OFDM Channel Estimation Based on Adaptive Thresholding for Sparse Signal Detection

0.09 MB·English

by  Mahdi Soltanolkotabi

OFDMCHANNELESTIMATION BASED ONADAPTIVETHRESHOLDINGFORSPARSE
SIGNALDETECTION
MahdiSoltanolkotabi,ArashAminiand FarokhMarvasti
Advanced CommunicationResearch Institute(ACRI)
EE Department,SharifUniversityofTechnology
{msoltan, arashsil}@ee.sharif.eduand [email protected]
9
0
0 ABSTRACT in[?]asparsity-basedestimationmethodemploying theMatching
2 Pursuitisdevised.
WirelessOFDMchannelscanbeapproximatedbyatimevaryingfil-
Amaindrawbackofthesemethodsisthattheydonotconsider
n terwithsparsetimedomaintaps.Recentachievementsinsparsesig-
zeropaddingintheirscenario.IncurrentOFDMstandardstheband-
a nalprocessing suchascompressed sensinghavefacilitatedtheuse
widthisnotfullyoccupied;anumberofthesubcarriersatbothedges
J of sparsity in estimation, which improves the performance signifi-
ofthebandwidtharesettozero(hencethenamezeropadding)toin-
6 cantly. Theproblemofthesesparse-basedmethodsistheneedfor
creasetheallowabletransitionbandoftheanalogbandpassfilterat
2 a stable transformation matrix which is not fulfilled in the current
thereceiver. Zeropaddingresultsinanunstablefrequencytotime
transmission setups. To assist the analog filtering at the receiver,
transformation(ill-conditionedtransformationmatrix). Inaddition,
] the transmitter leaves some of the subcarriers at both edges of the
T duetothelackofpilotinzeropaddedparts,commontimedomain
bandwidth unused whichresultsinanill-conditionedDFTsubma-
techniquesareimpractical. Inthispaperweproposeasparsechan-
I
trix. To overcome this difficulty we propose Adaptive Threshold-
. nelestimationmethodcalledAdaptiveThresholdingforDetectionof
s ingforSparseSignalDetection(ATSSD).Simulationresultsconfirm
c SparseSignals(ATSSD)thatworksadequatelyevenforzero-padded
thattheproposedmethodworkswellintime-invariantandspecially
[ OFDMsystems. SimulationresultsusingtheDVB-Hstandardcon-
time-varyingchannelswhereothermethodsmaynotworkaswell. firmthat ATSSDestimatesalmost theexact CFRintime-invariant
1
IndexTerms— Sparsechannel,Adaptivethresholding,OFDM channels. Furthermore, unlike the previous methods, the perfor-
v
channelestimation,MMSE manceundervaryingchannelsdegradesonlyslightlyastheDoppler
8
frequencyincreases.
4
9 1. INTRODUCTION
3 2. SPARSEOFDMCHANNELESTIMATION
1. Recent standards such as [?, ?, ?] introduce OFDM communica-
0 tionasoneofthebestoptionsforwirelesstransmissionofmultime- ToshowhowtheaforementionedchallengesinOFDMchanneles-
9 diasignalsincludingvideo. OFDMtransmission,althoughresistant timationcanbeovercomeusingtheinherentchannelproperties,we
0 againstmultipathfading,requiresaccurateestimationoftheChannel willbrieflyexplaintheproblemintwosubsections. In2.1OFDM
: FrquencyResponse(CFR)atthereceiver forappropriatedecoding systemcomponents relatedtochannel estimation arereviewed. In
v ofthedata. 2.2,inadditiontorestatingtheOFDMchannelestimationasasparse
i
X Formobilereception,thetransmitterreservessomeofthesub- signal processing problem, wewill explain thedifficultyof partial
carriersineach OFDMsymbol (whichmayvary indifferent sym- useofbandwidth(zeropadding)inestimatingsparsechannels.
r
a bols) for a predefined pattern of data, referred to as Pilot tones,
whichareaprioryknownatthereceiver. SinceinOFDMtransmis-
2.1. ReviewofOFDMSystemComponents
sionthedataaredirectlysent inthefrequency domain, thesepilot
tonesprovidethereceiverwithirregularnoisysamplesoftheCFR AttheOFDMtransmitter,codedbinarydataareinterleaved,grouped
in the time interval of that symbol. The main task of the channel andmapped toconstellationpointsbasedonaspecificmodulation
estimator block of the receiver istoestimate CFRat the non-pilot scheme. The resulting complex symbols which will form the fre-
subcarriersusingtheobtainedsamplesatpilottones. Conventional quencydomainofthetransmittingsignalarerearrangedintoblocks
methods involve some sort of interpolation between the obtained andforeachblockanumberofpilotsareinsertedamongthedataat
samples,rangingfromthesimpleLinearInterpolation(LI)tomore predefinedlocations. TofacilitatetheRFanalogfilteringatthere-
complicatedapproacheslikesplines.Underlimitedtimespreadand ceiver,theseblocksarezeropaddedtoformthefinalblocks(OFDM
Dopplerfrequency,thetimevaryingOFDMchannelcanbeviewed symbols, denoted by X(n) wheren istheblock number); i.e., the
asalowpass2Dsignal[?]. Inthissensetheproblemofchanneles- bandwidth is not fully occupied. The resulting symbols are con-
timationbecomesequivalenttothereconstructionofa2-Dlowpass vertedintothetimedomain(x(n))bytheuseoftheIFFTalgorithm,
signalfromitsirregularsamples[?]. where each symbol is extended by a Cyclic Prefix (CP, a copy of
Recent resultsinsparsesignal processingsuchasCompressed thelast partof eachOFDMsymbol) andthesefinalblocks arese-
Sensing[?],pavedthewaytoexploittheinherenttimedomainspar- riallytransmitted. Thecyclicprefixisinsertedtoprevent possible
sityofthechannelinitsestimation[?,?,?]. Usingl minimization Inter-SymbolInterference(ISI)inexistenceofmultipathchannels.
1
algorithms(BasisPursuit)introducedin[?],reductionofthenum- Atthereceiver,thecyclicprefixisremovedandthereceivedsig-
berofrequiredpilotsforchannelestimationisproposedin[?]while nals,y(n),aresenttoanFFTblock.Whenthedurationofthechan-
nelimpulseresponseisshorterthanthecyclicprefixlength,noISI 3. PROPOSEDMETHOD
isexpected between OFDM symbols. Assuming also no synchro-
nizationerror,theequationrelatingthenthtransmittedandreceived Aspreviouslystated,theOFDMChannelImpulseResponse(CIR)is
symbols,respectivelyX(n)andY(n),is: sparse.InthissectionwewillproposeanewschemecalledAdaptive
ThresholdingforSparseSignalDetection(ATSSD)thatcanexploit
Y(n)=X(n)⊙H(n)+W(n) (1)
thisinherentsparsityevenwithill-conditionedmatrices,F .
p,CP
where⊙representselement byelement multiplicationoftwovec- In this method, the spectrum of the channel is initially esti-
tors, H(n) is avector containing the samples of the Channel Fre- mated using linear interpolation between pilot subcarriers. The
quency Response (CFR) corresponding to the nth OFDM symbol zero-paddedpartsofthebandwidthwhicharepossiblymeasuredas
andWdenotesthesampledvector oftheAWGNnoiseinthefre- nonzeroduetothenoise,areinitiallyestimatedaszerowhileatthe
quencydomain.Sincethereceiverispriorlyawareofpilotpositions middle subcarriers which are equipped with comb-type pilots, the
andvalues, itcanobtainanoisyestimateofthechannelfrequency mentionedlinearinterpolationusingthenoisysamplesofthechan-
spectrumat thesesubcarriers. Inthissense, theOFDMchannel is nelatpilotlocationsisemployed. Thisinitialestimateisimproved
similar to a 2-D lattice in the discrete time-frequency plane from inaseriesofiterationsthatfindsthesparsesttimedomainresponse.
which certain points are known due to pilots; the goal of channel The initially estimated spectrum is passed through the IFFT
estimationistoestimatetherestviainterpolation. Theinterpolated block to obtain a crude version of the time domain impulse re-
channelestimateisthenusedintheequalizationblocktoobtainan sponse. This initial estimate is fed to the ATSSD block which
estimateofthetransmittedconstellationpoints(X(n)).Afterequal- consists of several iterations. In each iteration this method tries
ization, the approximated OFDM symbol is demodulated, deinter- to find the location of the taps via a thresholding scheme on the
leaved,anddecodedtoproducethebinaryoutputdata. estimatedchannel fromthepreviousiteration(orinthecaseofthe
firstiteration,thecrudeversiondiscussedabove). Thethresholding
2.2. RestatingOFDMChannelEstimationasaSparseProblem scheme isfurther described insec. 3.1. After findingthe location
of the taps ineach iteration (locations whose previously estimated
Considering the sparse distribution of the scattering objects, the amplitudes stay above the threshold), their respective amplitudes
OFDMchannel becomes sparse in thetime domain. Thus, by ex- are again found using the Minimum Mean Square Error (MMSE)
ploitingthissparsity, abetter estimatecould beobtainedusing the criteriondiscussedinsec.3.2.Ineachiteration,duetothresholding,
time domain. The resulting estimate is then transformed into the some of the fake taps which are noise samples whose amplitudes
frequency domain by use of the FFTalgorithm. In this sense, the were above the threshold in the previous iteration, are discarded.
problem of channel estimation becomes equivalent to finding the Thus, the new iteration starts with a lower number of fake taps.
sparseChannelImpulseResponse(CIR)(h)fromtheequation: Moreover,becauseoftheMMSEestimator,thevalidtapsapproach
theiractualvaluesineachnewiteration.Inthelastiteration,theac-
Hp =Fp·h+Wp (2) tualtapsaredetectedandtheMMSEestimatorgivestheirrespective
values.Themainstepsoftheproposedalgorithmare:
whereH isthevectorofobservedchannelcoefficientsatpilotsub-
p e
carriers,F isthesub-matrixoftheDFTmatrixobtainedbykeep-
p 1. Usinglinearinterpolation,setthecrudetime-domainchannel
ing theerows of the FFT matrix that correspond to pilot positions asdescribedabovefortheinitialestimate.
andW isthefrequency-domainnoisevectoratpilotpositions. As
p
statedearlier,inthecaseofnoISI,thelengthofthechannelcannot 2. Discardthetapswithamplitudesbellowthethreshold.
exceedthelengthofthecyclicprefix(NCP). Thus,(2)canbefur-
3. Estimate the value of the remaining taps using the MMSE
thersimplifiedinthezeroISIcase;onlythefirstNCP elementsof
criterion.
h(h )canhavenon-zerovalues.
CP
4. Stopiftheestimatedchannel isthesameasthepreviousit-
Hp =Fp,CP ·hCP +Wp (3)
erationorwhenamaximumnumberofiterationsisreached,
whereF isthesub-matrixoftheDFTmatrixobtainedbykeep- elsegotostep2.
p,CP e
ingonlythefirstNCP columnsofFp. Recentlytheideaofusing
timedomainsparsityinOFDMchannelestimationwasproposedin
[?]fordecreasingthenumberofpilots. Theauthorsproposedcom- 3.1. ChannelTapDetectionviaThresholding
pressivesensing algorithmstofindthesparsest timedomain chan-
Asmentionedearlier,ineachiteration,tapsbellowacertainthresh-
nel. Theyprovedthatincaseofuniformpilotinsertion,thematrix
F in(3)satisfiestheuniformuncertaintytheoremdescribedin oldarediscarded.Thethresholdintheithiterationissetas:
p,CP
[?]. Asaresult,linear-programming-basedalgorithmsusedincom-
pressedsensing,similartotheonesintroducedin[?]canbeapplied Threshold=βeαi (4)
toOFDMchannel estimation. However, theauthorsof [?]didnot
considerzero-paddingattheendpointsofthebandwidthintheirsce- whereirepresentstheiterationnumber;i.e.,thethresholdexponen-
nario,whichisanessentialpartofcurrentOFDMstandards.Thisas- tiallyincreasesastheiterationsproceed. αandβareconstantsthat
sumption,causesthematrixFp,CP tocontradicttheRestrictedIso- dependonthenumberoftapsandinitialpowersofnoiseandchannel
metricProperty(RIP)definedin[?]andthustheuseofCompressive taps;sincetheseparametersarehardlyknown,aroughapproxima-
Sensing(CS)algorithmsasdescribedin[?],unpractical.Inthepres- tionisused.Inthefirstiteration,thethresholdisasmallnumberand
enceofzero-padding,thematrixFp,CP becomesill-conditioned(a witheach iterationit isgradually increased. Intuitively, this grad-
smallvariationinH leadstoagreatchangeinh ).Duetozero- ualincreaseofthethreshold, resultsinagradual reductionoffake
p CP
paddingwedonothaveanypilotsinthezeropaddedparts,further taps(tapsthatarecreatedduetonoise). Thisisfurtherexplainedin
complicatingtheueseoftime-domaintechniques. section4fromamathematicalpointofview.
3.2. EstimationofChannelTapValuesviaMMSE Assumingηasthethresholdfortheithiteration,weexpecttheover-
allnoisepower(realandimaginary)ofthenextiterationtobeequal
Afterfindingtime-domainchanneltappositions,ineachiterationwe
toΣ . Althoughthisisonlyaroughstatisticalapproximation,
needtofindanestimateoftheirvalue. Thisisequivalenttosolving tap|n
simulationresultsconfirmthatafterMMSEestimation,thenoisein
alinearequationinthepresenceofnoise.Thatis,wewishtoobtain
theremainingtaps(locationsthatstayedabovethethresholdinthe
thevalueoftheCIRvectorhattappositions(T),fromtheequation
previous iteration) is almost normal with zero mean and variance
Hp =Fp,ThT +Wp (5) 0.5Σtap|n:
Fwhp,eTreisthtehevesuctbo-rmHatpreixisotfhethmeeDaFsuTremdaCtriFxR(Fve)cotobrtaaitnepdilobtypsoeslietciotinnsg, σn2,i+1 ≈ Σta2p|n =σn2,i 1+ 2ση22 (10)
therowsandcolumensthatpertaintothepilotpositions(p)andlikely ˛˛η ` n,i´
channel taps(T), respectively, andWp isthenoisevector atpilot Forthechoiceofη,in[?]iti˛˛sshownthatthethresholdwhichmaxi-
positions. TheMMSEestimatortriestofindtheestimatevectorhˆT mizesSNRofthesignalafterthresholding(consideringbothmissed
thatminimizesE{khT −hˆTk2},hencethenameMinimumMean andfaketaps)isfoundas:
SquaredError.Thisestimatevectorisgivenby:
wherehˆ.HT d=enRotheTsFthHpe,TH`eFrmp,TitRianhTopFeHpr,aTtio+nRanWd´R−h1Hpand RW a(6re) ηopt,i=σn2,is2σn2,iσ+t2apσt2ap ln„1−ptapptapσn2,iσ+n2,iσt2ap« (11)
Te
theauto-covariancematricesofh andW respectively. Whenthe Using(11)in(10)weget:
T p
noisevectorWisarandomcomplexwhiteGaussianprocess,RW
isequalto2σW2 I,wherevarianceofbothrealandimaginaryparts σn2,i+1 =1+ σn2,i+σt2ap ln 1−ptapσn2,i+σt2ap (12)
areassumedtobeσW2 . AlsoRhT canbeapproximatedwithPhI σn2,i σt2ap „ ptap σn2,i «
wherePh istheaverage power of thechannel. Thus, theestimate
vectorcanbewrittenas: which shows the increase inthe noise variance; the noise samples
whichstayabove thethreshold shouldhavehigher amplitudesand
hˆT ≈FHp,T Fp,TFHp,T + 2Pσhw2 I −1Hp (7) tnheel,reσfot2arpe,ihsilgahrgereveanroiuangchet.oFhoarveretahseonfoalblloewSinNgRapvparluoexsimoafttihoensc:han-
wherethechannelpow`erPhcaneasilybees´timaetedbydividingthe σn2,i
poweroftheto-be-equalizeddatabytheconstellationpower(power
oftheequalizeddataisthesameastheaverageconstellationpower). 8 ηopt,i≈σn2,is2ln„1−ptpatpap σσt2n2a,pi«
Inthissectionw4.epMreAseTnHtEaMmaAtTheImCaAtiLcaAlNinAtuLitYioSnISfortheproposed >>>>>>< σn2,i+1 ≈σn2,i„1+ln 1−ptapptapσσtn22a,0p −ln(σσn2n2,,0i)« (13)
` ´
cmtalupedtshianongdd.tthFheoeriansdiitmdiaiptlilviiteceinrtyaotiiosofentihfneotrhamneeatdliymbseiysd,loiwnmeeaaarisnisnuattmetrhepeothliaathttiobitnoe)trhaatrcieohnazne(nrione--l >>>>>>:Ascanbeseenin|(13), σn2,i{+kz1 < kσn2,i}; however, weassume
meannormalcomplexrandomvariableswithequalrealandimagi- theupperboundforthenoiseandthus:
naryvariances σ2 andσ2 , respectively. Therefore, theirampli-
tap n,i σ2 ≈ kσ2 ⇒σ2 ≈σ2 ki
tudeswhichareofmainconcerninATSSDhaveRayleighprobabil- n,i+1 n,i n,i n,0
itydistributionfunction. Inaddition,weassumethattheprobability
tohfealteinmgethsaomfptlheebceyincglicacphreafinxn,elwtaepeixsppetcatp;toi.eh.a,vifeNaCcPharnenperleswenitths ⇒ηopt,i ≈ σn2,0eiln(k)s2ln 1−ptapptapσt2ap −4iln(k)
` ´
ptap·NCP taps. Now, ifwesetathreshold(η)todistinguish the ≈ βeiln(k) (14)
noiseandvalidsamples,weprobablymisssomeoftheoriginaltaps
whiledetectinganumberoffakelocations.TheprobabilityofFalse whereβisaconstant. Similarto(4),(14)showsthesameexponen-
Alarm(FA);i.e.,detectionoffaketaps,usingtheRayleighdistribu- tialincreaseinthethresholdwithrespecttotheiterationnumber.
tionisgivenby:
pfa=Zη∞ σxn2,ie−2σxn22,i ·dx=e−2σηn22,i (8) Toverify the effic5a.cySoIfMthUeLpArTopIOosNedRcEhSanUnLeTl eSstimation, weper-
Moreover,theaveragepowerofthesefaketaps(Σ )is: formed computer simulations based on the DVB-H standard using
tap|n
MATLAB.Themainsimulationparametersandoptionsareshown
Σtap|n = E |w|2 |w|>η inTable1and2. Toshow therobustness of ATSSDinestimating
2 the CFR for time-invariant channels, we compared the BER after
= 1˘ ∞˛˛ x3 e−2σx¯n2,i ·dx Viterbi decoding of the proposed method to that of the hypotheti-
p σ2
fa Zη n,i calidealchannelinFig. 1; byidealchannelwemeanthatfordata
= p1 2σn2,i 1+ 2ση22 e−2σηn22,i ecqanuableizsaetieonni,nwFeigu.s1ed,ththeeBeExRactpecrhfaonrmnealnicnestoefadtheofpirtospeossteimdamtee.thAods
fa n,i almostcoincideswiththatoftheidealchannel. Thus,inthissense,
` η2 ´ theATSSDestimationisperfect. Moreover, theBERperformance
= 2σ2 1+ (9)
n,i 2σ2 ofLinearInterpolation(aconventionalchannelestimationmethod)
n,i
` ´
Table1.SimulationParameters
Parameter Specifications 10−1
DVB-Hmode 2K
Numberofcarriers 1705
Numberofpilotcarriers 146 10−2
OFDMsymbolduration 224µs
GaurdInterval 1/8(256)
R
SignalConstellation QAM-16 BE10−3
ConvolutionalEncoder 171OCT and133OCT Fd = 100
Codingrate 1/2 Fd = 75
[itermax, α, β] [5, 0.8, 0.008] 10−4 FFdd == 5300
No Doppler
10−5
10 15 20 25
Table2.Multi-PathProfile(BrazilChannelD) Carrier to Noise (dB)
Delay(µs) 0.0 0.48 2.07 2.90 5.71 5.78
Amp.(dB) −0.1 −3.9 −2.6 −1.3 0.0 −2.8
Fig. 2. Performance of the proposed method at different Doppler
frequenciesforBrazilDchannel.
thesameasifweusedtheexactchannelevenfortime-varyingchan-
isalsoshownfor comparison. Theproposed methodisalsoeffec- nels.
tiveintime-varyingchannels. ThisfactisverifiedinFig. 2where hamashalakie
theBERperformance (afterViterbidecoding) of ATSSDisshown
for different Doppler frequencies. As obvious in Fig. 2, ATSSD
shows littleperformance degradation withincrease of the Doppler
frequency.
100
10−1
10−2
R
E
B
10−3
Ideal channel
ATSSD
10−4 linear Interpolation
10−5
0 5 10 15 20
Carrier to Noise (dB)
Fig. 1. Performance of the ideal, linear-interpolated and the pro-
posedchannelestimatesundertime-invariantBrazilDchannel.
6. CONCLUSION
DuetothepartialuseofbandwidthinOFDMtransmission,there-
duced frequency to time transformation is no longer stable; there-
fore,mostofthesparsity-basedchannelestimationmethodsdiverge.
Wehaveproposedathresholdingmethodwhichsolvesthisproblem
whilebenefitingfromthesparsitycriterion.Simulationresultsshow
thattheperformanceofthismethodafterdataequalizationisalmost