Kaiser Window - Frequency Response

Frequency Response

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

\underbrace{\frac{I_0\left(\pi \alpha \sqrt{1 - \left(\frac{2t}{M}\right)^2}\right)} {I_0(\pi \alpha)}}_{w(t)} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \underbrace{\frac{M\cdot\sinh\left(\pi \sqrt{\alpha^2-\left(M\cdot f\right)^2}\right)}{I_0(\pi \alpha)\cdot\pi \sqrt{\alpha^2-\left(M\cdot f\right)^2}}}_{W(f)}.

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

for all integer values of t,

and where rect is the rectangle function.

The smaller the value of |α|, the narrower the window becomes; α = 0 corresponds to a rectangular window. Conversely, for larger |α| the main lobe of 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).

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