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Applications of the Canonical Representation to Estimation and Detection in Colored Noise
The representation of a random process as the output of a causal and causally invertible linear system driven by white noise is called canonical and specifies, quite simply, the whitening filter for the process. Whitening filter techniques replace the observation process, without loss of information, by a white noise process and allow simple formulation of the solutions of estimation and detection problems in terms of the equivalent process obtained by the whitening. Constructive methods based on the solution of a matrix Riccati equation are given for determining the canonical representation of differentiable observation processes which consist of a linear combination of the component process of a finite dimensional Markov process. Implementation of filtering solutions and likelihood ratios for detection are then obtained in a common formulation for a variety of signal with colored noise situations. The approach emphasizes the canonical representation of the observation process while requiring a minimum of attention to models for signal and noise components of the observation. Finite time interval problems for differentiable processes require attention to 'initial condition' random variables and the solutions discussed account for their contribution in a natural way.
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