##

Saddlepoint Approximation

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

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

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