Compensation of Adaptive Arrays_10

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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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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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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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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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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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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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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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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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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.


15/47


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


16/47


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;


17/47


18/47


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


19/47


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


20/47


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,


21/47


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].


22/47


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.


23/47


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.


24/47


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


25/47

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.


26/47


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


27/47

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.


28/47


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)


29/47


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.


30/47


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


31/47


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


32/47


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.


33/47


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


34/47


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


35/47

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)


36/47


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)


37/47


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.


38/47


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)


39/47


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.


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.


40/47


FIGURE 10-38


versus number of

taps for selected

amplitude mismatch

models withB = 0.75.


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


versus number of

taps for selected

amplitude mismatch

models with

B = 1.0.


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.


41/47


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).


42/47


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


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


43/47

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


44/47


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

Compensation of Adaptive Arrays_10

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