Y.T. Chan

A Kalman Filter Based Tracking Scheme with Input Estimation

Beginning with the derivation of a least squares estimator that yields an estimate of the acceleration input vector, this paper first develops a detector for sensing target maneuvers and then develops the combination of the estimator, detector, and a simple Kalman filter to form a tracker for maneuvering targets. Finally, some simulation results are presented. A relationship between the actual residuals, assuming target maneuvers, and the theoretical residuals of the simple Kalman filter that assumes no maneuvers, is first formulated. The estimator then computes a constant acceleration input vector that best fits that relationship. The result is a least squares estimator of the input vector which can be used to update the simple Kalman filter. Since typical targets spend considerable periods of time in the constant course and speed mode, a detector is used to guard against automatic updating of the simple Kalman filter. A maneuver is declared, and updating performed, only if the norm of the estimated input vector exceeds a threshold. The tracking sclheme is easy to implement and its capability is illustrated in three tracking examples.

Comparison of various periodograms for sinusoid detection and frequency estimation

With the advent of the fast Fourier transform (FFT) algorithm, the periodogram and its variants such as the Bartletts procedure and Welch method, have become very popular for spectral analysis. However, there has not been a thorough comparison of the detection and estimation performances of these methods. Different forms of the periodogram are studied here for single real tone detection and frequency estimation in the presence of white Gaussian noise. The threshold effect in frequency estimation, that is, when the estimation errors become several orders of magnitude greater than the Cramer-Rao lower bound (CRLB), is also investigated. It is shown that the standard periodogram gives the optimum detection performance for a pure tone while the Welch method is the best detector when there is phase instability in the sinusoid. As expected, since the conventional periodogram is a maximum likelihood estimator of frequency, it generally provides the minimum mean square frequency estimation errors.

A Kalman Tracker with a Simple Input Estimator

Two Kalman filter based schemes are proposed for tracking maneuvering targets. Both schemes use least squares to estimate a targets acceleration input vector and to update the tracker by this estimate. The first scheme is simpler and by an approximation to its input estimator the computation can be considerably reduced with insignificant performance degradation. The second scheme requires two Kalman filters and hence is more complex. However, since one of its two filters assumes input noise, it may outperform the first scheme when input noise is indeed present. A detector that compares the weighted norm of the estimated input vector to a threshold is used in each scheme. Its function is to guard against false updating of the trackers and to keep the error covariance small during constant velocity tracks. Simulation results for various target profiles are included. They show that in terms of tracking performance, both schemes are comparable. However, because of its computation simplicity, the first scheme is far superior.

Hybrid localization of an emitter by combining angle-of-arrival and received signal strength measurements

Emitter localization, using different types of sensors and integrating their output properly can potentially increase localization accuracy. It is relatively easy to measure the received signal strength (RSS), but RSS localization is generally less accurate than angle-of-arrival (AOA) localization in most practical environments. When both RSS and AOA are available, fusing their measurements can improve localization accuracy. This paper develops a set of linear equations that optimally combines both AOA and RSS measurements, and is suitable for solution by weighted least squares. The simulation results show that at sufficiently low noise, the hybrid scheme is near optimal.

Direction finding with a four-element Adcock-Butler matrix antenna array

The conventional analog Adcock-Butler matrix (ABM) antenna array direction finder suffers from systemic errors, component matching problems, and bandwidth limitations. Three digital bearing estimators are developed as candidates to replace the analog signal processing portion of the ABM. Using the same antenna array, they perform all signal processing in the frequency domain, thereby benefitting from the computational efficiency of the fast Fourier transform (FFT) algorithm. The first estimator requires two analog-to-digital converters (A-D) and three antenna elements. It multiplies the difference between the discrete Fourier transforms (DFTs) of the output signals from two antenna elements with that from a third antenna element. At each frequency component, the phase of this product is a function of the bearing. A weighted least squares (LS) fit through all the phase components then gives a bearing estimate. The second estimator is similar to the first but uses three A-D and all four antenna elements. The output signal from the additional antenna element provides an independent estimate of the weights for the LS fit, giving an improvement in accuracy. The third estimator applies the physical constraint existing between the time-difference-of-arrival (TDOA) of a signal intercepted by two perpendicular sets of antenna elements. This yields a better estimator than simple averaging of the bearing from each set of antenna elements. The simulation studies used sinusoids and broadband signals to corroborate the theoretical treatment and demonstrate the accuracy achievable with these estimators. All three direction finders have superior performance in comparison with the analog ABM.

Evaluation of various FFT methods for single tone detection and frequency estimation

The periodogram, implemented using the fast Fourier transform (FFT), is widely used for the detection and frequency measurement of single tones. This paper evaluates the detection and frequency estimation performance of the periodogram and its variants, such as the Welch and Bartlett methods and the polyphase-FFT. Performance results for the detection and frequency estimation performance of the periodogram and its variants are presented and compared. The standard periodogram generally gives the best detection performance and the minimum mean square frequency error for a fixed length of signal data. However, if the FFT length is fixed the Welch method gives the best performance.

Detection of random signals via spectrum matching

Using a priori knowledge of the signal power spectral density (PSD), a spectrum matching approach which effectively utilizes the available signal spectral shape is developed for random signal detection. Two spectrum matching detector (SMD) structures, which are implemented by correlogram and periodogram, respectively, are examined. Theoretical calculation of their false alarm rates is derived and confirmed by simulations. It is also demonstrated that the proposed detectors outperform the standard periodogram, Bartlett method, and energy detector under constant false alarm rate (CFAR) condition for two different random signals.

Frequency Estimation of Uncooperative Coherent Pulse Radars

RF Frequency estimation is required in many applications, such as Radar Electronic Warfare (REW) and telecommunications. For example, passive location estimation of uncooperative radar sites in a target area is an important military application and can be achieved by measuring the Doppler-shifted frequencies of trains of modulated pulses received by an Electronic Support Measure (ESM) receiver on-board of a moving platform such as an aircraft or Unmanned Aerial Vehicle (UAV). The accuracy of the location technique depends on the accuracy of Doppler frequency measurements. There are several techniques that can be used to estimate accurately the frequency of continuous wave signals. However, estimating the frequency of trains of modulated pulses is more challenging because the pulse durations are very short. Furthermore, the computational complexity required for accurate estimation may become impractical if the Pulse Repetition Frequency (PRF) becomes small. In this paper, an FFT-based approach is considered. Techniques to improve the frequency step size, such as the Zoom FFT technique and the secant method, will be presented. Simulation results show that the frequency estimation approaches presented here can closely approach the Cramer-Rao Lower Bound (CRLB) in most cases.

Doppler Frequency Geolocation of Uncooperative Radars

Passive geolocation of uncooperative radar emitters remains an important problem in radar electronic warfare. Several location estimation techniques have been investigated in the past. In this paper, we present a passive geolocation technique for radar emitters using Doppler frequency measurements. For uncooperative sources, neither the emitter location, nor its transmitted frequency is known a priori. The relationship between these unknowns and the measured Doppler frequencies is non-linear. In the special case where the moving receiver measures frequencies along a straight path at constant speed, the relationship becomes linear in the Cartesian location coordinates. A simple 1-D discrete search for the transmitted frequency is followed by a least squares (LS) estimator to provide a coarse estimate of the emitter coordinates. This is followed by Newtons algorithm to provide a maximum likelihood (ML) estimation. The simulation results demonstrate that the resulting ML estimator approximately meets the Cramer-Rao lower bound (CRLB).

Detection of stochastic signals in the frequency domain

The optimum detector for a random signal, the estimator-correlator, is difficult to implement. If the power spectral density (PSD) of a continuous time signal is known, a locally optimum detector is available. It maximizes the deflection ratio (DR), a measure of the detector output signal-to-noise ratio (SNR). A discrete version of this detector is developed here, called the discrete-MDRD, which takes a weighted sum of the spectral components of the signal data as the detection statistic. Its derivation is applicable to nonwhite noise samples as well. A comparison of this new detector against three other common types, through their DR values and simulation results, reveals that the discrete-MDRD is near optimal at low SNRs. When the PSD of a signal is not known, a common test statistic is the peak of the PSD of the data. To reduce spectral variations, the PSD estimator first divides the data sequence into several segments and then forms the averaged PSD estimate. The segment length affects the DR values; the length that maximizes the DR is approximately the reciprocal of the signal bandwidth. Thus for unknown signal PSD, a detector that approaches the maximum DR is realizable from just the knowledge of the signal bandwidth, which is normally available. Examples and simulation results are provided to illustrate the properties and performance of the new detector.