Window function

The Fourier transform 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.

Figure 2: Windowing a sinusoid causes spectral leakage. The same amount of leakage occurs whether there are an integer (blue) or non-integer (red) number of cycles within the window (rows 1 and 2). 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 DTFT, those samples are the outputs of the discrete Fourier transform (DFT). The red DTFT has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.

Windowing of a simple waveform like cos(ωt) causes its Fourier transform to develop non-zero values (commonly called spectral leakage) at frequencies other than ω. The leakage tends to be worst (highest) near ω and least at frequencies farthest from ω.

If the waveform under analysis comprises two sinusoids of different frequencies, leakage can interfere with our ability to distinguish them spectrally. Possible types of interference are often broken down into two opposing classes as follows: If the component frequencies are dissimilar and one component is weaker, then leakage from the stronger component can obscure the weaker one's presence. But if the frequencies are too similar, leakage can render them unresolvable even when the sinusoids are of equal strength. Windows that are effective against the first type of interference, namely where components have dissimilar frequencies and amplitudes, are called high dynamic range. Conversely, windows that can distinguish components with similar frequencies and amplitudes are called high resolution.

The rectangular window is an example of a window that is high resolution but low dynamic range, meaning it is good for distinguishing components of similar amplitude even when the frequencies are also close, but poor at distinguishing components of different amplitude even when the frequencies are far away. High-resolution, low-dynamic-range windows such as the rectangular window also have the property of high sensitivity, which is the ability to reveal relatively weak sinusoids in the presence of additive random noise. That is because the noise produces a stronger response with high-dynamic-range windows than with high-resolution windows.

At the other extreme of the range of window types are windows with high dynamic range but low resolution and sensitivity. High-dynamic-range windows are most often justified in wideband applications, where the spectrum being analyzed is expected to contain many different components of various amplitudes.

In between the extremes are moderate windows, such as Hamming and Hann. They are commonly used in narrowband applications, such as the spectrum of a telephone channel.

In summary, spectral analysis involves a trade-off between resolving comparable strength components with similar frequencies (high resolution / sensitivity) and resolving disparate strength components with dissimilar frequencies (high dynamic range). That trade-off occurs when the window function is chosen.[5]^{:p. 90}

When the input waveform is time-sampled, instead of continuous, the analysis is usually done by applying a window function and then a discrete Fourier transform (DFT). But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum. Figure 2, row 3 shows a DTFT for a rectangularly-windowed sinusoid. The actual frequency of the sinusoid is indicated as "13" on the horizontal axis. Everything else is leakage, exaggerated by the use of a logarithmic presentation. The unit of frequency is "DFT bins"; that is, the integer values on the frequency axis correspond to the frequencies sampled by the DFT. So the figure depicts a case where the actual frequency of the sinusoid coincides with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. In row 4, it misses the maximum value by ½ bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak). For a known frequency, such as a musical note or a sinusoidal test signal, matching the frequency to a DFT bin can be prearranged by choices of a sampling rate and a window length that results in an integer number of cycles within the window.

Figure 3: This figure compares the processing losses of three window functions for sinusoidal inputs, with both minimum and maximum scalloping loss.

The concepts of resolution and dynamic range tend to be somewhat subjective, depending on what the user is actually trying to do. But they also tend to be highly correlated with the total leakage, which is quantifiable. It is usually expressed as an equivalent bandwidth, B. It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B.[upper-alpha 1][6] The more the leakage, the greater the bandwidth. It is sometimes called noise equivalent bandwidth or equivalent noise bandwidth, because it is proportional to the average power that will be registered by each DFT bin when the input signal contains a random noise component (or is just random noise). A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect. The height of the noise floor is proportional to B. So two different window functions can produce different noise floors.

In signal processing, operations are chosen to improve some aspect of quality of a signal by exploiting the differences between the signal and the corrupting influences. When the signal is a sinusoid corrupted by additive random noise, spectral analysis distributes the signal and noise components differently, often making it easier to detect the signal's presence or measure certain characteristics, such as amplitude and frequency. Effectively, the signal to noise ratio (SNR) is improved by distributing the noise uniformly, while concentrating most of the sinusoid's energy around one frequency. Processing gain is a term often used to describe an SNR improvement. The processing gain of spectral analysis depends on the window function, both its noise bandwidth (B) and its potential scalloping loss. These effects partially offset, because windows with the least scalloping naturally have the most leakage.

Figure 3 depicts the effects of three different window functions on the same data set, comprising two equal strength sinusoids in additive noise. The frequencies of the sinusoids are chosen such that one encounters no scalloping and the other encounters maximum scalloping. Both sinusoids suffer less SNR loss under the Hann window than under the Blackman–Harris window. In general (as mentioned earlier), this is a deterrent to using high-dynamic-range windows in low-dynamic-range applications.

Figure 4: Two different ways to generate an 8-point Gaussian window sequence (σ = 0.4) for spectral analysis applications. MATLAB calls them "symmetric" and "periodic". The latter is also historically called DFT-even.

Figure 5: Spectral leakage characteristics of the functions in Figure 4

The formulas provided in this article produce discrete sequences, as if a continuous window function has been "sampled". (See an example at Kaiser window.) Window sequences for spectral analysis are either symmetric or 1-sample short of symmetric (called periodic,[7][8] DFT-even, or DFT-symmetric[9]^{:p. 52}). For instance, a true symmetric sequence, with its maximum at a single center-point, is generated by the MATLAB function hann(9,'symmetric'). Deleting the last sample produces a sequence identical to hann(8,'periodic'). Similarly, the sequence hann(8,'symmetric') has two equal center-points.[10]

Some functions have one or two zero-valued end-points, which are unnecessary in most applications. Deleting a zero-valued end-point has no effect on its DTFT (spectral leakage). But the function designed for N+1 or N+2 samples, in anticipation of deleting one or both end points, typically has a slightly narrower main lobe, slightly higher sidelobes, and a slightly smaller noise-bandwidth.[11]

The predecessor of the DFT is the finite Fourier transform, and window functions were "always an odd number of points and exhibit even symmetry about the origin".[9]^{:p. 52} In that case, the DTFT is entirely real-valued. When the same sequence is shifted into a DFT data window, [0 ≤ n ≤ N], the DTFT becomes complex-valued, except at frequencies spaced at regular intervals of 1/N.[lower-alpha 1] Thus, when sampled by an N-length DFT (see periodic summation), the samples (called DFT coefficients) are still real-valued.  Because of the periodic summation the last sample of the window function, w[N], is incorporated in the n = 0 term of the DFT:  exp{−i2πk0/N} · (w[0] + w[N]) = w[0] + w[N],  which is real-valued for all values of k (all DFT coefficients). So when the last sample of a symmetric sequence is truncated (w[N] = 0), the imaginary components remain zero.[upper-alpha 2] It does affect the DTFT (spectral leakage), but usually by a negligible amount (unless N is small, e.g. ≤ 20).[12][upper-alpha 3]

When windows are multiplicatively applied to actual data, the sequence usually lacks any symmetry, and the DFT is generally not real-valued. Despite this caveat, many authors reflexively assume DFT-symmetric windows.[9][13][14][15][16][17][lower-alpha 2] So it is worth noting that there is no performance advantage when applied to time domain data, which is the customary application. The advantage of real-valued DFT coefficients is realized in certain esoteric applications[upper-alpha 4] where windowing is achieved by means of convolution between the DFT coefficients and an unwindowed DFT of the data.[18][9]^{:p. 62}[5]^{:p. 85} In those applications, DFT-symmetric windows (even or odd length) from the Cosine-sum family are preferred, because most of their DFT coefficients are zero-valued, making the convolution very efficient.[upper-alpha 5][5]^{:p. 85}

Windows are sometimes used in the design of digital filters, in particular to convert an "ideal" impulse response of infinite duration, such as a sinc function, to a finite impulse response (FIR) filter design. That is called the window method.[19][20][21]

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

When analyzing a transient signal in modal analysis, such as an impulse, a shock response, a sine burst, a chirp burst, or noise burst, where the energy vs time distribution is extremely uneven, the rectangular window may be most appropriate. For instance, when most of the energy is located at the beginning of the recording, a non-rectangular window attenuates most of the energy, degrading the signal-to-noise ratio.[22]

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 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.

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:

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

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.

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).[25] Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows. A kth order B-spline basis function is a piece-wise polynomial function of degree k−1 that is obtained by k-fold self-convolution of the rectangular function.

Triangular window (with L = N + 1)

Triangular windows are given by:

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

where L can be N,[26] N + 1,[9] [27][28] or N + 2.[29]  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  L ≜ N + 1,  the Parzen window, also known as the de la Vallée Poussin window,[9] is the 4th order B-spline window given by:

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

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

Welch window

The Welch window consists of a single parabolic section:

${\displaystyle w[n]=1-\left({\frac {n-{\frac {N}{2}}}{\frac {N}{2}}}\right)^{2},\quad 0\leq n\leq N.}$[29]

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

Sine window

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

The corresponding ${\displaystyle w_{0}(n)\,}$ function is a cosine without the π/2 phase offset. So the sine window[30] is sometimes also called cosine window.[9] As it represents half a cycle of a sinusoidal function, it is also known variably as half-sine window[31] or half-cosine window.[32]

The autocorrelation of a sine window produces a function known as the Bohman window.[33]

These window functions have the form:[34]

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

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

This family is also known as generalized cosine windows.

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 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:

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

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

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

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

${\displaystyle w[n]=0.5\;\left[1-\cos \left({\frac {2\pi n}{N}}\right)\right]=\sin ^{2}\left({\frac {\pi n}{N}}\right),}$[7]

named after Julius von Hann, and sometimes 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, ${\displaystyle w_{0}(n),}$ 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.[35]

Setting  ${\displaystyle a_{0}}$  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.[9][36][37] The Hamming window is often called the Hamming blip when used for pulse shaping.[38][39][40]

Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,[9] to a nearly equiripple condition.[37] In the equiripple sense, the optimal values for the coefficients are a₀ = 0.53836 and a₁ = 0.46164.[37][5]

Hamming window is used for Audio Spectrum effect in Adobe After Effects.

Blackman window; α=0.16

Blackman windows are defined as:

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

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

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,[41] with a₀ = 7938/18608 ≈ 0.42659, a₁ = 9240/18608 ≈ 0.49656, and a₂ = 1430/18608 ≈ 0.076849.[42] These exact values place zeros at the third and fourth sidelobes,[9] 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.[9][43]

Nuttall window, continuous first derivative

The continuous form of Nuttall window, ${\displaystyle w_{0}(x),}$ 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.

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

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

Blackman–Nuttall window

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

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

Blackman–Harris window

A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels[44][45]

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

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

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.[13][46] 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,[46] or they may be of the usual cosine-sum variety:

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

The Matlab variant has these coefficients:

${\displaystyle a_{0}=0.21557895;\quad a_{1}=0.41663158;\quad a_{2}=0.277263158;\quad a_{3}=0.083578947;\quad a_{4}=0.006947368.}$

Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.[13]

Rife–Vincent windows[47] 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):  ${\displaystyle a_{0}=1;\quad a_{1}=1}$       Functionally equivalent to the Hann window.

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

Class I is defined by minimizing the high-order sidelobe amplitude. Coefficients for orders up to K=4 are tabulated.[48]

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.[48][49]

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.[50]

Since the log of a Gaussian produces a parabola, this can be used for nearly exact quadratic interpolation in frequency estimation.[51][50][52]

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

${\displaystyle \sigma \leq \;0.5\,}$

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

Confined Gaussian window, σ_t = 0.1

The confined Gaussian window yields the smallest possible root mean square frequency width σ_ω for a given temporal width  (N + 1) σ_t.[53] 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

Defining  L ≜ N + 1,  a confined Gaussian window of temporal width  L × σ_t  is well approximated by:[53]

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

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

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

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.[53]

A more generalized version of the Gaussian window is the generalized normal window.[54] Retaining the notation from the Gaussian window above, we can represent this window as

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

for any even ${\displaystyle p}$. At ${\displaystyle p=2}$, this is a Gaussian window and as ${\displaystyle p}$ approaches ${\displaystyle \infty }$, this approximates to a rectangular window. The Fourier transform of this window does not exist in a closed form for a general ${\displaystyle p}$. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the § Tukey window, this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window. See also [55] for a study on time-frequency representation of this window (or function).

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 samples) that is convolved with a rectangular window of width N(1 − α/2).

${\displaystyle \left.{\begin{array}{lll}w[n]={\frac {1}{2}}\left[1-\cos \left({\frac {2\pi n}{\alpha N}}\right)\right],\quad &0\leq n<{\frac {\alpha N}{2}}\\w[n]=1,\quad &{\frac {\alpha N}{2}}\leq n\leq {\frac {N}{2}}\\w[N-n]=w[n],\quad &0\leq n\leq {\frac {N}{2}}\end{array}}\right\}}$  [56][upper-alpha 7][upper-alpha 8]

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

Planck-taper window, ε = 0.1

The so-called "Planck-taper" window is a bump function that has been widely used[57] in the theory of partitions of unity in manifolds. It is smooth (a ${\displaystyle C^{\infty }}$ 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.[58] It is defined as a piecewise function:

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

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

The DPSS (discrete prolate spheroidal sequence) or Slepian window maximizes the energy concentration in the main lobe,[59] 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.[60]

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

The Kaiser, or Kaiser–Bessel, window is a simple approximation of the DPSS window using Bessel functions, discovered by James Kaiser.[61][62]

${\displaystyle w[n]={\frac {I_{0}\left(\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}-1\right)^{2}}}\right)}{I_{0}(\pi \alpha )}},\quad 0\leq n\leq N}$    [upper-alpha 9][9]^{:p. 73}

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

where ${\displaystyle I_{0}}$ is the zero-th order modified Bessel function of the first kind. Variable parameter ${\displaystyle \alpha }$ determines the tradeoff between main lobe width and side lobe levels of the spectral leakage pattern.  The main lobe width, in between the nulls, is given by  ${\displaystyle 2{\sqrt {1+\alpha ^{2}}},}$  in units of DFT bins,[69]  and a typical value of ${\displaystyle \alpha }$ is 3.

Dolph–Chebyshev window, α = 5

Minimizes the Chebyshev norm of the side-lobes for a given main lobe width.[70]

The zero-phase Dolph–Chebyshev window function ${\displaystyle w_{0}[n]}$ is usually defined in terms of its real-valued discrete Fourier transform, ${\displaystyle W_{0}[k]}$:[71]

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

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

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

and

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

is the unique positive real solution to ${\displaystyle T_{N}(\beta )=10^{\alpha }}$, where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.[70]

The window function can be calculated from W₀(k) by an inverse discrete Fourier transform (DFT):[70]

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

The lagged version of the window can be obtained by:

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

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

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

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

Variations:

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

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[72] and has application in antenna array design,[73] non-recursive filter design,[72] and spectrum analysis.[74]

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.[74][75]

The window can be expressed in the time-domain as follows:[74]

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

where ${\displaystyle C_{N}^{\mu }}$ is the Ultraspherical polynomial of degree N, and ${\displaystyle x_{0}}$ and ${\displaystyle \mu }$ control the side-lobe patterns.[74]

Certain specific values of ${\displaystyle \mu }$ yield other well-known windows: ${\displaystyle \mu =0}$ and ${\displaystyle \mu =1}$ give the Dolph–Chebyshev and Saramäki windows respectively.[72] See here for illustration of Ultraspherical windows with varied parametrization.

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 [76]). It is defined by

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

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

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

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

Bartlett–Hann window

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

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

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.[78] 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, α = 2

A Hann window multiplied by a Poisson window, which has no side-lobes, in the sense that its Fourier transform drops off forever away from the main lobe. It can thus be used in hill climbing algorithms like Newton's method.[79] The Hann–Poisson window is defined by:

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

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

The GAP window[80] is a family of adjustable window functions that are based on a symmetrical polynomial expansion of order ${\displaystyle K}$. It is continuous with continuous derivative everywhere. With the appropriate set of expansion coefficients and expansion order, the GAP window can mimic all the known window functions, reproducing accurately their spectral properties.

${\displaystyle w_{0}[n]=a_{0}+\sum _{k=1}^{K}a_{2k}\left({\frac {n}{\sigma }}\right)^{2k},\quad -{\frac {N}{2}}\leq n\leq {\frac {N}{2}},}$  [81]

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

Additionally, starting with a set of expansion coefficients ${\displaystyle a_{2k}}$ that mimics a certain known window function, the GAP window can be optimized by minimization procedures to get a new set of coefficients that improve one or more spectral properties, such as the main lobe width, side lobe attenuation, and side lobe falloff rate.[82] Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.

Sinc or Lanczos window

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

• used in Lanczos resampling
• for the Lanczos window, ${\displaystyle \operatorname {sinc} (x)}$ is defined as ${\displaystyle \sin(\pi x)/\pi x}$
• also known as a sinc window, because:

${\displaystyle w_{0}(n)=\operatorname {sinc} \left({\frac {2n}{N}}\right)\,}$ is the main lobe of a normalized sinc function