Kaiser window

The Kaiser window and its Fourier transform are given by:

${\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}{\frac {I_{0}\left[\pi \alpha {\sqrt {1-\left(2x/L\right)^{2}}}\right]}{I_{0}[\pi \alpha ]}},\quad &\left|x\right|\leq L/2\\0,\quad &\left|x\right|>L/2\end{array}}\right\}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {\sin {\bigg (}{\sqrt {(\pi Lf)^{2}-(\pi \alpha )^{2}}}{\bigg )}}{I_{0}(\pi \alpha )\cdot {\sqrt {(\pi Lf)^{2}-(\pi \alpha )^{2}}}}},}$   [3][upper-alpha 1]

Fourier transforms of two Kaiser windows

where:

• I₀ is the zeroth-order modified Bessel function of the first kind,
• L is the window duration, and
• 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 β, where β = πα.

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

${\displaystyle w[n]=L\cdot w_{0}\left({\tfrac {L}{N}}(n-N/2)\right)={\frac {I_{0}\left[\pi \alpha {\sqrt {1-\left({\frac {2n}{N}}-1\right)^{2}}}\right]}{I_{0}[\pi \alpha ]}},\quad 0\leq n\leq N,}$

where the length of the window is ${\displaystyle N+1,}$ and N can be even or odd. (see A list of window functions)

In the Fourier transform, the first null after the main lobe occurs at ${\displaystyle f={\tfrac {\sqrt {1+\alpha ^{2}}}{L}},}$ which is just ${\displaystyle {\sqrt {1+\alpha ^{2}}}}$ in units of N (DFT "bins"). As α increases, the main lobe increases in width, and the side lobes decrease in amplitude.  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 curve.  The Kaiser window is nearly optimal in the sense of its peak's concentration around frequency ${\displaystyle 0.}$[5]

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:

${\displaystyle d_{n}={\begin{cases}{\sqrt {\frac {\sum _{i=0}^{n}w[i]}{\sum _{i=0}^{N}w[i]}}}&{\mbox{if }}0\leq n<N\\{\sqrt {\frac {\sum _{i=0}^{2N-1-n}w[i]}{\sum _{i=0}^{N}w[i]}}}&{\mbox{if }}N\leq n\leq 2N-1\\0&{\mbox{otherwise}}.\\\end{cases}}}$

This defines a window of length 2N, where by construction d_n satisfies the Princen-Bradley condition for the MDCT (using the fact that w_N−n = w_n): d_n² + (d_n+N)² = 1 (interpreting n and n + N modulo 2N). The KBD window is also symmetric in the proper manner for the MDCT: d_n = d_2N−1−n.

• Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 73 (eq 46b). CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837.
• Kaiser, James F.; Schafer, Ronald W. (1980). "On the use of the I₀-sinh window for spectrum analysis". IEEE Transactions on Acoustics, Speech, and Signal Processing. 28: 105–107. doi:10.1109/TASSP.1980.1163349.
• Smith, J.O. (2011). "Spectral Audio Signal Processing, Kaiser and DPSS Windows Compared". ccrma.stanford.edu. Retrieved 2016-04-13.
• "Kaiser Window, R2018b". www.mathworks.com. Mathworks. Retrieved 2019-03-20.