Optimal Spectral Estimates.
NAVAL RESEARCH LAB WASHINGTON D C
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We consider the general problem of estimating the spectrum of a deterministic signal perturbed by noise. The emphasis of this report is on the reduction of the deterministic part of the error, and we wish to determine when equispaced sampling is optimal, and to what extent aliasing errors can be avoided by the use of nonequispaced sampling. It is shown that equispaced sampling is not always optimal so the problem reduces to determining whether any practical advantage would be gained by the use of optimal sampling schemes rather than equispaced sampling schemes when the latter are not optimal. The resolution of this question has been further reduced to a calculus problem, the minimization of an explicitly given function of N variables. This result has been achieved through the use of techniques borrowed from the theory of Sobolev spaces, which also have the advantage that they permit one to define the efficiency of a spectral estimator in terms of signal energy and signal bandwidth, whereas in the classical theories of numerical quadrature, quadrature errors are expressed in terms of bounds on the kth order derivatives of a function, parameters which have little significance in filter design. Author