Editing Kaiser window

{{Short description|Used in finite impulse response filter design and spectral analysis}}
[[Image:KaiserWindow.svg|right|thumb|401px|The Kaiser window for several values of its parameter]]

The '''Kaiser window''', also known as the '''Kaiser–Bessel window''', was developed by [[James Kaiser]] at [[Bell Laboratories]]. It is a one-parameter family of [[window function]]s used in [[finite impulse response]] [[filter design]] and [[spectral estimation|spectral analysis]].  The Kaiser window approximates the [[DPSS window]] which [[Spectral concentration problem|maximizes the energy concentration in the main lobe]]<ref>{{Cite web |url=https://ccrma.stanford.edu/~jos/sasp/Slepian_DPSS_Window.html |title=Slepian or DPSS Window |website=ccrma.stanford.edu |access-date=2016-04-13}}</ref> but which is difficult to compute.<ref>{{cite book |author1=Oppenheim, A. V. |author2=Schafer, R. W. |title=Discrete-time signal processing |location=Upper Saddle River, N.J. |publisher=Prentice Hall |year=2009 |page=541 |isbn=9780131988422}}</ref>

==Definition==

The Kaiser window and its Fourier transform are given by''':'''
:<math>
w_0(x) \triangleq \left\{
\begin{array}{ccl}
\tfrac{1}{L}\frac{I_0\left[\pi\alpha \sqrt{1 - \left(2x/L\right)^2}\right]}{I_0[\pi\alpha]},\quad &\left|x\right| \leq L/2\\
0,\quad &\left|x\right| > L/2
\end{array}\right\}
\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{\sin\bigg(\sqrt{(\pi L f)^2-(\pi \alpha)^2}\bigg)}
{I_0(\pi \alpha)\cdot \sqrt{(\pi L f)^2-(\pi \alpha)^2}},
</math> &nbsp; <ref>{{cite journal
 | doi =10.1109/TASSP.1981.1163506
 | last =Nuttall
 | first =Albert H.
 | title =Some Windows with Very Good Sidelobe Behavior
 | journal =IEEE Transactions on Acoustics, Speech, and Signal Processing
 | volume =29
 | issue =1
 | page =89 (eq.38)
 | date =Feb 1981
 | url =https://zenodo.org/record/1280930
}}</ref>{{efn-ua
|An equivalent formula is''':'''<ref>
{{cite web| url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_Window.html | website=ccrma.stanford.edu | title=Kaiser Window in Spectral Audio Signal Processing, eq.(4.40 & 4.42) | last=Smith | first=J.O. | date=2011 | access-date=2022-01-01}}
where <math>\beta \triangleq \pi \alpha,\ \omega \triangleq 2 \pi f,\ M=L.</math></ref>
:<math>\frac{\sinh\bigg(\sqrt{(\pi \alpha)^2 - (\pi L f)^2}\bigg)}
{I_0(\pi \alpha)\cdot \sqrt{(\pi \alpha)^2 - (\pi L f)^2}}.</math>
}}

[[Image:Kaiser-Window-Spectra.svg|right|thumb|401px|Fourier transforms of two Kaiser windows]]

where''':'''
* {{math|''I<sub>0</sub>''}} is the zeroth-order [[Modified_Bessel_function#Modified_Bessel_functions:_Iα,_Kα|modified Bessel function]] of the first kind,
* {{mvar|L}} is the window duration, and
* {{math|α}} is a non-negative real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design.
* Sometimes the Kaiser window is parametrized by {{math|β}}, where {{math|β {{=}} πα}}.

For [[digital signal processing]], the function can be sampled symmetrically as''':'''

:<math>w[n] = L\cdot w_0\left(\tfrac{L}{N} (n-N/2)\right) = \frac{I_0\left[\pi\alpha \sqrt{1 - \left(\frac{2n}{N}-1\right)^2}\right]}{I_0[\pi\alpha]},\quad 0 \leq n \leq N,</math>

where the length of the window is <math>N+1,</math> and N can be even or odd.  (see [[Window_function#Examples of window functions|A list of window functions]])

In the Fourier transform, the first null after the main lobe occurs at <math>f = \tfrac{\sqrt{1+\alpha^2}}{L},</math> which is just <math>\sqrt{1+\alpha^2}</math> in units of N ([[Normalized_frequency_(signal_processing)|DFT "bins"]]).  As ''α'' increases, the main lobe increases in width, and the side lobes decrease in amplitude.&nbsp; {{math|α}}&nbsp;=&nbsp;0 corresponds to a rectangular window.  For large {{math|α,}} the shape of the Kaiser window (in both time and frequency domain) tends to a [[Gaussian function|Gaussian]] curve.&nbsp; The Kaiser window is nearly optimal in the sense of its peak's concentration around frequency <math>0.</math><ref name=Oppenheim>
{{Cite book
 |last=Oppenheim
 |first=Alan V.
 |authorlink=Alan V. Oppenheim
 |last2=Schafer
 |first2=Ronald W.
 |author2-link=Ronald W. Schafer
 |last3=Buck
 |first3=John R.
 |title=Discrete-time signal processing
 |year=1999
 |publisher=Prentice Hall
 |location=Upper Saddle River, N.J.
 |isbn=0-13-754920-2
 |edition=2nd
 |chapter=7.2
 |page=[https://archive.org/details/discretetimesign00alan/page/474 474]
 |quote=a near-optimal window could be formed using the zeroth-order modified Bessel function of the first kind
 |url-access=registration
 |url=https://archive.org/details/discretetimesign00alan
 }}
</ref>
{{clear}}

==Kaiser–Bessel-derived (KBD) window==
[[File:Kbd-window.svg|right|401px]]
A related window function is the '''Kaiser–Bessel-derived (KBD)''' window, which is designed to be suitable for use with the [[modified discrete cosine transform]] (MDCT). The KBD window function is defined in terms of the Kaiser window of length ''N''+1, by the formula''':'''
:<math>
d_n =

\begin{cases}

\sqrt{\frac{\sum_{i=0}^{n} w[i]} {\sum_{i=0}^N w[i]}}
 & \mbox{if } 0 \leq n < N \\

\sqrt{\frac{\sum_{i=0}^{2N-1-n} w[i]} {\sum_{i=0}^N w[i]}}
 & \mbox{if } N \leq n \leq 2N-1 \\

0 & \mbox{otherwise}. \\

\end{cases}
</math>

This defines a window of length 2''N'', where by construction ''d''<sub>''n''</sub> satisfies the Princen-Bradley condition for the MDCT (using the fact that {{math|''w''{{sub|''N''−''n''}} {{=}} ''w''{{sub|''n''}}}}): {{math|''d''{{sub|''n''}}<sup>2</sup> + (''d''{{sub|''n''+''N''}})<sup>2</sup> {{=}} 1}} (interpreting ''n'' and ''n''&nbsp;+&nbsp;''N'' [[modular arithmetic|modulo]] 2''N''). The KBD window is also symmetric in the proper manner for the MDCT: ''d''<sub>''n''</sub>&nbsp;=&nbsp;''d''<sub>2''N''−1−''n''</sub>.

===Applications===
The KBD window is used in the [[Advanced Audio Coding]] digital audio format.

== Notes ==
{{notelist-ua}}

==References==
<references />

==Further reading==
* {{cite journal
  |doi=10.1109/PROC.1978.10837
  |last=Harris
  |first=Fredric J.
  |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform
  |journal=Proceedings of the IEEE
  |volume=66
  |issue=1
  |page=73 (eq 46b)
  |date=Jan 1978
  |url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880
  }}
* {{Cite journal |last1=Kaiser |first1=James F. |last2=Schafer |first2=Ronald W. |doi=10.1109/TASSP.1980.1163349 |title=On the use of the I<sub>0</sub>-sinh window for spectrum analysis |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=28 |pages=105–107 |year=1980}}
* {{Cite web|url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html|website=ccrma.stanford.edu| title =Spectral Audio Signal Processing, Kaiser and DPSS Windows Compared | last =Smith | first =J.O. | date =2011 |access-date=2016-04-13}}
* {{cite web| url =https://www.mathworks.com/help/signal/ug/kaiser-window.html | website=www.mathworks.com | title =Kaiser Window, R2018b | publisher =Mathworks | access-date =2019-03-20}}
<!--This is a dead link
{{cite web | url =http://ccrma.stanford.edu/courses/422/projects/kbd/Kaiser-Bessel | title =Music 422 / EE 367C: Perceptual Audio Coding | last =Sapp | first =Craig | date =2001 | publisher =Stanford University}}{{dead link|date=August 2020|bot=medic}}{{cbignore|bot=medic}}-->
<!--This very old book doesn't even have an ISBN that I can find
{{cite book | last1 = Kuo| first1 = Franklin F. | last2 = Kaiser| first2 = J. F. | title =System Analysis by Digital Computer | publisher =Wiley | date =1966 | location =New York | pages =438}}-->

[[Category:Digital_signal processing]]