5.25 FOURTH-ORDER GAUSSIAN MOMENT
A fourth-order moment for Gaussian random variables is provided in the following theorem, which is useful later in Chapter 12 on adaptive filtering.
Theorem 5.12 Let {X1, X2, X3, X4} be zero-mean jointly Gaussian random variables. Their joint moment is
(5.348)
Proof. Define the Gaussian random vector whose CF is given by (5.109) after substituting RXX and :
(5.349)
where . Because X has zero mean, this function is real valued (it is not a function of j as are most CFs). Typically such exponentials can be analyzed using a Maclaurin series:
(5.350)
where ··· represent high-order terms. The fourth moment is computed as follows:
(5.351)
which after differentiation and substituting , only the third term in (5.350) remains:
(5.352)
The autocorrelation matrix ...
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