where [J.sub.[mu]] the
Bessel function of the first kind of order [mu] and {[j.sub.[mu],n}.sup.[infinity].sub.n=1] 1 is the increasing sequence of positive zeros of [J.sub.[mu]] (cf.
One mathematical description of these structures is called the "
Bessel Function Model".
(iv) Several implementation details for algorithm optimization including Woodbury matrix identity for dimension-reduction, pruning a basis function and the third kind
Bessel function approximation are discussed, and the CRB of DOA estimation is derived.
A solution of the generalized Bessel differential equation is the generalized
Bessel function of order v and degree [mu] specified by the power series with infinite radius of convergence
If c = 1 and k [right arrow] 1, then the generalized k-Bessel function defined in (1) reduces to the classical
Bessel function [J.sub.v] defined in Erdelyi [3].
where R(p) > 0; [K.sub.v](x) is the
Bessel function of the second kind defined by ([18], p.
[I.sub.0]([[beta].sub.m]r) is the zero-order modified
Bessel function. The partial differentials of the function y to r and z are the two velocity solutions, [u.sub.r] and [u.sub.z], which are shown below:
Using the integral form of the
Bessel function and the limitation N [right arrow] [infinity], the normalized radiation vector [??]([theta], [phi], z) of the array can be deduced as:
(Here [J.sub.v] is
Bessel function of the first kind of order v [13].) Let
where [J.sub.[alpha]] denotes the usual
Bessel function of first kind and order [alpha].