adaptive space-time digital filters
DESCRIPTION
Adaptive space-time digitalfilters mti radarTRANSCRIPT
Chapter 7
Adaptive space-time digitalfilters
In Chapter 6 some transforms to reduce the signal vector space in the spatial dimensionhave been analysed. In this chapter we investigate the use of digital filters for space-time clutter suppression. In this way we try to reduce the signal vector space in thetemporal dimension. Temporal (i.e., pulse-to-pulse) digital filters have been used forclutter rejection in ground-based radar systems. For example, the usual two-pulse orthree-pulse clutter cancellers are typical temporal anti-clutter pre-filters.
An adaptive clutter filter based on the prediction error filter has been described byB UHRING and KLEMM [64]. It has been applied to a surveillance radar with rotatingantenna.
BARBAROSSA and PICARDI [30] discuss the use of predictive techniques foradaptive clutter rejection. The superposition of two types of clutter is addressed. TheLUD algorithm suggested by MAO et al. [336] allows one to compute the coefficientsof the prediction error filter more efficiently than by using the BuRG algorithm [67,68],GOLDSTEIN et al [164, 163, 165, 166] propose a multistage STAP Wiener filter withCFAR capability. Simulations (GOLDSTEIN et al. [166], GUERCI et al [187]) andexperiments with MCARM data (GUERCI et al. [185]) have demonstrated outstandingclutter rejection performance at high computational efficiency. HiMED and MiCHELS[210]achieved good suppression of hot1 and cold clutter by using 3D STAP (space-time-TIME) based on a multistage Wiener filter. JiANG et al. [222] and Li et al [306]analyse the 4VAR filter' (a least squares FIR filter) in comparison with DPCA withrespect to clutter rejection performance and target parameter estimation. It is shownthat the VAR filter is superior to the DPCA technique because it is adaptive.
The concept of space-time pre-whitening for suppression of airborne clutter hasbeen illustrated by WANG Z. and BAO [527]. GRIFFITHS [173] used a pre-DopplerSTAP architecture with MOUNTAINTOP data. The parametric clutter rejectiontechnique by SWINDLEHURST and PARKER [481] is strongly related to the leastsquares FIR filter described below.
terrain scattered jamming.
There are close relations between the optimum processor and the space-time FIRfilter. LOMBARDO [322] and in more detail in [323] has shown that for specialconditions:
• sidelooking linear equispaced array
• DPCA conditions
• exponential temporal clutter correlation2 pt(i) = pj.
• exponential spatial clutter correlation3 ps(i) = ps
the inverse of the space-time covariance matrix becomes a tridiagonal symmetricmatrix whose block rows describe a space-time FIR filter with temporal and spatialfilter length L = 3, K — 3. Under the given conditions this means a two-dimensionalspace-time FIR filter which operates in the time (pulse-to-pulse) domain as well as thespatial domain (across the array).4 It can be expected that for different clutter modelsadaptive FIR filters approximate well the optimum processor (see WANG Z. and BAO
[528]).From experience with ground-based radar several aspects of digital filters are
known:
1. In phased array radar the beam steering direction is kept constant during thedetection phase. If the PRF is constant clutter echoes can be considered to bestationary because no effect due to inhomogeneity of the clutter background northrough irregular sampling will occur. Digital filters applied to stationary signalshave constant coefficients independent of time.
2. In the case of staggered PRF the signal sequence is no longer stationary. Thenthe filter has to be trained anew with every PRI. Training has to be done on thebasis of data coming from range. As a result one obtains a FIR filter whosecoefficients vary during filtering.
3. The number of filter coefficients may be chosen independently of the Dopplerfilter length. This is of high importance for radar with changing modes ofoperation (low, medium, and high PRF).
4. The number of coefficients of a digital clutter filter may be small compared withthe Doppler filter length. This property is different from all processors treatedin the previous chapters. It may cause a dramatic reduction of the instantaneouscomputational load per pulse repetition interval during filtering the data throughall range increments. Also the amount of data required for updating the filtercoefficients will be reduced.
5. The filter can operate properly only if entirely filled with data. Therefore, itmust not slide over the edges of the data record. Consequently, the resulting data
2In contrast to the gaussian correlation assumed in our model (2.53).3Which differs considerably from our models (2.61) and (2.64).4Such a filter is depicted in Figure 7.17.
Figure 7.1: Space-time FIR filter processor
length after filtering is M — L + 1 where L is the filter length. That means,the length of the Doppler filters is shortened by L — 1. This shortening results insome signal loss. The trade-off between clutter filter length and signal integrationlength has been discussed by DiLLARD [95]. It is therefore essential that theclutter filter length is small compared with the total length of the data record.
6. If the clutter echoes are stationary the covariance matrix is Toeplitz and theprediction error filter can be calculated via the LEVINS ON algorithm (BURG
[67, 68], BUHRiNG and KLEMM [64]) from one row of the covariance matrix.That means for the adaptation that instead of a covariance matrix only a vectorof covariance coefficients has to be estimated.
7.1 Least squares FIR filters
The principle of least squares FIR filtering of stationary signals and its use for cluttercancellation was briefly addressed in Section 1.2.3.2, and in [64]. In this chapter wewant to extend this principle to space-time processing i.e., we have to add the spatialdimension.
7.1.1 Principle of space-time least squares FIR filters
In the following a multichannel space-time least squares FIR filter for adaptivesuppression of airborne clutter is derived. It is based on the principle of predictionerror filtering and is, therefore, closely connected with the maximum entropy method
shift register(space-timeecho data)
space-time FIR filter(weighted sum)
Doppler filter bank(FFT)
testfunction
Figure 7.2: Detailed block diagram of the space-time FIR filter
(BURG [67, 68]). The parametric adaptive matched filter (PAMF) by ROMAN et al[441] also belongs to this category of filters.
7.1.1.1 Some remarks on prediction error filters
It was shown in (1.91) that the prediction error filter is defined by
It can easily be verified that the matrix on the left side is the clutter covariance matrixof order L + 1 so that the prediction error filter becomes
i.e., the prediction error filter is proportional to the first column of the inverse of theclutter + noise covariance matrix.
(7.1)
(7.2)
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Figure 7.3: The FIR filter processor (full array, FL): o FIR filter processor (L = 5); *fully adaptive optimum processor
It should be mentioned that in this case a future value of the data sequence ispredicted, see (1.87). The prediction filter as defined by (1.89) can also be used toestimate a value inside the data segment, that means prediction of a value X{ withi = t + L — 1, . . . ,£. Then the i-th column of Q " 1 will play the role of a predictionerror filter. This case is called interpolation rather than prediction. If a value in the past(xt-L) is to be predicted the last column of Q " 1 represents the prediction error filter.This application is called smoothing.
We can conclude that each column of Q " 1 represents a least squares filter, thatmeans, a filter that minimises the output power of a stationary process defined by Q.Therefore basically each of the columns of Q - 1 can be used as a clutter rejection filter.In the following we use the first column of Q - 1 while keeping in mind that the otherswould work in a similar way. Since the outer columns are based on more correlationvalues than the inner columns the width of the clutter notch will be narrower. This isin accordance with the MEM resolution achievable with outer or inner columns of theToeplitz matrix inverse, respectively.
7.1.1.2 Extension to space-time radar data
The extension of the prediction error filter to space-time filtering is straightforward.The space-time clutter + noise covariance matrix of a data segment of length L has the
F
In analogy with the one-dimensional prediction error filters described by (1.91), aspace-time FIR filter operating in the temporal dimension only is a matrix given by
Since we assumed that the clutter echoes are stationary in the temporal dimension thecovariance matrix is block Toeplitz. Recall that the submatrices Qmp have dimensionNxN, i.e., the spatial dimension of the array.
The inverse of Q has the same form:
form (compare with (3.22))
(7.3)
(7.4)
Figure 7.4: Overlapping subarray processor with space-time FIR filter
shift register(space-timeecho data)
Doppler filter bank(FFT)
testfunction
space-time FIR filter(weighted sum)
The filtered output data matrix Y has the spatial dimension K x (M — L + 1), i.e.,the spatial dimension which is either given by the array size K = iV or by any of thepre-transforms described in Chapter 6.
(7.7)
where the x(r) are vectors containing shifted segments of the data sequence. Includingall vectors x(r) in a matrix X leads to
The filtering operation can be formulated as follows
(7.5)
(7.6)
Figure 7.5: Symmetric auxiliary sensor processor with space-time FIR filter (ASFF)
the first block column of K:
testfunction
Doppler filter bank(FFT)
I space-time FIR filter(weighted sum)
auxiliarychannels
mainchannel
auxiliarychannels
beamformer
shift register(space-timeecho data)
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Figure 7.6: Space-time FIR filtering and spatial transforms (FL, L = 5):o optimum processor; * symmetric auxiliary sensors K = 5; x disjoint subgroupsK = 6; + overlapping subarrays K — 5
Equation (7.7) describes the space-time data K moving over the filter matrix. Anequivalent version of this equation can be obtained by moving the filter matrix relativeto the data:
F
(7.8)
where
(7.9)The submatrices KJ^1 have been defined in (7.4). The filter operator H* has dimensionN(M - L + 1) x NM and the dimension of the vector of output signals y is TV(M -L + l) .
IF[C
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Figure 7.7: Space-time FIR filtering and spatial transforms (SL, L = 5):o optimum processor; * symmetric auxiliary sensors K — 5; x disjoint subgroupsK — 6; + overlapping subarrays K — 5
7.1.1.3 Pre-beamforming
The filter operation (7.7) or (7.8) has to be followed by signal match in both the spatialand Doppler dimension. By spatial matching the spatial dimension is removed. This isdone by multiplying the filtered data matrix with a beamformer vector
(7.10)
The same result is obtained by pre-multiplying the filter matrix with the beamformer
(7.11)
so that the space-time FIR filter is just a vector with dimension KL. This filter wasfirst proposed by KLEMM and ENDER [272, 274, 275] and later on by BARANOSKI
[28]. A mathematical derivation of this processor has been given by JAFFER [218].The numerical results presented in [218] agree well with those given below.
The filtering operation is given by
(7.12)
F
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Figure 7.8: Effect of spatial filter length (ASFF, FL, L = 5): o K = 3; * K = 5;x K = 7; +K = 9
The vector z contains the output signal sequence which has length M - L + 1.Usually the clutter filter is followed by a Doppler filter bank which has to be applied tothe shortened data sequence
(7.13)
where F is a matrix whose rows are Doppler filter vectors according to (2.33). Inpractice the Doppler filter bank is usually a DFT or FFT. The elements of f are fed intoa detection device.
7.1.1.4 Improvement factor
The efficiency of the filter technique described above can be judged by calculatingthe improvement factor as before. An equivalent way of formulating the FIR filteroperation is as follows
(7.14)
F
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where we assumed once again that the processor is matched to the expected targetsignal vector S(O;D)- Notice that the filter matrix H always appears twice, once forwhitening the received signal, and once for equalising the target signal reference (ormatched filter). This is in accordance with (1.71) and Figure 1.7b.
7.1.1.5 Filter length and eigenvalues
From the considerations in Chapter 3 we know that under ideal conditions (PRF= Nyquist frequency for clutter bandwidth, no additional bandwidth effects, clutter
Notice that the sub vectors h m of h have the spatial dimension TV, with N being thenumber of antenna elements. The matrix H contains replicas of the filter vector h thatare shifted by N between any two rows of H.
Then the improvement factor becomes in accordance with (1.93),
(7.16)
(7.15)
where
Figure 7.9: Effect of spatial filter length (ASFF, SL, L = 5): o K = 3; * K = 5;x K = 7; +K = 9
F
IF[C
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Figure 7.10: Effect of temporal filter length (ASFF, K = 5, FL): o L = 2 ;*L = 4 ;xL = 6;+ L = 8
echoes received from one semiplane only) the number of eigenvalues of Q isapproximately equal to iV + M — 1, see Figures 3.12 and 3.13 and equation (3.57).
The length of the FIR filter L has not yet been determined. It would be desirable tochoose L <C M in order to save arithmetic operations. This is of particular importancebecause the filtering operation has to be conducted in real-time for all range increments.How does a short filter length L cope with the number of eigenvalues of Q determinedby Ml Does the reduction M down to L cause lack of degrees of freedom? There aretwo ways to look at this problem:
• The FIR filter is calculated from a NL x NL space-time covariance matrixaccording to (7.11), where L is the length of a TV-channel data segment of thereceived echo data sequence of length M. While sliding over the data sequencethe filter operates at a certain instant of time only on an instantaneous datasegment of length L. Opposite to all processing schemes described in Chapters4-6 the FIR filter does not have to cope simultaneously with the complete datasequence of length M. Therefore, there is no lack of degrees of freedom whichotherwise would cause serious degradation of the clutter rejection performance.
• The full filter operation on the data sequence of length M has been describedin (7.14) by a matrix H. The column rank of H is TV(M — L + 1). SinceN(M - L + 1) is larger than the number of eigenvalues N + M — 1 if L issufficiently small no lack of degrees of freedom has to be taken into account.
F
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Figure 7.11: Effect of Doppler filter length (symmetric auxiliary sensors, FL, K =L = 5): o M = 12; * M = 24; x M = 48; + M = 96
7.1,2 Full antenna array
Let us start with the performance of a space-time FIR filter processor that uses alloutput signals of an array antenna. A block diagram is depicted in Figure 7.1. The Noutput signals of the antenna channels are stored in a N x M memory. The FIR filter isillustrated by a dashed rectangle. Each of the data inside this rectangle is weighted andsummed up according to (7.12). The filter moves through the memory as indicated bythe arrow. If the shift inherent in the filter operation is synchronised with the PRF, i.e.,the data flow is pipelined, then a shift register of size NxL may replace the N x Mmemory.5 A more detailed view of the space-time FIR filter is given in Figure 7.2.
The numerical example in Figure 7.3 shows a comparison of the FIR filter processorwith the optimum processor. As before the plot shows the normalised improvementfactor in the signal-to-clutter + noise ratio versus the normalised target Dopplerfrequency. The IF is normalised by its theoretical maximum while the target Dopplerfrequency is normalised by half the PRF.
It can be noticed that the IF achieved by the FIR filter processor runs closely to theoptimum curve. The difference between both curves of about 1-2 dB is caused by theloss in target signal energy due to shortening of the data sequence by the FIR filter.Recall that the final data length after FIR filtering is M — L -f-1 instead of M becausethe FIR filter should work only inside the data record. The correct design of the filterslength is a compromise between losses in signal power due to reduced integration timeand losses due to imperfect clutter cancellation. If the data length M is large compared
5Of course, this operation has to be carried out for all range increments.
F
IF[Gl
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Figure7.12: STAP FIR filter, effect of internal clutter motion (FL, <pL = 0°): o Bc = 0;* Bc = 0.01; x Bc = 0.1; + Bc = 0.3
with the temporal filter length L the FIR filter processor approximates the optimumsufficiently.
PARK et al. [405,406] optimise a linear predictor under the constraint that the targetDoppler frequency is unknown. Hence, the Doppler part of the signal covariance matrixis a unit matrix. The clutter cancellation performance achieved shows considerabledegradation compared with the optimum processor.
7.1.3 Spatial transforms and FIR filtering
Now we combine the spatial transforms treated in Chapter 6 with the concept of aspace-time FIR filter. Notice that thereby two different kinds of simplification of theprocessor scheme (spatial and temporal) are achieved. The mathematical descriptionof such pre-transform/FIR filter processors is in essence the same as above. All vectorsand matrices have to be replaced by quantities transformed with a spatial transformaccording to (6.1), and the number of antenna elements N has to be replaced by thereduced number of channels K.
7.1.3.1 Uniform overlapping subarrays
Figure 7.4 shows a block diagram for a processor in which the overlapping subarrayconfiguration has been combined with a FIR filter processor. The subarray transformwas given by (6.1), with the spatial submatrices being defined by (6.2).
F
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Figure 7.13: STAP FIR filter, filter length and clutter fluctuations (FL, <pL = 0°): oL = 2 ;*L = 3 ; x L = 5; + L = 7
7.1.3.2 Symmetric auxiliary sensors
The symmetric auxiliary sensor configuration has been described by (6.1), where thespatial submatrices have the form (6.17). A block diagram of an auxiliary sensor FIRfilter processor (ASFF) is given in Figure 7.5.
This processor has two beneficial properties. On the one hand the spatial transform(beamforming of centre elements) can be realised in the RF-domain as was noted inthe previous chapter. On the other hand, the product
(7.11), which incorporates the beamformer into the space-time FIR filter, can beomitted for the auxiliary sensor processor and can be replaced by just selecting thecolumn of K which belongs to the beamformer. The simplified processor becomes
(7.17)
where e(K+i)/2 is the centre column of the KxK unit matrix.
7.1.3.3 Disjoint subarrays
The matrix formulation for a processor with disjoint subarrays was again given by(6.1), but with submatrices of the form given by (6.10). The corresponding FIR filterprocessor is similar to the one shown in Figure 7.4 however with the antenna arraysubdivided into non-overlapping subarrays.
F
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Figure 7.14: STAP FIR filter, effect of range walk (FL, </?L — 0°): ° n o range walk; *AR = 100 m; x AR = 10 m; + AR = 1 m
7.1.3.4 Comparison of processors
The performance of the three processors is compared with the optimum processor(Chapter 4) in Figure 7.6, for a linear array in the forward looking configuration, andin Figure 7.7, for a sidelooking array. The filter length (data segment) was chosen to beL = 5 while the total length of the data record is M = 24. The length of the resultingdata sequence is then M — L + 1 = 20.
The following observations can be made:
• The auxiliary sensor/FIR filter and the overlapping subarray/FIR filterapproximate the optimum IF very well.
• The difference in IF between the optimum and the two suboptimum curves isdue to the shortening of the data record caused by the FIR filter of length L(M - L + 1 = 20). Therefore, the target signal energy is reduced by a factor of20/24.
• The disjoint subarray/FIR filter processor shows some losses at those frequencieswhich are associated with the maximum of the sensor directivity pattern (F = 0.5for the forward looking array, F = O for the sidelooking array). These lossesare due to spatial ambiguities caused by the displacement of the subarray phasecentres. This effect was already observed in Figures 6.3 and 6.4.
F
IF[C
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F i g u r e 7 . 1 5 : S T A P F I R filter, filter l e n g t h a n d r a n g e w a l k ( I m r a n g e r e s o l u t i o n , F L ,<pL = 0 ° ) : o L = 2 ; * L = 3 ; x L = 5 ; + L = 7
Table 7.1 Comparison of OAP and ASFF
Processor Adaptation Inversion Filtering/range incrementOAP 2N3M2 SN3M [NM)2
ASFF 2(Jf3L2) 8X3L KLfPRl9 KL(M - L + 1) total
7.1.3.5 Computational complexity
At this point we arrive at a space-time processing scheme which by virtue of aspatial transform combined with a space-time FIR filter offers a dramatic reduction inarithmetic operations while the performance is very close to the optimum. The amountof computations required for the various tasks associated with clutter rejection is givenin Tables 7.1.3.5 and 7.1.3.5. As can be seen the saving in arithmetic operations isextraordinary.
It should be noted that filtering has to be carried out for all range increments andis, therefore, the most critical operation. It is remarkable how little arithmetic effort isrequired for the FIR filter to accomplish this task. Notice that the total memory sizerequired is KLR if the filter operation is synchronised with the PRF (R is the numberof range increments). For the calculation of the FIR filter coefficients we assumed thatthe filter is calculated via the Akaike algorithm for block Toeplitz matrix inversion (seeAKAIKE [11]). A comparison of the computational load of all techniques discussed inthis book is given in Tables 12.1-12.4.
F
IF[C
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Figure 7.16: Symmetric auxiliary sensor processor with STAP FIR filter, effect of filterlength (1 m range resolution, FL, (pL = 0°, K = 5): o L = 2; * L = 3; x L = 5; +L = 7
Table 72 Numerical example: N = M = 24, K = L = 5
Processor Adaptation Inversion Filtering/range incrementOAP 16 • 106 2.6 • 10° 331776ASFF 6250 5000 25/PRI, 500 total
7.2 Impact of radar parameters
7.2.1 Sample size
In this section we briefly discuss the effect of spatial and temporal sample size, i.e.,antenna aperture and length of the echo sequence, on the performance of the symmetricauxiliary FIR filter processor.
7.2.1.1 Spatial filter dimension
The influence of the number of channels K of the symmetric auxiliary sensor processorwas already addressed in Chapter 6, Section 6.2.1, Figures 6.10 and 6.11. It was foundthat the performance of the processor was almost independent of the number of antennachannels, which is a highly desirable property. This means, in contrast to the processorsdiscussed in Chapter 5, that the spatial dimension K of the space-time processor can
F
be chosen independently of the antenna array size TV.The two subsequent figures are to confirm that this property is preserved even if
the pre-transform is followed by a suboptimum FIR filter instead of a fully adaptiveprocessor. Figure 7.8 shows IF curves for the forward looking linear array, Figure7.9, for the sidelooking configuration. As can be seen the sidelooking array is almostindependent of the number of antenna channels.
For the forward looking array we encounter some loss close to the clutter notch forK = S which is the minimum possible number of channels for this kind of processor.As was already discussed in Section 6.1.1.2, these losses have to do with the fact thatfor a forward looking antenna configuration there are always two clutter arrivals foreach Doppler frequency symmetrical to ip = 0°. Since the forward looking arraycan distinguish between left and right of the flight direction these symmetric clutterarrivals determine the number of degrees of freedom required. For two arrivals (foreach Doppler frequency) the choice of K = 3 channels is the absolute minimum. Ascan be seen for K = 5 we obtain an almost optimum IF curve.
This problem does not occur for a sidelooking array. A sidelooking array cannotdistinguish between left and right of the flight direction and therefore perceives bothsymmetric arrivals due to one Doppler frequency as one arrival only. Therefore, evenK = 3 leads to satisfactory performance.
7.2.1.2 Temporal filter dimension
In Figure 7.10 the dependence of the IF on the temporal dimension L of the space-timeFIR filter is illustrated. A forward looking array was assumed. It can be noticed thatthe performance is almost independent of the filter length. Some slight losses can be
Figure 7.17: 2-D space-time FIR filter processor
shift register(space-timeecho data)
Doppler filter bank(FFT)
testfunction
space-time FIR filter(weighted sum)
echo
puls
esantenna axis, platform motion
Figure 7.18: Projection of space-time echo samples onto a common time axis: clutter
recognised in the pass band that are due to the shortening of the data record through theFIR filter. Recall that the filtered data record has length M - L + 1. Shortening of thedata record leads to losses in target signal energy. The shorter the filter is the closer theIF curve runs along the theoretical optimum. The uppermost curve has been calculatedfor L — 2. However, L = 2 shows some slight deviations from the other curves in theclutter notch region.
The results obtained for a sidelooking array are very similar and have thereforebeen omitted.
7.2.1.3 Doppler filter length
Practical radar systems have different operational modes which may imply differentsignal integration lengths or, equivalently, different Doppler resolution. The length ofthe Doppler filter bank may vary considerably. The structure of all processors discussedin the previous chapters depends heavily on the length of the Doppler bank.
In Figure 7.11 the clutter filter dimension was kept constant (L = 5) while theDoppler filter length was varied. As the IF is proportional to the Doppler filter lengtheach curve has been normalised to its individual maximum. As can be seen, the longerthe Doppler filter is, the closer the IF approaches the maximum. For M > L theshortened length M — L + 1 approaches M. Furthermore, it can be noticed that thewidth of the clutter notch is practically independent of the Doppler filter length. Someslight losses can be recognised for very short Doppler filter lengths (in our example
echo
puls
esantenna axis, platform motion
Figure 7.19: Projection of space-time echo samples onto a common time axis: Dopplertarget
M = 8 which means that the effective Doppler filter length is only M — L + 1 = 4).The fact that the temporal and spatial dimensions of the auxiliary sensor FIR filter
are almost independent of the antenna size as well as of the Doppler filter length makesthis processor very attractive. The filter operation requires KL operations per PRIand range gate which can easily be accomplished in real-time with current digitaltechnology.
7.2.2 Decorrelation effects
7.2.2.1 System bandwidth
As carried out in Chapter 2, Section 2.5.2, the system bandwidth causes spatial(i.e., sensor-to-sensor) decorrelation of clutter arrivals. This decorrelation leads toan increase of the number of eigenvalues of the space-time clutter covariance matrix.Equivalently, we encounter a broadening of the clutter spectrum, see for exampleFigure 3.35, and a broadening of the clutter notch of the optimum clutter filter, seeFigure 4.23.
Similar results have been obtained for the suboptimum symmetric auxiliary sensorprocessor, see Figure 6.15. It should be emphasised that in the pass band no lossesoccur, which indicates that the number of degrees of freedom of the reduced vectorspace is still sufficient to cope with the increased number of clutter eigenvalues.
The spatial side of the FIR filter processors discussed in this chapter is the same asdescribed in Chapters 4 and 6. Therefore, we can expect that the results illustrated by
the above-quoted figures apply as well to the FIR filter processors of this chapter. Anumerical evaluation has, therefore, been omitted.
7.2.2.2 Clutter bandwidth
Clutter bandwidth owing to internal clutter motion causes a temporal decorrelation ofsubsequent clutter echoes which can also be expressed by an increase in the number ofclutter eigenvalues of the space-time clutter covariance matrix (see for example Figure3.25). The processors discussed in this chapter are characterised by their short temporalfilter dimension which has been obtained by exploiting the temporal stationarity ofclutter echoes. Therefore some effect of the clutter bandwidth (losses due to a lack ofdegrees of freedom) on the performance of this kind of processor can be expected.
The question to be answered is how far the increase of the number of cluttereigenvalues leads to a degradation in the clutter suppression performance of the space-time FIR filter. It should be kept in mind that normally an increase in the numberof degrees of freedom in the interference covariance matrix can be compensated forby an increase in degrees of freedom of the clutter filter. Since we are talking aboutthe temporal dimension of the problem (temporal decorrelation versus filter length) wecan expect that the filter length for obtaining the best possible clutter rejection has tobe increased.
Figure 7.12 shows the improvement factor versus the normalised Dopplerfrequency for different clutter bandwidths, corresponding to different strengths ofinternal clutter motion. As can be seen the clutter notch is broadened according tothe clutter bandwidth. In the pass band, however, almost the optimum improvement isreached. There is a slight ripple in the pass band which is typical for space-time FIRfilters. The loss compared with the theoretical optimum (0 dB) of the upper curve isdue to the fact that the coherent signal integration is shortened by L — 1 because theFIR filter must not move outside the echo sequence.
Figure 7.13 shows a comparison of different filter lengths of the space-time FIRfilter when applied to clutter with strong internal motion (Bc = 0.3). The broadenedclutter notch can be recognised clearly. For L = 2 some losses in the pass band can benoticed. For larger values of L the improvement factor becomes nearly independent ofthe filter length.
We refer here to the results presented by Liu and PENG [320] who analysed such'short-time processing' schemes. Their results indicate that the clutter notch of FIRfilter processors is broadened by the clutter bandwidth. However, there is no effect inthe remaining pass band due to a lack of temporal degrees of freedom of the clutterfilter. This is confirmed by our results (KLEMM [266]).
7.2.2.3 Range walk
Range walk is another temporal decorrelation effect (see Section 2.5.1.2). Figure 7.14shows the effect of range walk on the clutter rejection performance of the space-timeadaptive FIR filter. As pointed out earlier the forward looking case is the configurationwhich is most sensitive to decorrelation by range walk. Therefore, we focus in thefollowing on this case. The curves shown in Figure 7.14 are very similar to those of the
optimum processor as shown in Figure 4.20 except for pass band effects which havebeen explained before (loss in signal integration due to shortening of the data recordfrom M down to M - L + 1 by the FIR filter).
For strong range walk ( Im resolution) it is even more obvious that the filter lengthdoes not really play a significant role (Figure 7.15).
7,2.2.4 Spatial transforms and space-time FIR filters
Both the effects of internal motion as well as range walk are temporal by nature. Wecan expect that reducing the number of spatial degrees of freedom (as shown in theblock diagrams Figures 7.5 and 7.4) of the processor will not have any influence on theperformance of the FIR filter because the FIR filter length is temporal. This expectationis verified by the results shown in Figure 7.16 (calculated for a processor accordingto Figure 7.4) which look identical to the curves in Figure 7.15. For the processorthe number of antenna channels was chosen to be K — 5 (one main beam and fourauxiliary elements).
It should be noted that this processor has 5 x 5 adaptive coefficients (the numbercould be reduced even further) whereas the inverse covariance matrix in the optimumprocessor (1.3) would include (24 • 24) x (24 • 24) = 576 x 576 elements withoutoffering significant advantages in performance.
7.2.3 Computation of the filter coefficients
The use of digital space-time filters is justified by the temporal stationarity of echosequences. Because of the temporal stationarity space-time covariance matrices areblock Toeplitz. An effective recursive algorithm for block Toeplitz matrix inversion hasbeen presented by AKAIKE [H]. The Akaike algorithm has been extended by GOVER
and BARNETT [169] to block Toeplitz matrices that are not strongly non-singular. Thisvariant may be useful in cases where the clutter-to-noise ratio is large.
7.3 Other filter techniques
7.3.1 FIR filters for spatial and temporal dimension
In the case of a linear or rectangular equidistant antenna array the FIR filter principleso far used in the temporal dimension can be used as well in the spatial domain. Inthis case the FIR filter moves in both the temporal and spatial domain. This can beconsidered an alternative way of reducing the instantaneous vector space in the spatialdimension (KLEMM [274]).
The space-time subarray technique described by PILL Al et al [412] is stronglyrelated to two-dimensional space-time FIR filtering. It is shown that by formingsubarrays the required sample support can be significantly reduced. In fact, to adapt aFIR filter with KL coefficients requires about 2KL data vectors which is in generalmuch less than 2NM samples.6
6For the number of data samples required for adaptation see REED et al [421].
Figure 7.17 shows a block diagram of a space-time FIR filter processor, with theFIR filter operating in both the temporal and spatial domain. This requires a form ofthe FIR filter with no beamformer incorporated because the beamforming has to takeplace after moving the FIR filter over the antenna aperture.
In comparison with the other processors discussed in this chapter the concept of a2-D FIR filter processor has some disadvantages:
• The beamforming operation should be done simultaneously in order not to wasteoperation time. For temporal processing FIR filtering is particularly attractivebecause the data flow is determined by the PRF. In this case FIR filtering canbe carried out by pipelining the data. This option does not exist for the antennaarray.
• Spatial FIR filtering is restricted to linear or rectangular equidistant arrays. Manyoperational antennas are circular or elliptical.
• The array has to be fully digitised. This is expensive and should be avoided.
In view of these obvious disadvantages we omit further analysis of this principle.
7.3.2 The projection technique
In this section we describe briefly a space-time filtering technique which makes useof strict synchronisation of PRF, displacement of the sensors of a linear array, and theflight velocity. This idea has been developed by ENDER and KLEMM [109]. The use ofthis technique for detecting and imaging moving targets with SAR has been proposedby ZHOU and FENG [583].
The technique is based on the following conditions:
• A linear array moves at constant speed in the direction of the array axis(sidelooking configuration).
• The sensor spacing is subdivided into N subintervals.
• The PRF is adjusted to the platform velocity so that the spatial advance ofthe array's position between any two echoes is an entire multiple of half7 thesubinterval.
• The number of subintervals between two neighbouring sensors (identical to thenumber of sensors) and the number of half-subintervals travelled by the arrayduring one PRI have to be mutually prime.
Under these conditions all space-time echo samples can be projected onto a commontime (or space) axis in such a way that an equally spaced signal sequence is generated.This principle is illustrated in Figure 7.18. The various lines show the echo samples ofan array (horizontal) at different instants of time (vertical). Only one component of thecomplex signal is shown.
7Due to two-way propagation.
The signal in Figure 7.18 is a clutter echo coming from a point reflector at a certainangle relative to the array. Notice the spatial frequency which is a measure for theangle of arrival. After each PRI (from line to line) the array has moved by a certainamount. We chose PRF and the velocity such that the array moves 13 out of N = 15half-subintervals. Interpreting the half-subintervals in terms of phase (or time shift)leads to a shift pattern as given in Figure 7.18. Notice that the sine wave is fixed inspace while the array is moving.
If all of the space-time samples as depicted on the various lines are projected onone common time axis then we obtain an oversampled sine wave as can be recognisedin the centre of the lower line. The continuous sine wave is obtained when steady stateis reached. The gaps on the left and on the right are due to transient effects owing tothe limited space of the drawing. If we have realistic clutter coming from all directionswe obtain an oversampled signal which is band-limited by the platform velocity.
In the case of a moving target both the sine wave and the array are moving (Figure7.19). After projection on the common time axis (lower line) the resulting signal nowcontains higher-frequency components which are due to the target Doppler. Clutterrejection can easily be carried out by a simple (temporal) high pass filter operating onthe projected signal sequence, with the clutter bandwidth as cut-off frequency. A lowpass filter instead provides the echo components of the stationary background and maybe exploited for SAR imaging.
The beauty of this technique is the simplicity of clutter filtering which is enabledjust by data reordering. However, it can work properly only if the above mentionedconditions are precisely fulfilled. This is doubtful for airborne applications because theaircraft motion will perturbate the projected data sequence so that the samples of theprojected signal sequence are no longer equidistant. For more details of this techniquethe reader is referred to ENDER and KLEMM [109].
7.3.3 Space-time HR filters
Basically HR (infinite impulse response) filters may be designed for space-timeapplications. HR filters can be used for designing steep cut-off frequency slopes with arelatively small number of coefficients. However, IIR filters have long transient timeswhich may degrade the clutter suppression performance close to clutter edges or at thelimits of a data window.
IIR filters have been used in the past for MTI purposes in radar systems withrotating antennas. Such radar produces very long data sequences so that no transientproblems occur as far as limited data size is concerned. Application to phased arrayradar systems seems to be inappropriate because such radar uses normally pulse burstsof limited duration.
Finally, we have seen that even very short FIR filters accomplish almost optimumclutter suppression. Short FIR filters have short impulse responses so that transienteffects at clutter edges will be moderate. Furthermore, we have already taken in theabove analysis the limits of the pulse burst into account in that the filter moved onlyinside the data record.
7.3.4 Adaptive DPCA (ADPCA)
BLUM et al. [42] propose a suboptimum techniques which they call ADPCA (adaptiveDPCA).8 This technique is similar to FIR filtering in that the echo sequence issubdivided into segments. Then space-time adaptive processing is carried out for theindividual data segments. It has been shown by the authors (Gu et al [179]) that thesesuboptimum techniques may outperform the optimum (SMI) processor in the case ofnon-homogeneous clutter because the ADPCA converges more rapidly than the fullyadaptive SMI technique.
Other pre-Doppler techniques as described by WARD [530] and improved byBARANOSKl [28] use full submatrices according to (7.3) sliding in the time dimensionin the same way as the vectors in (7.15). The clutter notch produced by this technique,however, appears to be broader than the one generated by the FIR filter.
7.4 Summary
Space-time digital filters can be used to reduce the vector space of the clutter rejectionfunction in the time dimension. By using digital filters it is assumed that the echosequences are stationary so that the space-time covariance matrix is block Toeplitz. TheFIR filter is calculated from a short segment of length L of the received data record.
1. The space-time least squares FIR filter is given by the first block column of theinverse of the NxL space-time covariance matrix. This matrix is a submatrixof the NxM covariance matrix of the total data record.
2. Further simplification of the filter can be obtained by pre-multiplying the firstblock column of Q " 1 with a beamformer vector.
3. The space-time FIR filter approximates the optimum processor very well.
4. The clutter rejection performance is almost independent of the temporal filterlength. L = 3 , . . . ,5 is sufficient. The filter length is independent of the numberof clutter eigenvalues of the space-time clutter covariance matrix.
5. The aforementioned property is important especially when the number ofcoherent echoes is varied during the radar operation.
6. In conjunction with the spatial transforms the dimension clutter covariancematrix reduces to K x L which leads to very cost efficient clutter filters withthe capability of real-time processing.
7. Temporal decorrelation effects (internal clutter motion, range walk) lead to abroadened clutter notch, however, no additional losses owing to a lack of degreesof freedom of the FIR filter can be observed.
8. Two-dimensional FIR filters can designed which operate as FIR filters in boththe temporal and spatial dimension. This requires, however, that the antennaarray is
8This technique has been referred to by WARD [530] as element space pre-Doppler STAP.
• linear or planar rectangular;
• equispaced;
• fully digitised.
The last requirement is the most prohibitive one with current technology. Anarray whose channels are all fully equipped with receive channels including AfDconverters is not attractive for practical reasons (cost, power consumption, heat,etc.).
A comparison of all techniques in terms of computational complexity is presented inChapter 15.