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In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent
Discrete Fourier Transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
. Several ''
window functions A window is an opening in a wall, door, roof, or vehicle that allows the exchange of light and may also allow the passage of sound and sometimes air. Modern windows are usually glazed or covered in some other transparent or translucent mater ...
'' can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multiplying the given sampled signal by the window function.


Conventions

* w_0(x) is a zero-phase function (symmetrical about x=0), continuous for x \in N/2, N/2 where N is a positive integer (even or odd). * The sequence  \  is ''symmetric'', of length N+1. * \  is ''DFT-symmetric'', of length N. * The parameter B displayed on each spectral plot is the function's noise equivalent bandwidth metric, in units of ''DFT bins''. The sparse sampling of a DTFT (such as the DFTs in Fig 1) 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. Therefore, when choosing a window function, it is usually important to sample the DTFT more densely (as we do throughout this section) and choose a window that suppresses the sidelobes to an acceptable level.


Rectangular window

The rectangular window (sometimes known as the
boxcar A boxcar is the North American ( AAR) term for a railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simplest freight car design, is considered one of the most versatile since it can carry most ...
or
Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
window) is the simplest window, equivalent to replacing all but ''N'' values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off: : w = 1. Other windows are designed to moderate these sudden changes, which reduces scalloping loss and improves dynamic range, as described above (). The rectangular window is the 1st order ''B''-spline window as well as the 0th power power-of-sine window. The rectangular window provides the minimum mean square error estimate of the
Discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
, at the cost of other issues discussed.


''B''-spline windows

''B''-spline windows can be obtained as ''k''-fold convolutions of the rectangular window. They include the rectangular window itself (''k'' = 1), the (''k'' = 2) and the (''k'' = 4). Alternative definitions sample the appropriate normalized ''B''-spline
basis functions In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be repres ...
instead of convolving discrete-time windows.  A  ''k''th-order ''B''-spline basis function is a piece-wise polynomial function of degree ''k''−1 that is obtained by ''k''-fold self-convolution of the
rectangular function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname(t) = \Pi(t) = \left\{\begin{array}{r ...
.


Triangular window

Triangular windows are given by: : w = 1 - \left, \frac\,\quad 0\le n \le N where ''L'' can be ''N'', ''N'' + 1, or ''N'' + 2.  The first one is also known as Bartlett window or Fejér window. All three definitions converge at large ''N''. The triangular window is the 2nd order ''B''-spline window. The ''L'' = ''N'' form can be seen as the convolution of two ''N''/2-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.


Parzen window

Defining  ,  the Parzen window, also known as the de la Vallée Poussin window, is the 4th order ''B''-spline window given by: : w_0(n) \triangleq \left\ : w = \ w_0\left(n-\tfrac\right),\ 0 \le n \le N


Other polynomial windows


Welch window

The Welch window consists of a single parabolic section: : w 1 - \left(\frac\right)^2,\quad 0\le n \le N. The defining
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...
reaches a value of zero at the samples just outside the span of the window.


Sine window

: w = \sin\left(\frac\right) = \cos\left(\frac - \frac\right),\quad 0\le n \le N. The corresponding w_0(n)\, function is a cosine without the /2 phase offset. So the ''sine window'' is sometimes also called ''cosine window''. As it represents half a cycle of a sinusoidal function, it is also known variably as ''half-sine window'' or ''half-cosine window''. The
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
of a sine window produces a function known as the Bohman window.


Power-of-sine/cosine windows

These window functions have the form: : w = \sin^\alpha\left(\frac\right) = \cos^\alpha\left(\frac - \frac\right),\quad 0\le n \le N. The rectangular window (), the sine window (), and the
Hann window The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by: : w_0(x) \triangleq \left\.   For digital sign ...
() are members of this family. For even-integer values of these functions can also be expressed in cosine-sum form: : w a_0 - a_1 \cos \left ( \frac \right)+ a_2 \cos \left ( \frac \right)- a_3 \cos \left ( \frac \right)+ a_4 \cos \left ( \frac \right)- ... : \begin \hline \alpha & a_0 & a_1 & a_2 & a_3 & a_4 \\ \hline 0 & 1 \\ 2 & 0.5 & 0.5 \\ 4 & 0.375 & 0.5 & 0.125 \\ 6 & 0.3125 & 0.46875 & 0.1875 & 0.03125 \\ 8 & 0.2734375 & 0.4375 & 0.21875 & 0.0625 & 7.8125\times10^ \\ \hline \end


Cosine-sum windows

This family is also known as
generalized cosine windows
'. In most cases, including the examples below, all coefficients ''a''''k'' â‰¥ 0.  These windows have only 2''K'' + 1 non-zero ''N''-point DFT coefficients.


Hann and Hamming windows

The customary cosine-sum windows for case ''K'' = 1 have the form: : w = a_0 - \underbrace_\cdot \cos\left( \tfrac \right),\quad 0\le n \le N, which is easily (and often) confused with its zero-phase version: : \begin w_0(n)\ &= w\left n+\tfrac\right\ &= a_0 + a_1\cdot \cos \left ( \tfrac \right),\quad -\tfrac \le n \le \tfrac. \end Setting  a_0 = 0.5  produces a Hann window: : w = 0.5\; \left - \cos \left ( \frac \right) \right= \sin^2 \left ( \frac \right), named after
Julius von Hann Julius Ferdinand von Hann (23 March 1839 in Wartberg ob der Aist near Linz – 1 October 1921 in Vienna) was an Austrian meteorologist. He is seen as a father of modern meteorology. Biography He was educated at the gymnasium of Kremsmünster a ...
, and sometimes erroneously referred to as ''Hanning'', presumably due to its linguistic and formulaic similarities to the Hamming window. It is also known as raised cosine, because the zero-phase version, w_0(n), is one lobe of an elevated cosine function. This function is a member of both the cosine-sum and power-of-sine families. Unlike the
Hamming window In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several '' window functions'' can be defined, based on ...
, the end points of the Hann window just touch zero. The resulting side-lobes roll off at about 18 dB per octave. Setting  a_0  to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by Richard W. Hamming. That choice places a zero-crossing at frequency 5/(''N'' âˆ’ 1), which cancels the first sidelobe of the Hann window, giving it a height of about one-fifth that of the Hann window. The Hamming window is often called the Hamming blip when used for
pulse shaping In electronics and telecommunications, pulse shaping is the process of changing the waveform of transmitted pulses to optimize the signal for its intended purpose or the communication channel. This is often done by limiting the bandwidth of the tran ...
. Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes, to a nearly equiripple condition. In the equiripple sense, the optimal values for the coefficients are a0 = 0.53836 and a1 = 0.46164.


Blackman window

Blackman windows are defined as: : w = a_0 - a_1 \cos \left ( \frac \right) + a_2 \cos \left ( \frac \right) : a_0=\frac;\quad a_1=\frac;\quad a_2=\frac. By common convention, the unqualified term ''Blackman window'' refers to Blackman's "not very serious proposal" of (''a''0 = 0.42, ''a''1 = 0.5, ''a''2 = 0.08), which closely approximates the exact Blackman, with ''a''0 = 7938/18608 â‰ˆ 0.42659, ''a''1 = 9240/18608 â‰ˆ 0.49656, and ''a''2 = 1430/18608 â‰ˆ 0.076849. These exact values place zeros at the third and fourth sidelobes, but result in a discontinuity at the edges and a 6 dB/oct fall-off. The truncated coefficients do not null the sidelobes as well, but have an improved 18 dB/oct fall-off.


Nuttall window, continuous first derivative

The continuous form of the Nuttall window, w_0(x), and its first
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
are continuous everywhere, like the
Hann function The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by: : w_0(x) \triangleq \left\.   For digital sign ...
. That is, the function goes to 0 at unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows. The Blackman window () is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not. : w a_0 - a_1 \cos \left ( \frac \right)+ a_2 \cos \left ( \frac \right)- a_3 \cos \left ( \frac \right) : a_0=0.355768;\quad a_1=0.487396;\quad a_2=0.144232;\quad a_3=0.012604.


Blackman–Nuttall window

: w a_0 - a_1 \cos \left ( \frac \right)+ a_2 \cos \left ( \frac \right)- a_3 \cos \left ( \frac \right) : a_0=0.3635819; \quad a_1=0.4891775; \quad a_2=0.1365995; \quad a_3=0.0106411.


Blackman–Harris window

A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels : w a_0 - a_1 \cos \left ( \frac \right)+ a_2 \cos \left ( \frac \right)- a_3 \cos \left ( \frac \right) : a_0=0.35875;\quad a_1=0.48829;\quad a_2=0.14128;\quad a_3=0.01168.


Flat top window

A flat top window is a partially negative-valued window that has minimal scalloping loss in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components. Drawbacks of the broad bandwidth are poor frequency resolution and high . Flat top windows can be designed using low-pass filter design methods, or they may be of the usual cosine-sum variety: : \begin w = a_0 &- a_1 \cos \left ( \frac \right)+ a_2 \cos \left ( \frac \right)\\ &- a_3 \cos \left ( \frac \right)+a_4 \cos \left ( \frac \right). \end Th
Matlab variant
has these coefficients: : a_0=0.21557895;\quad a_1=0.41663158;\quad a_2=0.277263158;\quad a_3=0.083578947;\quad a_4=0.006947368. Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.


Rife–Vincent windows

Rife–Vincent windows are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied to , reflect that custom. Class I, Order 1 (''K'' = 1):  a_0=1;\quad a_1=1       Functionally equivalent to the
Hann window The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by: : w_0(x) \triangleq \left\.   For digital sign ...
. Class I, Order 2 (''K'' = 2):  a_0=1;\quad a_1=\tfrac;\quad a_2=\tfrac Class I is defined by minimizing the high-order sidelobe amplitude. Coefficients for orders up to K=4 are tabulated. Class II minimizes the main-lobe width for a given maximum side-lobe. Class III is a compromise for which order ''K'' = 2 resembles the .


Adjustable windows


Gaussian window

The Fourier transform of a
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either be truncated at the ends of the window, or itself windowed with another zero-ended window. Since the log of a Gaussian produces a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
, this can be used for nearly exact quadratic interpolation in
frequency estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signa ...
. : w \exp\left(-\frac \left ( \frac \right)^\right),\quad 0\le n \le N. : \sigma \le \;0.5\, The standard deviation of the Gaussian function is ''σ'' Â· ''N''/2 sampling periods.


Confined Gaussian window

The confined Gaussian window yields the smallest possible root mean square frequency width for a given temporal width  . 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 and the in the limiting cases of large and small , respectively.


Approximate confined Gaussian window

Defining  ,  a confined Gaussian window of temporal width    is well approximated by: : w = G(n) - \frac where G is a Gaussian function: :: G(x) = \exp\left(- \left(\cfrac\right)^2\right) The standard deviation of the approximate window is
asymptotically equal In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...
(i.e. large values of ) to    for  .


Generalized normal window

A more generalized version of the Gaussian window is the generalized normal window. Retaining the notation from the
Gaussian window In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several '' window functions'' can be defined, based on ...
above, we can represent this window as : w ,p\exp\left(-\left ( \frac \right)^\right) for any even p. At p=2, this is a Gaussian window and as p approaches \infty, this approximates to a rectangular window. The
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of this window does not exist in a closed form for a general p. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the , this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window. See also for a study on time-frequency representation of this window (or function).


Tukey window

The Tukey window, also known as the ''cosine-tapered window'', can be regarded as a cosine lobe of width (spanning observations) that is convolved with a rectangular window of width . : \left . \begin w = \frac \left -\cos \left(\frac \right) \right\quad & 0 \le n < \frac\\ w = 1,\quad & \frac \le n \le \frac\\ w -n= w \quad & 0 \le n \le \frac \end\right\}   At it becomes rectangular, and at it becomes a Hann window.


Planck-taper window

The so-called "Planck-taper" window is a
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
that has been widely used in the theory of
partitions of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
in
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
. It is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
(a C^\infty function) everywhere, but is exactly zero outside of a compact region, exactly one over an interval within that region, and varies smoothly and monotonically between those limits. Its use as a window function in signal processing was first suggested in the context of
gravitational-wave astronomy Gravitational-wave astronomy is an emerging branch of observational astronomy which aims to use gravitational waves (minute distortions of spacetime predicted by Albert Einstein's theory of general relativity) to collect observational data about ...
, inspired by the Planck distribution. It is defined as a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
function: : \left . \begin w = 0, \\ w = \left(1 + \exp\left(\frac - \frac\right)\right)^,\quad & 1 \le n < \varepsilon N \\ w = 1,\quad & \varepsilon N \le n \le \frac \\ w -n= w \quad & 0 \le n \le \frac \end\right\} The amount of tapering is controlled by the parameter ''ε'', with smaller values giving sharper transitions.


DPSS or Slepian window

The DPSS (discrete prolate spheroidal sequence) or Slepian window maximizes the energy concentration in the main lobe, and is used in multitaper spectral analysis, which averages out noise in the spectrum and reduces information loss at the edges of the window. The main lobe ends at a frequency bin given by the parameter ''α''. The Kaiser windows below are created by a simple approximation to the DPSS windows:


Kaiser window

The Kaiser, or Kaiser–Bessel, window is a simple approximation of the DPSS window using
Bessel functions Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
, discovered by
James Kaiser James Frederick Kaiser (Dec. 10, 1929 – Feb. 13, 2020) was an American electrical engineer noted for his contributions in signal processing. He was an IEEE Fellow and received many honors and awards, including the IEEE Centennial Medal, the I ...
. : w \frac,\quad 0\le n \le N     : w_0(n) = \frac,\quad -N/2 \le n \le N/2 where I_0 is the zero-th order modified Bessel function of the first kind. Variable parameter \alpha determines the tradeoff between main lobe width and side lobe levels of the spectral leakage pattern.  The main lobe width, in between the nulls, is given by  2\sqrt,  in units of DFT bins,  and a typical value of \alpha is 3.


Dolph–Chebyshev window

Minimizes the Chebyshev norm of the side-lobes for a given main lobe width. The zero-phase Dolph–Chebyshev window function w_0 /math> is usually defined in terms of its real-valued discrete Fourier transform, W_0 /math>: : W_0(k) = \frac = \frac,\ 0 \le k \le N. ''T''''n''(''x'') is the ''n''-th
Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
of the first kind evaluated in ''x'', which can be computed using : T_n(x) =\begin \cos\!\big(n \cos^(x) \big) & \text-1 \le x \le 1 \\ \cosh\!\big(n \cosh^(x) \big) & \textx \ge 1 \\ (-1)^n \cosh\!\big(n \cosh^(-x) \big) & \textx \le -1, \end and : \beta = \cosh\!\big(\tfrac \cosh^(10^\alpha)\big) is the unique positive real solution to T_N(\beta) = 10^\alpha, where the parameter ''α'' sets the Chebyshev norm of the sidelobes to −20''α'' decibels. The window function can be calculated from ''W''0(''k'') by an inverse
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
(DFT): : w_0(n) = \frac \sum_^N W_0(k) \cdot e^,\ -N/2 \le n \le N/2. The ''lagged'' version of the window can be obtained by: : w = w_0\left(n-\frac\right),\quad 0 \le n \le N, which for even values of ''N'' must be computed as follows: : \begin w_0\left(n-\frac\right) = \frac \sum_^ W_0(k) \cdot e^ = \frac \sum_^ \left \left(-e^\right)^k \cdot W_0(k)\righte^, \end which is an inverse DFT of  \left(-e^\right)^k\cdot W_0(k). Variations: * Due to the equiripple condition, the time-domain window has discontinuities at the edges. An approximation that avoids them, by allowing the equiripples to drop off at the edges, is
Taylor window
* An alternative to the inverse DFT definition is also availabl


Ultraspherical window

The Ultraspherical window was introduced in 1984 by Roy Streit and has application in antenna array design, non-recursive filter design, and spectrum analysis. Like other adjustable windows, the Ultraspherical window has parameters that can be used to control its Fourier transform main-lobe width and relative side-lobe amplitude. Uncommon to other windows, it has an additional parameter which can be used to set the rate at which side-lobes decrease (or increase) in amplitude. The window can be expressed in the time-domain as follows: : w = \frac \left C^\mu_N(x_0)+\sum_^ C^\mu_N \left(x_0 \cos\frac\right)\cos\frac \right where C^_ is the Ultraspherical polynomial of degree N, and x_0 and \mu control the side-lobe patterns. Certain specific values of \mu yield other well-known windows: \mu=0 and \mu=1 give the Dolph–Chebyshev and
Saramäki Saramäki (Finnish; ''Starrbacka'' in Swedish) is a district in the Maaria-Paattinen ward of the city of Turku, in Finland. It is located to the north of the city, and is a very sparsely populated area. The current () population of Saramäki i ...
windows respectively. Se
here
for illustration of Ultraspherical windows with varied parametrization.


Exponential or Poisson window

The Poisson window, or more generically the exponential window increases exponentially towards the center of the window and decreases exponentially in the second half. Since the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, 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 ). It is defined by : w e^, where ''Ï„'' is the time constant of the function. The exponential function decays as ''e'' â‰ƒ 2.71828 or approximately 8.69 dB per time constant. This means that for a targeted decay of ''D'' dB over half of the window length, the time constant ''Ï„'' is given by : \tau = \frac\frac.


Hybrid windows

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


Bartlett–Hann window

: w a_0 - a_1 \left , \frac-\frac \ - a_2 \cos \left (\frac\right ) : a_0=0.62;\quad a_1=0.48;\quad a_2=0.38\,


Planck–Bessel window

A multiplied by a
Kaiser window The Kaiser window, also known as the Kaiser–Bessel window, was developed by James Kaiser at Bell Laboratories. It is a one-parameter family of window functions used in finite impulse response filter design and spectral analysis. The Kaiser wi ...
which is defined in terms of a
modified Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
. This hybrid window function was introduced to decrease the peak side-lobe level of the Planck-taper window while still exploiting its good asymptotic decay. It has two tunable parameters, ''ε'' from the Planck-taper and ''α'' from the Kaiser window, so it can be adjusted to fit the requirements of a given signal.


Hann–Poisson window

A
Hann window The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by: : w_0(x) \triangleq \left\.   For digital sign ...
multiplied by a Poisson window. For \alpha \geqslant 2 it has no side-lobes, as its Fourier transform drops off forever away from the main lobe without local minima. It can thus be used in
hill climbing numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solutio ...
algorithms like
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
. The Hann–Poisson window is defined by: : w \frac\left(1-\cos\left(\frac\right)\right)e^\frac\, where ''α'' is a parameter that controls the slope of the exponential.


Other windows


Generalized adaptive polynomial (GAP) window

The GAP window is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order K. It is continuous with continuous derivative everywhere. With the appropriate set of expansion coefficients and expansion order, the GAP window can mimic all the known window functions, reproducing accurately their spectral properties. : w_0 = a_ + \sum_^ a_\left(\frac\right)^, \quad -\frac \le n \le \frac,   where \sigma is the standard deviation of the \ sequence. Additionally, starting with a set of expansion coefficients a_ that mimics a certain known window function, the GAP window can be optimized by minimization procedures to get a new set of coefficients that improve one or more spectral properties, such as the main lobe width, side lobe attenuation, and side lobe falloff rate. Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.


Lanczos window

w = \operatorname\left(\frac - 1\right) * used in
Lanczos resampling filtering and Lanczos resampling are two applications of a mathematical formula. It can be used as a low-pass filter or used to smoothly interpolate the value of a digital signal between its samples. In the latter case it maps each sample of t ...
* for the Lanczos window, \operatorname(x) is defined as \sin(\pi x)/\pi x * also known as a ''sinc window'', because: w_0(n) = \operatorname\left(\frac\right)\, is the main lobe of a normalized
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...


Notes


Page citations


References

{{reflist, 1, refs= {{Cite book , ref=Oppenheim , last1=Oppenheim , first1=Alan V. , author-link=Alan V. Oppenheim , last2=Schafer , first2=Ronald W. , author2-link=Ronald W. Schafer , last3=Buck , first3=John R. , title=Discrete-time signal processing , year=1999 , publisher=Prentice Hall , location=Upper Saddle River, N.J. , isbn=0-13-754920-2 , edition=2nd , chapter=7.2 , page
465
€“478 , url-access=registration , url=https://archive.org/details/discretetimesign00alan   url=https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf
{{cite web , url=http://www.mathworks.com/help/signal/ref/hann.html , title=Hann (Hanning) window - MATLAB hann , website=www.mathworks.com , access-date=2020-02-12 {{cite book , first=C.Britton , last=Rorabaugh , title=DSP Primer , series=Primer series , date=October 1998 , publisher=McGraw-Hill Professional , isbn=978-0070540040 , page=196 {{cite journal , last1=Toraichi , first1=K. , last2=Kamada , first2=M. , last3=Itahashi , first3=S. , last4=Mori , first4=R. , title=Window functions represented by B-spline functions , doi=10.1109/29.17517 , journal=IEEE Transactions on Acoustics, Speech, and Signal Processing , volume=37 , pages=145–147 , year=1989 {{cite journal , ref=Harris , doi=10.1109/PROC.1978.10837 , last=Harris , first=Fredric J. , title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform , journal=Proceedings of the IEEE , volume=66 , issue=1 , pages=51–83 , date=Jan 1978 , url=http://web.mit.edu/xiphmont/Public/windows.pdf, citeseerx=10.1.1.649.9880 , bibcode=1978IEEEP..66...51H , s2cid=426548 ''The fundamental 1978 paper on FFT windows by Harris, which specified many windows and introduced key metrics used to compare them.'' {{cite journal , last=Tukey , first=J.W. , title=An introduction to the calculations of numerical spectrum analysis , journal=Spectral Analysis of Time Series , year=1967 , pages=25–46 {{cite web , url=http://www.mathworks.com/help/signal/ref/triang.html , title=Triangular window – MATLAB triang , website=www.mathworks.com , access-date=2016-04-13 {{cite web , url=https://www.mathworks.com/help/signal/ref/bohmanwin.html , title=Bohman window – R2019B , website=www.mathworks.com , access-date=2020-02-12 {{cite web , url=https://www.mathworks.com/matlabcentral/fileexchange/81658-gap-generalized-adaptive-polynomial-window-function , title=Generalized Adaptive Polynomial Window Function , website=www.mathworks.com , access-date=2020-12-12 {{cite journal , last1=Welch , first1=P. , title=The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms , doi=10.1109/TAU.1967.1161901 , journal=IEEE Transactions on Audio and Electroacoustics , volume=15 , issue=2 , pages=70–73 , year=1967, bibcode=1967ITAE...15...70W {{cite book, last1=Bosi , first1=Marina , last2=Goldberg , first2=Richard E. , title=Introduction to Digital Audio Coding and Standards , chapter=Time to Frequency Mapping Part II: The MDCT , publisher=Springer US , series=The Springer International Series in Engineering and Computer Science , volume=721 , date=2003 , location=Boston, MA , page=106 , isbn=978-1-4615-0327-9 , doi=10.1007/978-1-4615-0327-9 {{cite journal , last1=Kido , first1=Ken'iti , last2=Suzuki , first2=Hideo , last3=Ono , first3=Takahiko , last4=Fukushima , first4=Manabu , date=1998 , title=Deformation of impulse response estimates by time window in cross spectral technique , journal=Journal of the Acoustical Society of Japan (E) , volume=19 , issue=5 , pages=349–361 , doi=10.1250/ast.19.349 , doi-access=free {{cite journal , last1=Landisman , first1=M. , last2=Dziewonski , first2=A. , last3=Satô , first3=Y. , date=1969-05-01 , title=Recent Improvements in the Analysis of Surface Wave Observations , journal=Geophysical Journal International , volume=17 , issue=4 , pages=369–403 , doi=10.1111/j.1365-246X.1969.tb00246.x , bibcode=1969GeoJ...17..369L , doi-access=free {{cite book , title=Programming and Analysis for Digital Time Series Data , first1=Loren D. , last1=Enochson , first2=Robert K. , last2=Otnes , publisher=U.S. Dept. of Defense, Shock and Vibration Info. Center , year=1968 , pages=142 , url=https://books.google.com/books?id=duBQAAAAMAAJ&q=%22hamming+window%22+date:0-1970 {{cite web , url=https://users.wpi.edu/~sunar/courses/ece3311/slides/ch16.pdf , title=A digital quadrature amplitude modulation (QAM) Radio: Building a better radio , website=users.wpi.edu , access-date=2020-02-12 , page=28 {{cite web , url=https://users.wpi.edu/~sunar/courses/ece3311/slides/ch08.pdf , title=Bits to Symbols to Signals and back again , website=users.wpi.edu , access-date=2020-02-12 , page=7 {{cite book , last1 =Johnson , first1 =C.Richard, Jr , last2 =Sethares , first2 =William A. , last3 =Klein , first3 =Andrew G. , title =Software Receiver Design , publisher =Cambridge University Press , date =2011-08-18 , chapter =11 , isbn =978-1139501453   Also https://cnx.org/contents/QsVBJjB4@3.1:6R_ztzDY@4/Pulse-Shaping-and-Receive-Filtering {{cite journal , ref=Nuttall , doi =10.1109/TASSP.1981.1163506 , last =Nuttall , first =Albert H. , title =Some Windows with Very Good Sidelobe Behavior , journal =IEEE Transactions on Acoustics, Speech, and Signal Processing , volume =29 , issue =1 , pages =84–91 , date =Feb 1981 , url =https://zenodo.org/record/1280930 ''Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.'' {{Cite book , author1=Rabiner, Lawrence R. , author2=Gold, Bernard , title=Theory and application of digital signal processing , year=1975 , publisher=Prentice-Hall , location=Englewood Cliffs, N.J. , isbn=0-13-914101-4 , chapter=3.11 , pag
94
, chapter-url-access=registration , chapter-url=https://archive.org/details/theoryapplicatio00rabi/page/94
{{cite book , last1=Crochiere , first1=R.E. , last2=Rabiner , first2=L.R. , title=Multirate Digital Signal Processing , year=1983 , chapter=4.3.1 , page=144 , publisher=Prentice-Hall , location=Englewood Cliffs, NJ , isbn=0136051626 , url=https://kupdf.net/download/multirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf {{cite web , url=https://www.dsprelated.com/freebooks/sasp/Kaiser_Window.html , title=Kaiser Window , website=www.dsprelated.com , access-date=2020-04-08 , quote=The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of {{math, α as the bin number of the edge of the critically sampled window main lobe. {{cite web , url=http://mathworld.wolfram.com/BlackmanFunction.html , title=Blackman Function , last=Weisstein , first=Eric W. , website=mathworld.wolfram.com , language=en , access-date=2016-04-13 {{cite web , url=http://zone.ni.com/reference/en-XX/help/371361E-01/lvanlsconcepts/char_smoothing_windows/#Exact_Blackman , title=Characteristics of Different Smoothing Windows - NI LabVIEW 8.6 Help , website=zone.ni.com , access-date=2020-02-13 {{cite book , url=https://smile.amazon.com/Measurement-Power-Spectra-Communications-Engineering/dp/B0006AW1C4 , title=The Measurement of Power Spectra from the Point of View of Communications Engineering , last1=Blackman , first1=R.B. , last2=Tukey , first2=J.W. , date=1959-01-01 , publisher=Dover Publications , isbn=9780486605074 , page=99 {{cite techreport , first=G. , last=Heinzel , last2=Rüdiger , first2=A. , last3=Schilling , first3=R. , title=Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows , id=395068.0 , institution=Max Planck Institute (MPI) für Gravitationsphysik / Laser Interferometry & Gravitational Wave Astronomy , year=2002 , url=http://edoc.mpg.de/395068 , access-date=2013-02-10 Also available at https://pure.mpg.de/rest/items/item_152164_1/component/file_152163/content {{cite book , last=Smith , first=Steven W. , title=The Scientist and Engineer's Guide to Digital Signal Processing , url=http://www.dspguide.com/ch9/1.htm , access-date=2013-02-14 , year=2011 , publisher=California Technical Publishing , location=San Diego, California, USA {{citation , last1=Rife , first1=David C. , first2=G.A. , last2=Vincent , title=Use of the discrete Fourier transform in the measurement of frequencies and levels of tones , journal=Bell Syst. Tech. J. , volume=49 , issue=2 , year=1970 , pages=197–228 , doi=10.1002/j.1538-7305.1970.tb01766.x {{citation , last1=Andria , first1=Gregorio , first2=Mario , last2=Savino , first3=Amerigo , last3=Trotta , title=Windows and interpolation algorithms to improve electrical measurement accuracy , journal=IEEE Transactions on Instrumentation and Measurement , volume=38 , issue=4 , year=1989 , pages=856–863 , doi=10.1109/19.31004 , bibcode=1989ITIM...38..856A {{citation , last1=Schoukens , first1=Joannes , first2=Rik , last2=Pintelon , first3=Hugo , last3=Van Hamme , title=The interpolated fast Fourier transform: a comparative study , journal=IEEE Transactions on Instrumentation and Measurement, volume=41 , issue=2 , year=1992 , pages=226–232 , doi=10.1109/19.137352 , bibcode=1992ITIM...41..226S {{cite journal , last1=Starosielec , first1=S. , last2=Hägele , first2=D. , title=Discrete-time windows with minimal RMS bandwidth for given RMS temporal width , journal=Signal Processing , volume=102 , pages=240–246 , date=2014 , doi=10.1016/j.sigpro.2014.03.033 {{cite book , doi=10.1109/ICASSP.2013.6638833 , chapter=Generalized normal window for digital signal processing , title=2013 IEEE International Conference on Acoustics, Speech and Signal Processing , pages=6083–6087 , year=2013 , last1=Chakraborty , first1=Debejyo , last2=Kovvali , first2=Narayan , isbn=978-1-4799-0356-6 , s2cid=11779529 {{cite journal , doi=10.1109/78.295214 , title=The generalized exponential time-frequency distribution , journal=IEEE Transactions on Signal Processing , volume=42 , issue=5 , pages=1028–1037 , year=1994 , last1=Diethorn , first1=E.J. , bibcode=1994ITSP...42.1028D {{cite book , last =Bloomfield , first =P. , title =Fourier Analysis of Time Series: An Introduction , publisher =Wiley-Interscience , date =2000 , location =New York {{cite book , last=Tu , first=Loring W. , title=An Introduction to Manifolds , chapter=Bump Functions and Partitions of Unity , year=2008 , publisher=Springer , location=New York , isbn=978-0-387-48098-5 , pages=127–134 , doi=10.1007/978-0-387-48101-2_13 , series=Universitext {{cite journal , last1=McKechan , first1=D.J.A. , last2=Robinson , first2=C. , last3=Sathyaprakash , first3=B.S. , title=A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries , journal=Classical and Quantum Gravity , date=21 April 2010 , volume=27 , issue=8 , pages=084020 , doi=10.1088/0264-9381/27/8/084020 , arxiv=1003.2939 , bibcode = 2010CQGra..27h4020M , s2cid=21488253 {{cite book , title=System Analysis by Digital Computer , last1=Kaiser , first1=James F. , last2=Kuo , first2=Franklin F. , publisher=John Wiley and Sons , year=1966 , pages=232–235 , quote=This family of window functions was "discovered" by Kaiser in 1962 following a discussion with B. F. Logan of the Bell Telephone Laboratories. ... Another valuable property of this family ... is that they also approximate closely the prolate spheroidal wave functions of order zero. {{cite journal , last=Kaiser , first=James F. , date=Nov 1964 , title=A family of window functions having nearly ideal properties , journal=Unpublished Memorandum {{cite journal , last1=Kaiser , first1=James F. , last2=Schafer , first2=Ronald W. , doi=10.1109/TASSP.1980.1163349 , title=On the use of the I0-sinh window for spectrum analysis , journal=IEEE Transactions on Acoustics, Speech, and Signal Processing , volume=28 , pages=105–107 , year=1980 {{cite web , url=https://www.mathworks.com/help/signal/ref/kaiser.html , website=www.mathworks.com , title=Kaiser Window, R2020a , publisher=Mathworks , access-date=9 April 2020 {{cite journal , last=Kabal , first=Peter , title=Time Windows for Linear Prediction of Speech , journal=Technical Report, Dept. Elec. & Comp. Eng., McGill University , year=2009 , issue=2a , page=31 , url=http://www-mmsp.ece.mcgill.ca/Documents/Reports/2009/KabalR2009b.pdf, access-date=2 February 2014 {{cite journal , last=Streit , first=Roy , title=A two-parameter family of weights for nonrecursive digital filters and antennas , journal=Transactions of ASSP , year=1984 , volume=32 , pages=108–118 , doi=10.1109/tassp.1984.1164275 , url=https://zenodo.org/record/1280988 {{cite journal , last1=Bergen , first1=S.W.A. , first2=A. , last2=Antoniou , title=Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics , journal=EURASIP Journal on Applied Signal Processing , volume=2004 , issue=13 , pages=2053–2065 , year=2004 , doi=10.1155/S1110865704403114 , bibcode=2004EJASP2004...63B , doi-access=free Bergen, Stuart W. A. (2005). {{cite web , url=https://dspace.library.uvic.ca/bitstream/handle/1828/769/bergen_2005.pdf?sequence=1 , title=Design of the Ultraspherical Window Function and Its Applications Dissertation, University of Viktoria. {{cite book , last=Deczky , first=Andrew , chapter=Unispherical Windows , year=2001 , volume=2 , pages=85–88 , doi=10.1109/iscas.2001.921012 , isbn=978-0-7803-6685-5 , title=ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196) , s2cid=38275201 {{cite web , url=https://ccrma.stanford.edu/~jos/filters/Zero_Phase_Filters_Even_Impulse.html , title=Zero Phase Filters , website=ccrma.stanford.edu , access-date=2020-02-12 {{cite web, url=https://ccrma.stanford.edu/~jos/sasp/Bartlett_Triangular_Window.html, title=Bartlett Window, website=ccrma.stanford.edu, access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Hann_Hanning_Raised_Cosine.html , title=Hann or Hanning or Raised Cosine , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Power_of_Cosine_Window_Family.html , title=Power-of-Cosine Window Family , website=ccrma.stanford.edu , access-date=10 April 2018 {{Cite web , url=https://ccrma.stanford.edu/~jos/sasp/Hamming_Window.html , title=Hamming Window, website=ccrma.stanford.edu , access-date=2016-04-13 {{Cite web , url=https://ccrma.stanford.edu/~jos/sasp/Blackman_Harris_Window_Family.html , title=Blackman-Harris Window Family , website=ccrma.stanford.edu , access-date=2016-04-13 {{Cite web , url=https://ccrma.stanford.edu/~jos/sasp/Three_Term_Blackman_Harris_Window.html , title=Three-Term Blackman-Harris Window , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Gaussian_Window_Transform.html , title=Gaussian Window and Transform , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Matlab_Gaussian_Window.html , title=Matlab for the Gaussian Window , website=ccrma.stanford.edu , access-date=2016-04-13 , quote=Note that, on a dB scale, Gaussians are quadratic. This means that parabolic interpolation of a sampled Gaussian transform is exact. ... quadratic interpolation of spectral peaks may be more accurate on a log-magnitude scale (e.g., dB) than on a linear magnitude scale {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Quadratic_Interpolation_Spectral_Peaks.html , title=Quadratic Interpolation of Spectral Peaks , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Slepian_DPSS_Window.html , title=Slepian or DPSS Window , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html , website=ccrma.stanford.edu , title=Kaiser and DPSS Windows Compared , last=Smith , first=J.O. , date=2011 , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_Window.html , quote=Sometimes the Kaiser window is parametrized by ''α'', where ''β'' = {{pi''α''. , website=ccrma.stanford.edu , title=Kaiser Window , last=Smith , first=J.O. , date=2011 , access-date=2019-03-20 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window.html , title=Dolph-Chebyshev Window , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window_Definition.html , title=Dolph-Chebyshev Window Definition , website=ccrma.stanford.edu , access-date=2019-03-05 {{cite web , url=https://ccrma.stanford.edu/~jos/sasp/Hann_Poisson_Window.html , title=Hann-Poisson Window , website=ccrma.stanford.edu , access-date=2016-04-13 {{cite web , url =https://ccrma.stanford.edu/~jos/sasp/Poisson_Window.html , title =Poisson Window , last =Smith , first =Julius O. III , date =2011-04-23 , website =ccrma.stanford.edu , access-date =2020-02-12 {{cite web , last1 =Gade , first1 =Svend , last2 =Herlufsen , first2 =Henrik , title =Technical Review No 3-1987: Windows to FFT analysis (Part I) , publisher =Brüel & Kjær , year =1987 , url =http://www.bksv.com/doc/Bv0031.pdf , access-date =2011-11-22 {{cite journal , last1=Berry , first1=C.P.L. , last2=Gair , first2=J.R. , title=Observing the Galaxy's massive black hole with gravitational wave bursts , journal=
Monthly Notices of the Royal Astronomical Society ''Monthly Notices of the Royal Astronomical Society'' (MNRAS) is a peer-reviewed scientific journal covering research in astronomy and astrophysics. It has been in continuous existence since 1827 and publishes letters and papers reporting orig ...
, date=12 December 2012 , volume=429 , issue=1 , arxiv=1210.2778 , pages=589–612 , doi=10.1093/mnras/sts360, bibcode=2013MNRAS.429..589B , s2cid=118944979
{{cite journal , last1=Lin , first1=Yuan-Pei , last2=Vaidyanathan , first2=P.P. , title=A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks , journal=IEEE Signal Processing Letters , volume=5 , issue=6 , pages=132–134 , date=June 1998 , url=http://authors.library.caltech.edu/6891/1/LINieeespl98.pdf , access-date=2017-03-16, doi=10.1109/97.681427 , bibcode=1998ISPL....5..132L , s2cid=18159105 {{cite journal , last1=Justo , first1=J. F. , last2=Beccaro , first2=W. , title= Generalized Adaptive Polynomial Window Function , journal=IEEE Access , date=2020-10-26 , volume=8 , pages=187584–187589 , doi=10.1109/ACCESS.2020.3030903, s2cid=225050036 , doi-access=free {{citation , website =mathworks.com , title =Generalized Adaptive Polynomial Window Function , author =Wesley Beccaro , date =2020-10-31 , access-date =2020-11-02 , url =https://www.mathworks.com/matlabcentral/fileexchange/81658-gap-generalized-adaptive-polynomial-window-function?s_tid=LandingPageTabfx&s_tid=mwa_osa_a Fourier analysis Signal estimation Digital signal processing Types of functions Mathematics-related lists