Chapter 9

Space-frequency processing

9.1 IntroductionIn the previous chapters several space-time processing algorithms were presentedwhich are capable of adaptive real-time processing. In this chapter some considerationon space-frequency domain processingx are made. We discuss a total of five differentprocessing schemes. Some of them are variants of the space-time processors describedin Chapters 5, 6, and 7. AU processors presented will be compared in terms ofperformance. A comparison in terms of computational complexity will be presentedin Chapter 15. WRIGHT and WELLS [563] compare pre-Doppler and post-Dopplerarchitectures for application with space-based radar.

The near-optimum techniques described in the previous chapters have been derivedfrom the optimum receiver (1.3), which in essence means pre-whitening of the cluttercomponent in the received echo signals and matching to the desired target signal. It hasbeen shown in Section 1.2.6 that the optimum test can be formulated in the frequencydomain as well as in the time domain. Now we have to extend this concept to space-time vector quantities.

The discrete Fourier transform can be written as a unitary matrix F. Since wewant to transform from time to frequency while not changing the space dimension thespace-time Fourier transform becomes

(9.1)

where I is the spatial NxN unit matrix.2 The coefficients wnm are given by (1.94)

1 Frequently referred to as post-Doppler processing.

2Or K x K according to the processors in Chapter 6.

(9.2)

which is obviously identical with y = x*Q 1S since F*F = FF* = I.

Then the optimum space-time processing becomes in the frequency domain

and the space-time vectors of the desired target signal and received echo signal become

By similarity transform the clutter covariance matrix Q becomes a power spectralmatrix

(9.4)

(9.5)

(9.3)

Figure 9.1: 2-D symmetric auxiliary sensor/echo processor (ASEP)

main channel dataauxiliary echo samples

weighted sums(doppler filter)

invers

e of

sp

ace-t

ime

cova

rianc

e m

atrix

spac

e-freq

uenc

y sign

al m

atch

testfunction

shift register(space-timeecho data)

mainchannel

auxiliarychannels

auxiliarychannels

beamformer

Figure 9.2: 2-D symmetric auxiliary sensor/echo processor, with signal contributionsby auxiliary sensors/data omitted (ASEP)

This well-known fact encourages us to look in more detail into Doppler frequencydomain approaches for real-time adaptive air- and spaceborne clutter suppression.While we are dealing with space-time vectors and matrices it should be noted that someof the techniques work in the space-frequency domain, i.e., a one-dimensional Fouriertransform will be applied in the time dimension of the space-time filter. Two othertechniques work in the angular-Doppler domain, which means the space-time data willbe transformed by a two-dimensional Fourier transform before adaptive processing.Five different techniques are described, two of which are new variants of the auxiliarysensor FIR filter (ASFF) processor described in Chapter 7.

9.2 The auxiliary space-time channel processor (ACP)This processor has already been discussed in Chapter 5, Section 5.3, and Figures 5.6and 5.7. Recall that a bunch of beams is distributed over the entire angular clutter

main channel dataauxiliary echo samples

weighted sums(doppler filter)

cente

r row

of

inv

erse o

fsp

ace-f

reque

ncy c

ovar

iance

m

atrix

testfunction

shift register(space-timeecho data)

mainchannel

auxiliarychannels

auxiliarychannels

beamformer

Figure 93: Frequency domain space-time FIR filter (FDFF)

domain so that all clutter is received by full gain beams.To match the clutter in angle and frequency each of the beams is cascaded with a

doppler filter matched to the respective clutter doppler frequency. A search channel (inpractice a beamformer cascaded with a doppler filter bank) is added for matching thedesired target.

Estimation of the clutter covariance matrix and clutter filtering takes place in thebeam/doppler channel domain. Therefore, this technique belongs to the family ofspace-frequency processors and has been mentioned here for the sake of completeness.

9.3 The symmetric auxiliary sensor/echo processorThe processor described below is based on auxiliary sensors and auxiliary echosamples. It is referred to as auxiliary sensor/echo processor (ASEP). Let us recall the

space-timeFIR filter spectrum

ordoppler dependant

spatial filtersspace-frequency spatial sums

echo datamemory

testfunction

estimation of space-time FIR filter coefficients

mainchannel

auxiliarychannels

beamformerauxiliarychannels

considerations on the symmetric auxiliary sensor processor made in Section 6.2.1. Theresults given there have shown that this structure is quasi-optimum in terms of slowtarget detection. In this section we try to extend this concept to the time dimension.

In Figure 7.5, the symmetric auxiliary sensor/space-time FIR filter processor wasshown. Let us now modify this scheme by replacing the FIR filter by an symmetricauxiliary data scheme as used in the spatial domain in the upper part of Figure 7.5.Recall that in (6.17) the symmetric auxiliary sensor transform was described by a

Figure 9.4: Frequency-dependent spatial filter (FDSP)

space-timeFIR filter spectrum

ordoppler dependant

spatial filtersspace-frequency spatial sums

echo datamemory

testfunction

estimation of doppler dependant spatial filters

mainchannel

auxiliarychannels

beamformerauxiliarychannels

L is the number of temporal channels after the transform, w / m is the Fourier phasecoefficient according to (9.2), with / denoting the Doppler frequency and m the time(index of successive PRIs). The processor and the improvement factor are given by(5.3), (5.4) and (5.5), respectively.

Figure 9.1 shows a block diagram of this processor. On the top we again havethe symmetric auxiliary sensor configuration which was discussed in some detail inSection 6.2.1. K-I auxilary sensors are arranged in a symmetric fashion on bothsides of a beamformer with beamformer weights b{. This step is described by thespatial transform (9.6).

Recall that the symmetric auxiliary channel structure as defined by (9.6) isequivalent to overlapping subarrays (6.8). The PRI-staggered post-Doppler processorby WARD and STEINHARDT [532] generates additional temporal degrees of freedomby forming several Doppler filters for each Doppler frequency, all being based on Ldifferent data windows. These windows are used to form overlapping data segmentswhich is a time domain equivalent to overlapping subarrays in the spatial domain. Theprocessor is equivalent to that in Figures 9.1 and 9.2. According to the authors this

(9.7)

where the b{ are beamformer weights. Extending this scheme to the time dimensionleads to a transform matrix of the form

(9.6)

spatial transform matrix of the form

technique is robust against tolerances of the element patterns bcause it belongs to theclass of post-Doppler processors.

The output data are stored in a K x M data memory. The data in the shadowedpart of the memory are fed into Doppler filters with equal Doppler frequency while thewhite data serve as temporal auxiliary channels. The sums in each of the columnsindicate the Doppler filters, that is, all the shadowed data in each of the columnsare weighted with Doppler filter coefficients before summation. Notice that wearranged here in the time domain the same auxiliary channel configuration as in thespatial dimension on top of the drawing. The Doppler filters assume the role of thebeamformer while echo data before and after the Doppler filter serve as 'auxiliarychannels'.

The number of output channels is now reduced in both the spatial and temporaldimension. At this level the adaptation (e.g. estimation of the space-frequencycovariance matrix) has to be carried out. The size of the space-frequency covariancematrix is KL x KL (in our example 25 x 25).

In practice we want to have the numbers of auxiliary sensors and auxiliary echodata small compared with the number of beamformer elements, or the length of theecho data record, respectively. Therefore, according to Section 6.2.1.3, the target signalcontributions of the auxiliary sensors and echo samples may be neglected. Then thetransformed (or secondary) signal matching vector

(9.8)

(9.9)

(9.10)

reduces to

The adaptive processor

reduces to the centre column of Q T X . This simplified processor is shown in Figure9.2. Since beamforming and Doppler filtering has been included in the pre-transformno further Doppler filter bank is required.

9.3.1 Computing the inverses of the spectral covariance matricesThe adaptive processing, that is, calculating the centre row of Q ^ 1 , requires theestimation and inversion of the space-time covariance matrices for all Dopplerfrequencies of interest.

Notice that only the main channel (shadowed) depends on the Doppler frequency.The auxiliary samples on the top and on the bottom of the memory (white) are the

IF[ClB]

Figure 9.5: Comparison of space-frequency processors (sidelooking array, y? L = 45):o OAP; * ASEP; x ACP; + ASFF; N = M = 24, K = L = 5

same for all Doppler frequencies. Therefore, the space-time clutter covariance matricesassociated with different Doppler frequencies differ only in the centre column and row.This property can be exploited by a dedicated algorithm for efficient calculation of thematrix inverses. This algorithm is based on matrix partitioning and has been describedin Chapter 5, Section 5.5.

9.4 Frequency domain FIR filter (FDFF)Let us again come back to the auxiliary sensors FIR filter (ASFF) described in (7.11),(7.14) and Figure 7.5. Notice that the FIR filter operation involves a convolution in thetime dimension. The temporal convolution of the space-time least squares FIR filterwith the space-time data sequence can basically be carried out in the frequency domainby multiplying the frequency response of the FIR filter with the spectra of the radardata (fast convolution).

Such a processor is depicted in Figure 9.3. The coefficients of the space-timeFIR filter are estimated in the time-domain, and a set of spectral filter coefficientsis calculated by applying a multichannel DFT (space-time filter spectrum). For eachfrequency the contributions of the K antenna channels are then summed up, and theresulting filtered spectrum is fed into the usual detection device.

On the one hand this approach needs a multichannel FFT instead of a single channelFFT; on the other hand the stationarity of the received echo sequences is exploitedin that only one set of filter coefficients must be estimated for all frequencies. This

F

IF[ClB]

Figure 9.6: Comparison of space-frequency processors (forward looking array, ip L =45): o OAP; * ASEP; x ACP; + ASFF; N = M = 24, K = L = 5

frequency domain processor is equivalent to the space-time FIR filter shown in Figure7.5, and described in (7.11) or (7.14).

It has been found in Chapter 7 that the clutter suppression performance is nearlyindependent of the temporal dimension of the space-time FIR filter. Under certainconditions additional degrees of freedom may be desirable (e.g. to compensate fortolerances of the sensor positions and for other array errors). The number of degreesof freedom can be increased by choosing a larger temporal filter length L. Increasingthe temporal filter dimension requires additional operations if the filtering is carriedout in the time domain. For a frequency domain FIR filter the computational expenseis independent of the filter length as long as the clutter filter is shorter than the numberof Doppler cells.

9.5 Frequency-dependent spatial processing (FDSP)A similar processor can be designed when the estimation of the filter coefficients fromclutter data is carried out in the frequency domain, that means, after the FFT, seeFigure 9.4. This filter is based on the frequency-dependent space-frequency covariancematrices which are based on adjacent Doppler bins and array elements. These matriceshave the dimension KL x KL where L means here a certain number of FFT channels.This kind of processing is similar to the JDL-GLR by WANG H. and CAI [512] whichwill be addressed further below.

Such post-Doppler processors tend to be more robust against errors of the sensor

F

Figure 9.7: Frequency-dependent spatial filtering (IF vs normalised Doppler; SL, CNR= 20 dB): a. M = 16; b. M = 64; c. M = 256; d. Af = 1048

patterns (see WARD and STEINHARDT [532]). The individual Doppler filters providenarrowband processing which is equivalent to using only a small sector of the elementpattern. Therefore, the Doppler spectral effects caused by different sensor patterns ismitigated.

An attractive processor is found for L 1 which means that spatial cluttercancellation filters are cascaded with the individual Doppler channels. The spectralcross-terms of the space-frequency clutter covariance matrix are not taken into accountfor calculating the space-frequency clutter filter. This is a significant simplification ofthe processing based on the fact that signals at the outputs of different Fourier channelstend to become mutually uncorrelated as the observation time MT approaches infinity.

This frequency-dependent spatial clutter cancellation technique has been appliedfor instance in the MCARM program by SURESH BABU et al [478] and FENNER etal? [132], GOLDSTEIN et al [159] using MOUNTAINTOP data T m [494], and in theAER II program by ENDER [104]. Several techniques based on a Doppler filter bankare presented by BAO et al. [26]. A comparison of frequency-dependent spatial cluttercancellation ('factored STAP') and DPCA for use on a spaceborne platform is givenby NOHARA [385]. A statistical analysis of this 'partially adaptive STAP detector' forDoppler target detection and azimuth estimation is given by REED et al. [422].

Some authors (WANG X. [519], WANG Y. et al. [522], WANG Y. and PENG3The authors refer to this as Doppler-factored STAR

c. d.

b.a.

Figure 9.8: Frequency-dependent spatial filtering (IF vs normalised Doppler; SL, CNR= 40 dB): a. M = 16; b. M = 64; c. M = 256; d. M = 1048

[520], Wu R. et al [565], XlONG et al [569]) combine frequency-dependent spatialfiltering with a temporal MTI pre-filter in all antenna channels. This pre-filter is toreduce mainbeam clutter. It can be anticipated that such a conventional temporal MTIfilter will not yield sufficient improvement in slow target detection. If the pre-filter ismatched to the clutter bandwidth determined by the main beam this leads to a broadstop band of the filter (compare with Figure 3.40). Alternatively, a narrow pre-filterwill not be able to cancel the mainbeam clutter sufficiently. As a consequence theclutter notch will be broadened which results in degraded detection of slow targets.Such results are shown in [520].

BAO et al. [25] proposed a STAP processor based on frequency-dependent spatialfiltering. However, they apply a conventional three-pulse canceller to the individualantenna channel prior to Doppler filtering for cancelling the main beam clutter. Asimilar receiver structure is proposed by WANG et al [520]. It can be expected that thebroad notch of the three-pulse canceller dominates the clutter rejection performance.Degradation in slow target detection can be anticipated.

The time-space cascaded STAP architecture by BRENNAN et al [57] is acombination of frequency-dependent spatial processing with the two-pulse delaycanceller principle. Experiments with NRL data (Section 1.1.6) have shown that thistechnique approximates the optimum processor quite well. Following this line the'post-Doppler STAP processor' analysed by COOPER [83] is a frequency-dependent

a. b.

c. d.

where now the indices of the spatial submatrices Qnm denote the Doppler frequency. Afrequency-dependent spatial filter is obtained by omitting the cross-variance matrices

4In this application the discrete Fourier transform might be efficiently computed by use of a recursivealgorithm (DlLLARD [94]).

(9.11)

Figure 9.9: Angle-Doppler adaptive processor

space-time processor with three delays.4

The space-frequency clutter covariance matrix as given by (9.3) has the same formas the space-time covariance matrix Q in (3.22)

testfunction

Angle-Doppler bin grouping

2-D discrete Fourier transform

temporalsamples

temporalsamples

temporalsamples

spatial samples

in (9.11) and taking the inverse

(9.12)

(9.13)

(9.14)

so that the processor becomes

where

is the Fourier spectrum of the target signal replica.If the target signal is a sine wave as assumed in (2.31), the transformed signal vector

has the form

where we assumed that the target signal frequency denoted by the index m coincideswith the frequency of the ra-th channel of the discrete Fourier transform matrix. Theprocessor then reduces to

(9.15)

(9.16)

(9.17)

According to (5.5) the improvement factor is

Inserting (9.16) into (9.17) leads to the following expression for the improvement factor

Since s m is a spatial signal vector except for a factor of \[M according to (1.94), anda complex phase factor it can be replaced by a beamformer vector

(9.18)

(9.19)

9.5.1 Spatial blocking matricesSince the space-time clutter filtering has been reduced to a sequence of spatial filtersassociated with different Doppler frequencies spatial blocking matrices according toSection 6.3.1 may be applied to each Doppler channel in order to mitigate the effect oftarget signal cancellation if the target signal is included in the adaptation.

9.6 Comparison of processorsIn Figures 9.5 and 9.6 the IF has been plotted versus the normalised target dopplerfrequency as in the previous chapters. The individual curves are associated withdifferent frequency domain processors. The curves in Figure 9.5 have been plottedfor a sidelooking antenna configuration, while Figure 9.6 shows the forward lookingcase.

As can be seen the performance of all processors is very similar. The usual clutternotch occurs at the clutter Doppler frequency in the look direction (clutter match of theprocessor). All four curves approximate the optimum IF very well.

Figures 9.7 and 9.8 show the performance of the frequency-dependent spatialfiltering technique (FDSF) in its dependence on the number of echo samples (M =16,64, 256,1048). Figure 9.7 has been calculated for CNR = 20 dB, Figure 9.8 forCNR = 40 dB. A sidelooking array was assumed. Similar results can be obtained forforward looking radar.

Recall that the theoretical maximum of the improvement factor is I F m a x = CNR NM9 i.e., it is proportional to the number of echoes. The four curves in Figure 9.7have been normalised to their individual I F m a x .

As can be seen from Figures 9.7 and 9.8 the optimum IF is quite well approximatedfor M > 256 (curves c). For shorter pulse sequences some losses can be observed.These losses are the penalty for assuming that the clutter power spectral matrix (9.3)of finite order is block diagonal. The block diagonal form is perfectly reached whenM > oo.

9.7 Angle-Doppler subgroups9.7.1 General descriptionThere is one more option of designing space-time adaptive clutter cancellers. Theprinciple is illustrated in Figure 9.9. The space-time echo samples are transformed bya two-dimensional Fourier transform into the angle-Doppler domain. The total numberof angle-Doppler cells is subdivided into two-dimensional groups of angle-Dopplercells. Clutter suppression will be carried out for each of the subgroups separately.

The individual way of grouping is at the choice of the designer. One straightforwardchoice is to use adjacent cells in both the angular and Doppler dimensions. In thespatial dimension subarray beamforming may be carried out in the RF domain. If afully digitised array is available subarray beamforming can be done digitally.

The principle of adaptive subgroup processing in the Doppler domain has beendiscussed for one-dimensional adaptive clutter suppression by KLEMM [230]. Thedoppler domain is subdivided into small subgroups of adjacent Doppler channels. Theadaptive processing is applied in parallel or successively to these subgroups. It wasfound that the achievable performance depends on the subgroup size. Groups of twoDoppler bins only are particularly attractive because the spectral covariance matriceshave size 2 x 2 so that matrix inversion does not require any arithmetic operations.The real-time STAP technique by MEYER-HILBERG [356] also belongs to the class offrequency domain approaches.

The Joint Domain Localised Generalised Likelihood Ratio Detector (JDL-GLR)by WANG H. and CAl [512] is a prominent example for an extension of this kindof processing to space-time signals received by an airborne radar. The authorsdemonstrate that joint space-time processing is superior to cascaded space-time ortime-space processing. In comparison with the optimum processor, however, somelosses are encountered. However, when operating on measured data (MCARM) theJDL-GLR showed favourable properties in terms of clutter suppression performanceand required secondary data support (MELVIN and HIMED [348]).

Some improvement can be obtained if the pre-calculated steering vectors arereplaced by measured sets of coefficients so as to compensate for sensor directivitypatterns and mutual coupling between sensors (ADVE and WiCKS [4, 7]).

The JDL space-time adaptive processor can be optimised so as to achieve good lowDoppler target detection performance with minimum support of secondary trainingdata (PADOS et al [398]). Two variants are shown. The first is suitable when theeigenspectrum is spread out whereas the second one is more adapted to cases wherethe eigenspectrum is concentrated on a few dominant eigenvalues.

A similar processing scheme has also been used by the authors for the evaluationof the S-A processor [515]. WICKS et #/.[544] compare the performance of theJDL-GLR with a DPCA based radar. It is shown that the JDL-GLR outperforms theDPCA processor, especially when the PRI is not matched to the platform velocity5 (seeChapter 4, Figure 4.17).

In the paper by COOPER [83] an 'adjacent Doppler bin processor' is discussed.This processor has an architecture similar to the JDL-GLR by H. WANG. The'M-CAP' processor by Y. WANG et al. [525] uses all array channels but only afew adjacent Doppler bins, which is a special case of the JDL-GLR. The authorsdemonstrate robustness against channel errors and good performance for variousantenna orientations.

9.7.2 Comparison with other techniquesComparing the angle-Doppler subgroup techniques a few observations can be made:

1. Let us compare the angle-Doppler subgroup techniques with the auxiliary sensorFIR filter (ASFF) approach, see Figure 7.5. It should be noted that the ASFFrequires fewer arithmetic operations for adaptation because only one small

5 Like other adaptive processors the JDL-GLR will work also in the forward looking arrangement. This

is not the case for a non-adaptive DPCA.

space-time covariance matrix has to be updated and inverted. This advantagefollows from the assumption of temporal stationarity which is given if the PRFis constant during the observation and the radar moves at constant speed on astraight course.

2. The ASFF seems to approximate the optimum processor better than the angle-Doppler subgroup technique (compare the results in Chapter 7 with those givenin [512]).6 On the other hand, as the angle-Doppler subgroup techniques donot assume temporal stationarity they might work also with staggered PRF. Ofcourse, in this case the Fourier transform cannot make use of the FFT algorithm.

3. The number of different spectral processors offers more degrees of freedom thanthe ASFF does. This may be an advantage over the ASFF if errors in the antennachannel have to be compensated for.

4. If the number of elements of each angle-Doppler subgroup is set equal to 1(one beamformer, one Doppler bin) then the angle-Doppler subgroup techniquebecomes identical to the frequency-dependent spatial filtering techniquedescribed in Section 9.5.

5. If the number of elements of each angle-Doppler subgroup is set equal toNM (full gain beamformers, full gain Doppler filters) then the angle-Dopplersubgroup technique becomes identical to the space-time auxiliary channeltechnique discussed in Chapter 5, Section 5.5.3.

6. We can conclude from 4 and 5 that the performance of the angle-Dopplersubgroup technique will be somewhere between the auxiliary space-time channelprocessor and the frequency-dependent spatial filtering technique. The firsttechnique is sensitive to additional degrees of freedom through bandwidth effectsor antenna channel errors. The second technique is close to optimum only forlarge data sequences (M > 256).

9.7.3 Other post-Doppler techniquesPost-Doppler STAP processors beyond those discussed in this chapter have beendescribed by several authors, for instance SHAW and MCAULEY [462], WARD [530,pp. 95 - 153] and DAY [92]. Su and ZHOU [473] describe a post-Doppler generalisedsidelobe canceller using eigenvectors of Doppler dependent spatial covariance matricesas auxiliary channels.

The processor suggested by SUN et al. [474, 475] includes a Doppler filter bankfor each antenna element and several Doppler filters cascaded with the main beam.The covariance matrix used for clutter cancellation includes the antenna elements ata certain target Doppler frequency plus the main beam Doppler filter outputs. Thetechnique is supposed to be more robust against array errors than the ACP (Section5.3).

6AcUmIIy this is difficult to compare. The authors calculate Ft, versus Doppler while we use the IFversus Doppler.

9.8 SummaryIn the previous chapters various techniques for space-time clutter rejection wereanalysed. Chapter 9 deals with some possible space-frequency approaches. Amonga large variety of possible receiver structures (there are no limits to the fantasy of thesystem designer) the following techniques have been identified:

1. The auxiliary channel processor described in Chapter 5 is in essence a space-frequency domain approach.

2. The concepts of overlapping subarrays or symmetric auxiliary sensors forreducing the signal vector space (number of antenna channels) in the spatialdimension treated in Chapter 6 can be applied in the time dimension as well.The clutter rejection performance of such processor is quasi-optimum.

3. Space-time FIR filtering after Chapter 7 can be implemented by using theproperties of the discrete Fourier transform. The Fourier transform is takenfor all channels along the time axis and so for the FIR filter coefficients. Thenthe convolution involved in the FIR filter operation is done by multiplication offrequency responses.

4. Frequency-dependent spatial filtering is a technique which is based onthe statistical independence of Fourier channels. This is satisfied for longdata sequences (typical > 256, increasing with the CNR). Applications arepredominantly in multichannel SAR systems. Spatial blocking matrices (Section6.3.1) can be applied for each Doppler frequency to avoid target signalcancellation by the clutter filters.

5. The JDL-GLR processor uses adjacent Doppler bins and beams as clutterreference.

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

Front MatterTable of Contents9. Space-frequency Processing9.1 Introduction9.2 The Auxiliary Space-time Channel Processor (ACP)9.3 The Symmetric Auxiliary Sensor/Echo Processor9.3.1 Computing the Inverses of the Spectral Covariance Matrices

9.4 Frequency Domain FIR Filter (FDFF)9.5 Frequency-dependent Spatial Processing (FDSP)9.5.1 Spatial Blocking Matrices

9.6 Comparison of Processors9.7 Angle-Doppler Subgroups9.7.1 General Description9.7.2 Comparison with Other Techniques9.7.3 Other Post-Doppler Techniques

9.8 Summary

Index