Linear Time-Invariant (LTI) Systems and WSS Processes

Correlation Matrices for Random Processes

Filterbank Interpretation

Toeplitz vs Circulant Matrix Diagonalization

Application of DKLT

The DKLT can be used to compress the representation of Eigenfaces

Strong vs Weak Properties

For Gaussian processes, weak and strong properties are equivalent (uncorrelated ⟺ independent).

Property	Strong (Strict-Sense)	Weak (Wide-Sense)
Stationarity	f_{X[k_1],\ldots,X[k_n]}(x_1,\ldots,x_n) = f_{X[k_1+\kappa],\ldots,X[k_n+\kappa]}(x_1,\ldots,x_n)	\mathcal{E}\{X[k]\} = \mu R_{XX}[k_1,k_2] = R_{XX}[k_2-k_1]
Ergodicity	\lim_{N\to\infty} \frac{1}{N}\sum_{k=0}^{N-1} g(X[k]) = \mathcal{E}\{g(X[k])\} for all functions g	\langle X[k] \rangle_k = \mathcal{E}\{X[k]\} \langle X[k]X[k+\kappa] \rangle_k = R_{XX}[\kappa]
White Noise	Independent for k_1 \neq k_2 f_{X[k_1],X[k_2]}(x_1,x_2) = f_{X[k_1]}(x_1) \cdot f_{X[k_2]}(x_2)	Uncorrelated R_{XX}[k_1,k_2] = \gamma \delta[k_1 - k_2]