Window function: Difference between revisions

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Window functions are used in spectral [[frequency spectrum#spectral analysis|analysis]]/modification/[[Overlap–add method#resynthesis|resynthesis]],<ref name=OLA/> the design of [[finite impulse response]] filters, merging multiscale and multidimensional datasets,<ref>{{Cite journal |last1=Ajala |first1=R. |last2=Persaud |first2=P. |title=Ground-Motion Evaluation of Hybrid Seismic Velocity Models |journal=The Seismic Record|date=2022 |volume=2 |issue=3 |pages=186–196 |doi=10.1785/0320220022 |s2cid=251504921 |doi-access=free |bibcode=2022SeisR...2..186A }}</ref><ref>{{Cite journal |last1=Ajala |first1=R. |last2=Persaud |first2=P. |title=Effect of Merging Multiscale Models on Seismic Wavefield Predictions Near the Southern San Andreas Fault |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2021JB021915 |journal=Journal of Geophysical Research: Solid Earth |date=2021 |language=en |volume=126 |issue=10 |doi=10.1029/2021JB021915 |bibcode=2021JGRB..12621915A |s2cid=239654900 |issn=2169-9313}}</ref> as well as [[beamforming]] and [[Antenna (radio)|antenna]] design.

[[File:Spectral_leakage_caused_by_%22windowing%22.svg|thumb|400px|Figure 2: Windowing a sinusoid causes spectral leakage. The same amount of leakage occurs whether there are an integer (blue) or non-integer (red) number of cycles within the window (rows 1 and 2). When the sinusoidalsinusoid is sampled and windowed, its [[discrete-time Fourier transform]] (DTFT) also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue DTFT, those samples are the outputs of the [[discrete Fourier transform]] (DFT). The red DTFT has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed.]]

=== Spectral analysis ===

{{Main|Spectral leakage}}

The [[Fourier transform]] of the function {{math|cos(''ωt'')}} is zero, except at frequency&nbsp;±''ω''. However, many other functions and waveformwaveforms 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 a similar transform) 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.