Radar Partial Coherence Theory: An Introduction

The nature of physical phenomena is such that scattering from portions of an object, a number of objects, or clutter, is not completely unrelated; the underlying environment causes some degree of order in the phenomenon. Radar partial coherence theory describes a structure for the general target, or clutter, and its relationship to radar cross section, waveform coding, and the radar output signal. The clutter ambiguity function is introduced for extended bodies and embraces the (Woodward) ambiguity function for a point target. Due to nonlinear effects caused by partial coherence within the general target, radar signals and targets are formulated in terms of mutual coherence functions. The basic quantities describing the radar output are 1) the radar mutual coherence function (formulated in terms of the radar waveform) and 2) the target mutual coherence function which depends upon target properties, physical environment, and viewing aspect. Random noise (independent point scatterers) and partially coherent portions of reflecting bodies are made accountable in the theory. Partial coherence effects are treated as patches of reflected energy: self-coherent energy patches plus mutually coherent energy among the patches.     

Ambiguity Functions for Quadratic Phase-Coded Pulse Trains

The matched filter ambiguity function is presented for a burst waveform composed of repeated subbursts, each one of which consists of N pulses in which the phase is varied quadratically from pulse to pulse. The resulting ambiguity function exhibits small residual ambiguities along the delay axis separated by the reciprocal of the pulse repetition frequency (PRF). A cross-ambiguity function is derived which reduces these ambiguities to zero amplitude. A third cross-ambiguity function is presented for a receiver matched to a generalized Hamming weighted repeated quadratic burst. The location in the delay/Doppler plane of the waveform ambiguities for these waveforms is compared with that of an uncoded pulse burst.     

A Generalized Clutter Computation Procedure for Airborne Pulse Doppler Radars

A general procedure for analyzing ground clutter effects in airborne pulse Doppler radars is described. The quantity computed is the expected clutter power at the output of any specified range gate/ Doppler filter processing cell. The procedure has been computerized and is quite general with respect to antenna gain pattern, clutter cross section variation, PRF, pulse and range gate shapes, and the various receiver processing functions. It is applicable only to distributed ground clutter and linear processing, and excludes the dynamic effects of continuous antenna scanning. To exemplify the use of the procedure, two studies conducted for a postulated high PRF radar are described, and the results are presented.     

Doppler ambiguity resolution using multiple PRF

An algorithm for velocity ambiguity resolution in coherent pulsed Doppler radar using multiple pulse repetition frequencies (PRF) is presented. It relies on the choice of particular values for the PRFs. The folded frequency of the target signal is obtained by averaging the folded frequency estimates for each PRF, and a quasi maximum likelihood criterion is maximized for ambiguity order estimation. The fast implementation of this nonambiguous estimation procedure is based on the fast Fourier transform (FFT), The proposed waveform allows full exploitation of any (even) number of PRFs, which appears to be important for estimation improvement. The effects of the waveform parameters and the folded frequency estimation variance on the performance of the ambiguity order estimation procedure are evaluated theoretically and through computer simulations. Mean square error (MSE) curves are given to assess the Doppler frequency estimation accuracy. Finally, the new method is compared with a classical technique and the implementation of the algorithm in a clutter environment is addressed.     

Synthesis of Ambiguity Functions for Prescribed Responses

The synthesis of radar ambiguity functions is approached using burst-pulse time-frequency waveform coding. Noting that the parameters that define the central response of the ambiguity function for these code classes also define the waveform code, a statistical decision procedure based upon the central response is employed to obtain Bayes-type codes. The selection of the code parameters is subject to restrictions imposed by the noncentral response of the ambiguity function. Three classes of random time-frequency codes are treated: 1) uniform amplitude, uniform pulsewidth matched codes; 2) uniform amplitude, nonuniform pulsewidth matched codes; and 3) uniform amplitude, uniform pulsewidth with receiver amplitude mismatch.     

Clutter Spectra of Low PRF AMTI Pulse-Doppler Radar

Approximate expressions are derived for the video clutter spectra in the receiver of a low pulse repetition frequency (PRF), airborne moving target indicator (AMTI), pulse-Doppler radar for both step-scanning and continuous-scanning antennas. The receiver is assumed to process the received waveform with a clutter-tracking oscillator and a window function is employed to obtain short-term spectra. Except for the broadening effects of the window function, it is shown that the clutter spectrum can be simply related to the antenna voltage-gain pattern. It is further shown, in the scanning antenna case, that the combined spectral broadening due to platform motion and antenna scanning cannot be assumed to be the result of the convolution of the separate effects unless the antenna gain pattern has a Gaussian shape. The approximate clutter expressions are illustrated by examples and are shown to agree well with the results of computer calculations.     

Clutter Suppression Properties of Weighted Pulse Trains

A common but troublesome requirement on radar sensors is the detection of a target in the interference from undesired scatterers, or clutter. Systems with coherent processing of pulse trains are uniquely suited for the purpose because, with pulse trains, it is possible to concentrate the receiver output for particular values of Doppler and thus suppress the clutter by Doppler filtering. This paper discusses to what degree the effectiveness of the method can be enhanced by tapering, or weighting, of the pulse amplitudes. The general results are illustrated by computer-plotted response functions for weighted pulse trains. The clutter suppression efficiency of weighting is calculated both for unilateral weighting in the receiver and for bilateral weighting in both receiver and transmitter. The significance of additional phase weighting is discussed and the results for pure amplitude weighting are compared with publishedwork on phase and amplitude weighting.     

Constrained bandwidth waveforms with minimal dilation sensitivity

Bandpass waveforms which have envelopes which are insensitive to this velocity-induced time dilation can be efficiently processed by narrowband receivers in which envelope correlation is fixed and Doppler tested using fast Fourier transform (FFT) processing. The peak level of the waveform ambiguity function (AF) can be used to gauge the distortion of the waveform induced by dilation. The degree of AF attenuation is shown to be proportional to the dilation parameter or velocity, waveform traveling wave (TW) product, and a sensitivity parameter which depends on the envelope function utilized. Classes of symmetric, constrained bandwidth, phase modulated envelope functions which are minimally dilation sensitive (Doppler tolerant) are derived. When the resulting waveforms are used with a simple correlation receiver structure and the echo data is derived from slowly fluctuating point scattering in white Gaussian noise, the receiver becomes an uncoupled joint estimator of delay and dilation (Doppler). In the case of the bandpass waveforms, only odd symmetry of the phase modulation (PM) yields an uncoupled estimator     

Effect of Particular Gain Changes upon LFM Sidelobes

The envelope variation of an LFM waveform due to transmitter droop or receiver STC tends to cause range sidelobes. A parametric analysis of the magnitude of the sidelobes has been performed. It is shown that the sidelobes can be quite high at the matched filter output, but are low at the output of the sidelobe reduction filter. 40-dB sidelobes can be achieved even with a 4-dB envelope droop. It is shown that these results are consistent with conventional paired-echo theory. Similar results are shown to hold for droop variations of the filter transfer function.     

Joint TDOA,FDOA and differential Doppler rate estimation: Method and its performance analysis

Considering the estimation accuracy reduction of Frequency Difference of Arrival(FDOA) caused by relative Doppler companding, a joint Time Difference of Arrival(TDOA),FDOA and differential Doppler rate estimation method is proposed and its Cramer-Rao low bound is derived in this paper.Firstly, second-order ambiguity function is utilized to reduce the dimensionality and estimate initial TDOA and differential Doppler rate.Secondly, the TDOA estimation is updated and FDOA is obtained using cross ambiguity function, in which relative Doppler companding is compensated by the existing differential Doppler rate.Thirdly, differential Doppler rate estimation is updated using cross estimator.Theoretical analysis on estimation variance and Cramer-Rao low bound shows that the final estimation of TDOA, FDOA and differential Doppler rate performs well at both low and high signal–noise ratio, although the initial estimation accuracy of TDOA and differential Doppler rate is relatively poor under low signal–noise ratio conditions.Simulation results finally verify the theoretical analysis and show that the proposed method can overcome relative Doppler companding problem and performs well for all TDOA, FDOA and differential Doppler rate estimation.     

Radar Waveforms for Suppression of Extended Clutter

A common problem in waveform design is to adapt the transmitted signal to the target environment in order that the interference from extended fields of scatterers is reduced. This problem is investigated here for the special case of detection of a single target in the ``vicinity' of an extended clutter space. The paper considers the possibility of confining the matched-filter response in delay and Doppler, or ambiguity function, to a narrow strip with arbitrary orientation in the delay-Doppler plane. It is shown that strict confinement of the response is achievable only with waveforms that are unlimited in both time and frequency domain. With practical waveforms, which are necessarily of finite extent, one merely can trade close-target separability against detectability in the background clutter. Thus, one form of the resolution problem is exchanged against the other. The paper examines these effects quantitatively.     

The effect of the clutter-to-noise ratio on Doppler filterperformance

The effect of the clutter-to-noise ratio on the performance of a Doppler filter is considered. Clutter is assumed to have a power level which is unknown and varies in range. The assessment of the performance of a Doppler filter is based on the gain of the filter, which is the normalized output signal-to-interference ratio improvement at a given Doppler. The gain is generally a complex function of the statistics of the clutter. New upper and lower bounds on the gain differential between the expected design point clutter-to-noise ratio and the actual clutter-to-noise ratio are found. These bounds are independent of the clutter covariance matrix and are only a function of the unknown clutter-to-noise ratio. The bounds are valid for both Gaussian and non-Gaussian noise and for arbitrary linear filters. The upper and lower bounds differ by the theoretical coherent integration gain, 10 logN dB, where N is the number of pulses. A tighter lower bound is found for the case when the filters are matched filters. A simple exact expression is found for matched filters assuming a Gaussian Markov clutter model as the clutter spectral width approaches zero. An easily implementable adaptive procedure is given which improves performance due to the unknown clutter-to-noise ratio. This work extends a previous result, valid for the Emerson filter, that shows the effect of clutter-to-noise ratio on performance in terms of an average quantity, the improvement factor     

A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar

The paper examines the problem of cancellation of direct signal, multipath and clutter echoes in passive bistatic radar (PBR). This problem is exacerbated as the transmitted waveform is not under control of the radar designer and the sidelobes of the ambiguity function can mask targets including those displaced in either (or both) range and Doppler from the disturbance. A novel multistage approach is developed for disturbance cancellation and target detection based on projections of the received signal in a subspace orthogonal to both the disturbance and previously detected targets. The resulting algorithm is shown to be effective against typical simulated scenarios with a limited number of stages, and a version with computational savings is also introduced. Finally its effectiveness is demonstrated with the application to real data acquired with an experimental VHF PBR system.     

Detection of chi-square fluctuating targets in arbitrary clutter

In a recent paper, general expressions were derived for the density and cumulative probability functions of the amplitude of a linear matched-filter output given a nonfluctuating target in a clutter-limited environment. These expressions were based on the clutter amplitude density function. The results are extended to calculate the cumulative probability function of the output of a linear matched filter used to detect a chi-square fluctuating target in a clutter-limited environment. The resulting method is applied to a common radar clutter model, and experimental sonar data.     

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     

Radar/Sonar Signal Design for Bounded Doppler Shifts

In many detection and estimation problems, Doppler frequency shifts are bounded. For clutter or multipath that is uniformly distributed in range and symmetrically distributed in Doppler shift relative to the signal, detectability of a point target or a communication signal is improved by minimizing the weighted volume of the magnitude-squared autoambiguity function. When clutter Doppler shifts are bounded, this volume is in a strip containing the range axis on the range-Doppler plane. For scattering function estimation, e.g., for weather radar, Doppler flow meters, and distributed target classifiers, it is again relevant to minimize ambiguity volume in a strip. Strip volume is minimized by using a pulse train, but such a signal has unacceptably large range sidelobes for most applications. Other waveforms that have relatively small sidelobe level within a strip on the range-Doppler plane, as well as small ambiguity volume in the strip, are obtained. The waveforms are composed of pulse pairs that are phase modulated with Golay complementary codes.     

Clutter Suppression by Complex Weighting of Coherent Pulse Trains

Uniform coherent pulse trains offer a practical solution to the problem of designing a radar signal possessing both high range and range-rate resolution. The Doppler sensitivity provides some rejection of off-Doppler (clutter) returns in the matched filter receiver. This paper considers the use of a processor in which members of the received pulse train are selectively weighted in amplitude and phase to improve clutter suppression. The techniques described are particularly suitable for rejecting interference entering the processor through ambiguous responses (range sidelobes) of the signal. The complex weights which are derived are optimum in the sense that they produce the maximum clutter suppression for a given detection efficiency. In determining these weights, it is assumed that the distribution of clutter in range and range rate relative to targets of interest is known. Thus, clutter suppression is achieved by reducing the sidelobe levels in specified regions of the receiver response. These techniques are directly applicable to array antennas; the analogous antenna problem would be to reduce sidelobe levels in a particular sector while preserving gain. Complex weighting is most successful when the clutter is limited in both range and velocity.     

Effect of earth?s rotation and range foldover on space-based radar performance

Space-based radar (SBR) by virtue of its motion generates a Doppler frequency component to the clutter return from any point on the Earth as a function of the SBR-Earth geometry. The effect of the rotation of the Earth around its own axis also adds an additional component to this Doppler frequency. The overall effect of the rotation of the Earth on the Doppler turns out to be two correction factors in terms of a crab angle affecting the azimuth angle, and a crab magnitude scaling the Doppler magnitude of the clutter patch. Interestingly, both these quantities depend only on the SBR orbit inclination and its latitude and not on the location of the clutter patch of interest. Further, the crab angle has maximum effect for an SBR on a polar orbit that is above the equator. The crab magnitude, on the other hand, peaks for an SBR on an equatorial orbit. Together with the range foldover phenomenon, their overall effect is to generate Doppler spread/splitting resulting in wider clutter notches that degrade the clutter nulling performance of adaptive processing techniques. A detailed performance analysis and methods to minimize these effects are discussed here     

Slope Coding of Coherent Pulse-Burst Waveforms

Useful new properties are obtained with a coherent pulse-burst waveform if the FM slopes are allowed to change from pulse to pulse. A design feature using such a "slope-coded" waveform is the ability to transfer, in a controllable manner, ambiguity volume from the range ambiguity peaks to regions located between the peaks (in the frequency direction). Slope coding thus provides a useful mechanism for matching the waveform to the environment; i. e., it permits the signal designer to make some compromise between range ambiguity-peak heights and low-velocity clutter rejection.     

Comparison between monostatic and bistatic antenna configurationsfor STAP

The unique characteristics of bistatic radar operation on the performance of airborne/spaceborne moving target indicator (MTI) radars that use space-time adaptive processing (STAP) are discussed. It has been shown that monostatic STAP radar has the following properties. 1) For a horizontal flight path and a planar Earth the curves of constant clutter Doppler (isodops) are hyperbolas. 2) For a sidelooking antenna geometry the clutter Doppler is range independent. 3) Clutter trajectories in the cosφ-F plane (F=normalized Doppler) are in general ellipses (or straight lines for a sidelooking array). We demonstrate that these well-known properties are distorted by the displacement between transmitter and receiver in a bistatic configuration. It is shown that even for the sidelooking array geometry the clutter Doppler is range-dependent which requires adaptation of the STAP processor for each individual range gate. Conclusions for the design of STAP processors are drawn