dB

Figure 3.28: Impact of system bandwidth on clutter eigenspectra (rectangular impulseresponse, sidelooking): o Bs = 0; * Bs = 0.05; x Bs = 0.1; + Bs = 0.2

looking array. It can easily be verified that there is no obvious relation to the rule(3.55). In fact, (3.55) has been created on the basis of a sidelooking linear array.

3.3 Power spectra

of the covariance matrix using the techniques described in Section 1.3. The question iswhich power estimator is best suited for this purpose.

As carried out in Section 1.3 power spectra are obtained by multiplying a steeringvector s(<£>, / D ) , i.e., a variable signal replica, with some representation of the measureddata, for instance the estimated clutter + noise covariance matrix or its inverse. Asstated above the steering vector s((p, / D ) can be considered a beamformer cascadedwith a Doppler filter. Steering s(<£>, / D ) over the whole range of azimuth and Dopplervalues is equivalent to cascading a set of beams with a Doppler filter bank. Notice thatthe transmit beam position is fixed. Therefore, only a cross-section of the spectra atip = ipi (i.e., where transmit and receive beams coincide) reflects real radar operation.

In all subsequent plots the Doppler frequency is normalised to the PRF

(3.59)

Previous Page

dB

Figure 3.29: Impact of system bandwidth on clutter eigenspectra (rectangular impulseresponse, forward looking): o B8 = 0; * Bs = 0.05; x B8 = 0.1; + B8 = 0.2

3.3.1 Fourier spectra

The 'signal match' power estimator (1.102)

(3.60)

becomes a two-dimensional Fourier transform if the array is equispaced and the PRFis constant. A one-dimensional example of a Fourier spectrum was shown in Figure1.13 (circles). In Figure 3.16, a typical Fourier clutter spectrum for a sidelookingarray has been plotted versus azimuth ip and the normalised Doppler frequency 6 F.The maximum at ip — 90° appears at the position of the transmit beam. In projectionon the Doppler axis it points at F = 0 which is the associated clutter Doppler frequency.

The calculated clutter spectrum extends from the upper right to the lower leftcorner. It is modulated by the transmit beam pattern and by the sensor pattern whichcan be recognised by the decay of the spectrum close to <p — 0 and <p — 180 ° (left andright corner).

There are two more 'sidelobe walls' one of which extends from ip — 0 , . . . ,180 ° atF = O, the other one from F = - 0 . 6 , . . . ,0.6 at ip = 90°. Notice that these sidelobesare not a part of the clutter model in Q but just a spurious reaction of the 2-D Fourierestimator. Similar cross-shaped sidelobe patterns can be observed in SAR images asresponse to strong targets. As can be seen the spurious sidelobes hide most of the true

Normalised to the PRF / P R .

dB

Figure 3.30: Impact of system bandwidth on clutter eigenspectra (rectangularfrequency response, sidelooking): o Bs = 0; * Bs = 0.05; x Bs = 0.1; + .B8 = 0.2

clutter spectrum. Moreover, these sidelobes are not modulated by the transmit andsensor patterns which indicates that they are responses of the Fourier estimator to themain beam clutter power rather than antenna or Doppler filter sidelobes.

In Figure 3.17, the array has been tapered with a Hamming window (cos2(.)-weighting) on receive and transmit. Moreover a Hamming window has been applied inthe time dimension. Now the main beam is clearly emphasised and broadened whileall kinds of sidelobes are attenuated. The 'true space-time sidelobes' can hardly benoticed along the diagonal of the plot. The decay due to transmit and sensor patterncan be seen clearly. Nevertheless, the spurious 'sidelobe walls' are still dominant overthe true sidelobe clutter.

In conclusion, the Fourier power estimator gives a wrong impression of thetwo-dimensional clutter spectrum because it exhibits sidelobes which have not beenincluded in the clutter model. If the 2-D Fourier transform is used as a basis forprocessor design the spurious sidelobe effects have to be taken into account.

3.3.2 High-resolution spectra

In the following we discuss briefly the use of high-resolution power estimators forspace-time clutter spectrum estimation.

dB

Figure 3.31: Impact of system bandwidth on clutter eigenspectra (rectangularfrequency response, forward looking): o ,B8 = 0; * Bs = 0.05; x 5 S = 0.1;+ Bs = 0.2

3.3.2.1 Minimum Variance Estimator

The minimum variance estimator (MVE) was (1.104)

(3.61)

where Q is the clutter + noise covariance matrix and s(<p, / D ) is a signal replica orsteering vector. An MV-clutter spectrum for a sidelooking linear array is shown inFigure 3.18. Like all high-resolution techniques the MVE tries to decompose a signalinto single peaks. This leads to very realistic spectra if the signal is composed ofsingle spectral lines (or point-shaped sources) but looks a bit awkward in the case of acontinuous spectrum.

The advantage of this high-resolution spectrum is that it comes closest to the trueclutter model. There are no spurious sidelobes, and the sidelobes of the transmit andreceive beam can be noticed as clusters of peaks.7

Computations have shown (see Figure 1.13) that maximum entropy or MUSICspectra exhibit even stronger peaks than the MV-spectrum. In some cases only one

7We made use of the fact that the shape of space-time MV spectra depends very much on the numberof points used in the plot. There is always the problem of 'graphical undersampling' when applying suchhigh-resolution techniques. However, the number of points is limited - not only by the computing time. Iftoo many points are taken the plot becomes entirely black. Therefore some of the 3-D plots are plotted with121 x 121 while others have 6 1 x 6 1 only. In any case, one has to be careful when drawing conclusionsfrom details of such two-dimensional MV spectra.

dB

Figure 3.32: MV spectrum for a sidelooking array (B c — 0.2)

high peak in the look direction is visible and no realistic impression of the sidelobestructure is produced. The MVE is certainly the best compromise between suppressionof spurious sidelobes and spikyness of the spectrum.

3.4 Effect of radar parameters on interference spectra

3.4.1 Array orientation

Figure 3.19 shows an MV clutter spectrum for a linear array in the forward lookingorientation. It can clearly be seen that the footprint of the spectrum is a circle whichis in accordance with the considerations made in Section 3.1.2. In this example weassumed that the sensors have forward looking directivity patterns according to (2.20)and (2.21). Therefore, the resulting spectrum is concentrated on a semicircle at positiveDoppler frequencies. There is no clutter response at negative Doppler frequencies.However, targets with negative Doppler frequencies can be detected. The spectralcomponents in the foreground are ambiguities due to the fact that F = [-0.6,0.6].

3.4.2 Temporal and spatial sampling

As pointed out in Chapter 2 we deal with two-dimensional signals S(/D?<£>) whichare sampled in both the temporal and spatial dimensions. By temporal sampling thesuccession of echoes as given by the PRF is understood while spatial sampling is done

F

cp/°

dB

Figure 3.33: Clutter power spectrum (FL array) with range walk ( A R = 10 m)

by the antenna array, specifically by the sensor positions. In Figures 3.20-3.23, theeffect of undersampling in space and time on cuv-p spectra is illustrated. In both cases(space and time) we assumed that the sampling frequency is half the Nyquist frequency.For details on space-time sampling see ENDER and KLEMM [HO].

3.4.2.1 Temporal undersampling

In all previous examples the PRF was chosen to be 12 kHz which is the Nyquistfrequency for the clutter bandwidth for the given parameters (see Table 2.1). In thefollowing two examples we assumed PRF = 6 kHz. Figure 3.20 shows the clutterspectrum for a sidelooking linear array. As can be seen there are ambiguous clutterresponses on both sides of the 'original' spectrum; these responses are 'copies' of theoriginal spectrum shifted along the Doppler axis.

In Figure 3.21, the same situation has been plotted for a forward looking lineararray. Again the temporal ambiguity produces a second 'replica' of the originalspectrum. Now we notice that clutter portions appear at both positive and negativeDoppler frequencies (compare with Figure 3.19).

Let us recall that each Doppler frequency is associated with a certain azimuth anddepression angle, see (2.35). It is obvious from this relation that any ambiguity inazimuth causes ambiguity in Doppler frequency and vice versa. This can be seenclearly in Figure 3.22.

F(p/°

dB

Figure 3.34: MV spectrum (FL array, rectangular impulse response, Bs = 0.2)

3.4.2.2 Spatial undersampling

Figure 3.22 shows an MV clutter spectrum for sidelooking radar. The array spacingd — xi+i - Xi has been set to d = A in equations (2.30) and (2.36). As can be seen thespectrum is repeated every 90° instead of 180° for Nyquist spacing (d — A/2).

Notice that for sidelooking radar the spatial ambiguities follow the same / D - y>trajectories as in case of temporal undersampling shown in Figure 3.22. The curvatureof the trajectories is caused by the non-linear cos (^-function in the steering vector (see2.31). If the spectrum were plotted versus cos ip instead of (p (as indicated in Figure3.18) the trajectories would become straight lines.

It is obvious that in Figure 3.20 the ambiguous responses are replicas of the originalspectrum shifted in Doppler frequency while in Figure 3.22, the ambiguous responsesare shifted in azimuth.

The spatial ambiguities in the case of a forward looking array give a quite confusingimpression (Figure 3.23). There are three spectral components following semicirculartrajectories which cross each other. The first one starts at ip = -85° and ends at about<p = -15° . The second one starts at tp — -75° and ends at about (f - 60°. The thirdone starts at ip = 0° and ends at tp = 75°.

Let us recall that the improvement factor is just the inverse of the MV spectrum. Inthe discussion of the optimum processor in Chapter 4 we will come back to the problemof undersampling. The IF plots give a much clearer impression of the ambiguities thanthe 3D spectral plots.

F

<p/°

dB

Figure 3.35: MV spectrum (FL array, rectangular frequency response, Bs = 0.2)

3.4.3 Decorrelation effects

3.4.3.1 Clutter bandwidth

In Figures 3.24 and 3.25 we see the influence of clutter fluctuations due to internalmotion on the eigenspectra of sideways and forward looking radar with a linear antennaarray. Only the first 200 eigenvalues are shown. The total number amounts to 24 x 24 =576. Bc is the Doppler bandwidth of clutter normalised by the unambiguous Dopplerfrequency range, see (2.53). As can be seen the effect of clutter bandwidth results in anincrease of the number of clutter eigenvalues. The increase is independent of the arrayorientation which is clear because we deal here with a purely temporal effect.

As pointed out earlier the number of clutter eigenvalues is a measure for thecomplexity of a clutter suppression filter. We will come back to this point in thefollowing chapters. Figure 3.32, shows the MV power spectrum for SL-radar with20% relative clutter bandwidth. Obviously the increase in clutter eigenvalues is herereflected by a broadening of the spectrum (compare with 3.18) for the main beam asfor the sidelobes.

3.4.3.2 Range walk

For a sideways looking array, under 'ideal' conditions (Nyquist sampling of the clutterDoppler band in space and time, no additional decorrelation effects) the number ofclutter eigenvalues is Ne — N -f- M — 1, see (3.53). In the following we give some

F

(p/°

dB

Figure 3.36: Space-time jammer eigenspectra (SL array): o J = l ; * J = 3 ; x J = 5

examples which illustrate the dependency of the eigenspectra on temporal decorrelationeffects due to range walk as described in Section 2.5.1.2.

Only the first 200 eigenvalues are shown in Figures 3.26 and 3.27 to highlight theclutter portion of the spectra. The pulse-to-pulse correlation in the presence of rangewalk as defined by (2.55) is proportional to the overlap area in Figure 2.2. As can beseen in Figure 2.2 the correlation is strongest in the broadside direction of a sidelookingarray and a minimum for a forward looking array. This is reflected by the associatedspace-time eigenspectra. Notice the difference in spreading of the eigenspectra inFigures 3.26 and 3.27.

An MV clutter spectrum illustrating the effect of range walk is shown in Figure3.33. The width of the range bin was chosen to be 10 m. We notice that the spectrumis broadened mainly in the look direction, i.e., in the direction of the transmit beam. Incontrast, internal motion causes a broadening of the entire spectrum (see Figure 3.32).Recall that the decorrelation caused by range walk is azimuth dependent. Obviously thetransmit beam selects the individual decorrelation associated with the look direction.

3.4.3.3 System bandwidth

In Figures 3.28 and 3.29 we find the effect of system bandwidth according to (2.63),on the eigenspectra of sideways and forward looking radar. In these examples weassumed a rectangular time response for the transmitted pulse and the band limitingmatched filter, see (2.7).

The decorrelation due to travel delays across the array aperture was given by (2.64),

dB

Figure 3.37: Space-time jammer + clutter eigenspectra (SL array)o J = 1; * J = 3; xJ = 5

for system characteristics with rectangular time response

(3.62)

For rectangular frequency response the correlation was given by (2.64).A wave coming from direction (ip, 9) travelling across the array is delayed between

any two sensors by (2.7):

(3.63)

If the array moves at speed vp we get instead

(3.64)

Under DPCA conditions, i.e., for a sidelooking arrays and Xi+\ — Xi — 2Tvp weget for an equispaced array certain combinations of x«, Xk and 2mTvp for which theargument X{ + 2mTvp — Xk in (3.64) becomes 0. In these cases of phase coincidencefull correlation is obtained. The points of full correlation are identical to the unityelements in (3.43).

Transmit pulses with rectangular impulse response are very common in pulseDoppler radars. The frequency response of such a pulse is a sin x/x-function. By'bandwidth' we understand here the width of the main lobe (from null to null) of this

dB

Figure 3.38: MV jammer + clutter spectrum (SL array, J = 1)

function. However, the sidelobes of the sin a;/#-function are rather high so that theband limits are not very distinct.

In Figure 3.28, the effect of phase coincidence on the eigenspectra for a sidelookingarray in DPCA mode is shown. As can be seen the number of clutter eigenvalues is notincreased beyond TV + M — 1. There are slight differences between eigenspectra belowN + M — 1 which however is not important for clutter rejection.8

In the case of a forward looking array all sensors have the same ^-coordinateand different y-coordinates. It follows from (3.64) that no DPCA effect (phaseconincidences through platform motion) can occur for forward looking arrays. Asa consequence the correlation between sensors is strongly effected by the systembandwidth (3.62). This is reflected in an increase of the number of clutter eigenvaluesbeyond the limit N + M - 1, see Figure 3.29.

Similar results are obtained for rectangular frequency response of the antennachannels. Again no increase of clutter eigenvalues can be noticed for the sidelookingarray while for forward looking radar there is a considerable increase with increasingsystem bandwidth. However, compared with rectangular impulse response the increaseof eigenvalues is much less severe because there are no significant sidelobes outside thebandwidth.

Figure 3.34, shows an MV spectrum for 20% relative system bandwidth withrectangular impulse response. As pointed out above the associated frequency response

8See Chapters 4 and 13.

F

<p/°

dB

Figure 3.39: MVjammer + clutter spectrum (FL array, J — 3)

has quite high sidelobes which leads to a rather irregular spectrum.9

If instead the system bandwidth is strictly rectangular no significant sidelobes showup outside the clutter ridge. The example in Figure 3.35, reflects clearly the rectangularshape of the system bandpass. Notice that the clutter ridge is quite narrow at broadside(ip = 0°) and broadens for off-broadside angles on both sides. This has to do with thefact that the travel delays are large at endfire and small at the broadside direction.

It should be pointed out that these considerations cover only the spatial propertiesof STAP used with broadband arrays. There is an additional spectral effect. The signalbandwidth produces a Doppler bandwidth which is largest in the forward direction andsmaller as the direction gets closer to endfire. This Doppler effect is large right wherethe spatial decorrelation effect is small, and vice versa.

For reasons of brevity we do not show the effect of system bandwidth on the clutterspectrum for sidelooking radar. As there is almost no system bandwidth effect on theMTI performance of sidelooking radar these spectra look like the one shown in Figure3.18.

9If such a spectrum were plotted by use of the two-dimensional Fourier transform (3.60) it would bedifficult to verify which of the sidelobes come from the system bandpass and which of them are spuriousresponses of the Fourier estimator.

F

(p/°

3.4.4 Clutter and jammer spectra

Jammers are assumed to be Doppler broadband and discrete in space. The jammercovariance matrix is block diagonal with spatial submatrices (see Figure 3.9, andequations (2.45-2.48)). If only one jammer is present each of the submatrices is adyadic and produces one eigenvalue of the covariance matrix. Accordingly, as we haveM submatrices in the space-time matrix the total number of eigenvalues due to onejammer amounts to M. For J jammers the number of jammer eigenvalues becomesJM.

This can be seen in the numerical example Figure 3.36. The number of temporalsamples is M — 24. Therefore, the number of jammer eigenvalues is 24 for onejammer, 72 for three jammers, and 120 for five jammers (the power was assumed to beequal for all jammers). If we have a superposition of clutter and jammers the numberof interference eigenvalues becomes for a sidelooking array and DPCA conditions

(3.65)

(3.66)

(RICHARDSON [430]), where 7 is the ratio of Doppler ambiguities to spatialambiguities (see (3.56)).

The fact that a broadband jammer produces M eigenvalues of a space-timecovariance matrix may have significant influence on the required number of degreesof freedom of the clutter canceller, in particular when suboptimum techniques basedon the interference subspace are under discussion.

In Figures 3.38 and 3.39 two examples for a superposition of clutter and jammersare given. The jammers appear as thin walls along the F-axis.

It should be noted that the considerations concerning system bandwidth effectsmade above hold for jammers as well. Normally, even in ground-based radarsystems, cancellation of off-broadside jammers is strongly degraded by a large systembandwidth. This can be compensated for by space-TIME processing where TIMEmeans the time associated with wave propagation. 10

3.5 Aspects of adaptive space-time clutter rejection

3,5.1 Illustration of the principle

The principle of space-time clutter rejection by space-time filtering is illustrated inFigure 3.40. In this example the sidelooking geometry was chosen. One recognisesthe clutter spectrum which runs along the diagonal of the azimuth-Doppler plane. Theclutter spectrum is modulated by the transmit directivity pattern and the sensor pattern.Let us consider some ways of suppressing this clutter spectrum.

10The fast radar time which corresponds to range.

see Figure 3.37.Equation 3.65 is a special case of

3.5.1.1 Optimum temporal filtering

In the background, filter characteristics can be seen; this is referred to as inversetemporal clutter filter. This filter has been obtained by projecting the clutter spectrumon to the Doppler axis and taking the inverse. Such an inverse filter is the optimumtemporal clutter canceller. Such temporal filters are used in ground-based radar whereno motion-induced Doppler spread of the clutter spectrum occurs. As one can see thereis a broad stop band which is determined by the transmit main beam. A fast target in themain beam may fall into the Doppler sidelobe domain of the inverse filter and, hence,may be detected. A slow target may fall into the stop band and will be rejected by thetemporal filter.

3.5.1.2 Optimum spatial filtering

On the left we have a similar situation in the spatial dimension. Here we find again afilter based on to the inverse clutter spectrum projected on to the azimuth axis. Againthe stop band is determined by the transmit beamwidth. However, as the radar beamlooks at the clutter background this stop band is located in the beam look direction.The adaptive spatial clutter filter makes the radar blind!

3.5.1.3 Space-time adaptive filtering

In the left lower corner a cross-section of the transmission characteristics of a space-time clutter filter is depicted. A space-time filter operates in the whole ip-uj plane andthereby recognises that the clutter spectrum is broad when looking from the lower rightcorner into the plot. It is quite narrow if one looks from the lower left side into theplot. Therefore, the filter forms a very narrow ditch along the trajectory of the clutterspectrum so that even slow targets may fall into the pass band and can be detected.

Another view of the problem is as follows: Clutter spectra of stationary radar arenarrowband (let us forget about clutter fluctuations for the time being). The platformvelocity causes a direction-dependent Doppler colouring of clutter echoes so that thespectrum is spread out along the diagonal of an azimuth-Doppler plane (instead ofalong a Doppler axis only). Since a space-time clutter suppression filter can operatein the (p-0Jr> plane the motion-induced Doppler effect has no influence on the clutterrejection performance. This may be considered as the motion compensation capabilityof the space-time filter. In other words, if the right processing is applied there is nolimitation in MTI performance caused by the radar platform motion.

3.5.2 Some conclusions

In the following some concluding remarks are summarised which may serve asguidelines for the further development of space-time processing concepts.

Figure 3.40: Principle of space-time clutter filtering (sidelooking array antenna)

3.5,2.1 Bandwidth limitations

Clutter echoes as defined by (2.38) are band limited in both the spatial and the Dopplerdomain. The Doppler frequency is defined by the platform velocity

where /3 was the look direction relative to the array axis, see Figure 3.2. The clutterDoppler bandwidth is therefore

The spatial frequency of clutter is

so that the spatial bandwidth becomes

Let us make a general remark on the processing of space-time signal fields. As longas such clutter echoes are sampled properly in space and time, i.e., at the Nyquist rateor higher, no information loss through the sampling process will occur and therefore

(3.69)

(3.70)

(3.68)

(3.67)

inverse temporal clutter filter

stopband

inversespatialclutterfilter

stopband fast target

slow target

space-timeclutter filter

clutter notch cosDoppler-azimuthspectrum

any kind of discrete signal processing (spectral analysis, filtering) can be principallyapplied to such data.

Historically the idea of motion compensation led to the development of DPCAsystems. It should be noted, however, that the DPCA effect (motion compensationthrough displaced phase centres, RICHARDSON [426], RICHARDSON and HAYWARD

[427]) is not required for explaining the principle of adaptive airborne cluttercancellation.

If the DPCA effect were necessary to explain the clutter cancellation operation thenonly sidelooking linear equispaced arrays would be suitable for airborne MTI radar.For example, there is no DPCA effect for a forward looking array. The spectra shownabove demonstrate clearly that this is not the case. Moreover, we will show in Chapter8 that the airborne MTI function can be combined with more complex antenna arraysfor which no DPCA conditions can be defined.

3.5.2.2 The role of DPCA in space-time adaptive processing

The DPCA effect plays an important role in space-time radar signal processing.It makes the sidelooking array configuration a preferable choice, but not from theviewpoint of motion compensation!

The nature of the complex envelope A(u) in (3.41) was not specified. We candistinguish between three cases in which arrivals from different directions are mutuallyuncorrelated, i.e., they fulfil the condition (3.40). The covariance matrix assumes theform (3.42).

• Clutter suppression for narrowband radar. We assume that by virtueof the reflection process the received amplitude is slowly varying so thatsubsequent echoes are coherent, however the arrivals from different directionsare independent on a long-term basis. The DPCA effect is not required forspace-time processing in narrowband systems. Since clutter echoes are band-limited in the spatial and temporal dimensions adaptive filtering will be capableof suppressing clutter returns for any kind of array geometry. Of course, cluttersuppression is limited if the number of degrees of freedom of the radar (JV, M)is smaller than the number of clutter eigenvalues of the space-time covariancematrix.

• Clutter suppression for broadband radar. The DPCA effect compensates forspatial decorrelation of clutter returns for broadband sidelooking radar. As theDPCA effect is based on the platform velocity in this respect the platform motionturns out to be an advantage compared with ground-based radar.

• Forward looking radar. Since for forward looking arrays no DPCA effectexists the effects of system bandwidth can be cancelled by space-TIMEprocessing for jammer cancellation and space-time-TIME processing for clutterrejection. Alternatively, a CPCT configuration (see Section 3.2.2.4, andSKOLNIK [468, pp. 16-20])) may be used for compensation for systembandwidth effects in forward looking antennas. Basically no losses have to beencountered if the appropriate antenna configuration and processing is applied.

• Clutter bandwidth. There is no way of compensating for temporal decorrelationcaused by fluctuations of the clutter background.

3.6 Summary

As an introduction, some general properties of airborne clutter such as isodops,azimuth-Doppler trajectories, and Doppler dependence of clutter are derived. Thevector quantities for signal, clutter, jamming, and noise are extended to space-timevector quantities. By means of the models derived in Chapter 2 space-time clutter+ noise covariance matrices can be developed. By use of the space-time models fortarget signals in Chapter 2 a steering vector can be defined by which spectral analysisof clutter echoes can be carried out. In detail the contents of this chapter can besummarised as listed below:

1. For horizontal flight and planar ground the isodops (curves of constant Dopplerfrequency on the ground) are hyperbolas.

2. For a sidelooking linear array the beam traces on the ground are hyperbolaswhich coincide with the isodops. Therefore, the clutter Doppler is rangeindependent.

3. For a forward looking linear array the beam trace hyperbolas are rotated by90° so that beam trace and isodop hyperbolas cross each other, especially at nearrange. Therefore, the clutter Doppler frequency is range dependent.

4. In practice the clutter covariance matrix is unknown and has to be estimated byuse of secondary data. Secondary data are usually obtained from neighbouringrange bins. If the clutter Doppler is range dependent (as is the case for all butsidelooking arrays) the clutter spectrum is broadened which results in degradeddetectability of low Doppler targets (BORSARI [49]). Some improvement canbe obtained by Doppler warping (compensation for the Doppler gradient) of thedata, see BORSARI [49] and KREYENKAMP [291]).

5. Space-time clutter covariance matrix. If

• the clutter background is homogeneous and

• illuminated with an omnidirectional antenna and

• the array has half-wavelength spacing between elements (spatial Nyquistsampling)

clutter is spatially white, and the spatial clutter covariance matrix becomesa identity matrix. No clutter suppression is possible. Higher correlationcomponents appear if a space-time covariance matrix is used.

6. The number of clutter eigenvalues of a linear sidelooking array is N + M - 1under DPCA conditions. For a forward looking array the number is about 2(TV +M - I ) . If only one half-plane is illuminated (or directive sensors are used) thenumber of clutter eigenvalues is about N + M - 1 for both configurations.

7. A 2-D Fourier azimuth-Doppler clutter spectrum exhibits high spurioussidelobes which might give a wrong impression of the true clutter spectrum.

8. The minimum variance estimator generates clutter spectra that are very closeto the clutter model used.

9. For a sidelooking array the clutter power is concentrated on the diagonal acrossthe cos Lp-F plane while for forward looking radar the clutter is on a circle. Thecos ip-F clutter trajectories are examples of the well-known Lissajou patterns.

10. Spatial and temporal undersampling lead to spatial and temporal ambiguities,respectively.

11. For sidelooking radar DPCA is a zero noise approximation of the optimumprocessor.

12. Clutter fluctuations lead to a broadening of the clutter spectrum that is constantfor all Doppler frequencies.

13. Range walk as occurs mainly in high-resolution radar leads to temporaldecorrelation and, hence, to a broadening of the clutter spectrum. The numberof clutter eigenvalues increases. This effect is stronger for forward than forsideways looking radar.

14. The system bandwidth effect causes a broadening of the clutter spectrumproportional to the off-boresight look angle.

15. The principles of spatial, temporal and space-time clutter rejections arecompared. Only space-time clutter filtering has the capability of slow targetdetection.

16. The DPCA effect is not the physical mechanism which makes space-time clutterfiltering possible. Clutter echoes are two-dimensional (spatial-temporal) signalswhich are strictly band-limited in both dimensions. When sampled at the Nyquistrate these signals can be filtered in arbitrary ways. The potential of slow targetdetection lies in the fact that the clutter trajectory is a narrow line (straight,elliptic, circular).

17. DPCA is not the basis of space-time processing. It is merely a special casewhich for sidelooking linear arrays and appropriate synchronisation of PRF, v p

and spacing d, may achieve near-optimum clutter rejection.

18. In contrast to the DPCA techniques, space-time adaptive processing is almostindependent of the PRF. Of course, if the PRF falls below the Nyquist rateambiguous clutter notches occur.