# Proof of equation involving fourth-order moments

How can you prove that

$$\mathrm{E}\left\lbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \mathrm{E}\left\lbrace \Delta \mathbf{w} (k) \Delta \mathbf{w}^{T} (k)\right\rbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \right\rbrace = 2 \mathbf{R}\mathrm{cov}\left[\Delta\mathbf{w}(k)\right]\mathbf{R} + \mathbf{R} \mathrm{tr}\left\lbrace \mathbf{R} \mathrm{cov}\left[ \Delta \mathbf{w}(k) \right] \right\rbrace$$

where $$\mathrm{E}\left\lbrace \cdot \right\rbrace$$ is the expected value, $$\mathbf{R} = \mathrm{E}\left\lbrace \mathbf{x}(k)\mathbf{x}^{T}(k) \right\rbrace$$, $$\mathrm{tr}\left\lbrace \cdot \right\rbrace$$ is the trace , and $$\mathrm{cov}\left[\Delta\mathbf{w}(k)\right] = \mathrm{E}\left\lbrace \Delta\mathbf{w}(k) \Delta\mathbf{w}^{T}(k) \right\rbrace$$.

This comes from Page 85 from Adaptive Filtering: Algorithms and Practical Implementation by Paulo S. R. Diniz (fourth edition). The book says that this calculations involves fourth-order moments and the results can be obtanied by expanding the matrix inside the operation $$\mathrm{E}\left\lbrace \cdot \right\rbrace$$ as describred in

4. J.E. Mazo, On the independence theory of equalizer convergence. Bell Syst. Tech. J. 58, 963–993 (1979)

and

13. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd edn. (McGraw Hill, New York, 1991)

for jointly Gaussian input signal samples.