Monopulse-radar tracking of swerling III-IV targets using multiple observations

This paper proposes a novel statistical prediction of monopulse errors (Levanon, 1988) for a radar Swerling III-IV target embedded in noise or noise jamming where multiple observations are available. First, the study of the maximum likelihood estimator (MLE) of the complex monopulse ratio for a Swerling III-IV target embedded in spatially white noise allows us to extend the use of the MLE practical approximate form introduced by Mosca (1969) for Swerling 0-I-II cases. Afterward, we derive analytical formulas for both the mean and variance of the MLE in approximate form conditioned by the usual detection step performed on the sum channel of a monopulse antenna. Last, we provide a comparison of target direction of arrival (DOA) estimation performance based on monopulse ratio estimation as a function of the Swerling model in the context of a multifunction radar.     

Adaptive Detection Algorithms for Multiple-Target Situations

The performance of a mean-level detector is considered for the case where one or more interfering target returns are present in the set of cells used in estimating the clutter-plus-noise level. A serious degradation of detection probability is demonstrated for all of the single-pulse Swerling target fluctuation models (i. e., cases 0, 2, and 4). Indeed, for fixed mean radar cross sections of the primary and interfering targets, the probability of detecting the primary target is asymptotic to values significantly less than unity as the signal-to-noise ratios of the returns approach infinity. A class of alternative adaptive detection procedures is proposed and analyzed. These procedures, based on ranking and censoring techniques, maintain acceptable performance in the presence of interfering targets, and require only a minor addition in hardware to a conventional mean-level detector.     

Radar measurement accuracy for fluctuating targets

The impact of target radar cross-section (RCS) fluctuations on the thermal noise limited accuracy of radar measurements of range, range rate, and angle are evaluated for the Swerling fluctuation models. For large signal-to-noise ratios (SNRs), the accuracy criterion used is the measurement error with the same probability span as the one corresponding to the standard deviation points of the measurement error for the nonfluctuating model.<>     

Simple Procedures for Radar Detection Calculations

The literature of radar contains results of Rice, Marcum, Swerling, and Schwartz in several families of curves, which permit radar engineersto estimate the signal energy ratio required for a given level of detectionperformance. The variety of radar problems, however, makes itimpractical to construct curves for all combinations of radar and targetparameters. The concept of detector loss is used here to evaluate lossesattributable to integration and collapsing, with an accuracy of ±0.3 dBon steady targets. This is added to a separate fluctuation loss, modifiedfor diversity effects, to obtain results on all Swerling target modelsand also on partially correlated targets. The accuracy of the combinedlosses is ±0.5 dB for a wide range of detection and false-alarm probabilities.Starting from the basic single-sample detection curves, onlythree additional graphs are needed to find the energy ratio for givendetection performance in any of these cases. Examples are given whichshow the ease with which different radar options may be compared asto performance on an arbitrary type of target.     

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.     

Radar Track Acquisition Range

The track acquisiton range of a track-while-scan radar is defined as the range at which the cumulative probability of M detections in N scans is 90 percent. Track acquisition ranges for 2 out of 3, 2 out of 4, and 3 out of 4 detection criteria are presented for Swerling 0, 1, and 3 target models.     

Performance of Sidelobe Blanking Systems

Sidelobe blanking systems are useful in preventing acquisition of strong targets in the antenna sidelobes and also in rejecting pulsed interference originating in the sidelobes. The analysis of a common two-channel system is presented in which the relationship between the probability of main-lobe detection and the probability of sidelobe detection are given in terms of false-alarm probability, signal-to-noise ratio, and the ratio of sidelobe levels of the two channels. The numerical results given provide a basis for the selection of the sidelobe blanking channel antenna gain and threshold levels.     

Impact of target RCS fluctuations on radar measurement accuracy

The impact of target radar cross section (RCS) fluctuations on the thermal noise limited accuracy of radar measurements of range, range rate, and angle is evaluated for Swerling fluctuation models. The impact is expressed as a modification of the large-signal approximation to the standard deviation σ of measurement error     

Hansen's Method Applied to the Inversion of the Incomplete Gamma Function, with Applications

Hansen's method for obtaining the threshold for a speciried false alarm probability following noncoherent integration after square law detection is applied to finding the inverse of the incomplete gamma function. The method is algebraic and direct, circumventing the necessity for repeated evaluation of an integral or of a finite series. The method is applied to finding any specified percentile of the chi-square distribution and to finding the required signal-to-noise ratio for a specified detection probability of a Swerling II target.     

Nonparametric rank detectors under K-distributed clutter in radar applications

This correspondence deals with a comparative analysis of parametric detectors versus rank ones for radar applications, under K-distributed clutter and nonfluctuating and Swerling II target models. We show that the locally optimum detectors (LODs) (optimum for very low signal-to-clutter ratio (SCR)) under K-distributed clutter are not practical detectors; on the contrary, asymptotically optimum detectors (optimum for high SCR) are the practical ones. The performance analysis of the parametric log-detector and the nonparametric (linear rank) detector is carried out for independent and identically distributed (IID) clutter samples, correlated clutter samples, and nonhomogeneous clutter samples. Some results of Monte Carlo simulations for detection probability (P/sub d/) versus SCR are presented in curves for different detector parameter values.     

Threshold Control for Automatic Detection in Radar Systems

In automatic detection in radar systems an estimate of background clutter power is used to set the detection threshold. Usually detection cells surrounding the cell under test for the presence of a target are used to estimate the clutter power. In the research reported herein, the target location is taken to be uncertain and thus returns from a target could corrupt this clutter power estimate. It is shown how the threshold should be varied to compensate for the resulting degradation in detection performance. The threshold control procedure is based on a priori information about target location that could be supplied by the radar's tracking system. In addition, a simple procedure for calculating detection and false alarm probabilities for Swerling II target models is presented.     

Detection of Slow Fluctuating Targets with Frequency Diversity Channels

In this paper an exact closed-form expression for the radar detection probability is derived and results are plotted for a frequency diversity radar receiver. The receiver model performs post-detection integration on all received pulses in all diversity channels. The target model assumed is the slow fluctuating Rayleigh-distributed (Swerling case I target) scatterer. Each of the M frequency diverse channels receives N amplitude-correlated returns to give a total of NM post square-law detection integrations. The tabulated data falls between the two extreme cases, that for which all the returns are amplitude-correlated and that for which each return is independent. The plotted results fall close to the figures obtained through simple empirical relationships.     

Performance analysis of linearly combined order statistic CFARdetectors

Linearly combined order statistic (LCOS) constant false-alarm rate (CFAR) detectors are examined for efficient and robust threshold estimation applied to exponentially distributed background observations for improved detection. Two optimization philosophies have been employed to determine the weighting coefficients of the order statistics. The first method optimizes the coefficients to obtain efficient estimates of clutter referred to the censored maximum likelihood (CML) and best linear unbiased (BLU) CFAR detectors. The second optimization involves maximizing the probability of detection under Swerling II targets and is referred to as the most powerful linear (MPL) CFAR detector. The BLU-CFAR detector assumes no knowledge of the target distribution in contrast to the MPL-CFAR detector which requires partial knowledge of the target distribution. The design of these CFAR detectors and the probability of detection performance are mathematically analyzed for background observations having homogeneous and heterogeneous distributions wherein the trade-offs between robustness and detection performance are illustrated     

Modified Generalized Sign Test Processor for 2-D Radar

The modified generalized sign test processor is a nonparametric, adaptive detector for 2-D search radars. The detector ranks a sample under test with its neighboring samples and integrates (on a pulse-to-pulse basis) the ranks with a two-pole filter. A target is declared when the integrated output exceeds two thresholds. The first threshold is fixed and yields a 10-6 probability of false alarm when the neighboring samples are independent and identically distributed. The second threshold is adaptive and maintains a low false-alarm rate when the integrated neighboring samples are correlated and when there are nonhomogeneities, such as extraneous targets, in the neighboring cells. Using Monte Carlo techniques, probability of false-alarm results, probability of detection curves, and angular accuracy curves have been generated for this detector. The detector was built and PPI photographs are used to indicate the detector's performance when the radar is operated over land clutter.     

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.     

Probability of Detecting a Swerling I Targetet on Two Correlated Observations

Modern radars characterized by electronically steered beams, frequency agility, and range-ambiguous waveforms can use a processing technique whereby each new detection is followed by a second dwell to verify the initial detection. The second dwell is used to minimize false alarms and to resolve range and/or Doppler ambiguities. Under the assumption of a Swerling I target model, the target cross section remains constant when both dwells occur on the same scan and the same transmission frequency is used. Analytic expressions have been developed for the probability of detecting a Swerling I target on both of the correlated dwells (same target cross section). These expressions are used to calculate the probability of a second dwell detection given a first dwell detection and the probability of at least one detection on two dwells. These probabilities are compared with those of independent dwells (independent target cross sections) which result when two transmission frequencies of sufficient separation are utilized.     

Performance Analysis of the Censored Mean-Level Detector

The censored mean-level detector (CMLD) is an alternative to the mean-level detector that achieves robust detection performance in a multiple-target environment by censoring several of the largest samples of the maximum likelihood estimate of the background noise level. Here we derive exact expressions for the probability of detection of the CMLD in a multiple-target environment when a fixed number of Swerling II targets are present. The primary target is modeled by Swerling case II, and only single-pulse processing is analyzed. Optimization of the CMLD parameters is considered, and a comparison to other detectors is presented.     

CFAR detection in clutter with unknown correlation properties

We develop a constant false-alarm rate (CFAR) approach for detecting a random N-dimensional complex vector in the presence of clutter or interference modeled as a zero mean complex Gaussian vector whose correlation properties are not known to the receiver. It is assumed that estimates of the correlation properties of the clutter/interference may be obtained independently by processing the received vectors from a set of reference cells. We characterize the detection performance of this algorithm when the signal to be detected is modeled as a zero-mean complex Gaussian random vector with unknown correlation matrix. Results show that for a prescribed false alarm probability and a given signal-to-clutter ratio (to be defined in the text), the detectability of Gaussian random signals depends on the eigenvalues of the matrix R_c^-1R_s. The nonsingular matrix R_c and the matrix R_s are the correlation matrices of clutter-plus-noise and signal vectors respectively. It is shown that the “effective” fluctuation statistics of the signal to be detected is determined completely by the eigenvalues of the matrix R_c^-1R_s. For example the signal to be detected has an effective Swerling II fluctuation statistics when all eigenvalues of the above matrix are equal. Swerling I fluctuation statistics results effectively when all eigenvalues except one are equal to zero. Eigenvalue distributions between these two limiting cases correspond to fluctuation statistics that lie between Swerling I and II models     

Data quantization effects in CFAR signal detection

In this paper, we investigate data quantization effects in constant false alarm rate (CFAR) signal detection. Exponential distribution for the input data and uniform quantization are assumed for the CFAR detector analysis. Such assumptions are valid in the case of radar for a Swerling I target in Gaussian clutter plus noise and a receiver with analog square-law detection followed by analog-to-digital (A/D) conversion. False alarm and detection probabilities of the cell averaging (CA) and order statistic (OS) CFAR detectors operating on quantized observations are analytically determined. In homogeneous backgrounds with 15 dB clutter power fluctuations, we show analytically that a 12-bit uniform quantizer is sufficient to achieve false alarm rate invariance. Detector performance characteristics in nonhomogeneous backgrounds, due to regions of clutter power transitions and multiple interfering targets, are also presented and detailed comparisons are given     

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.