near-fieldtesting of adaptive radar systems · near-fieldtesting of adaptive radar systems ... fig,...
TRANSCRIPT
A.J. Fenn
Near-Field Testing ofAdaptive Radar Systems
Large phased-array antennas with multiple displaced phase centers are applied toradar applications that require adaptive suppression of jamming and clutter. Beforethe deployment of this adaptive radar, tests must veritY how well the system detectstargets and suppresses clutter and jammer signals. This article discusses a recentlydeveloped focused near-field testing technique that is suitable for implementation inan anechoic chamber. With this technique, phased-array near-field focusing providesfar- field equivalent performance at a range distance ofone aperture diameter from theadaptive antenna under test. The technique is applied theoretically to a dual-phasecenter sidelobe-canceler antenna with multiple near-field sources within the mainbeam and sidelobes. Numerical simulations indicate that near-field and far-fieldtesting can be eqUivalent.
The development of any radar system requires tests of the associated hardware andsoftware at various levels ofthe design. Before aradar system can be deployed, design specifications mustbe verified at the subsystem development level, system prototype level, and finalsystem level. For ground-based or airborneradar systems, the final system-can be tested in
the field and modified orupgraded as necessary.For a spaceborne radar, however, where thedeployed system hardware is not accessible,comprehensive prelaunch testing and modifications must be performed on the ground. Thisarticle describes a recently developed focusednear-field technique that measures the performance characteristics of adaptive radar systems within a ground-test facility. Although thetechnique is especially suitable for space-basedradar systems [1-3], it applies to most adaptiveground-based and airborne radar systems andcommunications systems as well.
The important subsystems of an adaptiveradar system consist of an antenna, a multichannel receiver, and a signal processor. Theradar receives desired target signals along withinterference consisting of noise, backgroundclutter, and sidelobe jamming (Fig. 1). The antenna collects signals that the receiver filters,downconverts, and digitizes. The digitized data
The Lincoln Laboratory Jouma1. Volwne 3. Nwnber 1 (1990)
are then processed by the signal processor,which suppresses undeSired interference signals and produces desired target reports.
The radar system typically utilizes a deployable planar phased-array antenna structurewith a largest dimension of 10 to 50 m and anominal operating range of several thousandkilometers. Because of this long operatingrange, the antenha receives planar-shapedwavefronts from targets, clutter, and jamming.To approximate plane-wavefront conditions atmicrowave frequencies, a conventional far-fieldtest distance is on the order of 2 to 10 km. Theminimum far-field distance is determined by2rY/ A.., where D is the antenna diameter andA.. is the wavelength. For low-sidelobe antennapattern measurements, a longer far-field minimum test distance is often used. It is difficult ifnot impossible to place many radiating testsources on a far-field ground-test range over awide field of view, and demonstrate low sidelobes together with jammer/clutter suppression. Thus an alternate ground-test configuration is necessary. Near-field testing, with theradar system positioned in a high-quality controlled-environment anechoic chamber, is adesirable method for evaluating radar systemperformance. Figure 2 shows the proposed testmethod. The antenna and test sources Uammer,
23
Fenn - Near-Field Testing ojAdaptive Radar Systems
Fig, 1-The important subsystems of an adaptive radarsystem. Noise, clutter, jamming, and desired target signals are received by the antenna. These signals aredownconverted and digitized by the receiver and thenprocessed within the signal processor to provide desiredtarget information.
clutter. and target) are placed within the anechoic chamber. and the adaptive nulling receiver and signal processor are placed outsidethe anechoic chamber. The details of how thistechnique naturally develops are describedbelow.
Figure 3 depicts the various regions in frontof a test antenna. The horizontal "axis is theantenna dimension D/ltand the vertical axis isthe normalized test distance z/D. The figureindicates the far- field test region, but this regionis not considered for potential testing for the sizereasons cited above. Planar scanning with a
Clutter Jamming
Antenna
AdaptiveNulling
Receiver
SignalProcessor
Target
Noise
probe antenna at a distance of a few wavelengths (typically 31t) from the test antenna is aconventional near-field technique for calibration and far-field radiation pattern measurement [4.5]. This form ofplanar near-field scanning is a non-real-time measurement techniquein which near-field data are collected and farfield data are computed. However. an adaptivenulling test requires real-time signalwavefronts. which precludes using the planarnear-field scanning technique. The Fresnelregion [6] (or transition region between the nearfield and far field). which extends from about0.6 .JD3 / It to 2D2 / It, offers reduced-range testing but is still too distant for large antennas.Ranges less than 0.6.JD 3 / It define the near-fieldregion.
A compact-range reflector [7] can reduce thetest distance to two to four aperture diameters(2Dto 4D). which is within the near-field region.This technique utilizes a parabolic reflector toconvert the spherical wavefrontfrom a feed hominto a planar wavefront. A compact-range reflector can be used for adaptive nulling testsprovided that the wavefront is sufficiently freeof multipath. The reqUired reflector diameter islarge. however (approximately 2D at L-band
.' ".: frequencies and below). and sources widely:"'~paced 'in.,the test-antenna field of view are
difficult to achieve. Thus the compact-rangereflector is also not of interest here. All of theabove techniques create plane-wave illumination for the antenna, which results in impractical test geometries for adaptive nulling. Todevelop a suitable testing method, the planewave constraint must be dropped and sphericalwaves must be considered instead.
A recently developed technique called focused near-field adaptive nulling implementsconventional near-field focusing to establish aninstantaneous or real-time antenna radiationpattern that is eqUivalent to its far-field pattern[8]. This technique appears to be ideal forground-based testing. The test distance variesfrom one to two aperture diameters (D to 2D) ofthe adaptive antenna under evaluation, asshown in Fig. 3. For large antenna diametersthis test distance is located well within the nearfield boundary. The incident wavefront from
24 The Lincoln Laboratory Journal. Volume 3, Number 1 (1990)
Fenn - Near-Field Testing ofAdaptive Radar Systems
Ground Test FacilityAnechoic Chamber
AdaptiveNulling
Receiver
SignalProcessor Output
Fig. 2-Focused near-field nulling concept for a ground test of an adaptive radar system. The antenna under testandradiating sources (clutter, jammer, targets) arepositioned within a high-quality anechoic chamber. Focusing thetest antenna in the near field produces conditions equivalent to the far field. By utilizing proper timing and control, asignal environment comparable to fielded-radar system co'ncjitions can be achieved. The adaptive nulling receiver and signal processor operate in real time. This configuratipn provides a thorough test of the important radarsubsystems. .-. ~ " .; "
radiating sources in the near field is sphericalrather than planar. which allows the radiatingsource antennas to be simple horns or dipoles.Four principal papers describing focused nearfield adaptive nulling have been published bythe author [8-11). The eqUivalence betweenconventional far-field adaptive nulling and focused near-field adaptive nulling has beendemonstrated for sidelobe canceler [8) and fullyadaptive arrays [9). Near-field clutter and jamming for a sidelobe canceler have also beenaddressed [10). In Refs. 8 through 10. theanalysis assumes that the array elem~ntsand radiating sources are isotropic. Reference 1Lstudiesthe effects of array polarization and mutualcoupling. and demonstrates that the equivalence between near-field and far-field adaptivenulling still holds for single and multiple jammers. The present article expands the mutual
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
. coupling formulation to include clutter andjamming.
In the next section the characteristics of anincident signal wavefront are investigated as afunction ofsource distance and angle of arrival.This investigation is followed by a description ofhow focusing is used to establish appropriatequiescent conditions for near-field adaptivenulling. An applicati'op of the technique to adisplaced phase-center antenna (DPCA) ismade, and details ofthe theoretical formulationare given. Antenna modeling is accomplishedby using the method of moments, includingarray mutual-coupling effects. The theory isapplied to a linear array of dipole elements withdipole near-field sources (clutter andjamming).The results ~how that focused near-fieldadaptive nulling is a viable approach to testingfull-scale adaptive radar systems.
25
Fenn - Near-Field Testing ojAdaptive Radar Systems
where the quantity (r1
- r2)1L is the nonnalized
range difference. Equations 1 and 2 show thatthe far-field and near-field dispersions have acommon factor BLIc. If near-field nulling is
(1)
(2)
BL . eYFF = - sm l
c
and time delay as
where B is the nulling bandwidth. c is the speedof electromagnetic wave propagation. and e
iis
the angle of incidence. Note that the dispersionis maximum for endfire incidence (e. = 90°) and
l
zero for broadside incidence (e. = 0°). Next.l
consider a point source at a constant distanceZ = z. and variable angle e= e.. which produces
l l
an incident spherical wavefront. The dis-tances between the source and the two endpoints are denoted r
1and r
2. The near-field
dispersion YNF is given by
.. BL h - r2 )
YNF = c L
Near-Field/Far-fteIA SoUltceWavefront Dispersion
This section explains why near-field nullingcan be equal to far-field nulling. Signalwavefront dispersion (time-bandwidth product)is an effect that limits the depth of null (orcancellation) achieved by an adaptive antenna[12. 13). The amount of wavefront dispersion Yobserved by a linear array is a function of thebandwidth. array length. source range. andangle of incidence. A simple but effective dispersion model for spherical-wave incidenceand plane-wave incidence considers thewavefront dispersion observed across the endpoints of an adaptive array. This calculationgains some initial insight into how near-fieldnulling relates to far-field nulling.
Consider a plane wave arriving from infinityand an array of length L. The far-field dispersion for this case is denoted YFF and is computed according to the product of bandwid~h
1000100
~oree
"::u....
.--.:::-=--------..".- 15 = 4
~~~~~~~~ 15= 2
~~~~~~~Q-~ 15 = 1
100.1 LL....---!.l..-~J......L..I...L.I.I.l..____:~--I...u...:...JU:...L.il.l.J'_____l_WI..ll.l1 U-L..LLJ
1
QNQ)ucell(j)o(j)Q)
I"0Q)
.!::!""iiiEoz
Antenna Dimension Of?.
Fig. 3-Summaryofantenna test regions. The hemispherical volume in frontof an antenna can be divided into a number of test regions. The three basicregions are far field, Fresnel, and near field. The near-field region (shown inred) can be further divided into compact range (a reflector-based method),focused near field (which is addressed in this article), and planar near field.The focused near-field region is located between one and two aperturediameters from the antenna under test.
26 The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
Fig. 4-Normalized wavefront dispersion as a function ofsource angle of arrival. The wavefront dispersion (timebandwidth product), observed relative to the end points ofan antenna, is a simple but effective model for comparingthe characteristics of near-field and far-field sources.Maximum dispersion occurs when the source (either nearfield or far field) illuminates the antenna from endfire(e. = 90°). For source distances of one aperture diameterof more, the difference between the near-field and far-fielddispersion is small.
possibly eqUivalent to far-field nulling. then YFF
must be eqUivalent to YNF
• This condition isclearly satisfied when (r
1- rz)/L = sinei' Figure
4 shows a plot of the normalized dispersiony!(BL/ c) as a function of the angle of incidence for values of source distance from 0.25 to2 aperture lengths (Le., the normalized sourcedistance is varied from z/L = 0.25 to 2). Thisfigure shows that the near-field. dispersionapproaches the value of the far-field dispersionfor source distances greater than approximately one aperture diameter. or z/L= z/D'2 1(in this article, aperture length L and aperturediameter Dare eqUivalent and interchangeable).At one diameter. the percent difference betweennear-field and far-field dispersion is less than10%. At two diameters. the percent differenceis less than 3%. Clearly. at source distancessuch that z/D '2 0.5 (one-half aperture diameter), the near-field dispersion is significantly different (by as much as 30%) com-
~ 1.0-JCO-- 0.8?--
c::0
0.6'wCDa-li)
0.40uCD.!:! 0.2(ijE0 0.0Z 0 30 60 90
Source Angle (Jj (deg)
Fenn - Near-Field Testing ojAdaptive Radar Systems
pared to the far-field dispersion.At source distances of one to two aperture
diameters, the incident near-field wavefrontdispersion appears to be comparable to that ofa far-field wavefront. However. before the radarsystem attempts to perform near-field adaptivenulling, the receive-antenna quiescent conditions must be made to appear the same as farfield quiescent conditions. This requirement isachieved by focusing the antenna under test asdescribed in the next section.
Focused Near-Field Testing Concept
The near-field testing technique described inthis article assumes that the array quiescentnear-field radiation pattern has the same characteristics as the quiescent far-field radiationpattern. This assumption requires the formation of a main beam and sidelobes in the nearfield. The dynamic range of received signalsfrom sources distributed across the radar fieldof view depends upon the antenna quiescentconditions. Phase focusing can be used to produce an array near-field pattern that approximately equals the far-field pattern [14].·.Figure 5 shows a continuous wave (CW)
. "cfi1.ibfati,9p. source located at a desired focalp:bint· of the' array. The array maximizes thesignal received from the calibration source byadjusting its phase shifters so that the spherical wavefront phase variation is removed. Thefirst step to determine this adjustment is tochoose a reference element; this reference isusually the center element of the array. Thevoltage received at the nth array element relative to the center element of the array is computed in this article by using the method ofmoments [15]. To maXimize the received voltage at the array output, the phase conjugate ofthe incident wavefront must be applied at thearray elements. The resulting instantaneousarray radiation pattern seen at the source testplane z= zolooks similar to a far-field pattern[11]. In this p~ttern a main beam points at thearray focal poiht. and sidelobes exist at anglesaway from the main beam. Interferers are thenplaced on near-field sidelobes in the test (orfocal) plane. as shown in Fig. 5. Similarly.
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990) 27
Fenn - Near-Field Testing ojAdaptive Radar Systems
Auxiliary Channels
Array Plane z = 0
/ Focused Array Pattern
z
AdaptiveWeights
y\
W\\\ X\\ Planar
W W ••• W Array
····Adaptive :Beamformer :
•..··(CW-Caiibratio;;Source~CiUtte(:Targeif·-----N~~:R~I~jT~~tPi~~-;::D--:::··..•.....W ~ Jammers--+-W •.•..•••.
~ I . ~•••• Focal ;- .... ~.... Point \, '+' ••••.~--------------------- ._-~---~ ---------------_._----------------------------~,,,,,,,,,,,
Focusing
. Output._-_._------------------:------~-------~~~
Fig. 5-Focused near-field adaptive nulling test concept. A CW radiating source is used as a calibration signalfor a phased-array antenna. The antenna-elementphase shifters focus the receive antenna radiation pattern inthe direction of the calibration source. This focusing creates an antenna radiation pattern that is similar to a farfield radiation pattern. Clutterand targetsources can beplacedon the main beam as desired. Similarly,jammerscan be positioned on sidelobes.
clutter sources are positioned on the near-fieldmain beam. Near-field focusing that uses onlyphase control results in some distortion of themain beam and first sidelobes. Amplitude control can make the near-field pattern main beamand first sidelobes more closely resemble a farfield pattern [16]. For simplicity, this articledescribes phase focusing only. "
The minimum size of the required groundtest facility can be determined by consideringthe near-field geometry. For compatibility withconventional planar near-field scanning equipment, a flat test plane is assumed. Let ()maxdenote the maximum angle of interest for the
antenna radiation pattern. The reqUired nearfield scan length D for pattern coverage of±()x maxis given in terms of the F/L ratio as
Dx = 2L( ~) tan ()max'
Figure 6 depicts the reqUired scan lengths for600 and 1200 field-of-view coverage with F/ Lratios ofLand 2L. To reduce the scan length (orsource deployment length) the F/Lratio must bekept as close to unity as possible. Clearly, theground-test facility must be large enough toencompass both the desired scan length D and
xthe focal length F.
28 The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
Fenn - Near-Field Testing ojAdaptive Radar Systems
Fig. 6-Examples of source deployment and scan lengtho for (a) 60 0 field of view and (b) 120 0 field of view. Thesefi§ures illustrate the importance of a close test distance.
Adaptive DPCA Radar
are not analyzed here.
The DPCA technique can be applied to airborne or spaceborne radar systems that requireadaptive suppression of jamming and clutter(2). A DPCA array cancels stationary groundclutterfrom a moving platform by employing twoor more independent receive phase centers withwell-matched main beams. Figure 7 shows amoving target and a moving radar in a twophase-center DPCA system. On transmit, thefull aperture ofthe array sends a burst ofpulses;on receive, two displaced portions of the aperture record the returned signals. Because of theplatform motion, two consecutive transmitpulses occur corresponding to the transmitphase-center positions AT and BT. The radartransmit signal illuminates both moving targetsand fixed ground terrain that, because ofthe radar cross section, produces the desired signalsand clutter. Ground-based emitters also represent a source of interference or jamming.
On reception, the antenna phase-center displacement between the two receive apertures isadjusted to compensate for the platform veloc-
.'it}i::;For two transmit pulses separated in time byone pulse-repetition interval (PRJ), the first reception occurs at the forward receive phasecenter position, denoted A in Fig. 7. A secondreception occurs at the trailing receive phasecenter position B. This bistatic radar system iseqUivalent to a monostatic radar system thatmakes two independent observations of thesignal environment at a single point in space.This common point, denoted AB, is located atthe midpoint of the line joining either points ATand A or BT and B. During a' PRJ, the targetmoves while the clutter is effectively stationary.As a result the target produces a relative phaseshift dUring this time, while the clutter has nophase shift. The clutter is assumed to be correlated between the two phase centers. In contrast, widebanp noise jamming is assumed to beuncorrelated between the two phase centersdue to the one-PRJ delay imposed in the signalprocessing. When the signals received by thetwo phase centers are adaptively combined, the
FIL =2FIL = 1
".
10=OmaxI.,,
.. x
z
+;--0Ox= 2.4L
I
zA:-0
Ox= l.OLI
I
" I ",OX = 1.2L ,,. I .!
, I,A'T"*>.,,60~
'II
,,,, ..
II
Ox=3.5L........,.... 1... ..... "
.... I ".
.... ........--r---- ".... ..-.... 1200;~'"
.... ,.... • X
(b)
(a)
.....
The application ofthe focused near-field testing concept to a specific radar system requiresthe following important assumptions:(1) the incident near-field wavefront must be
reasonably well matched to a far-fieldwavefront;
(2) the adaptive antenna under test must befocused at the range of the test sources;
(3) the antenna characteristics, beamformercharacteristics, and receiver characteristics, such as channel mismatch, must beindependent of the type of wavefront (nearfield or far field); and
(4) the technique must be applicable to bothanalog and digital adaptive nullingsystems.
The analysis in this article accounts for assumptions 1, 2, and part of 3 (antenna characteristics). The remaining assumptions,which are not expected to limit the technique,
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990) 29
Fenn - Near-Field Testing ojAdaptive Radar Systems
Clutter andMoving Target
Fig. 7-Displacedphase-center antenna (DPCA) radar system concept. Receive phase centers (or subarrays) A and Bcompensate for the radar motion by creating a phase-center displacement. To cancel stationary main-beam clutter andsidelobe jamming, and to detect the target, the radar system effectively makes two measurements of the signalenvironment at the common point AB and adaptively combines the phase centers.
and
clutter and jammer are significantly canceled. ments is frrst split into two paths that areThe corresponding target signal depends on the weighted and summed in separate power com-amount of target phase shift dUring one PRJ biners to form two independent subarray maininterval (0° phase shift produces no target signal,' -:', channels (or movable phase centers). In eachwhile 180° phase shift produces maximum tar- - element channel is a TIR module that hasget signal). The DPCA quiescent main-beam amplitude and phase control. The amplitudepattern match is a function of array geometry control provides the desired low-sidelobe arrayand scan conditions (due to array-element illumination function and phase-center dis-mutual coupling), and hardware tolerances placement. The modules utilize phase shifters(such as the quantization and random errors of that steer the main beam to a desired angle.the transmit/receive [T/R] modules). Both T/R Letmodule effects and array mutual coupling aretaken into account in the next two sections ofthis article.
Adaptive DPCA Array Formalism
Consider the DPCA array and adaptivebeamformer as shown in Fig. 8. The arraycontains N elements that form the receive mainchannel. Included within these elements is aguard band of passively terminated elementsthat provides impedance matching to the activeelements and isolation from ground-planeedges. The output from each of the array ele-
30
denote the array-element weight vectors (including quantization and random errors) ofphase centers Aand B, respectively (superscriptT means transpose). To effect phase-centerdisplacement, a portion of each subarray isturned off by applying a large value of attenuation for a group of antenna elements. Thisaction moves the electrical phase center to thecenter of gravity for the remaining elements.
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
Fenn - Near-Field Testing ojAdaptiveRadar Systems
•••
N
•••Phase Center A
AAuxiliaries
VN-2VN-1
•••
Array Elements
V··· V
Auxiliaries
•••Phase Center B
AdaptiveWeights
ArrayModules
Adaptive Beamformer Output
Fig. 8-Adaptive beamformer arrangement for DPCA operation. The output from each antenna element is split into twopaths independently weighted with the array modules that contain amplitude andphase control. The outputs ofthe mainchannels andauxiliary channels are adaptively weighted to null the interference. The vertical arrow entering the adaptiveprocessor refers to the input data vector consisting ofsamples.ofp}e main and auxiliary channels. The horizontalarrowsexiting the adaptive processor refer to the adaptive weight commands. The jamming signals in phase centers A and Bare canceled at points A . and B', while the clutter is cancele.d qf the final outpUt of the adaptive beamformer.
adaptive weights (no quantization or randomerrors) are assumed. with
where * means complex conjugate and E(·)means mathematical expectation. (For notational convenience. note that the superscript)in v and v in Eq. 3 has been omitted.) Since
m n
(3)
denoting the adaptive-channel weight vector.The fundamental quantities required to characterize the incident field for adaptive nullingpurposes are the adaptive-«;::hannel crosscorrelations.
The cross-correlation RJ of the receivedmnvoltages in the mth and nth adaptive channels,due to thejth source, is given by
Thus the effective number of elements actuallyused to receive signals in phase centers A and Bare denoted by N
Aand N
B• respectively.
When a wavefront (either planar or spherical)due to the)th source (either clutter or jammer)passes across the array, the result is a set ofarray-element received voltages denoted by
Let M be the number of adaptive channelsper phase center. For a sidelobe cancelerM = 1 + N where N is the number of auxili-
a= a= "ary channels in each phase center". This adap-tive system has M degrees of freedom in eachphase center and thus a total of 2M degrees offreedom for the combined phase centers. Foreach phase center, the main- and auxiliarychannel voltages are derived from the above setof array received voltages. In this article. ideal
The Lincoln Laboratory Journal, Volume 3. Number 1 (1990) 31
Fenn - Near-Field Testing ojAdaptive Radar Systems
Wq = (1, 0, 0, ... , -1, 0, 0, ... , O)T.
where W is the quiescent weight vector (12].q
For a dual-phase-center side10be canceler, thequiescent weight vector is chosen to be
(7)
(6)
C = Pa.Pq
Substituting Eq. 6 into Eq. 7 yields
The adaptive-array cancellation ratio, denoted C, is defmed here as the ratio of interference output power after adaption to the interference output power before adaption,
'TC = w a RWa
'T .w q RWq
Next, the covariance matrix defined by the elements in Eq. 3 is Hermitian (that is, R =RoT). Bythe spectral theorem, R can be decomposed ineigenspace as
where *T means complex conjugate transpose.The interference-plus-noise-to-noise ratio.denoted INR, is computed as the ratio of theoutput power with the interferer present (defined in Eq. 6) to the output power with onlyreceiver noise present; that is.
*TRINR = W *T W.
W W
Thus the main-channel weights are ± 1 and theauxiliary-channel weights are zero.
The output power at the adaptive-arraysumming junction is given by
Prior to generation of an adaptive null, theadaptive-channel weight vector W is chosen tomaintain a desired quiescent radiation pattern.When undesired signals are present, the optimum set of weights w a to form one or moreadaptive nulls is computed by
.' ....,.
jthTransmitting
Antenna
~sourcel
where B =12 - 11 is the nulling bandwidth. Notethat Eq. 4 accounts for the spherical or planarshape of the wavefront.
Let the channel or source covariance matrixbe denoted R. If J uncorrelated broadband interference sources exist, then the J-source covariance matrix is the sum of the covariancematrices for the individual sources. Thus
+v..rf!cn,]
12
Rinn = ~ f Um (1) u~ (1) dl (4)
JI
U and u represent voltages of the samem n .
waveform, but at different times, RJ is alsomn
called an autocorrelation function.In the frequency domain, assuming the
source has a band-limited white-noise powerspectral density, Eq. 3 can be expressed as thefrequency average
J
R = I R) + I (5))=1
where R. is the covariance matrix of the jthJ
source, and I is the identity matrix that repre-sents the thermal noise level of the receiver.
Fig. 9-Receive array and near-field source antennamodel. The quantityZ represents the mutual impedancebetween array elemerp,fs, while Z . represents the mutualimpedance between the nth elemJnt and the jth transmitting antenna.
2M
R = 1: Akeke~Tk=l
where Ak
, k = I, 2, .... 2M are the eigenvalues
32 The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
ofR, and ek
, k = 1,2, .... 2M are the associatedeigenvectors of R (17). The multiple-sourcecovariance matrix eigenvalues (AI' ,1,2' .••• A2M)
are a convenient quantitative measure of theutilization ofthe degrees offreedom of the adaptive array. Because the identity matrix was added to the covariance matrix. the minimum amplitude that an eigenvalue can have is 0 dB. (thereceiver noise level). The number ofeigenvaluesabove the receiver noise level directly indicateshow many degrees of freedom are used to suppress the undesired signals [13. 18).
Array Antenna/Source Modeling
This section applies the method of moments.already mentioned on p. 25, to compute thearray-element received voltages (given in Eq. 3)due to near-field or far-field sources. The farfield formulation in this article is similar to theformulation considered by I.J. Gupta and A.A.Ksienski (19). Assume that each element isterminated in a known load impedance ZL(Fig. 9). Let v . represent the open-circuit
nJvoltage in the nth array element due to thejth source. The jth source can denote eitherthe CW calibrator-a movable source probe forsampling the near-field radiation pattern-orone of the jammer or clutter sources. Next. let Zbe the open-circuit mutual-impedance matrixfor the N-element array. The array elementsare assumed to be dipoles over an infiniteground plane. The array received-voltagematrix, denoted v~ec. due to thejth source. can
Jbe expressed as
(20). In Eq. 8 the nth element ofv. is computed.J
for near-field sources, by the relation
where i. is the terminal current for thejth sourceJ
and Z . is the open-circuit mutual impedancenJ
between the jth source and the nth array ele-ment. For thin-wire array antenna elements,the moment-method expansion and testing
The Lincoln Laboratory JournaL Volume 3. Number 1 (1990)
Fenn - Near-Field Testing ojAdaptilJe Radar Systems
functions are assumed to be sinusoidal. Theabove open-circuit mutual impedances arecomputed on the basis of modified subroutinesfrom a well-known moment-method computer code (21). In the modified subroutines.double-precision computations are necessaryto evaluate Z .for thejth interferer. For far-field
nJsources. V
nJis evaluated by assuming plane-
wave incidence. The main-charmel output iscomputed by using WA'Tv ilj for receive phasecenter A-and W;TVBJ for receive phase center B.where vAJ and v BJ are the received voltages ofphase centerA and B, respectively. duetothejthsource.
As mentioned earlier, each phase center ofthe array is initially calibrated (phase focused)by a CW radiating dipole. To accomplish thiscalibration numerically. after computation ofthe CW received voltage the receive-array weightvector WA (or WB) has its phase commands setequal to the conjugate of the correspondingreceived phases. Receive-antenna radiationpatterns are obtained by scarming (moving) adipole with half-length l in either the far- field ornear-field region and computing the antennaresponse. Far-field receive patterns are computed by using a 8-polarized dipole source atir'finity to generate plane-wave illumination ofthe array. The open-circuit voltage is then setequal to the amplitude and phase ofthe incidentfar-field wavefront. For a far-field source. theincident wavefront amplitude is a constant andthe phase varies linearly from element to element. The coordinates (x. y. z) specifY a nearfield point in front of the test antenna. Principalplane near-field radiation pattern cuts (versusangle) are obtained by computing the nearfield on the line (x;. O. z) with the relation8(x) = tan,l(x/z). The near-field source is anx-polarized dipole with half-length l. LetvxNF(e) denote the voltage received by the arraydue to the x-directed near-field dipole. and letpie) denote the 8 component of the dipoleprobe pattern. Then the probe-compensated array near-field received pattern is expressed as
(9)
33
Fenn - Near-Field Testing ojAdaptive Radar Systems
PhaseCenter
A
PhaseCenter
B
J = JammerIC = Clutter Digitally Controlled SourcesT = Target
Fig. 1Q-Near-field source positioning for a displacedphase-center antenna. Two sets of sources, "A Hand UB, "are used one set ata time to illuminate the test antenna.
(10)
the impedance matrix Z is computed at K frequencies and the system of equations given byEq. 8 is solved at each frequency. The interference covariance matrix elements are computedby numerically integrating Eq. 4 according toSimpson's rule. For multiple sources, the covariance matrix is evaluated by using Eq. 5.Adaptive-array radiation patterns are computed by superimposing the quiescent radiationpattern with the weighted sum of auxiliarychannel received voltages.
where the value
(8) = cos(f3l sin 8) - cos(f3l) .Po cos 8
The propagation constant f3 is 2n/A.The array received-voltage matrix for thejth
source (denoted v:ec) is computed at K frequencies across the nulling bandwidth. To obtain thereceived voltages
DPCA Near-FieldSource Distribution
Figure 10 shows the position of near-field- .-"
sources for two-phase-center DPCA operation;'~"
Two sets of sources exist, one set for phase'centerA and one set for phase center B. Multipieclutter sources are distributed across the mainbeam of both phase centers. The figure alsoshows a desired target signal embedded withinthe clutter signals. Jammer signals are assumed to radiate from antennas located withinthe sidelobe region. As mentioned before. theclutter is correlated between phase centers Aand B. As an example, denote the first cluttersource in phase center A as CAl (8
1) and the first
clutter source in phase center B as Cal (81),
Clearly. in theory CAl (81) equals Cal (81), Similarequalities exist for the remaining cluttersources. To achieve this correI'ation in an experimental configuration requires digitally controlled arbitrary-waveform generators. Thesesignal generators can also create the jammersand desired target signals. The two sets ofsources must be operated dUring different time
intervals separated by the radar PRJ delay.Thus, for example, a measurement ofthe phasecenter A signals is performed with only the MAWgroup of sources radiating. The next measurement (o~e PRJ later) is for the phase center Bsignals with only the MB" group of sources radiating. This switching is implemented by usingtiming and control, as suggested earlier inFig. 2.
The near-field source antennas do not interact through mutual coupling in such a way thatthe adaptive nulling performance is modified.For example, with one source radiating, thesurrounding source antennas represent possible multipath" The near-field sources areknown to couple. but this coupled interferencedoes not influence the adaptive weights, provided that the coupled signal is reradiated andreceived at below the receiver noise level. Themultipath contribution between two antennas(one radiating and the other passively terminated in~ a load) is accurately computed asfollows. Let II be the terminal current generatedon the active source antenna. Similarly, let 1
2denote the parasitic current generated on the
34 The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
Fenn - Near-Field Testing ojAdaptive Radar SystemsTT<ansm;n;ng Source D;pole
Dipole Array
TerminatedElements
TerminatedElements
I m (Receive Current)
---..-+- -+- -0- -0- ••• -0- ••• -0-... -0- -0- -+- -+-
1 23m n 146 147 148
777777777777777777777777777777777777Infinite Ground Plane
1.....·1--------------0-----------·..1
Fig. 11-Geometry for dipole receive array and dipole source antenna. The transmitting source antenna re~·
resents clutter, jammer, the target, and noise.
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990)
Near-Field/Far-Field Simul~tions
passive antenna. From circuit theory the ratio ofthe parasitic current to active current is given by
This section analyzes a specific adaptiveDPCA array and demonstrates the eqUivalencebetween near-field clutter andjammer suppression and far-field clutter and jammer suppression. Consider a corporate-fed phased-array16-m antenna that consists of a single row of
receive dipole elements. The array, which has atotal of N = 148 elements with two elements ateach end used as passive terminations, has 144active receive antenna elements. The antennaelements are one-half-wavelength-long electrically thin dipoles that are center fed and spacedone-quarter wavelength above an infiniteground plane (Fig. 11). The center frequency is
·,'ch'osen to be 1.3 GHz (L-band) and the interele~ent'spaCing' is 10.922 cm, or 0.473 wave-
, lengths. Thus the active portion of the arrayspans 15.61 m. The output from each activereceive antenna element is divided into twopaths to form two independent phase centers.The T/R modules are chosen to have 5 bits ofamplitude and phase control with rms errors of0.3 dB and 3.0°. The load impedance ZL isassumed to be 50 Q resistive at each array element.
The near-field test distance' z is chosen tobe 15.61 m, which corresponds to one activereceive aperture diameter. Seven auxiliarychannels. randomly selected from the elementoutputs of one ofthe phase centers, form a multiple-sidelobe canceler configuration. This random pattern repeats in the second phase center.Since the value ofN is 7 in each phase center,
awethe total number ofdegrees offreedom is 16. Thechannel covariance matrix is dimensioned16 x 16 and has 16 eigenvalues. The receiver
35
(11)1121 IZ 211TIJ = IZ22 + ZL I
where Z21 is the open-circuit mutual impedancebetween the two antennas, Z22 is the self-impedance of the passive antenna, and ZL is the loadimpedance of the passive antenna. Clearly, asmall value of mutual impedance is desired.Equation 11 will be used later to verify that themutual coupling between source antennas issufficiently small for a particular near-fieldsource configuration. An important point to bestressed here is that mutual coupling betweensource antennas in a particular near-field testconfiguration needs to be carefully evaluated.However, with proper source antenna design.and judicious placement of anechoic materialbetween source antennas, mutual couplingshould not be a problem.
FenD - Near-Field Testing ojAdaptive Radar Systems
60-30 0 30
Probe Angle 9 (deg)
-80 L..-__..L-__......l...__--I.__---J
-60
Or----~--,.---__,_-----,
synthesize a -40-dB uniform-sidelobe-levelChebyshev radiation pattern (in the absence ofT/R module errors) with a scan angle Osequal to-300
• Assume that the phase centers are fullysplit apart. so that the effective number ofreceive elements per phase center (N
Aand N
B) is
one-half of 144, or 72. This number gives aphase-center separation of 7.86 m. Figure 12shows the subarray amplitude illuminationfunction for phase centers A and B. The expected random amplitude error of the T/R
"OffState"-
10
1+-----16m----..
-10
0.15
OJ
~ 0.10C.~ 0.05
o
-5 0 5
Element Position x (m)
Fig. 12-SimulatedOPCA illumination functions for the 16m linear test array. The phase-center displacement t:. iscreated by turning off the left half of the array for phasecenter A and the right half of the array for phase center B.
bandwidth (also called the nullingbandwidthl is1 MHz. The auxiliary channels are attenuatedby 20 dB to have a signal output power comparable to the main channels. In a practical radar.auxiliary-channel attenuation will be implemented to keep the dynamic range of signalswithin the limits of the adaptive nullingreceiver.
Let the array illumination be chosen to
0.20 r---.-.:..----r---'-~--_,
O.---.-....,.....--r----..,~--r--__r_-___,
! !ArrayB A
Range = D..........~ .
~ ~'·9: ·9:'" .,., .,
-- Phase Center A
-- Phase Center B
Fig. 14-Near-field radiation patterns at one-aperture-diameter test distance for phase centers A and B, as afunction ofobservation angle. The patterns are computedfrom Eqs. 9 and 10 by using the simulated near-field datain Fig. 13.
. '-. ~ ..::., - ...
105o-80 L...-_..l-_--l..._---l._--I.__..l-_...J
-20 -15 -10 -5
CD"-;; -20
".~C.E -40~
.~iii -60a;a:
Probe Position x (m)
Fig. 13-Simulatednear-fieldprobe scan at one-aperturediameter test distance for the OPCA 16-m linear testarray.The source frequency is 1.3 GHz and the receive-arrayscan angle is -30°.
-- Phase Center A
-- Phase Center B
CW Probe Scan•••••••• , ••••••••x
FIO= 1
~rrayB A
modules makes the illumination functionsslightly different from one another. Notice howeach illumination function is equal to zero over7.86 m. This illumination shifts the apparentphase center by 3.93 m to the left of the antennacenter for.phase center B and 3.93 m to the rightof the antenna center for phase center A
To phase-focus the DPCA array in the nearfield to the distance z = 15.61 m and angles° = -300 (with respect to each phase center).sa CW radiating dipole source is positionedat x = -12.95 m for phase center B and at
36 The Lincoln Laboratory Jownal. Volume 3. Nwnber I (l990)
Fenn - Near-Field Testing ofAdaptive Radar Systems
O.-------r---.......---r-----,Figure 13 shows the resulting near-field received amplitude distribution for both phasecenters. Notice that the peak amplitudes occurat the desired locations. The main beams arefully separated so that one phase center has apeak when the other phase center has a sidelobe. While the test distance is specified as oneaperture diameter for the full length ofthe array.the test distance effectively appears to be twosubarray. diameters for the displaced phasecenters. Figure 14 shows the near-field datareplotted as a function of angle with respect toeach phasecenter. The near-field main beams ofphase centers A and B are clearly well matchedas desired in a DPCA system. Because ofphasecenter displacement. the near-field patternscover different angular sectors. Figure 15 showsthe corresponding far-field radiation patterns ofphase centers A and B. A good main-beammatch is also apparent in this figure.
Figure 16 compares near-field and far-fieldradiation patterns. Figures 16(a) and 16(b) showthe radiation patterns for phase centers B andA, respectively. The near-field main beam agreeswith the far-field main beam down to -20 dB.Amplitude calibration can compensate for adefocusing ofthe near-field main beamand first
._~g:telobes. as mentioned earlier. Although thenear-field sidelobes do not match the far-field
60
Range = 00......~'~~...-.0: -.0:., .,I, I,
b bArrayB A
-30 0 30Probe Angle () (deg)
-80 L..-__.....L.. I....-__-'-__--J
-60
-- Phase Center A
-- Phase Center B
CD:£. -20Gl"0
E0..-40E«Gl>~-60Qia:
Fig. 15-Simulated far-field radiation pattern for the OPCA16-m linear array. The focal distance and observationdistance are both set to infinity.
x = -5.09 m for phase center A. To focus thesubarrays. the conjugate of the momentmethod-calculated element phases is applied tothe receive modules. The CW source is scannedacross 26.29 m and the array output is computed at uniformly spaced probe positions.
Far Field (F10 = 00 )
Near Field (FlO = 1)
O,....---~r;----.-----.-------,
CD~ -20OJ"0
.-Ea.E -40<:OJ>.~
(jj -60a:
(a)-80 L- .L.- L- L-__---'
-60 -30 0 30 60 -60
Probe Angle e (deg)
(b)
-30 0 30
Probe Angle e (deg)
60
Fig. 16-Comparison of simulated near-field and far-field OPCA radiation patterns (before adaptive nulling) previouslyshown in Figs. 14and 15, respectively. The (a) phase centerB radiationpatterns and (b) phase centerA radiationpatternsestablish the quiescent conditions for the adaptive antenna.
The Lincoln Laboratory Journal. Volume 3. Number 1 (1990j 37
Fenn - Near-Field Testing ojAdaptive Radar Systems
sidelobes on a point-by-point basis. the averagesidelobe levels are equal.
When proper quiescent conditions are established. seven clutter sources are uniformly distribu ted across the main beam ofboth near-fieldand far-field patterns. For this dual-phasecenter example. seven clutter sources exist perphase center. In each phase center. all squrcesare assumed to have equal power and allsources are uncorrelated. Note that equalpower sources are chosen for convenience. Thesources are distributed in angle over a 50 sectorcentered at the beam peak. and they cover themain beam down to -20 dB. The total power thatthese sources produce in one phase centerequals 40 dB relative to receiver noise. Anincrease in the number of clutter sources beyond seven does not significantly influence theadaptive nulling results that follow. Finally. letone jammer be positioned at () = -;-200 to produce an output power for the combined phasecenters of 50 dB above noise.
As mentioned earlier. mutual couplingamong near-field signal sources can be animportant consideration. In the current example. since the sidelobe jammer power is largethe coupling between the radiating jammerantenna and a clutter antenna is the most
important case to consider. The sidelobe level atthe jammer position is approximately -35 dBdown from the main-beam peak. The mainbeam level at the nearest clutter antenna position is -20 dB. Thus a parasitically generatedjammer signal at the clutter antenna is effectively increased by 15 dB at the test antennabecause of the pattern directivity increase.Without the contribution of mutual coupling.the parasitic jammer power in the main beamwould be 65 dB above noise. The parasiticjammer current at the clutter antenna iscomputed by using Eq. 11. The self-impedance of a one-half-wavelength clutter dipoleis ~2 = 73 + j42 Q. The separation betweenthe jammer and the nearest clutter antenna is approximately lOA.. For this spacing. themutual impedance is computed to beZ21 =-0.03748 - jO.000618 n. Substitutingthese values and the load impedance (ZL= 50 Q)into Eq. 11 yields
11211111 = 0.000289.
In decibels. the parasitic jammer current isdown by -71 dB. The parasitic jammer signal isthen the difference between 65 dB and 71 dB.
: ~or -6 dB below receiver noise. According to. '-
Far Field (F /0 = 00 )
Near Field (F /0 = 1)
Or-----.......----,.----r------.
60
Moving Target(Phase Shift = 180°)
-30 0 30
Probe Angle (J (deg)
(b)
Cancellation
49.3 dB Far Field48.0 dB Near Field
(a)-80 L-- -'----L__-J-_~_-J... ......I
-60 -30 0 30 60 -60
Probe Angle e (deg)
Clutter
ro -+- ...-~ -20QJ"0
.Ea.E -40~QJ>~Qi -60a:
Fig. 17-Adapted radiation patterns for the combinedphase centers of the OPCA 16-m lineararray for the (a) stationarytarget and (b) moving target. The near-field simulation is ata test distance of one aperture diameter and the far-fieldsimulation is ata testdistance ofinfinity. Seven white-noisecluttersources are distributeduniformlyacrossthe main beam,and one white-noise jammer is in a sidelobe. The nulling bandwidth is 1 MHz.
38 The Lincoln Laboratory Journal. Volwne 3. Nwnber 1 (1990)
theoretical simulations. this power level doesnot affect the computation of the adaptiveweights.
For the signal environment. the covariancematrix was computed and the adapted weightswere derived and then applied to cancel theinterference. Figure 17 shows the near-field andfar-field adapted radiation patterns. In Fig.17(a). both the clutter and jammer are clearlysuppressed by the pattern nulls. and the patterns are very similar. The total cancellation ofjamming and clutter power is 48.0 dB in thenear field and 49.3 dB in the far field. As theamount ofcancellation is large. this small difference (1.3 dB) is insignificant. The radiationpatterns in Fig. 17(a) show a stationary targetwhose signal is effectively canceled because ofzero phase shift in one PRI. In contrast. areceived signal from a moving target that produces a 1800 phase shift in one PRI sees theantenna radiation pattern shown in Fig. 17(b).where full antenna gain is available in the mainbeam direction. Finally. Figure 18 shows a plotof the covariance matrix eigenvalues. A total ofeight eigenvalues above receiver noise indicatesthat eight degrees offreedom are consumed. Thenear-field and far-field eigenvalues are in goodagreement over a large dynamic range. By taking all ofthe above results into consideration. wecan now state that. for all practical purposes.near-field nulling is eqUivalent to far-fieldnulling.
Fenn - Near-FY.eld Testing ofAdaptiveRadar Systems
Conclusion
A theory for analyzing sources radiating inthe near field of an adaptive radar system isdeveloped. Conventional phase focusing of anarray is used to create antenna far-field patternconditions in the near-field region. Cluttersources are distributed across the main beam ofthe focused antenna pattern. and jammers arepositioned within the sidelobes. The method ofmoments is used to analyze a displaced phasecenter antenna linear array with near-field andfar-field clutter and jamming. Focused nearfield adaptive nulling is shown to be eqUivalentto conventional far-field adaptive nulling. Thenear-field range distance can be one aperturediameter. which opens the possibility for indooranechoic chamber testing. Thus a radar systemdesigned for far-field conditions can potentiallybe evaluated by using near-field sources. Thistechnique is particularly attractive for spacebased radar systems for which prelaunchground testing is desirable. With this method.integrated testing of a phased-array antenna.receiver. and adaptive signal processor can beperformed. Array calibration. antenna radiation patterns. adaptive cancellation. and targ~t detection can be verified. Experimental
. \i.etification ,of this technique is currentlyunder investigation.
Acknowledgements
Far Field (F 10 = 00 )
80
60CD~Q) 40u.~
20i3..E~
0
-201
ReceiverNoiseLevel
16
Initial development of the near-field testingtechnique was encouraged by V. Vitto and H.Kottler. and their support is appreciated. Technical discussions with G.N. Tsandoulas. R.W.Miller. J.R. Johnson. H.M. Aumann. F.G. Willwerth. E.J. Kelly. D.H. Temme. and S.C. Pohligwere valuable in the development of this work.The software support of S.E. French is alsoappreciated.
ReferencesIndex
Fig. 18-Covariance matrix eigenvalues for near-field andfar-field source distributions. Eigenvalues 1 to 8 are abovethe receiver noise level and represent the consumption ofeight degrees of freedom.
The Uncoln Laboratory Journal. Volume 3. Number 1 (1990)
1. L.J. Cantafio ... ed., Space-Based Radar Handbook (Artech House. Dedham, MA, 1989).
2. E.J. Kelly and C.N. Tsandoulas, ~A Displaced PhaseCenter Antenna Concept for Space Based RadarApplications: IEEE 16th Ann. Electron. and Aerospace Con] and Expo.• Washington. DC. 19-21 Sept.
39
Fenn - Near-Field Testing ojAdaptive Radar Systems
1983. p.141.3. G.N. Tsandoulas. "Space-Based Radar." Science 237.
257 (1987).4. AD. Yaghjian. "An Overview of Near-Field Antenna
Measurements." IEEETrans. Antennas Propag.AP-34.30 (1986).
5. AJ. Fenn. F.G. Willwerth. and H.M. Aumann. "Displaced Phase Center Antenna Near Field Measurements for Space-Based Radar Applications: PhasedArrays 1985 Symp. Proc.• BedJord. MA. 15-18 Oct.1985. p. 303. RADC-TR-85-171,ADA-169316.
6. C.H. Walter, Traveling Wave Antennas (Dover, NewYork. 1970). p. 38.
7. RC. Johnson. H.A Ecker. and RA Moore, "CompactRange Techniques and Measurements." IEEE nuns.Antennas Propag. AP-17, 568 (1969).
8. AJ. Fenn. "Theory and Analysis ofNear-Field AdaptiveNulling." 1986 IEEE AP-S Symp. Digest. VoL 2 (IEEE.New York. 1986). p. 579.
9. AJ. Fenn. "Evaluation of Adaptive Phased Array Antenna Far-Field Nulling Performance in the Near-FieldRegion," IEEE Trans. Antennas Propag. AP-38. 173(1990)
10. AJ. Fenn. "Theoretical Near-Field Clutter and Interference Cancellation for an Adaptive Phased ArrayAntenna." 1987 IEEE AP-S Symp. Digest. VoL I (IEEE.New York. 1987). p. 46.
11. A.J. Fenn. "Moment Method Analysis of Near-FieldAdaptive Nulling." lEE 6th Inti. Con] on Antennas andPropagation. ICAP 89. Coventry. UK. 4-7 Apr. 1989. p.295.
12. RA Monzingo and T.W. Miller. Introduction toAdaptiveArrays (Wiley, New York. 1980). p. 253.
ALAN J. FENN is a staffmember in the Space RadarTechnology group. wherehis research is in the development of near-field testingtechniques for adaptive ra
dar systems. He has a B.S. degree from the University ofIllinois at Chicago, and M.S. and Ph.D. degrees from OhioState University, all in electrical engineering. Before comingto Lincoln Laboratory in 1981, Alan worked for MartinMarietta Aerospace Corporation in Denver, Colo. He iscurrently an associate editor in the area of adaptive arraysfor the IEEE Transactions on Antennas and PropagatiorL
40
13. J.T. Mayhan, "Some Techniques for Evaluating theBandwidth Characteristics of Adaptive NullingSystems," IEEE Trans. An1l?nnas Propag. AP-27, 363(1979).
14. W.E. Scharfman and G. August. "Pattern Measurements of Phased-Arrayed Antennas by Focusing intothe Near Zone." Phased Array Antennas (Proc. oj the1970 Phased Array Antenna Symp.J. eds. AA Olinerand C.H. Knittel (Artech House. Dedham. MA. 1972).p.344.
15. W.L. Stutzman and G.A. Thiele. Antenna Theory andDesign (Wiley, New York., 1981). p. 306.
16. H.M.AumannandF.G. Willwerth, "Synthesis ofPhasedArrayFar-FieldPatterns byFocusingin the Near-Field."Proc. oj the 1989 IEEE NaiL Radar Con] Dallas. TX.29-30Mar. 1989. p. 101.
17. G. Strang. Linear Algebra and. Its Applications (Academic Press, New¥ork, 1976). p. 213.
18. AJ. Fenn. "Maxim.izing Jammer Effectiveness forEvaluating the Performance ofAdaptive Nulling ArrayAntennas: IEEETrans. Antennas Propag. AP-33, 1131(1985).
19. I.J. GuptaandA.A. Ksienski. "EtTectofMutual Couplingon the Performance of Adaptive Arrays." IEEE Trans.Antennas Propag. AP-31. 785 (1983).
20. AJ. Fenn. "Moment Method Analysis of Near-FieldAdaptive Nulling: Technical Report842. Lincoln Laboratory (7 Apr. 1989). AD-A208-228.
21. J.H. Richmond. "Radiationand ScatteringbyThin-WireStructures in a Homogeneous Conducting Medium(Computer Program Description)," IEEE Trans. Antennas Propag.AP-22. 365 (Mar. 1974).
The Lincoln Laboratory Journal, Vo[wne 3. Number 1 (1990)