Window function
Encyclopedia
In signal processing
, a window function (also known as an apodization function or tapering function) is a mathematical 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 part where they overlap; the "view through the window". Applications of window functions include spectral analysis, filter design
, and beamforming
.
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, that is, that the function goes sufficiently rapidly toward zero.
In typical applications, the window functions used are non-negative smooth "bell-shaped" curves, though rectangle and triangle functions and other functions are sometimes used.
Applications of window functions include spectral analysis
and the design of finite impulse response
filters.
of the function cos ωt is zero, except at frequency ±ω. 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 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.
) at frequencies other than ω. The leakage tends to be worst (highest) near ω and least at frequencies farthest from ω.
If the signal under analysis is composed of two sinusoids of different frequencies, leakage can interfere with the ability to distinguish them spectrally. If their frequencies are dissimilar and one component is weaker, then leakage from the larger component can obscure the weaker’s presence. But if the frequencies are similar, leakage can render them unresolvable even when the sinusoids are of equal strength.
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 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 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.
(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 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 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.
, caused by this effect. The height of the noise floor is proportional to B. So two different window functions can produce different noise floors.
, 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.
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 decibel
s as:
. 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.
(FIR) filter design. Window choice considerations are related to those described above for spectral analysis, or can alternatively be viewed as a tradeoff between "ringing" and frequency-domain sharpness.
.png)
The rectangular window is sometimes known as a Dirichlet window. It is the simplest window, taking a chunk of the signal without any other modification at all, which leads to discontinuities at the endpoints (unless the signal happens to be an exact fit for the window length, as used in multitone testing, for instance). The first side-lobe is only 13 dB lower than the main lobe, with the rest falling off at about 6 dB per octave.
.png)
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, a window function (also known as an apodization function or tapering function) is a mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is zero-valued outside of some chosen interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
. 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 part where they overlap; the "view through the window". Applications of window functions include spectral analysis, filter design
Filter design
Filter design is the process of designing a filter , often a linear shift-invariant filter, that satisfies a set of requirements, some of which are contradictory...
, and beamforming
Beamforming
Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in the array in a way where signals at particular angles experience constructive interference and while others experience destructive...
.
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, that is, that the function goes sufficiently rapidly toward zero.
In typical applications, the window functions used are non-negative smooth "bell-shaped" curves, though rectangle and triangle functions and other functions are sometimes used.
Applications
Spectral analysis
Spectral analysis or Spectrum analysis may refer to:* Spectrum analysis in chemistry and physics, a method of analyzing the chemical properties of matter from bands in their visible spectrum...
and the design of finite impulse response
Finite impulse response
A finite impulse response filter is a type of a signal processing filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response filters, which have internal feedback and may continue to respond indefinitely...
filters.
Spectral analysis
The Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the function cos ωt is zero, except at frequency ±ω. 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 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.
Windowing
Windowing of a simple waveform, like cos ωt causes its Fourier transform to develop non-zero values (commonly called spectral leakageSpectral leakage
Spectral leakage is an effect in the frequency analysis of finite-length signals or finite-length segments of infinite signals where it appears as if some energy has "leaked" out of the original signal spectrum into other frequencies....
) at frequencies other than ω. The leakage tends to be worst (highest) near ω and least at frequencies farthest from ω.
If the signal under analysis is composed of two sinusoids of different frequencies, leakage can interfere with the ability to distinguish them spectrally. If their frequencies are dissimilar and one component is weaker, then leakage from the larger component can obscure the weaker’s presence. But if the frequencies are similar, leakage can render them unresolvable even when the sinusoids are of equal strength.
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 amplitudes.
In between the extremes are moderate windows, such as Hamming
Richard Hamming
Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
and Hann
Julius von Hann
Julius Ferdinand von Hann was an Austrian meteorologist. He is seen as the father of modern meteorology.-Biography:...
. 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.
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 transformDiscrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
(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 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 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.
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 floorNoise floor
In signal theory, the noise floor is the measure of the signal created from the sum of all the noise sources and unwanted signals within a measurement system, where the noise is defined as any signal other than the one being monitored....
, 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 processingSignal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, 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.
For example, the worst possible scalloping loss from a Blackman–Harris window (below) is 0.83 dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
, 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 decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
s as:
. 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.Filter design
Windows are sometimes used in the design of digital filters, for example to convert an "ideal" impulse response of infinite duration, such as a sinc function, to a finite impulse responseFinite impulse response
A finite impulse response filter is a type of a signal processing filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response filters, which have internal feedback and may continue to respond indefinitely...
(FIR) filter design. Window choice considerations are related to those described above for spectral analysis, or can alternatively be viewed as a tradeoff between "ringing" and frequency-domain sharpness.
Window examples
Terminology:
represents the width, in samples, of a discrete-time, symmetrical window function.
is an integer, with values 0 ≤ n ≤ N-1.- In practice, N is typically an odd number. An even-length FFT window is generated by using only the coefficients corresponding to 0 ≤ n ≤ N-2.This produces an asymmetric form known as "DFT-even" or "periodic".
- The examples below are the time-shifted forms of the windows:
, where 
is maximum at n=0. - 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
, and the other group comprises 
. The Gauss and Kaiser windows are families that span both groups, though only one or two examples of each are shown.
Rectangular window
The rectangular window is sometimes known as a Dirichlet window. It is the simplest window, taking a chunk of the signal without any other modification at all, which leads to discontinuities at the endpoints (unless the signal happens to be an exact fit for the window length, as used in multitone testing, for instance). The first side-lobe is only 13 dB lower than the main lobe, with the rest falling off at about 6 dB per octave.
Hann window
- Note that:
The ends of the cosine just touch zero, so the side-lobes roll off at about 18 dB per octave.
The Hann and Hamming windows, both of which are in the family known as "raised cosine" or "generalized Hamming" windows, are respectively named after Julius von HannJulius von HannJulius Ferdinand von Hann was an Austrian meteorologist. He is seen as the father of modern meteorology.-Biography:...
and Richard HammingRichard HammingRichard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
. The term "Hanning window" is sometimes used to refer to the Hann window.
Hamming window
The "raised cosine" with these particular coefficients was proposed by Richard W. Hamming. The window is optimized to minimize the maximum (nearest) side lobe, giving it a height of about one-fifth that of the Hann window, a raised cosine with simpler coefficients..png)
- Note that:
Tukey window
The Tukey window, also known as the tapered cosine window, can be regarded as a cosine lobe of width
that is convolved with a rectangle window of width 
At α=0 it becomes rectangular, and at α=1 it becomes a Hann window.
Cosine window
.png)
- also known as sine window
- cosine window describes the shape of
Lanczos window
.png)
- used in Lanczos resamplingLanczos resamplingLanczos resampling is an interpolation method used to compute new values for sampled data. It is often used in multivariate interpolation, for example for image scaling , but can be used for any other digital signal...
- for the Lanczos window, sinc(x) is defined as sin(πx)/(πx)
- also known as a sinc window, because:
-
is the main lobe of a normalized sinc function
Triangular windows
Bartlett window with zero-valued end-points:.png)
With non-zero end-points:.png)
Can be seen as the convolution of two half-sized rectangular windows, giving it a main lobe width of twice the width of a regular rectangular window. The nearest lobe is -26 dB down from the main lobe.
Gaussian windows
The frequency response of a Gaussian is also a Gaussian (it is an eigenfunction.png)
EigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
of the Fourier Transform). Since the 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 parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, this can be used for exact quadratic interpolation in frequency estimationFrequency estimationFrequency estimation is the process of estimating the complex frequency components of a signal in the presence of noise. The most common methods involve identifying the noise subspace to extract these components...
.
Bartlett–Hann window
.png)
Blackman windows
Blackman windows are defined as:.png)
By common convention, the unqualified term Blackman window refers to α=0.16.
Kaiser windows
.png)
A simple approximation of the DPSS window using Bessel functions, discovered by Jim Kaiser..png)
where
is the zero-th order modified Bessel function of the first kind, and usually 
.
- Note that:
Nuttall window, continuous first derivative
.png)
Blackman–Harris window
A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels.png)
Blackman–Nuttall window
.png)
Flat top window
.png)
Dolph-Chebyshev window
Minimizes the Chebyshev norm of the side-lobes for a given main lobe width.
Hann-Poisson window
A Hann window multiplied by a Poisson window, which has no side-lobes, in the sense that the frequency response drops off forever away from the main lobe. It can thus be used in hill climbingHill climbingIn computer science, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by incrementally changing a single element of the solution...
algorithms like Newton's methodNewton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
.
Exponential or Poisson window
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 functionExponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
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
where
is the time constant of the function. The exponential function decays as 
or approximately 8.69 dB per time constant.
This means that for a targetted decay of
dB over half of the window length, the time constant 
is given by
DPSS or Slepian window
The DPSS (digital prolate spheroidal sequence) or Slepian window is used to maximize the energy concentration in the main lobe.
Comparison of windows
When selecting an appropriate window function for an application, this comparison graph may be useful. The graph shows only the main lobe of the window's frequency response in detail. Beyond that only the envelope of the sidelobes is shown to reduce clutter. The frequency axis has units of FFT "bins" when the window of length N is applied to data and a transform of length N is computed. For instance, the value at frequency ½ "bin" is the response that would be measured in bins k and k+1 to a sinusoidal signal at frequency k+½. It is relative to the maximum possible response, which occurs when the signal frequency is an integer number of bins. The value at frequency ½ is referred to as the maximum scalloping loss of the window, which is one metric used to compare windows. The rectangular window is noticeably worse than the others in terms of that metric..png)
Other metrics that can be seen are the width of the main lobe and the peak level of the sidelobes, which respectively determine the ability to resolve comparable strength signals and disparate strength signals. The rectangular window (for instance) is the best choice for the former and the worst choice for the latter. What cannot be seen from the graphs is that the rectangular window has the best noise bandwidth, and despite its 3 dB potential scalloping loss, it is the best choice for detecting a sinusoid at low signal-to-noise ratios.
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 methodWelch methodIn physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal at different frequencies: that is, is is an approach to spectral density estimation. The method is based on the concept of using periodogram spectrum estimates,...
of power spectral analysis and the Modified discrete cosine transformModified discrete cosine transformThe modified discrete cosine transform is a Fourier-related transform based on the type-IV discrete cosine transform , with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset,...
.
See also
- MultitaperMultitaperIn signal processing, the multitaper method is a technique developed by David J. Thomson to estimate the power spectrum SX of a stationary ergodic finite-variance random process X, given a finite contiguous realization of X as data....
- ApodizationApodizationApodization literally means "removing the foot". It is the technical term for changing the shape of a mathematical function, an electrical signal, an optical transmission or a mechanical structure.- Apodization in signal processing :...
- Welch methodWelch methodIn physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal at different frequencies: that is, is is an approach to spectral density estimation. The method is based on the concept of using periodogram spectrum estimates,...
- Short-time Fourier transformShort-time Fourier transformThe short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
- Window design method
Other references
Article on FFT windows which introduced many of the key metrics used to compare windows.
Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.
- LabView Help, Characteristics of Smoothing Filters, http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/
- Evaluation of Various Window Function using Multi-Instrument, http://www.multi-instrument.com/doc/D1003/Evaluation_of_Various_Window_Functions_using_Multi-Instrument_D1003.pdf
- Multitaper
- Note that: