Advanced Probability and Statistics

Chapter 2 Remarks and Problems on Statistical Signal Processing

[1] Construct the Lattice filter order recursion for an ℝ^M-valued vector stationary stochastic process X(t), t ∈ ℤ by minimizing

$$\mathbb{E}[\Vert X(t)+{\displaystyle \sum }_{k=1}^{p}A(k)X(t-k){\Vert }^2]$$

with respect to the M × M prediction coefficient matrices A (k), k = 1, 2, ..., p.

hints: Setting the variational derivatives of the above energy w.r.t the A(m)′s to zero gives us the optimal normal equations in the form of block structured matrix equations

$$R\left(m\right)+{\displaystyle \sum _{k=1}^p{A}_p\left(k\right)R\left(m-k\right)}=0,m=1,2,\dots ,p$$

where

$$R\left(m\right)=\mathbb{E}\left(X\left(t\right)X{\left(t-m\right)}^T\right)\in {ℝ}^{M\times M}$$

Note that

$$R\left(-m\right)=R{\left(m\right)}^T$$

Note also that the optimal prediction error covariance is given by

$$R_e\left(p\right)=BbE(e_p\left(t\right)e_p{\left(t\right)}^{}$$