Chapter 5

Space-time subspace techniques

The optimum space-time processor as analysed in Chapter 4 suffers from a high degreeof computational complexity so that its use in practical applications, especially forreal-time operation, is unlikely.

Transforming the covariance matrix by the discrete wavelet transform (DWT)may be a way of reducing the computational load for matrix inversion, exploiting thesparsening property of the DWT (BRAUNREITER et al. [52]).

In Chapters 5, 6, 7 and 9 we discuss several ways of reducing the signal vectorspace and, hence, the computational workload associated with space-time adaptiveMTI processing. All of these techniques are based on linear subspace transforms ashas been described already in Section 1.2.3.

The principle of linear transforms to reduce the signal subspace1 has beenaddressed by several authors (KLEMM [240], WARD [530, p. 81], HAIMOVICH etal [194], SELIKTAR et al. [459], BARANOSKI [28]). WANG Y. and PENG [524]present a unified approach for various kinds of transform processors (element-pulse,beamspace-pulse, element-Doppler, beamspace-Doppler). A similar overview of thesetechniques has been given by WARD [531, 530].

B OJANCZYK and MELVIN [46] analyse a least squares STAP technique based onthe preconditioned conjugate gradients iterative method.

HIMED and MELVIN [208] give a brief overview of different subspace processors:factored time-space (frequency-dependent spatial processing), extended factored time-space (frequency dependent spatial processing, with additional auxiliary Dopplerchannels involved), adaptive displaced phase centre (ADCPA), eigencanceller(orthogonal projection), and eigen-based cross-spectral metric (CSM, GOLDSTEIN andREED [160], GUERCI et al. [187]). Some of them are compared using a set of clutterdata measured with the MCARM system, and with an artificial target signal inserted.It is remarkable that processing based on a subspace order Ne — N + M — 1 does notappear to be sufficient to detect the target in clutter.

'Referred to as rank deflation by some authors, e.g., BARANOSKI [28].

Figure 5.1: The auxiliary eigenvector processor (AEP)

5.1 Principle of space-time subspace transforms

In this chapter we consider space-time transforms, i.e., transforms that are effective inboth the time and space dimensions. Such a transform T is a TVM x C matrix, whereC is the dimension of the reduced vector space.2

(5.1)

Notice that each of the spatial subvectors am* has the dimension NxI of the array.The transform has to be in accordance with the criteria presented in Chapter 1 in

section 1.2.3. Following criterion 1, one of the columns of T should be matched tothe expected signal in space and time. We call this the search channel. The associatedspace-time vector a(y?L? ^rad) includes beamformer and Doppler filter coefficients asgiven by (2.31).

technically speaking, C is the number of channels.

clutter subspace transformone space-time search channel

C - 1 eigenvectors

1. row of inverse ofclutter covariance matrix

(for all target Doppler frequencies)

testfunction

Principle of space-time subspace transforms 153

-0.6

Figure 5.2: Comparison of eigenvector and optimum processor (N = M = 24; C —48, FL): o auxiliary eigenvector processor; * optimum processor

The remaining C-I space-time vectors describe auxiliary channels for measuringthe clutter covariance matrix in the reduced vector space. Then the space-timetransform becomes

A ) (5.2)T = ( s((/?L, / D ) R2 • - &L ) = ( S(<£L, j

After transforming the data according to (1.53),

q T = T*q; s T = T*s; x T = T*x; Q T = T*QT (5.3)

the optimum processor in the transformed domain becomes, according to (1.54),

WT = T Q T 1 S T (5.4)

and the improvement factor according to (1.55) becomes

tr(Q)IF =

The transformed signal vector is

• s*s(5.5)

s T = T*s = (5.6)

IF[C

lB]

Figure 5.3: Reduction of the number of channels (FL, AEP): o C = 4 8 ; * C = 4 0 ; xC = 32; + C = 24

Notice that the first row of T* is matched to the signal reference while the auxiliarychannels have to be matched in some way to the interference. Therefore we can write

(5.7)

so that the transformed signal reference becomes approximately

(5.8)

Then the subspace processor becomes

(5.9)

In this processor only the signal component in the search channel s is evaluated whilethe signal contributions of the auxiliary channels are neglected. In (5.4) the signalcontributions of all channels (search and auxiliary) are included. Notice that (5.9) is afurther simplification of (5.4). The processor (5.9) approximates the one in (5.4) wellif the condition (5.7) is satisfied.

The auxiliary channels can be chosen in several ways. In any case they should bematched as well as possible to the clutter so as to produce reference signals with highCNR. In the following two Sections we discuss two techniques which lead basically tonear-optimum clutter rejection performance.

F

IF[C

lB]

Figure 5.4: Influence of clutter bandwidth (FL, AEP): o Bc = 0; * Bc = 0.01; xBc = 0.03; + £ c = 0.1

5.2 The auxiliary eigenvector processor (AEP)

One way of focusing the auxiliary channels a 2 . . . a c on the clutter echoes is touse the clutter eigenvectors as auxiliary channels (KLEMM [241]). This principlewas discussed in Section 1.2.3. It was shown (see also NICKEL, [378]) that thisauxiliary eigenchannel processor is identical to the orthogonal projection processor(OPP) described in Section 4.2.2, if there is a perfect match between the eigenvectorsof the covariance matrix and the channels of the sidelobe canceller transform. Thesidelobe canceller transform matrix is in this case

(5.10)

where E is the TVM x ( C - I ) matrix of clutter eigenvectors of Q. The number ofauxiliary channels C - I must be chosen so that the total clutter subspace is included.For a sidelooking linear array we have to take (3.53) into account and postulate

(5.11)

According to GUERCI et al [187] the inclusion of the signal vector s in thetransform leads to higher compression of the subspace of the cross-spectral metricbased auxiliary eigencanceller technique which is a similar kind of processor asthe AEP. The transform (5.10) belongs to the class of data dependent transforms(PECKHAM^a/. [408]).

A block diagram of the auxiliary eigenvector processor (AEP) based on (5.9)and (5.10) is given in Figure 5.1. The signals received by N antenna elements are

F

IF[C

lB]

Figure 5.5: Influence of system bandwidth (FL, AEP, rectangular frequency response):o B8=O;* B8 = 0.01; x Bs = 0.03; + Bs = 0.1

demodulated and digitised (not shown) and stored in shift registers of length M. TheNM space-time samples are then transformed by the auxiliary eigenvector transformaccording to (5.10).

After the transform the C x C clutter covariance matrix is estimated (not shown).The remaining C channels are multiplied with the first column3 of Q^ 1 for cluttercancellation. This operation has to be carried out for all range increments of the visiblerange.

Notice that the search channel includes both a beamformer and a Doppler filter.That means, beamforming and Doppler filtering is carried out before clutter rejection.Therefore, clutter rejection in the processor in Figure 5.1 has to be carried out for alltarget Doppler frequencies of interest. The Doppler channel with the maximum poweris selected for detection and Doppler estimation.

In the following we analyse the performance of the auxiliary eigenvector processorby calculating the improvement factor (IF) as a function of the normalised targetDoppler frequency F. For the numerical examples we assumed a forward looking lineararray. Similar results can be obtained for a sidelooking array configuration.

5.2.1 Comparison with the optimum adaptive processor (OAP)

The performance of the auxiliary eigenvector processor in comparison with theoptimum fully adaptive processor described in Section 4.2.1 is illustrated in Figure 5.2.

3We assume that the first column denotes the search channel as in (5.10).

F

Figure 5.6: Principle of the auxiliary channel processor (ACP)

The IF has been plotted versus the normalised Doppler frequency. The look directionwas again 45° as stated in the parameter list on page 63. This is reflected in the off-zeroDoppler shift of the clutter notch. The number of channels was chosen to be C = 48which is in accordance with the number of eigenvalues since TV = M = 24.

In Figure 5.2 ideal conditions (identical receive channels, no bandwidth effects)have been assumed. It can be noticed that under these conditions the two curves for theAEP and the OAP coincide perfectly. Using the full subspace of clutter eigenvectors isobviously an optimum choice.

5.2.2 Reduction of the number of channels

Now we try to answer the question of how far the number of channels can be reducedeven further. The motivation for further reduction of the degrees of freedom of theclutter filter (5.9) lies in the fact that usually some of the clutter eigenvalues arerelatively small so that the associated eigenvectors do not play a significant role.

Figure 5.3 shows four curves with different numbers of channels C. It can be seenthat for C = 32 still the optimum IF curve is obtained. For C = 24 some small lossescan be noticed. Obviously, under the conditions assumed, the number of channels canbe divided by 2 which means that the computational expense for calculating the matrixinverse4 is reduced by a factor of 8.

4Matrix inversion needs oc C 3 complex operations.

auxiliary channels

search channels

search channels

auxiliary channels

Figure 5.7: Block diagram of the auxiliary channel processor (ACP)

It should be noted that the required number of channels depends heavily on theactual number of dominant eigenvalues which in turn depends on parameters such astransmit pattern, sensor pattern, system and clutter bandwidth, etc. Let us keep inmind that the auxiliary eigenvector processor tends to be tolerant against increase inthe number of eigenvalues.

5.2.3 Bandwidth effects

In the following two examples we consider again the case C = N + M — 48,which means that the number of auxiliary channels has been matched to the numberof eigenvalues. Let us recall that the number of eigenvalues can be increased bybandwidth effects (see Chapter 3, Figures 3.24-3.31).

The increase in the number of eigenvalues may have two effects on the clutter

testfunction

1. row of inverse of clutter covariance matrix(for each target doppler)

Dopplerfilter

fcauxiliarychannels

Dopplerfilter

f2

Dopplerfilterbank

searchchannel

beamformers

IF[C

lB]

Figure 5.8: Comparison of auxiliary channel and optimum processor (forwardlooking): o auxiliary channel receiver (N — M = 24; Nc — 48); * optimum processor

rejection performance of the space-time processor:

• the clutter notch is broadened;

• losses in IF may occur due to lack of degrees of freedom of the clutter filter.

While the first effect occurs mainly close to the clutter Doppler frequency the secondone may also influence the IF at Doppler frequencies far away from the clutter notch.Experience has shown that a lack in degrees of freedom results in some sidelobe ripplein the IF curves due to Doppler filter sidelobes.

5.2.3.1 Clutter bandwidth

In Figure 5.4 the effect of clutter Doppler bandwidth due to clutter fluctuations (for theclutter fluctuation model see (2.53)) is shown. It can be seen that for Bc = 0 , . . . ,0.03the IF curves run smoothly while for Bc = 0.1 some ripple can be noticed. Obviouslythe number of degrees of freedom C = 48 is sufficient for a relative clutter bandwidthup to Bc = 0.03.

5.2.3.2 System bandwidth

In Figure 5.5, the effect of spatial decorrelation due to the system bandwidth isillustrated for a forward looking array (for sidelooking arrays there is no spatialdecorrelation due to the system bandwidth). A rectangular frequency response

F

IF[C

lB]

Figure 5.9: Reduction of the number of channels (FL, ACP): o C = 48; * C = 40; xC = 32; + C = 24

according to (2.64) was assumed. As can be seen the spatial decorrelation due tothe system bandwidth leads to similar degradation in clutter rejection performanceas the temporal decorrelation due to internal clutter motion. For Bs = 0.1 someslight ripple can be recognised: This indicates that the processor is short of degreesof freedom, which means that the number of channels C has been chosen too small.Some considerations on the impact of the system bandwidth on reduced rank space-time processing have been made by Z ATM AN [573].

5.2.3.3 Related techniques

In the previous discussion we assumed that the space-time transform matrix containsiVg — N -\- M — 1 or fewer eigenvectors associated with the largest eigenvalues of theclutter covariance as auxiliary channels plus an additional search channel. GOLDSTEIN

and REED [160] propose a generalised sidelobe canceller in which those eigenvectorsare selected as auxiliary channels which maximise the cross-spectral metric (CSM)

(5.12)

where Vi are the eigenvectors of the auxiliary subspace, Xi the associated eigenvalues,and rxs is the cross-correlation between search channel and auxiliary channels. Ashas been shown by BERGER and WELSH [37, 38] this criterion does not necessarilyselect those eigenvectors associated with the largest eigenvalues. The effect of limitedsecondary data support for updating the clutter covariance in the CSM algorithm has

F

IF[C

lB]

Figure 5.10: Influence of clutter bandwidth (FL, ACP): o Bc = 0; * Bc = 0.01; xBc = 0.03; + £ c - 0 . 1

been analysed by HALE [197]. The performance of the CSM and related techniqueshas been analysed by simulations and MOUNTAINTOP data (Tm [494]) by GUERCI

etal. [187].Starting from the above-mentioned CSM technique, BERGER and WELSH [37]

propose a SINR (signal-to-interference + noise ratio) metric

(5.13)

which allows the selection of eigenvectors for a space-time transform based onmaximisation of the SINR (f are eigenvectors to be selected). It is shown that thistechniques provides graceful degradation when the number of channels is reducedbelow the number of eigenvalues of the clutter covariance matrix. In this respect theSINR method is superior to the CSM method. It should be noted that the straight-forward method analysed above (one search channel, eigenvectors belonging to thelargest eigenvalues) is quite tolerant against reduction of the number of channels (seeFigure 5.3).

5.3 Auxiliary channel processor (ACP)

The auxiliary eigenvector processor described above has the disadvantage that theeigenvectors of the covariance matrix are not known but have to be calculated from

F

IF[C

lB]

Figure 5.11: Influence of system bandwidth (FL, ACP): rectangular frequencyresponse): o Bs = 0; * Bs = 0.01; x £ s = 0.03; + Bs = 0.1

an estimate of the clutter covariance matrix, in other words, the transform is datadependent.

A more intuitive concept for a space-time auxiliary channel processor whichcircumvents these problems by being data independent (PECKHAM et al. [408]) hasbeen proposed by KLEMM [240] and is described in the following.

Let us consider first the one-dimensional problem of jammer cancellation by spatialnulling. Suppose a number of jamming sources are radiating on a sensor array andassume that the jammer positions are known. Under these conditions a subspaceprocessor can be designed by steering one beam on each of the jammers and havingone more beam for target search. It has been shown numerically that such a processoris practically optimal (KLEMM [228]). NICKEL [378] has shown analytically thatthis kind of sidelobe canceller is identical to the orthogonal projection processor (seeSection 4.2.2). The problem with this concept is that the positions of the jammers arenormally unknown.5 For some more details on multibeam configurations see Section1.2.3.2.

In this respect airborne clutter suppression is easier because the positions of clutterin space (normally homogeneous) as well as the direction dependant clutter Dopplerfrequencies are known. We can, therefore, design a space-time processor according to(5.9) with a transform matrix of the form

(5.14)

F

5In fact, it is the task of an adaptive processor to cope with unknown jammer positions.

The column S((^L, / D ) denotes again the space-time search channel while a2 . . . s.care auxiliary space-time channels matched to the clutter in direction and Dopplerfrequency.

The principle of the auxiliary channel processor6 is illustrated in Figure 5.6. Weconsider the case of a sidelooking array configuration as in Figure 3.40. One recognisesfirst of all the clutter power spectrum extending along the diagonal of the plot.

The ellipses in the /D-COS (p plane denote footprints of space-time receive channels.There are auxiliary channels (dotted ellipses) covering the whole clutter azimuthrange, each of them matched to the Doppler frequency associated with the individualdirection.

In the look direction we have a number of search channels for all possible targetDoppler frequencies (solid ellipses). Notice that there must be an auxiliary channelin the look direction in order to receive the transmit main beam clutter. No searchchannel should be matched exactly to the clutter Doppler frequency (centre of plot)because then the transform matrix would become singular.7

Figure 5.7 shows a block diagram of the auxiliary channel processor. The outputsignals of the N antenna channels are transformed by a multiple beamformer network.Each of the auxiliary beams points into a different direction so as to cover the wholeazimuth range. For instance, this multiple beam network has been implemented in theOLPI radar (WiRTH [558, 556]) and is used frequently in sonar systems.

Each of the auxiliary beams is followed by a Doppler filter matched to the Dopplerfrequency associated with the look direction of the individual beam (dotted ellipses inFigure 5.6). The search beam is cascaded with Doppler filter bank (solid ellipses). Theoutput signals are multiplied with the first column of the inverse covariance matrix andfed into a detection device.

5.3.1 Comparison with optimum processor

First the performance of the auxiliary channel processor is compared with the optimumprocessor treated in Chapter 4. Figure 5.8 shows that the IF curves of both processorscoincide perfectly, as in case of the auxiliary eigenvector processor, see Figure 5.2.

5.3.2 Reduction of the number of channels

A substantial difference between the auxiliary clutter channel processor and theauxiliary eigenvector processor (AEP) described in Section 5.2 is that the AEP hasimportant and less important channels, depending on the magnitude of the associatedeigenvalue. In the auxiliary channel processor all channels have the same priority sothat reducing the number of channels is more critical.

Figure 5.9 shows the effect of reducing the number of auxiliary channels on theperformance of the auxiliary channel processor. Comparing Figure 5.9 with Figure5.3, it is obvious that the ACP is much more sensitive to a reduction of channels. We

6In KLEMM [240] this processor was referred to as an auxiliary channel receiver (ACR).7This problem can be avoided by adding 'artificial noise' to the covariance matrix. If no transform is

applied one adds Q^L) = Q + /xI (diagonal loading). This is a well-known method to improve the conditionof the covariance matrix. In the transformed domain one has to add instead Q^ = Q T + /^T* T.

notice strong ripple even for a slight reduction by eight channels (asterisks). As statedin the previous paragraph this is an expected result.

It can be noticed that on the one hand there is a degradation close to the clutterDoppler frequency (clutter notch). On the other hand, significant losses show up inthe pass band. The fact that the sidelobe structure of the Doppler filter bank becomesapparent indicates a lack of degrees of freedom (or channels).

5.3.3 Bandwidth effects

In the following examples we assumed again that the number of auxiliary channelsis matched to the number of eigenvalues of the ideal clutter covariance matrix: C =N -f M — 48. As in Section 5.2.3 we want to find out what the effect of additionaleigenvalues of Q caused by bandwidth effects are.

5.3.3.1 Clutter bandwidth

Again we use the clutter fluctuation model given by (2.53). The IF curves areshown in Figure 5.10. Comparing the curve Bc = 0 with the others shows that theauxiliary channel processor is short of degrees of freedom because the actual number ofeigenvalues of Q has been increased through the clutter fluctuation model. The clutternotch is broadened which results in degraded detection capability of slow targets.Even in the pass band some significant IF losses occur for larger clutter bandwidth.Comparing this plot with Figure 5.4 shows that the auxiliary eigenvector processor ismuch more tolerant against an increase in the number of clutter eigenvalues of Q thanthe ACP.

5.3.3.2 System bandwidth

Similar effects are caused by the system bandwidth. Like in Figure 5.5, a rectangularfrequency response according to (2.64) was assumed. Again a forward looking arraygeometry was assumed. Figure 5.11 shows some numerical results. Basically the effectof system bandwidth (spatial decorrelation) on the clutter suppression performance issimilar to the influence of the clutter bandwidth (temporal decorrelation).

5.4 Other space-time transforms

In the preceding sections two techniques (AEP and ACP) have been described whichpromise to approximate the clutter rejection performance optimally because theirauxiliary channels are well matched to the clutter. It was found that under idealconditions (no bandwidth decorrelation effects) the improvement factor achieved bythe optimum processor (OAP, see Section 4.2.1) was reached perfectly by both of thespace-time auxiliary channel processors.

Instead of using auxiliary eigenvectors (AEP) or auxiliary beams and Dopplerfilters (ACP) one might think of a variety other space-time auxiliary channelconfigurations.

5.4.1 Single auxiliary elements and echo samples transform

One possiblity is to choose single array elements at single instants of time as auxiliarychannels. The advantage over the previously described techniques is that no auxiliarybeams and Doppler filters have to be formed. This concept is a space-time analogue ofthe usual sidelobe canceller commonly used for jammer suppression.

There are disadvantages, however. As stated in Chapter 1, Section 1.6, the clutterrejection performance of the sidelobe canceller is degraded whenever the interference-to-noise ratio in the auxiliary channels is smaller than that in the search channel. Thiscan happen if the sidelobe level of the search beam is higher than the gain of theauxiliary element. This effect was shown in the numerical example Figure 1.6.

5.4.2 Space-time sample subgroups

The idea of this technique is to subdivide the total of spatial8 and temporal9 samplesinto space-time sample subgroups (KLEMM, [242]). The samples of each space-timesubgroups are combined by a primary beamformer and a primary Doppler filter sothat all sample subgroups are focused in the same direction and on the same Dopplerfrequency. Then the space-time covariance matrix is estimated and inverted at thesubgroup level.

In a second stage a secondary beamformer can be steered inside the primary beampattern, and a secondary Doppler filter bank can be designed to perform Doppleranalysis in the limits of the primary Doppler filter main lobe. The shapes of the primarybeamformer and the primary Doppler filter determine the required number of primary<£-/D positions.

There is a variety of possibilities of designing the space-time subgroups. They maybe different in the spatial and temporal dimensions, they may overlap or not, adjacent ordisplaced samples can be chosen. The appropriate design of such processors is a wideplayground for system designers. It should be noted that not all transforms performequally well. We confine our considerations to the two examples treated above (AEP,ACP) whose performance is near optimum.

5.4.3 Space-time blocking matrices

All adaptive processors are based on the assumption that the clutter covariance matrixdoes not contain any portion of the desired signal. That means that the covariancematrix should be estimated from signal-free data. This can be a problem, especially ifthe amount of data available is small. In some applications (SAR, passive sonar) thesignal (if there is any) is continuously present.

Inclusion of the desired signal in the adaption leads to severe signal suppressionbecause the processor treats the signal as interference. No degradation in signal powermay be obtained by the optimum processor (Chapter 4) if the processor (i.e., thesteering vector) is perfectly matched to the signal (Cox [88]). In a radar search mode

8 Array elements.9Echoes.

the angular (Doppler) cells are determined by the beamwidth (Doppler filter) so that amismatch between steering vector and signal happens intentionally.

A way to mitigate the effect of signal inclusion in sidelobe canceller types ofprocessors (such as the AEP, ACP) is to apply a blocking matrix so that the transformed('blocked') auxiliary channels

(5.15)

become insensitive to the signal. B is called a space-time blocking matrix. It can becomposed for example of some columns of a space-time projection matrix orthogonalto the signal vector

(5.16)

(5.17)

Notice that the column rank of A is reduced by 1. Therefore, the number of auxiliarychannels has to be reduced at least by 1. Details on the use of blocking matricescan be found in the papers by SCOTT and MULGREW [455], Su and ZHOU [473],GOLDSTEIN and REED [161], and GOLDSTEIN etal [162].

5.5 Aspects of implementation

5.5.1 General properties

5.5.1.1 Number of operations

Space-time auxiliary channel processors can reach the performance of the optimumreceiver (see Chapter 4) at greatly reduced computational expense. A comparison ofall processors treated in this book in terms of computational complexity will be givenin Chapter 15.

The number of space-time auxiliary channels depends on the number of cluttereigenvalues which is approximately N + M — 1. Therefore, the complexity of space-time auxiliary channel receivers increases with increasing number of antenna elementsand the length of the pulse burst. Such solutions are useful for small numbers ofTV and M. In practice inhomogeneity of the clutter background (dominating clutterdiscretes) may lead to a reduced number of clutter eigenvalues and, hence, to lesscomplex receiver structures.

5.5.1.2 System bandwidth

The system bandwidth effect does not occur for linear or planar sidelooking arrays, seethe comments at the end of Chapter 3, page 112.

For other than linear or planar sidelooking arrays the system bandwidth effectscan be compensated for by means of space-time-TIME or space-time-FREQUENCYprocessing (see the remark on page 114).

so that

5.5.1.3 Calculation of the matrix inverse

As mentioned earlier the search channel in the transform T includes a Doppler filter.That means that the clutter filter (first column of Q^1) has to be calculated for allpossible target Doppler frequency. The inverse covariance matrix can be calculatedefficiently in the following way:

The transformed covariance matrix assumes the form

(5.18)

where m is the Doppler frequency index, P(m) = s(m) * Qs(ra) is the clutter power inthe search channel, h(ra) = A*Qs(ra) is the cross-variance between search channeland auxiliary channels (A is the matrix of auxiliary channels, see (5.2)), and D =A* QA is the covariance matrix of auxiliary channels. Notice that D does not dependon the Doppler frequency.

Then the inverse becomes

(5.19)

where C = D - -p^yhh*(m), and o and 0 denote the zero vector and matrix,

respectively. If D " 1 is known C""1 is obtained as follows:

(5.20)

The following steps have to be carried out:

1. Estimate D,P(ra) ,h(m) form = 1,. . . ,M

2. Compute D " 1

3. Compute C " 1 (m) for m = 1 , . . . ,M (5.20)

4. Compute Q^ 1 form = 1 , . . . , M (5.19).

The computational expense for inverting the matrix is about

5.5.2 Auxiliary eigenvector processor

5.5.2.1 Generation of auxiliary channels

The eigendecomposition of the clutter covariance matrix requires more computationsthan taking the inverse of the matrix. In this view the auxiliary eigenvector processorhas only theoretical value. However, this problem might be circumvented by selecting

PQ = s*Qs is the clutter power in the search channel, pi = s*e^Ai are the cross-variances between search and auxiliary channels, e^ and A are clutter eigenvectorsand eigenvalues of Q, respectively. The special form of (5.21) may offer somesimplification in calculating the matrix inverse. However, if pre-calculated transformmatrices are used there will always be some mismatch between the transform and theactual covariance matrix so that the lower right (C — 1) x (C — 1) submatrix is nolonger diagonal.

5.5.3 Auxiliary channel processor

5.5.3.1 Generation of auxiliary channels

Compared with the auxiliary eigenvector prosessor the auxiliary channel processor hasthe advantage that the pre-transform consisting of clutter beams and Doppler filters areknown a priori while the eigenvectors of the AEP have to be calculated beforehand.

5.5.3.2 Number of channels

It was shown that the number of channels of the ACP must not be reduced belowN + M. On the other hand it might be of interest to increase the number of channelsbeyond N H- M in order to cope with decorrelation effects caused by system or clutterbandwidth, see Section 5.3.3, or by channel errors, see Chapter 15. It should be notedthat for DPCA conditions the pre-transform T cannot have more than C = N + M —1 auxiliary space-time vectors. This follows from the fact that a DPCA covariancematrix without corruption by stochastic effects has only C = N + M — 1 eigenvalues.Therefore, more than C — N + M — 1 space-time clutter matched vectors becomelinearly dependent so that Q becomes singular. It is not possible to match the numberof channels to the actual number of eigenvalues of Q. Diagonal loading of Q T with'artificial noise' (see footnote on p. 163) might be a way out of this dilemma.

5.5.3.3 Related concept

WANG Y. and PENG [521, 523] propose a similar processor, however with taperedDoppler filter weights so as to reduce the filter sidelobes. This results in a reducednumber of degrees of freedom of the space-time clutter covariance matrix. Also LiuQ.-G. et al. [319] found that the auxiliary channel processor is quite sensitive to array

I0For example, for homogeneous clutter and, if necessary, for various flight velocities.

the appropriate transform from a set of 'typical'10 pre-calculated eigenvector matricesaccording to the actual flight conditions (KLEMM [241]).

For the actual eigenvalues of Q the transformed clutter covariance matrix assumesthe form

(5.21)

channel errors. They propose a modified version by applying temporal and spatialtaper weights11 to the data before estimating the space-time clutter covariance matrix.Such weightings reduce the sidelobe level and, hence, the number of significant cluttereigenvalues. This results in less sensitivity to channel errors.

5.6 Summary

1. Space-time transform processors basically offer a possibility of cluttersuppression in a reduced signal vector space. They are based on a lineartransform whose columns are space-time vectors. One of these vectors is a signalmatched search channel while the others serve as clutter reference channels.

2. The number of channels (dimension of the subspace) has to be about N +M. Therefore, these techniques have the disadvantage that the computationalworkload increases with increasing dimension of the signal vector space, that is,number of sensors and the number of coherent echoes M.

3. The auxiliary eigenvector transform achieves near-optimum clutter rejectionperformance without degradation in slow target detection. In detail thistechnique is characterised by the following properties:

• The transform requires the eigendecomposition of the clutter covariancematrix. This needs more arithmetic operations than the inversion involvedin the optimum processor.

• The transform may be pre-calculated off-line. Since the transform isfollowed by an adaptive processor in the clutter subspace some mismatchbetween the transform and the actual clutter subspace may be tolerated.

• This processor technique is relatively robust against bandwidth effects.

• The AEP requires a fully digitised array, at least in the horizontaldimension.

4. The CSM technique by GOLDSTEIN and REED [160] and the SINR metrictechnique by BERGER and WELSH [37] belong to the class of eigenvectortransform techniques.

5. The auxiliary channel processor (ACP) transform uses a bunch of parallelbeamformers which cover the entire angular domain. Each of the beamformers iscascaded with a receiver chain and a Doppler filter which is matched to the clutterDoppler frequency of the individual beam. This processor has the followingproperties:

• The total number of channels must not exceed N + M, otherwise theassociated space-time vectors become linearly dependent which results ina singular transformed covariance matrix.

1 ] 40 dB Dolph-Chebychev.

• The ACP is sensitive to bandwidth effects and to channel errors. Thenumber of degrees of freedom cannot increased beyond N + M, see above.

• The problem associated with the linear dependence of channel might becircumvented by adding 'artificial' noise in the transformed domain (seethe footnote on p. 163).

• The multibeam auxiliary channels can be implemented in the RF domain.However, each beam has to be followed by a receiver chain. Therefore, thenumber of required digital array channels is about the same as for the AEP.

6. A variety of other transforms may be used, for instance forming space-timesample subgroups. It is up to the fantasy of the designer to create different kindsof sample subgroups. It should be noted, however, that not all transforms showsatisfactory performance.

A comparison of all techniques in terms of computational complexity is presented inChapter 15.