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     

Tracking multiple targets with correlated measurements andmaneuvers

The problem of tracking N targets with correlation in both measurement and maneuver statistics is solved by transforming to a coordinate frame in which the N targets are decoupled. For the case of N identical targets, the decoupling is shown to coincide with a transformation to a set of nested center-of-mass coordinates. Absolute and differential tracking accuracies are compared with suboptimal results to show the improvement that is achieved by properly exploiting the correlation between targets     

Periodic ambiguity function of CW signals with perfect periodicautocorrelation

A periodic ambiguity function (PAF) is discussed which describes the response of a correlation receiver to a CW signal modulated by a periodic waveform, when the reference signal in the receiver is constructed from an integral number N, of periods T, of the transmitted signal. The PAF is a generalization of the periodic autocorrelation function, to the case of non-zero Doppler shift. It is shown that the PAF of N periods is obtained by multiplying the PAF of a single period by the universal function sin(Nπν T)/N sin(πνT), where ν is the Doppler shift, to phase-modulated signals which exhibit perfect periodic autocorrelation when there is no Doppler shift. The PAF of these signals exhibits universal cuts along the delay and Doppler axes. These cuts are functions only of t, N and the number M, the modulation bits in one period     

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     

Cascaded detector for multiple high-PRF pulse Doppler radars

A postdetection design methodology for a multiple high-pulse-repetition frequency (PRF) pulse Doppler radar has been developed. The postdetection processor consists of an M out of N detector where range and target ambiguities are resolved, followed by a square-law detector which enhances the minimum signal-to-noise (S/N) power-ratio per pulse burst performance. For given probabilities of false alarm and detection, formulas are derived from which the three thresholds associated with the cascaded detector can be found. Fundamental tradeoffs between the minimum S/N required, number of ghosts, and the number of operations (NOPs) that the cascaded detector must perform are identified. It is shown that the NOPs and the number of ghosts increase and the minimum S/N required decreases as the binary M out of N detector passes more detections to the square-law detector     

Convergence bounds of an SMI/Gram-Schmidt canceler in colored noise

The performance of the sampled matrix inversion (SMI) adaptive algorithm in colored noise is investigated using the Gram-Schmidt (GS) canceler as an analysis tool. Lower and upper bounds of average convergence are derived, indicating that average convergence slows as the input time samples become correlated. When the input samples are uncorrelated, the fastest SMI algorithm convergence occurs. When the input samples are correlated then the convergence bounds depend on the number of channels N, the number of samples per channels K , and the eigenvalues associated with K×K correlation matrix of the samples in a given channel. This matrix is assumed identical for all channels     

Multiple target angle tracking using sensor array outputs

The use of the output of an array of sensors to track multiple independently moving targets is reported. The output of each sensor in the array is the sum of signals received from each of the targets. The results of direction-of-arrival estimation by eigenvalue analysis are extended to derive a recursive procedure based on a matrix quadratic equation. The solution of this matrix quadratic equation is used to provide updated target positions. A linear approximation method for estimating the solution of the matrix equation is presented. The algorithm is demonstrated by the simulated tracking of two targets. The main advantage of the algorithm is that a closed-form solution for updating the target angle estimates has been obtained. Also, its application is straightforward, and the data association problem due to uncertainty in the origin of the measurements is avoided. However, it requires the inversion of an N×N as well as other linear operations, so that the computational burden becomes substantial as N becomes very large     

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     

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     

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     

Adaptive canceler and pulse compressor interactions

Performance results for the sidelobe level of a compressed pulse that has been preprocessed through an adaptive canceler are obtained. The adaptive canceler is implemented using the sampled matrix inversion algorithm. Because of finite sampling, the quiescent compressed pulse sidelobe levels are degraded due to the preprocessing of the main channel input data stream (the uncompressed pulse) through an adaptive canceler. It is shown that if N is the number of input canceler channels (main and auxiliaries) and K is the number of independent samples per channel, then K/N can be significantly greater than one in order to retain sidelobes that are close to the original quiescent sidelobe level (with no adaptive canceler). Also it is shown that the maximum level of degradation is independent of whether pulse compression occurs before or after the adaptive canceler if the uncompressed pulse is completely contained within the K samples that are used to calculate the canceler weights. This same analysis can be used to predict the canceler noise power level that is induced by having the desired signal present in the canceler weight calculation     

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     

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     

CW alternatives to the coherent pulse train-signals and processors

Continuous wave (CW) signals phase modulated by a periodic waveform, and their corresponding receivers, are discussed. The combined response in delay and Doppler is almost identical to the (ideal) response of the coherent pulse train. The receivers are matched to an integral number N of modulation periods of the transmitted signal. CW implies a duty cycle of 100%. However, the signal duration need not be longer than N+2 periods. The CW signals have the advantage that their peak power is equal to the average power. Their disadvantages are more complicated receiver processing and the need for two antennas     

False alarm control using Doppler estimation

A technique which uses maximum-likelihood estimates (MLEs) of target Doppler and target amplitude is developed for rejecting clutter residues. Multiple estimates are made and consistency checks are applied to the estimates. Simulation results indicate that for large clutter-to-noise ratios (C/N⩾55 dB) the probability of false alarm from clutter residues is reduced from 1.0 to below 0.01     

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     

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     

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     

Collapsing losses of peak signal integrators

Collapsing losses are computed for systems in which the peak return of K samples of noise plus one sample of signal-plus-noise are integrated over N looks. The statistical approach, collapsing losses, and an application are described. The peak integrator is found to have substantially lower collapsing losses than conventional systems in which the average, not the peak, is integrated     

Distributed detection by a large team of sensors in tandem

The problem of decentralized binary hypothesis testing by a team consisting of N decision makers (DMs) in tandem is considered. Each DM receives an observation and transmits a binary message to its successor; the last DM has to decide which hypothesis is true. The necessary and sufficient condition for the probability of error to go asymptotically to zero as N→∞ is that the log-likelihood ratio of the observation of each DM be unbounded. The result is generalized for multiple hypotheses and multiple messages. An easily implementable suboptimal decision scheme is also considered; in this case, the necessary and sufficient condition for the probability of error to asymptotically go to zero is that the log-likelihood ratio of the observation of each DM be unbounded from both above and below. The tradeoff between the complexity of the decision rules and their performance is examined, and numerical results are presented, in order to demonstrate that the performance of both decision schemes is comparable