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The lattice [A.sup.*.sub.n] is an important lattice because of its covering properties in low dimensions. Clarkson described an algorithm to compute the nearest lattice point in [A.sup.*.sub.n] that requires O(n log n) arithmetic operations. In this correspondence, we describe a new algorithm. While the complexity is still O(n log n), it is significantly simpler to describe and verify. In practice, we find that the new algorithm also runs faster. Index Terms--Channel coding, direction-of-arrival estimation, frequency estimation, lattice theory, nearest point algorithm, quantization, synchronization.

A frequency estimator employing only three Fourier coefficients have presented, which has asymptotic variance of order T(super -3). A similar procedure of Rife and Vincent is observed to have asymptotic variance of larger order. A new estimator that has asymptotic variance less than 1.65 times the CRLB is presented.

An extended Kalman filter (EKF) efficiently tracks very low signal-to-noise ratio (SNR) signals. Previous knowledge of the nature of the signal is integrated into the structure of the filter so as to increase the SNR. The EKF accurately tracks the slowly changing frequency of a signal with SNR between 0 and -30 dB. The effectiveness of the filter on simulated sonar data is discussed.

Rife and Boorstyn performance equations used in maximum likelihood estimation of frequency are simplified accurately. The simplified equation enables faster calculation of the initiation of the threshold as a function of signal to noise ratio and sample size. Computer simulation analysis reveals the accuracy of the new equations.

A theoretical approximation derived for the variance of Kay's weighted linear predictor frequency estimator is valid at medium to high signal-to-noise ratios or for large numbers of samples. Theoretical studies help derive a simple expression to describe the variance threshold of the estimator. Optimization of window weights leads to improved variance and signal-to-noise ratio. Number of samples influences the optimal weight.

A technique was developed to estimate the frequency of a sample through interpolation on three Fourier coefficients obtained successively. The order of the mean square error of the estimated frequency was similar to that of the Cramer-Rao lower bound. This technique was better than other conventional techniques such as Pisarenko's, associated with less number of sample autocorrelations.

A recent study on weighted linear predictor frequency estimator demonstrated the need for care in obtaining bounds to the variances of estimators. A new study was able to calculate an approximation to the variance of the estimator for fixed N and signal-to-noise ratio (SNR). Moreover, the new findings indicate that the relative error tended to zero as either N or the SNR increased.