This paper examines the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the problem of n signals closely spaced in frequency. The main results presented are simple analytic expressions for the CR bound in terms of the maximum frequency separation, delta omega , SNR, and the number of data vectors, N, that are valid for small delta omega . The results are applicable to the conditional (deterministic) signal model. The results show that the CR bound on frequency estimates is proportional to ( delta omega )/sup -2(n-1)//N*SNR. Therefore, the bound increases rapidly as the signal separation is reduced. Examples indicate that the expressions closely approximate the exact CR bounds whenever the signal separation is smaller than one resolution cell. Based upon the results, it is argued that the threshold SNR at which an unbiased estimator can resolve n closely spaced signals is at least proportional to ( delta omega )/sup -2n//N. The results are quite general and apply to many different types of temporal and spatial sampling grids. >

The Cramer-Rao bound on frequency estimates of signals closely spaced in frequency

We propose a beamsplitting-like approach to estimate the directions of arrival (DOA) of multiple radar targets present in the mainlobe of a rotating antenna. The proposed method is based on the maximum likelihood (ML) technique and it avoids the need for a difference channel by exploiting knowledge of the antenna main beam pattern. Two scenarios are considered: multiple targets with unknown deterministic complex amplitudes and multiple targets with Gaussian distributed random complex amplitudes. The performance of the proposed estimator is investigated through Monte Carlo simulation and it is compared with the Cramer-Rao lower bound (CRLB).

DOA estimation by exploiting the amplitude modulation induced by antenna scanning

A generalized theory of radar resolution has been developed to facilitate understanding of the fundamental resolution limitations of radar systems. Previous work by Woodward and Elspas to determine limitations on radar resolution led to the concept of an ambiguity function which is a quantitative measure of radar resolution in range and range rate. This theory has been extended to include simultaneous resolution in range, range rate, azimuth and elevation and led to the derivation of a four-dimensional ambiguity function. Resolution constants derived from the ambiguity function show clearly the trade-offs between system parameters and resolution. A new concept, "angular dispersion and compression," has been evolved from the theoretical development. An angular compression system, analogous to a pulse compression system, employs a pseudo-randomly dispersed pattern which is compressed in angle at the receiver with a correlation technique, to produce the effect of a narrow beam without having a physically narrow beam. The signal bandwidth's effect on the pattern of an antenna and on its angular resolution has been found to be slight, except for very large bandwidths. It has been concluded that trade-offs between signal complexity and antenna complexity have no practical advantage. Because of the presence of noise, radar measurement of target parameters is essentially equivalent to statistical estimation. Woodward and Elspas have shown that the range, range-rate ambiguity function is the natural quantity to use in making a maximum likelihood estimate of range and range rate. The technique of maximum likelihood estimation has been extended to angular measurement.

Generalized Resolution in Radar Systems

This work deals with the problem of estimating complex amplitudes, Doppler frequencies, and directions of arrival (DOA) of multiple targets present in the same range-azimuth resolution cell of a surveillance radar. The maximum likelihood (ML) and the asymptotic (large sample size) ML (AML) estimators are derived. To reduce the computational complexity of the maximization of the nonlinear two-dimensional criterion function of the AML estimator, we propose a computationally efficient algorithm based on the RELAXation method. It allows decoupling the problem of jointly estimating the parameters of the signal components into a sequence of simpler problems, where the parameters of each component are separately and iteratively estimated. The proposed method overcomes the resolution limitation of the classical monopulse technique and resolves multiple targets exhibiting an arbitrarily small difference in azimuth as long as their Doppler frequencies differ at least by the inverse of the number of integrated pulses, provided that enough signal-to-noise ratio (SNR) per pulse is available. The performance of the proposed AML-RELAX estimator is numerically investigated through Monte Carlo simulation and Cramer-Rao lower bound (CRLB) calculation.

Multiple radar targets estimation by exploiting induced amplitude modulation

Formulas are presented for the maximum accuracy with which waveform parameters can be estimated in the presence of Gaussian noise. These formulas are presented in a form which is not widely known and which is very simple to use, especially in the white noise case. The unknown parameters may include the conventional ones, such as range, range rate, and acceleration, as well as others depending on target motion and configuration, properties of the transmission medium, or even sensor bias errors. A discussion is given of the conditions of validity of the formulas. The relationship between the approach followed here and other approaches to the parameter estimation problem is discussed.

Parameter estimation accuracy formulas

An investigation is made of the limits imposed by receiver noise on the accuracy with which the angular position of a target can be determined by a pulsed search radar. Using a result in the theory of statistical estimation, a lower bound is derived for the standard deviation of regular unbiased estimates of target angular position, for a large class of methods of angular position determination; the lower bound depends on scan rate, pulse repetition rate, beamwidth, beam shape, and signal-to-noise ratio. A similar analysis is made of the limits on angular accuracy imposed by a combination of receiver noise and one particular type of target cross section fluctuation. Operations which can be performed on the received signal to form an estimate of target angular position, the standard deviation of which approximately attains the theoretical lower bound, are discussed. The relation between the estimation of angular position and the problem of target detection is discussed. A graphical presentation of the main results is given.

Maximum Angular Accuracy of a Pulsed Search Radar

This paper discusses paradoxes related to the possibility of infinite information capacity of certain types of channels. First, a paradox of this type is derived which shows that such paradoxes are not necessarily dependent on the assumption of Gaussian statistics. Next, in the case where signal and noise are assumed to be Gaussian, a different example of this type of paradox is derived; also, a necessary and sufficient condition for the avoidance of this form of the paradox is derived. This condition is shown to be satisfied in a class of plausible physical situations.

Peter Swerling

Papers

Maximum Angular Accuracy of a Pulsed Search Radar

Paradoxes related to the rate of transmission of information