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C H A P T E
10Compensation of
Adaptive Arrays
'
&
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Chapter Outline
10.1 Array Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
10.2 Array Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
10.3 Broadband Signal Processing Considerations. .. .. .. .. .. .. . .. .. .. .. . .. .. . .. .. .. . 38010.4 Compensation for Mutual Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.5 Multipath Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
10.6 Analysis of Interchannel Mismatch Effects . .. .. .. .. . .. .. .. .. . .. .. . .. .. .. .. . .. .. . 406
10.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
10.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Narrowbandadaptivearraysneedonly onecomplexadaptiveweightineachelementchan-nel. Broadband adaptive arrays, however, require tapped delay lines (transversal filters)in each elementchannel to makefrequency-dependentamplitudeand phaseadjustments.
The analysis presented so far assumes that each element channel has identical electron-ics andno reflected signals. Unfortunately, the electrical characteristics of each channelareslightly differentandlead to channel mismatching in whichsignificantdifferencesin frequency-response characteristics from channel to channel may severely degradeanarraysperformancewithoutsomeformof compensation. Thischapter startswithananal-ysis of array errors and then addresses array calibration and frequency-dependent mis-matchcompensation usingtapped delay lineprocessing, whichis importantfor practicalbroadbandadaptivearray designs.
The number of taps used in a tapped delay line processor depends on whether the
tapped delay linecompensates for broadbandchannel mismatcheffectsor for theeffectsof multipathandfinitearray propagation delay. Minimizingthe number of tapsrequiredfor aspecifiedset of conditionsisanimportantpractical designconsideration, sinceeachadditional tap (andassociated weighs) increases the cost andcomplexity of the adaptivearray system.
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374 C H A P T E R 10 Compensation of AdaptiveArrays
10.1 ARRAY ERRORS
Array errorsresultfromthemanufacturingtolerancesdefinedbythematerials,processes,andconstructionof thecomponentsinanarray. Thesesmall errorsarerandom,becausethe
manufacturing techniques employed havevery tight tolerances. Therandomdifferencesbetween any components distort the signal path by adding phase and amplitude errorsas well as noiseto each signal. These types of errors are static, because oncemeasuredthey remainrelatively unchangedover thelifeof thecomponent. Higher frequencieshavetighter tolerances for phase distortion than lower frequencies, because the errors are afunction of wavelength. Not only are the accuracy of thedimensions of thecomponentsimportant, buttheaccuracyof thevaluesof theconstitutiveparametersof thecomponentsare also important. For instance, the dielectric constant determines the wavelength andhence the phase of the signal passing through it, so an error in the dielectric constantproduces aphaseerror.
Dynamic errors change with time and are primarily due to changes in temperature.Onlinecalibrationcorrectsfor thesedynamicerrorsalsotakescareof anydrift inthestatic
errors. Thedynamic errorsarealso frequency dependent. Theeffects of temperaturearesmallest at the center frequency and increase as the frequency migrates away from thecenter frequency.
10.1.1 Error Analysis
Randomerrorsthat affect arraysfall into four categories:
1. Randomamplitudeerror, an2. Randomphaseerror, pn3. Randompositionerror, sn
4. Randomelementfailure, Pe
n = 1 elementfunctioningproperly0 element failureThefirst threetypes of randomerrors fit into thearray factor as perturbationsto thearrayweightsandelementlocations
AFerr =N
n=1
an + an
ej(pn+
pn )ej k(sn+
sn)u (10.1)
Elementfailures result when an elementnolonger transmitsor receives. Theprobabilitythatanelementhasfailed,1 Pe, is thesameasarootmeansquare(rms)amplitudeerror,a
2
n . Position errors are not usually a problem, so a reasonable formula to calculatethermssidelobelevel of thearray factor for amplitudeandphaseerrorswithelementfailures
is [1]
sllrms =(1 Pe) + a2n + Pe p
2
n
Pe
1 p2n
t N
(10.2)
Figure10-1 isanexampleof atypical corporate-fedarray. A randomerrorthatoccursat one element is statistically uncorrelated with a random error that occurs in anotherelement in the array as long as that error occurs after the last T junction and beforean
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10.1 Array Errors
1 2 3 4 5 6 7 8
A
B
C
FIGURE 10-1
Corporate-fed ar
with random erro
50 0 5030
25
20
15
10
5
0
5
10
q (degrees)
Directivity(dB)
Errors
Error free
FIGURE 10-2
Array factor
with random,
uncorrelated erro
superimposed on
the error-free arra
factor.
element. If arandomerror occursprior toA, for instance, thentherandomerror becomescorrelatedbetweentheelementsthatsharetheerror. For instance, arandomerror betweenA and B results in a randomcorrelated error shared by elements 1 and 2. Likewise, arandomerrorbetweenB andC resultsinarandomcorrelated error sharedbyelements1,2, 3, and 4.
As an example, consider an eight-element, 20dB Chebyshev array that has elementsspaced /2 apart. If therandomerrorsarerepresented by an = 0.15and pn = 0.15, thenan example of thearray factor witherrorsis shownin Figure10-2. Notethat therandomerrorslower the main beamdirectivity, induce a slight beam-pointingerror, increasethesidelobelevels, and fill in someof thenulls.
10.1.2 Quantization Errors
PhaseshiftersandattenuatorshaveNbp control bitswiththeleastsignificantbitsgivenby
a = 2Nba (10.3) p = 2 2Nbp (10.4)
If thedifferencebetweenthedesiredandquantizedamplitudeweightsisauniformly dis-tributedrandomnumber withtheboundsbeingthemaximumamplitudeerror ofa/2,then the rms amplitude error is an = a/
12. The quantization error is randomonly
when no two adjacent elements receive the same quantized phaseshift. The difference
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376 C H A P T E R 10 Compensation of AdaptiveArrays
between the desired and quantized phase shifts is treated as uniform randomvariablesbetween p/2.Aswiththeamplitudeerror, therandomphaseerrorformulainthiscaseis pn = p/
12. Substitutingthis error into(b) yieldstherms sidelobelevel.
Thephasequantization errors becomecorrelated when thebeamsteering phaseshift
issmall enoughthatgroupsof adjacentelementshavetheir beamsteeringphasequantizedto the same level. This means that N/NQ subarrays of NQ elements receive the samephaseshift. Thegratinglobes dueto thesesubarraysoccur at [2]
sinm = sins m
NQde= sins
1 m(N 1) 2
Nbp
N
sins
1 m2Nbp (10.5)
The approximation in (10.5) assumes that the array has many elements. For large scanangles, quantization lobes do not form, becausetheelement-to-elementphasedifferenceappears random. Therelativepeaks of thequantization lobes aregiven by [1]
AF QLN = 12Np1 sin2
1 sin2s(10.6)
Figure 10-3 shows an array factor with a 20 dB n = 3 Taylor amplitude taper for a20-element, d = 0.5 array with its beam steered to = 3 when the phase shiftershave three bits. Four quantization lobes appear. The quantization lobes decrease whenhigher-precision phaseshifters areused andwhen thebeamis steered tohigher angles.
Significant distortion also results from mutual coupling, variation in group delaybetweenfilters,differencesinamplifiergain,toleranceinattenuatoraccuracy,andaperture
jitter in a digital beamforming array. Aperture jitter is the timing error between samplesin an analog-to-digital (A/D) converter. Without calibration, beamformingor estimation
of the direction of arrival (DOA) of the signal is difficult, as the internal distortion isuncorrelated with thesignal. As aresult, theuncorrelated distortion changes theweightsat eachelementandthereforedistorts thearray pattern.
FIGURE 10-3
Array factor steered
to 3 degrees with
three-bit phase
shifters compared
with phase shifters
with infinite
precision.
90 45 0 45 90
30
20
10
0
Arrayfa
ctor(dB)
3 bit phase shifter
q (degrees)
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10.2 Array Calibration
10.2 ARRAY CALIBRATION
A phasedarrayneedscalibratedbeforeitcangenerateanoptimumcoherentbeam.Calibra-tion involvestuning,forexample,thephaseshifters,attenuators, or receiverstomaximize
the gain andto createthe desired sidelobe response. Offlinecalibration takes careof thestatic errorsandisdoneatthefactoryorondeployment. Narrowbandcalibrationisappliedatthecenter frequencyof operation.Broadbandcalibrationisappliedoverthewholeoper-atingbandwidthof thearray. Thecalibratedphasesettingsarestoredfor all beamsteeringangles. Temperaturecauses drift in thecomponentcharacteristics over time, sothearrayrequiresperiodic recalibration. Thegainof theradiofrequency (RF) channelsmust beac-curately controlledtoavoidnonlinearitiesarisingfromsaturation of components,becausethesenonlinearities cannot beremoved.
Thetop vector inFigure10-4showstheresultinguncalibratedarray output whentheindividual five-elementvectorshaverandomamplitudeandphaseerrors.Whenthearrayiscalibrated(bottomvectorinFigure10-4), thentheindividual elementvectorsarethesamelengthandalign. As aresult, thecalibrated array outputvector magnitudeis maximized,
and its phase is zero. Methods for performing array calibration use a calibrated source,signal injection, or near-field scanning. Theseapproaches arediscussed in the followingsections.
10.2.1 Calibrated Source
A known calibration source radiates a calibration signal to all elements in the array [3].Figure10-5 showsacalibrationsourcein thefar field of anarray. At regularintervals, themain beamis steered to receive the calibration sourcesignal. Alternatively, amultibeamantenna can devoteone beamto calibration. Calibration with near-field sources requiresthat distanceand angular differences betaken intoaccount. If thecalibrationsourceis inthefarfield, thenthephaseshiftersareset tosteer thebeamin thedirectionof thesource.
Ineither case, eachelementtogglesthroughall of itsphasesettingsuntil theoutputsignalis maximized. The difference between the steering phase and the phase that yields themaximumsignal is thecalibration phase.
Element 1 Element 2 Element 3 Element 4 Element 5
Calibrated array output
Uncalibrated array output
FIGURE 10-4 The uncalibrated array output is less than the calibrated array output, becauseerrors in the uncalibrated array do not allow the signal vectors from the elements to align.
Target
Calibration source FIGURE 10-5
Far-field calibrati
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378 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-6
Layout of the smart
antenna test bed.
Making power measurements for every phase setting at every element in an arrayis extremely time-consuming. Calibration techniques that measure both amplitude andphase of the calibrated signal tend to be much faster. Accurately measuring the signalphaseis reasonable in an anechoic chamber butdifficult in the operational environment.Measurements at four orthogonal phase settings yield sufficient information to obtaina maximumlikelihood estimateof the calibration phase [4]. The element phase error iscalculatedfrompowermeasurementsatthefourphasestates,andtheprocedureisrepeatedfor eachelementinthearray. Additional measurementsimprovesignal-to-noiseratio, and
theprocedurecan berepeated to achievedesired accuracy within resolutionof thephaseshifters, sincethealgorithmis intrinsically convergent.
Another approachusesamplitude-only measurementsfrommultipleelementstofindthe complex field at an element [5]. The first step measures the power output from thearray when the phases of multiple elements are successively shifted with the differentphase intervals. Next, the measured power variation is expanded into a Fourier series toderive the complex electric field of the corresponding elements. Themeasurement timereduction comes attheexpenseof increased measurementerror.
Transmit/receive module calibration is an iterative process that starts with adjustingtheattenuatorsforuniformgainattheelements[6]. Thephaseshiftersarethenadjustedtocompensatefor theinsertion phasedifferences at eachelement. Ideally, when calibrating
thearray, thephaseshiftersgainremainsconstantasthephasesettingsarevaried,buttheattenuators insertion phase can vary as a function of the phase setting. This calibrationshould bedoneacrossthebandwidth, rangeof operatingtemperatures,andphasesettings.If the phase shifters gain varies as a function of setting, then the attenuators need to becompensatedaswell. Afteriteratingover thisprocess, all thecalibrationsettingsaresavedandapplied at theappropriatetimes.
Figure 10-6 shows an eight-element uniform circular array (UCA) in which a cen-ter element radiates a calibration signal to the other elements in the array [7]. Since thecalibrationsourceis in thecenter of thearray, thesignal pathfromthecalibrationsourceto each element is identical. As previously noted, random errors are highly dependenton temperature [8]. An experimental model of the UCA in Figure 10-6 was placed in-sideatemperature-controlled roomandcalibrated at 20C. Themeasured amplitudeand
phaseerrorsatthreetemperaturesareshowninFigure10-7 andFigure10-8, respectively.Increasing the temperature of the roomto 25C then to 30C without recalibration in-creases the errorsshown in Figure 10-7 andFigure 10-8. This experiment demonstratestheneed of dynamic calibration in asmart antennaarray.
10.2.2 Signal Injection
Calibratingwitharadiatingsourceisdifficult,becausethecalibrationsignal transmission/reception depends on the environment. One technique commonly used in digital
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10.2 Array Calibration
2 3 4 5 6 7 8
40
20
0
Signal path
Phaseerror(degrees)
20 C
25 C
30 C
FIGURE 10-7
Amplitude error f
the UCA antenna
as the system
temperature
changes from 20with calibration to
25C withoutrecalibration and
30C withoutrecalibration.
2 3 4 5 6 7 8
0.6
Signal path
Amp
litudeerror(Vrms
) 20 C
25 C
30 C
0.8
1.0
FIGURE 10-8
Phase error for th
UCA antenna as
system temperat
changes from 20
with calibration to
25C withoutrecalibration and
to 30C withoutrecalibration.
beamformingarrays is injectinga calibration signal into the signal path of each elementin the array behind each element as shown in Figure 10-9 [9]. This technique provides
a high-quality calibration signal for the circuitry behind the element. Unfortunately, itdoes not calibratefor the element patterns that havesignificant variations due to mutualcoupling, edgeeffects, andmultipath.
10.2.3 Near-Field Scan
A planar near-field scanner positioned very close to the array moves a probe directly infront of each element to measure the amplitudeand phaseof all the elements [10]. Themeasured field is transformed back to the aperture to recreatethe field radiated at each
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380 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-9
Inserting a
calibration signal
into the signal paths
in a digital
beamformer.
Computer
Calibration
signal
Arrayelements
Receiver A/D
Receiver A/D
Receiver A/D
FIGURE 10-10 Alignment results (measured phase deviation from desired value).
a: Unaligned. b: After single alignment with uncorrected measurements. c: After alignment
with fully corrected measurements. From W. T. Patton and L. H. Yorinks, Near-field alignment
of phased-array antennas, IEEE Transactions on Antennas and Propagation, Vol. 47, No. 3,
March 1999, pp. 584591.
element. The calibration algorithm iterates between the measured phase and the arrayweightsuntil thephaseatall theelementsisthesame. Figure10-10showstheprogressionof thephasecorrectionalgorithmfromleft toright.Thepictureontheleftis uncalibrated,the center picture is after one iteration, and the picture on the right is after calibrationis completed. This techniques is exceptionally good at correcting static errors prior todeployingan antennais not practical for dynamic errors.
10.3 BROADBAND SIGNAL PROCESSINGCONSIDERATIONS
Broadbandarraysusetappeddelaylinesthathavefrequency-dependenttransferfunctions.Arrayperformanceisafunctionof thenumberoftaps,thetapspacing,andthetotal delayineach channel. Theminimumnumber of tapsrequired to obtain satisfactory performancefor a given bandwidth may be determined as discussed in Section 2.5. The discussionof broadband signal processing considerations given here follows the treatment of thissubjectgivenbyRodgersandCompton[1113]. Theideal (distortionless)channel transferfunctionsarederived; adaptivearrayperformanceusingquadraturehybridprocessingandtwo-, three-, andfive-tapdelaylineprocessingareconsidered;andresultsandconclusionsfor broadbandsignal processingarethen discussed.
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10.3 BroadbandSignal ProcessingConsiderations
10.3.1 Distortionless Channel Transfer Functions
The element channels of the two-element array in Figure 10-11 are represented by thetransfer functions H1() and H2(). Let thedesired signal arriveats, measuredrelativetothearray facenormal. Thearray carrier frequency is 0, andthepoint sources spacing
is d = 0/2 = b/0, where is thewavefront propagation velocity.Fromthepointof view of thedesiredsignal, theoverall transferfunctionencountered
in passingthrough thearray of Figure10-11 is
Hd() = H1() + H2() exp
j d
sins
(10.7)
andtheoverall transfer functionseen bytheinterferencesignal is
HI () = H1() + H2() exp
j d
sini
(10.8)
Now requirethat
Hd() = exp(j T1) (10.9)and
HI () = 0 (10.10)By choosing Hd() according to (10.9), the desired signal is permitted to experience atimedelayT1 in passingthroughthe array but otherwise remains undistorted. ChoosingHI () = 0resultsincompletesuppressionof theinterferencesignal fromthearrayoutput.
H2(w) H1(w)
Interference
Signal
qi
qs
dsin
qs
dsin
qi
d
Array
output
FIGURE 10-11
Two-element arra
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382 C H A P T E R 10 Compensation of AdaptiveArrays
Todeterminewhetheritispossibletoselect H1() and H2() tosatisfy (10.9)and(10.10),solve(10.9)and(10.10) for H1() and H2(). Setting H1() = |H1()| exp[j 1()] andH2() = |H2()| exp[j 2()] resultsin
|H1()| exp[j 1()] + |H2()| expj2()
0 sins = exp(j T1) (10.11)
|H1()| exp[j 1()] + |H2()| exp
j
2()
0sini
= 0 (10.12)
To satisfy (10.9) and(10.10), it followsfrom(10.11) and (10.12) (as shownby thedevel-opmentoutlined in theProblems section) that
H1() = H2() =1
2
1 cos
0
(sini sins) (10.13)
2() =
2
0 [sins + sini ] n
2 T1 (10.14)
1() =
2
0
[sin s sini ] n
2 T1 (10.15)
where n is any odd integer. This result means that the amplitude of the ideal transferfunctions areequal andfrequency dependent. Equations(10.14) and(10.15) furthermoreshowthatthephaseof eachfilter isalinearfunctionof frequencywiththeslopedependenton the spatial arrival anglesof the signals as well as on the time delayT1 of the desiredsignal.
Plots of theamplitudefunctionin (10.13) areshownin Figure10-12 for two choicesof arrival angles (s = 0 and s = 80), where it is seen that the amplitude of thedistortionlesstransferfunctionisnearlyflatovera40%bandwidthwhenthedesiredsignalis at broadside(s
=0) and the interference signal is 90 frombroadside (i
=90).
Examinationof (10.13)showsthatwhenever (sinI sins) is in theneighborhoodof1,
FIGURE 10-12
Distortionless
transfer function
amplitude versus
normalized
frequency for
d= 0/2. FromRodgers and
Compton, Technical
Report ESL 3832-3,
1975 [12].
00
10
20
30
40
50
60
70
80
90
100
0.5 1 1.5 2
Bandwidth
40%|H(w)|
w/w0
s = 80
i = 90
s = 0
i = 90
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10.3 BroadbandSignal ProcessingConsiderations
then the resulting amplitude function will be nearly flat over the 40% bandwidth region.If, however, both the desired and interference signals are far from broadside (as whend = 80 and i = 90), then theamplitudefunction is no longer flat.
Thedegreeof flatness of thedistortionlessfilter amplitudefunctionisinterpretedin
termsof thesignal geometrywithrespecttothearray sensitivity pattern. Ingeneral, whenthephasesof H1() and H2() areadjustedtoyieldthemaximumundistortedresponsetothedesired signal, thecorrespondingarray sensitivity pattern will havecertain nulls. Thedistortionlessfilter amplitudefunction is then the most flat when the interference signalfalls intooneof thesepatternnulls.
Equation(10.13) furthermoreshowsthatsingularitiesoccurinthedistortionlesschan-nel transfer functions whenever (/0)(sini sins) = n2 where n = 0, 1, 2, . . ..
Thecasewhenn = 0occurswhenthedesiredandinterferencesignalsarrivefromexactlythe same direction, so it is hardly surprising that the array would experience difficultytrying to receive one signal while nulling the other in this case. The other cases whenn = 1, 2, . . ., occur when thesignals arrivefromdifferent directions, but thephaseshiftsbetween elements differ by a multiple of 2 radians at some frequency in the signal
band.The phase functions 1() and 2() of (10.14) and (10.15) are linear functionsof
frequency. When T1 = 0, the phase slope of H1() is proportional to sins sini ,whereas that of H2() is proportional to sini + sins. Consequently, when the desiredsignal is broadside, 1() = 2(). Furthermore, the phasedifference between 1()and2() is also alinear functionof frequency, aresult that would beexpectedsincethisallows the interelement phase shift (which is also a linear function of frequency) to becanceled.
10.3.2 Quadrature Hybrid and Tapped Delay Line Processingfor a Least Mean Squares Array
Consider atwo-elementadaptivearray usingtheleastmean squares (LMS) algorithm. Ifwis thecolumn vector of array weights,Rxx is thecorrelation matrix of input signals toeach adaptive weight, andrxd is thecross-correlation vector between thereceivedsignalvector x(t) andthe reference signal d(t), then as shown in Chapter 3 the optimumarrayweight vector that minimizes E{2(t)} (where(t) = d(t)array output) is given by
wopt = R1xxrxd (10.16)
If thesignal appearingattheoutputof eachsensorelementconsistsof adesiredsignal, aninterferencesignal, andathermal noisecomponent(whereeachcomponentisstatisticallyindependent of the others and has zero mean), then the elements ofRxx can readily be
evaluated in terms of thesecomponentsignals.Consider the tapped delay lineemploying real (instead of complex) weights shown
in Figure10-13. Sinceeachsignal xi (t) is just atime-delayed version ofx1(t), it followsthat
x2(t) = x1(t )x2(t) = x1(t 2)...
xL (t) = x1[t (L 1)]
(10.17)
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384 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-13
Tapped delay line
processor for a
single-element
channel having real
adaptive weights.
w1 wL
xL(t)x4(t)x3(t)x2(t)x1(t)
Channel
output
Sensor
element
w2 w4w3
Now sincetheelements ofRxx aregiven by
rxi xj= E{xi (t)xj (t)} (10.18)
it follows from(10.17) that
rxi xj = rx1x1(ij) (10.19)whererx1x1(ij) is the autocorrelation function ofx1(t), and ij is the time delay betweenxi (t) andxj (t). Furthermore,rxi xi (ij) isthesumof threeautocorrelationfunctionsthoseof thedesired signal, theinterference, andthethermal noiseso that
rx1x1(ij) = rdd(ij) + rII (ij) + rnn(ij) (10.20)For theelementsofRxx correspondingto xi (t) and xj (t) fromdifferentelementchannels,rxi xj consistsonly of thesumof theautocorrelationfunctionsof thedesiredsignal andtheinterference signal (withappropriate delays) but not the thermal noisesince the elementnoise from channel to channel is uncorrelated. Thus, for signals in different elementchannels
rxi xj (ij) = rdd(dij ) + rII (Iij ) (10.21)where dij denotes the timedelay between xi (t) and xj (t) for the desired signal, and Iijdenotesthetimedelay between xi (t) and xj (t) for theinterferencesignal (thesetwo timedelayswill in general bedifferent duetothedifferent anglesof arrival of thetwo signals).Only when xi (t) and xj (t) arefromthesamearray elementchannel will dij = Iij (whichmay then bedenoted by ij ).
Next, consider the quadrature hybrid array processor depicted in Figure 10-14. Letx1(t) and x3(t) denote the in-phase signal components and x2(t) and x4(t) denote thequadrature-phase signal components of each of the elements output signals. Then thein-phaseand quadraturecomponentsarerelated by
x2(t) = x1(t)x4(t) = x3(t)
(10.22)
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10.3 BroadbandSignal ProcessingConsiderations
w1
x3(t) x4(t)x1(t) x2(t)
Array output
Quadrature
hybrid
w2 w4w3
Quadrature
hybrid
FIGURE 10-14
Quadrature hybri
processing for a
two-element arra
Thesymbol denotes theHilbert transform
x(t)= 1
x( )
t d (10.23)
wherethepreviousintegral is regarded as aCauchy principal valueintegral. Thevariouselements of thecorrelation matrix
rxi xj = E{xi (t)xj (t)} (10.24)
canthenbefoundbymakinguseof certainHilberttransformrelationsasfollows[14,15]:
E{x(t)y(s)} = E{x(t)y(s)} (10.25)E{x(t)y(s)} = E{x(t)y(s)} (10.26)
sothat
E{x(t)x(t)} = 0 (10.27)E{x(t)y(s)} = E{x(t)y(s)} (10.28)
where E{
x(t)y(s)}
denotes the Hilbert transformofrxy
( ) where =
s
t. With theprevious relationsand from(10.22) it then followsthat
rx1x1 = E{x1(t)x1(t)} = rx1x1(0) (10.29)rx1x2 = E{x1(t)x2(t)} = E{x1(t)x1(t)} = 0 (10.30)rx2x2 = E{x2(t)x2(t)} = E{x1(t)x1(x)} (10.31)
= E{x1(t)x1(t)} = rx1x1(0)
whererx1x1( ) is theautocorrelation function ofx1(t) given by(10.20).
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386 C H A P T E R 10 Compensation of AdaptiveArrays
Whentwodifferentsensorelementchannelsareinvolved[aswithx1(t) and x3(t), forexample], then
E{x1(t) x3(t)} = rdd(d13) + rII (I13) (10.32)
where d13 and I13 represent the spatial timedelays between the sensor elements of Fig-ure10-14for thedesired andinterferencesignals, respectively. Similarly
E{x1(t)x4(t)} = E{x1(t)x3(t)} = E{x1(t)x3(t)}= rdd(d13) + rII (I13) (10.33)
E{x2(t)x3(t)} = E{x1(t)x3(t)} = E{x1(t)x3(t)}= E{x1(t)x3(t)} = rdd(d13) rII (I13) (10.34)
E{x2(t)x4(t)} = E{x1(t)x3(t)} = x{x1(t)x3(t)}= rdd(d13) + rII (I13) (10.35)
Now consider thecross-correlation vector rxd defined by
rxd= E
x1(t)d(t)x2(t)d(t)...
x2N(t)d(t)
(10.36)
where N is the number of sensor elements. Each element of rxd, denoted by rxi d, isjust the cross-correlation between the reference signal d(t) and signal xi (t). Since thereference signal is just a replica of the desired signal and is statistically independentof the interference and thermal noise signals, the elements of rxd consist only of theautocorrelationfunctionof thedesired signal sothat
rxi d = E{xi (t)d(t)} = rdd(di ) (10.37)where di represents the time delay between the reference signal and the desired signalcomponent of xi (t). For an array with tapped delay line processing, eachelement ofrxdis the autocorrelation function of the desired signal evaluated at a time-delay valuethatreflectsboththespatial delaybetweensensorelementsandthedelaylinedelaytothetapofinterest. For anarraywithquadraturehybridprocessing, theelementsofrxd correspondingto an in-phasechannel yield the autocorrelation function of the desired signal evaluatedat thespatial delay appropriatefor that elementas follows:
rxi d(in-phase channel) = E{xi (t)d(t)} = rdd(di ) (10.38)
The elements ofrxd corresponding to quadrature-phasechannels can beevaluated using
(10.27) and (10.28) as follows:
rxi+1d(quadrature-phasechannel) = E{xi+1(t)d(t)}= E{xi (t)d(t)} = E{xi (t)d(t)} (10.39)= E{xi (t)d(t)} = rxi d(di )
OnceRxx andrxd havebeen evaluated for a given signal environment, the optimal LMSweights can becomputed from(10.16), andthe steady-stateresponseof the entire arraycan then beevaluated.
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10.3 BroadbandSignal ProcessingConsiderations
The tapped delay line in the element channel of Figure 10-13 has a channel transferfunction given by
H1() = w1 + w2ej + w3ej2 + . . . + wL ej (L1) (10.40)
Likewise, thequadraturehybridprocessorof Figure10-14hasachannel transferfunction
H1() = w1 j w2 (10.41)
Thearray transfer functionfor thedesiredsignal andtheinterferenceaccountsfor theeffects of spatial delays between array elements. A two-element array transfer functionfor thedesired signal is
Hd() = H1() + H2()ejd (10.42)
whereas thetransfer functionfor theinterferenceis
HI () = H1() + H2() ej I (10.43)Thespatial timedelaysassociatedwiththedesiredandinterferencesignalsarerepresentedby d and I , respectively, betweenelement1 [withchannel transfer function H1()] andelement2[with channel transferfunction H2()]. Withtwosensor elementsspacedapartby adistanced as in Figure10-11, thetwo spatial timedelays aregiven by
d =d
sin s (10.44)
I =d
sin I (10.45)
Theoutput signal-to-total-noiseratio is defined as
SNR= Pd
PI + Pn(10.46)
where Pd, PI , and Pn representtheoutputdesiredsignal power, interferencesignal power,andthermal noise power, respectively. Thearray output power for each of theforegoingthree signals may now beevaluated. Let dd() and II () represent thepower spectraldensities of the desired signal and the interference signal, respectively; then the desiredsignal output power is given by
Pd =
dd()|Hd()|2d (10.47)
whereHd() istheoverall transferfunctionseenbythedesiredsignal, andtheinterferencesignal output power is
PI =
II ()|H1()|2d (10.48)
where HI () is theoverall transfer functionseen by theinterferencesignal. Thethermalnoisepresentin eachelementoutputis statistically independentfromoneelementto thenext. Let nn() denote the thermal noise power spectral density; then the noisepower
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388 C H A P T E R 10 Compensation of AdaptiveArrays
contributed to thearray outputby element1 is
Pn1 =
nn()|H1()|2d (10.49)
whereas that contributed by element2 is
Pn2 =
nn()|H2()|2d (10.50)
Consequently, thetotal thermal noiseoutputpower fromatwo-elementarray is
Pn =
nn()[|H1()|2 + |H2()|2] d (10.51)
Theforegoingexpressionsmay nowbeusedin (10.46) toobtaintheoutputsignal-to-total-noiseratio.
10.3.3 Performance Comparison of Four Array Processors
Inthissubsection, four adaptivearraysonewithquadraturehybridprocessingandthreewith tapped delay lineprocessing (using real weights)are compared for signal band-widths of 4, 10, 20, and 40%. Tapped delay lines use real weights to preserve as muchsimplicity as possible in the hardwareimplementation, althoughthis sacrifices the avail-able degrees of freedomwithaconsequentdegradation in tapped delay lineperformancerelativeto combined amplitudeandphaseweighting. Theresultsobtained will neverthe-less serve as an indication of the relative effectiveness of tapped delay line processingcompared withquadraturehybrid processingfor broadbandsignals.
The four array processors to be compared are shown in Figure 10-15, where eacharray has two sensor elements andthe elements are spaced one-half wavelength apart at
thecenter frequencyof thedesiredsignal bandwidth.Figure10-15ashowsanarray havingquadrature hybrid processing, whereas Figure 10-15b10-15d exhibit tapped delay lineprocessing. Theprocessor of Figure10-15b has onedelay elementcorrespondingtoone-quarter wavelength at the center frequency and two associated taps. The processor ofFigure10-15c has two delay elements, each correspondingto one-quarter wavelength atthecenter frequency, andthree associated taps. Theprocessor of Figure10-15d has fourdelay elements, each corresponding to one-eighth wavelength at the center frequency,and five associated taps. Note that the total delay present in the tapped delay line ofFigure10-15dis thesameasthatof Figure10-15c, sotheprocessorinFigure10-15dmayberegarded as amorefinely subdivided versionof theprocessor in Figure10-15c.
Assumethat thedesired signal is biphasemodulated of theform
sd(t) = A cos[0t + (t) + ] (10.52)where (t) denotes a phase angle that is either zero or over each bit interval, and isan arbitrary constantphaseangle(within therange[0, 2]) for theduration of any signalpulse. Thenthbit interval is defined overT0 + (n 1)T t T0 + nT, wheren is anyinteger,T is thebit duration, andT0 isaconstantthat determineswherethebit transitionsoccur, as shownin Figure10-16.
Assume that (t) is statistically independent over different bit intervals and is zeroor with equal probability and that T0 is uniformly distributed over one bit interval;
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390 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-16
Bit transitions for
biphase modulated
signal.
f(t)
p
T T
T0
T T T
t
FIGURE 10-17
Desired signal power
spectral density.
Sd(w)
w0 w1 w0 +w1w0
Thereferencesignal equalsthedesiredsignal component ofx1(t) andistimealignedwith thedesired componentof x2(t). Thedesired signal bandwidth will betaken tobethefrequency rangedefined by thefirst nulls of thespectrumgiven by (10.53). With thisdefinition, thefractional bandwidth then becomes
desiredsignal bandwidth = 210
(10.54)
where1 is thefrequency separation between thecenter frequency 0 and thefirst null
1 =2
T(10.55)
Assumethattheinterferencesignal isaGaussianrandomprocesswithaflat,bandlim-itedpower spectral density over therange0 1 < < 0 + 1; thentheinterferencesignal spectrum appears in Figure 10-18. Finally, the thermal noise signals present ateach element are statistically independent between elements, having a flat, bandlimited,Gaussian spectral density over the range 0 1 < < 0 + 1 (identical with theinterferencespectrumof Figure10-18).
With the foregoingdefinitionsof signal spectra, the integrals of (10.48) and(10.51)yieldinginterferenceandthermal noisepoweraretakenonlyoverthefrequencyrange01 < < 0 + 1. Thedesired signal power also is considered only over thefrequencyrange0 1 < < 0 + 1 to obtain a consistent definition of signal-to-noise ratio(SNR). Therefore, theintegral of (10.47) is carriedoutonly over 0
1 < < 0
+1.
FIGURE 10-18
Interference signal
power spectral
density.
SI(w)
w0 w1 w0 +w1w0
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10.3 BroadbandSignal ProcessingConsiderations
To compare the four adaptive array processors of Figure 10-15, the output SNRperformance is evaluated for the aforementioned signal conditions. Assume the elementthermal noisepower pn is10dB belowtheelementdesiredsignal power ps sothat ps/ pn =10dB. Furthermore,supposethattheelementinterferencesignal power pi is20dB stronger
than the element desired signal power so that ps/ pi = 20 dB. Now assume that thedesiredsignal isincidentonthearrayfrombroadside.TheoutputSNR givenby(10.46)canbeevaluatedfrom(10.47), (10.48), and(10.49) byassumingtheprocessorweightssatisfy(10.16) for eachof thefour processor configurations. Theresultingoutputsignal-to-totalnoiseratio thatresultsusingeachprocessorisplottedinFigures10-1910-22asafunctionof theinterferenceangleof arrival for 4,10, 20, and40% bandwidthsignals, respectively.
In all cases, regardless of the signal bandwidth, when the interference approachesbroadside (near the desired signal) the SNR degrades rapidly, and the performance of
3 Taps
20
10
5
0
5
10
15
3 & 5 Taps
5 Taps
2 Taps
Quadrature hybrid
40
Interference angle
60 80
Pi
Ps= 20 dB
Pn
Ps = 10 dB
Pd
PI
+PN
(dB)
Output
FIGURE 10-19
Output signal-to-
interference plus
noise ratio
interference angl
for four adaptive
processors with 4
bandwidth signa
From Rodgers an
Compton, IEEE
Trans. Aerosp.
Electron. Syst.,
January 1979 [13
3 Taps
20
5
0
5
10
15
3 & 5 Taps
5 Taps
2 Taps
Quadrature hybrid
40
Interference angle
60 80
Pi
Ps= 20 dB
Pn
Ps= 10 dB
Pd
PI
+PN
(dB)
Output
FIGURE 10-20
Output signal-to-
interference plus
noise ratio versus
interference angl
for four adaptive
processors with
10% bandwidth
signal. FromRodgers and
Compton, IEEE
Trans. Aerosp.
Electron. Syst.,
January 1979 [13
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392 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-21
Output signal-to-
interference plus
noise ratio versus
interference angle
for four adaptiveprocessors with
20% bandwidth
signal. From
Rodgers and
Compton, IEEE
Trans. Aerosp.
Electron. Syst.,
January 1979 [13].
3 Taps
20
10
5
0
5
10
15
3 & 5 Taps
5 Taps
2 Taps
Quadrature hybrid
40
Interference angle
60 80
Pi
Ps= 20 dB
Pn
Ps= 10 dB
Pd
PI
+PN
(dB)
Output
FIGURE 10-22
Output signal-to-
interference plus
noise ratio versus
interference angle
for four adaptive
processors with
40% bandwidth
signal. From
Rodgers andCompton, IEEE
Trans. Aerosp.
Electron. Syst.,
January 1979 [13].
3 Taps
20
10
5
0
5
10
15
5 Taps
2 Taps
Quadrature hybrid
40
Interference angle
60 80
Pi
Ps= 20 dB
Pn
Ps= 10 dB
Pd
PI
+PN
(dB)
Output
all four processorsbecomes identical. This SNR degradation is expected since, when theinterference approaches the desired signal, the desired signal falls into the null providedto cancel the interference, and the output SNR consequently falls. Furthermore, as theinterferenceapproachesbroadside, theinterelementphaseshift for thissignal approacheszero. Consequently, the need to provide a frequency-dependent phaseshift behind eacharray elementto deal with theinterferencesignal is less, andtheperformanceof all fourprocessors becomes identical.
When the interference signal is widely separated from the desired signal, then theoutput SNR is different for the four processors being considered, and this differencebecomesmorepronouncedasthebandwidthincreases.For20and40%bandwidthsignals,
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10.3 BroadbandSignal ProcessingConsiderations
for example, neither the quadrature hybrid processor nor the two-tap delay line proces-sor provides good performance as the interference signal approaches endfire. The per-formance of boththe three- andfive-tap delay lineprocessors remains quitegood in theendfireregion, however.If 20%ormorebandwidthsignalsareaccommodated,thentapped
delay lineprocessingbecomesanecessity. Figure10-22showsthat thereisnosignificantperformance advantage provided by the five-tap processor compared with the three-tapprocessor,soathree-tapprocessorisadequatefor upto40%bandwidthsignalsinthecaseof atwo-elementarray.
Figures 10-21 and 10-22 show that the output SNR performance of the two-tapdelay line processor peaks when the interference signal is 30 off broadside, becausethe interelement delay time is /4 (sincethe elements are spaced apart by /2). Conse-quently, thesingle-delay elementvalueof/4providesjusttherightamountof timedelayto compensateexactly for theinterelementtimedelay andtoproducean improvementintheoutput SNR.
The three-tap and five-tap delay line processors both produce a maximum SNR ofabout 12.5dB at wideinterferenceanglesof 70 or greater. For ideal channel processing,
theinterferencesignal iseliminated,thedesiredsignal ineachchannel isaddedcoherentlyto produce Pd = 4ps, and the thermal noise is added noncoherently to yield PN = 2pn.
Thus, thebestpossibletheoretical outputSNR for atwo-elementarray withthermal noise10 dB below thedesired signal and no interferenceis 13 dB. Therefore, thethree-tap andfive-tap delay lineprocessorsaresuccessfully rejectingnearly all the interference signalpower at wideoff-boresight angles.
10.3.4 Processor Transfer Functions
Ideally, the array transfer function for the desired signal should be constant across thedesired signal bandwidth, thereby preventing desired signal distortion. Theinterferencetransfer function should bealowarray responseover theinterferencebandwidth.
The transfer functions for the four processors and the two-element array are evalu-ated using (10.40)(10.45). Using the same conditions adopted in computing the SNRperformance, Figures 10-2310-26show |Hd()| and |HI ()| for thefour processors of
20
90
90
10
Interference angleInterference signal
10
135
0.98 0.984 0.988 0.992 0.996 1.001 1.004 1.008 1.012 1.016 1.02
120
105
90
75
Amplitude(dB)
60
45
30
15
0
Desired signal
Frequency (w/w0)
FIGURE 10-23
Quadrature hybri
transfer functions
4% bandwidth.
From Rodgers an
Compton, Techn
Report ESL 3832
1975 [12].
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394 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-24
Two-tap delay line
transfer functions
at 4% bandwidth.
From Rodgers and
Compton, TechnicalReport ESL 3832-3,
1975 [12].
2090
9010
Interference angle
Interference signal
10
135
120
105
90
75
Amplitude(dB)
60
45
30
15
0
Desired signal
0.98 0.984 0.988 0.992 0.996 1.001 1.004 1.008 1.012 1.016 1.02
Frequency (w/w0)
Figure 10-15 with a 4% signal bandwidth and various interference signal angles. Theresultsshown in thesefigures indicatethat for all four processorsandfor all interferenceangles thedesired signal responseis quiteflat over thesignal bandwidth. As theinterfer-enceapproaches the desired signal angle at broadside, however, the (constant) responselevel of thearray tothedesiredsignal dropsbecauseof thedesiredsignal partially fallingwithin thearray pattern interferencenull.
The results in Figure 10-23 for quadrature hybrid processing show that the arrayresponse to the interference signal has a deep notch at the center frequency when theinterferencesignal iswell separated(i > 20
) fromthedesiredsignal. Astheinterferencesignal approaches thedesired signal (i < 20
), thenotch migrates away fromthecenterfrequency, because the processor weights must compromise between rejection of theinterferencesignal andenhancementof thedesiredsignal whenthetwo signalsareclose.Migrationof thenotchimprovesthedesiredsignal response(sincethedesiredsignal powerspectral density peaksat thecenter frequency) whileaffectinginterferencerejectiononlyslightly (since the interference signal power spectral density is constant over the signalband).
Thearray responsefor thetwo-tapprocessor isshowninFigure10-24. Theresponsetoboththedesired andinterferencesignalsis very similar tothat obtained for quadraturehybrid processing. Themost notable changeis theslightly differentshapeof thetransferfunction notch presented to the interference signal by the two-tap delay line processorcompared withthequadraturehybrid processor.
Figure 10-25 shows the three-tap processor array response. Theinterference signalresponse is considerably reduced, with a minimumrejection of the interference signalof about 45 dB. When the interference signal is close to the desired signal, the arrayresponse has a single mild dip. As the separation angle between the interferencesignaland the desired signal increases, the single dip becomes more pronounced and finallydevelopsintoadouble dip atvery wideangles. Itis difficult toattributemuchsignificanceto the double-dip behavior since it occurs at such a low response level (of more than75dB attenuation). Thefive-tap processor responseof Figure10-26is very similar tothethree-tapprocessorresponseexceptslightlymoreinterferencesignal rejectionisachieved.
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10.3 BroadbandSignal ProcessingConsiderations
20
70
10
90
10
Interference angle
Interference signal
90
135
120
105
90
75
Amplitude(dB)
60
45
30
15
0
Desired signal
0.98 0.984 0.988 0.992 0.996 1.001 1.004 1.008 1.012 1.016 1.02
Frequency (w/w0)
FIGURE 10-25
Three-tap delay l
transfer functions
at 4% bandwidth
From Rodgers an
Compton, TechnReport ESL 3832
1975 [12].
20
50
10
90
10
Interference angle
Interference signal
90
135
120
105
90
75
Amp
litude(dB)
60
40
30
15
0
Desired signal
0.98 0.984 0.988 0.992 0.996 1.001 1.004 1.008 1.012 1.016 1.02
Frequency (w/w0)
FIGURE 10-26
Five-tap delay lin
transfer functions
at 4% bandwidth
From Rodgers an
Compton, Techn
Report ESL 3832
1975 [12].
As the signal bandwidth increases, theprocessor responsecurves remain essentially
thesameas in Figures 10-2310-26exceptthefollowing:1. As theinterferencesignal bandwidthincreases, it becomes moredifficult to reject the
interferencesignal overtheentirebandwidth,sotheminimumrejectionlevel increases.
2. Thedesired signal responsedecreases becausethearray feedback reduces all weightstocompensatefor thepresenceof agreater interferencesignal componentat thearrayoutput, thereby resulting in greater desiredsignal attenuation.
The net result is that as the signal bandwidth increases, the output SNR performancedegrades, as confirmed by theresultsof Figures 10-1910-22.
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396 C H A P T E R 10 Compensation of AdaptiveArrays
10.4 COMPENSATION FOR MUTUAL COUPLING
In many applications, the limited spaceavailable for mountingan antenna motivates theuseof asmall array. As thearray sizedecreases, thearray elementspacing becomes less
than a half-wavelength, andmutual couplingeffects become moreof a factor in degrad-ingthearray performance. Whenanarray consistsof single-modeelements(meaningthatthe element aperturecurrentsmay change in amplitude but not in shape as a function ofthesignal angleof arrival), thenit ispossibletomodify theelementweightstocompensatefor thepattern distortion caused by the mutual couplingat aparticular angle [16]. Theseweight adjustments may work for morethan oneangle.
Let the vector v denote the coupling perturbed measured voltages appearing at theoutputof the array elements, andlet vd represent the couplingunperturbed voltages thatwould appear at thearray elementoutputs if nomutual couplingwerepresent. Theeffectof mutual couplingonsingle-modeelements is written as
v(u)=C v
d(u) (10.56)
whereu = sin, is theangle of arrival, andthematrixC describestheeffectsof mutualcouplingandisindependentofthesignal scanangle.If thearrayiscomposedof multimodeelements, then thematrixC would bescan angledependent.
It follows that theunperturbed signal vector,vd can berecovered fromtheperturbedsignal vector byintroducingcompensation for themutual coupling
vd = C1v (10.57)
Introducing the compensation network C1 as shown in Figure 10-27 then allows allsubsequent beamforming operations to be performed with ideal (unperturbed) elementsignals, as arecustomarily assumed in pattern synthesis.
Thismutual couplingcompensationis appliedtoaneight-elementlinear array havingelement spacingd = 0.517 consistingof identical elements. Figure10-28(a) showstheeffects of mutual couplingbydisplayingthedifferencein elementpatternshapebetweenacentral andan edgeelementin thearray.
Figure10-28displaysasynthesized30dBChebyshevpatternbothwithout(a)andwith(b) mutual coupling compensation. It is apparent fromthis result that the compensationnetwork gives abouta10dB improvementin thesidelobelevel.
FIGURE 10-27
CouplingCompensation and
Beamforming in an
Array Antenna. From
Steyskal & Herd,
IEEE Trans. Ant &
Prop., Dec. 1995.
1
2
N
1d
2d
Nd
Arrayelements
1
2
N
Coupling
perturbedmeasured
voltages
Couplingcompensation Unperturbedvoltages Complexweights
Output
beamC C1
W1
W2
WN
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10.4 Compensation for Mutual Coupling
Angle, deg
(a)
(b)
80 60 30 0 30 60 80
80 60 30 0 30 60 80
40
30
20
10
0
Power,dB
Measured
Theory
Angle, deg
40
30
20
10
0
Power,dB
Measured
Theory
FIGURE 10-28 30 dB Chebyshev pattern (a) without and (b) with Coupling Compensation
with a Scan Angle of 0. From Steyskal & Herd, IEEE Trans. Ant. & Prop. Dec. 1995.
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398 C H A P T E R 10 Compensation of AdaptiveArrays
10.5 MULTIPATH COMPENSATION
In many operating environments, multipath rays impinge on the array shortly after thedirect path signal arrives at the sensors. Multipath distorts any interference signal that
may appear in the various element channels, thereby severely limiting the interferencecancellation.A tappeddelay lineprocessorcombinesdelayedandweightedreplicasof theinputsignal toformthefilteredoutputsignal andtherebyhasthepotential tocompensatefor multipath effects, since multipath rays also consist of delayed andweighted replicasof thedirectpathray.
10.5.1 Two-Channel Interference Cancellation Model
Consider an ideal two-element adaptive array with one channels (called the auxiliarychannel) responseadjustedsothat anyjammingsignal enteringtheother channel throughthe sidelobes (termed the main channel) is canceled at the array output. A systemde-signedtosuppresssidelobejamminginthismanner iscalledacoherentsidelobecanceller
(CSLC), and Figure 10-29 depicts a two-channel CSLC system in which the auxiliarychannel employs tapped delay line compensation involving L weights and L 1 delayelementsof value secondseach. A delay elementof value D = (L 1)/2is includedin the main channel so the center tap of the auxiliary channel corresponds to the outputof the delay D in the main channel, thereby permitting compensation for both positiveand negative values of the off-broadside angle . This ideal two-element CSLC systemmodel exhibitsall thesalientcharacteristicsthatamorecomplex systeminvolvingseveralauxiliary channels would have, so the two-element systemserves as a convenient modelfor performanceevaluation of multipathcancellation [17].
The systemperformance measure is the ability of the CSLC to cancel an undesiredinterference signal throughproper designof thetapped delay line. In actual practice, an
adaptivealgorithmadjuststheweight settings. To eliminatetheeffectof algorithmselec-tion fromconsideration, only thesteady-stateperformanceisevaluated. Sincethesteady-state solution can be found analytically, it is necessary to determine only the resultingsolutionfor the output residue power. This residue power is then a direct measureof theinterferencecancellation ability of thetwo-elementCSLC model.
FIGURE 10-29
Ideal two-element
CSLC model with
auxiliary channel
compensation
involving L weights
and L
1 delay
elements.
w1
wL
xL(t)
Residue
x2(t)
x1(t)
Main channel
Auxiliary channel
w2
x0(t)
DL 1
2=
+
q
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10.5 MultipathCompensation
Let x0(t), x1(t), and e(t) representthecomplex envelopesignals of themain channelinputsignal, theauxiliarychannel inputsignal, andtheoutputresiduesignal, respectively.Definethecomplex signal vector
xT
=[x1(t), x2(t) , . . . , xL (t)] (10.58)
where
x2(t)= x1(t )
...
xL (t)= x1 [t (L 1)]
Also, definethecomplex weight vector
wT = [w1, w2, . . . , wL ] (10.59)
Theoutput of thetapped delay linemay then beexpressed as
filter output=L
i=1x1[t (i 1)]wi = wx(t) (10.60)
Theresidue(complex envelope) signal is given by
e(t) = x0(t D) +wx(t) (10.61)Theweightvectorwminimizestheresiduesignal inameansquareerror(MSE) sense.
For stationary randomprocesses, this is equivalent to minimizing theexpression
Ree(0) = E {e(t)e(t)} (10.62)From(10.61) and thefactthat
E{x0(t D)x0(t D)} = rx0x0(0) (10.63)E{x(t)x0(t D)} = rxx0(D) (10.64)
E{x(t)x(t)} = Rxx(0) (10.65)it follows that
Ree(0) = rx0x0(0) rxx0(D)R1xx (0)rxx0(D)+ [rxx0(D) +wRxx(0)]R1xx (0) [rxx0(D) +Rxx(0)w] (10.66)
Minimize (10.66) by appropriately selecting the complex weight vectorw. Assume thematrixRxx(0) is nonsingular: thevalueofwfor which this minimumoccursis given by
wopt = R1xx (0)rxx0(D) (10.67)
Thecorresponding minimumresiduesignal power then becomes
Ree(0)min = rx0x0(0) rxx0(D)R1xx (0)rxx0(D) (10.68)Interferencecancellation performanceof theCSLC model of Figure10-27is determinedby evaluating (10.66) usingselected signal environmentassumptions.
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400 C H A P T E R 10 Compensation of AdaptiveArrays
10.5.2 Signal Environment Assumptions
Let s1(t, 1) represent the interference signal arriving from direction 1, and letsm(t, m, Dm, m+1) form= 2,..., M representthemultipathstructureassociatedwiththeinterferencesignal that consists of acollectionofM 1correlated planewavesignals of
thesamefrequency arrivingfromdifferentdirections sothat m+k = 1 and m+k = m+lfor k = l. Themultipathrayseachhavean associatedreflection coefficientm andatimedelay with respect to the direct ray Dm. The structure of the covariance matrix for thismultipathmodel can then beexpressed as [18]
Rss = VsAVs (10.69)whereVs is the N M signal matrix given by
Vs = | | |vs1 vs2 vsM| | |
(10.70)
whosecomponents aregiven by the N 1 vectors
vsm =
Psm
1exp[j2(d/0) sinm]exp[j2(d/0)2sinm]...
exp[j2(d/0) (N 1) sinm]
(10.71)
where Psm = 2m denotes the power associated withthesignal sm, andA is themultipathcorrelation matrix. WhenA= I, thevarioussignal componentsareuncorrelatedwhereasforA = U (the M M matrix of unity elements) the various components are perfectlycorrelated. For purposes of numerical evaluation the correlation matrix model may beselected as [18]
A=
1
2
M
1
1 M1...
M1 1
0 1 (10.72)
Note that channel-to-channel variations in m, Dm, and m cannot beaccommodated bythis simplified model. Consequently, amoregeneral model must bedeveloped to handlesuchvariations, whichtendtooccurwherenear-fieldscatteringeffectsaresignificant. Theinputsignal covariancematrix may bewritten as
Rxx = Rnn +VsAVs (10.73)whereRnn denotes thenoisecovariancematrix.
If only a single multipathray is present, then s(t, 1) denotes the direct interferencesignal, and sm(t, m, Dm, 2) represents the multipath ray associated with the directinterferencesignal. Thereceived signal at themain channel elementis then given by
x0(t) = s(t, 1) + sm(t, m, Dm, 2) (10.74)Denote s(t, 1) by s(t); then sm(t, m, Dm, 2) can be written as ms(t Dm) exp(j 0Dm) sothat
x0(t) = s(t) + ms(t Dm) exp(j 0Dm) (10.75)
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10.5 MultipathCompensation
where0 is thecenter frequency of theinterferencesignal. It then followsthat
x1(t) = s(t 12) exp(j 012)+ ms(t Dm 22) exp[j0(Dm + 22)] (10.76)
where 12 and 22 represent the propagation delay between the main channel elementand the auxiliary channel element for the wavefronts of s(t, 1) and sm(t, m, dm, 2),respectively.
Assuming the signals s(t, 1) and sm(t, m, Dm, 2) possess flat spectral densityfunctions over the bandwidth B, as shown in Figure 10-30a, then the correspondingauto- andcross-correlation functions of x0(t) and x1(t) can beevaluated by recognizingthat
Rxx( ) = 1{xx()} (10.77)
where1{} is the inverse Fourier transform, andxx() denotes the cross-spectraldensity matrix ofx(t).
From(10.74), (10.76), and (10.77) it immediately followsthat
rx0x0(0) = 1+ |m|2 +sin BDm
BDm
me
j 0Dm + mej 0Dm
(10.78)
Likewise, defining f[, sgn1, sgn2]= sin B [ + sgn1 (i 1) + sgn2 D]
B [ + sgn1 (i 1) + sgn2 D]and g[, sgn]
= sin B [ + sgn (i k)] B[ + sgn (i k) ] , then
rxi x0(D) = f[12, +, ] exp{j0[12 + (i 1)]}+ f[Dm + 22, +, ]mexp{j 0[22 + (i 1) + Dm]} (10.79)+ f[Dm 12, , +]mexp{j 0[12 + (i 1) Dm]}+ f[22, +, ]|m|2 exp{j 0[22 + (i 1)]}
xx(w)
w= 2fB0
(a)
B
Rxx()
t0
1
B
(b)
2
B
FIGURE 10-30
Flat spectral den
function and
corresponding
autocorrelation
function for
interference sign
a: Spectral densi
function.b: Autocorrelatio
function.
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402 C H A P T E R 10 Compensation of AdaptiveArrays
rxi xk(0) = g[0, +][1+ |m|2] exp[j0(i k)]+ g[12 22 Dm, ]mexp{j0[12 22 Dm (i k)]} (10.80)+ g[12 22 Dm, +]mexp{j0[12 22 Dm + (i k)]}
Thevector rxx0(D) is then given by
rxx0(D) =
rx1x0(D)rx2x0(D)
...
rxN x0(D)
(10.81)
andthematrixRxx(0) is given by
Rxx(0) =
rx1x1(0) rx1x2(0) rx1xN (0)... rx2x2(0)
. . .
rx1xN (0) rxN xN (0)
(10.82)
To evaluate (10.68) for the minimum possible value of output residue power (10.78),(10.79), and(10.80), showthat it is necessary tospecify thefollowingparameters:
N = number of tapsin thetransversal filterm = multipathreflection coefficient0 = (radian) center frequency of interferencesignal
Dm = multipathdelay timewith respect todirect ray12 = propagation delay between themain antennaelementandtheauxiliary antenna
elementfor thedirect ray22
=propagation delay between themain antennaelementandtheauxiliary antenna
elementfor themultipathray = transversal filter intertap delayB = interferencesignal bandwidthD = main channel receiver timedelay
Thequantities 12 and 22 arerelated totheCSLC array geometry by
12 =d
sin1
22 =d
sin2
(10.83)
where
d = interelementarray spacing = wavefront propagationspeed
1 = angle of incidenceof directray2 = angle of incidenceof multipathray
10.5.3 Example: Results for Compensation of Multipath Effects
Aninterferencesignal hasadirectray angleof arrival is 1 = 30, themultipathray angleof arrival is 2 = 30, and the interelement spacing is d = 2.250. Some additional
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10.5 MultipathCompensation
signal andmultipathcharacteristicsare
center frequency f0 = 237 MHzsignal bandwidth B = 3 MHz (10.84)
multipathreflection coefficient m = 0.5Referring to (10.76), (10.79), and (10.80), we see that the parameters 0, 12, 22,
Dm, and enter the evaluation of the output residue power in the formof the products012, 022, 0Dm, and0. Theseproductsrepresentthephaseshiftexperiencedatthecenter frequency0 asaconsequenceof thefourcorrespondingtimedelays.Likewise, theparameters B, D, Dm, 12, 22, and enter theevaluation of theoutputresidue power intheformof theproducts BD, BDm, B12, B22, and B; thesetimebandwidth productsarephaseshiftsexperiencedbythehighestfrequencycomponentof thecomplex envelopeinterferencesignal asaconsequenceof thefivecorrespondingtimedelays.Boththeintertapdelay andthemultipathdelay Dm areimportantparametersthataffecttheCSL C systemperformance through their correspondingtimebandwidth products; thus, the results are
givenherewiththetimebandwidthproductstakenasthefundamental quantity of interest.Sincefor this example 1 = 2, theproduct 012 is specified as
then the product
012 =
4
022 =
4
(10.85)
Furthermore, let theproducts0Dm and 0 begiven by
0Dm = 0 2k, k any integer0 = 0 2l, l any integer
(10.86)
For theelementspacing d=
2.250 and 1=
30, then specify
B12 = B22 =1
P, P = 72 (10.87)
Finally, specifying themultipathdelay timetocorrespond to46meters yields
BDm = 0.45 (10.88)
Since
D = N 12
(10.89)
Only N and B need to be specified to evaluate the output residue power by way of
(10.68).To evaluate the output residue power by way of (10.68) resulting from the array
geometry and multipath conditions specifiedby (10.84)(10.89) requires that the cross-correlation vectorrxx0(D), the N N autocorrelation matrixRxx(0), andtheautocorre-lation function rx0x0(0) beevaluated by way of (10.78)(10.80). A computer programtoevaluate(10.68) for the multipath conditions specified was written in complex, double-precisionarithmetic.
Figure10-31showsaplot of theoutput residuepower wheretheresulting minimumpossible valueof canceled power output in dB is plotted as afunction ofB for various
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404 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-31
Decibel cancellation
versus B for
multipath.
Evaluated
with
B=
BDm= 0.45
Rm= 0.5
1
78
N= 3
N= 1
N= number of taps
N= 5
N= 7
500.1 0.2 0.4 0.6 0.8 10.3
40
30
Cancellation
(dB)
20
10
0
B
specified values of N. It will be noted in Figure 10-31 that for N = 1 the cancellationperformanceisindependentof B sinceno intertapdelaysarepresent withonly asingletap.AsexplainedinAppendix B, thetransferfunctionof thetappeddelay linetransversalfilter has a periodic structurewith (radian) frequency period 2 Bf, which is centered atthe frequency f0. It should be noted that the transversal filter frequency bandwidth Bfis not necessarily the same as the signal-frequency bandwidth B. The transfer functionof a transversal filter within the primary frequency band (| f f0| < Bf/2) may beexpressed as
F ( f)=
N
k=1
[Akej k] exp[
j2(k
1) f] (10.90)
where Akejk represents thekth complex weight, f = f f0, f0 = center frequency,
andthetransversal filter frequency bandwidthis
Bf =1
(10.91)
Sincethetransversal filter shouldbecapableof adjustingthecomplex weightstoachieveappropriateamplitudeandphasevaluesover theentiresignal bandwidthB, it followsthatBf should satisfy
B f B (10.92)Consequently, themaximumintertap delay spacing is given by
max =1
B(10.93)
It follows that values of B that are greater than unity should not be considered forpractical compensation designs; however, values of B f > B (resultingin 0 < B < 1)aresometimes desirable.
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10.5 MultipathCompensation
Figure10-31showsthat,as B decreasesfrom1,for valuesofN > 1thecancellationperformance rapidly improves (the minimum canceled residue power decreases) untilB = BDm (0.45 for this example), after which very little significant improvementoccurs. As B becomesvery muchsmaller thanBDm (approachingzero), thecancellation
performancedegradessincetheintertapdelayiseffectively removed.Thesimulationcouldnot computethis result sinceas B approaches zerothematrixRxx(0) becomes singularandmatrixinversionbecomesimpossible.Cancellationperformanceof30dB isvirtuallyassured if thetransversal filter has at leastfivetapsand is selected sothat = Dm.
Supposeforexamplethatthetransversal filter isdesignedwith B = 0.45. Usingthesamesetof selectedconstantsasforthepreviousexample,wefindituseful toconsiderwhatresultswouldbeobtainedwhentheactual multipathdelay isdifferentfromtheanticipatedvaluecorrespondingtoBDm = 0.45. Fromtheresultsalready obtainedin Figure10-31, itmay beanticipatedthat, if BDm > B, thenthecancellation performancewoulddegrade.If, however, BDm B, then thecancellation performancewould improvesincein thelimit as Dm 0 thesystemperformancewith nomultipathpresentwould result.
10.5.4 Results for Compensation of Array Propagation Delay
Intheabsenceofamultipathray,theanalysispresentedintheprecedingsectionincludesallthefeaturesnecessary toaccountfor array propagationdelay effects. Whenwesetm = 0andlet12 = representtheelement-to-elementarray propagationdelay, (10.78)(10.80)permit(10.68) tobeusedtoinvestigatetheeffectsof array propagationdelay oncancella-tion performance. Onthebasis of thebehavioralreadyfoundfor multipathcompensation,it would bereasonabletoanticipatethat with B = B thenmaximumcancellationper-formance would obtain, whereas if B > B then the cancellation performance woulddegrade. Figure 10-32 gives the resulting cancellation performance as a function of Bfor fixed B. Thenumber of taps N is an independent parameter, and all other system
constantsarethesameasthoseintheexampleof Section10.4.3. It isseenthat theresultsconfirmtheanticipated performancenoted already.
Evaluate with
no multipath
N=
3
N= 1
N= number of taps
N=
5
700.1 0.2 0.4 0.6 0.8 10.3
60
50
Cancella
tion
(dB)
40
30
20
B
B=1
78
FIGURE 10-32
Decibel cancellat
versus B for ar
propagation dela
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406 C H A P T E R 10 Compensation of AdaptiveArrays
10.6 ANALYSIS OF INTERCHANNELMISMATCH EFFECTS
Any adaptive array processor is susceptible to unavoidable frequency-dependent vari-
ations in gain and phase between the various element channels. Additional degrees offreedomprovidedbyatappeddelaylinecompensateforsuchfrequency-dependentchan-nel mismatch effects. Since a simple two-element CSLC systemexhibits all thesalientcharacteristicsof channel mismatchingpresentinmorecomplexsystems,thetwo-elementmodel is again adopted as the example for performanceevaluation of channel mismatchcompensation.
Figure10-33is asimplifiedrepresentationof asingleauxiliarychannel CSLC systemin which the single complex weight is a function of frequency. The transfer functionT0(,) reflects all amplitude and phase variations in the main beam sidelobes as afunction of frequency as well as any tracking errors in amplitude and phase between themain andauxiliary channel electronics. Likewise, theequivalenttransfer function for the
auxiliary channel (includingany auxiliary antennavariations)isdenotedbyT1(, ).Thespectral power density of a wideband jammer is given by J J (). The signal fromtheauxiliary channel is multiplied by thecomplex weight w1 = ej , andthecancelledoutputof residuepower spectral density is represented by rr (, ).
Theobjectiveof theCSLC isto minimizetheresiduepower, appropriately weighted,over the bandwidth. Since the integral of the power spectral density over the signal fre-quency spectrumyieldsthesignal power, therequirementto minimizetheresiduepoweris expressed as
Minw1
rr (,)d (10.94)
where
rr
(,)= |
T0
(,)
w1T
1(,)
|2
J J() (10.95)
Now replace the complex weight w1 in Figure10-33by a tapped delay line having2N + 1 adaptively controlled complex weights each separated by a time delay as inFigure10-34. A delay elementof valueN is includedin themain channel (just asin thepreceding section) so that compensation for both positive andnegativeangles of arrivalis provided. Themain and auxiliary channel transfer functionsarewritten in terms of theoutputof themainchannel, sonodelay termsoccur intheresultingmain channel transfer
FIGURE 10-33
Simplified model of
single-channel
CSLC.
w1
S(w)
d
T0 (w, q)+
q
Main channel
T1 (w, q)
R(w, q)
Auxiliary channel
Adaptive
electronics
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10.6 Analysis of Interchannel Mismatch Effects
T0 (w) ejwN
ejw ejw
+
Main
channelResidue
Auxiliary
channel
Ndelay elements
w1 w1 wN+1 w2N+1
A()
F(w) =A(w)
Ndelay elements
ejw
F() = wN+1+k ejwkN
k =N
FIGURE 10-34
Single-channel
CSLC having ma
channel distortio
and tapped delay
line auxiliary chancompensation.
function, A(). Assumefor analysispurposesthatall channel distortion isconfinedtothemainchannel andthatT1(, ) = 1.Thetransversal filter transferfunction, F (), canbeexpressed as
F () =N
k=NwN+1+ke
j k (10.96)
wherethewN+1+ks arenonfrequency-dependent complex weights.Wewantto minimizethe outputresidue power over the signal bandwidthby appro-priately selecting the weight vectorw. Assuming the jammer power spectral density isconstantover thefrequency regionof interest, then minimizingtheoutputresiduepoweris equivalent to selectingthe F () that provides the best estimate(denoted by A())of themain channel transferfunctionover that frequency range. If theestimate A() istobe optimal in the MSE sense, then the error in this estimatee() = A() F () mustbeorthogonal to A () = F (), that is,
E{[A() F ()]F ()} = 0 (10.97)wheretheexpectation E{} is taken over frequency andis thereforeequivalentto
E{}= 12 B
B B
{ } d (10.98)
where all frequency-dependent elements in theintegrandof (10.98) arereduced to base-band. Letting A() = A0()ej0(), substituting(10.96) into (10.97), andrequiring theerror tobeorthogonal toall tap outputstoobtain theminimumMSE estimate A() thenyieldsthecondition
E{[A0() exp[j 0()] F ()] exp(j k)} = 0 for k = N, . . . , 0, . . . , N(10.99)
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408 C H A P T E R 10 Compensation of AdaptiveArrays
Equation (10.99) can berewritten as
E{A0() exp[j (k 0()]} E
N
l=NWN+1+l exp(j l)
exp(j k)
= 0 for k = N, . . . , 0, . . . , N
(10.100)
Notethat
E{exp[j (l k)]} = sin[ B(l k)] B(l k) (10.101)
it follows that
E
N
l=N
WN+1+l exp(jl)
exp(j l)
=
N
l=N
WN+1+lsin[ B(l k)]
B(l
k)
(10.102)so that (10.100) can berewritten in matrix formas
v= Cw (10.103)where
vk = E{A0() exp[j (k 0())]} (10.104)Ck,l =
sin[ B(l k)] B(l k) (10.105)
Consequently, thecomplex weight vector must satisfy therelation
w= C1v (10.106)Using (10.106) to solvefor the optimumcomplex weight vector, wecan find the outputresiduesignal power by using
Ree(0) =1
2 B
B B
|A() F ()|2J J ()d (10.107)
whereJ J () istheconstantinterferencesignal power spectral density. Assumetheinter-ference power spectral density is unity acrossthe bandwidthof concern; then the outputresiduepower dueonly tomain channel amplitudevariations is given by
ReeA =1
2 B B
B |A0() F ()|2
d (10.108)
Since A() F () is orthogonal to F (), it followsthat [15]
E{|A() F ()|2} = E{|A()|2} E{|F ()|2} (10.109)
and hence
ReeA =1
2 B
B B
[A20() |F ()|2] d (10.110)
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10.6 Analysis of Interchannel Mismatch Effects
It likewisefollows from(10.107) that the output residue power contributed by mainchannel phasevariationsis given by
Reep=
1
2 B
B
B |ej 0()
F ()
|2J J () d (10.111)
where 0() represents the main channel phase variation. Once again assuming thatthe input signal spectral density is unity across the signal bandwidth and noting that[ej0() F ()] must beorthogonal to F (), it immediately followsthat
Reep =1
2 B
B B
[1 |F ()|2] d
= 1N
j=N
Nk=N
wkwj
sin[ B(k j )] B(k j ) (10.112)
wherethecomplex weight vector elements must satisfy (10.103)(10.106).If it is desired to evaluate the effects of both amplitude and phase mismatchingsimultaneously, then the appropriate expression for the output residue power is givenby (10.107), which(becauseof orthogonality) may berewritten as
Ree(0) =1
2 B
B B
{|A()|2 |F ()|2}J J () d (10.113)
where the complex weights used to obtain F () must again satisfy (10.102)(10.106),whichnow involvebothamagnitudeandaphasecomponentanditisassumedthatJ J ()is aconstant.
10.6.1 Example: Effects of Amplitude Mismatching
To evaluate(10.110) it isnecessary to adopt achannel amplitudemodel corresponding toA(). Onepossible channel amplitudemodel is given in Figure10-35for which
A() =
1+ R cosT0 for || B0 otherwise
(10.114)
B0
1
Array bandwidth
B
A(w)
R
FIGURE 10-35
Channel amplitud
model having 3 12
cycles of ripple fo
evaluation of
amplitude misma
effects.
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410 C H A P T E R 10 Compensation of AdaptiveArrays
where
T0 =2n+ 1
2Bfor n = 0, 1, 2, . . .
andtheinteger n correspondsto (2n+
1)/2 cyclesof amplitudemismatchingacrossthebandwidthB. Lettingthephaseerror 0() = 0, it follows from(10.104) that
vk =1
2 B
B B
[1+ R cosT0]ejk d (10.115)
or
vk =sin( Bk)
Bk+ R
2
sin( B[T0 + k])
B [T0 + k]+ sin( B[T0 k])
B[T0 k]
for k = N, . . . , 0, . . . , N (10.116)
Evaluationof (10.116) permitsthecomplex weight vectortobefound, whichin turnmay
beused todeterminetheresiduepower by way of (10.110).Now
|F ()|2 = F () F () = ww (10.117)
where
=
ejN
ej( N1) ...
ejN
(10.118)
Carrying out thevector multiplications indicated by (10.117) then yields
|F ()|2 =2N+1
i=1
2N+1k=1
wi wke
j (ki ) (10.119)
Theoutput residuepower is thereforegiven by [seeequation (10.110)]
ReeA = B
B[1+ R cosT0]2 d
B B
2N+1i=1
2N+1k=1
wi wke
j (ki) d (10.120)
Equation (10.120) may beevaluated usingthefollowing expressions:
1
2 B B
B[1+ R cosT0]2
d = 1+ R2
2+ 2Rsin[(2n+ 1)/2][(2n+ 1)/2]
+ R2
2
sin (2n+ 1) (2n+ 1) (10.121)
1
2 B
B B
2N+1i=1
2N+1k=1
wi wke
j (ki ) d =2N+1
i=1
2N+1k=1
wi wk
sin(k i )B(k i)B
(10.122)
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10.6 Analysis of Interchannel Mismatch Effects
10.6.2 Results for Compensation of Selected
Amplitude Mismatch Model
Theevaluationof (10.120) requiresknowingtherippleamplitudeR, thenumber of cyclesof amplitudemismatchingacrossthe bandwidth, andthe product of B (where B is thecancellation bandwidth and is the intertap delay spacing). The results of a computerevaluation of theoutputresiduepower aresummarizedinFigures10-3610-39for B =0.25, 0.5, 0.75, and 1, and R = 0.09. Each of the figures presents a plot of the decibelcancellation (of theundesired interference signal) achieved as a function of thenumberof taps in the transversal filter and the number of cycles of ripple present across thecancellation bandwidth.No improvement(over thecancellation that canbeachievedwithonly one tap) is realized until a sufficient number of taps is present in the transversalfilter to achievethe resolution required by the amplitude versus frequency variations in
ForB= 0.25 andRm= 0.09
N= number of taps
700 5 10 15
60
50
Cancellation(dB)
40
30
20
3 Cycles1
2
2 Cycles1
2
1 Cycles1
2Cycle
1
2
FIGURE 10-36
Decibel cancellat
versus number o
taps for selected
amplitude misma
models with
B = 0.25.
ForB= 0.5 andRm= 0.09
N= number of taps
700 5 10 15
60
50Cancella
tion(dB)
40
30
10
3 Cycles1
2
2 Cycles1
2
1 Cycles1
2Cycle1
2
20
FIGURE 10-37
Decibel cancellat
versus number o
taps for selected
amplitude misma
models with
B = 0.5.
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412 C H A P T E R 10 Compensation of AdaptiveArrays
FIGURE 10-38
Decibel cancellation
versus number of
taps for selected
amplitude mismatch
models withB = 0.75.
ForB= 0.75 andRm= 0.09
N= number of taps
700 5 10 15
60
50Cancellation
(dB)
40
30
10
3 Cycles1
2
2 Cycles1
2
1 Cycles1
2Cycle1
2
20
FIGURE 10-39
Decibel cancellation
versus number of
taps for selected
amplitude mismatch
models with
B = 1.0.
ForB= 1.0 andRm= 0.09
N= number of taps
700 5 10 15
60
50Cancellation
(dB)
40
30
10
3 Cycles1
2
2 Cycles1
2
1 Cycles1
2
Cycle1
2
20
theamplitudemismatch model. Thesufficient number of tapsfor theselected amplitudemismatch model was found empirically tobegiven by
Nsufficient
Nr 12
[7 4(B)] + 1 (10.123)
where Nr is thenumber of half-cyclesof ripple appearingin themismatch model.If thereareasufficientnumber of tapsinthetransversal filter,thecancellationperfor-
manceimproves when moretapsareadded dependingonhow well theresulting transferfunction of the transversal filter matches the gain and phase variations of the channelmismatch model. Since the transversal filter transfer function resolution depends in parton the product B, a judicious selection of this parameter ensures that providing addi-tional tapsprovides a better match (andhence a significant improvement in cancellationperformance), whereasapoor choiceresultsinvery poor transfer functionmatchingevenwiththeadditionof moretaps.
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10.6 Analysis of Interchannel Mismatch Effects
ForN= 3 andRm= 0.9
B
0 0.5 1 1.5
150
100
50
20
10
Cancellation
(dB)
0 FIGURE 10-40
Decibel cancellat
versus B for
one-half-cycle
amplitude misma
model.
Taking the inverse Fourier transform of (10.114) 1{A()} yields a time functioncorrespondingto an autocorrelation function f(t) that can beexpressed as
f(t) = s(t) + Ks(t T0) (10.124)Theresultsof Section10.5.3andequation(10.124)implythat = T0 (orequivalently,
B = number cycles of ripplemismatch) if theproduct B is tomatch theamplitudemismatch model. This result is illustrated in Figure 10-40 where decibel cancellation isplottedversus B for aone-half-cycleripple mismatch model. A pronounced minimumoccurs at B = 12 for N = 3 and Rm = 0.9.
When the number of cycles of mismatch ripple exceeds unity, the foregoing rule
of thumb leads to the spurious conclusion that B should exceed unity. Suppose, forexample,thereweretwocyclesof mismatchrippleforwhichitwasdesiredtocompensate.By settingB = 2(correspondingto Bf = 12 B), two completecyclesfor thetransversalfilter transferfunctionarefoundtooccur acrossthecancellationbandwidth. By matchingonly one cycle of the channel mismatch, quite good matching of the entire mismatchcharacteristic occurs but at thepriceof sacrificing theability to independently adjust thecomplex weights across the entire cancellation bandwidth, thereby reducing the abilityto appropriately process broadband signals. Consequently, if the number of cycles ofmismatch ripple exceeds unity, it is usually best to set B = 1 and to accept whateverimprovementincancellation performancecanbeobtainedwiththat value, or increasethenumber of taps.
10.6.3 Example: Effects of Phase Mismatching
Let () correspondingto thephaseerror becharacterizedby
() =
A cosT0 for || B0 otherwise
(10.125)
where A represents the peak number of degrees associated with the phaseerror ripples.This model corresponds to the error ripple model of (10.112) (with zero average valuepresent).
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414 C H A P T E R 10 Compensation of AdaptiveArrays
Since
vk =1
2 B
B B
exp(j{A cosT0 + k}) d for k = N, . . . , 0, . . . , N(10.126)
it can easily beshownby defining
f(K, sgn)= sin [K + sgn (i (N + 1))B]
[K + sgn (i (N + 1))B]
and g(K)= f(K, +) + f(K, ) that
vi = J 0(A) f(0, +) + j J 1( A)g
2n+ 12
(10.127)
+
k=1(1)k
J 2k( A)g[k(2n+ 1)] + j J 2k+1(A)g
(2k + 1)
2n+ 1
2
whereJ n() denotes aBessel function of thenthorder for i = 1,2,...,2N + 1.
10.6.4 Results for Compensation of Selected Phase
Mismatch Model
The computer evaluation of the output residue power resulted in the performance sum-marized in Figures 10-4110-43 for B = 0.2, 0.45, and 1.0 and A = 5. Thesefigures present the decibel cancellation achieved as a function of the number of tapsin the transversal filter andthe number of cycles of phaseripple present acrossthe can-cellation bandwidth. Thegeneral natureof thecurves appearing in Figures 10-4110-43isthesameasthat of Figures 10-3610-39for amplitudemismatching. Furthermore, just
as in the amplitudemismatch case, a better channel transfer function fit can beobtainedwiththetransversal filter whenthemismatchcharacteristic hasafewer number of ripples.
FIGURE 10-41
Decibel cancellation
versus number of
taps for selected
phase mismatch
models with
B = 0.2.
ForB= 0.2 andA= 5
N= number of taps
700 531 7
60
50Cancella
tion
(dB)
40
30
10
6 &1
210 Cycles
1
2
2 Cycles1
2
Cycle1
2
20
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10.7 Summary andConclusions
ForB= 0.45 andA= 5
N= number of taps
700 531 7
60
50Cancellation
(dB)
40
30
10
6 &1
210 Cycles
1
2
2 Cycles1
2
Cycle1
2
20
FIGURE 10-42
Decibel cancellat
versus number o
taps for selected
phase mismatch
models withB = 0.45.
ForB= 1 andA= 5
N= number of taps
0 531 760
50
Cancellation(dB)
40
30
10
6 &1
210 Cycles
1
2
2 Cycles1
2
Cycle1
2
20
FIGURE 10-43
Decibel cancellat
versus number o
taps for selected
phase mismatch
models with
B = 1.0.
10.7 SUMMARY AND CONCLUSIONS
Array errorsduetomanufacturing tolerances distortthearray pattern. To minimize theseerrors, thearray must becalibrated at thefactory andat regular intervals oncedeployed.
The transversal filter consisting of a sequence of weighted taps with intertap delayspacingoffersapractical meansfor achievingthevariableamplitudeandphaseweightingas a function of frequency that is required if an adaptive array system is to performwell against wideband interference signal sources. The distortionless channel transferfunctionsfor atwo-elementarray werederived.It wasfoundthat toensuredistortion-freeresponsetoabroadbandsignal thechannel phaseisalinearfunctionof frequency, whereasthe channel amplitude function is nearly flat over a 40% bandwidth. Quadrature hybridprocessing provides adequate broadband signal responsefor signals having as much as20% bandwidth. Tapped delay line processing is a practical necessity for 20% or more
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416 C H A P T E R 10 Compensation of AdaptiveArrays
bandwidth signals. A transversal filter provides an attractivemeansof compensating thesystemauxiliary channels for theundesirableeffects of thefollowing:
1. Multipathinterference
2. Interchannel mismatch
3. Propagation delay across thearray
For multipath interference, the value of the intertap delay is in the neighborhood of thedelay time associated with the multipath ray. If the intertap delay time exceedsthe mul-tipath delay time by morethan about 30% and the multipath delay time is appreciable,a severe loss of compensation capability is incurred. If the intertap delay is too small,thenanexcessivenumber of tapswill berequiredfor effectivecancellationtooccur.Sincemultipathdelayhavingsmall valuesof associatedtimedelaydonotseverelydegradethearray performance, it is reasonabletodeterminethemost likely valuesof multipathdelaythat will occurfor thedesiredapplicationandbasethemultipathcompensationdesignonthosedelay times(assuming B 1). For reflectioncoefficientsof 0.5andBDm = 0.45,
theuseof fivetaps will ensurea30dB cancellation capability.The results shown in Figures
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