Since no closed-form expression for the joint PDF of in (8.1) is known, we apply the Saddlepoint approximation [47],[16]. To obtain the SPA, we need the joint cumulant generating function (CGF) of , namely,

where is the joint moment-generating function (MGF) of . Also, we need the first and second-order partial derivatives of . Once these are known, the formulas in reference [16] may be used to obtain the SPA.

It is shown in [48] that

 (8.5)

where

with

and

The first-order partial derivatives are

for , and the second-order partial derivatives are

where

and we drop the dependence from , , and , for simplicity. The third and fourth derivatives, necessary for the first-order correction term of the SPA have also been worked out [48].

The equations simplify considerably if we assume that and are all zero and compute the PDF under the WGN assumption . We then have

 (8.6)

The first order partial derivatives reduce to

and the second order partial derivatives become

The SPA algorithm is provided in software/pdf_quadspa.m, which assumes that and are all zero and computed the PDF under the WGN () assumption.

Baggenstoss 2017-05-19