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Encyclopedia > Window function

In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or a signal (data) is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the "view" through the window. Applications of window functions include spectral analysis, filter design and beamforming. Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... Filter design is the process of working out a filter (in the sense in which the term is used in signal processing, statistics, and applied mathematics), often a linear shift-invariant filter, which satisfies a set of requirements, some of which are contradicting. ... Beamforming is a signal processing technique used with arrays of transmitting or receiving transducers that control the directionality of, or sensitivity to, a radiation pattern. ...

Contents

Spectral analysis

The Fourier transform of the function: cos(omega t), is zero, except at frequency pm omega,. However, many other functions and data (that is, waveforms) do not have convenient closed form transforms. Alternatively, one might be interested in their spectral content only during a certain time period. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...


In either case, the Fourier transform (or something similar) 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 1: Zoom view of spectral leakage
Figure 1: Zoom view of spectral leakage

Image File history File links Spectral_leakage_from_a_sinusoid_and_rectangular_window. ... Image File history File links Spectral_leakage_from_a_sinusoid_and_rectangular_window. ...

Windowing

Windowing of a simple waveform, like cos(omega t), causes its Fourier transform to have non-zero values (commonly called leakage) at frequencies other than omega,. It tends to be worst (highest) near omega, and least at frequencies farthest from omega,.


If there are two sinusoids, with different frequencies, leakage can interfere with the ability to distinguish them spectrally. If their frequencies are dissimilar, then the leakage interferes when one sinusoid is much smaller in amplitude than the other. That is, its spectral component can be hidden by the leakage from the larger component. But when the frequencies are near each other, the leakage can be sufficient to interfere even when the sinusoids are equal strength; that is, they become unresolvable.


The rectangular window has excellent resolution characteristics for signals of comparable strength, but it is a poor choice for signals of disparate amplitudes. This characteristic is sometimes described as low-dynamic-range.


At the other extreme of dynamic range are the windows with the poorest resolution. These high-dynamic-range low-resolution windows are also poorest in terms of sensitivity; this is, if the input waveform contains random noise close to the signal frequency, the response to noise, compared to the sinusoid, will be higher than with a higher-resolution window. In other words, the ability to find weak sinusoids amidst the noise is diminished by a high-dynamic-range window. High-dynamic-range windows are probably most often justified in wideband applications, where the spectrum being analyzed is expected to contain many different signals of various strengths.


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 tradeoff between resolving comparable strength signals with similar frequencies and resolving disparate strength signals with dissimilar frequencies. That tradeoff occurs when the window function is chosen. Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was a mathematician whose work had many implications for computer science and telecommunications. ... Julius von Hann (1839-1921) was an Austrian meteorologist. ...


Discrete-time signals

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 coarse sampling of the actual DTFT spectrum. Figure 1 shows a portion of the DTFT for a rectangularly-windowed sinusoid. The actual frequency of the sinusoid is indicated as "0" on the horizontal axis. Everything else is leakage. The unit of frequency is "DFT bins"; that is, the integer values are the frequencies sampled by the DFT. So the figure depicts a case where the actual frequency of the sinusoid happens to coincide with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. When it misses the maximum value by some amount [up to 1/2 bin], the measurement error is referred to as scalloping loss (inspired by the shape of the peak). But the most interesting thing about this case is that all the other samples coincide with nulls in the true spectrum. (The nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.) So in this case, the DFT creates the illusion of no leakage. Despite the unlikely conditions of this example, it is a popular misconception that visible leakage is some sort of artifact of the DFT. But since any window function causes leakage, its apparent absence (in this contrived example) is actually the DFT artifact. In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... The Fourier transform of a function of a discrete time variable (where is an integer) is called the discrete-time Fourier transform (or DTFT), and is mathematically the inverse of a Fourier series expansion. ...


Noise bandwidth

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. Think of it as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B. The more 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.


Processing gain

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. Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... The phrase signal-to-noise ratio, often abbreviated SNR or S/N, is an engineering term for the ratio between the magnitude of a signal (meaningful information) and the magnitude of background noise. ...


For example, the worst possible scalloping loss from a Blackman–Harris window (below) is 0.83 dB, compared to 1.42 dB for a Hann window. But the noise bandwidth is larger by a factor of 2.01/1.5, which can be expressed in decibels as:   10 log10(2.01 / 1.5) = 1.27. Therefore, even at maximum scalloping, the net processing gain of a Hann window exceeds that of a Blackman–Harris window by:  1.27 +0.83 -1.42 = 0.68 dB. And when we happen to incur no scalloping (due to a fortuitous signal frequency), the Hann window is 1.27 dB more sensitive than Blackman–Harris. In general (as mentioned earlier), this is a deterrent to using high-dynamic-range windows in low-dynamic-range applications. For other uses, see Decibel (disambiguation). ... For other uses, see Decibel (disambiguation). ...


Window examples

Terminology:

  • N, represents the width, in samples, of a discrete-time window function. Typically it is an integer power-of-2, such as 210 = 1024.
  • n, is an integer, with values 0le ; nle ; N-1,. So these are the time-shifted forms of the windows:  w(n-begin{matrix} frac{N-1}{2}end{matrix}),, where w(n), is maximum at n=0,.
    • Some of these forms have an overall width of N−1, which makes them zero-valued at n=0 and n=N−1. That sacrifices two data samples for no apparent gain, if the DFT size is N. When that happens, an alternative approach is to replace N−1 with N in the formula.
  • Each figure label includes the corresponding noise equivalent bandwidth metric (B), in units of DFT bins. As a guideline, windows are divided into two groups on the basis of B. One group comprises 1 le B le 1.8, and the other group comprises B ge 1.98. The Gauss and Kaiser windows are families that span both groups, though only one or two examples of each are shown.

High- and moderate-resolution windows

Rectangular window

Rectangular window; B=1.00
Rectangular window; B=1.00
w(n) = 1,

The rectangular window is sometimes known as a Dirichlet window. Image File history File links Download high resolution version (1038x419, 6 KB) source: http://en. ... Image File history File links Download high resolution version (1038x419, 6 KB) source: http://en. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...

Gauss windows

Gauss window, σ=0.4; B=1.45
Gauss window, σ=0.4; B=1.45
w(n)=e^{-frac{1}{2} left ( frac{n-(N-1)/2}{sigma (N-1)/2} right)^{2}}
sigma le ;0.5,

Image File history File links Download high resolution version (1046x419, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1046x419, 7 KB) source: http://en. ...

Hamming window

Hamming window; B=1.37
Hamming window; B=1.37
w(n)=0.53836 - 0.46164; cos left ( frac{2pi n}{N-1} right)

Image File history File links Download high resolution version (1037x419, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1037x419, 7 KB) source: http://en. ...

Hann window

Hann window; B = 1.50
Hann window; B = 1.50
w(n)= 0.5; left(1 - cos left ( frac{2 pi n}{N-1} right) right)

The Hann and Hamming windows, both of which are in the family known as "raised cosine" windows, are respectively named after Julius von Hann and Richard Hamming. The term "Hanning window" is sometimes used to refer to the Hann window, but is ambiguous as it is easily confused with Hamming window. Image File history File links Download high resolution version (1052x420, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1052x420, 7 KB) source: http://en. ... Julius von Hann (1839-1921) was an Austrian meteorologist. ... Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was a mathematician whose work had many implications for computer science and telecommunications. ...

Bartlett window (zero valued end-points)

Bartlett window; B=1.33
Bartlett window; B=1.33
w(n)=frac{2}{N-1}cdotleft(frac{N-1}{2}-left |n-frac{N-1}{2}right |right),

Image File history File links Download high resolution version (1025x420, 9 KB) source: http://en. ... Image File history File links Download high resolution version (1025x420, 9 KB) source: http://en. ...

Triangular window (non-zero end-points)

Triangular window; B=1.33
Triangular window; B=1.33
w(n)=frac{2}{N}cdotleft(frac{N}{2}-left |n-frac{N-1}{2}right |right),

Image File history File links Download high resolution version (1030x419, 8 KB) source: http://en. ... Image File history File links Download high resolution version (1030x419, 8 KB) source: http://en. ...

Bartlett–Hann window

Bartlett-Hann window; B=1.46
Bartlett-Hann window; B=1.46
w(n)=a_0 - a_1 left |frac{n}{N-1}-frac{1}{2} right| - a_2 cos left (frac{2 pi n}{N-1}right )
a_0=0.62;quad a_1=0.48;quad a_2=0.38,

Image File history File links Download high resolution version (1044x422, 8 KB) source: http://en. ... Image File history File links Download high resolution version (1044x422, 8 KB) source: http://en. ...

Blackman window

Blackman window; B=1.73
Blackman window; B=1.73
w(n)=a_0 - a_1 cos left ( frac{2 pi n}{N-1} right) + a_2 cos left ( frac{4 pi n}{N-1} right)
a_0=0.42;quad a_1=0.5;quad a_2=0.08,

Image File history File links Download high resolution version (1039x424, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1039x424, 7 KB) source: http://en. ...

Kaiser windows

Main article: Kaiser window
Kaiser window, α =2π; B=1.5
Kaiser window, α =2π; B=1.5
Kaiser window, α =3π; B=1.8
Kaiser window, α =3π; B=1.8
w(n)=frac{I_0Bigg (alpha sqrt{1 - (begin{matrix} frac{2 n}{N-1} end{matrix}-1)^2}Bigg )} {I_0(alpha)}

The Kaiser window is a nearly optimal window function wk used for digital signal processing, and is defined by the formula: Kaiser window function for n=100 and α= 0. ... Image File history File links Size of this preview: 800 × 271 pixelsFull resolution (1761 × 597 pixel, file size: 81 KB, MIME type: image/png) N=; k=:N; dr = ; alpha = *; w = ,alpha*-*k/N . ... Image File history File links Size of this preview: 800 × 271 pixelsFull resolution (1761 × 597 pixel, file size: 81 KB, MIME type: image/png) N=; k=:N; dr = ; alpha = *; w = ,alpha*-*k/N . ... Image File history File links Size of this preview: 800 × 272 pixelsFull resolution (1785 × 607 pixel, file size: 84 KB, MIME type: image/png) N=; k=:N; dr = ; alpha = *; w = ,alpha*-*k/N . ... Image File history File links Size of this preview: 800 × 272 pixelsFull resolution (1785 × 607 pixel, file size: 84 KB, MIME type: image/png) N=; k=:N; dr = ; alpha = *; w = ,alpha*-*k/N . ...

Low-resolution (high-dynamic-range) windows

Nuttall window, continuous first derivative; B=2.02
Nuttall window, continuous first derivative; B=2.02

Image File history File links Download high resolution version (1048x420, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1048x420, 7 KB) source: http://en. ...

Nuttall window, continuous first derivative

w(n)=a_0 - a_1 cos left ( frac{2 pi n}{N-1} right)+ a_2 cos left ( frac{4 pi n}{N-1} right)- a_3 cos left ( frac{6 pi n}{N-1} right)
a_0=0.355768;quad a_1=0.487396;quad a_2=0.144232;quad a_3=0.012604,
Blackman–Harris window; B=2.01
Blackman–Harris window; B=2.01

Image File history File links Download high resolution version (1056x421, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1056x421, 7 KB) source: http://en. ...

Blackman–Harris window

w(n)=a_0 - a_1 cos left ( frac{2 pi n}{N-1} right)+ a_2 cos left ( frac{4 pi n}{N-1} right)- a_3 cos left ( frac{6 pi n}{N-1} right)
a_0=0.35875;quad a_1=0.48829;quad a_2=0.14128;quad a_3=0.01168,
Blackman–Nuttall window; B=1.98
Blackman–Nuttall window; B=1.98

Image File history File links Download high resolution version (1047x422, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1047x422, 7 KB) source: http://en. ...

Blackman–Nuttall window

w(n)=a_0 - a_1 cos left ( frac{2 pi n}{N-1} right)+ a_2 cos left ( frac{4 pi n}{N-1} right)- a_3 cos left ( frac{6 pi n}{N-1} right)
a_0=0.3635819; quad a_1=0.4891775; quad a_2=0.1365995; quad a_3=0.0106411,
Flat top window; B=3.77
Flat top window; B=3.77

Image File history File links Download high resolution version (1041x422, 7 KB) source: http://en. ... Image File history File links Download high resolution version (1041x422, 7 KB) source: http://en. ...

Flat top window

w(n)=a_0 - a_1 cos left ( frac{2 pi n}{N-1} right)+ a_2 cos left ( frac{4 pi n}{N-1} right)- a_3 cos left ( frac{6 pi n}{N-1} right)+a_4 cos left ( frac{8 pi n}{N-1} right)
a_0=1;quad a_1=1.93;quad a_2=1.29;quad a_3=0.388;quad a_4=0.032,

Other windows

Sine window; B=1.24
Sine window; B=1.24

Image File history File links Size of this preview: 800 × 329 pixelsFull resolution (1017 × 418 pixel, file size: 19 KB, MIME type: image/png) N=; k=:N; dr = ; w = *k/N; B = N*w. ... Image File history File links Size of this preview: 800 × 329 pixelsFull resolution (1017 × 418 pixel, file size: 19 KB, MIME type: image/png) N=; k=:N; dr = ; w = *k/N; B = N*w. ...

Sine window

w(n)=sinleft(frac{pi n}{N-1}right)

Bessel window

Exponential window

Tukey window

Comparison of windows

Stopband attenuation of different windows

When selecting an appropriate window function for an application, this comparison graph may be useful. The most important parameter is usually the stopband attenuation close to the main lobe. However, some applications are more sensitive to the stopband attenuation far away from the cut-off frequency.
Image File history File links Size of this preview: 800 × 588 pixelsFull resolution (988 × 726 pixel, file size: 80 KB, MIME type: image/png) Frequency response of the window functions. ... Image File history File links Size of this preview: 800 × 588 pixelsFull resolution (988 × 726 pixel, file size: 80 KB, MIME type: image/png) Frequency response of the window functions. ...


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 Welch method of power spectral analysis. In 1967 P.D. Welch wrote a paper titled The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms that appeared in IEEE Trans. ...


See also

Apodization literally means removing the foot. To apodize is the technical term for changing the shape of a mathematical function, an electrical signal, an optical transmission or a mechanical structure to remove or smooth a discontinuity at the edges. ... In 1967 P.D. Welch wrote a paper titled The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms that appeared in IEEE Trans. ...

References

  • Oppenheim, A.V., and R.W. Schafer, Discrete-Time Signal Processing, Upper Saddle River, NJ: Prentice-Hall, 1999, pp 468–471.
  • Albert H. Nuttall, Some Windows with Very Good Sidelobe Behavior, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.ASSP-29, No.1, February 1981, pp 84–91.
  • Frederic J. Harris, On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proceedings of the IEEE, Vol.66, No.1, January 1978, pp 51–83.
  • S.W.A. Bergen and A. Antoniou, Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics, EURASIP Journal on Applied Signal Processing, vol. 2004, no. 13, pp. 2053–2065, 2004.
  • S.W.A. Bergen and A. Antoniou, Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function, EURASIP Journal on Applied Signal Processing, vol. 2005, no. 12, pp. 1910–1922, 2005.
  • LabView Help, Characteristics of Smoothing Filters, http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/

  Results from FactBites:
 
Window function - Wikipedia, the free encyclopedia (1494 words)
In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval.
For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation.
The Hann window is sometimes called the "Hanning" window, in analogy to the Hamming window.
Window Features (6979 words)
The coordinates of a top-level window are relative to the upper left corner of the screen; the coordinates of a child window are relative to the upper left corner of the parent window.
The CascadeWindows function cascades the windows on the desktop or cascades the child windows of the specified parent window.
The TileWindows function tiles the windows on the desktop or tiles the child windows of the specified parent window.
  More results at FactBites »

 
 

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