List of window functions

This list also appears at Window function § A list of window functions (its original home), where the missing lead information can also be found. Therefore this article is unnecessary.

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 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.

Figure 1: Windowing a sinusoid causes spectral leakage, even if the sinusoid has an integer number of cycles within a rectangular window. The leakage is evident in the 2nd row, blue trace. It is the same amount as the red trace, which represents a slightly higher frequency that does not have an integer number of cycles. When the sinusoid is sampled and windowed, its discrete-time Fourier transform also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue sinusoid DTFT (3rd row of plots, right-hand side), those samples are the outputs of the discrete Fourier transform (DFT). The red sinusoid DTFT (4th row) has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.

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  $\{w[n]=w_{0}(n-N/2),\quad 0\leq n\leq N\}$  is symmetric, of length $N+1.$
• $\{w[n],\quad 0\leq n\leq N-1\}$  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

Rectangular window

The rectangular window (sometimes known as the boxcar or Dirichlet 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:

$w[n]=1.$

Other windows are designed to moderate these sudden changes, which reduces scalloping loss and improves dynamic range, as described above (§ Spectral analysis).

The rectangular window is the 1st order B-spline window as well as the 0th power power-of-sine window.

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

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 § Triangular window (k = 2) and the § Parzen window (k = 4). Alternative definitions sample the appropriate normalized B-spline basis functions 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.

Triangular window

Triangular window (with L = N + 1)

Triangular windows are given by:

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

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

Parzen window

Not to be confused with Kernel density estimation.

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

$w_{0}(n)\triangleq \left\{{\begin{array}{ll}1-6\left({\frac {n}{L/2}}\right)^{2}\left(1-{\frac {|n|}{L/2}}\right),&0\leq |n|\leq {\frac {L}{4}}\\2\left(1-{\frac {|n|}{L/2}}\right)^{3}&{\frac {L}{4}}<|n|\leq {\frac {L}{2}}\\\end{array}}\right\}$

$w[n]=\ w_{0}\left(n-{\tfrac {N}{2}}\right),\ 0\leq n\leq N$

Welch window

Other polynomial windows

Welch window

The Welch window consists of a single parabolic section:

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

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

Sine window

Sine window

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

The corresponding $w_{0}(n)\,$ function is a cosine without the π/2 phase offset. So the sine window is sometimes also called cosine window. As it represents half a cycle of a sinusoidal function, it is also known variably as half-sine window or half-cosine window.

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

Power-of-sine/cosine windows

These window functions have the form:

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

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

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

$w[n]=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)-a_{3}\cos \left({\frac {6\pi n}{N}}\right)+a_{4}\cos \left({\frac {8\pi n}{N}}\right)-...$

${\begin{array}{l|llll}\hline \alpha &a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\\hline 0&1\\2&0.5&0.5\\4&0.375&0.5&0.125\\6&0.3125&0.46875&0.1875&0.03125\\8&0.2734375&0.4375&0.21875&0.0625&7.8125\times 10^{-3}\\\hline \end{array}}$

Cosine-sum windows

This family is also known as generalized cosine windows.

$w[n]=\sum _{k=0}^{K}(-1)^{k}a_{k}\;\cos \left({\frac {2\pi kn}{N}}\right),\quad 0\leq n\leq N.$

(Eq.1)

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

Hann and Hamming windows

Main article: Hann function

Hann window

Hamming window, a₀ = 0.53836 and a₁ = 0.46164. The original Hamming window would have a₀ = 0.54 and a₁ = 0.46.

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

$w[n]=a_{0}-\underbrace {(1-a_{0})} _{a_{1}}\cdot \cos \left({\tfrac {2\pi n}{N}}\right),\quad 0\leq n\leq N,$

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

${\begin{aligned}w_{0}(n)\ &=w\left[n+{\tfrac {N}{2}}\right]\\&=a_{0}+a_{1}\cdot \cos \left({\tfrac {2\pi n}{N}}\right),\quad -{\tfrac {N}{2}}\leq n\leq {\tfrac {N}{2}}.\end{aligned}}$

Setting  $a_{0}=0.5$  produces a Hann window:

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

named after Julius von Hann, and sometimes erroneously referred to as Hanning, presumably due to its linguistic and formulaic similarities to the Hamming window. It is also known as raised cosine, because the zero-phase version, $w_{0}(n),$ is one lobe of an elevated cosine function.

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

Setting  $a_{0}$  to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by Richard W. Hamming. That 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. The Hamming window is often called the Hamming blip when used for pulse shaping.

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

Blackman window

Blackman window; α = 0.16

Blackman windows are defined as:

$w[n]=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)$

$a_{0}={\frac {1-\alpha }{2}};\quad a_{1}={\frac {1}{2}};\quad a_{2}={\frac {\alpha }{2}}.$

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, with a₀ = 7938/18608 ≈ 0.42659, a₁ = 9240/18608 ≈ 0.49656, and a₂ = 1430/18608 ≈ 0.076849. These exact values place zeros at the third and fourth sidelobes, but result in a discontinuity at the edges and a 6 dB/oct fall-off. The truncated coefficients do not null the sidelobes as well, but have an improved 18 dB/oct fall-off.

Nuttall window, continuous first derivative

Nuttall window, continuous first derivative

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

$w[n]=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)-a_{3}\cos \left({\frac {6\pi n}{N}}\right)$

$a_{0}=0.355768;\quad a_{1}=0.487396;\quad a_{2}=0.144232;\quad a_{3}=0.012604.$

Blackman–Nuttall window

Blackman–Nuttall window

$w[n]=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)-a_{3}\cos \left({\frac {6\pi n}{N}}\right)$

$a_{0}=0.3635819;\quad a_{1}=0.4891775;\quad a_{2}=0.1365995;\quad a_{3}=0.0106411.$

Blackman–Harris window

Blackman–Harris window

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

$w[n]=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)-a_{3}\cos \left({\frac {6\pi n}{N}}\right)$

$a_{0}=0.35875;\quad a_{1}=0.48829;\quad a_{2}=0.14128;\quad a_{3}=0.01168.$

Flat top window

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 § Noise bandwidth.

Flat top windows can be designed using low-pass filter design methods, or they may be of the usual cosine-sum variety:

${\begin{aligned}w[n]=a_{0}&{}-a_{1}\cos \left({\frac {2\pi n}{N}}\right)+a_{2}\cos \left({\frac {4\pi n}{N}}\right)\\&{}-a_{3}\cos \left({\frac {6\pi n}{N}}\right)+a_{4}\cos \left({\frac {8\pi n}{N}}\right).\end{aligned}}$

The 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 Eq.1, reflect that custom.

Class I, Order 1 (K = 1):  $a_{0}=1;\quad a_{1}=1$       Functionally equivalent to the Hann window.

Class I, Order 2 (K = 2):  $a_{0}=1;\quad a_{1}={\tfrac {4}{3}};\quad a_{2}={\tfrac {1}{3}}$

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

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

Class III is a compromise for which order K = 2 resembles the § Blackman window.

Adjustable windows

Gaussian window

Gaussian window, σ = 0.4

The Fourier transform of a Gaussian is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either be truncated at the ends of the window, or itself windowed with another zero-ended window.

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

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

$\sigma \leq \;0.5\,$

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

Confined Gaussian window, σ_t = 0.1

Confined Gaussian window

The confined Gaussian window yields the smallest possible root mean square frequency width σ_ω for a given temporal width  (N + 1) σ_t. These windows optimize the RMS time-frequency bandwidth products. They are computed as the minimum eigenvectors of a parameter-dependent matrix. The confined Gaussian window family contains the § Sine window and the § Gaussian window in the limiting cases of large and small σ_t, respectively.

Approximate confined Gaussian window, σ_t = 0.1

Approximate confined Gaussian window

Defining  L ≜ N + 1,  a confined Gaussian window of temporal width  L × σ_t  is well approximated by:

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

where $G$ is a Gaussian function:

$G(x)=\exp \left(-\left({\cfrac {x-{\frac {N}{2}}}{2L\sigma _{t}}}\right)^{2}\right)$

The standard deviation of the approximate window is asymptotically equal (i.e. large values of N) to  L × σ_t  for  σ_t < 0.14.

Generalized normal window

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

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

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 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 § Tukey window, this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window. See also for a study on time-frequency representation of this window (or function).

Tukey window

Tukey window, α = 0.5

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).

$\left.{\begin{array}{lll}w[n]={\frac {1}{2}}\left[1-\cos \left({\frac {2\pi n}{\alpha N}}\right)\right],\quad &0\leq n<{\frac {\alpha N}{2}}\\w[n]=1,\quad &{\frac {\alpha N}{2}}\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}$  

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

Planck-taper window

Planck-taper window, ε = 0.1

The so-called "Planck-taper" window is a bump function that has been widely used in the theory of partitions of unity in manifolds. It is smooth (a $C^{\infty }$ function) everywhere, but is exactly zero outside of a compact region, exactly one over an interval within that region, and varies smoothly and monotonically between those limits. Its use as a window function in signal processing was first suggested in the context of gravitational-wave astronomy, inspired by the Planck distribution. It is defined as a piecewise function:

$\left.{\begin{array}{lll}w[0]=0,\\w[n]=\left(1+\exp \left({\frac {\varepsilon N}{n}}-{\frac {\varepsilon N}{\varepsilon N-n}}\right)\right)^{-1},\quad &1\leq n<\varepsilon N\\w[n]=1,\quad &\varepsilon N\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}$

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.

DPSS window, α = 2

DPSS window, α = 3

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

Kaiser window, α = 2

Kaiser window, α = 3

Kaiser window

Main article: Kaiser window

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

$w[n]={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}-1\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad 0\leq n\leq N$    

$w_{0}(n)={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad -N/2\leq n\leq N/2$

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  $2{\sqrt {1+\alpha ^{2}}},$  in units of DFT bins,  and a typical value of $\alpha$ is 3.

Dolph–Chebyshev window

Dolph–Chebyshev window, α = 5

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

The zero-phase Dolph–Chebyshev window function $w_{0}[n]$ is usually defined in terms of its real-valued discrete Fourier transform, $W_{0}[k]$:

$W_{0}(k)={\frac {T_{N}{\big (}\beta \cos \left({\frac {\pi k}{N+1}}\right){\big )}}{T_{N}(\beta )}}={\frac {T_{N}{\big (}\beta \cos \left({\frac {\pi k}{N+1}}\right){\big )}}{10^{\alpha }}},\ 0\leq k\leq N.$

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

$T_{n}(x)={\begin{cases}\cos \!{\big (}n\cos ^{-1}(x){\big )}&{\text{if }}-1\leq x\leq 1\\\cosh \!{\big (}n\cosh ^{-1}(x){\big )}&{\text{if }}x\geq 1\\(-1)^{n}\cosh \!{\big (}n\cosh ^{-1}(-x){\big )}&{\text{if }}x\leq -1,\end{cases}}$

and

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

is the unique positive real solution to $T_{N}(\beta )=10^{\alpha }$, where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.

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

$w_{0}(n)={\frac {1}{N+1}}\sum _{k=0}^{N}W_{0}(k)\cdot e^{i2\pi kn/(N+1)},\ -N/2\leq n\leq N/2.$

The lagged version of the window can be obtained by:

$w[n]=w_{0}\left(n-{\frac {N}{2}}\right),\quad 0\leq n\leq N,$

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

${\begin{aligned}w_{0}\left(n-{\frac {N}{2}}\right)={\frac {1}{N+1}}\sum _{k=0}^{N}W_{0}(k)\cdot e^{\frac {i2\pi k(n-N/2)}{N+1}}={\frac {1}{N+1}}\sum _{k=0}^{N}\left[\left(-e^{\frac {i\pi }{N+1}}\right)^{k}\cdot W_{0}(k)\right]e^{\frac {i2\pi kn}{N+1}},\end{aligned}}$

which is an inverse DFT of  $\left(-e^{\frac {i\pi }{N+1}}\right)^{k}\cdot W_{0}(k).$

Variations:

• Due to the equiripple condition, the time-domain window has discontinuities at the edges. An approximation that avoids them, by allowing the equiripples to drop off at the edges, is a Taylor window.
• An alternative to the inverse DFT definition is also available.[1].

Ultraspherical window

The Ultraspherical window's μ parameter determines whether its Fourier transform's side-lobe amplitudes decrease, are level, or (shown here) increase with frequency.

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

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

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

$w[n]={\frac {1}{N+1}}\left[C_{N}^{\mu }(x_{0})+\sum _{k=1}^{\frac {N}{2}}C_{N}^{\mu }\left(x_{0}\cos {\frac {k\pi }{N+1}}\right)\cos {\frac {2n\pi k}{N+1}}\right]$

where $C_{N}^{\mu }$ 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 windows respectively. See here for illustration of Ultraspherical windows with varied parametrization.

Exponential or Poisson window

Exponential window, τ = N/2

Exponential window, τ = (N/2)/(60/8.69)

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

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

where τ is the time constant of the function. The exponential function decays as e ≃ 2.71828 or approximately 8.69 dB per time constant. This means that for a targeted decay of D dB over half of the window length, the time constant τ is given by

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

Hybrid windows

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

Bartlett–Hann window

Bartlett–Hann window

$w[n]=a_{0}-a_{1}\left|{\frac {n}{N}}-{\frac {1}{2}}\right|-a_{2}\cos \left({\frac {2\pi n}{N}}\right)$

$a_{0}=0.62;\quad a_{1}=0.48;\quad a_{2}=0.38\,$

Planck–Bessel window

Planck–Bessel window, ε = 0.1, α = 4.45

A § Planck-taper window multiplied by a Kaiser window which is defined in terms of a modified Bessel function. This hybrid window function was introduced to decrease the peak side-lobe level of the Planck-taper window while still exploiting its good asymptotic decay. It has two tunable parameters, ε from the Planck-taper and α from the Kaiser window, so it can be adjusted to fit the requirements of a given signal.

Hann–Poisson window

Hann–Poisson window, α = 2

A Hann window 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 algorithms like Newton's method. The Hann–Poisson window is defined by:

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

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

Other windows

GAP window (GAP optimized Nuttall window)

Generalized adaptive polynomial (GAP) window

The GAP window is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order $K$. It is continuous with continuous derivative everywhere. With the appropriate set of expansion coefficients and expansion order, the GAP window can mimic all the known window functions, reproducing accurately their spectral properties.

$w_{0}[n]=a_{0}+\sum _{k=1}^{K}a_{2k}\left({\frac {n}{\sigma }}\right)^{2k},\quad -{\frac {N}{2}}\leq n\leq {\frac {N}{2}},$  

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

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

Sinc or Lanczos window

Lanczos window

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

• used in Lanczos resampling
• for the Lanczos window, $\operatorname {sinc} (x)$ is defined as $\sin(\pi x)/\pi x$
• also known as a sinc window, because:
  $$w_{0}(n)=\operatorname {sinc} \left({\frac {2n}{N}}\right)\,$$
  is the main lobe of a normalized sinc function