8. Analysis of a Linear Time Invariant Relation between Two Vector-Valued Stochastic Series
Society for Industrial and Applied Mathematics eBooks · 2001
8.1 INTRODUCTION Consider an (r + s) vector-valued stationary series [X (t) Y (t) ] 8.1.1 with X(t) r vector-valued and Y(t) s vector-valued. We assume the series (8.1.1) satisfies Assumption 2.6.1 and we define the means EX (t)= cX EY (t)= cY , 8.1.2 the covariances E { [X (t+u) − cX ] [X (t) − cX ]τ } = cXX (u) E { [X (t+u) − cX ] [Y (t) − cY ]τ } = cXY (u) E {[Y (t+u) − cY ] [Y (t) − cY ]τ } = cYY (u) u=0,±1,… , 8.1.3 and the second-order spectral densities fXX (λ) = (2π)−1 ∑ u=−∞ ∞ cXX (u) exp {−iλu} fXY (λ) = (2π)−1 ∑ u=−∞ ∞ cXY (u) exp {−iλu} fYY (λ) = (2π)−1 ∑ u=−∞ ∞ cYY (u) exp {−iλu} for−∞<λ<∞. 8.1.4