Variable Projection Methods with Application to Sums of Exponentials in White Noise
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We examine Nonlinear Least-Squares NLS methods for estimating the pole and amplitude parameters for a sum of damped and undamped sinusoids observed in white noise. A special case of NLS, which takes advantage of the semilinear structure of this signal model, involves minimization of the Variable Projection Functional VPF. The VPF offers computational efficiency by providing a means of estimating the signal poles independently of the amplitudes. We extend the existing set of algorithms for optimizing the VPF by deriving the exact Hessian matrix and introducing a Newton algorithm based on the exact Hessian. We also develop algorithms for optimizing the prediction filter VPF, which is a special case of VPF especially suited for estimating the parameters of sums of exponentials, and introduce methods for constraining each error functional to contain known signal poles. In addition, we provide detailed background material covering computational linear algebra, optimization, and variable projection theory.
- Statistics and Probability