Kaiser window

From Academic Kids

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

Missing image
Kaiser-window.png
Kaiser window function for n=100 and α= 0.5,1,2,4,8,16.
<math>

w_k =

\left\{ \begin{matrix}

\frac{I_0(\pi\alpha \sqrt{1 - (2k/n-1)^2})} {I_0(\pi\alpha)}

     & \mbox{if } 0 \leq k \leq n \\  \\

0 & \mbox{otherwise} \\

\end{matrix} \right.

<math>

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 is the length of the window.

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 wk, while the side lobes decrease in amplitude. Thus, this parameter controls the tradeoff between main-lobe width and side-lobe area. For large α, the shape of the Kaiser window tends to a Gaussian curve.

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 dk is defined in terms of the Kaiser window wk by the formula:

Missing image
Kbd-window.png
KBD window function for n=100 and α= 0.5, 2, 8, 32.
<math>

d_k =

\left\{ \begin{matrix}

\sqrt{\frac{\sum_{j=0}^{k} w_j} {\sum_{j=0}^{n} w_j}}

     & \mbox{if } 0 \leq k < n \\ \\

\sqrt{\frac{\sum_{j=0}^{2n-1-k} w_j} {\sum_{j=0}^{n} w_j}}

     & \mbox{if } n \leq k < 2n \\ \\

0 & \mbox{otherwise} \\

\end{matrix} \right.

<math>

This defines a window of length 2n, where by construction dk satisfies the Princen-Bradley condition for the MDCT (using the fact that wnk = wk): dk2 + dk + n2 = 1 (interpreting k and k + n modulo 2n). The KBD window is also symmetric in the proper manner for the MDCT: dk = d2n−1−k.

References

  • A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).
  • J. F. Kaiser, "Digital Filters," System Analysis by Digital Computer chap. 7 (Wiley: New York, 1966); F. F. Kuo and J. F. Kaiser, eds.
  • Marina Bosi, Kaiser-Bessel Derived Window (http://ccrma-www.stanford.edu/courses/422/projects/kbd/), Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2005).ja:カイザー窓
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