In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, a
window function (also known as an
apodization function or
tapering function) is a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
that is zero-valued outside of some chosen
intervalIn 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 "view" through the window. Applications of window functions include spectral analysis,
filter designFilter design is the process of designing a filter , often a linear shift-invariant filter, which satisfies a set of requirements, some of which are contradictory...
, and
beamformingBeamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This spatial selectivity is achieved by using adaptive or fixed receive/transmit beampatterns. The improvement compared with an omnidirectional reception/transmission is known as the...
.
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 times 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
Illustrative applications of window functions include:
Spectral analysis, such as the
discrete Fourier transformIn mathematics, the discrete Fourier transform is a specific kind of Fourier 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...
: If one takes a finite time segment of a sampled signal and takes the discrete Fourier transform (DFT) of it, one suffers
spectral leakageSpectral 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....
: wavelengths that do not exactly divide the window size leak into a range of frequencies. This can be interpreted as due to the frequency response of the rectangular filter, which corresponds to the truncation of the signal.
If, instead of simply truncating, the signal is multiplied by a window function so that it tapers off to zero, the effects of spectral leakage are mitigated. Ideally each frequency would be precisely captured, with no leakage – a frequency response of a
Dirac delta functionThe Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
. This is not possible – the infinitely many frequencies cannot fit into the finitely many bins of the DFT – and instead one must trade off
frequency precision: as measured by the
main lobe width, with
noise suppression: as measured by
side lobe level (height of tallest side lobe, generally the first side lobe) and
side lobe fall-off (how rapidly the peaks of the side lobes fall off).
One speaks of "high resolution" windows versus "high dynamic range" windows.
Finite impulse responseA finite impulse response filter is a type of a digital filter. The impulse response, the filter's response to a Kronecker delta input, is finite because it settles to zero in a finite number of sample intervals. This is in contrast to infinite impulse response filters, which have internal...
(FIR) Filter design, such as a
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
: In FIR filter design, one has filters with
finite impulse response. One method for designing and understanding them is the window design method, considering them as windowed versions of ideal infinite impulse response (IIR) filters, which are realized as finite filters for implementation, and because actual signals are finite in time. Truncating or windowing an IIR filter yields a FIR filter, and the frequency response of the FIR filter is the frequency response of the IIR filter, convolved with the frequency response of the window.
Thus the ideal frequency response of a window is a Dirac delta function, as that results in the frequency response of the FIR filter being identical to that of the IIR filter, but this is not attainable for finite windows: the Fourier transform of the Dirac delta is a constant (flat) function, corresponding to the infinite window: all samples. Deviations of the frequency response of the window from the Dirac delta yield differences between the FIR response and the IIR response.
An important example is a
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
: the ideal low-pass filter (
brick-wall frequency response) is a
sinc filterIn signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase...
, which has infinite support. Truncating or windowing convolves this brick wall, yielding an imperfect low-pass filter; here main lobe width corresponds to the width of the
transition bandThe transition band is a range of frequencies, that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner frequency....
, the side lobes correspond to frequency domain
ringingIn electronics, signal processing, and video, ringing is unwanted oscillation of a signal, particularly in the step response...
(assuming their sign alternates), with the side lobe level corresponding to the maximum magnitude of ringing, and side lobe fall-off corresponding to frequency domain settle time.
The sinc filter also has time domain artifacts, namely
time domain
ringing artifactsIn signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal...
and overshoot – truncating the side lobes eliminates these, while using different windows that do not cut off all negative lobes (such as the Lanczos window) reduces these.
Spectral analysis
The
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
of the function
: is zero, except at frequency . 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 causes its Fourier transform to have non-zero values (commonly called
spectral leakageSpectral 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 . It tends to be worst (highest) near and least at frequencies farthest from .
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
HammingRichard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
and
HannJulius Ferdinand von Hann was an Austrian meteorologist. He is seen as the father of modern meteorology.He studied mathematics, chemistry and physics at the University of Vienna...
. 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 transformIn mathematics, the discrete Fourier transform is a specific kind of Fourier 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 floorIn 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....
, 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 is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, 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
dBThe decibel is a logarithmic unit of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit...
, 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
decibelThe decibel is a logarithmic unit of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit...
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 functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
, to a
finite impulse responseA finite impulse response filter is a type of a digital filter. The impulse response, the filter's response to a Kronecker delta input, is finite because it settles to zero in a finite number of sample intervals. This is in contrast to infinite impulse response filters, which have internal...
(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 window function. Typically it is an integer power-of-2, such as .
- is an integer, with values . So these are the time-shifted forms of the windows: , where is maximum at .
- 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 , 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.
High- and moderate-resolution windows
Rectangular window
The rectangular window is sometimes known as a Dirichlet window.
Hamming window
The "raised cosine" with these particular coefficients was proposed by Richard W. Hamming. The height of the maximum side lobe is about one-fifth that of the Hann window, a raised cosine with simpler coefficients.
Hann window
The Hann and Hamming windows, both of which are in the family known as "raised cosine" windows, are respectively named after
Julius von HannJulius Ferdinand von Hann was an Austrian meteorologist. He is seen as the father of modern meteorology.He studied mathematics, chemistry and physics at the University of Vienna...
and
Richard 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.
Cosine window
- also known as sine window
- cosine window describes the shape of
Lanczos window
- used in Lanczos resampling
Lanczos resampling is a multivariate interpolation method used to compute new values for any digitally sampled data. It is often used for image scaling , but could 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
In mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
Bartlett window (zero valued end-points)
Triangular window (non-zero end-points)
Gauss windows
Bartlett–Hann window
Blackman windows
Blackman windows are defined as:
By common convention, the unqualified term
Blackman window refers to α=0.16.
Low-resolution (high-dynamic-range) windows
Nuttall window, continuous first derivative
Blackman–Harris window
Blackman–Nuttall window
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 worst than the others in terms of that metric.
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 methodIn physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal vs. frequency, reducing noise compared to the methods it is based upon. Welch's method is based on the concept of using periodograms, which converts a signal from...
of power spectral analysis and the
Modified 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
- Multitaper
In 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....
- Apodization
Apodization 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...
- Welch method
In physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal vs. frequency, reducing noise compared to the methods it is based upon. Welch's method is based on the concept of using periodograms, which converts a signal from...
- Short-time Fourier transform
The 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