Embedding of array processing into full radar data processing

Target parameter estimation and array features

1.9 Embedding of array processing into full radar data processing

If the objective is to improve a radar system one has to recognize that e.g. max- imizing the SNR or determining just a direction is not the task of a radar. The improvement with new processing has to be judged evaluating the resulting system features. As a starting point this means that when implementing refined methods we have also to consider the effect on the subsequent processing. Known properties of the advanced methods should be exploited in subsequent processing stages to get optimum performance. This requires some mutual communication between the different kinds of processing, in particular between the signal processing and data processing stage. Also, the system management (beam steering, waveform control etc.) should be adapted according to super-resolution or ABF results. We have studied these problems for the case of adaptive interference suppression for the case of three special tasks: detection, angle estimation and tracking. With the mod- ifications introduced below significant improvements for the system could be demonstrated.

1.9.1 Adaptive monopulse

For large arrays angle estimation is usually implemented by monopulse estimation which is an established technique for rapid and precise angle estimation. It is based on two beams formed in parallel, a sum beam and a difference beam. The differ- ence 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 direction from the sum beam pointing direction. In fact, it can be shown that the monopulse

estimator is an approximation of the Maximum-Likelihood angle estimator, see Section 1.7 and [14], which is considered as the optimum estimator. The monopulse estimator has been generalized in [14] to arrays with arbitrary sub- arrays and arbitrary sum and difference beams.

With irregular sub-arrays many different forms of difference and sum beams can be formed resulting in different monopulse functions. In particular, adaptive beams will distort the shape of the sum beam due to the interference that is to be suppressed. The difference beam, which must adaptively suppress the interference as well, will suffer another and different distortion. The ratio of both beams which is used in conventional monopulse as described in Section 1.2.4 will then no more indicate the target direction. Several techniques have been proposed to counter this problem, e.g. [61–63], to mention a few. The generalized monopulse procedure described in [14] provides a general approach by providing correction values to compensate these distortions.

The generalized monopulse formula for estimating angles ð^u;^vÞ^T with a planar array and a sum and two difference beams formed into directionðu0;v₀Þ^Tis given by:

^ u

^

v ¼ u0

v0 c_xx c_xy c_yx c_yy

Rxm_x Rym_y

(1.75) C¼ cxx cxy

cyx cyy

is a slope correction matrix andm¼ m_x

m_y is a bias correction.

R¼RefD=Sgis the monopulse ratio formed with the measured difference beam D_x¼d^H_xzfor azimuth beamforming weightdx, or D_y¼d^H_yzfor elevation beam- forming weight d_y, divided by the sum beam output S¼w^Hz. For standard beamforming one usesw¼a, d_x¼diagf gaxi andd_y¼diagf ga. However, theyi

formula can be applied as well for any kind of beams, in particular adapted beams.

The point is that we can determine the correction values for all kinds of beams.

The monopulse ratio is a function of the unknown target directions (u,v). Let the vector of monopulse ratios be denoted byR(u,v)¼(Rx(u,v),Ry(u,v))^T. The correction quantities are determined such that the expectation of the error is zero and such that a linear function with slope one is approximated. More precisely, for the function of the unknown target direction:

Mðu;vÞ ¼C ðEfRðu;vÞg mÞ (1.76)

we require:

Mðu₀;v₀Þ ¼0 and @M

@u ðu₀;v₀Þ ¼ 1 0 !

; @M

@v ðu₀;v₀Þ ¼ 0 1 !

or C @R

@u

@R

@v

ðu0;v0Þ ¼I

(1.77)

These conditions can only approximately be fulfilled for sufficiently high SNR. One obtains for the bias correction m for a pointing direction a0 ¼ a(u0,v0), [14]

m_a¼Re d^H_aa₀ w^Ha₀

; for a¼x;y (1.78)

For the elements of the inverse slope correction matrix C¹¼ ðc^a;hÞ_a¼x;y

h¼u;v

one obtains:

c^a;h¼Re d^H_aa_h;0a^H₀wþd^H_aa₀a^H_h;0w

n o

w^Ha₀

j j² m_a2Re w^Ha_h;0 w^Ha₀

(1.79) witha¼xor yanda_h,0 denoting the derivativeð@a=@hÞ_ðu0;v0Þ, whereh¼uorv.

In the non-adaptive case the slope corrections are fixed quantities determined by the antenna configuration. For example, for omni-directional antenna elements and phase steering at the elements we havea₀¼Ge(1, . . . ,1)^T, whereGeis the antenna element gain, anda^T_u;0¼Geðj2pf=cÞðx1; :::;xNÞ. For sub-arrayed antennas we have to calculate the correction values for the sub-array manifolds as in (1.20). The important point is that for the adapted case the correction values can be calculated from known quantities: the adaptive weight vectors and array transmission vectors.

Also 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.

This formula can also be extended to finding the maximum of the non- coherently averaged scan pattern P_K

k¼1j jSk². This leads to an averaged monopulse ratio also called Mosca-monopulse which is formed as R_K¼ Re PK

k¼1D_kS_k=PK k¼1j jSk²

n o

.

The statistical performance of this generalized monopulse estimator has been described in [64] for the case of Rayleigh fluctuating targets by calculation of the mean and covariance. This fluctuation model is also known as the Swerling-I model for slow fluctuations from scan to scan, or as the Swerling-II model for rapidly fluctuating targets from pulse to pulse. Moreover, in [65] the corresponding mean and covariance have been given for the case of targets with deterministic amplitude variation, sometimes also called the Swerling-0 case. These results have been extended in [66] to a mixture of Swerling-0, I, II models and to extended targets. To indicate the general performance the results for the Rayleigh targets (Swerling-I case), which is the simplest solution, is given here. We are looking for the statistical description of:

Ef g ¼u u₀CðEf g R mÞ and covf g ¼u Ccovf gCR ^T (1.80) For the expectations of the monopulse ratio Rwe consider only those measure- ments which have an integrated sum beam output above a certain threshold h,

i.e. for whichP_s¼PK

k¼1j jS_k²>h. Then one obtains, [14], EfRjPs>hg ¼Re G_DS

GS

covfRjPs>hg ¼

0:5VE₁ h GS

e^h=GS; for K ¼1 0:5V 1

K1

e_K2ðh=G_SÞ

eK1ðh=GSÞ; for K >1 8>

>>

<

>>

>:

(1.81)

The matrixVis given by:

V¼Re G_DG_DSG_S¹G^H_DS

=G_S (1.82)

E1 denotes the exponential integral E₁ðxÞ ¼R₁

x e^t=t dt which can be well approximated for high SNR by logð1þ1=xÞ, and en (x) is defined as enðxÞ ¼Pn

k¼0x^k=k!. The matrices G_D, G_DS, GS denote the blocks of the joint covariance matrix of the sum and difference beams. More precisely, if we assume the vector of beam outputs Dx;Dy;ST

¼ D

S to be complex Gaussian dis- tributed with mean m_DS and covariance matrix G, then we partition G¼ G_D G_DS

G^H_DS GS

.

Clearly, the detection thresholdhcontrols the magnitude of the variance. If we consider the classical monopulse withK¼1 andh¼0 we obtain the well-known result that the variance is infinite.

To indicate the performance of this approach we consider the generic array of Figure 1.7 with 902 elements and 32 sub-arrays and a scenario with a jammer slightly less than a beamwidth away from the antenna look direction. Figure 1.27 shows the resulting biases and variances for different target directions within the beamwidth. A jammer is located at the asterisk symbol direction with JNR¼27 dB.

The hypothetical target has a sum beam output SNR of 22.8 dB. The 3 dB contour of the unadapted sum beam is shown by a dashed circle. The 3 dB contour of the adapted beam will be of course different. No averaging was performed (true monopulse) and the detection threshold was set to the classical level of 13 dB SNR.

The biases are shown by arrows for different possible target positions with the standard deviation ellipses at the tip corresponding to (1.81). One can see that in the beam pointing direction (0,0) the bias is zero by construction and the variance is small. The errors increase for target directions on the skirt of the main beam and close to the jammer.

To get an impression of the relevance of this accuracy we show the corre- sponding results with the Crame´r–Rao bound (CRB) for comparison. We use the CRB formulation of [67]. As we are using sum and difference beams, we have to take the CRB based on the distribution of the three input quantitiesðDx;Dy;SÞ. The result is seen in Figure 1.28. This would be the asymptotic accuracy of an ideal unbiased estimator using the three beam outputs. One can see that the uncertainty

ellipses are somewhat larger than with our generalized monopulse. However, the CRB is a rather meaningless measure in this case because the asymptotic case which the CRB describes is not attained for finite SNR and finite element number.

In realistic scenarios the estimates are biased and the real overall error is a com- bination of bias and variance. So, angle estimation with ABF is a good example that the CRB is not always a good indicator of the performance of an estimator. In fact, if we consider the CRB for biased estimates then the results will be more similar to our generalized monopulse estimator performance of Figure 1.27, see the plots in [14]. The variances are a bit larger because the CRB does not account for the detection threshold. The important point is that for this result with modified CRB we need to know the bias which we have taken from our result (1.81).

The large bias may not be satisfying. However, one may repeat the monopulse estimate with a look direction steered at sub-array level into the new estimated direction. This is an all-offline processing with the given sub-array data. No new transmit pulse is needed. We have called this themulti-step monopulse procedure [14]. Figure 1.29 shows that the multi-step monopulse procedure with only one additional iteration can reduce the bias considerably. The variances appearing in Figure 1.27 are virtually not changed with the multi-step monopulse procedure and they are omitted here for better visibility.

–0.05 0 0.05

–0.05 –0.04 –0.03 –0.02 –0.01 0 0.01 0.02 0.03 0.04 0.05

u

v

Figure 1.27 Bias and standard deviation ellipses of generalized monopulse with adapted sum and difference beams for different target positions, (from [4])

–0.05 0 0.05 –0.05

–0.04 –0.03 –0.02 –0.01 0 0.01 0.02 0.03 0.04

u

v

Figure 1.28 Crame´r–Rao bound corresponding to scenario of Figure 1.27

–0.05 0 0.05

–0.05 –0.04 –0.03 –0.02 –0.01 0 0.01 0.02 0.03 0.04 0.05

u

v

Figure 1.29 Bias for 2-step monopulse for different target positions and jammer scenario of Figure 1.27, (from [4])

An application of this adaptive monopulse technique to space-time adaptive processing (STAP) has been given in [68], where the case of ground target detec- tion from an airborne radar was considered. Another application to tracking in jammed scenarios will be given below in Section 1.9.3.

1.9.2 Adaptive detection

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. The pioneering first test statistics accounting for this two data set structure were the Generalized Likelihood test (GLRT) of Kelly, [69], the Adaptive Matched Filter (AMF) detector, [70], and the Adaptive Cosine Estimator (ACE) detector, [71]. In our sub-array output formulation (1.20) these tests have the form:

TGLRTð~zÞ ¼ ~a^H₀Q^¹_SMI~z²

~

a^H₀Q^¹_SMI~a₀1þ_K¹~z^HQ^¹_SMI~z (1.83)

TAMFð~zÞ ¼~a^H₀Q^¹_SMI~z²

~

a^H₀Q^¹_SMI~a₀

(1.84)

T_ACEð~zÞ ¼ ~a^H₀Q^¹_SMI~z²

~

a^H₀Q^¹_SMI~a₀~z^HQ^¹_SMI~z

(1.85) This formulation admits that the quantities ~z;~a₀;Q^ are all generated at the sub- array outputs.Q^_SMIis the SMI estimate of the covariance matrix (1.52) applied at the subarray outputs. ~a₀ denotes the plane wave model for a direction u₀. Basic relations between these tests are:

(i) T_GLRT¼ T_AMF

ð1þ_K¹~z^HQ^¹_SMI~zÞ and T_ACE¼ T_AMF

~ z^HQ^¹_SMI~z

(1.86)

(ii) TAMF¼ w~^H~z²

~

w^HQ^_SMIw~ if we set w~ ¼Q^¹_SMI~a₀ (1.87) Property (ii) shows that the AMF detector represents an estimate of the signal-to- noise ratio and this provides a meaningful physical interpretation. A complete statistical description of these tests has been given in a very compact form in [72,73].

These results are valid as well for planar arrays with irregular sub-arrays and also with a mismatched weighting vector (not pointing in the true target direction).

Actually, all these detectors use the adaptive weight of the SMI algorithm which has unsatisfactory performance as mentioned in Section 1.6.2. The unsa- tisfactory finite sample performance is just the motivation for introducing advanced

weight estimators like LSMI, LMI or CAPS. Clutter, insufficient adaptive sup- pression and surprise interference are the motivation for requiring low sidelobes.

Recently several more complicated adaptive detectors have been introduced with the aim of achieving additional robustness properties, [74–78]. In some cases well know adaptive techniques like diagonal loading have been re-invented by this laborious detour. In particular it has already been shown in [75] that diagonal loading provides significantly better detection performance.

In this context it would be desirable to generalize the tests of (1.83)–(1.85) to arbitrary weight vectors with the aim of inserting the already well-known and robust weights derived in Section 1.6.1. This has been done in [79]. The first observation is that the formulation of the AMF test (1.87) can be used for any weight vectorw. Secondly, one can observe that ACE and GLRT have the form of~ a sidelobe blanking device because the detector T_ACE>h is equivalent to a detectorT_AMF>h~z^HQ^¹_SMI~zand we have a similar relation forTGLRT. These two observations allow the generalization to arbitrary weights. In the following we show that the forms~z^HQ^¹_SMI~zand 1þ_K¹~z^HQ^¹_SMI~zcan be considered as generalized guard channels.

Adaptive guard channel and adaptive sidelobe blanking.A guard channel is implemented in radar systems to eliminate impulsive interference, may be hostile or from other neighbouring radars, using the sidelobe blanking (SLB) device. The guard channel receives data from a separate omni-directional 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. If the received signal power in the guard channel is above the power of the main channel, this must be a signal coming via the sidelobes. Such a signal will be blanked. If the guard channel power is below the main channel power it must result from a signal in the main beam and this is considered as a valid detection. We call the property of the guard channel pattern to be above the sidelobes and below the main beam the SLB condition. The principle of sidelobe blanking has been described by Farina in [7 p. 24.11] and has been studied in detail in [25]. Also, in [7 p. 24.17] a concept is described how for an adaptive antenna in a SLC configuration the SLB device can be realized.

The first observation is that with phased arrays it is not necessary to provide a separate and external omni-directional guard channel. Such a channel can be gen- erated from the antenna itself; all the required information is in the antenna. In fact, one can use the non-coherent sum of the sub-arrays as a guard channelG. This is the same as the average omni-directional power. Some additional shaping of the guard pattern may be achieved by using a weighting for the non-coherent sum:

G¼X^L

i¼1

~

g_ij j~z_i² (1.88)

The directivity pattern of such guard channel is given byS_GðuÞ ¼PL

i¼1~g_ij~a_iðuÞj². If all sub-arrays are equal, a uniform weighting ~g¼ð1; ::;1Þ^T may be suitable.

For unequal irregular sub-arrays as for the generic array (Figure 1.7) the different contributions of the sub-arrays may be compensated. Recall that the notion of sub- arrays is fairly general and includes the case of fully digital arrays, arrays with auxiliary antennas and multi-beam antennas.

More generally, we may use a combination of coherent and non-coherent sums of the sub-arrays. For example, one can also take differences of pairs of sub-arrays to generate a guard channel with a kind of difference beam pattern. Mathematically this can be written with a diagonal matrix D containing the weights for non- coherent integration and by a beamforming matrixKfor the coherent integration.

This results in a generalized guard channel

G¼~z^HKDK^H~z with the directivity pattern SGðuÞ ¼~a^Hð ÞKDKu ^H~a uð Þ (1.89) These two kinds of weightings allow a precise matching of the guard channel to fulfil the SLB condition. Examples of such generalized guard channels are shown in Figure 1.30 for the generic array with 35 dB Taylor element tapering.

Obviously the SLB condition is always fulfilled. The nice feature of these guard channels is that

(i) they automatically scan together with the antenna look direction, and (ii) they can be easily made adaptive.

If we use adaptive beams, we must also use an adaptive guard channel. A CW jammer would make the SLB blank all range cells, i.e. would just switch off the radar. The simultaneous presence of continuous interference with impulsive interference requires an adaptive SLB device. To generate an adaptive guard channel one can replace in (1.89) the data vector of the cell under test (CUT) by the pre-whitened data vector~z_prew¼Q^¹⁼²~z. Then the test statistic can be written as

T¼ T_AMFð~zÞ

Gadaptð~zÞ with Gadaptð~zÞ ¼~z^HQ^¹⁼²KDK^HQ^¹⁼²~z for ACE and Gadaptð~zÞ ¼1þ1

K~z^HQ^¹⁼²KDK^HQ^¹⁼²~z for GLRT

(1.90)

HenceTACEcan be modified to an AMF detector with a generalized adaptive guard channel and similarlyTGLRTwith a generalized adaptive guard channel on a ped- estal. Figure 1.31 shows examples of some generalized adapted guard channels generated with the generic array of Figure 1.7 with35 dB Taylor weighting and a jammer at the first sidelobes of the main beam. The unadapted patterns are shown by dashed lines.

Typically the SLB device is used as a second detection step only in range or angle cells where a detection has occurred, i.e. whereTAMF>h. For the adaptive case this two-step detection procedure with the AMF, ACE and GLR tests has been introduced in [72], called there the 2D adaptive sidelobe blanking (ASB) detector. The generalization of the 2D ASB to arbitrary weights has been intro- duced in [79]. The generalized AMF is testing the presence of a potential target and

(a)

(b)

(c)

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –60

–50 –40 –30 –20 –10

u

dB

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –60

–50 –40 –30 –20 –10 0

u

dB

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –60

–50 –40 –30 –20 –10 0

u

dB

Figure 1.30 Generalized guard channels patterns and sum beam patterns for generic array (from [4]). (a) Uniform sub-array weighting, (b) non- coherent weighting for equal sub-array power and (c) power equalized plus sub-array difference beam weighting

(a)

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

–60 –50 –40 –30 –20 –10

u

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

u (b)

dB

–60 –50 –40 –30 –20 –10 0

dB

27

SumAdapt GuardAdapt SumClear GuardClear Jammer

SumAdapt GuardAdapt SumClear GuardClear Jammer

Figure 1.31 Adapted guard patterns for a jammer with JNR of 34 dB at u¼ 0.27 (15.7). The unadapted patterns are shown by dashed lines.

(a) Weighting for equal sub-array power and (b) difference type guard with weighting for equal sub-array power

the generalized ACE or GRL tests are confirming this target or blank it. By using two different thresholds for the two tests one can control the properties of the overall performance. An analytical statistical analysis of the features of the 2D ASB test with the original AMF, ACE and GLR test of (1.83)–(1.85) has been given in [72,73].

It turns out that for adapted beams and adapted guard channel it is a challenge to fulfil the SLB condition and to define the suitable thresholds for the modified tests. For an arbitrary weight vector it is nearly impossible to determine this analytically. In [79] the detection margin has been introduced as an empirical tool for judging a good balance between the AMF and ASB thresholds for given jammer scenarios. The detection margin is defined as the difference between the expecta- tion of the AMF statistic and the guard channel, where the expectation is taken only over the interference entering the adaptation for a known interference scenario.

This conditional expectation can be calculated in case studies for different scenarios. More precisely, let n~ denote the interference that is present in the training data with a covariance matrix Q. Let~ ~s be a test signal, e.g. a surprise jammer or a target to be detected. Then we calculate:

TAMF¼E_~n w~^Hð~s~s^Hþn~n~^HÞw~

~

w^HQ^w~ ¼w~^H~s~s^Hw~

~

w^HQ^w~ þw~^HQ~w~

~

w^HQ^w~ w~^H~s~s^Hw~

~

w^HQ^w~ þ1 (1.91) Using the approximationQ~ Q. Similarly, we take this expectation for the guard^ channel which results as shown in detail in [79]:

G_ACE¼En_~ tr KDK^HQ^¹⁼²ð~s~s^Hþn~n~^HÞQ^¹⁼²

n o

n o

~s^HQ^¹⁼²KDK^HQ^¹⁼²~sþtr KDK^H GGLR1þ1

K~s^HQ^¹⁼²KDK^HQ^¹⁼²~sþ1

Ktr KDK^H

(1.92)

The detection margin is nowTAMF=h_ACEGACE orTAMF=h_GLRGGLR and this indicates the fulfilment of the SLB condition on the average. If~s is in the main beam it should be detected, if it is in the sidelobe region it should be blanked. In addition, we can calculate the variance ofTAMF,GACE andGGLR. Inspecting the mean levels together with standard deviations allows characterizing the probability of a random threshold excess if the estimated weights are applied and this char- acterizes the SLB condition.

The critical feature that has to be checked is the performance against jammers close to the main beam. An example of the detection margin is shown in Figure 1.32 (same antenna and weighting as in Figure 1.30) for the AMF and ACE guard channel. The scenario for which the antenna (the generic array) and the guard channel are adapted consists of a CW jammer at azimuthu¼ 0.4 with JNR at the elements of 40 dB and an intermittent interference with 50% duty cycle at azimuth u¼ 0.8 with JNR of 35 dB. We have used the LSMI algorithm for adaptation with diagonal load of 3s². All patterns are normalized to the ACE thresholdh_ACE.