4.1 Introduction

In this chapter we focus on two space-time processors which are fully adaptive. 'Fully'adaptive means that the number of degrees of freedom as given by the number of arrayelements and echo pulses will be preserved in the clutter rejection process. 'Adaptive'means that clutter suppression is based in some way on the received clutter data, forinstance on an estimate of the clutter covariance matrix.

Why do we need adaptive clutter suppression? In principle the space-time cluttercharacteristics are well known a priori through the angle-Doppler relation. However,various kinds of errors in the receiving instrument or perturbations of the flight paththrough platform motion may degrade the performance of non-adaptive techniquessuch as DPCA. For example, a comparison between adaptive and non-adaptive DPCAprocessing (SHAW and MCAULEY [462]) in the presence of beam squint errors revealsthat adaptivity can compensate for errors in the receiver mechanism. The use ofadaptive space-time filtering to compensate for various kinds of radar inherent errorshas been pronounced by SuRESH BABU et al. [477].

In contrast to subspace techniques (see Chapters 5, 6, 7) which exploit the subspaceproperties of space-time clutter data the processors discussed in this chapter are basedon the full space-time covariance matrix of the available data vector space. Theapplication of the fully adaptive space-time processor for clutter rejection has beendiscussed by KLEMM [238] and WARD [530, p. 57]. For small data size TV, Mthe optimum processor is easy to handle and has been frequently used in theoreticalconsiderations, for instance by BARBAROSSA and FARINA [31].

Fully adaptive processing techniques are normally not useful for practicalapplications. The reason is the high computational complexity involved for:

Adaptation. This involves the estimation of the inverse of the clutter + noisespace-time covariance matrix. The number of data vector samples required

Chapter 4

Fully adaptive space-timeprocessors

for estimating the covariance matrix increases with the matrix dimension (forspace-time processing at least 2NM), see REED et al, [421]. Adaptation is amoderately fast operation which should be able to cope with any changes of thedata statistics. Such changes may be caused by inhomogeneity of the clutterbackground and perturbations caused by platform motion.

Filtering. Optimum filtering means multiplying the vectors of received datawith the covariance matrix inverse, with a beamformer vector and a Dopplerfilter bank for all possible Doppler frequencies. This is a fast operation whichhas to be carried out sequentially for all range bins. Typical clock rate in therange direction is 1 to several MHz.

A realistic radar array antenna may include TV = 1000 elements or more.Moreover, the number of coherent echoes involved in the detection process may varyfrom M = 10,...,1000. This results in dimensions of the space-time covariance matrixup to TVM = 106! It is obvious that such huge vector spaces cannot be handled byreasons of computing time, numerical stability and cost.

In this chapter we analyse fully adaptive processors on the basis of a linear arraywith aperture of moderate size (N = 24) and a moderate number of echoes (M = 24).We have to keep in mind that linear arrays will rarely be used in practice. However,analysing the behaviour of linear arrays can give considerable insight into the problemswith moderate computing time. Furthermore, on the basis of the fully adaptiveprocessor we can study the various effects of radar parameters on space-time clutterrejection without caring about additional effects caused by suboptimum processing. Inthat sense Chapter 4 has some tutorial value. In the sequel the results obtained with theoptimum processor will serve as a reference when we analyse suboptimum techniquesfor real-time processing, see Chapters 5, 6, 7.

4.2 General description

4.2.1 The optimum adaptive processor (OAP)

The optimum (likelihood ratio) processor was introduced in (1.3):

Now we deal with space-time vector quantities according to (3.21). The covariancematrix will have the form shown in (3.22). We assume that sequences of echo samplesare stationary during the observation time. This assumption has been verified bynumerous radar experiments and is valid as long as the radar transmits at constantPRF and the motion of the radar is such that the relative angle between the radarand incoming signals does not change significantly (~ 1/100th of the beamwidth, seeHAYWARD [206]). Then the clutter + noise covariance matrix Q = i{qq*} becomes

(4.1)

Figure 4.1: The optimum adaptive space-time processor

block Toeplitz:

(4.2)

The submatrices Q m are spatial covariance matrices and have the dimensions NxN.For instance, Q0 is the spatial covariance matrix measured at the same instant of timewhile Qi is a spatial cross-covariance matrix between one echo and the next one. Thetemporal information is included in the relations between different submatrices.

A block diagram of the optimum processor is depicted in Figure 4.1. The N

spatial samples

temporalsamples

clutter and jammer suppressioninverse of space-time covariance matrix

orprojection matrix

space-timematchedfilter

testfunction

1 2 N

dB

Figure 4.2: Improvement factor for a sidelooking array

antenna elements provide spatial sampling of the backscattered wavefield. Each ofthe array channels includes amplification, complex demodulation and digitisation (notshown). Each channel is followed by a shift register to store subsequent echo samples.This is the temporal dimension of the space-time processor. All the spatial-temporaldata are filtered by the inverse of the space-time covariance matrix. This is followedby a space-time matched filter including the coefficients of the signal reference(beamformer and Doppler filter coefficients) according to (2.31). In practice thespace-time matched filter has to be implemented for all possible Doppler frequenciessimultaneously (Doppler filter bank).

A test function is then calculated based on the actual output signals of the Dopplerfilter bank and is fed into a detection and indication device. One possible test functionconsists of a search for the maximum magnitude Doppler channel output

(4.4)

F

cp/

(4.3)

'target + noise'otherwise

This test function is called a Doppler processor and is widely used in practicalapplications. NAYEBI et al. [366] have shown that a discrete form of the likelihoodfunction for gaussian noise and deterministic signals:

'target + noise'otherwise

dB

Figure 4.3: Improvement factor for a forward looking array

is near optimum for both white and coloured noise. Io is the modified Besselfunction. This processor is called the Constrained Averaged Likelihood Ratio (CALC)./i determines the number of Doppler channels which usually assumes the values 1 or2. As shown by NAYEBI [366] the Doppler processor (4.3) is near-optimum for whitenoise but suffers some considerable losses in the case of coloured noise. In his paperthe author presents an overview of a large number of test functions (which in essenceare different approximations of the optimum likelihood ratio processor).

As pointed out in Chapter 1 the efficiency of the processor can be jugded bycalculating the improvement factor (IF). For the optimum processor (4.1) the IF isgiven by

(4.5)

where the elements of the signal reference vector (or steering vector) are given by(2.31). cpi, is the look direction while / D = 2^rad/A is the target Doppler frequencydue to the radial target velocity vr&&. In contrast to the spectra shown in Chapter 3 thecalculation of the optimum IF requires that the transmit beam is steered in the samedirection as the receive beam (perfect signal match).

In Figures 4.2 and 4.3 the IF as given by (4.5) has been plotted versus the ip-Fplane where F is again the target Doppler frequency normalised to the PRF. It can benoticed that along the ip-F clutter trajectories (see Figures 3.18 and 3.19), now a narrowtrench has been formed by the optimum processor. In fact, as pointed out in Chapter1 (compare (1.16) and (1.105)), the optimum IF is just the inverse of the minimum

F

cp/

IF[C

lB]

Figure 4.4: The potential of space-time adaptive processing: o optimum processing; *beamformer + Doppler filter; x beamformer + adaptive temporal filter

variance spectrum. Everywhere outside the clutter trench we find an IF plateau wheredetection of moving targets is optimum.

4.2.1.1 The potential of space-time adaptive processing

In Figure 3.40 the principle of space-time adaptive clutter rejection was illustrated.In particular a comparison of space-time processing with pure temporal or spatialprocessing has been done. The comments given there are supported by the numericalexample shown in Figure 4.4.

Like the examples given in Chapter 1 the plot shows the IF (improvement factor) inSCNR versus the normalised target Doppler frequency. The IF has been normalised bythe theoretical maximum which is approximately CNRxTV x M. Notice that this kindof plot is just a cross-section through a 3-D IF plot as shown for example in Figure 4.2.The cross-section runs along the Doppler axis in the look direction.

Figure 4.4 shows a comparison between space-time adaptive filtering, beamformingand Doppler filtering (but no clutter rejection), and beamforming cascaded with anoptimum temporal clutter FIR filter as has been described by BUEHRING and KLEMM[64], and (1.90).

The curve for beamforming and Doppler filtering reflects the sidelobe response ofthe Doppler filter bank to clutter. Notice that there is a kind of inverse main lobe in thelook direction. The losses compared with the optimum curve are dramatic.

A slightly better performance is achieved with a beamformer cascaded with anoptimum temporal FIR filter, however there are still considerable losses, especially in

F

IF[d

B]

Figure 4.5: Comparison of optimum and orthogonal projection processing

the look direction (F 0.35). These losses degrade particularly the detectability ofslow targets.

We showed the comparison of these three ways of processing to demonstrate whatthe motivation for space-time adaptive processing is. In Chapters 5, 6 and 7 we tryto approximate the performance of the optimum processor as much as possible bysuboptimum processors with reduced computational expense and real-time capability.The IF curves of such processors will appear somewhere between the optimum space-time and the optimum temporal processor. Favourable solutions will have to be closerto the upper (o) than the lower (x) curve.

4.2.2 The orthogonal projection processor (OPP)

The orthogonal projection operator is given by (1.47)

where E is the matrix of eigenvectors belonging to the interference component in Q.The IF becomes, according to (1.15),

where s = s((/?L, / D ) is again the steering vector with elements given by (2.31), whoseparameters assume all possible values of the (^L / D plane. The block diagram is the

(4.7)

(4.6)

F

same as for the optimum processor except that the inverse of the covariance matrix isreplaced by the projection matrix (4.6).

As we know from Figure 1.3, the IF of the orthogonal projection is very close tothe optimum in one-dimensional applications. Figure 4.5 shows a similar example forairborne clutter cancellation (sidelooking array, N = 12 sensors, M = 12 echoes).In fact, these curves are cross-sections through 3-D IF plots like those in Figures 4.2and 4.3. As can be seen the clutter rejection performance is almost the same for theoptimum processor and the orthogonal projection method. The curves differ in thatthe clutter notch caused by the orthogonal projection method is an exact null while theoptimum processor suppresses the clutter only down to the noise level.

Ts AO et al. [502] discussed the choice of degrees of freedom in rank reducedSTAP processors. It was found that a slight overestimate of the degrees of freedomdoes not degrade the performance. Underestimating is more critical, in particular whendecorrelation effects such as clutter fluctuations come into play.

4.2.2.1 Sidelooking array, DPCA conditions

LiU Q.-G. et al [318] analysed the performance of an orthogonal projection processorbased on a sidelooking array under DPCA conditions. In this case the cluttercomponent of the space-time covariance matrix was Q = Pc)5]* (see (3.42)) whereE is the shift operator defined by (3.38). Then the orthogonal projection operator isgiven by the pseudoinverse

(4.8)

Since X) is a sparse matrix the calculation of the projection matrix needs only asmall amount of computation. Even under array errors near optimum performanceis obtained, provided that DPCA conditions are satisfied.

The interrelation between optimum processing, DPCA and orthogonal projectionhave been exploited by RICHARDSON [426] for designing a suboptimum simpleprocessor:

The zero noise approximation of the space-time covariance matrix inverse Q ~x

is a projection matrix of the form

where the columns of B span the clutter subspace of Q. For instance, B mayinclude the clutter eigenvectors of Q.

Under DPCA conditions the matrix may be composed of identity vectors of thefollowing kind (N > M)

(4.9)

(4.10)

where e is a TV x 1 identity vector with a ' 1' in the i-th position and 0 elsewhere.0 is a N x 1 zero vector.

A similar set of vectors can be found for iV < M [426]. It can be seen by inspectionthat Richardson's basis vector system (4.9), (4.10) is identical to (4.8), (3.38), i.e., it isin essence a multichannel/multipulse DPCA system. The use of DPCA for spaceborneradar has been proposed by TAM and FAUBERT [485] and WANG H. S. C. [510].

A procedure for adaptive calculation of the eigenvalues and eigenvectors belongingto a subspace of the covariance matrix (for instance, the clutter subspace) has beendeveloped by MATHEW and REDDY [343].

Two techniques based on eigendecomposition of the sample covariance matrix havebeen compared with the 'joint domain' optimum processor ('JDO') by GOLDSTEIN etal [159] on the basis of MOUNTAINTOP data (Tm [494]).

4.3 Optimum processing and motion compensation

In this Section we try to reveal the interrelation between motion compensation andoptimum processing.

4.3.1 Principle of RF motion compensation

First let us recall the principle of platform motion compensation, see SKOLNIK [468,pp. 17-7]. Let us assume an antenna looking in the cross-flight direction (sidelooking).In Figure 4.6a, a phasor diagram for two subsequent clutter echoes c(t) and c(t + r)coming from a single clutter scatterer is depicted. Because of the platform motion thereis a phase advance

(4.11)

between the two arrivals. Lp denotes azimuth. The amplitude of c(t) is proportional tothe two-way antenna field intensity.

The principle of platform motion compensation is shown in Figure 4.6b. Two so-called correction patterns yci(t) and yC2(t) &

re added to the clutter signals in such away that both echoes c(t) and c(t + r) are projected on a common phase angle 0/2.Now both echoes are in phase so that they can easily be cancelled by subtraction, that

Figure 4.6: Motion compensation with correction patterns

means, by use of a conventional two-pulse RF clutter canceller which is available inany coherent radar.

This principle dates back to the times where digital technology was not yetavailable. Motion-compensated clutter cancellation was carried out in the RF domain.The correction patterns are given by

(4.12)

see SKOLNiK [468, pp. 17-7]. S(^?) is the one-way antenna sum beam pattern. Thecorrection pattern is very similar to the difference pattern of a monopulse trackingradar. Obviously no correction is applied in the look direction (ip = O) while forany angle ( / 9 / 0 the difference pattern follows the sidelobe structure of the sumbeam. Therefore, the main channel is associated with an auxiliary channel for cluttercancellation in such a way that the CNR is always the same in both the main andauxiliary channel. 1

In fact, the output signal of a difference pattern is in quadrature with the sumbeam. Therefore, with the two channels (sum and difference) of a standard monopulseantenna, motion-compensated airborne MTI can be realised.

'Recall the considerations concerning the sidelobe canceller, Section 1.2.3 and Figure 1.6.

a. b.

c.

Figure 4.7: Motion compensation by the optimum processor

It should be noted that this motion compensation technique works only in asidelooking configuration where the main beam center Doppler frequency is zero. Inthe case where the antenna is horizontally tilted so as to look into a squint angle theclutter Doppler frequency in the beam center has to be compensated for because thestandard two-pulse canceller is based on the subtraction of two equiphased signals.This compensation has been done by the TACCAR2 loop, see SKOLNIK [468, pp. 17-32]. TACCAR is a control loop which adjusts the local oscillator frequency in such away that the phases of subsequent echoes become equal.

4.3.2 Correction patterns

The above described motion compensation technique was developed for radar withreflector antenna. ANDREWS [17] proposed a correction pattern optimisation techniquefor use with antenna arrays which follows tightly the motion compensation principleillustrated before. The correction patterns yc\ (t) and yC2(t) according to Figure 4.6 areoptimised in such a way that the squared magnitude of the difference between the idealand the actual corrected echoes is a minimum. The resulting optimised weightings for

2Acronym for 'time-averaged-clutter coherent airborne radar'.

a. b.

c.

spatialwhitening

motioncorrection

motioncorrection

IF[C

lB]

where b is a beamformer vector and Qo and Qi are spatial submatrices of the space-time covariance matrix, see (4.2), for the case M = 2.

Notice that the submatrix Qi is a cross-variance matrix between the clutter echovector c at time t and at time t+r/2 and tr/2 respectively. They have to be calculatedbased on an analytical clutter model as has been done by ANDREWS [17]. An adaptiverealisation is not easily achieved because at time t + r/2 no data are available.

This problem could be overcome by doubling the PRF which, however, mighthave unwanted effects on the radar operation. For instance, the unambiguous rangeis reduced by a factor of 2 when the PRF is doubled. The unambiguous Doppler rangeis doubled. However, if the number of pulses is kept constant the Doppler resolution islowered by a factor of 2.

4.3.3 Interrelation with the optimum processor

4.3.3.1 Example: the inverse of Q for M 2

Some interesting relations between motion compensation after ANDREWS [17] andoptimum processing has been shown by KLEMM [239]. Similar to the considerations in

Figure 4.8: Influence of transmit beamwidth (SL): o Nt = 1;* Nt = 24 (like receivearray); x Nt = 48

the array outputs are

(4.13)

F

IF[C

lB]

F

Figure 4.9: Effect of array size (SL):

Chapter 1 let us consider the space-time covariance matrix for the special case M = 2.

(4.14)

where ci = c(t) and c2 = c(t + r) denote the first and second echo vector. Using theFrobenius-Schur relation (BODEWIG [45, p. 217]) the inverse of Q becomes

with

(4.15)

(4.16)

being the partial covariance matrix of the clutter vector c 2 given ci (ANDERSON [14,pp. 27]). An equivalent version of (4.15) is given by (BODEWIG [45, p. 217])

with

(4.17)

(4.18)

being the partial covariance matrix of the clutter vector c i given C2. Comparing (4.15)with (4.18) one obtains

(4.19)

dB

Figure 4.10: Array size versus sample size (FL), eigenspectra: o N = 48, M = 3; *N = 24, M = 6; x TV = 12, M = 12; + N = 24, M = 6; no symbol JV = 48, M = 3

4.3.3,2 Correction patterns

Let us now recall the principle of optimising correction patterns according toANDREWS [17]. Instead of correcting for just half the phase advance 0/2 (see Figure4.6b) we now consider the case of correction for the total phase as depicted in Figure4.7b. The beamformer responses to a clutter echo vector are

(4.20)

The signal at time t is to be phase corrected by adding a correction vector

(4.21)

where y is the vector of correction coefficients. The correction vector is to be chosenso that the difference between the corrected echo and the following echo becomes aminimum.

Hence we minimise the expression

(4.22)

IF[C

lB]

Figure 4.11: Array size versus sample size (FL), IF: o N = 48, M = 3; * N =24, M = 6; x N = 12, M = 12; + N = 24, M = 6; no symbol TV = 48, M = 3

F

Using

(4.22) becomes

The minimum is obtained by differentiation of (4.23) with respect to y and setting theresult equal to the zero vector. The solution is

(4.23)

The corrected beamformer vector is then

(4.24)

(4.25)

4.3.3.3 Comparison

Comparing (4.25) with (4.19) we find the correction factor Q ^ 1 Qi in the off-diagonalsubmatrices. This indicates that in the second column of Q " 1 the vector c(t) iscorrected while the vector c(t + r) is not. The same happens in the first column, butthe other way around. The minus signs in both the off-diagonal submatrices take care

dB

Figure 4.12: Effect of temporal under sampling (SL, / P R = 0.5 Nyquist)

of the clutter cancellation like the minus sign occurring in the well-known two-pulsecanceller.

The inverses of the partial covariance matrices Af21 and A^1

1 provide the spatialpart of the space-time clutter cancellation.3 Now we can summarise our insights intothe nature of the optimum processor. It includes three components:

Motion compensation. The phase advance of one out of two subsequent clutterechoes due to the platform motion is corrected for by a correction matrix (4.25).

Spatial cancellation. The spatial part of the clutter cancellation is carried out bythe inverses of the partial covariance matrices A 1 2 and A 2 i . Notice that this part of theoptimum processor is not included in the DPCA techniques nor in ANDREWS' motioncompensation technique.

It should be noted that in the DPCA case Q 0 becomes the identity matrix and alloff-diagonal submatrices Q m are identity matrices with the main diagonal shifted bym steps in the cross-diagonal direction.4 For example, for two sensors (N 2) we get

F

(p/

(4.26)

Therefore, the partial covariance matrices A 1 2 and A 2 i according to (4.18) and (4.18),respectively, become identity matrices. By looking at the covariance matrix (3.43),it becomes obvious that this holds for arbitrary values of N. Under ideal DPCA

3For details on partial correlation see MARDIA et al. [337, pp. 169-170] and ANDERSON [14, p. 29].4Compare with (3.43).

dB

Figure 4.13: Effect of temporal under sampling (FL, / P R = 0.5 Nyquist)

conditions no spatial decorrelation is necessary. This can be seen by inserting (4.26)into (4.18). In this case only motion compensation plus temporal cancellation isoptimum.

If ideal DPCA conditions are not fulfilled, i.e., the zero elements in the covariancematrix (3.43)5 assume values unequal to zero, the partial covariance matrices will playa role in spatial clutter rejection:

Non-ideal DPCA conditions may be caused by several reasons:

The PRF is not precisely the Nyquist frequency of the clutter bandwidth asdetermined by the platform velocity.

The PRF is not harmonised with the sensor spacing and the platform speed.

The antenna array is different from an equispaced, linear or planar, array in thesidelooking configuration.

The clutter background is inhomogeneous. Singular peaks of high intensity mayoccur.

Directive illumination through the transmit beam (compare the effect of directiveillumination in Figures 3.7 and 3.8).

The sensor directivity patterns of the receive array cause some additionaldirectivity.

5See also Figure 3.8.

F

q>/

dB

Figure 4.14: Effect of sensor spacing (SL, d/X = 1)

Under all these circumstances the optimum adaptive processor is superior to the motioncompensation and subtraction method proposed by ANDREWS [17].

Temporal cancellation The temporal clutter cancellation component is given bysubtraction of the clutter vectors after spatial decorrelation and motion compensation(multiplication with the correction matrix, see (4.25)).

A block diagram of the optimum processor is shown in Figure 4.7c. The processorhas to be completed with a signal matching network (see Figure 4.1) of the form

(4.27)

where the factors attached to the beamformer vectors are the coefficients of a two-pointDoppler filter and ut is the Doppler frequency of the target.

6

6For M = 2 the length of the Doppler filter bank is only 2. In practice a MTI FIR filter as given by oneof the columns of Q~ x is cascaded with a Doppler filter bank whose length is chosen independent of theclutter filter length M 2. The operation of the two columns is marked in Figure 4.7b by solid and dashedlines.

It should be noted that this way of processing (pre-filtering for clutter rejection followed by a Doppler filterbank) is a suboptimum approach which is based on a much larger data size than required for the adaptiveclutter filter. The optimum processor for this data size would have the same temporal dimension, i.e. muchlarger than M = 2. We come back to this point in Chapter 7.

F

(p/

dB

Figure 4.15: Effect of sensor spacing (FL, d/X = 1)

4.4 Influence of radar parameters

In the following we discuss some properties of the optimum processor by variation ofseveral radar parameters.

4.4.1 Transmit beamwidth

Let us consider first the influence of the directivity of the transmit antenna. For thatpurpose all clutter arrivals are weighted with a transmit directivity pattern as definedby (2.39) and (2.40).

In Figure 4.8 the improvement factor (IF) has been plotted versus the normalisedDoppler frequency. The look direction has been kept constant (45 ). Such curves arecross-sections through a three-dimensional plot as shown for example in Figures 4.2and 4.3. We will use this kind of plot throughout the following Chapters.

The three curves in Figure 4.8 have been plotted for Nt = 1 sensor7

(omnidirectional transmission), Nt = 24 (transmit array = receive array), Nt = 48(transmit array = 2 x receive array). The improvement factor has been normalised bythe maximum IF which amounts approximately to

(4.28)

see (1.35).7Nt is the number of elements of the transmit array, see (2.39).

F

cp/

IF[C

lB]

Figure 4.16: Effect of the CNR (SL): o CNR = 40 dB; * CNR = 30 dB; x CNR = 20dB; +CNR= 1OdB

We notice the following facts:

The maximum IF is reached for all target Doppler frequencies except for theclutter notch. This notch appears at that clutter Doppler frequency which isassociated with the look direction. In other words, the target Doppler equalsthe clutter Doppler. In the given example we have ipi, = 45; cos 45 = 0.707;therefore the notch appears at F = 0.35. Actually it is located slightly below F =0.35 because of the depression angle given by the radar-ground geometry.8.

The width of the clutter notch is a measure for the detectability of slow targets.

The IF is practically independent of the transmit beamwidth. The width of theclutter notch is merely determined by the clutter resolution capability of theadaptive receiver.

4.4.2 Array and sample size

The effect of the array size (number of elements N) is illustrated in Figure 4.9. Thefour curves are plotted for N 3,6,12,24. In accordance with doubling the number ofsensors the improvement factor increases in increments of 3 dB in the antenna sideloberegion, i.e., far away from the clutter notch.

Moreover, it can be noticed that the clutter notch becomes narrower as the arraysize increases. This fact is well known from one-dimensional processing (spatial or

8See Table 2.1

F

IF[C

lB]

Figure 4.17: Effect of non-DPCA sampling (SL): o / P R = Nyquist; * / P R =0.7-Nyquist; x / P R = 1.43-Nyquist

temporal), see, for example, Figure 1.2. Now, as we increased the number of arrayelements the Doppler response shows the same behaviour as the spatial response!This, however, follows directly from the equivalence between azimuth and Doppler,see (2.35). It is impossible to make a trench like the one in Figure 4.2 narrower in onedimension only without doing it in the other dimension as well.

Moreover, without giving further numerical examples, we can state that the clutternotch depends on the sample size M in a similar way. In summary, whicheverdimension (N, M) is smaller limits the clutter resolution, i.e., the width of the clutternotch.

Let us recall that under DPCA conditions the number of clutter eigenvalues of theclutter covariance matrix for a sidelooking equidistant array is

Ne = N + M - 1

For a given total sample size ./VM the minimum number of eigenvalues is obtained if

N = M (4.29)

The reader is reminded that the number of eigenvalues is a measure of the degrees offreedom of a clutter suppression filter.

In Figure 4.10 the total space-time sample size was assumed to be NM = 144. Theplots show curves for different ratios NjM. It is clearly seen that for Af = M 12the minimum number of clutter eigenvalues is obtained.9

9Notice that only 70 out of 144 eigenvalues are shown.

F

IF[C

lB]

Figure 4.18: Influence of clutter bandwidth (SL): o Bc = 0; * Bc = 0.05; x Bc = 0.1;+ Bc = 0.2

Figure 4.11, shows the corresponding IF curves. In comparison with Figure 4.10it can be noticed that the clutter notches are broadened as the number of cluttereigenvalues increases.

These examples were calculated for a forward looking array. Similar results can beobtained for a sidelooking array.

4.4.3 Sampling effects

As mentioned before the IF of the optimum processor is proportional to the inverse ofthe MV spectrum (compare (1.16) and (1.104)). Therefore, the following four plotscorrespond closely to Figures 3.20, 3.21, 3.22 and 3.23.

4.4.3.1 Temporal undersampling

Figure 4.12 shows the IF versus the azimuth angle cp and the normalised Dopplerfrequency F for a sidelooking linear array. The PRF has been set to one half ofthe Nyquist frequency corresponding to the clutter bandwidth (6 KHz). Therefore,ambiguous responses can be noticed on both sides of the main clutter trench.

In Figure 4.13, we see an example for forward looking radar with again halfthe Nyquist frequency as PRF. The clutter trench, originally located only at positiveDoppler frequencies, is now repeated at negative target Doppler frequencies.

F

IF[C

lB]

Figure 4.19: Effect of range walk (SL, ipL = 90): o no range walk; * A R = 100 m;x AR = 10 m; H- AR = 1 m

4.4.3.2 Spatial undersampling

In the following two figures we assume that the sensor spacing is d = A, i.e., twicethe spatial Nyquist frequency. In Figure 4.14, we see again the clutter trench andthe associated ambiguous responses on both sides for sidelooking radar. Notice thatthe clutter ambiguities run along the same trajectories as in the case of temporalundersampling (Figure 4.12). Therefore, if the clutter echoes are undersampled in bothtime and space in the same way the ambiguities fall onto the same trajectories. Thishas to do with the fact that for a sidelooking array the array motion is aligned with thearray axis.

For a forward looking array with d A the IF becomes as shown in Figure 4.15.Notice that there are multiples of the original semicircle which are shifted along thespatial (^-coordinate. Opposite to the sidelooking array case there is no coincidenceof the trajectories for temporal and spatial undersampling (compare Figure 4.15 withFigure 4.13).

4.4.3.3 Non-DPCA sampling

Let us recall the remarks made on the nature of the inverse of the clutter covariancematrix on page 132. It was found that the DPCA technique performs motioncompensation and temporal clutter cancellation, but does not include partial spatialwhitening as does the optimum processor. Let us further recall that the partialspatial covariance matrices in (4.16) and (4.18) become identity matrices only if

F

IF[C

lB]

Figure 4.20: Effect of range walk (FL,

IF[C

lB]

where Ac is a diagonal matrix whose diagonal has the inverse clutter eigenvalues onthe first 7VC positions and zeroes elsewhere. I is a diagonal matrix with zeroes on thefirst Nc positions and unity elsewhere. Nc is the number of clutter eigenvalues. Exceptfor the small contribution of the clutter subspace E ACE*, i.e., for large CNR, IF o p tbecomes inversely proportional to the noise power Pw. This is illustrated in Figure4.16. The four curves have been plotted for CNR = 10, 20, 30, 40 dB. Notice that thedifference between the various curves is constant (10 dB), except for the two lowerones (low CNR) where the clutter subspace still has little influence.

4.4.5 Bandwidth effects

4.4.5.1 Clutter bandwidth

The following results are based on the clutter fluctuation model (2.53). As we knowfrom Figures 3.24 and 3.25, temporal decorrelation due to internal clutter motionis reflected by an increase in the number of clutter eigenvalues. This in turn hasconsequences for the width of the clutter notch and, hence, on the detection of slowtargets.10

10Compare, for example, Figures 4.10 and 4.11.

Figure 4.21: Influence of system bandwidth (SL, rectangular impulse response): oBs = 0; * Bs = 0.05; x Bs = 0.1; + B8= 0.2

so that (4.30) becomes

(4.31)

F

IF[d

B]

Figure 4.22: Influence of system bandwidth (FL, rectangular impulse response): o,B8 = 0; * B3 = 0.05; x B8 = 0.1; + Bs = 0.2

Figure 4.18, p. 138, shows the effect of clutter fluctuations on the detectabilityof slowly moving targets. As can be seen, a target whose Doppler frequency fallsinside the clutter bandwidth will be attenuated by the clutter suppression process so thatdetection is degraded. Notice that outside the area covered by the clutter bandwidth themaximum gain is obtained.

According to BARILE et al. [32] internal clutter motion may lead to a strongdecrease in improvement factor as well as to a rise of antenna sidelobes.

4.4.5.2 Effect of system bandwidth: temporal decorrelation throughrange walk

In Figures 4.19 and 4.20 the effect of range walk (for a model see Section 2.5.1.2) isillustrated for sideways and forward looking array configurations. Both arrays look intothe broadside direction (90 and 0, respectively). Both figures show that the higher therange resolution is, the larger the clutter notch of the IF curves becomes. A comparisonof both figures confirms that range walk has more impact on the clutter notch in the inflight' direction than in the cross-flight direction.

These results indicate that the received data might have to be range correctedprior to adaptation and clutter filtering. This depends on parameters such assignal integration time (M) and range resolution. For details on range migrationcompensation techniques see CURLANDER and McDONOUGH [91, pp. 189-196].

F

IF[C

lB]

Figure 4.23: Influence of system bandwidth (FL, rectangular frequency response): oB8 = 0; * B8 = 0.05; x B8 = 0.1; + B8 = 0.2

4.4.5.3 Effect of system bandwidth: spatial decorrelation throughtravel delay

The effect of system bandwidth on space-time clutter suppression has been discussedby KLEMM and ENDER [273], KLEMM [254], and HERBERT [207]. For thesidelooking array we know from the considerations in Chapter 3 (see the eigenspectrain Figure 3.28, p. 99 and Figure 3.30, p. 101) that the number of clutter eigenvaluesis independent of the system bandwidth. We recall that the DPCA effect compensatesfor travel delays of incoming waves across the array, see Section 3.4.3. The followingresults are based on the system bandwidth models (2.60) and (2.64). Figure 4.21 showsthe improvement factor of a sidelooking array under DPCA conditions (see Table 2.1).The curves have been calculated for different values of the relative system bandwidthB8. A rectangular frequency response of the receive channels has been assumed.Recall that the associated eigenspectra in Figure 3.30, p. 101, are almost identical.Accordingly, the IF curves coincide perfectly. There is no degradation in SCNR due tothe system bandwidth.

The DPCA effect which compensates for system bandwidth effects in sidelookingarrays does not apply to forward looking arrays or any other array configuration.Therefore, as we know from the clutter spectra shown in Chapter 3 (Figure 3.34, p. 105and Figure 3.35, p. 106) forward looking arrays are sensitive to spatial decorrelationcaused by the system bandwidth.

In Figure 4.22, IF curves for different system bandwidth (rectangular impulseresponse according to (2.60) are shown. It can be noticed that the clutter notch is

F

Figure 4.24: Superposition of system bandwidth and range walk effect (IF vs F, FL,rectangular frequency response): a. No bandwidth effect; b. system bandwidth effectonly; c. range walk effect only; d. superposition of both bandwidth induced effects.Upper to lower curves: B8 = 0.0001 ( A R = 150 m); B8 = 0.001 ( A R = 15 m);Bs = 0.01 (AR = 1.5 m)

severely broadened. Even in the pass band the optimum gain is not reached. This canbe explained with the relatively high sidelobes of the sin x/x autocorrelation functionof a rectangular impulse.

For a channel response with rectangular spectrum according to (2.64), p. 62, thebandwidth effect on the improvement factor is mitigated in that outside the clutterfrequency region the optimum IF is reached (Figure 4.23, p. 143). However, somebroadening of the clutter notch (loss in SCNR for slow targets) has to be taken intoaccount.

If required system bandwidth effects can be compensated for by space-time-TIME or space-time-FREQUENCY processing (by TIME is meant the echo delaytime equivalent to range). FREQUENCY is the associated frequency after Fouriertransform. It should be noted that the temporal decorrelation due to clutter motioncannot be compensated for.

In Figure 4.27, the effect of system bandwidth (rectangular frequency response)for a sidelooking antenna configuration is shown. The three curves have been plottedfor different PRFs. This plot corresponds to Figure 4.17, p. 137, however with B8 =0.1. The figure is to illustrate that the optimum space-time processor compensates forspatial decorrelation effects caused by the system bandwidth even under non-DPCAsampling conditions (that means, the array elements do not assume the positions oftheir predecessors while the array is moving). Obviously the system bandwidth has no

a. b.

d.c.

IF[C

lB]

Figure 4.25: Doppler spread within range gate (FL, H = 1 km): o R8 = 2 km,AR = 1500 m; * R5 = 2 km, AR = 150 m; x Rs = 20 km, A R = 1500 m; +R8 = 20 km, A R = 150 m

effect on the clutter notch.The spatial decorrelation due to the system bandwidth occurs only when

narrowband array steering is applied. ZATMAN and BARANOSKI [575] have shownthat the effect of system bandwidth can be compensated by using time-delay steering inangle and Doppler. TECH AU [489] analyses the effect of different characteristics of thereceive filter on the hot clutter mitigation performance. RABIDEAU and KOGON [417]discuss the impact of the system bandwidth in the context of space-based STAP radar.They propose to use sufficiently narrow subband filters to reduce the system bandwidtheffect. Some considerations on STAP with wideband radar have been examined byHERBERT [207]. He concludes that by using true delay lines rather than phase shiftersfor beamsteering the performance of the STAP processor can be improved even forforward looking arrays. For application in communications, BOCHE and SCHUBERT[44] give an upper bound for signal errors caused by the system bandwidth.

4.4.5.4 Superposition of range walk and travel delay effects

Both of the two above discussed effects (temporal decorrelation due to range walk,spatial decorrelation due to travel delays of incoming waves) have the same origin,namely the bandwidth of the radar system. Although both effects have been treatedseparately in the previous Sections they occur in practice simultaneously.

This principle is illustrated by Figure 4.24. In Figure 4.24a we see the optimum IFcurve without any decorrelation effects. Figure 4.24b shows the system bandwidth

F

Figure 4.26: Doppler spread and range walk (IF vs. F, FL, H = I km; left clutternotch: Rs = 3 km, right clutter notch: Rs = 10 km): a. A R = 1500 m; b. A R = 150m; c. A R = 15 m; d. A R = 1.5 m

effect as would occur in a stationary radar with the associated bandwidths (rangeresolutions A R ) . The impact of platform velocity is shown in Figure 4.24c while dshows a superposition of both effects. As can be seen the curves in c and d lookidentical. Obviously range walk is the dominating effect.

4.4.5.5 Doppler spread within range gate

It was shown in the previous subsections that temporal decorrelation due to range walkbecomes stronger if the range gate is getting narrower (high range resolution). Forlarge range gates some Doppler spread within the range gate may occur, provided theclutter Doppler frequency is range dependent. As was demonstrated in Chapter 3 thisapplies to all array configurations other than sidelooking.n A model for the Dopplerspread was given in Section 2.5.3.

Figure 4.25 shows a numerical example for the effect of Doppler spread on the IF.It can be noticed that a significant degradation of the improvement factor takes placeonly at short range (i.e., Rs is of the same order of magnitude as H), and if the width ofthe range gate is of the order of magnitude of Rs and H.

n These are quite exceptionalconditions. We can conclude that Doppler spread within the range gate does not limitthe performance of the adaptive space-time processor in most applications.

1 ' For bistatic radar operation even sidelooking arrays may exhibit range dependence of the clutter Doppler,see Chapter 12.

12For the definitions of Rs and H see Figure 2.1.

a. b.

c. d.

IF[C

lB]

Figure 4.27: Effect of non-DPCA sampling (SL, system bandwidth Bs = 0.1): o/ P R ^Nyquist; * / P R = 0.7-Nyquist; x / P R = 1.43-Nyquist

There is obviously a trade-off between the Doppler spread and the range walkeffect. In the example given in Figure 4.26 both effects have been superimposed. Thetwo curves in each of the subplots have been calculated for Rs = 3 km and for Rs = 10km, respectively. Notice that the clutter notches for the two distances are Dopplershifted due to different depression angles (sin 0 = H/Rs).

For a large range bin ( A R = 1500 m, Figure 4.26a) the Doppler spread dominates.Notice that at short range the clutter notch is considerably broadened whereas at farrange the effect is not as significant. As the range bin gets smaller ( A R = 150 m,Figure 4.26b) the clutter notch becomes narrower even for short range. In this case theimpact of both Doppler spread and range walk is rather weak. For even smaller rangegates (Figures 4.26c and d) decorrelation through range walk dominates. This effect isnearly range dependent.

4,4.6 Moving clutter

The mathematical description of the phase relations for Doppler clutter was given in(2.44), p. 57. It is obvious from (2.44) that the radial clutter velocity causes just aDoppler shift of the clutter spectrum. That means in turn that the resulting IF plotsare identical to those shown in Figure 4.2, p. 120, or Figure 4.3, p. 121, but shiftedby a Doppler term corresponding to the clutter velocity v c. Adding a clutter velocitycomponent results in a shift of the clutter notch according to the radial clutter velocity.Since this is obvious we omit any numerical example.

F

R

F

Figure 4.28: Range-Doppler matrix (greytones denote IF/dB, R = range/m, SL, if L 90)

4.5 Range-Doppler IF matrix

A common means of illustrating the reactions of a radar to clutter, interference andtargets is the range-Doppler matrix. The range-Doppler matrix normally shows theFourier transforms of radar echo sequences for various range increments. It can,however, be used as well to illustrate the performance of signal processing techniques.

The principle of clutter suppression by use of the optimum processor is illustratedin Figures 4.28, p. 148, and 4.29, p. 149. The figures show the improvement factoraccording to (4.5), p. 121, versus range and normalised Doppler frequency. Weassumed that the CNR is 20 dB at 15000 m range.

In Figure 4.28 the clutter notch appears as a vertical straight line in black. Itindicates that the clutter Doppler frequency is range independent for a sidelookingarray (compare with (3.19), p. 74). Notice that the improvement factor decreaseswith range. This is in accordance with the radar range equation which states that thebackscattered radar power due to a single point scatterer is Pc oc -^.

For the forward looking radar case13 the clutter notch runs according to the rangelaw of the clutter Doppler frequency given by (3.20), p. 75. Since we assumed a radarplatform altitude of H = 3000 m, Rs 3000 m means a look perpendicular onto theground. Therefore there is no radial velocity component between radar and ground so

13For the parameter configuration used in Figure 4.29 the clutter is range ambiguous. For simplificationthis aspect has been neglected here. We will come back to this aspect in Chapter 15.

R

Figure 4.29: Range-Doppler matrix (greytones denote IF/dB, R = range/m, FL,

4. The principle of clutter rejection by the optimum processor can be interpretedas a three-step function:

compensation for motion-induced phase advances between subsequentclutter echoes;

spatial clutter cancellation through the inverses of partial covariancematrices;

temporal clutter cancellation through subtraction of subsequent echovectors.

5. The clutter rejection performance of the optimum processor depends on thesampling rates as follows:

Spatial undersampling (sensor spacing larger than A/2) leads to spatialambiguities.

Temporal undersampling (PRF below the Nyquist rate of clutterbandwidth) leads to temporal ambiguities.

The optimum adaptive processor functions optimally even if the PRF is notchosen at the DPCA (Nyquist) rate.

6. Decorrelation effects degrade the clutter rejection performance in the followingway:

The clutter bandwidth due to internal clutter fluctuation causes a temporaldecorrelation which results in a broadening of the clutter notch (equivalentto degradation in slow target detection). These losses cannot becompensated for.

Range walk is another temporal decorrelation effect which causesbroadening of the clutter notch.

The system bandwidth causes spatial decorrelation of clutter echoes andsignals because of delays of arriving waves travelling over the antennaaperture. This decorrelation results also in a broadening of the clutter notch.This effect can be compensated for by space-time-TIME processing.

Temporal decorrelation due to range walk and spatial decorrelation dueto travel delays across the aperture have the same origin (the systembandwidth) and, therefore, occur simultaneously. It appears that the rangewalk effect dominates over the spatial decorrelation.

Decorrelation may occur due to Doppler spread within the individual rangegate if the clutter Doppler is range dependent. This does not apply tosidelooking arrays. The effect of Doppler spread becomes significant onlyfor large range gates at short distance.

7. The range-Doppler matrix is a useful way of illustrating the range dependenceof the clutter Doppler frequency.

Front MatterTable of Contents4. Fully Adaptive Space-time Processors4.1 Introduction4.2 General Description4.2.1 The Optimum Adaptive Processor (OAP)4.2.2 The Orthogonal Projection Processor (OPP)

4.3 Optimum Processing and Motion Compensation4.3.1 Principle of RF Motion Compensation4.3.2 Correction Patterns4.3.3 Interrelation with the Optimum Processor

4.4 Influence of Radar Parameters4.4.1 Transmit Beamwidth4.4.2 Array and Sample Size4.4.3 Sampling Effects4.4.4 Influence of the CNR4.4.5 Bandwidth Effects4.4.6 Moving Clutter

4.5 Range-Doppler IF Matrix4.6 Summary

Index