FFT 各种窗函数说明

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For the term used in SQL statements, see .

A popular window function, the . Most popular window functions are similar bell-shaped curves.

In  and , a window function (also known as an apodization function or tapering function) is a  that is zero-valued outside of some chosen , normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, , not segmentation, is the main purpose of window functions.

The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution. But that method also changes the frequency content of the signal by an effect called . Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.

In typical applications, the window functions used are non-negative, smooth, "bell-shaped" curves. Rectangle, triangle, and other functions can also be used. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is , and, more specifically, that the function goes sufficiently rapidly toward zero.

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Window functions are used in spectral /modification/, the design of  filters, as well as  and  design.

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.

Spectral analysis[]

Main article: 

The  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.

Filter design[]

Main article: 

Windows are sometimes used in the design of , in particular to convert an "ideal" impulse response of infinite duration, such as a , to a  (FIR) filter design. That is called the .

Statistics and curve fitting[edit]

Main article: 

Window functions are sometimes used in the field of  to restrict the set of data being analyzed to a range near a given point, with a  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 .

Rectangular window applications[]

Analysis of transients[]

When analyzing a transient signal in , 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.

Harmonic analysis[]

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.

Overlapping windows[]

When the length of a data set to be transformed is larger than necessary to provide the desired frequency resolution, a common practice is to subdivide it into smaller sets and window them individually. To mitigate the "loss" at the edges of the window, the individual sets may overlap in time. See  of power spectral analysis and the .

Two-dimensional windows[]

Main article: 

Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform. They can be constructed from one-dimensional windows in either of two forms. The separable form, ?(?,?)=?(?)?(?) is trivial to compute. The  form, ?(?,?)=?(?), which involves the radius ?=(???/2)2+(???/2)2, is , independent on the orientation of the coordinate axes. Only the  function is both separable and isotropic. The separable forms of all other window functions have corners that depend on the choice of the coordinate axes. The isotropy/ of a two-dimensional window function is shared by its two-dimensional Fourier transform. The difference between the separable and radial forms is akin to the result of  from rectangular vs. circular apertures, which can be visualized in terms of the product of two  vs. an , respectively.

A list of window functions[]

Conventions:

• ?0(?) is a zero-phase function (symmetrical about ?=0), continuous for ?∈[??/2,?/2], where ? is a positive integer (even or odd).
• The sequence  {?[?]=?0(???/2),0≤?≤?}  is symmetric, of length ?+1.
• {?[?],0≤?≤??1}  is DFT-symmetric, of length ?.

• The parameter B displayed on each spectral plot is the function's  metric, in units of DFT bins.^{:?p.56 eq.(16)?}

The sparse sampling of a DTFT (such as the DFTs in Fig 2) only reveals the leakage into the DFT bins from a sinusoid whose frequency is also an integer DFT bin. The unseen sidelobes reveal the leakage to expect from sinusoids at other frequencies. 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  or  window) is the simplest window, equivalent to replacing all but N consecutive values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off:

?[?]=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 .

The rectangular window provides the minimum mean square error estimate of 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   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 .

Triangular window[]

Triangular window (with L = N + 1)

Triangular windows are given by:

?[?]=1?|???2?2|,0≤?≤?

where L can be N, N + 1,  or N + 2.  The first one is also known as  window or  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 .

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:

?[?]= ?0(???2), 0≤?≤?

Other polynomial windows[]

Welch window[]

The Welch window consists of a single  section:

?[?]=1?(???2?2)2,0≤?≤?.

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

Sine window[]

Sine window

?[?]=sin?(???)=cos?(?????2),0≤?≤?.

The corresponding ?0(?) 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  of a sine window produces a function known as the Bohman window.

Power-of-sine/cosine windows[]

These window functions have the form:

?[?]=sin??(???)=cos??(?????2),0≤?≤?.

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

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

?[?]=?0??1cos?(2???)+?2cos?(4???)??3cos?(6???)+?4cos?(8???)?... ??0?1?2?3?40120.50.540.3750.50.12560.31250.468750.18750.0312580.27343750.43750.218750.06257.8125×10?3

Cosine-sum windows[]

This family is also known as .

?[?]=∑?=0?(?1)???cos?(2????),0≤?≤?.

(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 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:

?[?]=?0?(1??0)??1?cos?(2???),0≤?≤?,

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

?0(?) =?[?+?2]=?0+?1?cos?(2???),??2≤?≤?2.

Setting  ?0=0.5  produces a Hann window:

?[?]=0.5[1?cos?(2???)]=sin2?(???),

named after , 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, ?0(?), is one lobe of an elevated cosine function.

This function is a member of both the  and  families. Unlike the , the end points of the Hann window just touch zero. The resulting  roll off at about 18 dB per octave.

Setting  ?0  to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by . 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 .

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:

?[?]=?0??1cos?(2???)+?2cos?(4???) ?0=1??2;?1=12;?2=?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, ?0(?), and its first  are continuous everywhere, like the . 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.

?[?]=?0??1cos?(2???)+?2cos?(4???)??3cos?(6???) ?0=0.355768;?1=0.487396;?2=0.144232;?3=0.012604.

Blackman–Nuttall window[]

Blackman–Nuttall window

?[?]=?0??1cos?(2???)+?2cos?(4???)??3cos?(6???) ?0=0.3635819;?1=0.4891775;?2=0.1365995;?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

?[?]=?0??1cos?(2???)+?2cos?(4???)??3cos?(6???) ?0=0.35875;?1=0.48829;?2=0.14128;?3=0.01168.

Flat top window[]

Flat-top window

A flat top window is a partially negative-valued window that has minimal  in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components. However, its broad bandwidth results in high  and wider frequency selection, which depending on the application could be a drawback.

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

?[?]=?0??1cos?(2???)+?2cos?(4???)??3cos?(6???)+?4cos?(8???).

The  has these coefficients:

?0=0.21557895;?1=0.41663158;?2=0.277263158;?3=0.083578947;?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):  ?0=1;?1=1       Functionally equivalent to the .

Class I, Order 2 (K = 2):  ?0=1;?1=43;?2=13

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[]

Gaussian window, σ = 0.4

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

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

?[?]=exp?(?12(???/2??/2)2),0≤?≤?. ?≤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  and the  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  of temporal width  L × σ_t  is well approximated by:

?[?]=?(?)??(?12)[?(?+?)+?(???)]?(?12+?)+?(?12??)

where ? is a Gaussian function:

?(?)=exp?(?(???22???)2)

The standard deviation of the approximate window is  (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  above, we can represent this window as

?[?,?]=exp?(?(???/2??/2)?)

for any even ?. At ?=2, this is a Gaussian window and as ? approaches ∞, this approximates to a rectangular window. The  of this window does not exist in a closed form for a general ?. 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  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).

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Window function - Wikipedia.7z

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  that has been widely used in the theory of  in . It is  (a ?∞ 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 , inspired by the . It is defined as a  function:

ε, with smaller values giving sharper transitions.

DPSS or Slepian window[]

The DPSS (discrete prolate spheroidal sequence) or Slepian window , and is used in  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: 

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

?[?]=?0(??1?(2???1)2)?0(??),0≤?≤?    ^{:?p. 73?} ?0(?)=?0(??1?(2??)2)?0(??),??/2≤?≤?/2

where ?0 is the zero-th order modified Bessel function of the first kind. Variable parameter ? 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  21+?2,  in units of DFT bins,  and a typical value of ? is 3.

Dolph–Chebyshev window[]

Dolph–Chebyshev window, α = 5

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

The zero-phase Dolph–Chebyshev window function ?0[?] is usually defined in terms of its real-valued discrete Fourier transform, ?0[?]:

?0(?)=??(?cos?(???+1))??(?)=??(?cos?(???+1))10?, 0≤?≤?.

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

??(?)={cos(?cos?1?(?))if ?1≤?≤1cosh(?cosh?1?(?))if ?≥1(?1)?cosh(?cosh?1?(??))if ?≤?1,

and

?=cosh(1?cosh?1?(10?))

is the unique positive real solution to ??(?)=10?, 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  (DFT):

?0(?)=1?+1∑?=0??0(?)???2???/(?+1), ??/2≤?≤?/2.

The lagged version of the window can be obtained by:

?[?]=?0(???2),0≤?≤?,

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

?0(???2)=1?+1∑?=0??0(?)???2??(???/2)?+1=1?+1∑?=0?[(?????+1)???0(?)]??2????+1,

which is an inverse DFT of  (?????+1)???0(?).

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 .
• An alternative to the inverse DFT definition is also available..

Ultraspherical window[]

The Ultraspherical window's ? 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:

?[?]=1?+1[???(?0)+∑?=1?2???(?0cos????+1)cos?2????+1]

where ??? is the  of degree N, and ?0 and ? control the side-lobe patterns.

Certain specific values of ? yield other well-known windows: ?=0 and ?=1 give the Dolph–Chebyshev and  windows respectively. See  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  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

?[?]=??|???2|1?,

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

?=?28.69?.

Hybrid windows[]

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

Bartlett–Hann window

Bartlett–Hann window[]

?[?]=?0??1|???12|??2cos?(2???) ?0=0.62;?1=0.48;?2=0.38

Planck–Bessel window[]

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

A  multiplied by a  which is defined in terms of a . This hybrid window function was introduced to decrease the peak side-lobe level of the Planck-taper window while still exploiting its good asymptotic decay. 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  multiplied by a . For ??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  algorithms like . The Hann–Poisson window is defined by:

?[?]=12(1?cos?(2???))???|??2?|?

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 ?. 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.

?0[?]=?0+∑?=1??2?(??)2?,??2≤?≤?2,  

where ? is the standard deviation of the {?} sequence.

Additionally, starting with a set of expansion coefficients ?2? 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[]

?[?]=sinc?(2???1)

• used in 
• for the Lanczos window, sinc?(?) is defined as sin?(??)/??
• also known as a sinc window, because:
  ?0(?)=sinc?(2??)
  is the main lobe of a normalized 

Asymmetric window functions[]

The ?0(?) form, defined above, is symmetric around ?=0. However, there are window functions that are asymmetric like  used in FIR implementation of . These asymmetries are used to reduce the delay when using large window sizes or to emphasize the initial transient of a decaying pulse.^[] There are ways to make normally symmetric windows asymmetric like this below by applying this formula:

?=??

to the time function (x position is shifted so is 0 to 1 to apply this formula above before mapping it back to -1 to 1 range), where the window is skewed towards the latest samples where ?>1 and conversely to the earliest samples when 

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Window function - Wikipedia.7z

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