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The Kaiser window is a window function w_k used for digital signal processing, and is defined by the formula:

Kaiser window function for N=100 and α= 0.5,1,2,4,8,16.

where I₀ is the zeroth order modified Bessel function of the first kind, α is an arbitrary real number that determines the shape of the window, and the integer N gives the length of the window (N+1 points).

By construction, this function peaks at unity for k = N/2, i.e. at the center of the window, and decays exponentially towards the window edges.

The larger the value of |α|, the narrower the window becomes; α = 0 corresponds to a rectangular window. Conversely, for larger |α| the width of the main lobe increases in the Fourier transform of w_k, while the side lobes decrease in amplitude. Thus, this parameter controls the tradeoff between main-lobe width and side-lobe area, as is illustrated in the plot of the frequency spectra below. 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 ω=0 (Oppenheim et al., 1999).

Frequency spectra of Kaiser windows for α=2 and α=4. The sharp minima in the side lobes are places where the amplitude goes all the way to zero, but does not here because of the finite plotting resolution.

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 d_n is defined in terms of the Kaiser window w_k by the formula:

KBD window function for N=100 and α= 0.5, 2, 8, 32.

This defines a window of length 2N, where by construction d_k satisfies the Princen-Bradley condition for the MDCT (using the fact that w_N−k = w_k): d_k² + d_k + N² = 1 (interpreting k and k + N modulo 2N). The KBD window is also symmetric in the proper manner for the MDCT: d_k = d_2N−1−k.

• Oppenheim, A. V.; Schafer, R. W.; and Buck J. R. (1999). Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall. 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.
• Marina Bosi, Kaiser-Bessel Derived Window, Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2005).