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 and filter design, as well as audio data compression such as Vorbis.

Spectral analysis

It can be shown theoretically that the Fourier transform of the function: $\backslash cos(\backslash omega\; t)\backslash ,$ is zero, except right at frequency $\backslash pm\; \backslash omega\backslash ,$. But many other functions and data (that is, waveforms) do not have convenient closed form transforms. Or we might be interested in their spectral content only during a certain time period. In either case, we have to perform the Fourier transform (or something similar) on one or more finite intervals of the waveform. So in general, the transform is applied to the product of the waveform and a window function. But any window (including rectangular) affects the spectrum that we are trying to measure.

The effect is easiest to understand in terms of a simple waveform, like $\backslash cos(\backslash omega\; t)\backslash ,$. Windowing causes its Fourier transform to have non-zero values (commonly called leakage) at frequencies other than $\backslash omega\backslash ,$. It tends to be worst (highest) near $\backslash omega\backslash ,$ and least at frequencies farthest from $\backslash omega\backslash ,$. If there are two sinusoids, with different frequencies, leakage can interfere with our 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. I.e., 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. And they are also poorest in terms of sensitivity. I.e., if the input waveform contains random noise, its spectral amplitude, compared to the sinusoid, will appear higher than with a less extreme 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. The figure above 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 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.

Total leakage

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.

Although we have depicted leakage as interference that one component imposes on other frequencies, the effect is reciprocal. Thus, if frequency $f\_1\backslash ,$ leaks a percentage of its energy into frequency $f\_2\backslash ,$, then a frequency component at $f\_2\backslash ,$ returns the favor in the same proportion, as does every other frequency component of the input signal in varying amounts. So the spectral measurement at frequency $f\_1\backslash ,$ is perturbed by all the other components. The greater the noise bandwidth (B), the greater the effect.

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:   $10\backslash mbox\{\; \}\backslash log\_\{10\}(2.01/1.5)\; =\; 1.27$. 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.

Window examples

Terminology:

• $N\backslash ,$ represents the width, in samples, of a discrete-time window function. Typically it is an integer power-of-2, such as $2^\{10\}\; =\; 1024$.
• $n\backslash ,$ is an integer, with values $0\backslash le\; \backslash ;\; n\backslash le\; \backslash ;\; N-1\backslash ,$. So these are the time-shifted forms of the windows:  $w(n-\backslash begin\{matrix\}\; \backslash frac\{N-1\}\{2\}\backslash end\{matrix\})\backslash ,$, where $w(n)\backslash ,$ is maximum at $n=0\backslash ,$.
  • 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 $1\; \backslash le\; B\; \backslash le\; 1.8$, and the other group comprises $B\; \backslash ge\; 1.98$. 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

$w(n)\; =\; 1\backslash ,$

Gauss windows

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

$\backslash sigma\; \backslash le\; \backslash ;0.5\backslash ,$

Hamming window

$w(n)=0.53836\; -\; 0.46164\backslash ;\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

Hann window

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

The Hann window is sometimes called the "Hanning" window, in analogy to the Hamming window. However, this is incorrect, because the windows were named after Julius von Hann and Richard Hamming, respectively.

Bartlett window (zero valued end-points)

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

Triangular window (non-zero end-points)

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

Bartlett-Hann window

$w(n)=a\_0\; -\; a\_1\; \backslash left\; |\backslash frac\{n\}\{N-1\}-\backslash frac\{1\}\{2\}\; \backslash right|\; -\; a\_2\; \backslash cos\; \backslash left\; (\backslash frac\{2\; \backslash pi\; n\}\{N-1\}\backslash right\; )$

$a\_0=0.62;\backslash quad\; a\_1=0.48;\backslash quad\; a\_2=0.38\backslash ,$

Blackman window

$w(n)=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)\; +\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

$a\_0=0.42;\backslash quad\; a\_1=0.5;\backslash quad\; a\_2=0.08\backslash ,$

Kaiser windows

Window_function_(kaiser 3).png

$w(n)=\backslash frac\{I\_0\backslash Bigg\; (\backslash pi\backslash alpha\; \backslash sqrt\{1\; -\; (\backslash begin\{matrix\}\; \backslash frac\{2\; n\}\{N-1\}\; \backslash end\{matrix\}-1)^2\}\backslash Bigg\; )\}\; \{I\_0(\backslash pi\backslash alpha)\}$

See Kaiser window.

Low-resolution (high dynamic range) windows

Nuttall window, continuous first derivative

$w(n)=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N-1\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

$a\_0=0.355768;\backslash quad\; a\_1=0.487396;\backslash quad\; a\_2=0.144232;\backslash quad\; a\_3=0.012604\backslash ,$

Blackman-Harris window

$w(n)=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N-1\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

$a\_0=0.35875;\backslash quad\; a\_1=0.48829;\backslash quad\; a\_2=0.14128;\backslash quad\; a\_3=0.01168\backslash ,$

Blackman-Nuttall window

$w(n)=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N-1\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

$a\_0=0.3635819;\; \backslash quad\; a\_1=0.4891775;\; \backslash quad\; a\_2=0.1365995;\; \backslash quad\; a\_3=0.0106411\backslash ,$

Flat top window

$w(n)=a\_0\; -\; a\_1\; \backslash cos\; \backslash left\; (\; \backslash frac\{2\; \backslash pi\; n\}\{N-1\}\; \backslash right)+\; a\_2\; \backslash cos\; \backslash left\; (\; \backslash frac\{4\; \backslash pi\; n\}\{N-1\}\; \backslash right)-\; a\_3\; \backslash cos\; \backslash left\; (\; \backslash frac\{6\; \backslash pi\; n\}\{N-1\}\; \backslash right)+a\_4\; \backslash cos\; \backslash left\; (\; \backslash frac\{8\; \backslash pi\; n\}\{N-1\}\; \backslash right)$

$a\_0=1;\backslash quad\; a\_1=1.93;\backslash quad\; a\_2=1.29;\backslash quad\; a\_3=0.388;\backslash quad\; a\_4=0.032\backslash ,$

Other windows

Bessel window

Sine window

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.

See also

• Apodization
• Welch method

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.

Statistics | Fourier analysis | Signal processing | Digital signal processing

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