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Applications

Typical window function frequency response, showing main lobe, side lobe level, and side lobe fall-off.

Illustrative applications of window functions include:

Spectral analysis, such as the discrete Fourier transform: If one takes a finite time segment of a sampled signal and takes the discrete Fourier transform (DFT) of it, one suffers spectral leakage:^[4] 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 function. 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).^[5]

One speaks of "high resolution" windows versus "high dynamic range" windows.

Sine integral, showing part of frequency response for a truncated low-pass filter.

Finite impulse response (FIR) Filter design, such as a low-pass filter^[6]: 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 filter: the ideal low-pass filter (brick-wall frequency response) is a sinc filter, 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 band, the side lobes correspond to frequency domain ringing (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 artifacts 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 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 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

Windowing

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

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.

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 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 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.

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: $10 log 10 (2.01 / 1.5) = 1.27$ . 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.

Filter design

Main article: 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 response (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.^[7]

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 $210 = 1024$ .
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 $1 \le B \le 1.8$ , 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; B=1.00

Rectangular window

$w(n) = 1\,$

The rectangular window is sometimes known as a Dirichlet window.

Hamming window; B=1.37

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.^[8]

$w(n) = 0.54 - 0.46\; \cos \left ( \frac{2\pi n}{N-1} \right)$ ^{[note 1]}

Note that:

$\begin{align} w_0(n)\ &\stackrel{\mathrm{def}}{=}\ w(n+\begin{matrix} \frac{N-1}{2}\end{matrix})\\ &= 0.54 + 0.46\; \cos \left ( \frac{2\pi n}{N-1} \right) \end{align}$

Hann window; B = 1.50

Hann window

Main article: Hann function

$w(n) = 0.5\; \left(1 - \cos \left ( \frac{2 \pi n}{N-1} \right) \right)$ ^{[note 1]}

Note that:

$w_0(n) = 0.5\; \left(1 + \cos \left ( \frac{2 \pi n}{N-1} \right) \right)$

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.

Cosine window; B=1.24

Cosine window

$w(n) = \cos\left(\frac{\pi n}{N-1} - \frac{\pi}{2}\right) = \sin\left(\frac{\pi n}{N-1}\right)$ ^{[note 1]}

also known as sine window
cosine window describes the shape of $w_0(n)\,$

Sinc or Lanczos window; B=1.31

Lanczos window

$w(n) = \mathrm{sinc}\left(\frac{2n}{N-1}-1\right)$

used in Lanczos resampling
for the Lanczos window, sinc(x) is defined as sin(πx)/(πx)
also known as a sinc window, because:

$w_0(n) = \mathrm{sinc}\left(\frac{2n}{N-1}\right)\,$ is the main lobe of a normalized sinc function

Bartlett window; B=1.33

Bartlett window (zero valued end-points)

$w(n)=\frac{2}{N-1}\cdot\left(\frac{N-1}{2}-\left |n-\frac{N-1}{2}\right |\right)\,$

Triangular window; B=1.33

Triangular window (non-zero end-points)

$w(n)=\frac{2}{N}\cdot\left(\frac{N}{2}-\left |n-\frac{N-1}{2}\right |\right)\,$

Gauss window, σ=0.4; B=1.45

Gauss windows

$w(n)=e^{-\frac{1}{2} \left ( \frac{n-(N-1)/2}{\sigma (N-1)/2} \right)^{2}}$

$\sigma \le \;0.5\,$

Bartlett-Hann window; B=1.46

Bartlett–Hann window

$w(n)=a_0 - a_1 \left |\frac{n}{N-1}-\frac{1}{2} \right| - a_2 \cos \left (\frac{2 \pi n}{N-1}\right )$

$a_0=0.62;\quad a_1=0.48;\quad a_2=0.38\,$

Blackman window; α = 0.16; B=1.73

Blackman windows

Blackman windows are defined as:^{[note 1]}

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right) + a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)$

$a_0=\frac{1-\alpha}{2};\quad a_1=\frac{1}{2};\quad a_2=\frac{\alpha}{2}\,$

By common convention, the unqualified term Blackman window refers to α=0.16.

Kaiser window, α =2; B=1.5

Kaiser window, α =3; B=1.8

Kaiser windows

Main article: Kaiser window

$w(n)=\frac{I_0\Bigg (\pi\alpha \sqrt{1 - (\begin{matrix} \frac{2 n}{N-1} \end{matrix}-1)^2}\Bigg )} {I_0(\pi\alpha)}$

where e.g. $α = 3$ .

Note that:

$w_0(n) = \frac{I_0\Bigg (\pi\alpha \sqrt{1 - (\begin{matrix} \frac{2 n}{N-1} \end{matrix})^2}\Bigg )} {I_0(\pi\alpha)}$

Low-resolution (high-dynamic-range) windows

Nuttall window, continuous first derivative; B=2.02

Nuttall window, continuous first derivative

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$ ^{[note 1]}

$a_0=0.355768;\quad a_1=0.487396;\quad a_2=0.144232;\quad a_3=0.012604\,$

Blackman–Harris window; B=2.01

Blackman–Harris window

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$ ^{[note 1]}

$a_0=0.35875;\quad a_1=0.48829;\quad a_2=0.14128;\quad a_3=0.01168\,$

Blackman–Nuttall window; B=1.98

Blackman–Nuttall window

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$ ^{[note 1]}

$a_0=0.3635819; \quad a_1=0.4891775; \quad a_2=0.1365995; \quad a_3=0.0106411\,$

Flat top window; B=3.77

Flat top window

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)+a_4 \cos \left ( \frac{8 \pi n}{N-1} \right)$ ^{[note 1]}

$a_0=1;\quad a_1=1.93;\quad a_2=1.29;\quad a_3=0.388;\quad a_4=0.032\,$

Other windows

This section requires expansion.

Bessel window

Dolph-Chebyshev window

Exponential window

Tukey window

The Tukey window is also known as the tapered cosine window. [1]

Comparison of windows

Stopband attenuation of different 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 method of power spectral analysis and the Modified discrete cosine transform.

Notes

^ ^a ^b ^c ^d ^e ^f ^g ^h Windows of the form:

$w(n)=\sum_{k=0}^{K} a_k \cos \left(\frac{2\pi k n}{N-1}\right)$

have only 2K+1 non-zero DFT coefficients, which makes them good choices for applications that require windowing by convolution in the frequency-domain. In those applications, the DFT of the unwindowed data vector is needed for a different purpose than spectral analysis. (see Overlap-save method)

References

^ Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics. CRC Press. ISBN 1584883472. http://books.google.com/books?id=aFDWuZZslUUC&pg=PA97&dq=apodization+function&lr=&as_brr=0&ei=27ycSPCdJZDwsgOdxIieBQ&sig=ACfU3U1EhHuuq88rRYF2W01Jj1o2Ab-6wA#PPA95,M1.
^ Carlo Cattani and Jeremiah Rushchitsky (2007). Wavelet and Wave Analysis As Applied to Materials With Micro Or Nanostructure. World Scientific. ISBN 9812707840. http://books.google.com/books?id=JuJKu_0KDycC&pg=PA53&dq=define+%22window+function%22+nonzero+interval&lr=&as_brr=3&ei=iyGbSKX_OJCKtAPXwaD2CQ&sig=ACfU3U1zZzq20w9GO1fbmNcvD3MlQFevsg#PPA53,M1.
^ Curtis Roads (2002). Microsound. MIT Press. ISBN 0262182157.
^ FFT window functions: Limits on FFT analysis, Bores Signal Processing
^ Power system harmonics, by J. Arrillaga, N. R. Watson, John Wiley and Sons, 2003, ISBN 978 0 47085129 6, p. 41
^ FIR Filter Design: The Window Design Method, by Nguyen Huu Phuong
^ Mastering Windows: Improving Reconstruction
^ Loren D. Enochson and Robert K. Otnes (1968). Programming and Analysis for Digital Time Series Data. U.S. Dept. of Defense, Shock and Vibration Info. Center. pp. 142. http://books.google.com/books?id=duBQAAAAMAAJ&q=%22hamming+window%22+date:0-1970&dq=%22hamming+window%22+date:0-1970&lr=&as_brr=0&ei=4LEASfHtJoXWsgOcz7WdDA&pgis=1.

Other references

harris, fredric j. (January 1978). ""On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform"". Proceedings of the IEEE 66 (1): 51–83. Article on FFT windows which introduced many of the key metrics used to compare windows.

Nuttall, Albert H. (February 1981). "Some Windows with Very Good Sidelobe Behavior". IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1): 84-91. Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.

Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John A. (1999). Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall. pp. 468–471. ISBN 0-13-754920-2.

Bergen, S.W.A.; A. Antoniou (2004). "Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics". EURASIP Journal on Applied Signal Processing 2004 (13): 2053–2065. doi:10.1155/S1110865704403114.

Bergen, S.W.A.; A. Antoniou (2005). "Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function". EURASIP Journal on Applied Signal Processing 2005 (12): 1910–1922. doi:10.1155/ASP.2005.1910.

Park, Young-Seo, "System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation", US patent 7065150, published 2003, issued 2006

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

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Window function

Contents

Applications

Spectral analysis

Windowing

Discrete-time signals

Noise bandwidth

Processing gain

Filter design

Window examples

High- and moderate-resolution windows

Rectangular window

Hamming window

Hann window

Cosine window

Lanczos window

Bartlett window (zero valued end-points)

Triangular window (non-zero end-points)

Gauss windows

Bartlett–Hann window

Blackman windows

Kaiser windows

Low-resolution (high-dynamic-range) windows

Nuttall window, continuous first derivative

Blackman–Harris window

Blackman–Nuttall window

Flat top window

Other windows

Bessel window

Dolph-Chebyshev window

Exponential window

Tukey window

Comparison of windows

Overlapping windows

See also

Notes

References

Other references

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