# A Priori Bounds

• R. S. Bucy
Part of the Signal Processing and Digital Filtering book series (SIGNAL PROCESS)

## Abstract

We want to study the equation
$${P_{n + 1}} = {\phi _{n + 1}}\left( {{P_n}{{H'}_n}{{({H_n}{P_n}{{H'}_n} + {R_n})}^{ - 1}}{H_n}{P_n}} \right){\phi '_{n + 1}} + {G_{n + 1}}{Q_n}{G'_{n + 1}}$$
where Pno = Г. Recall that Pn+1 represents the prediction error covariance at time n + l given data to time n, while $${P_n} - {P_n}{H'_n}{\left( {{H_n}{P_n}{{H'}_n} + {R_n}} \right)^{ - 1}}{H_n}{P_n}$$ represents the filter error covariance, the error covariance matrix of the signal process x at time n given data to time n. We have and will assume the matrix Φ n is invertible for all n. When the processes arise from sampling of continuous time diffusion processes this assumption is fulfilled. Define the mapping τn+1 as
$${\tau _{n + 1}}({P_n})\mathop = \limits^\vartriangle {\phi _{n + 1}}{\left( {{P_n} - {P_n}{{H'}_n}({H_n}{P_n}{{H'}_n}) + {R_n}} \right)^{ - 1}}{H_n}{P_n}){\phi '_{n + 1}} + {G_{n + 1}}{Q_n}{G'_{n + 1}}$$

## Keywords

Riccati Equation Error Covariance Matrix Nonlinear Difference Equation Gaussian Random Vector Joint Information
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