查看“︁凯泽窗”︁的源代码

[[Image:KaiserWindow.jpg|right|thumb|凯泽窗]]
'''凯泽窗'''（Kaiser window）<!--also known as the '''Kaiser–Bessel window--->是由[[贝尔实验室]]的James Kaiser所提出的。凯泽窗是一個單參數的[[窗函数]]群，用在[[数字信号处理]]中，其定義如下<ref name="f.harris">{{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
  | pages = 73–74
  | date = Jan 1978
  | url = http://web.mit.edu/xiphmont/Public/windows.pdf
  | access-date = 2017-01-29
  | archive-date = 2017-03-19
  | archive-url = https://web.archive.org/web/20170319024811/http://web.mit.edu/xiphmont/Public/windows.pdf
  | dead-url = no
  }} Article on FFT windows which introduced many of the key metrics used to compare windows.</ref><ref>{{cite journal|last=Kaiser|first=James F.|author2=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>

其中：
* ''N'' 為序列的長度
* ''I''<sub>0</sub> 是零階的第一類[[修正貝索函數]]
* ''α'' 是任意非負實數，用來調整凯泽窗的外形。在[[頻域]]上可以在主瓣（main-lobe）寬度及旁瓣（side lobe）大小中取拾，這是窗函數設計的重要考量因素。

若''N''為奇數，窗函數最大值會在&nbsp;<math>\scriptstyle w[(N-1)/2] = 1,</math>&nbsp;。若''N''為偶數，窗函數最大值會在&nbsp;<math>\scriptstyle w[N/2-1]\ =\ w[N/2]\ <\ 1.</math>。

==傅立葉變換==
若將上述離散數列視為是連續函數，並進行[[傅立葉變換]]：

:<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|兩個不同α參數凯泽窗的傅立葉變換]]
''w''<sub>''0''</sub>(''t'')的最大值為''w''<sub>''0''</sub>(0)&nbsp;=&nbsp;1.&nbsp; 上述的''w''[n]數列為以下函收的取様：

:<math>w_0\left(t-\tfrac{(N-1)T}{2}\right)\cdot \operatorname{rect}\left(\tfrac{t-(N-1)T/2}{NT}\right),</math>&nbsp; &nbsp;，在間隔T的時間進行取樣。

而且rect()為[[矩形函数]].&nbsp;''W''<sub>''0''</sub>(''f'')主瓣後的第一個零點在：

:<math>f = \frac{\sqrt{1+\alpha^2}}{NT},</math>&nbsp; &nbsp; &nbsp;<!-- which in units of DFT bins is just &nbsp;<math>\scriptstyle \sqrt{1+\alpha^2}.</math>--><ref>{{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 | pmid =  | pmc = }}</ref>

調整''α''可以在主瓣的寬度及旁瓣大小中進行取捨。若''α''增加，''W''<sub>''0''</sub>(''f'')主瓣的寬度增加，而旁瓣的大小減小，如右圖所示。''α''&nbsp;=&nbsp;0會對應長方形的窗函數。若''α''增加，時域及頻率下凯泽窗的形狀都會接近[[高斯函數|高斯]]曲線。凯泽窗在頻率0附近的集中程度是幾乎最佳化的（Oppenheim ''et al.'', 1999）。

==腳註==
{{reflist}}

==參考資料==
* {{cite book |author1=Oppenheim, A. V. |author2=Schafer, R. W. |author3=Buck J. R. | title=Discrete-time signal processing |url=https://archive.org/details/discretetimesign00alan | 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, [https://web.archive.org/web/20090327144432/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).
[[Category:信号处理]]