Bill Correll

A Closed Form Expression for the Number of Costas Arrays of Arbitrary Order

We give a closed form expression for the number of Costas arrays of arbitrary order by counting them as lattice points in the chambers of a related hyperplane arrangement and suggest how recent algorithmic breakthroughs might be used to find them.

Mean-Squared-Error Prediction for Bayesian Direction-of-Arrival Estimation

In this article, we study the mean-squared-error performance of Bayesian direction-of-arrival (DOA) estimation in which prior belief about the target location is incorporated into the estimation process. Our primary result is an extension of the method of interval errors (MIE) to the case of maximum a posteriori (MAP) direction-of-arrival estimation. We work in a general framework in which the prior information used in the MAP estimation may not match the actual target distribution. In particular, when the prior is incorrect, the MAP estimator degrades relative to the performance of a MAP estimator with the correct prior. Our methods are able to accurately predict the performance of a MAP estimator in this more general situation. We apply our methods to investigate the sensitivity of MAP direction-of-arrival estimation to mismatches between the chosen prior and the actual angular distribution of the target.

Prior mismatch in Bayesian direction of arrival estimation for sparse arrays

We study the mean-squared-error (MSE) performance of Bayesian direction-of-arrival (DOA) estimation for sparse linear arrays in which prior belief about the target location is incorporated into the estimation process. We utilize a recent extension of the method of interval errors (MIE) to the case of maximum a posteriori (MAP) direction-of-arrival estimation to more accurately predict low-medium MSE values in the presence of prior mismatch. We also develop a misspecified Cramér-Rao bound on MAP estimation that can improve the performance of MIE. We specialize to log-periodic arrays to conduct a notional trade study in which we consider the trade in improved estimation performance potentially possible with larger sparser arrays vs the increased sensitivity to incorrectly specified priors.

Costas arrays and the Lovász Local Lemma

We propose a fundamentally new approach to studying the existence of Costas arrays. We explore the consequences of applying new formulations of the Lovász Local Lemma in an effort to apply the probabilistic method to show that Costas arrays of a given order n must exist. Our applications of the Lovász Local Lemma make use of a characterization of all O(n6) configurations of ones within a permutation matrix whose existence violates the definition of a Costas array. We show that this approach can at least be used to establish the existence of arrays of orders 3 - 5 and also can be applied to prove the existence of frequency-hopping waveforms of all orders satisfying many subsets of at least n(n - 1)(n - 2)2 - 2 of the violating constraints.

The density of Costas arrays and three-free permutations

We describe how 3-free permutations might be used to improve Daviess density result for Costas arrays. We associate both Costas arrays and 3-free permutations with a pair consisting of a lattice point and a hyperplane arrangement to establish relationships between the two and compare how the exponential decay predicted by bounds on the number of 3-free permutations compares with a claim of Drakakis.

Waveform-dependent Bayesian Cramér-Rao angle-estimation bounds and threshold SNR estimates for MIMO radars

Multiple-input multiple-output (MIMO) radars operate by simultaneously transmitting multiple independent waveforms. This facilitates improved angle-estimation performance by enabling the use of sparse antenna arrays without the ambiguities that occur when sparse arrays are used in conventional radars. Angle-estimation performance can be characterized in terms of the local error-performance bound given by the Cramér-Rao bound and in terms of the threshold point given by the SNR at which the estimator deviates significantly from the Cramér-Rao bound. In this paper, we extend results of Bliss, Forsythe, and Richmond on angle-estimation performance as a function of transmit waveform covariance for a MIMO radar. The analysis described in the above work is dependent upon an estimate or test location of a target position. Here, we provide a framework for a Bayesian extension that incorporates knowledge of the priors on the target position probability density. This information affects both the Cramér-Rao bound and the threshold SNR. Consequently, it affects waveform and system optimization.

Selecting appropriate Costas arrays for target detection

In this paper we consider requirements-driven selection of Costas arrays for detection of closely-spaced targets. We review order selection strategies and treat selection of specific Costas arrays. Our analyses prioritize a clear central region in discrete ambiguity functions and address maximal frequency hops. We announce and use a new database of all known Costas arrays out to order 1030.