Window function

The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra.

The duration of the segments is determined in each application by requirements like time and frequency resolution.

But that method also changes the frequency content of the signal by an effect called spectral leakage.

Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application.

In typical applications, the window functions used are non-negative, smooth, "bell-shaped" curves.

A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.

Any window (including rectangular) affects the spectral estimate computed by this method.

[7][8][9] 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.

See Welch method of power spectral analysis and the modified discrete cosine transform.

[13] The separable forms of all other window functions have corners that depend on the choice of the coordinate axes.

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

[19] Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows.

For even-integer values of α these functions can also be expressed in cosine-sum form: This family is also known as generalized cosine windows.

The customary cosine-sum windows for case K = 1 have the form which is easily (and often) confused with its zero-phase version: Setting

[33][34][35] Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,[16] to a nearly equiripple condition.

[38] These exact values place zeros at the third and fourth sidelobes,[16] but result in a discontinuity at the edges and a 6 dB/oct fall-off.

A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels[40][41] A flat top window is a partially negative-valued window that has minimal scalloping loss in the frequency domain.

Flat top windows can be designed using low-pass filter design methods,[42] or they may be of the usual cosine-sum variety: The Matlab variant has these coefficients: Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.

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

The so-called "Planck-taper" window is a bump function that has been widely used[53] in the theory of partitions of unity in manifolds.

Its use as a window function in signal processing was first suggested in the context of gravitational-wave astronomy, inspired by the Planck distribution.

[54] It is defined as a piecewise function: 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,[55] and is used in multitaper spectral analysis, which averages out noise in the spectrum and reduces information loss at the edges of the window.

:[67] Tn(x) is the n-th Chebyshev polynomial of the first kind evaluated in x, which can be computed using and is the unique positive real solution to

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.

[75] 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.

it has no side-lobes, as its Fourier transform drops off forever away from the main lobe without local minima.

[78] Therefore, a GAP window function can be developed with designed spectral properties depending on the specific application.

However, there are window functions that are asymmetric, such as the Gamma distribution used in FIR implementations of Gammatone filters.

These asymmetries are used to reduce the delay when using large window sizes, or to emphasize the initial transient of a decaying pulse.

A popular window function, the Hann window . Most popular window functions are similar bell-shaped curves.
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 (DTFT) also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue DTFT, those samples are the outputs of the discrete Fourier transform (DFT). The red DTFT has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.
Rectangular window
Triangular window (with L = N + 1)
Parzen window
Welch window
Sine window
Hann window
Hamming window, a 0 = 0.53836 and a 1 = 0.46164. The original Hamming window would have a 0 = 0.54 and a 1 = 0.46.
Blackman window; α = 0.16
Nuttall window, continuous first derivative
Blackman–Nuttall window
Blackman–Harris window
Flat-top window
Gaussian window, σ = 0.4
Confined Gaussian window, σ t = 0.1
Approximate confined Gaussian window, σ t = 0.1
Tukey window, α = 0.5
Planck-taper window, ε = 0.25
DPSS window, α = 2
DPSS window, α = 3
Kaiser window, α = 2
Kaiser window, α = 3
Dolph–Chebyshev window, α = 5
The Ultraspherical window's μ parameter determines whether its Fourier transform's side-lobe amplitudes decrease, are level, or (shown here) increase with frequency.
Exponential window, τ = N /2
Exponential window, τ = ( N /2)/(60/8.69)
Bartlett–Hann window
Planck–Bessel window, ε = 0.1, α = 4.45
Hann–Poisson window, α = 2
GAP window (GAP optimized Nuttall window)
Sinc or Lanczos window