The array factor of the Bessel planar arrays is given in (1) and (4) for even and odd numbers of elements, respectively, where the excitation currents are as defined in (12). For θ0 = ϕ0 = 0, the highest sidelobe in the array pattern is in the ϕ = 0 plane, and due to the symmetry of the pattern, it also occurs in the ϕ = π/, ϕ = π, and ϕ = 3π/2 planes. The equation of the array factor for ϕ = 0 becomes
Using the properties of Chebyshev polynomials [Balanis, 1997], equation (13) can be written as
where z = cos[πdx/λsinθ], and Tp(·) denotes a Chebyshev polynomial of order p. Clearly, F(z) is a polynomial in z. The normalized version of F(z) is
The top points of the sidelobes in the ϕ = 0 planes correspond to real values of θ, and consequently z, at which the derivative of Fn(z) with respect to z vanishes. Since the highest sidelobe in a Bessel planar array is one that is directly next to the main lobe, the z value at which the maximal sidelobe level occurs is the maximum real root of dFn(z)/dz. Calling it zm, then
The corresponding θ value is
The dB value of the maximal sidelobe level (MSLL) is then
For every β value, a set of Imn values is obtained, and the corresponding MSLL is computed from (14), (15), (16), and (18). This makes it possible to plot the MSLL versus β, and from the obtained plots, to find the β value needed to set a desired maximal sidelobe level. Through experimental results, it was found out that this value is independent of dx and dy, and from θ0 and ϕ0, which were set to zero for simplification, and depends solely on L. A closed-form expression for β as a function of the MSLL is not feasible. In the Kaiser window design of FIR filters, only an empirical formula exists that relates β to the filter's ripple, and not even to the sidelobe levels of the Fourier transform of the window.