凱澤窗

凱澤窗（Kaiser window）是由貝爾實驗室的James Kaiser所提出的。凱澤窗是一個單參數的窗函數群，用在數碼訊號處理中，其定義如下^[1]^[2]：

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：

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

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：

\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參數凱澤窗的傅立葉變換

w₀(t)的最大值為w₀(0) = 1. 上述的w[n]數列為以下函收的取様：

w_{0}\left(t-{\tfrac {(N-1)T}{2}}\right)\cdot \operatorname {rect} \left({\tfrac {t-(N-1)T/2}{NT}}\right),

，在間隔T的時間進行取樣。

而且rect()為矩形函數. W₀(f)主瓣後的第一個零點在：

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，W₀(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，時域及頻率下凱澤窗的形狀都會接近高斯曲線。凱澤窗在頻率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. （原始內容存檔 (PDF)於2017-03-19）. Article on FFT windows which introduced many of the key metrics used to compare windows.
^ 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).