CFAR data fusion center with inhomogeneous receivers

Detection systems with distributed sensors and data fusion are increasingly used by surveillance systems. A system formed by N inhomogeneous constant false alarm rate (CFAR) detectors (cell-averaging (CA) and ordered statistic (OS) CFAR detectors) is studied. A recursive formulation of an algorithm that permits a fixed level of false alarms in the data fusion center is presented, to set the optimum individual threshold levels in the CFAR receivers and the optimum `K out of N' decision rule in order to maximize the total probability of detection. The algorithm also considers receivers of different quality or with different communication channel qualities connecting them with the fusion center. This procedure has been applied to several hypothetical networks with distributed CA-CFAR and OS-CFAR receivers and for Rayleigh targets and interference, and it was seen that in general the fusion decision OR rule is not always the best     

Performances of order statistics CFAR

A previous analysis of order-statistics constant-false-alarm-rate (OS-CFAR) radar receiving a single pulse from a Rayleigh fluctuating target in a Rayleigh background is extended to a Rayleigh-plus-dominant target. The analysis includes effects of a multitarget environment. A detailed comparison of OS-CFAR, cell-averaging (CA) CFAR, and censored CA-CFAR is provided for a Rayleigh target in the presence of strongly interfering targets. The false-alarm analysis of OS-CFAR is extended to the more general case of a Weibull background. The deterioration of the CFAR property of OS as the shape factor, C, of a Weibull probability density function changes from Rayleigh (C=2) to a longer-tailed one (C<2) is evaluated. The analytic comparison between CA-CFAR and OS-CFAR is extended to an integration of pulses reflected from a Swerling II target. The OS-CFAR performance (with and without interfering targets) yields an integral equation that is solved numerically     

Explicit solutions for some simple decentralized detection problems

A decentralized detection problem is considered in which a number of identical sensors transmit a finite-valued function of their observations to a fusion center which makes a final decision on one of M alternative hypotheses. The authors consider the case in which the number of sensors is large, and they derive (asymptotically) optimal rules for determining the messages of the sensors when the observations are generated from a simple and symmetrical set of discrete distributions. They also consider the tradeoff between the number of sensors and the communication rate of each sensor when there is a constraint on the total communication rate from the sensors to the fusion center. The results suggest that it is preferable to have several independent sensors transmitting low-rate (coarse) information instead of a few sensors transmitting high-rate (very detailed) information. They also suggest that an M-ary hypothesis testing problem can be viewed as a collection of M(M-1)/2 binary hypothesis testing problems. From this point of view the most useful messages (decision rules) are those that provide information to the fusion center that is relevant to the largest possible numbers of these binary hypothesis testing problems     

Optimal serial distributed decision fusion

The problem of distributed detection involving N sensors is considered. The configuration of sensors is serial in the sense that the Jth sensor decides using the decision it receives along with its own observation. When each sensor uses the Neyman-Pearson test, the probability of detection is maximized for a given probability of false alarm, at the Nth stage. With two sensors, the serial scheme has a performance better than or equal to the parallel fusion scheme analyzed in the literature. Numerical examples illustrate the global optimization by the selection of operating thresholds at the sensors     

Distribution of general noncentral positive definite quadratic formin K-dimensions

The authors present a series solution using Hermite polynomials to the long-standing problem of computing the probability P that positive definite noncentral quadratic form d(x) of a Gaussian random vector x∈R satisfies d( x)⩽r² for any given r∈R. This problem has wide applications in radar, tracking, air traffic control, etc. The fast-converging series solution presented is very accurate and can be performed rapidly using the recursion relations for Hermite polynomials     

Estimation of radar detection and false alarm probability

The accuracy with which detection and false alarm probabilities can be estimated with a limited amount of measured radar data is addressed. A simple simulation method for estimating the statistical performance of a radar detection system is presented. Confidence limits and a rule of thumb for accuracy for the estimated probabilities are presented along with procedures for calculating them. It is concluded that the minimum value of N used in a detection radar signal simulation should be 10/P_FA when the simple simulation method is used, where P_FA is the probability of false alarm, and that a value closer to 100/P _FA is preferable     

Low-altitude target detection by coastline operated marine radar

Relevant to a Richian family of fluctuating targets with a composite background of sea-plus-land clutter, the performance prediction of a radar operating in near-coastal regions is elucidated by assuming noncoherent integration of the pulses. Considering the dominance of land clutter, a modified K-distributed statistic is indicated for the overall clutter envelope; and the corresponding probability of false alarm and probability of detection are deduced for fixed threshold detection (s) based on N pulses integrated in the presence of the sea-plus-land clutter and the noise. Even when the target offers a dominant scattered echo, the worst situations of the land clutter affecting the detection performance are indicated     

Nonparametric rank detectors on quantized radar video signals

Uniform randomization of ties is required for defining distribution-free ranks of independent and identically distributed quantized samples. Formulas of rank probabilities are given and applied to radar detection under quantized video samples. For some detectors, and assuming Gaussian noise, the asymptotic loss L_∞(q) is calculated versus the normalized quantization step q, and the loss L(q) is estimated by Monte Carlo simulations. Both of these resulted in monotonic functions of q (0<q<1.1) that are independent of the other parameters. Furthermore, L(q)≈L_∞(q )⩽0.45 dB, as q<0.8. The quantization step q is normalized with respect to the noise standard deviation     

Performance of receivers with linear detectors

The false-alarm and detection probabilities of a receiver summing M independent outputs of a linear detector are calculated by numerical saddlepoint integration. The saddlepoint approximation is also considered. Both constant-amplitude and Rayleigh-fading signals are treated, and the relative efficiency of the quadratic and the linear detectors for these is calculated for a broad range of values of M . The numerical integration method is the more efficient, the smaller the false-alarm probability or the false-dismissal probability, that is, under just those conditions for which the terms in the Gram-Charlier series oscillate most violently and the series becomes least reliable. The simpler saddlepoint approximation yields values that in those same regions have been found close enough to the exact probabilities to be adequate for most engineering purposes. The larger the number M of samples, the more efficient methods are     

Tracking multitarget in cluttered environment

A method for multitarget tracking and initiating tracking in a cluttered environment is proposed. The algorithm uses a sliding window of length uT (T is the sampling time) to keep the measurement sequence at time k. Instead of solving a large problem, the entire set of targets and measurements is divided into several clusters so that a number of smaller problems are solved independently. When a set of measurements is received, a new set of data-association hypotheses is formed for all the measurements lying in the validation gates within each cluster from time K-u+1 to K. The probability of each track history is computed, and, choosing the largest of these histories, the target measurement is updated with an adaptive state estimator. A covariance-matching technique is used to improve the accuracy of the adaptive state estimator. In several examples, the algorithm successfully tracks targets over a wide range of conditions     

Detection probabilities for correlated Rayleigh fading signals

Both the method of saddlepoint integration and its associated saddlepoint approximation are applied to calculating the probability of detecting correlated Rayleigh-fading signals in Gaussian noise by means of a detector that integrates M samples of the output of a quadratic rectifier. The quadrature components of the signal samples are modeled as an autoregressive moving-average process, and specific results are exhibited for a first-order Markov process. By these methods the fluctuation loss can be computed for much larger values of M and for larger values of the detection probability than previously. Values calculated by the saddlepoint approximation prove to be close enough to the exact values to be useful over a broad range of signal parameters     

Optimum gain control for A/D conversion using digitized I/Q data inquadrature sampling

Calculation of optimum gain for minimum distortion due to A/D (analog-to-digital) conversion requires the estimation of the input signal strength. To use a common AGC (automatic gain control) for both the I/Q (in-phase and quadrature) signals, it is efficient to estimate the input signal strength using the quantized A/D output from both channels. Assuming a Gaussian input, the relationship between σ of the input of the A/D converter and E(|x|+|y|) and E(max(|x|,|y|)+1/2 min (|x|,|y|)) for t quantized I/Q output x and y is derived. Numerical results obtained using the derived expression and the statistical data obtained through simulation show excellent agreement. It is concluded that, because of its simplicity, the cubic equation obtained by fitting the numerical results should be useful     

Optimal speckle reduction in polarimetric SAR imagery

Speckle is a major cause of degradation in synthetic aperture radar (SAR) imagery. With the availability of fully polarimetric SAR data, it is possible to use the three complex elements (HH, HV, VV) of the polarimetric scattering matrix to reduce speckle. The optimal method for combining the elements of the scattering matrix to minimize image speckle is derived, and the solution is shown to be a polarimetric whitening filter (PWF). A simulation of spatially correlated, K-distributed, fully polarimetric clutter is then used to compare the PWF with other, suboptimal speckle-reduction methods. Target detection performance of the PWF, span, and single-channel |HH|² detectors is compared with that of the optimal polarimetric detector (OPD). A novel, constant-false-alarm-rate (CFAR) detector (the adaptive PWF) is as a simple alternative to the OPD for detecting targets in clutter. This algorithm estimates the polarization covariance of the clutter, uses the covariance to construct the minimum-speckle image, and then tests for the presence of a target. An exact theoretical analysis of the adaptive PWF is presented; the algorithm is shown to have detection performance comparable with that of the OPD     

On the uniform sampling of a sinusoidal signal

It is generally accepted that a monochromatic signal such as sin 2πWt (W>0) must be sampled at a uniform rate greater than the ostensible Nyquist rate of 2W samples per to effect a reconstruction of the signal. It is shown that a sinusoid of frequency W Hz is completely determined by its samples taken at the uniform rate of 2r samples per second, where r>0 is arbitrary subject only to the restriction that W ≠kr for any positive integer k. In particular, a pure sinusoid may be sampled uniformly without loss of information at arbitrarily small rates     

A parallel square-root algorithm for modified extended Kalmanfilter

A parallel square-root algorithm and its systolic array implementation are proposed for performing modified extended Kalman filtering (MEKF). The proposed parallel square-root algorithm is designed based on the singular value decomposition (SVD) and the Faddeev algorithm, and a very large scale integration (VLSI) systolic array architecture is developed for its implementation. Compared to other square root Kalman filtering algorithms, the proposed method is more numerically stable. The VLSI architecture described has good parallel and pipelining characteristics in applying to the MEKF and achieves higher efficiency. For n-dimensional state vector estimations, the proposed architecture consists of O(2n²) processing elements and uses O ((s+17)n) time-steps for a complete iteration at each instant, in contrast to the complexity of O((s+6) n³) time-steps for a sequential implementation, where s≈log n     

Identification of rational transfer function from frequencyresponse sample

A method for identifying a transfer function, H(z)=A(z)/B(z), from its frequency response values is presented. Identifying the transfer function involves determining the unknown degrees and coefficients of the polynomials A(z) and B( z), given the frequency response samples. The method for finding the parameters of the transfer function involves solving linear simultaneous equations only. An important aspect of the method is the decoupled manner in which the polynomials A(z) and B(z) are determined. The author presents two slightly different derivations of the linear equations involved, one based on the properties of divided differences and the other using Vandermonde matrices or, equivalently, Lagrange interpolation. A matrix synthesized from the given frequency response samples is shown to have a rank equal to the number of poles in the system     

The autocorrelation of m-sequences over nonprime finitefields

Nonbinary m-sequences (maximal length sequences) for spread-spectrum communication systems that have a two-level autocorrelation are presented. The autocorrelation function of an m -sequence over the Galois field of q elements GF(q), where q=p^k, for p a prime and k an integer greater than 1, is developed and shown to be bilevel when the elements of GF(q) are expressed as elements of a vector space over the pth roots of unity     

Detection performance in K-distributed and correlatedRayleigh clutters

Performance prediction for a detection system employing noncoherent integration is carried out for a chi-square family of fluctuating targets in K-distributed clutter plus noise. The detection performance for Swerling 11 targets in the K-distributed clutter plus noise is compared with that in exponentially correlated Rayleigh clutter. The results show that the performance prediction based on N pulses integrated in clutter plus noise using the K-distributed clutter model may be approximately equivalent to that using the exponentially correlated Rayleigh-distributed clutter model     

Detection probability for partially correlated chi-square targets

The probability of detection of the sum of N square-law-detected pulses is derived for the case where the signal fluctuation obeys chi-square statistics with four degrees of freedom. P. Swerling's (1960) case III and IV represent the cases where the signal is completely correlated and completely decorrelated, respectively, from pulse to pulse. An exact expression for probability of detection is derived for the condition of partial signal correlation. The results given are compared with the approximate technique commonly used to handle partial signal correlation     

Finding the best set of K paths through a trellis withapplication to multitarget tracking

A solution is presented to the problem of finding the best set of K completely unmerged paths through a trellis with M _i⩾K states at depth i in the trellis, i=0, 1, 2, . . ., N. Here, `best set' means that the sum of the metrics of all K paths in the set is minimized, and `completely unmerged' means that no two paths pass through a common state. The solution involves using the Viterbi algorithm on an expanded trellis. This result is then used to separate the tracks of K targets optimally in a simplified model of a multitarget radar system. The model includes measurement errors and false alarms, but it does not include the effects of missing detections or merged measurements