Window function: Difference between revisions

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[[File:Hanning.svg|thumb|A popular window function, the [[Hann function|Hann window]]. Most popular window functions are similar bell-shaped curves.]]

In [[signal processing]] and [[statistics]], a '''window function''' (also known as an '''apodization function''' or '''tapering function'''<ref name=Weisstein/>) is a [[function (mathematics)|mathematical function]] that is zero-valued outside of some chosen [[interval (mathematics)|interval]]. Typically, normallywindows functions are symmetric around the middle of the interval, usually approachingapproach a maximum in the middle, and usually taperingtaper away from the middle. Mathematically, when another function or waveform/data-sequence 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". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.

The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution. But that method also changes the frequency content of the signal by an effect called [[spectral leakage]]. Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.

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=== Asymmetric window functions ===

The <math>w_0(x)</math> form, definedaccording to the convention above, is symmetric around <math>x = 0</math>. However, there are window functions that are asymmetric, likesuch as the [[Gamma distribution]] used in FIR implementationimplementations of [[Gammatone filter]]s. These asymmetries are used to reduce the delay when using large window sizes, or to emphasize the initial transient of a decaying pulse.{{cn|date=January 2023}} There are ways to make normally symmetric windows asymmetric like this below by applying this formula:

Any [[bounded function]] with [[compact support]], including asymmetric ones, can be readily used as a window function. Additionally, there are ways to transform symmetric windows into asymmetric windows by transforming the time coordinate, such as with the below formula

:<math>

x \leftarrow N\left( \frac{x}{N}+\frac{1}{2} \right)^\alpha-\frac{N}{2}\,,

x = x^\alpha

</math>

to the time function (''x'' position is shifted so is 0 to 1 to apply this formula above before mapping it back to -1 to 1 range), where the window isweights skewedmore towardshighly the latestearliest samples wherewhen <math>\alpha > 1</math>, and conversely toweights more highly the earliestlatest samples when <math>\alpha < 1</math>.<ref>{{cite journal |last1=Luo |first1=Jiufel |last2=Xie |first2=Zhijiang |last3=Li |first3=Xinyi |title=Asymmetric Windows and Their Application in Frequency Estimation |journal=Chongqing University |date=2015-03-02 |volume=9 |issue=Algorithms & Computational Technology |pages=389–412 |doi=10.1260/1748-3018.9.4.389 |s2cid=124464194 |doi-access=free }}</ref>

{{expand section|more different ways to make asymmetric window functions, not just making normally symmetric ones asymmetric|date=January 2023}}

== See also ==

* [[Apodization]]