Array processing for radar is well established in the literature, but only few of these algorithms have been implemented in real systems. The reason may be that the impact of these algorithms on the overall system must be well understood. For a successful implementation of array processing methods exploiting the full potential, the desired radar task has to be considered and all processing necessary for this task has to be eventually adapted. In this tutorial paper, we point out several viewpoints which are relevant in this context: the restrictions and the potential provided by different array configurations, the predictability of the transmission function of the array, the constraints for adaptive beamforming, the inclusion of monopulse, detection and tracking into the adaptive beamforming concept, and the assessment of superresolution methods with respect to their application in a radar system. The problems and achieved results are illustrated by examples from previous publications.

Array processing is well established for radar. Publications of this topic have appeared for decades, and one might question what kind of advances we may still expect now. On the other hand, if we look at existing radar systems we will find very few methods implemented from the many ideas discussed in the literature. The reason may be that all processing elements of a radar system are linked, and it is not very useful to simply implement an isolated algorithm. The performance and the property of any algorithm will have an influence on the subsequent processing steps and on the radar operational modes. Predictability of the system performance with the new algorithms is a key issue for the radar designer. Advanced array processing for radar will therefore require to take these interrelationships into account and to adapt the related processing in order to achieve the maximum possible improvement. The standard handbooks on radar [

In this tutorial paper, viewpoints are presented which are relevant for the implementation of array processing methods. We do not present any new sophisticated algorithms, but for the established algorithms we give examples of the relations between array processing and preceding and subsequent radar processing. We point out the problems that have to be encountered and the solutions that need to be developed. We start with spatial sampling, that is, the antenna array that has to be designed to fulfill all requirements of the radar system. A modern radar is typically a multitasking system. So, the design of the array antenna has to fulfill multiple purposes in a compromise. In Section

Array processing starts with the array antenna. This is hardware and is a selected construction that cannot be altered. It must therefore be carefully designed to fulfill all requirements. Digital array processing requires digital array outputs. The number and quality of these receivers (e.g., linearity and number of ADC bits) determine the quality and the cost of the whole system. It may be desirable to design a fully digital array with AD-converters at each antenna element. However, weight and cost will often lead to a system with reduced number of digital receivers. On the other hand, because of the

Thinned arrays: the angular discrimination of an array with a number of elements can be improved by increasing the separation of the elements and thus increasing the aperture of the antenna. Note that the thinned array has the same gain as the corresponding fully filled array.

Subarrays: element outputs are summed up in an analog manner into subarrays which are then AD-converted and processed digitally. The size and the shape of the subarrays are an important design criterion. The notion of an array with subarrays is very general and includes the case of steerable directional array elements.

Antenna elements may be arranged on a line (1-dimensional array), on a plane (a ring or a 2-dimensional planar array), on a curved surface (conformal array), or within a volume (3D-array, also called Crow’s Nest antenna, [

Antenna element design and the need for fixing elements mechanically lead to element patterns which are never omnidirectional. The elements have to be designed with patterns that allow a unique identification of the direction. Typically, a planar array can only observe a hemispherical half space. To achieve full spherical coverage, several planar arrays can be combined (multifacetted array), or a conformal or volume array may be used.

For linear, planar, and volume arrays, elements with nearly equal patterns can be realized. These have the advantage that the knowledge of the element pattern is for many array processing methods not necessary. An equal complex value can be interpreted as a modified target complex amplitude, which is often a nuisance parameter. More important is that, if the element patterns are really absolutely equal, any cross-polar components of the signal are in all channels equal and fulfill the array model in the same way as the copolar components; that is, they produce no error effect.

This occurs typically by tilting the antenna elements as is done for conformal arrays. For a planar array, this tilt may be used to realize an array with polarization diversity. Single polarized elements are then mounted with orthogonal alignment at different positions. Such an array can provide some degree of dual polarization reception with single channel receivers (contrary to more costly fully polarimetric arrays with receivers for both polarizations for each channel).

Common to arrays with unequal patterns is that we have to know the element patterns for applying array processing methods. The full element pattern function is also called the array manifold. In particular, the cross-polar (or short

To save the cost of receiving modules, sparse arrays are considered, that is, with fewer elements than the full populated

If a high antenna gain with low sidelobes is desired one has to go back to the fully filled array. For large arrays with thousands of elements, the large number of digital channel constitutes a significant cost factor and a challenge for the resulting data rate. Therefore, often subarrays are formed, and all digital (adaptive) beamforming and sophisticated array processing methods are applied to the subarray outputs. Subarraying is a very general concept. At the elements, we may have phase shifters such that all subarrays are steered into a given direction and we may apply some attenuation (tapering) to influence the sidelobe level. The sum of the subarrays then gives the sum beam output. The subarrays can be viewed as a superarray with elements having different patterns steered into the selected direction. The subarrays should have unequal size and shape to avoid grating effects for subsequent array processing, because the subarray centers constitute a sparse array (for details, see [

Principle of forming subarrays.

Beamforming using subarrays can be mathematically described by a simple matrix operation. Let the complex array element outputs be denoted by

Figure

2D generic array with 902 elements grouped into 32 subarrays.

An important feature of digital beamforming with subarrays is that the weighting for beamforming can be distributed between the element level (the weighting incorporated in the matrix

Coherent processing of a time series

Conventional beamforming is the same as coherent integration of the spatially sampled data; that is, the phase differences of a plane wave signal at the array elements are compensated, and all elements are coherently summed up. This results in a pronounced main beam when the phase differences match with the direction of the plane wave and result in sidelobes otherwise. The beam shape and the sidelobes can be influenced by additional amplitude weighting.

Let us consider the complex beamforming weights

Low sidelobes by amplitude tapering.

Low sidelobe patterns of equal beamwidth

Taper functions for low sidelobes

The rationale for low sidelobes is that we want to minimize some unknown interference power coming over the sidelobes. This can be achieved by solving the following optimization problem, [

An example for reducing the sidelobes in selected areas where interference is expected is shown in Figure

Antenna patterns of planar array with reduced sidelobes at lower elevations for different beam pointing directions (by courtesy of W. Bürger of Fraunhofer (FHR)).

Deterministic pattern shaping is applied if we have rough knowledge about the interference angular distribution. In the sidelobe region, this method can be inefficient because the antenna response to a plane wave (the vector

In the sequel, we formulate the ABF algorithms for subarray outputs as described in (

Let us first suppose that we know the interference situation; that is, we know the interference covariance matrix

This formulation for weight vectors applied at the array elements can be easily extended to subarrays with digital outputs. As mentioned in Section

Sometimes interference suppression is realized by minimizing only the jamming power subject to additional constraints, for example,

Single unit gain directional constraint:

Gain and derivative constraint:

Gain and norm constraint:

Norm constraint only:

As we mentioned before, fulfilling the constraints may imply a loss in SNIR. Therefore, several techniques have been proposed to mitigate the loss. The first idea is to allow a compromise between power minimization and constraints by introducing coupling factors

The performance of ABF is often displayed by the adapted antenna pattern. A typical adapted antenna pattern with 3 jammers of 20 dB SNR is shown in Figure

Antenna and normalized SNIR patterns for a three jammer configuration and generic array.

Adapted antenna pattern

SNIR

Plots of the SNIR are better suited for displaying this effect. The SNIR is typically plotted for varying target direction while the interference scenario is held fixed, as seen in Figure

The effect of target and steering direction mismatch is not accounted for in the SNIR plot. This effect is displayed by the scan pattern, that is, the pattern that arises if the adapted beam scans over a fixed target and interference scenario. Such a plot is rarely shown because of the many parameters to be varied. In this context, we note that for the case that the training data contains the interference and noise alone the main beam of the adapted pattern is fairly broad similar to the unadapted sum beam and is therefore fairly insensitive to pointing mismatch. How to obtain an interference-alone covariance matrix is a matter of proper selection of the training data as mentioned in the following section.

Figure

Antenna and normalized SNIR patterns for a three jammer configuration for antenna with low sidelobes (−40 dB Taylor weighting).

Adapted antenna pattern

SNIR

In reality, the interference covariance matrix is not known and must be estimated from some training data

SNIR for SMI, LSMI (

One may go even further and ignore the small eigenvalue estimates completely; that is, one tries to find an estimate of the inverse covariance matrix based on the dominant eigenvectors and eigenvalues. For high SNR, we can replace the inverse covariance matrix by a projection matrix. Suppose we have

Figure

For EVP, we have to know the dimension of the jammer subspace (dimJSS). In complicated scenarios and with channel errors present, this value can be difficult to determine. If dimJSS is grossly overestimated, a loss in SNIR occurs. If dimJSS is underestimated the jammers are not fully suppressed. One is therefore interested in subspace methods with low sensitivity against the choice of the subspace dimension. This property is achieved by a “weighted projection,” that is, by replacing the projection by

One of the most efficient methods for pattern stabilization while maintaining a low desired sidelobe level is the constrained adaptive pattern synthesis (CAPS) algorithm, [

Subspace methods require an estimate of the dimension of the interference subspace. Usually this is derived from the sample eigenvalues. For complicated scenarios and small sample size, a clear decision of what constitutes a dominant eigenvalue may be difficult. There are two principle approaches to determine the number of dominant eigenvalues, information theoretic criteria and noise power tests.

The information theoretic criteria are often based on the sphericity test criterion; see, for example, [

For small sample size, AIC and MDL are known for grossly overestimating the number of sources. In addition, bandwidth and array channels errors lead to a leakage of the dominant eigenvalues into the small eigenvalues, [

Comparison of tests for linear array with

A more thorough study of the small sample size dimJSS estimation problem considering the “effective number of identifiable signals” has been performed in [

The objective of radar processing is not to maximize the SNR but to detect targets and determine their parameters. For detection, the SNR is a sufficient statistic (for the likelihood ratio test); that is, if we maximize the SNR we maximize also the probability of detection. Only for these detected targets we have then a subsequent procedure to estimate the target parameters: direction, range, and possibly Doppler. Standard radar processing can be traced back to maximum likelihood estimation of a single target which leads to the matched filter, [

The resolution limit for classical beamforming is the 3 dB beamwidth. An antenna array provides spatial samples of the impinging wavefronts, and one may define a multitarget model for this case. This opens the possibility for enhanced resolution. These methods have been discussed since decades, and textbooks on this topic are available, for example, [

From the many proposed methods, we mention here only some classical methods to show the connections and relationships. We have spectral methods which generate a spiky estimate of the angular spectral density like.

Capon’s method:

An alternative group of methods are parametric methods, which deliver only a set of “optimal” parameter estimates which explain in a sense the data for the inserted model by

Deterministic ML method (complex amplitudes are assumed deterministic):

A typical feature of the MUSIC method is illustrated in Figure

MUSIC spectra with a planar array of 7 elements.

Simulated data

Measured real data

A result with real data with the deterministic ML method is shown in Figure

Superresolution of multipath propagation over sea with deterministic ML method (real data from vertical linear array with 32 elements, scenario illustrated above).

The problems of superresolution methods are described in [

Another problem is the exact knowledge of the signal model for all possible directions (the vector function

For an array with digital subarrays, superresolution has to be performed only with these subarray outputs. The array manifold has then to be taken at the subarray outputs as in (

More refined parametric methods with higher asymptotic resolution property have been suggested (e.g., COMET, Covariance Matching Estimation Technique, [

Superresolution is a combined target number and target parameter estimation problem. As a starting point all the methods of Section

These methods may lead to a possibly overestimated target number. To determine the power allocated to each target a refined ML power estimate using the estimated directions

The deterministic ML method (

Combined target number and direction estimation for 2 targets with 7-element planar array.

The problem of resolution of two closely spaced targets becomes a particular problem in the so called threshold region, which denotes configurations where the SNR or the separation of the targets lead to an angular variance departing significantly from the Cramer-Rao bound (CRB). The design of the tests and this threshold region must be compatible to give consistent joint estimation-detection resolution result. These problems have been studied in [

As mentioned in Section

In particular, subarraying in time domain is an important tool to reduce the numerical complexity for space-time adaptive processing (STAP) which is the general approach for adaptive clutter suppression for airborne radar, [

Figure

Symmetric auxiliary sensor/echo processor of Klemm [

A key problem that has to be recognized is that the task of a radar is not to maximize the SNR, but to give the best relevant information about the targets after all processing. This means that for implementing refined methods of interference suppression or superresolution we have also to consider the effect on the subsequent processing. To get optimum performance all subsequent processing should exploit the properties of the refined array signal processing methods applied before. In particular it has been shown that for the tasks of detection, angle estimation and tracking significant improvements can be achieved by considering special features.

Monopulse is an established technique for rapid and precise angle estimation with array antennas. It is based on two beams formed in parallel, a sum beam and a difference beam. The difference beam is zero at the position of the maximum of the sum beam. The ratio of both beams gives an error value that indicates the offset of a target from the sum beam pointing direction. In fact, it can be shown that this monopulse estimator is an approximation of the Maximum-Likelihood angle estimator, [

When adaptive beams are used the shape of the sum beam will be distorted due to the interference that is to be suppressed. The difference beam must adaptively suppress the interference as well, which leads to another distortion. Then the ratio of both beams will no more indicate the target direction. The generalized monopulse procedure of [

The generalized monopulse formula for estimating angles

It is important to note that this formula is independent of any scaling of the difference and sum weights. Constant factors in the difference and sum weight will be cancelled by the corresponding slope correction. Figure

Bias and standard deviation ellipses for different target positions.

The large bias may not be satisfying. However, one may repeat the monopulse procedure by repeating the monopulse estimate with a look direction steered at subarray level into the new estimated direction. This is an all-offline procedure with the given subarray data. No new transmit pulse is needed. We have called this the multistep monopulse procedure [

Bias for 2-step monopulse for different target positions and jammer scenario of Figure

For detection with adaptive beams, the normal test procedure is not adequate because we have a test statistic depending on two different kinds of random data: the training data for the adaptive weight and the data under test. Various kinds of tests have been developed accounting for this fact. The first and basic test statistics were the GLRT, [

Basic properties of these tests are

Actually, all these detectors use the adaptive weight of the SMI algorithm which has unsatisfactory performance as mentioned in Section

A guard channel is implemented in radar systems to eliminate impulsive interference (hostile or from other neighboring radars) using the sidelobe blanking (SLB) device. The guard channel receives data from an omnidirectional antenna element which is amplified such that its power level is above the sidelobe level of the highly directional radar antenna, but below the power of the radar main beam, [

With phased arrays it is not necessary to provide an external omnidirectional guard channel. Such a channel can be generated from the antenna itself; all the required information is in the antenna. We may use the noncoherent sum of the subarrays as guard channel. This is the same as the average omnidirectional power. Some shaping of the guard pattern can be achieved by using a weighting for the noncoherent sum:

Guard channel and sum beam patterns for generic array of Figure

Uniform subarray weighting

Power equalized weighting

Power equalized + difference weighting

Adapted guard patterns for jammer at

Weighting for equal subarray power

Difference type guard with weighting for equal subarray power

This is the adaptive generalization of the usual sidelobe blanking device (SLB) and the AMF, ACE and GLRT tests can be used as extension of the SLB detector to the adaptive case, [

A problem with these modified tests is to define a suitable threshold for detection. For arbitrary weight vector it is nearly impossible to determine this analytically. In [

Detection margin (gray shading) between AMF and nonweighted adapted guard pattern with confidence bounds for ACE with LSMI (64 snapshots and

Comparing the variances of the ACE and GLRT guard channels in [

A key feature of ABF is that overall performance is dramatically influenced by the proximity of the main beam to an interference source. The task of target tracking in the proximity of a jammer is of high operational relevance. In fact, the information on the jammer direction can be made available by a jammer mapping mode, which determines the direction of the interferences by a background procedure using already available data. Jammers are typically strong emitters and thus easy to detect. In particular, the Spotlight MUSIC method [

Let us assume here for simplicity that the jammer direction is known. This is highly important information for the tracking algorithm of a multifunction radar where the tracker determines the pointing direction of the beam. We will use for angle estimation the adaptive monopulse procedure of Section

All these techniques have been implemented in a tracking algorithm and refined by a number of stabilization measures in [

Look direction stabilization: the monopulse estimate may deliver measurements outside of the 3 dB contour of the sum beam. Such estimates are also heavily biased, especially for look directions close to the jammer, despite the use of the multistep monopulse procedure. Estimates of that kind are therefore corrected by projecting them onto the boundary circle of sum beam contour.

Detection threshold: only those measurements are considered in the update step of the tracking algorithm whose sum beam power is above a certain detection threshold (typically 13 dB). This guarantees useful and valuable monopulse estimates. It is well known that the variance of the monopulse estimate decreases monotonically with this threshold increasing.

Adjustment of antenna look direction: look directions in the jammer notch should generally be avoided due to the expected lack of good measurements. In case that the proposed look direction lies in the jammer notch, we select an adjusted direction on the skirt of the jammer notch.

Variable measurement covariance: a variable covariance matrix of the adaptive monopulse estimation according to [

QuadSearch and Pseudomeasurements: if the predicted target direction lies inside the jammer notch and if, despite all adjustments of the antenna look direction, the target is not detected, a specific search pattern is initiated (named QuadSearch) which uses look directions on the skirt of the jammer notch to obtain acceptable monopulse estimates. If this procedure does not lead to a detection, we know that the target is hidden in the jammer notch and we cannot see it. We use then the direction of the jammer as a pseudobearing measurement to maintain the track file. The pseudomeasurement noise is determined by the width of the jammer notch.

LocSearch: in case of a permanent lack of detections (e.g., for three consecutive scans) while the track position lies outside the jammer notch, a specific search pattern is initiated (named LocSearch) that is similar to the QuadSearch. The new look directions lie on the circle of certain radius around the predicted target direction.

Modeling of target dynamics: the selection of a suitable dynamics model plays a major role for the quality of tracking results. In this context, the so-called interacting multiple model (IMM) is a well-known method to reliably track even those objects whose dynamic behavior remains constant only during certain periods.

Gating: in the vicinity of the jammer, the predicted target direction (as an approximation of the true value) is used to compute the variable angle measurement covariance. Strictly speaking, this is only valid exactly in the particular look direction. Moreover, the tracking algorithm regards all incoming sensor data as unbiased measurements. To avoid track instabilities, an acceptance region is defined for each measurement depending on the predicted target state and the assumed measurement accuracy. Sensor reports lying outside this gate are considered as invalid.

Illustration of different stabilization measures to improve track stability and track continuity.

In order to evaluate our stabilization measures we considered a realistic air-to-air target tracking scenario [

Target tracking scenario with standoff jammer: geographic plot of platform trajectories.

Exemplary azimuth measurements and models for a specific time window for tracking scenario of Figure

From these investigations, it turned out that tracking only with adaptive beamforming and adaptive monopulse nearly always leads to track loss in the vicinity of the jammer. With additional stabilization measures that did not require the knowledge of the jammer direction (projection of monopulse estimate, detection threshold, LocSearch, gating) still track instabilities occurred culminating finally in track loss. An advanced tracking version which used pseudomeasurements mitigated this problem to some degree. Finally, the additional consideration of the variable measurement covariance with a better estimate of the highly variable shape of the angle uncertainty ellipse resulted in significantly fewer measurements that were excluded due to gating. In this case all the stabilization measures could not only improve track continuity, but also track accuracy and thus track stability, [

In this paper, we have pointed out the links between array signal processing and antenna design, hardware constraints and target detection, and parameter estimation and tracking. More specifically, we have discussed the following features.

Interference suppression by deterministic and adaptive pattern shaping: both approaches can be reasonably combined. Applying ABF after deterministic sidelobe reduction allows reducing the requirements on the low sidelobe level. Special techniques are available to make ABF preserve the low sidelobe level.

General principles and relationships between ABF algorithms and superresolution methods have been discussed, like dependency on the sample number, robustness, the benefits of subspace methods, problems of determining the signal/interference subspace, and interference suppression/resolution limit.

Array signal processing methods like adaptive beamforming and superresolution methods can be applied to subarrays generated from a large fully filled array. This means applying these methods to the sparse superarray formed by the subarray centers. We have pointed out problems and solutions for this special array problem.

ABF can be combined with superresolution in a canonical way by applying the pre-whiten and match principle to the data and the signal model vector.

All array signal processing methods can be extended to space-time processing (arrays) by defining a corresponding space-time plane wave model.

Superresolution is a joint detection-estimation problem. One has to determine a multitarget model which contains the number, directions and powers of the targets. These parameters are strongly coupled. A practical joint estimation and detection procedure has been presented.

The problems for implementation in real system have been discussed, in particular the effects of limited knowledge of the array manifold, effect of channel errors, eigenvalue leakage, unequal noise power in array channels, and dynamic range of AD-converters.

For achieving best performance an adaptation of the processing subsequent to ABF is necessary. Direction estimation can be accommodated by using ABF-monopulse; the detector can be accommodated by adaptive detection with ASLB, and the tracking algorithms can be extended to adaptive tracking and track management with jammer mapping.

With a single array signal processing method alone no significant improvement will be obtained. The methods have to be reasonably embedded in the whole system, and all functionalities have to be mutually tuned and balanced. This is a task for future research. The presented approaches constitute only a first ad hoc step, and more thorough studies are required. Note that in most cases tuning the functionalities is mainly a software problem. So, there is the possibility to upgrade existing systems softly and step-wise.

The main part of this work was performed while the author was with the Fraunhofer Institute for High Frequency Physics and Radar Techniques (FHR) in Wachtberg, Germany.