Kaiser window

The Kaiser window is a window function used for digital signal processing, and is defined by the formula [James F. Kaiser and Ronald W. Schafer, "On the Use of the Io-Sinh Window for Spectrum Analysis", IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-28, No. 1, February 1980, pp 105-107.] :

:$$

w_n =

left{ egin{matrix}

frac{I_0left(alpha sqrt{1 - left(frac{2n}{N}-1 ight)^2} ight)} {I_0(alpha)} & mbox{if } 0 leq n leq N \ \

0 & mbox{otherwise} \

end{matrix} ight.

where "I"₀ 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 "n" = "N"/2, i.e. at the center of the window, and decays exponentially towards the window edges.

The discrete-time Fourier transform of the sequence $\{w\_n\}$ is given by:

:$W\_K(omega)=frac\{(N+1)cdotsinhleft(sqrt\{alpha^2-left(frac\{(N+1)cdotomega\}\{2\}\; ight)^2\}\; ight)\}\{I\_0(alpha)cdotsqrt\{alpha^2-left(frac\{(N+1)cdotomega\}\{2\}\; ight)^2$

for the normalized frequency $-pi\; leq\; omega\; leq\; pi$.

The larger the value of |α|, the narrower the window becomes; α = 0 corresponds to a rectangular window. Conversely, for larger |α| the main lobe of $W\_K(omega)$ increases in width, 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).

Kaiser-Bessel derived (KBD) window

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 $\{w\_n\}$ by the formula:

:$d\_n\; =$

left{ egin{matrix}

sqrt{frac{sum_{j=0}^{n} w_j} {sum_{j=0}^{N} w_j & mbox{if } 0 leq n < N \ \

sqrt{frac{sum_{j=0}^{2N-1-n} w_j} {sum_{j=0}^{N} w_j & mbox{if } N leq n < 2N \ \

0 & mbox{otherwise} \

end{matrix} ight.

This defines a window of length 2"N", where by construction "d"_"n" satisfies the Princen-Bradley condition for the MDCT (using the fact that "w"_"N"−"n" = "w"_"n"): "d"_"n"² + "d"_{"n" + "N"}² = 1 (interpreting "n" and "n" + "N" modulo 2"N"). The KBD window is also symmetric in the proper manner for the MDCT: "d"_"n" = "d"_{2"N"−1−"n"}.

References

*
* 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, [http://ccrma-www.stanford.edu/courses/422/projects/kbd/ Kaiser-Bessel Derived Window] , "Music 422 / EE 367C: Perceptual Audio Coding" (Stanford University course page, 2005).