A Method for System Identification with Random Inputs |
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and/or arbitrary basis basis choices basis elements c₁ causal system coefficients of h(t column vector convolution integral equation correlation functions D₁ Denote deterministic input deterministic problem dimensional vector error expansion of S(jw F(jw finite matrix equation Fourier series Fourier transform function R(T Gram-Schmidt process half plane pole identification of H(p impulse response h(T infinite matrix inner product input x(t integrand j,k jk Jordan's lemma LHP H(jw matrix equation 2.44 method for system orthogonal filters orthonormal basis orthonormal set output P₁ P₁₂ P₂ parameters pole locations poles of S(P poles per half random input problem rational function real exponentials residue evaluation respect RHP S(jw right-half plane poles right-half plane zeros Rxy(T s₁ shown spectrum stationary random input Sx(p Sxy(jw Sy(p symmetric matrix system identification problem transform of R(T triangular matrix TT jw unitary transformation unknown impulse response zero of H(p