Kaiser window

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The Kaiser window is a one-parameter family of window functions used for digital signal processing, and is defined by the formula ^[1]:

Kaiser window function for M = 128 and πα = 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. 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.
• M is an integer, and the length of the sequence is N=M+1.

Some definitions of the Kaiser window combine π and α into a single parameter. For instance, MATLAB names the shape factor β, where β = π * α.

When N is an odd number, the peak value of the window is w_M/2 = 1.  And when N is even, the peak values are w_N/2-1 = w_N/2 < 1.

[edit] Frequency response

Underlying the discrete sequence is this continuous-time function and its Fourier transform:

The maximum value of w(t) is w(0) = 1. The w_n sequence defined above are the samples of:

      for all integer values of t,

and where rect() is the rectangle function.

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(f) 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).

Frequency spectra of Kaiser windows for πα = 4 and 8.

[edit] 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} of length M+1, by the formula:

KBD window function for M = 128 and πα = 2, 8, 24, 100.

This defines a window of length 2M, where by construction d_n satisfies the Princen-Bradley condition for the MDCT (using the fact that w_M−n = w_n): d_n² + d_n + M² = 1 (interpreting n and n + M modulo 2M). The KBD window is also symmetric in the proper manner for the MDCT: d_n = d_2M−1−n.

[edit] Applications

The KBD window is used in the Advanced Audio Coding digital audio format.

[edit] Notes

1. ^ 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.

[edit] References

• 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.
• Craig Sapp, Kaiser-Bessel Derived Window Examples and C-language Implementation, Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2001).

[edit] External links

• FIR Filter Design: The Window Design Method – The Kaiser window, by Nguyen Huu Phuong