Parameter estimation and super-resolution

Target parameter estimation and array features

1.7 Parameter estimation and super-resolution

The objective of radar processing is not to maximize the SNR, but to detect targets and to determine their parameters. If we consider the detection problem as the first step of radar processing, then it can be shown that the common likelihood ratio test for a single target contains the beam output SNR as a sufficient statistic. That means, if we maximize the output SNR this will also give the maximum probability of detection. The radar detection problem as a likelihood ratio test has been dis- cussed in detail in [13 Chapter 2].

The detection procedure is the first step of data reduction and it is applied on a course grid of the parameter values of interest in direction, range and possibly Doppler. Only for the few bins with detected targets the target parameters are then subsequently estimated in a refined procedure.

1.7.1 Maximum likelihood estimation and monopulse

Standard radar parameter estimation can be traced back to ML estimation of a single target. Single target ML estimation leads to the matched filter in different domains [13 Section 3.2.1]. The properties of the matched filter can be judged by the correlation of the signal model used in the likelihood function and the received data. This leads to the beam pattern for angle estimation and to the ambiguity function for range and Doppler estimation. If the ambiguity function has a narrow beam and sufficiently low sidelobes, then the model of a single target is a good approximation as other targets are attenuated by the sidelobes.

Let us illustrate this fact for the simple case of direction estimation based on a single snapshotz2C^N with an Nelement array. We assume zN_C^NðaðuÞb;IÞ with some complex amplitude b which may be random or deterministic.

We assume the receiver noise is normalized to a standard deviation one. The ML estimate of the target direction then is given by

^

u ¼arg max

u;b pðzju;bÞ

¼arg max

u;b

1

p^Ne^{ðzaðuÞbÞHðzaðuÞbÞ} (1.64)

Maximization of this expression is equivalent to minimizing the exponent and this is just a least-squares problem. For all values of u the minimum over bis attained for^b¼ ðaðuÞ^HaðuÞÞ¹aðuÞ^Hzwhich is the conventional sum beam output

normalized by the length of the beamforming vector. Inserting this expression for

^b into the exponent of (1.64) we find that the ML estimate will minimize kzaðuÞaðuÞ^Hz=aðuÞ^HaðuÞk². Multiplying this out and omitting constant terms this is equivalent to maximizing jaðuÞ^Hzj²=aðuÞ^HaðuÞ. This means that the ML estimate is given by the maximum of the magnitude pattern by scanning the antenna beam over the target (the scan pattern). If there is only one target present, z¼aðu₀Þ, we see that this is just the same as an ambiguity function. The shape of this function determines the accuracy and resolution.

The resolution limit for classical beamforming, i.e. the single target ML esti- mator, is the 3 dB beamwidth. This is the separation of two closely spaced point targets at which the ambiguity function (beam pattern) begins to produce two local maxima. If we have targets at closer spacing or patterns with high sidelobes, this simple criterion is inadequate and multiple target models have to be used for parameter estimation. A variety of such multiple target estimation methods have been introduced and we denote these assuper-resolution methods. These methods will be considered in the next section.

Maximization of the scan pattern for a single snapshotzmay be impossible for analogue beamforming and can be time consuming for digital beamforming depending on the number of directions. A rapid procedure for this task is desired.

One could approximate the measured sum beam output by a parabola by approx- imating jaðuÞ^Hzj²aþbuþcu² (for simplicity we note this here only for one angle). The beam parameters a,band c could be measured from a set of beam directions or from derivatives of the beam and one can then determine the location of the maximum. If we maximize instead the functionFðuÞ ¼lnðjaðuÞ^Hzj²Þ, then one can obtain an approximation in the form of a linear equation. Assuming that the target direction^uis close to the antenna look directionu0such thatFð^uÞ Fðu0Þ, one can approximate the derivative ofFin a first order Taylor series at^uby:

F⁰ðu0Þ F⁰ð^uÞ þF⁰⁰ð^uÞðu0^uÞ

As the derivative vanishes at the maximum, we obtain the linear approximation

^

uu0 ðF⁰⁰Þ¹ð^uÞF⁰ðu0Þ (1.65)

Writing this explicitly shows that this leads to the well-known monopulse formula for angle estimation:

F⁰¼ ða^0Hzz^Haþa^Hzz^Ha⁰Þ=a^Hzz^Ha¼2Re a^0Hz=a^Hz

n o

(1.66) In fact, this is the monopulse ratio because the weighting with @ai=@u¼ jð2pf=cÞxiai produces a difference pattern. This also tells us how to form the optimum difference pattern with phased arrays: with an amplitude weighting xi. A simplified weighting with sign{xi} would give the classical difference beam and is a rough approximation. The first derivative is taken for the antenna look direction with the measured data inserted. The second derivative is taken at the unknown

target direction and this is approximated by a constant value. For vanishing receiver noise one obtains:

F⁰⁰ð^uÞ a^00Haþa^Ha⁰⁰

a^Ha ð Þ^u (1.67)

ThusF⁰⁰is a fixed slope correction quantity and is only determined by the antenna configuration. For a centred array we have a^0Ha¼ j2pf=c PN

i¼1xi¼0 and a^00Ha¼ ð2pf=cÞ² PN

i¼1x²_i. As the Taylor expansion is the best linear approx- imation this formula also tells us the optimum slope correction factor. This pro- cedure can be extended to planar and volume arrays, see [14]. The optimal slope correction formula also gives an indication in which way good angle estimates can be obtained: the array should be dynamically balanced by (a) PN

i¼1xi¼0, PN

i¼1y_i¼0, (b)PN

i¼1x_iy_i¼0, and (c)PN

i¼1x²_i ¼PN

i¼1y²_i. Condition (a) ensures unbiasedness and condition (b) gives independence of the azimuth and elevation estimates.

Let us emphasize that the monopulse technique and all other kinds of inter- or extra-polation of the scan pattern are numerical methods to approximate the single target ML-estimate, i.e. are in general not adequate for multiple targets.

1.7.2 Super-resolution

An antenna array provides spatial samples of the impinging wavefronts and one may define a multi-target model for this case. Such refined target models are the basis for enhanced resolution beyond the classical resolution limit (the 3 dB width of the beam or of the ambiguity function). Historically these methods have often been introduced to improve the limited resolution of the matched filter. We call these super-resolution methods and distinguish it from high resolution, which may be achieved also by the classical method of using a very large antenna or time aperture. Super-resolution methods have been discussed since decades, and text- books on this topic are available, e.g. [11].

We confine our development to the angle parameter estimation problem (spatial domain), but corresponding versions can be applied in the time domain as well. The spatial resolution is described by the classical Rayleigh limit which is the 3 dB beamwidth and this is determined by the antenna aperture. The antenna is a once defined hardware design resulting from several electrical, mechanical and operational constraints. Therefore there is a high interest to overcome this limita- tion by signal processing techniques. In the spatial domain the super-resolution is faced with the challenges of irregular sampling and sub-array processing.

The intention of this chapter is to point out the principles, inter-relations and aspects of implementation of these methods. Therefore we mention here from the many proposed methods only some most popular methods which are also applic- able to sub-arrays and irregular arrays. To give an overview we like to classify the methods into two categories: spectral methods, which generate a spiky estimate of the angular spectral density, and parametric methods, which deliver only a set of

‘optimal’ parameter estimates which explain in a sense best the data for the inserted model. This classification is not exclusive; there are methods in between (e.g. the ESPRIT method mentioned below).

The most popular spectral methods with an angular power density estimate S(u) are (see [13 Sections 3.2.4 and 3.2.5])

Capon’s method(Capon 1969):

SCðuÞ ¼aðuÞ^HR^¹_MLaðuÞ1

with R^_ML¼ 1 K

X^K

k¼1

z_kz^H_k (1.68)

MUSIC method (Multiple Signal Classification, Bienvenu/Kopp 1986, Schmidt 1987):

SMUSICðuÞ ¼aðuÞ^HP^?aðuÞ1

(1.69) with P^?¼IXX^H, and Xspanning the dominant sub-space, which is usually determined by the eigenvectors corresponding to the dominant eigenvalues of the estimated array data covariance matrixR^_ML. As we have shown in (1.55) this projection is an estimate of the inverse covariance matrix for high SNR.

Therefore the MUSIC method incorporates an artificial SNR enhancement compared with Capon’s method. If the projection would be built with true steering vectors of the targets the power density would have poles in these directions. Parameter estimation, i.e. the target directions, is then done by find- ing theM highest maxima of these spectra (M1-dimensional maximizations for a linear array orM2-dimensional maximizations for a planar array). For special arrays this numerical maximization can be replaced by an analytic solution of the maxima: for a ULA by solving for the roots of a complex polynomial (Root- MUSIC, [46], Pisarenko method, [47]) and for an array consisting of two sub- arrays separated by a fixed shift vector by solving a certain eigen-decomposition problem (ESPRIT, [48]). Of course, an estimate of the signal sub-space can also be obtained without eigen-decomposition, e.g. by rank revealing QR decom- position or by a suitable transformation of the data, see [49] for the Hung-Turner projection (HTP), Yeh–Brandwood projection (YBT) and Matrix Transform projection (MTP).

The Capon density function can be generalized to the form SCðuÞ ¼ ðaðuÞ^HR^^r_MLaðuÞÞ^rwhich results in an enhancement of the peaks as with MUSIC, if we normalize the noise eigenvalues to one. If the true covariance matrixRcan be written as R¼U_sdiagf gUli ^H_s þU_nU^H_n, then it can be shown that R^r¼ IU_sdiag 1_l¹r

i

n o

U^H_s. This estimate has been introduced by Pisarenko [50].

An LMI-version as in (1.56) instead of MUSIC would also be possible, but this is not very common. A weighting of the sub-space components is counter- productive to the desired high peaks of the density. The spiky density is a desired feature for super-resolution. Adding some small eigenvalues to a given MUSIC projection is a good way to check the angular relevance of these eigenvalues and can be used to confirm the selected sub-space dimension, see also Section 1.7.4.

The most popular parametric methods are:

Deterministic ML (detML) method

For ML estimation with a multiple target model with directions u₁;. . .u_M with deterministic complex amplitudes b under Gaussian assumptions we assume that the array output data z_k; k¼1. . .K; are assumed to be distributed as z_kN_C^NðAðqÞbk;IÞ. Similar to the maximization problem in (1.64) this leads to minimizing the objective function:

FdetðqÞ ¼ 1 K

X^K

k¼1

kzkAðA^HAÞ¹A^Hz_kk² ¼trðP^?_AR^_MLÞ (1.70) withP^?_A¼IAðA^HAÞ¹A^HandA¼ ðaðu1Þ; ;aðuMÞÞ, [13 p. 74]. Actually, the measured data enter here only via the estimated covariance matrix. One could replace this estimated covariance matrix by the direction information carrying part, i.e. by the dominant (signal) sub-spaceXX^H. One may additionally apply a positive weighting for the sub-space components with a matrixW¼diagf g. This results inwi

the weighted sub-space fitting (WSSF) methodFWSSFðqÞ ¼trðP^?_AXWX^HÞ, [51].

The detML method has some intuitive interpretations which give rise to some efficient numerical procedures:

1. FdetðqÞ¼_K¹ PK

k¼1z^H_kP^?_Az_k¼_K¹PK

k¼1kP^?_Az_kk²¼_K¹PK

k¼1jjzkAðA^HAÞ¹A^Hz_k

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

¼b^

jj²

means that the mean squared residual error after signal extraction is minimized, which is a useful criterion for target number estimation.

2. WritingFdetðqÞ ¼CPK

k¼1z^H_kAðA^HAÞ¹A^Hz_k shows that this can be inter- preted as maximizing a set of decoupled sum beamsða^Hðu1Þzk;. . .;a^HðuMÞzkÞ with a decoupling matrixðA^HAÞ¹. Maximization of these decoupled beams could numerically be realized by a gradient iteration which results in decou- pled difference beams similar to the monopulse technique. A refined iteration of this kind is used in Figure 1.24.

3. One can write the detML criterion asFdetðqÞ ¼Ca^H_nullR a^ _null=a^H_nulla_nullwith a_null ¼P^?_AaðuÞ, where we have partitioned the matrix of steering vectors into A¼ ðaðuÞ;AÞ. This property is valid due to the projection decomposition lemma which says that for any partitioning A¼(F,G) we can write P^?_A¼P^?_GP^?_GFðF^HP^?_GFÞ¹F^HP^?_G. If we keep the directions in A fixed, this relation says that one may maximize the scan pattern overuwhile the source directions inA are deterministically nulled (see (1.44)). One can now perform the multi-dimensional maximization by alternating one-dimensional max- imizations of this kind while keeping the remaining directions fixed. This is the basis of the alternating projection (AP) method [52] which is equivalent to the IMP (Incremental Multi-Parameter) method, [11 p. 105], which was developed independently. AP and IMP are iterative numerical minimizations along alternating slices parallel to the co-ordinate axes. For these methods inspection of the residual scan patternsa^H_nullR a^ _nullis also useful to check the relevance of some additional target directions, see also Section 1.7.4.

There are other methods similar the AP/IMP method to solve the detML estimation by a sequence of 1-dimensional maximizations: the Expectation Maximization (EM) method and the Space Alternating Generalized EM (SAGE) algorithm ([1, p. 316], [53]). These methods are based on a decomposed signal model z_m¼aðumÞbmþn= ffiffiffiffiffi

pM

;m¼1. . .M. The associated data and likelihood functions are sequentially estimated and maximized.

Stochastic ML (stoML) method

For ML estimation with a multiple target model with directions u₁;. . .u_M with complex normal distributed amplitudesbunder Gaussian assumptions we assume that the array output data are distributed i.i.d. as zN_C^Nð0;AðqÞBAðqÞ^HþIÞ.

One can show that this leads to minimizing the objective function:

FstoðqÞ ¼log detðRð ÞqÞ þtr Rð Þq ¹R^_ML

(1.71) whereR(q) denotes the completely parameterized covariance matrix, w.r.t. angles, powers and correlations. A formulation depending only on the unknown directions can be given by, [54],

FstoðqÞ ¼det A^BðqÞA^Hþ^s²ðqÞI

with (1.72)

Bð^ qÞ ¼ ðA^HAÞ¹A^HR^_ML^s²ðqÞI

AðA^HAÞ¹ and ^s²ðqÞ ¼ 1

NMtr P^?_AR^_ML (1.73) and forA¼A(q). The behaviour of the function (1.72) is quite difficult to check.

A first general comment concerning all super-resolution methods is that these methods in different ways try to interpret the fine structure of the signal. To achieve any super-resolution effect with these methods we have therefore to require a higher SNR than for the detection of a single target.

To characterize the performance of these methods we show three examples.

A typical feature of the MUSIC method is illustrated in Figure 1.22 obtained with the experimental system DESAS of Fraunhofer FHR, [1 p. 309], or [13 p. 81].

Sub-plot (a) shows the excellent resolution in simulations for an azimuth cut for 2 targets separated at 0.6 BW while for real data in sub-plot (b) with the same sce- nario the pattern looks almost the same as with Capon’s method. This degradation is due to the channel errors in the used experimental system leading to a mismatch between the real and the assumed signal model. Note that the used calibration method has also a decisive effect. In this case we focused the array on each of the two sources using each emitted signal alone and averaged the correction quantities.

Clearly, one can improve the results by more clever calibration procedures.

A result with the deterministic ML method with real data is shown in Figure 1.24. The scenario consisted of a vertical 32 element ULA observing an approaching aircraft at constant low height as indicated in Figure 1.23. The pro- blem of this scenario over sea is that the electromagnetic energy will be reflected from the sea surface which may create a highly correlated image target. This result

has also been presented in [1 p. 307] under the name ‘Parametric Target Model Fitting’ (PTMF).

The numerical minimization of the decoupled sum beam function (interpreta- tion 2) was performed by a stochastic approximation method, i.e. by using in each iteration step a new measurementzk[55]. However, the standard Robbins–Monro iteration converged unacceptably slow and this has also been reported by other

–0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 –10

0 10 20 30 40 50 60 70 80

u

dB

10 10

BW MUSIC

–0.5

0

0.5

–0.5 0

0.5 0 5 10 15 20 25

v u

dB

(b) (a)

Figure 1.22 MUSIC spectra of two targets at 0.6BW separation with a planar array of 8 elements (7 elements irregularly distributed on a ring and with one in the centre). (a) Simulated data and (b) measured data

researchers. Therefore several modifications were implemented. Besides the reduction of the iteration step size by the classical Robbins–Monro factor ak¼m=ðbþkÞ in the kth step (for suitable constants m, b), a modified update vector was introduced because we observed strong fluctuations in the length of the gradient. Instead of the gradient of the function (1.70) we used an objective function which is oriented along the derivation of the monopulse estimator in [14], namely the logarithm of the decoupled sum beam function lnðS_kðqÞÞ ¼ lnðz^H_kAðA^HAÞ¹A^Hz_kÞ. This leads to an iteration with an update vector consisting of the gradient divided by the decoupled sum beam, i.e. gradfSkðqÞg=SkðqÞ, which is similar to the monopulse ratio. Also a clipping of the length of this gradient was introduced. The combination of these modifications resulted in an accelerated convergence. We also mention here that any modification of the stochastic approximation by using in addition an estimate of the Hessian (the matrix of second derivatives) of the objective function did not result in an improvement because the fluctuations of this matrix were so strong that they rather distorted the convergence.

Figure 1.23 Scenario of low-angle tracking over sea with target flying at constant height and with multi-path (dashed lines)

Monopulse estimate

detML estimate

True elevation

2.57

1.00.60.2–0.2

Elevation (BW) –0.6

3.08 3.58 4.08 4.59 5.09 5.60 6.10 Range (KM)

6.61 7.11 7.62 8.12 8.62

Figure 1.24 Super-resolution of multi-path propagation over sea with

deterministic ML method, real data from vertical linear array with 32 elements, scenario of Figure 1.23, (from [4])

This real data result shows that the detML-method is able to resolve highly correlated targets which arise due to the reflections on the sea surface for low angle tracking. The behaviour of the classical monopulse estimates in Figure 1.24 reflects the variation of the phase differences between the direct and reflected path from 0 to 180 (glint effect). For a phase difference of 0 the monopulse points into the centre, for 180 it points outside the 2-target configuration. These phase differences can be well tracked by the associated complex amplitude estimates

^b_k¼ ðA^HAÞ¹A^Hz_kand this is an advantage of the detML-method.

The general problems of super-resolution methods are described in [13,56].

One of the key problems is the numerical effort of finding the M maxima (oneM-dimensional optimization or M 1-dimensional optimizations for a linear antenna). For the deterministic ML method the stochastic approximation algorithm and the IMP or alternating projection method have been successfully proven with real data. The IMP method is an iteration of maximizations of an adaptively formed beam pattern. Therefore the generalized monopulse method of (1.75) can be used for this purpose, see Section 1.9.1 and [14].

Another problem is the exact knowledge of the signal model for all possible directions, i.e. the vector function a(u). The co-domain of this function is some- times called the array manifold. The correctness of this model is mainly a problem of the receiving system accuracy and of proper calibration. While the transmission of a plane wave in the main beam direction can be locally quite accurately mod- elled (using calibration) this can be difficult in the sidelobe region. In particular this can lead to problems if super-resolution methods are applied to sub-arrayed arrays.

More refined parametric methods with higher asymptotic resolution property have been suggested (e.g. COMET, Covariance Matching Estimation Technique [57]). However, application of such methods to real data often revealed no improvement (similar to the case with MUSIC in Figure 1.22). The reason is that these methods are much more sensitive to the signal model than the accuracy of the system provides. We have observed that a very sensitive matching criterion with a very sharp ideal minimum of the objective function may lead to a measured data objective function where the minimum has completely disappeared.

1.7.3 Super-resolution applied to sub-arrays

For an array with digital sub-arrays super-resolution has to be performed only with the sub-array outputs. The array manifold taken at the sub-array outputs has then to be considered, see Section 1.3.3. This manifold is essentially described by the sub-array patterns which can be well modelled in the main beam region. However, in the sidelobe region the random sub-array sidelobes produce in general no reason- able results. If e.g. we consider the sidelobes of the generic array as in Figure 1.8(b) it is quite reasonable that beams formed into the sub-array sidelobe region will lead to patterns with terrible sidelobes and an insignificant main beam.

In that case it is advantageous to use a simplified array manifold model which is only locally valid in the mainbeam of the sub-arrays. The simplest and most effective model is to use the sub-array centres (the super-array) and consider these