Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag

which is in closed-form.

When the number of emitters is unknown, we may apply the following two procedures based on the pseudo maximum likelihood estimate to detect them, under the assumption that the signals of the emitters are narrowband Gaussian, and not fully correlated with each other.

a) The Multiple Alternative Hypothesis Testing Problem

Let H K be the hypothesis that K emitters exist. The log-likelihood function for HK when 8 K equals its pseudo maximum likelihood estimate is obtained from (2.160) as

42 Z. Zhu and S. Haykin

Setting K = 0, 1, ... , P, where P < M -1, we perform a sequence of binary hypothesis tests:

H1 :HM - 1 ,

(2.166)

At the Kth step, the generalized log-likelihood ratio may be found from (2.164) and (2.165) as

i=K+l

The value of K, for which HK is first accepted, is selected as the estimate K of the number of emitters.

b) The Model-Order Selection Problem

When using the pseudo maximum likelihood estimate, the unknown parameter vector @K' where the number of emitters is K, contains Al ,· .. ,AK; VI" .. , VK and (]'2. Obviously, the number of adjustable parameters, mK , is larger than that of (2.121) by 1, due to the unknown (]'2 in the present case. Thus, we have

Substituting (2.164) and (2.169) into (2.51), and omitting terms that are independent of K, we get for Akaike's information criterion

Both of these criteria are in closed-form. The criteria (2.170) and (2.171) first appeared in [2.8].

2.5 Discussion

In this chapter, we have derived the structural basis of radar detection using array processing from the standpoint of statistical decision theory. The performance analysis and evaluation, however, have not been performed here. In simple cases, such as the detection of a single target with known direction, as described in Sects. 2.2.2 and 2.3.2, the well-known results of signal detection theory can be applied directly. This is evident from the test statistic of(2.23) and (2.65). In other cases of interest, it may not be easy to obtain an explicit analytical solution of the detection performance.

In the discussion, we have assumed that in the active radar case, the noise covariance (f2eo is known a priori. The justification for making this assumption is not only to simplify the analytical derivation, but also to gain insight into the basic structure ofradar detection using an array ofsensors. Of course, in real-life situations, we have to estimate the noise covariance by using reference noise samples, and the quality of the background estimate may strongly affect the detection performance. A pursual of this line of thinking leads us naturally into adaptive detection and parameter estimation for multidimensional signal models; for more details, see Kelly and Forsythe [2.18].

We have also assumed that the direction vector a(O) of an array is a known function of direction. The direction vector a(O) plays an important role in spatial-domain operations. Even when a pseudo maximum likelihood estimate based on an eigendecomposition method is used, a(O) will still appear in the estimation of directions and signals of sources. We have seen that the pseudo maximum likelihood estimate decouples the combined detection-estimation problem, and in the detection step, the processing structure is not dependent on a(O) . Nevertheless, the detection performance is intimately related to a(O). The direction vector a(O) of an array has to be determined by using a calibration scheme. Errors in calibration cannot be avoided. Efficient methods of calibrating an array and the effect of array calibration errors on the radar performance need further investigation.

The high computational complexity, especially the multi-dimensional search of directions of sources when a true maximum likelihood estimate is used, is the main shortcoming of such a method. Hence, the development of efficient algorithms and hardware implementations is extremely important for practical realizations of these array processing methods. In this context, the use of neural networks merits special attention.

The approaches we have described for estimating the number of sources have their own requirements/shortcomings. The approach, involving a sequence of binary hypothesis tests, requires that a threshold for a specified false alarm

44 Z. Zhu and S. Haykin

probability be determined. The method based on model-order selection cannot freely control the false alarm probability. In the discussion of a generalized likelihood ratio test using a pseudo maximum likelihood estimate based on the eigendecomposition method, we have assumed that the sources are not fully correlated with each other. Otherwise, this approach breaks down. A partial solution of overcoming the problem of fully correlated sources, applicable only to the case of a uniform linear array, involves the use of pre-processing in the form of spatial smoothing [2.19].

Finally, the discussion presented herein has been confined to the case of narrowband sources (passive or active). When the sources of radar or emitters are wideband, we have another level of complexity. Some aspects of the wideband passive detection problem using array processing have been considered in the literature [2.20-22]. A uniform treatment of the different hierarchies of wideband radar detection using array processing, along similar lines to that presented in this chapter, would be desirable.

Acknowledgements. The authors of this chapter would like to express their deep gratitude to Dr. M. Wax, Center for Signal Processing, Israel; Dr. Allan O. Steinhardt, Cornell University; and Dr. A. Nehorai, Yale University, for reviewing early versions of the manuscript of this chapter and for suggesting many valuable recommendations.

References

2.1A.A. Ksienski, R.B. McGhee: Radar signal processing for angular resolution beyond the Rayleigh limit. Radio Electron. Eng. 34, 161-174 (1967)

2.2G.O. Young, J.E. Howard: Application of space-time decision and estimation theory to antenna processing system design. Proc. IEEE 58, 771-778 (1970)

2.3W.S. Hodgkiss, L.W. Nolte: Bayes optimum and maximum-likelihood approaches in an array processing problem. IEEE Trans. AES-11, 913-916 (1975)

2.4R.O. Schmidt: A signal subspace approach to multiple emitter location and spectral estimation. Ph.D. Thesis, Stanford University (1981)

2.5U. Nickel: Superresolution using an active antenna array. lEE Conf. Pub!. 216, 87-91 (1982) (RADAR '82)

2.6J.F. Bohme: Estimation of spectral parameters of correlated signals in wavefields. Signal Processing 11, 329-337 (1986)

2.7R. Kumaresan, D.W. Tufts: Estimating the angles of arrival of multiple plane waves. IEEE Trans. AES-19, 134-139 (1983)

2.8M. Wax: Detection and estimation ofsuperimposed signals. Ph.D. Thesis, Stanford University (1985)

2.9S. Haykin (ed.): Array Signal Processing (Prentice-Hall, Englewood Cliffs, NJ 1985)

2.10C.W. Helstrom: Statistical Theory 0/Signal Detection, 2nd edn. (Pergamon, Oxford 1968)

2.11H.L. VanTrees: Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York 1968)

2.12J.V. DiFranco, V.L. Rubin: Radar Detection (Prentice-Hall, Englewood Cliffs, NJ 1968)

2.13A.D. Whalen: Detection o/Signals in Noise (Academic, New York 1971)

2.14M.I. Skolnik: Introduction to Radar Systems, 2nd edn. (McGraw-Hill, New York 1980)

2.15G.M. Stewart: Introduction to Matrix Computations (Academic, New York 1973)

2.16E.L. Lehmann: Testing Statistical Hypotheses (Wiley, New York 1959) Chap. 3

2.17T.W. Anderson: Asymptotic theory for principal component analysis, Ann. Math. Statist. 34, 122-148 (1963)

2.18E.J. Kelly, K.M. Forstythe: Adaptive detection and parameter estimation for multidimensional signal models, MIT Lincoln Laboratory, Technical Report 848 (April 1989)

2.19T.J. Shan, M. Wax, T. Kailath: On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE Trans. ASSP-33, 806-811 (1985)

2.20G. Su, M. Morf: The signal subspace approach for multiple wide-band emitter location. IEEE Trans. ASSP-31, 1502-1522 (1983)

2.21H. Wang, M. Kaveh: Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources. IEEE Trans. ASSP-33, 823-831 (1985)

2.22K.M. Buckley, L.J. Griffiths: Broad-band signal-subspace spatial-spectrum (BASS-ALE) estimation. IEEE Trans. ASSP-36, 953-964 (1988)

Additional References

W.P. Ballance, A.G. Jaffer: Low-angle direction finding based on maximum likelihood: A Unification. Conf. Record of 21st Asilomar Corn. on Signals, Systems and Computers, Pacific Grove, CA (1987)

J.F. Biihme: Source-parameter estimation by approximate maximum likelihood and nonlinear regression. IEEE J. Oceanic Engineering OE-IO, 206-212 (1985)

J. Clark: Robustness ofeigen based analysis techniques versus iterative adaptation. lEE Conf. Publ. 281 (Radar '87) 84-88 (1987)

M.H. EI Ayadi: Generalized likelihood adaptive detection of signals deformed by unknown linear filtering in noise with slowly fluctuating power. IEEE Trans. ASSP-33, 401-405 (1985)

H.M. Finn, R.S. Johnson: Adaptive detection mode with threshold control as a function ofspacially sampled clutter level estimates. RCA Rev. 29, 414-464 (1968)

S. Haykin: Array Processing: Application to Radar (Dowden, Hutchinson & Ross, Stroudsburg, PA 1980)

E.J. Kelly: An adaptive detection algorithm. IEEE Trans. AES-22, 115-127 (1986)

A.A. Ksienski, R.B. McGhee: A decision theoretic approach to the angular resolution and parameter estimation problem for multiple targets. IEEE Trans. AES-4, 443-455 (1968)

U. Nickel: Super-resolution by spectral line fitting, in signal processing II: Theories and applications, ed. by H.W. Schussler (Elsevier, Amsterdam 1983)

U. Nickel: Angular superresolution with phased array radar: A review of algorithm and operational constraints, lEE Proc. 134, Pt. F, 53-59 (1987)

R.M. O'Donnell, C.E. Muehe, M. Labitt: Advanced signal processing for airport surveillance radars. EASCON'74 (1974)

W.L. Root: Radar resolution of closely spaced targets. IRE Trans. MIL-6, 197-204 (1962)

R.O. Schmidt: Multiple emitter location and signal parameter estimation. IEEE Trans. AP-34, 276-280 (1986)

J.A. Stuller: Generalized likelihood signal resolution. IEEE Trans. IT-21, 276-282 (1975)

J.B. Thomas, J.K. Wolf: On the statistical detection problem for multiple signals. IRE Trans. IT-S, 274-280 (1962)

D.W. Tufts, R. Kumaresan: Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood. Proc. IEEE 70, 975-989 (1982)

M. Wax, T. Kailath: Optimum localization of multiple sources by passive arrays. IEEE Trans. ASSP-31, 1210-1217 (1983)

M. Wax, T. Kailath: Detection of signals by information theoretic criteria. IEEE Trans. ASSP-33, 387-392 (1985)

A.J. Weiss, A.S. Willsky, B.C. Levy: Eigenstructure approach for array processing with unknown intensity coefficients. IEEE Trans. ASSP-36, 1613-1617 (1988)

46 Z. Zhu and S. Haykin

G.O. Young, J.E. Howard: Array processing for target detection in a nonuniform clutter back-

ground. Proc. of the Symp. on Computer Processing in Communication, New York (1969) pp. 727-743

L.C. Zhao, P.R. Krishnaiah, Z.D. Bai: On detection of the number of signals in presence of white noise. Multivar. Anal. 20, 1-25 (1986)

L.C. Zhao, P.R. Krishnaiah, Z.D. Bai: On detection of the number of signals when the noise covariance matrix is arbitrary. J. Multivar. Anal. 20, 2~9 (1986)

I.Ziskind, M. Wax: Maximum likelihood localizatien of multiple sources by alternating projection. IEEE Trans. ASSP-36, 1553-1560 (1988)

3.Radar Target Parameter Estimation with Array Antennas

U. Nickel

With 18 Figures

The final processing in a radar reception chain is the estimation of target parameters. These estimates are then displayed, or further processed, by a radar data processor. The quality ofthe estimated parameters depends not only on the algorithm used, but also on the whole hardware and signal processing employed. The question as to how to do radar parameter estimation is indeed very complex, and not simply a matter of choosing some algorithm. Many publications are available on this problem, starting with the classic Radar Handbook of Skolnik [3.1].

In this chapter, we are not concerned with classical procedures and hardware issues. We want to point out the new potentials and extensions made possible by the use of new array signal processing methods. Description ofclassical methods is included to show the relationships between them and the new methods.

Section 3.1 briefly reviews the features ofa phased array antenna, introduces the terminology, and gives an introduction to the special problems and constraints that arise in parameter estimation with a phased array radar. The most important parameter estimation problem, angle estimation, is treated in Sect. 3.2. Starting with the general description of monopulse estimation for phased arrays, we then give a review of modem superresolution techniques; this section contains the main contribution of this chapter. Section 3.3 describes to what extent these ideas can be applied to the frequency estimation problem. Many of the modem superresolution methods have their origin in spectral analysis oftime series. However, the practical constraints in array processing are very different from those in time series analysis. Section 3.4 mentions, for completeness, how the remaining radar target parameters are estimated. For range resolution, the application of superresolution methods is also possible. The waveform to be estimated is different in this case: it is the filtered target echo, which is a time limited function. This limits the possibilities to obtain larger sample numbers of the waveform, which is necessary for a reasonable application of superresolution methods.

3.1 Radar Parameter Estimation Problem

The history of phased array radars starts with the invention of antennas that allow an almost infinitely fast switching of the direction of the antenna beam.

48 U. Nickel

This is achieved, in principle, by synthesizing a plane wavefront by a large number of spherical elementary waves that are controlled in phase. The early interest in phased array radar research was, therefore, focused on the construction of phase-shifters and on the arrangement of antenna elements over the antenna aperture.

Consequences of fast beam switching were soon recognized: The search patterns for the observation space were no longer limited to conventional scanning, e.g., a rotation of the antenna; rather, they can be variable, e.g., regions of high interest can be searched more intensively. The dwell time for one direction (integration time) can be variable. This allows the use of different detectors and filters for different directions, depending on the interference background.

The sequential detector is a good example for improving the solution to the detection problem [3.2J. By using two thresholds, the sequential detector checks if the available data are sufficient to make a decision; if not, more samples are taken. This scheme can produce a saving in transmit power (or dwell time for the given direction) by a significant factor, compared to a fixed sample size test. The choice of different detectors, filters (MTI, Doppler filter), and search patterns is determined by the objective to save transmit energy. Thus, one can produce a higher signal-to-noise ratio, and, therefore, obtain a higher probability of detection and a more accurate parameter estimation. We call this flexible, economic use of the transmit power 'energy management'.

A second consequence offast beam-switching is that different radar tasks can be done in a time-multiplexed manner, e.g., long-range search, search in clutter regions, target acquisition, target tracking, etc., all with different transmit signal waveforms. Modern phased array radars can track up to 40 targets while performing the search task at the same time [3.3J; they can thus incorporate the signal processing functions of many conventional radars. Energy management and multifunction operation are, indeed, typical features of modern phased array radars.

For the time being, the main advantage of a phased array radar lies in the multifunction operation; however, future phased array radars will employ modern array signal processing methods, i.e., algorithms based on temporal and spatial samples of the wavefront. For radar, array signal processing methods offer two interesting applications: adaptive interference suppression and resolution enhancement. In this chapter, we will focus on the application of methods for resolution enhancement. Because these methods can yield a resolution capability beyond the classical Rayleigh resolution limit, simply by signal processing, we term this type of processing 'superresolution'.

3.1.1 Range and Angle Estimation

The most important radar target parameters are range and angle, i.e., the location of the target. In the search mode, a modern multifunction phased array first scans the observation space in azimuth and elevation on a grid ofdirections,

and in range on a sequence of range bins that have the size of a conventional resolution limit. For angular resolution the conventional resolution limit is set by the 3dB-antenna beamwidth; for range resolution the resolution limit is set by the 3dB-width of the receiving (matched) filter output of the transmit waveform. (For uncoded pulses this is approximately equal to the transmitted pulse length; for coded pulses it is approximately equal to a subpulse length).

Mter a target has been detected, this detection is, in general, validated by a second look of the antenna in this direction. In this case, the radar will also determine the location of the target more accurately. Therefore, one will choose the transmit parameters so as to produce as high a signal-to-noise ratio (SNR) as possible, e.g., by using a suitable waveform or coherent integration. With this high SNR echo, one uses sensitive parameter estimation techniques for angle (monopulse) and range. This signal processing is done only for the range cells of interest.

Mter the presence of a target has been validated, a multi-function radar will initiate a track processor, e.g., a Kalman filter [3.4]. The look direction of the antenna into the target direction and the update intervals will then be controlled by this track processor. Radar parameters that produce high SNR to obtain accurate parameter estimation are used for this type of operation. Superresolution methods will, in general, only be applied for validated targets on track, because superresolution is time and/or energy consuming. The update intervals for targets on track depend on the maneuvers of the target. Typically, these are in the range of 1-2 sec, which is the time frame superresolution methods have to work with.

Advanced superresolution methods are computationally intensive and, therefore, time-consuming, and time is a quantity we are always short of in the use of radar. So, even for targets on track, superresolution methods will not replace conventional estimation. In addition, conventional beamforming is rather robust against measurement errors. This is in contrast to superresolution methods which, even for a single target, are sensitive, and in many cases give more noisy angle estimates. For these reasons, superresolution methods are presently of only practical interest for special applications, where a high signal- to-noise ratio is available, or can be achieved by coherent integration. It is always a second-stage processing; that is, after having done conventional detection and estimation (monopulse), we may then switch to the use of a superresolution method.

a)Possible Applications of Superresolution Methods

i)Resolution of Clusters of Targets. This requires two-dimensional angular resolution and, hence, a planar array. The targets are within a limited angular sector, and we have a good initial guess of the target directions. Target angles, together with the number of targets, have to be determined.

ii)Reduction of Multipath Error of Low Flying Targets. This occurs mainly at short ranges over sea, and is only a one-dimensional (elevation) resolution