Peter M. Schultheiss

Array shape calibration using sources in unknown locations--Part I: Far-field sources

This paper deals with source localization using a two-dimensional array of sensors whose locations are not known precisely. If only a single source is observed, uncertainties in sensor location increase errors in source bearing and range by an amount which is independent of signal-to-noise ratio and which can easily dominate over-all localization accuracy. Major performance gains could therefore result from successful calibration of array geometry. The paper derives Cramer-Rao bounds on calibration and source location accuracies achievable with far-field sources whose bearings are not initially known. The sources are assumed to radiate Gaussian noise and to be spectrally disjoint of each other. When the location of one sensor and the direction to a second sensor is known, three noncollinear sources are sufficient to calibrate sensor positions with errors which decrease to zero as calibrating source strength or time-bandwidth products tend to infinity. The sole exception to this statement is a nominally linear array for which such calibration is not possible. When one sensor location is known but no directional reference is available, three noncollinear sources can determine array shape, but there remains a residual error in angular orientation which is irremovable by the calibration procedure. When no sensor locations are known a priori, one adds to the residual error in rotation a translational component. In the far field, the latter should be unimportant. In addition to the asymptotic results, Cramer-Rao bounds are computed for finite signal-to-noise ratios and observation times. One finds that calibration permits significant reductions in localization errors for parameter values well within the practical range.

Array shape calibration using sources in unknown locations--Part II: Near-field sources and estimator implementation

Uncertainty concerning sensor locations can seriously degrade the ability of an array to estimate the location of radiating sources. Array calibration then becomes an important issue. This paper deals with array calibration using spectrally disjoint near-field sources whose locations are not known a priori. It calculates Cramer-Rao bounds on source location and array shape errors. It shows that, with mild restrictions on source-array geometry, the array shape errors can be made arbitrarily small by using a sufficient number of sufficiently strong calibrating sources. The required number of calibrating sources is five for a four-sensor array, four for a five-sensor array, and three for an array of six or more sensors. Accurate calibration is not possible for a three-sensor array. While calibration can establish array shape with great accuracy, it cannot resolve a rotational uncertainty in array orientation. This uncertainty translates into a residual error in source bearings, but not in source ranges. Thus, incremental errors in target range due to sensor location uncertainty can be reduced to any desired extent. The paper also proposes an actual calibration procedure and presents simulation results indicating that useful calibration can be obtained with calibrating sources of very moderate signal-to-noise ratio.

Optimum range and bearing estimation with randomly perturbed arrays

The absolute minimum attainable values of incremental mean‐square error in bearing and range estimation caused by random perturbations of sensor locations are determined and compared with the performance of conventional processors. At low signal‐to‐noise ratios the optimum estimator realizes no improvement over the conventional systems; at high signal‐to‐noise ratios it realizes only slight improvement.

Estimation of differential Doppler shifts

If the acoustic signal radiated by a moving source is observed at two or more locations, the received signals exhibit differential Doppler shifts which provide information about source motion. This paper calculates minimum mean‐square errors for the estimates of differential Doppler shifts, which can be obtained in a given observation interval. Both Gaussian and sinusoidal signals are considered. The noise is assumed to be Gaussian and independent from sensor to sensor. Dependence of the estimation errors on observation time, signal‐to‐noise ratio, and size of the receiving array are studied. The estimation of differential Dopplers is found to be uncoupled from the estimation of differential delays and from the estimation of signal parameters, such as center frequency and bandwidth. A comparison is made between two possible procedures of differential Doppler estimation: coherent processing of the signals received at two sensors and subtraction of separate frequency estimates obtained at each sensor. The t...

Realizable lower bounds for time delay estimation

The accuracy of time delay estimates obtainable in active localization systems is studied, focusing on the effect of ambiguities in the time delay estimates. Such ambiguities occur when the transmitted signal has small relative bandwidth. Then, for signal to noise ratios below a certain threshold, the commonly used Cramer-Rao lower bound is not realizable. The study concentrates on the region of intermediate SNR values, where the Cramer-Rao bound is no longer achievable, but useful information on time delays can still be obtained from the measurement. Realizable bounds for the single and two echo cases are obtained by deriving a new form of the Barankin (1949) bound for active time delay estimation. This form maintains the realizability property of the most general form, but is of reasonable complexity. New bounds are derived for the multiple echo case. Examples are presented to illustrate the dependence of the bound on parameters such as SNR, relative bandwidth, and echo separation. >

Realizable lower bounds for time delay estimation. 2. Threshold phenomena

In time delay estimation of narrowband signals the mean square error (MSE) plotted as a function of the signal to noise ratio (SNR) exhibits threshold phenomena. The thresholds divide the domain of SNR values into several disjoint segments. For very low SNR values the observations are dominated by noise and are essentially useless for delay estimation. Here the MSE is determined largely by the available a priori information. For intermediate SNR values, time delay estimates are possible, but are subject to ambiguities resulting from the oscillatory nature of the signal sample correlation. For very high SNR values, these ambiguities are resolvable and the Cramer-Rao bound (CRB) yields a realistic bound for the attainable performance. A previous paper by Zeira and Schultheiss (see IEEE Trans. Signal Processing, vol.41, p.3102-3113, Nov. 1993) used the Barankin (1949) bound to obtain a realizable lower bound for the intermediate SNR region but left open the question where the transitions from low to medium and medium to high SNR operation occur. Since the attainable MSE in the three regions can be very different, the location of the SNR thresholds is of considerable practical interest. The present paper addresses this issue. It obtains threshold values for the single and two echo problems, with emphasis on the effect of echo separation in the two echo case. It concludes that the location of the thresholds is only weakly dependent on echo separation, even though the attainable MSE in both the intermediate and high SNR regions varies drastically as the echo separation changes. >

Lower bounds on the localization errors of a moving source observed by a passive array

The Cramer-Rao inequality is used to set absolute bounds on the accuracy of location and velocity estimates obtainable by observing the signal radiated from a moving acoustic source at an array of stationary sensors. The source radiates a zero mean Gaussian random process and the observations are made in a background of spatially incoherent Gaussian noise. Results are first obtained for the error covariance matrix of a set of parameters characterizing the time variable differential delays observed at various sensor pairs. These are then translated into bounds on the error covariance matrix of a set of parameters describing source location and track. Numerical results are presented for the specific case of a source moving in a straight line course at constant velocity.

Short‐Time Frequency Measurement of Narrow‐Band Random Signals by Means of a Zero Counting Process

Any instrumentation for measuring the mean frequency of a narrow‐band random signal presents an output quantity which inherently fluctuates about that value which represents the true mean frequency, due to the finite measuring time. In comparing the accuracy of various instrumentations for short‐time frequency measurement a useful figure of merit is variance of the output/(sensitivity)2. In this paper the figure of merit is obtained for an instrumentation which measures frequency by determining the average number of zero crossings of the signal in a short time. The significant problem is the determination of the mean square number of zeros of a random function in time T. The general result is presented in the form of an integral. For an assumed Gaussian power spectrum the integral is integrated graphically to obtain a figure of merit which is compared with previously published figures of merit of an autocorrelator and frequency discriminator.

Passive Ranging in Multipath Dominant Environments: Part II-Unknown multipath parameters

Passive ranging accuracy using arrays deteriorates at long ranges because of the vanishing wavefront curvature observable by an array. The idea of including a priori knowledge of propagation conditions in ranging procedures for the purpose of improving the range estimate has received considerable attention in recent years. The interest is based on an observation that a large propagation delay measured between two coherent arrivals at a single receiver is equivalent geometrically to a delay generated by a single arrival in propagating to two widely spaced receivers. If the equivalent sensor spacing is on the order of or larger than the dimensions of an actual array, multipath ranging appears to be an attractive alternative to conventional curvature measurement approaches. This paper considers the potential performance improvements attainable in the process of exploiting a second path in long range ranging. The performance of line arrays is pursued in detail for a problem setting in which all of the parameters describing the signal and channel parameters are known a priori. >

Static and Sliding Friction in Feedback Systems

One of the most common nonlinearities encountered in servomechanisms design is the friction phenomenon in electromechanical systems. Conventional linear theory fails to predict its effect upon system performance. This paper extends familiar techniques to the treatment of friction nonlinearity in servosystems. Frequency‐response methods are employed throughout and the theoretical results are verified by means of an analog computer. Sliding friction and static friction are represented by describing functions which form the critical factors in determining system stability. The analysis indicates that certain series equalizers designed from linear theory may fail to achieve effective compensation in the presence of sliding and static friction. On the other hand, a subsidiary loop may avoid the stability problem while still realizing an essentially equivalent loop gain function.