|
|
| Line 1: |
Line 1: |
| [[Image:KaiserWindow.jpg|right]]
| | Gabrielle Straub is what you can call me although it's not the quite a few feminine of names. Guam has always been home. Filing offers been my profession smoothly time and I'm doing pretty good financially. Fish keeping is what I provide every week. See what is new on my world wide web site here: http://prometeu.net/<br><br>Also visit my web blog: [http://prometeu.net/ clash of clans hack ipad download] |
| 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 for [[digital signal processing]], and is defined by the formula
| |
| ,<ref name="f.harris">
| |
| {{cite journal
| |
| | doi = 10.1109/PROC.1978.10837
| |
| | last = Harris
| |
| | first = Fredric j.
| |
| | coauthors =
| |
| | title = On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform
| |
| | journal = Proceedings of the IEEE
| |
| | volume = 66
| |
| | issue = 1
| |
| | pages = 73–74
| |
| |date=Jan 1978
| |
| | url=http://web.mit.edu/xiphmont/Public/windows.pdf}} Article on FFT windows which introduced many of the key metrics used to compare windows.</ref><ref>{{cite journal|last=Kaiser|first=James F.|coauthors=Ronald W. Schafer|title=On the Use of the I0-Sinh Window for Spectrum Analysis|journal=IEEE Transactions on Acoustics, Speech and Signal Processing|date=February 1980|volume=ASSP-28|issue=1|pages=105–107}}</ref>''':'''
| |
| :<math>
| |
| | |
| w[n] =
| |
| | |
| \left\{
| |
| \begin{matrix}
| |
| | |
| \frac{I_0\left(\pi \alpha \sqrt{1 - \left(\frac{2n}{N-1}-1\right)^2}\right)} {I_0(\pi \alpha)},
| |
| & 0 \leq n \leq N-1 \\ \\ | |
| | |
| 0 & \mbox{otherwise}, \\
| |
| | |
| \end{matrix}
| |
| \right.
| |
| </math>
| |
| | |
| where:
| |
| * ''N'' is the length of the sequence.
| |
| * ''I''<sub>0</sub> is the zeroth order [[Modified Bessel function]] of the first kind.
| |
| * ''α'' is an arbitrary, 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.
| |
| | |
| When ''N'' is an odd number, the peak value of the window is <math>\scriptstyle w[(N-1)/2] = 1,</math> and when ''N'' is even, the peak values are <math>\scriptstyle w[N/2-1]\ =\ w[N/2]\ <\ 1.</math>
| |
| | |
| ==Fourier transform==
| |
| Underlying the discrete sequence is this continuous-time function and its Fourier transform''':'''
| |
| | |
| :<math>\underbrace{\frac{I_0\left(\pi \alpha \sqrt{1 - \left(\frac{2t}{(N-1)T}\right)^2}\right)} {I_0(\pi \alpha)}}_{w_0(t)}
| |
| \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
| |
| \underbrace{\frac{(N-1)T\cdot\sinh\left(\pi \sqrt{\alpha^2-\left((N-1)T\cdot f\right)^2}\right)}{I_0(\pi \alpha)\cdot\pi \sqrt{\alpha^2-\left((N-1)T\cdot f\right)^2}}}_{W_0(f)}.
| |
| </math>
| |
| | |
| [[Image:Kaiser-Window-Spectra.jpg|right|thumb|452px|Fourier transforms of Kaiser windows for typical values of parameter α]]
| |
| The maximum value of ''w''<sub>''0''</sub>(''t'') is ''w''<sub>''0''</sub>(0) = 1. The ''w''[n] sequence defined above are the samples of''':'''
| |
| | |
| :<math>w_0\left(t-\tfrac{(N-1)T}{2}\right)\cdot \operatorname{rect}\left(\tfrac{t-(N-1)T/2}{NT}\right),</math> sampled at intervals of '''T''',
| |
| | |
| and where rect() is the [[rectangle function]]. The first null after the main lobe of ''W''<sub>''0''</sub>(''f'') occurs at''':'''
| |
| | |
| :<math>f = \frac{\sqrt{1+\alpha^2}}{NT},</math> which in units of DFT bins is just <math>\scriptstyle \sqrt{1+\alpha^2}.</math><ref>{{cite doi|10.1109/TASSP.1980.1163349}}</ref> | |
| | |
| ''α'' controls the tradeoff between main-lobe width and side-lobe area. As ''α'' increases, the main lobe of ''W''<sub>''0''</sub>(''f'') increases in width, and the side lobes decrease in amplitude, as illustrated in the figure at right. ''α'' = 0 corresponds to a rectangular window. For large ''α'', the shape of the Kaiser window (in both time and frequency domain) tends to a [[Gaussian function|Gaussian]] curve. The Kaiser window is nearly optimal in the sense of its peak's concentration around frequency ''0'' (Oppenheim ''et al.'', 1999).
| |
| | |
| ==Kaiser-Bessel derived (KBD) window==
| |
| [[File:Kbd-window.jpg|right]]
| |
| 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 ''M''+1, by the formula''':'''
| |
| :<math>
| |
| d_n =
| |
| | |
| \left\{\begin{matrix}
| |
| | |
| \sqrt{\frac{\sum_{i=0}^{n} w[i]} {\sum_{i=0}^M w[i]}}
| |
| & \mbox{if } 0 \leq n < M \\ \\
| |
| | |
| \sqrt{\frac{\sum_{i=0}^{2M-1-n} w[i]} {\sum_{i=0}^M w[i]}}
| |
| & \mbox{if } M \leq n < 2M \\ \\
| |
| | |
| 0 & \mbox{otherwise}. \\
| |
| | |
| \end{matrix}
| |
| \right.
| |
| </math>
| |
| | |
| This defines a window of length 2''M'', where by construction ''d''<sub>''n''</sub> satisfies the Princen-Bradley condition for the MDCT (using the fact that ''w''<sub>''M''−''n''</sub> = ''w''<sub>''n''</sub>): ''d''<sub>''n''</sub><sup>2</sup> + ''d''<sub>''n'' + ''M''</sub><sup>2</sup> = 1 (interpreting ''n'' and ''n'' + ''M'' [[modular arithmetic|modulo]] 2''M''). The KBD window is also symmetric in the proper manner for the MDCT: ''d''<sub>''n''</sub> = ''d''<sub>2''M''−1−''n''</sub>.
| |
| | |
| ===Applications===
| |
| The KBD window is used in the [[Advanced Audio Coding]] digital audio format.
| |
| | |
| ==Notes==
| |
| <references />
| |
| | |
| ==References==
| |
| * {{cite book | author=Oppenheim, A. V.; Schafer, R. W.; and Buck J. R. | title=Discrete-time signal processing | location=Upper Saddle River, N.J. | publisher=Prentice Hall | year=1999 | isbn=0-13-754920-2}}
| |
| * Kaiser, J. F. (1966). Digital Filters. In Kuo, F. F. and Kaiser, J. F. (Eds.), ''System Analysis by Digital Computer'', chap. 7. New York, Wiley.
| |
| * Craig Sapp, [http://ccrma.stanford.edu/courses/422/projects/kbd/ Kaiser-Bessel Derived Window Examples and C-language Implementation], ''Music 422 / EE 367C: Perceptual Audio Coding'' (Stanford University course page, 2001).
| |
| | |
| ==External links==
| |
| | |
| [[Category:Signal processing]]
| |
Gabrielle Straub is what you can call me although it's not the quite a few feminine of names. Guam has always been home. Filing offers been my profession smoothly time and I'm doing pretty good financially. Fish keeping is what I provide every week. See what is new on my world wide web site here: http://prometeu.net/
Also visit my web blog: clash of clans hack ipad download