Detection Statistics for Laser Radar in Atmospheric Turbulence with Fluctuating Targets

The probability of detection and false-alarm rates are developed for laser radar systems perturbed by background radiation, fluctuating targets, and atmospheric turbulence. In particular, some results on the decibel loss due to atmospheric turbulence are presented.     

Probability Densities and Characteristic Functions for Fluctuating Targets

Probability density functions and characteristic functions for the projected area of a tumbling target object are derived together with the densities and characteristic functions for the corresponding signal-plus-noise voltage output of an infrared sensor. Several cases are considered in which the signal pulse arrival time is either known or unknown, a matched filter or a suboptimum filter is used; the signal pulse shape is either Gaussian or is unspecified; and the target orientation angles are either uniformly or nonuniformly distributed.     

The Detection of Moderately-Fluctuating Rayleigh Targets

Pulse train detection of fluctuating targets whose coherence time is long compared to the time between pulses and short compared to the pulse train duration is compared for two systems: (1) a recursive digital optimal receiver operating in conjunction with a coherent pulse train transmitter, and (2) a good (but not optimal) receiver operating with a block coherent frequency-hopping transmitter. The equivalence of this type of problem to noise-in-noise detection problems is demonstrated, the performance relations for both systems are derived and comparisons are made.     

Bayesian Detection of Targets of Unknown Location

In this paper the array detection problem for certain deterministic signals in spatially uncorrelated Gaussian noise is formulated in Bayesian terms and the required likelihood ratios and resulting performance characteristics are obtained. Primary emphasis is on detection of targets of unknown spatial location using a likelihood ratio approach, with the target location treated as a random parameter, rather than an estimator-correlator approach. The tradeoff among array parameters and signal parameters is demonstrated. It is shown that the dominant uncertainty affecting detection performance seems to be that of location when compared with uncertainty on energy and/or phase.     

An Upper Bound on Detection Probability for Fluctuating Signals

In the theory of signal detectability, the signal-to-noise ratio (SNR), defined as the quotient of the average received signal energy and the spectral density of the white Gaussian noise, is a fundamental parameter. For a signal which is exactly known, or known except for a random phase, this ratio uniquely defines the detection performance which can be achieved with a matched filter receiver. However, when the signal amplitude is a random parameter, the detection performance is changed and must be determined from the probability density function (pdf) of the amplitude. Relative to the case of a constant signal amplitude, such signal amplitude fluctuation usually degrades performance when a high probability of detection (Pd) is required, but improves performance at low values of Pd; the corresponding change in the required SNR is the so-called signal fluctuation loss Lf. Thus, since Lf in some cases represents an improvement in performance for low values of Pd, a question of at least theoretical interest is: how large might this improvement be, when the class of all signal amplitude pdf's is considered. The solution, presented here, results in a lower bound on the signal fluctuation loss Lf as a function of Pd, or equivalently an upper bound on Pd as a function of SNR. The corresponding most favorable pdf was determined using the Lagrange multiplier technique and results of a numerical maximization are included to provide insight into the general properties of the solution.     

Radar Detection of Correlated Targets in Clutter

This paper provides general models of radar echoes from a target. The rationale of the approach is to consider the echoes as the output of a linear dynamic system driven by white Gaussian noise (WGN). Two models can be conceived to generate N target returns: samples generated as a batch, or sequentially generated one by one. The models allow the accommodation of any correlation between pulses and nonstationary behavior of the target. The problem of deriving the optimum receiver structure is next considered. The theory of "estimator-correlator" receiver is applied to the case of a Gaussian-distributed time-correlated target embedded in clutter and thermal noise. Two equivalent detection schemes are obtained (i. e., the batch detector and the recursive detector) which are related to the above mentioned procedures of generating radar echoes. A combined analytic-numeric method has been conceived to obtain a set of original detection curves related to operational cases of interest. Finally, an adaptive implementation of the proposed processor is suggested, especially with reference to the problem of on-line estimation of the clutter covariance matrix and of the CFAR threshold. In both cases detection loss due to adaptation has been evaluated by means of a Monte Carlo simulation approach. In summary, the original contributions of the paper lie in the mathematical formulation of a powerful model for radar echoes and in the derivation of a large set of detection curves.     

Fluctuating Target Detection in Clutter Using Sidelobe Blanking Logic

In an earlier paper, Maisel [6] considered two-channel detection systems using a sidelobe blanking logic when a nonfluctuating target was present. This paper is an extension of the earlier work to include fluctuating targets. The Swerling I, II, III, and IV models are considered when single-pulse detection is of interest. An adaptive threshold procedure is also briefly discussed whereby the probability of false alarm at any given resolution cell is maintained constant, even though the input clutter level may vary from cell to cell or from beam position to beam position. Useful data are presented for detection probabilities in the range 0.5 to 0.9, for false alarm probabilities in the range 104 to 10-8, and for a false detection probability of 0.1 for a sidelobe target yielding an apparent signal to total noise power density ratio of 13.0 dB in the main beam receiver.     

Losses for Frequency Diversity Waveform Systems

The use of a frequency diversity waveform to improve target detectability by reducing the fluctuation loss is a well-established technique. The diversity gain is commonly computed under the assumptions of a matched filter processor and the independent fluctuation of the target from channel to channel. Sometimes these conditions are not met and the full diversity gain cannot be obtained. In order to accurately compute the attainable diversity gain, it is necessary to calculate losses that occur only for diversity waveforms. The losses that have been considered are 1) unknown target velocity loss, 2) dependent samples due to small channel-to-channel frequency spacing, 3) linear envelope detector law loss, and 4) constant false alarm rate (CFAR) lpss.     

Frequency Diversity in Low-Angle Radar Tracking

Elevation angle errors due to sea-reflected multipath are evaluated theoretically for a radar operating in an off-boresight monopulse tracking mode. The computer simulation accounts only for specular reflection. Angle estimates at three frequencies are assumed to be available. It is shown that one can improve upon a simple average of the three indicated angles by unequally weighting them according to rank (lowest, middle, or highest). Some sample computations show that there is an optimum difference between the three frequencies.     

Maximum-Likelihood Detection of Unresolved Radar Targets and Multipath

Interference in the form of multipath or uncooperative targets can seriously degrade the angle-of-arrival estimation accuracy of mutiplebeam processors. In this paper, the generalized likelihood ratio test is used to derive a test to detect the presence of interference for multiple beam processors. The detector performance is then analyzed in detail with respect to its dependence on signal-to-noise ratio (SNR), signal-to-interference ration (SIR), and on the relative phase between the target and interfering signals. It is shown that good detection performance can be obtained unless the phase difference between the target and interference signals is either in or out of phase.     

A Detection Algorithm for Optical Targets in Clutter

There is active interest in the development of algorithms for detecting weak stationary optical and IR targets in a heavy opticalclutter background. Often only poor detectability of low signal-to-noise ratio (SNR) targets is achieved when the direct correlation method is used. In many cases, this is partly obviated by using detection with correlated reference scenes [1, 2].This paper uses the experimentally justified assumption that most optical clutter can be modeled as a whitened Gaussian randomprocess with a rapidly space-varying mean and a more slowlyvarying covariance [2]. With this assumption, a new constant falsealarm rate (CFAR) detector is developed as an application of the classical generalized maximum likelihood ratio test of Neyman and Pearson. The final CFAR test is a dimensionless ratio. This test exhibits the desirable property that its probability of a false alarm(PFA) is independent of the covariance matrix of the actual noiseencountered. When the underlying noise processes are complex intime, similar considerations can yield a sidelobe canceler CFARdetection criterion for radar and communications. Performance analyses based on the probability of detection (PD)versus signal-to-noise ratio for several given fixed false alarm probabilities are presented. Finally these performance curves are validated by computer simulations of the detection process which use real image data with artificially implanted signals.     

Exact Detection Probability for Partially Correlated Rayleigh Targets

The detection probability PD of a radar receiver which postdetection integrates N pulses of an expqnentially correlated signal from a Rayleigh target in thermal noise is determined. At the limiting correlation coefficients, p = 1 and p = 0, the analysis yields, respectively, the well known Swerling case 1 and case 2 formulas. The effect of partial (0 ? p ? 1) correlation is exhibited in a set of curves of PD versus signal-to-noise ratio, X, for various N and p. Additional curves compare the exact fluctuation loss determined from the above analysis with an approximate expression universally employed by radar system engineers.     

On the Statistical Representation of Targets for Detection Studies

Statistical distribution functions were determined ined for the radar cross section for flying aircraft. Most distributions could be adequately represented by one or another member of the family of chi-square functions. The degree of freedom parameter varied widely.     

Adaptive Detection Probabilities for Fluctuating Target Models in Nonhomogeneous Gaussian Noise

Under the assumption that the average noise power may vary from cell to cell, new, more easily computed expressions are given for the probability of detecting a fluctuating target by means of a cell-averaging CFAR test. The generalized chi-square family of fluctuating targets is considered with the Swerling I and III models given as special cases.     

Polygonal Detection Thresholds for In-Phase and Quadrature Channels

A joint detection threshold corresponding to a circle around the origin of the in-phase and quadrature components of a radar IF process is equivalent to envelope thresholding. Square and octagonal approximations to the circle are evaluated for prescribed false-alarm probabilities and the variation of detection probability with signal phase.