poisson_window">Hann–Poisson window</h4></div>

A <a href="#Hann_and_Hamming_windows">Hann window</a> multiplied by a <a href="#Exponential_or_Poisson_window">Poisson window</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \geqslant 2}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha \geqslant 2}</annotation>

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</math><img src="./d526628b57d6dd5de5623d15471302560aa6c707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.749ex; height:2.343ex;" alt="{\displaystyle \alpha \geqslant 2}" loading="lazy"> 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 <a href="Hill_climbing" title="Hill climbing">hill climbing</a> algorithms like <a href="Newton's_method" title="Newton's method">Newton's method</a>.<a href="#cite_note-HannPoisson-82">[77]</a> The Hann–Poisson window is defined by:

<mfrac>

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<annotation encoding="application/x-tex">{\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}}\,}</annotation>

</semantics>

</math><img src="./4e7d44df8e90604b099629079cb1ed07faa045a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.77ex; height:6.176ex;" alt="{\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}}\,}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Other_windows">Other windows</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Generalized_adaptive_polynomial_(GAP)_window">Generalized adaptive polynomial (GAP) window</h4></div>

The GAP window is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. 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.

<msub>

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<annotation encoding="application/x-tex">{\displaystyle w_{0}[n]=a_{0}+\sum _{k=1}^{K}a_{2k}\left({\frac {n}{\sigma }}\right)^{2k},\quad -{\frac {N}{2}}\leq n\leq {\frac {N}{2}},}</annotation>

</semantics>

</math><img src="./0985efd235d3dcfbd7c9a2631aa173cd5c2d4cdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.632ex; height:7.343ex;" alt="{\displaystyle w_{0}[n]=a_{0}+\sum _{k=1}^{K}a_{2k}\left({\frac {n}{\sigma }}\right)^{2k},\quad -{\frac {N}{2}}\leq n\leq {\frac {N}{2}},}" loading="lazy"> <a href="#cite_note-Beccaro-83">[78]</a></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is the standard deviation of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{n\}}</annotation>

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</math><img src="./335b6ca8393d83e990d21e637d1fd829b73f3971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle \{n\}}" loading="lazy"> sequence.

Additionally, starting with a set of expansion coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2k}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{2k}}</annotation>

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</math><img src="./8b3e4ff1e038f8c7a475cdce10d8aa8cd1907dd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.009ex;" alt="{\displaystyle a_{2k}}" loading="lazy"> 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.<a href="#cite_note-MWGAP-84">[79]</a> Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.

<div class="mw-heading mw-heading4"><h4 id="Lanczos_window">Lanczos window</h4></div>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle w[n]=\operatorname {sinc} \left({\frac {2n}{N}}-1\right)}</annotation>

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</math>

<ul><li>used in <a href="Lanczos_resampling" title="Lanczos resampling">Lanczos resampling</a></li>

<li>for the Lanczos window, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)}</annotation>

</semantics>

</math><img src="./35b9704e41871579eb37ddd542a66366875e2133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.027ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (x)}" loading="lazy"> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi x)/\pi x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\pi x)/\pi x}</annotation>

</semantics>

</math><img src="./d8a323d078d4ee7b6f4d88375ee047b473cf5414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.151ex; height:2.843ex;" alt="{\displaystyle \sin(\pi x)/\pi x}" loading="lazy"></li>

<li>also known as a sinc window, because: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(n)=\operatorname {sinc} \left({\frac {2n}{N}}\right)\,}">

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{0}(n)=\operatorname {sinc} \left({\frac {2n}{N}}\right)\,}</annotation>

</semantics>

</math> is the main lobe of a normalized <a href="Sinc_function" title="Sinc function">sinc function</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Asymmetric_window_functions">Asymmetric window functions</h3></div>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{0}(x)}</annotation>

</semantics>

</math><img src="./6e2e240f4a8bf7796a983ab6c130d39e64c6d73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.857ex; height:2.843ex;" alt="{\displaystyle w_{0}(x)}" loading="lazy"> form, according to the convention above, is symmetric around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">. However, there are window functions that are asymmetric, such as the <a href="Gamma_distribution" title="Gamma distribution">gamma distribution</a> used in FIR implementations of <a href="Gammatone_filter" title="Gammatone filter">gammatone filters</a>, or the <a href="Beta_distribution" title="Beta distribution">beta distribution</a> 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.

Any <a href="Bounded_function" title="Bounded function">bounded function</a> with <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a>, 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

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\leftarrow N\left({\frac {x}{N}}+{\frac {1}{2}}\right)^{\alpha }-{\frac {N}{2}}\,,}</annotation>

</semantics>

</math><img src="./f0e2da5d5f2508fc280d669af1a4e6fceb920a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.226ex; height:6.176ex;" alt="{\displaystyle x\leftarrow N\left({\frac {x}{N}}+{\frac {1}{2}}\right)^{\alpha }-{\frac {N}{2}}\,,}" loading="lazy"></dd></dl>

where the window weights more highly the earliest samples when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha >1}</annotation>

</semantics>

</math><img src="./17d81dbbc4786493c7b8548cc324a978d7cf5dbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}" loading="lazy">, and conversely weights more highly the latest samples when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <1}</annotation>

</semantics>

</math><img src="./4769a8dab3c8a2045bc128b9000da5d661f7dab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <1}" loading="lazy">.<a href="#cite_note-85">[80]</a>

<ul><li><a href="Apodization" title="Apodization">Apodization</a></li>

<li><a href="Kolmogorov%E2%80%93Zurbenko_filter" title="Kolmogorov–Zurbenko filter">Kolmogorov–Zurbenko filter</a></li>

<li><a href="Multitaper" title="Multitaper">Multitaper</a></li>

<li><a href="Short-time_Fourier_transform" title="Short-time Fourier transform">Short-time Fourier transform</a></li>

<li><a href="Spectral_leakage" title="Spectral leakage">Spectral leakage</a></li>

<li><a href="Welch_method" class="mw-redirect" title="Welch method">Welch method</a></li>

<li><a href="Weight_function" title="Weight function">Weight function</a></li>

<li><a href="Window_design_method" class="mw-redirect" title="Window design method">Window design method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-upper-alpha">

<li id="cite_note-18"><a href="#cite_ref-18">^</a> Some authors limit their attention to this important subset and to even values of N.<a href="#cite_note-Harris-16">[16]</a><a href="#cite_note-Heinzel2002-17">[17]</a> But the window coefficient formulas are still the ones presented here.

</li>

<li id="cite_note-56"><a href="#cite_ref-56">^</a> This formula can be confirmed by simplifying the cosine function at <a rel="nofollow" class="external text" href="http://www.mathworks.com/help/signal/ref/tukeywin.html">MATLAB tukeywin</a> and substituting r=α and x=n/N.

</li>

<li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#Harris">Harris 1978</a> (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.

</li>

<li id="cite_note-70"><a href="#cite_ref-70">^</a> The Kaiser window is often parametrized by β, where β = πα.<a href="#cite_note-Rabiner-64">[60]</a><a href="#cite_note-Crochiere-65">[61]</a>

<a href="#cite_note-Vaidyanathan-66">[62]</a><a href="#cite_note-JOSKaiser-67">[63]</a><a href="#cite_note-KaiserDPSS-61">[57]</a><a href="#cite_note-MWkaiser-68">[64]</a><a href="#cite_note-Oppenheim-7">[7]</a>: p. 474 The alternative use of just α facilitates comparisons to the DPSS windows.<a href="#cite_note-Kaiser_Window.html-69">[65]</a>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Page_citations">Page citations</h2></div>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#Harris">Harris 1978</a>, p 57, fig 10.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.labbookpages.co.uk/audio/firWindowing.html">"FIR Filters by Windowing – The Lab Book Pages"</a>. www.labbookpages.co.uk. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/filters/Zero_Phase_Filters_Even_Impulse.html">"Zero Phase Filters"</a>. ccrma.stanford.edu. Retrieved 2020-02-12.</cite>

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<cite id="Harris" class="citation journal cs1">Harris, Fredric J. (Jan 1978). <a rel="nofollow" class="external text" href="http://web.mit.edu/xiphmont/Public/windows.pdf">"On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform"</a> (PDF). Proceedings of the IEEE. 66 (1): 51–83. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978IEEEP..66...51H">1978IEEEP..66...51H</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.649.9880">10.1.1.649.9880</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPROC.1978.10837">10.1109/PROC.1978.10837</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:426548">426548</a>.</cite> The fundamental 1978 paper on FFT windows by Harris, which specified many windows and introduced key metrics used to compare them.

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<cite id="CITEREFHeinzelRüdigerSchilling2002" class="citation techreport cs1">Heinzel, G.; Rüdiger, A.; Schilling, R. (2002). <a rel="nofollow" class="external text" href="http://edoc.mpg.de/395068">Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows</a> (Technical report). Max Planck Institute (MPI) für Gravitationsphysik / Laser Interferometry & Gravitational Wave Astronomy. 395068.0. Retrieved 2013-02-10.</cite> Also available at <a rel="nofollow" class="external free" href="https://pure.mpg.de/rest/items/item_152164_1/component/file_152163/content">https://pure.mpg.de/rest/items/item_152164_1/component/file_152163/content</a>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Bartlett_Triangular_Window.html">"Bartlett Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFTukey1967" class="citation journal cs1">Tukey, J.W. (1967). "An introduction to the calculations of numerical spectrum analysis". Spectral Analysis of Time Series: 25–46.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathworks.com/help/signal/ref/triang.html">"Triangular window – MATLAB triang"</a>. www.mathworks.com. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFWelch1967" class="citation journal cs1">Welch, P. (1967). "The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms". IEEE Transactions on Audio and Electroacoustics. 15 (2): 70–73. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967ITAE...15...70W">1967ITAE...15...70W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAU.1967.1161901">10.1109/TAU.1967.1161901</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13900622">13900622</a>.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathworks.com/help/signal/ref/hann.html">"Hann (Hanning) window - MATLAB hann"</a>. www.mathworks.com. Retrieved 2020-02-12.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Hann_Hanning_Raised_Cosine.html">"Hann or Hanning or Raised Cosine"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFEnochsonOtnes1968" class="citation book cs1">Enochson, Loren D.; Otnes, Robert K. (1968). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=duBQAAAAMAAJ&q=%22hamming+window%22+date:0-1970">Programming and Analysis for Digital Time Series Data</a>. U.S. Dept. of Defense, Shock and Vibration Info. Center. p. 142.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Hamming_Window.html">"Hamming Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://users.wpi.edu/~sunar/courses/ece3311/slides/ch16.pdf">"A digital quadrature amplitude modulation (QAM) Radio: Building a better radio"</a> (PDF). users.wpi.edu. p. 28. Retrieved 2020-02-12.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://users.wpi.edu/~sunar/courses/ece3311/slides/ch08.pdf">"Bits to Symbols to Signals and back again"</a> (PDF). users.wpi.edu. p. 7. Retrieved 2020-02-12.</cite>

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<cite id="CITEREFJohnsonSetharesKlein2011" class="citation book cs1">Johnson, C.Richard Jr; Sethares, William A.; Klein, Andrew G. (2011-08-18). "11". Software Receiver Design. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-139-50145-3</bdi>.</cite> Also <a rel="nofollow" class="external free" href="https://cnx.org/contents/QsVBJjB4@3.1:6R_ztzDY@4/Pulse-Shaping-and-Receive-Filtering">https://cnx.org/contents/QsVBJjB4@3.1:6R_ztzDY@4/Pulse-Shaping-and-Receive-Filtering</a>

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<cite id="Nuttall" class="citation journal cs1">Nuttall, Albert H. (Feb 1981). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1280930">"Some Windows with Very Good Sidelobe Behavior"</a>. IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTASSP.1981.1163506">10.1109/TASSP.1981.1163506</a>.</cite> Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.

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<cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BlackmanFunction.html">"Blackman Function"</a>. mathworld.wolfram.com. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://zone.ni.com/reference/en-XX/help/371361E-01/lvanlsconcepts/char_smoothing_windows/#Exact_Blackman">"Characteristics of Different Smoothing Windows - NI LabVIEW 8.6 Help"</a>. zone.ni.com. Retrieved 2020-02-13.</cite>

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<cite id="CITEREFBlackmanTukey1959" class="citation book cs1"><a href="R._B._Blackman" class="mw-redirect" title="R. B. Blackman">Blackman, R.B.</a>; Tukey, J.W. (1959-01-01). <a rel="nofollow" class="external text" href="https://smile.amazon.com/Measurement-Power-Spectra-Communications-Engineering/dp/B0006AW1C4">The Measurement of Power Spectra from the Point of View of Communications Engineering</a>. Dover Publications. p. 99. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-60507-4</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Blackman_Harris_Window_Family.html">"Blackman-Harris Window Family"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Three_Term_Blackman_Harris_Window.html">"Three-Term Blackman-Harris Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFSmith2011" class="citation book cs1">Smith, Steven W. (2011). <a rel="nofollow" class="external text" href="http://www.dspguide.com/ch9/1.htm">The Scientist and Engineer's Guide to Digital Signal Processing</a>. San Diego, California, USA: California Technical Publishing. Retrieved 2013-02-14.</cite>

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<cite id="CITEREFRifeVincent1970" class="citation cs2">Rife, David C.; Vincent, G.A. (1970), "Use of the discrete Fourier transform in the measurement of frequencies and levels of tones", Bell Syst. Tech. J., 49 (2): 197–228, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1970.tb01766.x">10.1002/j.1538-7305.1970.tb01766.x</a></cite>

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<cite id="CITEREFAndriaSavinoTrotta1989" class="citation cs2">Andria, Gregorio; Savino, Mario; Trotta, Amerigo (1989), "Windows and interpolation algorithms to improve electrical measurement accuracy", IEEE Transactions on Instrumentation and Measurement, 38 (4): 856–863, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989ITIM...38..856A">1989ITIM...38..856A</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F19.31004">10.1109/19.31004</a></cite>

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<cite id="CITEREFSchoukensPintelonVan_Hamme1992" class="citation cs2">Schoukens, Joannes; Pintelon, Rik; Van Hamme, Hugo (1992), "The interpolated fast Fourier transform: a comparative study", IEEE Transactions on Instrumentation and Measurement, 41 (2): 226–232, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ITIM...41..226S">1992ITIM...41..226S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F19.137352">10.1109/19.137352</a></cite>

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<cite id="CITEREFBosiGoldberg2003" class="citation book cs1">Bosi, Marina; Goldberg, Richard E. (2003). "Time to Frequency Mapping Part II: The MDCT". Introduction to Digital Audio Coding and Standards. The Springer International Series in Engineering and Computer Science. Vol. 721. Boston, MA: Springer US. p. 106. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4615-0327-9">10.1007/978-1-4615-0327-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4615-0327-9</bdi>.</cite>

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<cite id="CITEREFKidoSuzukiOnoFukushima1998" class="citation journal cs1">Kido, Ken'iti; Suzuki, Hideo; Ono, Takahiko; Fukushima, Manabu (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1250%2Fast.19.349">"Deformation of impulse response estimates by time window in cross spectral technique"</a>. Journal of the Acoustical Society of Japan E. 19 (5): 349–361. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1250%2Fast.19.349">10.1250/ast.19.349</a>.</cite>

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<cite id="CITEREFLandismanDziewonskiSatô1969" class="citation journal cs1">Landisman, M.; Dziewonski, A.; Satô, Y. (1969-05-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-246X.1969.tb00246.x">"Recent Improvements in the Analysis of Surface Wave Observations"</a>. Geophysical Journal International. 17 (4): 369–403. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969GeoJ...17..369L">1969GeoJ...17..369L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-246X.1969.tb00246.x">10.1111/j.1365-246X.1969.tb00246.x</a>.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/signal/ref/bohmanwin.html">"Bohman window – R2019B"</a>. www.mathworks.com. Retrieved 2020-02-12.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Power_of_Cosine_Window_Family.html">"Power-of-Cosine Window Family"</a>. ccrma.stanford.edu. Retrieved 10 April 2018.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Matlab_Gaussian_Window.html">"Matlab for the Gaussian Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13. <q>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</q></cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Gaussian_Window_Transform.html">"Gaussian Window and Transform"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Quadratic_Interpolation_Spectral_Peaks.html">"Quadratic Interpolation of Spectral Peaks"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFStarosielecHägele2014" class="citation journal cs1">Starosielec, S.; Hägele, D. (2014). "Discrete-time windows with minimal RMS bandwidth for given RMS temporal width". Signal Processing. 102: 240–246. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014SigPr.102..240S">2014SigPr.102..240S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.sigpro.2014.03.033">10.1016/j.sigpro.2014.03.033</a>.</cite>

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<cite id="CITEREFChakrabortyKovvali2013" class="citation book cs1">Chakraborty, Debejyo; Kovvali, Narayan (2013). "Generalized normal window for digital signal processing". 2013 IEEE International Conference on Acoustics, Speech and Signal Processing. pp. 6083–6087. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FICASSP.2013.6638833">10.1109/ICASSP.2013.6638833</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4799-0356-6</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11779529">11779529</a>.</cite>

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<cite id="CITEREFDiethorn1994" class="citation journal cs1">Diethorn, E.J. (1994). "The generalized exponential time-frequency distribution". IEEE Transactions on Signal Processing. 42 (5): 1028–1037. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994ITSP...42.1028D">1994ITSP...42.1028D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F78.295214">10.1109/78.295214</a>.</cite>

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<cite id="CITEREFBloomfield2000" class="citation book cs1">Bloomfield, P. (2000). Fourier Analysis of Time Series: An Introduction. New York: Wiley-Interscience.</cite>

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<cite id="CITEREFTu2008" class="citation book cs1">Tu, Loring W. (2008). "Bump Functions and Partitions of Unity". An Introduction to Manifolds. Universitext. New York: Springer. pp. 127–134. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-48101-2_13">10.1007/978-0-387-48101-2_13</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-48098-5</bdi>.</cite>

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<cite id="CITEREFMcKechanRobinsonSathyaprakash2010" class="citation journal cs1">McKechan, D.J.A.; Robinson, C.; Sathyaprakash, B.S. (21 April 2010). "A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries". Classical and Quantum Gravity. 27 (8): 084020. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1003.2939">1003.2939</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010CQGra..27h4020M">2010CQGra..27h4020M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F27%2F8%2F084020">10.1088/0264-9381/27/8/084020</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21488253">21488253</a>.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Slepian_DPSS_Window.html">"Slepian or DPSS Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFSmith2011" class="citation web cs1">Smith, J.O. (2011). <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html">"Kaiser and DPSS Windows Compared"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite id="CITEREFKaiserKuo1966" class="citation book cs1">Kaiser, James F.; Kuo, Franklin F. (1966). System Analysis by Digital Computer. John Wiley and Sons. pp. 232–235. <q>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.</q></cite>

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<cite id="CITEREFKaiser1964" class="citation journal cs1">Kaiser, James F. (Nov 1964). "A family of window functions having nearly ideal properties". Unpublished Memorandum.</cite>

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<cite id="CITEREFRabiner,_Lawrence_R.Gold,_Bernard1975" class="citation book cs1">Rabiner, Lawrence R.; Gold, Bernard (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00rabi/page/94">"3.11"</a>. Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. p. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00rabi/page/94">94</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-914101-4</bdi>.</cite>

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<cite id="CITEREFCrochiereRabiner1983" class="citation book cs1">Crochiere, R.E.; Rabiner, L.R. (1983). "4.3.1". <a rel="nofollow" class="external text" href="https://kupdf.net/download/multirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf">Multirate Digital Signal Processing</a>. Englewood Cliffs, NJ: Prentice-Hall. p. 144. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-605162-6</bdi>.</cite>

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<cite id="CITEREFLinVaidyanathan1998" class="citation journal cs1">Lin, Yuan-Pei; Vaidyanathan, P.P. (June 1998). <a rel="nofollow" class="external text" href="http://authors.library.caltech.edu/6891/1/LINieeespl98.pdf">"A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks"</a> (PDF). IEEE Signal Processing Letters. 5 (6): 132–134. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998ISPL....5..132L">1998ISPL....5..132L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F97.681427">10.1109/97.681427</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18159105">18159105</a>. Retrieved 2017-03-16.</cite>

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<cite id="CITEREFSmith2011" class="citation web cs1">Smith, J.O. (2011). <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Kaiser_Window.html">"Kaiser Window"</a>. ccrma.stanford.edu. Retrieved 2019-03-20. <q>Sometimes the Kaiser window is parametrized by α, where β = πα.</q></cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/signal/ref/kaiser.html">"Kaiser Window, R2020a"</a>. www.mathworks.com. Mathworks. Retrieved 9 April 2020.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.dsprelated.com/freebooks/sasp/Kaiser_Window.html">"Kaiser Window"</a>. www.dsprelated.com. Retrieved 2020-04-08. <q>The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of α as the bin number of the edge of the critically sampled window main lobe.</q></cite>

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<cite id="CITEREFKaiserSchafer1980" class="citation journal cs1">Kaiser, James F.; Schafer, Ronald W. (1980). "On the use of the I0-sinh window for spectrum analysis". IEEE Transactions on Acoustics, Speech, and Signal Processing. 28: 105–107. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTASSP.1980.1163349">10.1109/TASSP.1980.1163349</a>.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window.html">"Dolph-Chebyshev Window"</a>. ccrma.stanford.edu. Retrieved 2016-04-13.</cite>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window_Definition.html">"Dolph-Chebyshev Window Definition"</a>. ccrma.stanford.edu. Retrieved 2019-03-05.</cite>

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<cite id="CITEREFKabal2009" class="citation journal cs1">Kabal, Peter (2009). <a rel="nofollow" class="external text" href="http://www-mmsp.ece.mcgill.ca/Documents/Reports/2009/KabalR2009b.pdf">"Time Windows for Linear Prediction of Speech"</a> (PDF). Technical Report, Dept. Elec. & Comp. Eng., McGill University (2a): 31. Retrieved 2 February 2014.</cite>

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<cite id="CITEREFStreit1984" class="citation journal cs1">Streit, Roy (1984). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1280988">"A two-parameter family of weights for nonrecursive digital filters and antennas"</a>. IEEE Transactions on Acoustics, Speech, and Signal Processing. 32: 108–118. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftassp.1984.1164275">10.1109/tassp.1984.1164275</a>.</cite>

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<cite id="CITEREFDeczky2001" class="citation book cs1">Deczky, Andrew (2001). "Unispherical Windows". ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No. 01CH37196). Vol. 2. pp. 85–88. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fiscas.2001.921012">10.1109/iscas.2001.921012</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7803-6685-5</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:38275201">38275201</a>.</cite>

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<li id="cite_note-85"><a href="#cite_ref-85">^</a> <cite id="CITEREFLuoXieLi2015" class="citation journal cs1">Luo, Jiufel; Xie, Zhijiang; Li, Xinyi (2015-03-02). <a rel="nofollow" class="external text" href="https://doi.org/10.1260%2F1748-3018.9.4.389">"Asymmetric Windows and Their Application in Frequency Estimation"</a>. Chongqing University. 9 (Algorithms & Computational Technology): 389–412. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1260%2F1748-3018.9.4.389">10.1260/1748-3018.9.4.389</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124464194">124464194</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFHarris1976" class="citation web cs1">Harris, Frederic J. (September 1976). <a rel="nofollow" class="external text" href="https://apps.dtic.mil/dtic/tr/fulltext/u2/a034956.pdf">"Windows, Harmonic Analysis, and the Discrete Fourier Transform"</a> (PDF). apps.dtic.mil. Naval Undersea Center, San Diego. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190408141816/https://apps.dtic.mil/dtic/tr/fulltext/u2/a034956.pdf">Archived</a> (PDF) from the original on April 8, 2019. Retrieved 2019-04-08.</cite></li>

<li><cite id="CITEREFAlbrecht2012" class="citation book cs1 cs1-prop-long-vol">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. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7795%2F110.20121022aa">10.7795/110.20121022aa</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-86918-281-0</bdi>.</cite></li>

<li><cite id="CITEREFBergenAntoniou2005" class="citation journal cs1">Bergen, S.W.A.; Antoniou, A. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FASP.2005.1910">"Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function"</a>. EURASIP Journal on Applied Signal Processing. 2005 (12): 1910–1922. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005EJASP2005...44B">2005EJASP2005...44B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FASP.2005.1910">10.1155/ASP.2005.1910</a>.</cite></li>

<li><cite id="CITEREFPrabhu2014" class="citation book cs1">Prabhu, K. M. M. (2014). Window Functions and Their Applications in Signal Processing. Boca Raton, FL: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4665-1583-3</bdi>.</cite></li>

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</style><a rel="nofollow" class="external text" href="https://worldwide.espacenet.com/textdoc?DB=EPODOC&IDX=US7065150">US patent 7065150</a>, Park, Young-Seo, "System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation", published 2003, issued 2006<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&rft.number=7065150&rft.cc=US&rft.title=System+and+method+for+generating+a+root+raised+cosine+orthogonal+frequency+division+multiplexing+%28RRC+OFDM%29+modulation&rft.inventor=Park&rft.date=2006&rft.pubdate=2003"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Window_function" class="extiw external" title="commons:Category:Window function">Window function</a> at Wikimedia Commons</li>

<li>LabView Help, Characteristics of Smoothing Filters, <a rel="nofollow" class="external free" href="http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/">http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/</a></li>

<li>Creation and properties of Cosine-sum Window functions, <a rel="nofollow" class="external free" href="http://electronicsart.weebly.com/fftwindows.html">http://electronicsart.weebly.com/fftwindows.html</a></li>

<li><a rel="nofollow" class="external text" href="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</a></li></ul></div><div><div class="zim-footer">

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