凯泽窗 凯泽窗 （Kaiser 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[ 1] [ 2]
w
[
n
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=
{
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π π -->
α α -->
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− − -->
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2
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N
− − -->
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π π -->
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≤ ≤ -->
n
≤ ≤ -->
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otherwise
,
{\displaystyle 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.}
其中：
N 為序列的長度I 0 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
w
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(
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1
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{\displaystyle \scriptstyle w[(N-1)/2]=1,}
N 為偶數，窗函數最大值會在
w
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N
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− − -->
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=
w
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{\displaystyle \scriptstyle w[N/2-1]\ =\ w[N/2]\ <\ 1.}
若將上述離散數列視為是連續函數，並進行傅立葉變換 ：
I
0
(
π π -->
α α -->
1
− − -->
(
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t
(
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I
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π π -->
α α -->
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⏟ ⏟ -->
w
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t
)
⟺ ⟺ -->
F
(
N
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T
⋅ ⋅ -->
sinh
-->
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π π -->
α α -->
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(
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I
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π π -->
α α -->
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⋅ ⋅ -->
π π -->
α α -->
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f
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.
{\displaystyle \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)}.}
兩個不同α參數凯泽窗的傅立葉變換 w 0 t )的最大值為w 0 w [n]數列為以下函收的取様：
w
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t
− − -->
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⋅ ⋅ -->
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{\displaystyle w_{0}\left(t-{\tfrac {(N-1)T}{2}}\right)\cdot \operatorname {rect} \left({\tfrac {t-(N-1)T/2}{NT}}\right),}
而且rect()為矩形函数 . W 0 f )主瓣後的第一個零點在：
f
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1
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α α -->
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N
T
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{\displaystyle f={\frac {\sqrt {1+\alpha ^{2}}}{NT}},}
[ 3] 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 0 f )主瓣的寬度增加，而旁瓣的大小減小，如右圖所示。α = 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 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）。
^ Harris, Fredric j. On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform (PDF) . Proceedings of the IEEE. Jan 1978, 66 (1): 73–74 [2017-01-29 ] . doi:10.1109/PROC.1978.10837 存档 (PDF) 于2017-03-19）. ^ Kaiser, James F.; Ronald W. Schafer. On the Use of the I0-Sinh Window for Spectrum Analysis. IEEE Transactions on Acoustics, Speech and Signal Processing. February 1980,. ASSP-28 (1): 105–107. ^ Kaiser, James F.; Schafer, Ronald W. On the use of the I0 -sinh window for spectrum analysis. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1980, 28 : 105–107. doi:10.1109/TASSP.1980.1163349
Oppenheim, A. V.; Schafer, R. W.; Buck J. R. Discrete-time signal processing . Upper Saddle River, N.J.: Prentice Hall. 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, Kaiser-Bessel Derived Window Examples and C-language Implementation , Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2001). 
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a81b40939275883d17aec37311fa6417bac74d7f)
![{\displaystyle \scriptstyle w[(N-1)/2]=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9d2b63c9d36d6abe8019c816e3d8c05ef31284)
![{\displaystyle \scriptstyle w[N/2-1]\ =\ w[N/2]\ <\ 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7835fbb025165c645caa9504efdd368392964bcd)
