The Kaiser window, also known as the Kaiser–Bessel window, was developed by James Kaiser at Bell Laboratories. It is a one-parameter family of window functions used in finite impulse response filter design and spectral analysis. The Kaiser window approximates the DPSS window which maximizes the energy concentration in the main lobe but which is difficult to compute.
Definition The Kaiser window and its Fourier transform are given by:
w 0 ( x ) ≜ { 1 L I 0 [ π α 1 − ( 2 x / L ) 2 ] I 0 [ π α ] , | x | ≤ L / 2 0 , | x | > L / 2 } ⟺ F sin ( ( π L f ) 2 − ( π α ) 2 ) I 0 ( π α ) ⋅ ( π L f ) 2 − ( π α ) 2 , {\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}{\frac {I_{0}\left[\pi \alpha {\sqrt {1-\left(2x/L\right)^{2}}}\right]}{I_{0}[\pi \alpha ]}},\quad &\left|x\right|\leq L/2\\0,\quad &\left|x\right|>L/2\end{array}}\right\}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {\sin {\bigg (}{\sqrt {(\pi Lf)^{2}-(\pi \alpha )^{2}}}{\bigg )}}{I_{0}(\pi \alpha )\cdot {\sqrt {(\pi Lf)^{2}-(\pi \alpha )^{2}}}}},}
where:
I0 is the zeroth-order modified Bessel function of the first kind, L is the window duration, and α is a non-negative 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. Sometimes the Kaiser window is parametrized by β, where β = πα. For digital signal processing, the function can be sampled symmetrically as:
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