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Examination of the Effect ofArray Weighting Function onRadar Target Detectability

C. M. ALABASTERE. J. HUGHES, Member, IEEECranfield University

A methodology is described to assess the detectability of

targets by an airborne fire control radar (FCR) operating in a

medium pulse-repetition frequency (PRF) mode in the presence

of strong ground clutter as a function of the transmitting and

receiving antenna array weighting functions and proportion

of failed array elements. It describes the radar, antenna, and

clutter modelling processes and the method by which target

detectability is quantified. The detectability of targets in clutter is

described using a detectability map, which provides a useful means

of comparing target detectability as clutter conditions change.

It concludes that the best target detectability is to be achieved

using those weighting functions on transmit and receive which

result in the lowest average sidelobe levels but that the margins

between the more highly tapered weighting functions were small.

Furthermore, it concludes that target detectability degrades as

the proportion of failed elements increases. A failure of 5% of the

elements gave modest, though meaningful, degradations in target

detectability and would therefore form a suitable upper limit.

Manuscript received September 12, 2007; revised July 23, 2008,and January 19 and March 19, 2009; released for publication March26, 2009.

IEEE Log No. T-AES/46/3/937977.

Refereeing of this contribution was handled by P. Lombardo.

This work was carried out as part of the UK MOD funded Output3 Research Programme. The array antenna data was suppplied byQuinetiQ, Malvern.

Authors’ address: Cranfield University, Shrivenham, UK, E-mail:([email protected]).

0018-9251/10/$26.00 c° 2010 IEEE

I. INTRODUCTION

This paper describes simulation work to assessthe detectability of targets by an airborne fire controlradar (FCR) operating in a medium pulse repetitionfrequency (PRF) mode in the presence of strongground clutter as a function of transmitting andreceiving array weighting functions. This paper alsomodels the compromise in target detectability throughthe graceful degradation of up to 5% of failed arrayelements. It describes the radar, antenna, and cluttermodelling for a system operating a three of eightmedium PRF schedule waveform. Medium PRFwaveforms and the selection of PRFs are describedin the authors’ previous papers [1—5].Target detectability depends on the number

of PRFs in which any target is visible and onthe probability of detection (Pd) in each PRF[6]. The Pd in each PRF is determined by thesignal-to-noise-plus-clutter ratio (SNCR), amongstother factors, and varies across the range and velocity(Doppler) detection space of the radar due to theambiguous repetition of clutter across this detectionspace. Minimizing sidelobe clutter (SLC) through theminimization of antenna sidelobe level is a designpriority for such systems. This may be achieved byapplying a tapered illumination function across theantenna aperture and can be implemented readily byappropriate amplitude and phase weightings of theelements of an active electronically scanned array(AESA) antenna. However, tapered illuminationfunctions result in a reduction in main beam boresightgain together with a broadening of the main beam,both of which are further degraded when thebeam is phase steered away from its mechanicalboresight. Furthermore, phase steering tends togenerate increased sidelobes. Thus there appearsto be a conflict of interests in applying taperedillumination across an array antenna as far as targetdetection is concerned; on the one hand the taperedillumination reduces the sidelobe level, but on theother it leads to a loss of main beam gain. Thusboth clutter and target signal strengths are reducedthrough the use of a tapered antenna illumination,or conversely, both are maximized for a uniformlyilluminated antenna. The question arises as to whethertapered illumination actually leads to increased targetdetectability or not in scenarios in which targetdetection is likely to be clutter limited (i.e., low flying,look-down).This question has been addressed by modelling

the clutter scene in an airborne FCR for variouscombinations of transmitting and receiving arrayweighting functions, azimuth and elevation steeringangles, platform altitudes, and probabilities offailed array elements. For each combination ofconditions, target detectability is derived over thefull range/velocity detection space of the radar.

1364 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 3 JULY 2010

TABLE IRadar Model Parameters

Parameter Value

Frequency 10 GHz (fixed)PRI (=1/PRF) 35.5, 38.5, 44.5, 49.5, 56.0,

64.5, 69.0 and 94.0 ¹sSpace charging time 1.7 ms

Target illumination time 42.5 msDuty ratio 10% (fixed)

(Transmitted pulse width 10% of PRI)Peak transmitted power 10 kWPulse compression Yes (variable with PRF)Range resolution 75 m (0.5 ¹s in time) (fixed)Doppler processing 64 point FFTEclipsing blindness transmitted pulse width+0:5 ¹sMaximum range 185 km (100 nmi)Maximum velocity 1500 ms¡1 (Doppler = 100 kHz)

System noise figure, F 5 dB

Comparisons between the target detectability ofthe various conditions are evaluated in order todetermine the optimum transmitting and receivingarray weighting functions. As a secondary aim,the degradations resulting from a loss of up to5% of the array elements were also modelled andquantified.Section II of this paper describes the radar, antenna

and clutter modelling processes and the method bywhich target detectability is derived and compared. InSection III, the results are presented and discussed.Finally, Section IV draws conclusions.

II. MODELLING PROCESSES

A. RADAR MODEL

The radar model is intended to be representativeof a modern FCR. It has been assumed that theradar operates on a medium PRF schedule of eightPRFs and requires target data in a minimum of threePRFs for ambiguity resolution. It is further assumedthat range and Doppler ambiguities are resolvedusing the coincidence algorithm. The selection ofthe eight pulse-repetition intervals (PRI = 1=PRF)was made in a separate exercise as described in[1—5]. It is commonplace to use a filter to reject mainbeam clutter (MBC) over a narrow bandwidth in theDoppler domain prior to fast Fourier transform (FFT)processing. It is also commonplace to apply platformmotion compensation (PMC) such that the velocity ofmainbeam boresight detections are ground referenced.In this way, MBC is centred at zero Doppler and atmultiples of the PRF. PMC is assumed in this study;however, no MBC filtering is assumed. This ensuresthat target detectability may be evaluated even inregions of strong MBC. The radar platform altitudesconsidered were 1000 m and 5000 m and the platformvelocity was taken as 300 ms¡1. Other parameters ofthe radar model are summarised in Table I.

TABLE IICombinations of Array Weighting Functions

Transmit Weighting Receive Weightingpatterns Function Function

1 Uniform Uniform2 Uniform Taylor 35 dB3 Uniform Taylor 45 dB4 RTT Taylor 35 dB5 RTT Taylor 45 dB6 SPTN Taylor 35 dB7 SPTN Taylor 45 dB

B. Antenna Model

A planar AESA antenna comprised of 1041elements distributed in a diamond lattice overa circular area of nominal diameter 56 cm wasmodelled. The element spacings were nominally ahalf wavelength. Three possible transmitting arrayweighting functions were considered: uniform, radialtransmit taper (RTT) [7, 8], and successive projectiontransmit nulling (SPTN) [8, 9]; and two possiblereceiving array weighting functions were considered:Taylor 35 dB and Taylor 45 dB (n= 2). In additionto these six combinations of weighting functions, aseventh, that of uniform on transmit and uniform onreceive, was also considered for comparative purposes.The seven combinations of the transmitting andreceiving array weighting functions (named patterns)are defined in Table II.The weighting function data defined the magnitude

and phase of the current exciting each element.Furthermore, each element of the array was defined ashaving a power gain pattern which varies as the cosineof the angle off the mechanical boresight. The phaseof each element was under the control of a six bitphase shifter. The magnitude (power) of each elementwas subject to a tolerance of 0.3 dB (Gaussian ofzero-mean and ¾ = 0:3 dB) and a phase tolerance of2± (Gaussian of zero-mean and ¾ = 2±). It was alsonecessary to account for the random failure of 0%,2%, and 5% of the elements. This was modelled byincluding a function which set the probability of eachelement having zero amplitude to 0.00, 0.02, and 0.05,respectively. This ensured that the selection of failedelements was randomized but that each element wasequally likely to fail. The complete loss of elements,i.e., zero transmitted and received power, was theonly failure mode considered in this study since itis the most damaging to the array radiation pattern.The authors acknowledge the possibility of a host ofother possible failures (e.g., partial loss of powers,loss in either transmit or receive modes, and increasedreceiver noise figure), but these were consideredoutside the scope of this study. Notwithstandingthis, it would be easy to accommodate such failuremodes within the model. The simulation has beenconducted in MATLAB; it loads in element amplitude

ALABASTER & HUGHES: EXAMINATION OF THE EFFECT OF ARRAY WEIGHTING FUNCTION 1365

TABLE IIISummary of Array Radiation Patterns

Weighting Function Peak Gain (dBi) Gain Loss wrt Uniform (dB) 3 dB Beamwidth (deg) Peak SLL (dB) rms SSL (dB)

Uniform 33.53 0 3.08 ¡17:50 ¡32:12Taylor 35 dB 32.41 1.111 3.74 ¡34:82 ¡47:20Taylor 45 dB 31.70 1.822 4.04 ¡43:63 ¡54:82

RTT 32.99 0.538 3.46 ¡20:63 ¡35:13SPTN 33.30 0.228 3.33 ¡16:68 ¡32:73

(lower hemisphere) (¡18:86) (¡35:90)

and phase data from a large data file matrix andresults in the antenna gain being expressed as alarge two-dimensional array. It would be possible tooverwrite specific elements of the array (i.e., specificelements of the array data matrix) to adjust theiramplitude and/or phase responses as required. Theorganisation of such matrices also enables wholerows, columns, or subarrays to be readily adjusted asrequired. Simple phase gradients were derived whichprovided the necessary steering in both azimuth andelevation. Simulations were run for azimuth steeringangles of 0± (dead ahead), 30±, and 56± and forelevation angles of 0± (towards horizon, both platformaltitudes) and 5:5± down (5000 m altitude only).Therefore, there were a total of nine combinationsof altitude, azimuth, and elevation steering anglestogether with three probabilities of failed elementsgiving rise to 27 differing conditions for each of theseven patterns, i.e., 189 total simulations.The far-field radiation pattern was derived by

computing the two-dimensional Fourier transformover the array surface. Only the lower hemisphereneed be derived since only this portion illuminatesthe ground. Furthermore, only the forward-lookinghalf-hemisphere was considered since the rearwardlooking pattern is likely to be dominated by theinteraction with the radome which was outside thescope of this study. The rearward pattern results innegligibly low levels of clutter in the negative Dopplerdomain. Later analysis (Section IIE) supports thisassumption; results are dominated by the far higherantenna gains in the forward half-hemisphere.Each element produces 10 W of RF power, giving

rise to approximately 10 kW of total power. The peakmain beam boresight gain for the uniform weightingfunction (assuming no magnitude and phase errorsand zero failed elements) was normalised to 33.5 dBiby an appropriate scaling factor. All other radiationdiagrams were scaled by the same factor to ensurethat the computed radiation diagrams representedthe true effective radiated power (ERP). Exampleradiation diagrams are reproduced in Figs. 1 and 2.Fig. 1 illustrates the best case radiation diagram of theuniform weighting function having zero phase steeringangles, no magnitude or phase errors, and zero failedelements, whereas Fig. 2 illustrates the worst caseradiation diagram of the uniform weighting function

having phase steering angles of ¡5:5± in elevation and56± in azimuth, 0.3 dB magnitude and 2± phase errors,and 5% failed elements.The radiation pattern of the ideal antenna

uniformly weighted illustrated in Fig. 1 indicatesa main beam boresight gain of 33.53 dB, a 3 dBbeamwidth of 3:08±, and a peak sidelobe level (SLL)of 17.50 dB below the main beam. When elementalmagnitude and phase tolerances of 0.3 dB and 2±,respectively, and a proportion of failed elements of5% are all applied, the changes in these parametersare barely noticeable. However, when the beam issteered well away from its mechanical boresight,as in Fig. 2, the antenna parameters degrade to thefollowing: main beam boresight gain = 30:21 dB, peakSLL =¡16:86 dB, and azimuth 3 dB beamwidth =5:71±. Similar plots affirm the peak SLL of ¡35 dBfor the Taylor 35 dB function. However, the Taylor45 dB function results in a peak SLL of around¡43 dB, which is slightly above the expected levelof ¡45 dB. This discrepancy is due to the magnitudeand phase tolerances of each element, since whenthese were set to zero, a peak SLL of ¡44 dB wasobtained. The radiation patterns for the RTT andSPTN functions indicate that the RTT has a maximumSLL of around ¡20:6 dB, whereas the SPTN hasa maximum SLL of around ¡16:7 dB, although itssidelobes at large angular offsets from the mainbeamboresight decay away more quickly than for the RTT.The SPTN function also gives rise to unusually largesidelobes some 11:5± below the mainbeam boresight;these are at a level of ¡23:0 dB. A summary of theradiation patterns for each array weighting functionis given at Table III under the ideal cases of zeromagnitude/phase errors, zero failed elements, and zerosteering angles. Note that the SPTN pattern has beenoptimised to reduce lower hemisphere sidelobes at theexpense of larger sidelobes in the upper hemisphere.The authors acknowledge that subarray processingand radome effects may limit the integrity to whichthese idealised patterns may be reproduced, however,the use of idealised radiation patterns in this work issufficient to demonstrate the principle.

C. Clutter Model

In modelling the clutter and noise, all statisticalvariation has been eliminated in order to permit


Fig. 1. Ideal array for uniform weighting function. No steering angle, zero magnitude and phase errors, zero failed elements.

Fig. 2. Worst case array for uniform weighting function. 56± azimuth and ¡5:5± elevation steering angles, 0.3 dB magnitude and 2±phase errors, 5% failed elements.

small changes in target detectability to be resolved.Clutter modelling in the forward hemisphere onlyis required. The method previously described in[1] is used. Clutter is mapped by considering thesurface under the radar to be marked out by a gridalong orthogonal x and y coordinates centred at 0,0directly under the radar. The model steps throughincrements in the x and y coordinates in the forwardhalf space (i.e., positive y) out to the maximum rangeof interest. At each location the model computes theslant and ground ranges and the resolved Dopplershift along the line of sight to the radar together withthe grazing angle, clutter backscatter coefficient,and clutter radar cross section (RCS). An importantaspect of the clutter mapping process is the resolutionof the increments along the x=y coordinate system.

At each location the clutter RCS is computed onthe basis of a clutter area equal to the square of thex=y resolution. Ideally, the x=y resolution should befiner than the radar range resolution since otherwisethere will be large clutter patches appearing in someresolution cells and nothing in neighbouring cells.However, very fine x=y resolution is unnecessaryand increases the computational time. Since the radarrange resolution is 75 m, an x=y resolution of 50 mhas been used.The clutter backscatter coefficient (BSC) is a

function of the grazing angle μg which is computedfor each point in the clutter modelling process. TheBSC was defined as

BSC= ¾0 sin(μg) +¾0V expμ¡(90¡ μg)

μ

¶(1)


Fig. 3. Surface clutter BSC versus grazing angle.

where ¾0 =¡15 dBm2 and ¾0V =¡5 dBm2 (in linearunits), and μ =¡20= loge(sin70±=A) and A= ¾0V=¾0.¾0V defines the BSC at normal incidence, and

¾0 defines the BSC at a mid-grazing angle. Thedependence of BSC on grazing angle is depictedin Fig. 3. The power of the clutter returns wascalculated using the following form of the radar rangeequation in which the clutter RCS is cascaded with thetransmitting and receiving antenna gains along the lineof sight to the radar by reference to the appropriateantenna radiation pattern data.

clutter power =PT:GT:GR:¸

2:RCSC64:¼3:R4S

(2)

where PT = peak transmitted power (= 10 kW),GT = transmitting antenna gain,GR = receiving antenna gain,RS = slant range,¸=wavelength = 0:03 m,RCSC = clutter radar cross section = BSC£

(x=y resolution)2.The clutter power calculated from (2) was

then range and Doppler gated and added into theappropriate range/Doppler cell on top of the noisepower and any previously calculated clutter signals.PMC is applied by precalculating the Doppler gate ofthe centre of the main beam and applying this as anoffset to the Doppler gated clutter. The total clutterand noise power is then stored in a two-dimensionalmatrix (range cell versus Doppler cell) and displayedon a folded clutter map. Folded clutter maps in eachPRF are derived which are subsequently required toproduce the maps of target detectability. A foldedclutter map is one in which the clutter amplitude fromthe full detection space of the radar is folded intoone ambiguous range and Doppler interval. Thus, aclutter map always has 64 equal intervals in Doppler,since this is the FFT size, but a variable number ofrange cells, which is equal to the number of rangecells in one PRI. A constant fixed noise power level

Fig. 4. Folded clutter map. PRI = 56 ¹s.

of k:T0:Bn:F is included in every cell of the map inwhich k is Boltzmann’s constant = 1:38£ 10¡23 J/K;T0 is a standard temperature of 290K, Bn is the noisebandwidth = (transmitted pulse width)¡1; and F is thenoise figure = 3:16 (5 dB). An example of a foldedclutter map is given in Fig. 4. The clutter map maybe unfolded to cover the complete range/velocitydetection space specified by the radar model by tilingthe folded clutter map as many times as necessary.

D. Target Detectability

Target detectability over the full range/Dopplerdetection space of interest is conveniently representedby a detectability map [1]. The folded clutter mapof Fig. 4 is replicated in range and Doppler overthe full detection space of interest (i.e., 185 km inrange by 1500 m/s in velocity) due to the repetitionof data in the time and frequency domains. Thisresults in an unfolded clutter map. Each PRF in theschedule has a similar, though different, unfoldedclutter map. The probability of detection of a discretetarget at any range/Doppler cell of interest dependson the number of PRFs in which the range/Dopplercell is not eclipsed and the probability of detectionin each PRF, as determined by the SNCR of thecell. Blindness results from eclipsing, with no MBCblanking being assumed. A detectability map cantherefore be derived over the full range and Dopplerdetection space of the radar and denotes the minimumtarget RCS required for detection at each range andDoppler cell in an appropriate number of PRFs. Thedetectability map may be thresholded at a given fixedRCS to indicate regions where a target of the givenRCS would be visible/not visible. This thresholdingforms the classic blind zone map for a medium PRFschedule. An example detectability map is given inFig. 5 based on a required SNCR= 0 dB in at leastthree PRFs from the total of eight. Similar criteriahave been used in the generation of all detectabilitymaps used in this study. Should a more realisticSNCR of, for example, +13 dB be required, one needonly apply a 13 dB offset to the detectability map


Fig. 5. Detectability map.

data. Therefore a target with RCS of 13 dB greaterthan the level read from the detectability map wouldbe detected with a Pd commensurate with the SNCRof 13 dB. Whilst every effort has been maintained toensure that the minimum target RCS requirement ofthe detectability maps is properly calibrated, there areinevitable sources of error. The radar range equationof (2) omits the integration gain and filter shapelosses associated with the FFT process and also theatmospheric and system losses. Both static clutter anddiscrete targets would be subject to similar processinggains; subtle differences would arise depending onthe temporal statistics of each, whereas losses affectboth equally. For identical processing gains (clutterand discrete targets) the SCR would be independentof the processing gain. Integration gain has thereforebeen omitted from equation (2). As far as noise isconcerned, the appropriate integration gain is appliedin the generation of the detectability map. However,detection is generally clutter limited rather thannoise limited. There is no distinction between clutterand discrete targets. There are no discrete targetswithin the (simulated) scene, just surface clutter. Adetectability map simply plots the RCS that a targetwould have to possess if it were to be detected inthe requisite number of PRFs (three from eight inthis case) with a sufficient SCR (0 dB in this case).This is the threshold value of RCS that triggers thetransition from blindness to visibility in a classicalblind zone map. The threshold RCS varies for eachrange/velocity cell within the detection space ofthe radar [1]. Without intimate knowledge of everyaspect of the system design, it would be impossibleto calibrate the detectability maps. However, eachdetectability map is valid given the assumptions madein the radar, antenna, and clutter models, and thereforecomparisons between detectability maps are also valid.Furthermore, comparisons remain valid irrespective ofany offsets which may be applied to the detectabilitymaps (such as may be required to depict a detectioncriterion of SNCR¸+13 dB) so long as a constant

offset is applied to all detectability maps. In this workthe benefits of different array weighting functionswere derived through direct comparisons betweendetectability maps. Detectability maps are a usefulmeans of characterizing relative performances inclutter.

E. Evaluating Target Detectability

Each of the 189 simulations results in eightfolded clutter maps, one for each PRF. However, theresolution differs in each of the eight. A folded cluttermap always has a fixed number of Doppler cells eachof width = PRF=FFT size, thereby fixing the numberof Doppler cells to the FFT size (= 64, in this case)but yielding a Doppler resolution which varies withPRF. The range cell width, however, is fixed by thecompressed pulse width (= 0:5 ¹s or 75 m, in thiscase), and so the number of range cells in the foldedclutter map = PRI=0:5 ¹s and is therefore a functionof the PRF. For an example PRF of 10 kHz, oneobtains a folded clutter map of 64£ 200 = 12800range/Doppler cells; each cell forming a pixel ofthe clutter map. For the case of a 10 kHz PRF, thetransmitted pulse width is 10 ¹s and so the numberof blind (eclipsed) range cells = 21. Thus the totalnumber of blind range/Doppler cells is 21£ 64 = 1344or 10.5% of the clutter map.In constructing the detectability map, the eight

folded clutter maps corresponding to the eight PRFsin the schedule are unfolded to occupy the wholerange and Doppler detection space of the radar. Thisrequires that they be read at a common resolutionwhich can be no finer than that of the coarsest map.Hence the resolution of the detectability map ismarginally coarser than the original clutter maps.The eight maps are then overlaid, and the RCSlevel which toggles blindness is established. Sinceall detectability maps are plotted with a commonresolution, it is a simple exercise to compare twomaps pixel by pixel since each pixel relates to aconsistent range/Doppler cell. Due to the unfoldingprocess, each detectability map is comprised of some560000 pixels (range/Doppler cells).In order to assess the dependence of target

detectability on array weighting functions, a teststrategy was developed in which the 27 detectabilitymaps of one set of transmitting and receiving arrayweighting function conditions (corresponding tothe 27 combinations of azimuth/elevation steeringangles, proportion of failed elements, and platformaltitude) were compared with the corresponding27 detectability maps for each of the other sixsets of transmitting and receiving array weightingfunctions. This progressed until all sets of transmittingand receiving array weighting functions had beencompared with all the other sets. This test strategyhas been found to be necessary due to the complexity


of the optimization problem posed by this study. Asan optimization problem, this work seeks to optimizea single objective: the detection performance of theradar via the selection of an optimal combinationof transmitting and receiving antenna weightingfunctions. However, the complexity arises becausetarget detectability is quantified over several hundredsof thousands of range/Doppler cells (i.e., it is highlymulti-dimensional) which must be distilled intosimpler metrics. These optimization problems cantypically yield several optimal solutions, collectivelyforming a Pareto optimal front [10], otherwise knownas a trade-off surface. Optimal solutions identifiedby each set of comparisons may differ in each casedepending on what metric is used to define targetdetectability and what baseline standard is adoptedfor each comparison. Two metrics (X and Y) havebeen derived which compare the data in pairs ofdetectability maps, A and B.Both the X and Y metrics were derived to give

a comparison of the detectability levels betweentwo detectability maps. Since each detectability mapcomprises around 560000 pixels, the comparisonof two maps is not trivial. The X metric gives animpression of the relative area of the range/Dopplerspace for which the detectability of one test is greaterthan the detectability of another. The X metric isrelated to the median of the differences betweenthe two detectability maps. However, the X metricis insufficient on its own to convey the generalsuperiority of one map over another because it doesnot indicate the margin of any such superiority.Hence the Y metric is also used. The Y metric sumsthe differences (cubed) between the magnitudes ofcorresponding pixels of two detectability maps. TheY metric is the third-order moment of the differencedata and is therefore representative of the skew inthe distribution. The two metrics convey differentaspects of the superiority of one map over another.The derivation of the two metrics is explained in theparagraphs below.1. Ratio of Comparisons, X: The algorithm runs

as follows:

1) Exclude all the elements (i.e., range-Dopplercell) of A and B which are in regions of eclipsedblindness. This avoids corrupting the statisticalcomparisons which follow.2) Derive a logical comparison matrix for which

A > B, and sum all its elements. (The comparisonmatrix consists of elements = 1 or 0 depending onwhether A > B or not. In summing all the elements,one derives the total number of elements for whichA > B.)3) Derive a logical comparison matrix for which

B > A and sum all its elements. (The comparisonmatrix consists of elements = 1 or 0 depending onwhether B > A or not. In summing all the elements,

one derives the total number of elements for whichB > A.)4) These two sums do not necessarily sum to the

total number of elements in the detectability map sincethe case of A= B has not been computed. A= B inblind regions, and elsewhere, A= B = noise in regionsof very low clutter.5) Derive the ratio of the two sums and express it

on a decibel scale, i.e.,

X = 10: log10

μ§(A > B)§(B > A)

¶: (3)

If the detectability levels of A are generally higherthan those of B, then

P(A > B) is large and

P(B >

A) is small and their ratio> 1, so X is a (large)positive quantity. If the reverse is true, then X isa (large) negative quantity. The X metric gives animpression of the relative area of the range/Dopplerspace for which the detectability of one test isgreater than the detectability of another. The Xmetric is related to the median of the differencesbetween the two detectability maps. The X metric isa first-order statistic, similar to the mean; however,unlike the mean, it is less sensitive to highly skeweddistributions. Regions of dominance of one over theother may be obtained by mapping the comparisonmatrices. The X metric gives no information on themargin by which one is greater than the other.

2. Sum of Difference Comparison, Y: Thealgorithm runs as follows:

1) Exclude all the elements of A and B whichare in regions of eclipsed blindness. This avoidscorrupting the statistical comparisons which follow.2) Derive the matrix for A¡B. This matrix yields

signed difference values.3) Cube the difference matrix element by element.

This accentuates the differences and preserves theirsign.4) Sum all the elements of the cubed difference

matrix. This returns the net difference over the wholerange/Doppler detection space.

Y =§(A¡B)3: (4)

The Y metric is the third-order moment of thedifference data and is therefore representative ofthe skew of the difference data distribution. If thedetectability levels of A are generally higher thanthose of B then the Y metric will be a (large) positivenumber, whereas if the reverse is true, the Y metricwill be a (large) negative number. The Y metric givesan impression of the “aggregate” level by which thedetectability of one test is greater than the detectabilityof another. However, one cannot distinguish betweenthe case where a few elements in one matrix aresignificantly higher than those of the other matrix andthe case where most of the elements in one matrix aremarginally higher than those of the other. Thus the Y


metric indicates the margin of superiority but not itsextent in area.The combination of the X and Y metrics therefore

indicate both the area extent of superiority of onedetectability map over another and also on theaggregate margin of this superiority. (Note: in all testsit was necessary to exclude the first column of the Aand B matrices since these were dominated by MBCand mask the subtle effects of the sidelobe clutter.This provides a partial filtering of MBC by excludingthe very central region but retaining the peripheraldata.)Of secondary interest is the effect of an increasing

number of failed elements in the array. One expectsfailed elements to result in increased sidelobes and areduction in main beam boresight gain and thereforea loss of detectability of targets in clutter; however, itis worth quantifying these changes in order to judgean acceptable number of failed elements. To evaluatethis, it is necessary to make comparisons betweendetectability maps defined by a common patterns butdiffering in the proportion of failed elements Pfe. Asbefore, the detectability maps were compared usingthe same X and Y metrics previously described.

III. RESULTS AND DISCUSSION

A. Array Weighting Function

A series of comparisons have been made betweenpairs of detectability maps and the X and Y metrics ofeach comparison derived. Batches of 27 comparisonsare made for each combination of transmitting andreceiving array weighting functions, and therefore,the means X and Y over the 27 comparisons havebeen derived. A uniformly weighted mean has beenused in this study, however, one might considerderiving a weighted mean in order to give preferentialtreatment of certain conditions e.g., zero steeringangle. Note that the use of the mean of these metricsis not intended to imply that they have a Gaussianspread. Each detectability map has dimensionsof 2467£228 = 562476. However, since the firstcolumn of each map (matrix) is excluded from theanalysis, the processed data has dimensions 2467£227 = 560009. In some cases it is possible to obtainP(B > A) = 0 and therefore an infinite value of X

results in which case the mean of X would also beinfinite. The maximum finite value of X arises whenP(A > B) = 560008 and

P(B > A) = 1 and results in

X = 57:48. This is unlikely to arise since there existsthe possibility that A= B in some elements, however,some of the results approach to within 0.02 of thisvalue. It was therefore decided to cap values of Xwhich would otherwise be infinite to a value of 57.50.Similarly, values of X which would otherwise be ¡1are capped at ¡57:50. In this way, the calculationof the mean is not confounded. The X results are

TABLE IVX Results

A patterns

B patterns 1 2 3 4 5 6 7

1 0 ¡49:8 ¡51:8 ¡45:9 ¡47:1 ¡40:6 ¡44:02 49.8 0 ¡7:2 ¡13:7 ¡15:7 ¡7:7 ¡11:53 51.8 7.2 0 ¡5:8 ¡12:5 ¡0:4 ¡6:14 45.9 13.7 5.8 0 ¡5:5 5.7 0.45 47.1 15.7 12.5 5.5 0 8.8 5.16 40.6 7.7 0.4 ¡5:7 ¡8:8 0 ¡6:07 44.0 11.5 6.1 ¡0:4 ¡5:1 6.0 0

TABLE VY Results

A patterns

B patterns 1 2 3 4 5 6 7

1 0 -3e32 -4e32 -5e32 -5e32 -6e32 -6e322 3e32 0 -3e29 -6.e29 -2e30 -3e30 -5e303 4e32 3e29 0 -3e27 -2e29 -4e29 -1e304 5e32 6e29 3e27 0 -9e28 -3e29 -8e295 5e32 2e30 2e29 9e28 0 -2e28 -1e296 6e32 3e30 4e29 3e29 2e28 0 -3e287 6e32 5e30 1e30 8e29 1e29 3e28 0

given in Table IV and the Y results in Table V.Tables IV and V offer mean results over a number ofcomparisons. The A patterns defines the transmit andreceive weighting function used as a baseline againstwhich all the other weighting functions are compared,as defined by the B patterns. Firstly, the A patterns isset to 1 (i.e., uniform on transmit and receive) andthe B patterns is set to 2, 3, 4, 5, 6, and 7 in turn.Next the A patterns is set to 2 (uniform on transmitand Taylor 35 dB on receive) and the B patternsis set to 1, 3, 4, 5, 6, and 7 in turn. This continueswith A patterns being set to 3, 4, 5, 6, and 7 in theirturn and for each value of A patterns, the B patternscycles through all the other values. In this way, eachcombination of tranmit and receive weighting functionis compared with all the other ones. (Actually, each iscompared twice, e.g., 1 versus 3 and 3 versus 1, hencethe symmetrical/inverted nature of Tables IV and V.)This exhaustive method of comparison is necessarydue to the high dimensionality of the test function,each detectability map being represented by about halfa million pixels.Both results tables are matrices with a leading

diagonal of zeros and both are symmetrical andinverted about the leading diagonal. It is worthrecalling that X and Y results which are positivemean and that the minimum target RCS requirementsdefined by the A detectability maps are greater thanthose of the B detectability maps. Therefore, positiveX and Y results across the B patterns rows denote theability to detect smaller targets using the B patternswhen compared with the respective A patterns. The


larger the positive results, the greater this margin.Negative results indicate the opposite.The rank order of the B patterns from highest

Y (best target detectability) to lowest Y (lowesttarget detectability) for all seven sets of results isconsistently 7, 6, 5, 4, 3, 2, 1. The consistency of thisresult is believed to be due to the simple arithmeticexpression for Y. The rank order of the B patternsfor the X results is not consistent across all seven setsof results. However, in all but one set of results, i.e.,those of A patterns= 1, the highest X (best targetdetectability) is obtained for B patterns= 5. WhenA patterns= 1, the highest X (best target detectability)is obtained for B patterns= 3. In all seven sets ofresults the lowest X (worst target detectability) isconsistently obtained for B patterns= 1 and it isobvious that patterns= 1 is far removed from theoptimal solution. The inconsistency in the rank orderof the B patterns for the X results when A patterns=1 is believed to be due to the fact that the baseline(A patterns= 1) is very distant from the bettersolutions. The small degree of inconsistency in therank order of the patterns for the other X results isbelieved to be due to the nonarithmetical nature ofthe expression for X. If one sums the X results overthe seven sets of results (i.e.,

PX) one obtains the

rank order of the patterns from highest (PX) (best

target detectability) to lowest (PX) (lowest target

detectability): 5, 4, 7, 3, 6, 2, 1.The preferred solution depends on the metric

used to quantify target detectability and the baselineagainst which comparisons are made. This is a typicaldilemma associated with optimisation problems whoseoptimisation goal has high dimensionality, and there isas yet no known metric which avoids the inconsistentbehaviour observed here. One method to arrive at anoptimum solution based on equal weightings of the Xand Y results may be to award points to each of thepatterns based on the rank order of their seven sets ofX and Y results, i.e., a nonparametric normalizationusing rank ordering. The following scoring method isproposed here: 7 points are awarded for a first placein the rank order, 6 points for second place, 5 pointsfor third and so on down to one point for a seventhplace finish. Since also: maximum

PX points =

maximumPY points = 49, each patterns has an

associated distance from the best possible solutiongiven by

DISTANCE =r³49¡

XX points

´2+³49¡

XY points

´2:

(5)

The points are displayed in Fig. 6 in which “Best”indicates the maximum utopia solution.The ascending rank order of DISTANCE

determines the rank order of the solutions defined by

Fig. 6. Points positions of solutions.

patterns. The results of the points scoring are (frombest to worst): 7, 5, 6, 4, 3, 2, and 1.From Fig. 6 it is evident that patterns= 5 and

7 are almost equal solutions which fall on a Paretosurface, i.e., no one solution is better on both metricssimultaneously. However, of the two, patterns=7 offers the slightly better target detectability.Furthermore, it is believed that the rank order ofthe various results could alter if realistic statisticalfluctuations were admitted in the various modellingprocedures. It may be noted that the better targetdetectability is generally obtained for the Taylor45 dB weighting function to be applied to thereceiving array over the corresponding Taylor 35 dBfunction. The worst X and Y results were consistentlyobtained for patterns= 1; indeed all solutions whichentail the transmission using the uniform weightingfunction (patterns= 1, 2, and 3) exhibit poor targetdetectability.

B. Proportion of Failed Elements

A further series of comparisons betweendetectability maps has been carried out to comparethe cases of the probability of failed elements,Pfe = 0:00 versus Pfe = 0:02 and Pfe = 0:00 versusPfe = 0:05, for the various combinations of steeringangles, altitude, and transmitting and receiving arrayweighting functions. In this case, the A detectabilitymaps were taken to be those for Pfe = 0:00, whereasthe B detectability maps were those of for whichPfe = 0:02 and 0.05. Thus there are two comparisonsto be made (Pfe = 0 versus Pfe = 0:02 and Pfe = 0:00versus Pfe = 0:05) for the nine different combinationsof altitude, azimuth, and elevation steering anglesat each of the seven combinations of transmittingand receiving array weighting functions (the sevenpatterns). The results are given in Table VI, in whichthe X and Y results have been averaged (=X and Y,respectively) over all seven patterns. As before, the


TABLE VIX and Y versus Pfe Results

Pfe(A) Pfe(B) Azimuth Steering Angle (deg) Elevation Steering Angle (deg) Altitude (m) X Results Y Results

0.00 0.02 0 0 5000 ¡19:5 -3e210.00 0.05 0 0 5000 ¡38:6 -2e220.00 0.02 30 0 5000 ¡18:0 1e220.00 0.05 30 0 5000 ¡21:9 3e230.00 0.02 56 0 5000 ¡14:1 -1e200.00 0.05 56 0 5000 ¡19:2 -2e220.00 0.02 0 ¡5:5 5000 ¡20:9 3e240.00 0.05 0 ¡5:5 5000 ¡32:2 9e250.00 0.02 30 ¡5:5 5000 ¡14:2 2e290.00 0.05 30 ¡5:5 5000 ¡19:6 2e300.00 0.02 56 ¡5:5 5000 ¡12:4 3e290.00 0.05 56 ¡5:5 5000 ¡13:9 2e300.00 0.02 0 0 1000 ¡17:4 -7e260.00 0.05 0 0 1000 ¡33:7 -9e270.00 0.02 30 0 1000 ¡13:9 1e290.00 0.05 30 0 1000 ¡17:6 3e300.00 0.02 56 0 1000 ¡9:4 6e280.00 0.05 56 0 1000 ¡14:4 7e30

values of X which would otherwise be infinite havebeen capped at 57.50, and the use of the means inTable VI is not intended to imply that the metrics havea Gaussian spread.All the X metric values are negative, indicating

that detectability levels increase (i.e., targets needto be larger to be detected) for Pfe > 0:00, whichis unsurprising. The larger negative magnitude ofX is consistently obtained for Pfe(B) = 0:05, asopposed to Pfe(B) = 0:02, which, again, is to beexpected. Comparing the cases of zero elementfailures with 2% element failures results in Xranging from ¡20:9 to ¡9:4, mean =¡15:5, whichis comparable to the difference in target detectabilitybetween patterns 2 (uniform on transmit andTaylor 35 dB on receive) and patterns 5 (RTT ontransmit and Taylor 45 dB on receive), see Table IV.Comparing the cases of zero element failures to5% element failures results in X ranging from¡38:6 to ¡14:4, mean =¡23:5. In this case thereis no near comparison with the margins in targetdetectability between the combinations of patternsfrom Table IV. Nevertheless, it is clear that the effectsof 2% and 5% element failures is comparable tosome of the more significant differences in targetdetectability between some of the best and worstarray weighting functions, as quantified by theratio of comparisons metric. The Y metrics varybetween positive and negative values for differentcombinations of conditions; there are no consistenttrends, and the Y metric data is rather inconclusive.However, the magnitudes of Y are typically severalorders of magnitude lower than those of Table V,Section IIIA, indicating typically smaller margins ofsuperiority/inferiority in detectability performance.The changes in target detectability due to 2% and5% failed elements are, on the whole, less significant

than the effects of array weighting functions.The combination of the two metrics suggests thatfailed elements do result in large regions of therange/velocity detection space of the radar wheretarget detectability has degraded. However, over thewhole range/velocity detection space, the aggregatemargin by which target detectability changes issmall and inconsistent. Therefore, the regions ofdegraded target detectability exhibit relatively smallmargins of degradation. It is also reasonable toassume that partial failures of elements would resultin smaller regions and margins of degraded targetdetectability.

C. Sensitivity of Results to Conditions

The data of Tables IV, V, and VI and the scales ofFig. 6 are difficult to calibrate. It is difficult to deriveany absolute level of performance, and these tablesand figures only yield comparative performances.It is in the nature of the problem that the marginsof one scenario over another cannot be reduced toa single figure; it is not possible to claim that targetdetectability in one scenario is x dB better thananother because of the variation across the scene. Thisstudy is based on a sample of typical combinationsof realistic operating conditions which the authorsbelieve to be representative of most clutter limitedsituations, and hence, the conclusions drawn fromthese results are valid in this context.With regard to the sensitivity of the results to the

model parameters, some comments on each of these isoffered below:

Distortion of the beam patterns: It can be seenthat the margins between patterns 5 and 7 is marginal.Clearly, the difference between these two cases is sosmall that it would probably be masked by noise,


clutter statistical variation, and target fluctuation.Therefore, it is quite clear that target detectability isnot particularly sensitive to small variations in beampattern.Clutter: All statistical variation in clutter has

been deliberately eliminated as it was feared that suchrandom variations might mask small changes due tothe array weighting functions. Given the near identicalperformances of patterns 5 and 7, this seems to havebeen justified since the introduction of clutter statisticswould probably cloud the judgement between similarbeam patterns; it would introduce another unwantedand unknown factor. Variations in clutter statisticshave been considered in [1].Failure Probabilities: Three probabilities of

failure have been considered. The results have beenpresented and discussed and enable the reader tomake comparisons between these three cases. Theseresults have also been compared with the differencesdue to the use of differing array weighting functions.In the ensuing discussion it was noted that the Xand Y metrics did not reveal consistent trends butthat in the worst case (i.e., the X metric), the effectsof 2% and 5% element failures are comparable tosome of the more significant differences in targetdetectability between some of the best and worstarray weighting functions. However, more typically,the consequences of up to 5% element failureson target detectability are relatively small andinconsistent.PRFs: Previous work concludes that target

detectability is highly sensitive to the exact choice ofPRF values, number of PRFs used, and criterion fordetection (e.g., three of eight). This is an importantsubject area and has been the subject of much ofthe authors’ previous work extending over severalyears which is reported in [1—5]. The techniquesreported in these works have been used to derive anear-optimum PRF set for the radar assumed in thisstudy.Blindness: The sensitivity of target detectability

to eclipsing blindness has not specifically beenconsidered in this work; however, the blindnessproblem has been considered alongside the authors’earlier work on the exact choice of PRF values,number of PRFs used, and criterion for detection (e.g.,three of eight) in [1—5].In summary, there is no simple metric to relate the

sensitivity of results to other parameters. These issuesare far from trivial; however, several of these havebeen major research topics in their own right and havebeen discussed in previous papers.

IV. CONCLUSIONS

Clearly, the differing metrics which one may useto quantify target detectability result in differingsolutions with very little to choose between them.

However, by combining the means of both theX and Y metrics in a points scoring system, thebest overall solution was identified as being thecombination of the SPTN function on transmissionand the Taylor 45 dB function on reception. Thiswas very closely followed by the combination of theRTT function on transmission and the Taylor 45 dBfunction on reception. The overall preference forthe former may well be due to lower rms sidelobelevels in the lower hemisphere. Nevertheless, itought to be stressed that the margins between thesetwo cases are very small and may very well bemasked by statistical variations in noise, clutter,and target RCS. It may also be worth noting thatthe RTT function results in an effective radiatedpower (ERP) some 0.6 dB higher than that of theSPTN function and so enjoys a small advantagein detection performance in noise limited cases.Furthermore, the RTT function (and its resultingbeam pattern) is circularly symmetrical and soremains constant irrespective of the platform rollangle. The worst target detection performancewas obtained when using the uniform weightingfunction on the transmitting array. Indeed the testcase of the uniform function on both transmissionand reception was found by both metrics to yieldthe worst target detection capability by a largemargin.Target detectability degraded as the proportion of

failed elements increased from zero to 5%. Failureof the elements contributes towards increases inthe sidelobes, reduction in main beam boresightgain, and hence reduction in detection performance.Failed elements result in significant regions ofthe range/velocity detection space of degradedtarget detectability; however, the margins by whichdetectability is degraded tended to be less than themargins between detectability using the best and worstarray weighting functions. A failure of 5% of thearray elements resulted in modest, though meaningful,degradations in target detectability. Therefore, 5%would be an appropriate upper limit on the proportionof failed elements.These conclusions pertain to a reasonable sample

of operating scenarios which were designed to resultin clutter-limited detection conditions for medium PRFoperation. The authors believe that these conclusionsremain valid for different, though similar, scenariosresulting in clutter-limited detection conditions.Should the radar operate at substantially higheraltitudes and/or in look-up attitudes and/or in highPRF modes, then detection is quite likely to becomenoise limited. Noise-limited detection conditionswill result in different solutions for optimal PRFvalues, FFT sizes, and numbers of coherent processingintervals and may also lead to different solutions foroptimal array weighting functions.


REFERENCES

[1] Alabaster, C. M. and Hughes, E. J.The design of medium PRF radar schedules for optimumdetectability in diverse clutter scenes.Presented at the IEEE Waveform Diversity & DesignConference, Lihue, Kaua’i, HI, Jan. 22—27, 2006.

[2] Alabaster, C. M. and Hughes, E. J.Novel PRF schedules for medium PRF radar.Presented at the Radar 2003 Conference, Adelaide,Australia, Sept. 3—5, 2003.

[3] Alabaster, C. M., Hughes, E. J., and Matthew, J. H.Medium PRF radar PRF selection using evolutionaryalgorithms.IEEE Transactions on Aerospace and Electronic Systems,39, 3 (July 2003), 990—1001.

[4] Davies, P. G. and Hughes, E. J.Medium PRF set selection using evolutionary algorithms.IEEE Transactions on Aerospace and Electronic Systems,38, 3 (July 2002), 933—939.

[5] Wiley, D. A., Parry, S. M., Alabaster, C. M., and Hughes,E. J.Performance comparison of PRF schedules for mediumPRF radar.IEEE Transactions on Aerospace and Electronic Systems,42, 2 (Apr. 2006), 601—611.

Clive M. Alabaster received his B.Sc. degree in physics with microelectronicsfrom the University College Swansea, Wales, in 1985 and his Ph.D. fromCranfield University, Shrivenham in 2004.From 1985 to 1992 he worked as a microwave design and development

engineer on airborne radar systems with GEC Marconi, Milton Keynes, England.From 1992 to 1998 he worked as a lecturer in radar techniques at ArborfieldGarrison, near Reading, England. He joined Cranfield University, Shrivenham,UK in 1998 as a lecturer and now works in the Sensors Group within theDepartment of Informatics and Sensors. His research interests include the pulseDoppler radar, radar waveform design and the dielectric properties of materials,particularly in the millimetre wave band.Dr. Alabaster is a member of the Institute of Physics and is a Chartered

Engineer.

Evan J. Hughes (M’98) received his B.Eng. and M.Eng. degrees in electrical andelectronic engineering from the University of Bradford, England, in 1993 and1994 respectively.From 1993 to 1995 he worked as a design engineer with GEC Marconi,

Leicester. He received his Ph.D. in 1998 from Cranfield University, Shrivenham,England. His primary research interests include noisy multi-objective evolutionaryalgorithms, swarm guidance, data fusion, artificial neural networks, fuzzysystems, intelligent agents, and radar systems.Dr. Hughes is a member of the IET and is a chartered engineer. He received

the prize for best paper at GALESIA ’97, Glasgow, UK, won the EvolutionaryCheckers Competitions at CEC 2001, Seoul, S. Korea and WCCI 2002, Honolulu,HI, and won the Time Series Prediction Competition at WCCI 2002. He iscurrently working as a senior lecturer for the Sensors group in the Departmentof Informatics and Sensors.

[6] Alabaster, C. M. and Hughes, E. J.The dependence of radar target detectability on arrayweighting function.Presented at the Radar 2007 Conference, Edinburgh, UK,Oct. 15—18, 2007.

[7] Iglehart, S. C.Design of weighting functions under peak amplitude andeffective radiated voltage constraints.IEEE Transactions on Aerospace and Electronic Systems,15, 2 (Mar. 1979), 267—274.

[8] Mellor, I. M. and Adams, F. J.Benefits of transmit beam pattern synthesis for airbornephased array radar.In Proceedings of the International Radar SocietySymposium 2005, Berlin, Germany, Sept. 6—8, 2005.

[9] Poulton, G.Power pattern synthesis using the method of successiveprojections.In Proceedings of the Antennas and Propagation SocietyInternational Symposium, 1986, 667—670.

[10] Kalyanmoy, D.Multi-Objective Optimization Using EvolutionaryAlgorithms.Hoboken, NJ: Wiley, 2001; ISBN 0-471-87339-X.


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