walsh变换matlab仿真word.docx

walsh变换matlab仿真word.docx

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DiscreteWalsh-HadamardTransform

Contents

∙Introduction

∙Walsh（orHadamard）Functions

∙DiscreteWalsh-HadamardTransform

∙Walsh-TransformApplications

∙References

Introduction

TheWalsh-Hadamardtransform（WHT）isasuboptimal,non-sinusoidal,orthogonaltransformationthatdecomposesasignalintoasetoforthogonal,rectangularwaveformscalledWalshfunctions.ThetransformationhasnomultipliersandisrealbecausetheamplitudeofWalsh（orHadamard）functionshasonlytwovalues,+1or-1.

WHTsareusedinmanydifferentapplications,suchaspowerspectrumanalysis,filtering,processingspeechandmedicalsignals,multiplexingandcodingincommunications,characterizingnon-linearsignals,solvingnon-lineardifferentialequations,andlogicaldesignandanalysis.

ThisdemoprovidesanoverviewoftheWalsh-Hadamardtransformandsomeofitspropertiesbyshowcasingtwoapplications,communicationsusingspreadspectrumandprocessingofECGsignals.

Walsh（orHadamard）Functions

Walshfunctionsarerectangularorsquarewaveformswithvaluesof-1or+1.AnimportantcharacteristicofWalshfunctionsissequencywhichisdeterminedfromthenumberofzero-crossingsperunittimeinterval.EveryWalshfunctionhasauniquesequencyvalue.

Walshfunctionscanbegeneratedinmanyways（see[1]）.HereweusethehadamardfunctioninMATLAB®togenerateWalshfunctions.LengtheightWalshfunctionsaregeneratedasfollows.

N=8;%LengthofWalsh（Hadamard）functions

hadamardMatrix=hadamard（N）

hadamardMatrix=

11111111

1-11-11-11-1

11-1-111-1-1

1-1-111-1-11

1111-1-1-1-1

1-11-1-11-11

11-1-1-1-111

1-1-11-111-1

Therows（orcolumns）ofthesymmetrichadamardMatrixcontaintheWalshfunctions.TheWalshfunctionsinthematrixarenotarrangedinincreasingorderoftheirsequenciesornumberofzero-crossings（i.e.'sequencyorder'）butarearrangedin'Hadamardorder'.TheWalshmatrix,whichcontainstheWalshfunctionsalongtherowsorcolumnsintheincreasingorderoftheirsequenciesisobtainedbychangingtheindexofthehadamardMatrixasfollows.

HadIdx=0:

N-1;%Hadamardindex

M=log2（N）+1;%Numberofbitstorepresenttheindex

Eachcolumnofthesequencyindex（inbinaryformat）isgivenbythemodulo-2additionofcolumnsofthebit-reversedHadamardindex（inbinaryformat）.

binHadIdx=fliplr（dec2bin（HadIdx,M））-'0';%Bitreversingofthebinaryindex

binSeqIdx=zeros（N,M-1）;%Pre-allocatememory

fork=M:

-1:

2

%Binarysequencyindex

binSeqIdx（:

k）=xor（binHadIdx（:

k）,binHadIdx（:

k-1））;

end

SeqIdx=binSeqIdx*pow2（（M-1:

-1:

0）'）;%Binarytointegersequencyindex

walshMatrix=hadamardMatrix（SeqIdx+1,:

）%1-basedindexing

walshMatrix=

11111111

1111-1-1-1-1

11-1-1-1-111

11-1-111-1-1

1-1-111-1-11

1-1-11-111-1

1-11-1-11-11

1-11-11-11-1

DiscreteWalsh-HadamardTransform

TheforwardandinverseWalshtransformpairforasignalx（t）oflengthNare

Fastalgorithms,similartotheCooley-Tukeyalgorithm,havebeendevelopedtoimplementtheWalsh-HadamardtransformwithcomplexityO（NlogN）（see[1]and[2]）.SincetheWalshmatrixissymmetric,boththeforwardandinversetransformationsareidenticaloperationsexceptforthescalingfactorof1/N.ThefunctionsfwhtandifwhtimplementtheforwardandtheinverseWHTrespectively.

Example1PerformWHTontheWalshmatrix.Theexpectedresultisanidentitymatrixbecausetherows（orcolumns）ofthesymmetricWalshmatrixcontaintheWalshfunctions.

y1=fwht（walshMatrix）%FastWalsh-Hadamardtransform

y1=

10000000

01000000

00100000

00010000

00001000

00000100

00000010

00000001

Example2ConstructadiscontinuoussignalbyscalingandaddingarbitrarycolumnsoftheHadamardmatrix.ThissignalisformedusingweightedWalshfunctions,sotheWHTshouldreturnnon-zerovaluesequaltotheweightsattherespectivesequencyindices.WhileevaluatingtheWHT,theorderingisspecifiedas'hadamard',becauseaHadamardmatrix（insteadoftheWalshmatrix）isusedtoobtaintheWalshfunctions.

N=8;

H=hadamard（N）;%Hadamardmatrix

%ConstructasignalbyaddingafewweightedWalshfunctions

x=8.*H（1,:

）+12.*H（3,:

）+18.*H（5,:

）+10.*H（8,:

）;

y=fwht（x,N,'hadamard'）

y=

80120180010

WHTisareversibletransformandtheoriginalsignalcanberecoveredperfectlyusingtheinversetransform.Thenormbetweentheoriginalsignalandthesignalobtainedfrominversetransformationequalszero,indicatingperfectreconstruction.

xHat=ifwht（y,N,'hadamard'）;

norm（x-xHat）

ans=

0

TheWalsh-Hadamardtransforminvolvesexpansionusingasetofrectangularwaveforms,soitisusefulinapplicationsinvolvingdiscontinuoussignalsthatcanbereadilyexpressedintermsofWalshfunctions.BelowaretwoapplicationsofWalsh-Hadamardtransforms.

Walsh-TransformApplications

ECGsignalprocessingOften,itisnecessarytorecordelectro-cardiogram（ECG）signalsofpatientsatdifferentinstantsoftime.Thisresultsinalargeamountofdata,whichneedstobestoredforanalysis,comparison,etc.atalatertime.Walsh-HadamardtransformissuitableforcompressionofECGsignalsbecauseitoffersadvantagessuchasfastcomputationofWalsh-Hadamardcoefficients,lessrequiredstoragespacesinceitsufficestostoreonlythosesequencycoefficientswithlargemagnitudes,andfastsignalreconstruction.

AnECGsignalanditscorrespondingWalsh-Hadamardtransformisevaluatedandshownbelow.

x1=ecg（512）;%Singleecgwave

x=repmat（x1,1,8）;

x=x+0.1.*randn（1,length（x））;%Noisyecgsignal

y=fwht（x）;%FastWalsh-Hadamardtransform

figure（'Color','white'）;

subplot（2,1,1）;

plot（x）;

xlabel（'Sampleindex'）;

ylabel（'Amplitude'）;

title（'ECGSignal'）;

subplot（2,1,2）;

plot（abs（y））

xlabel（'Sequencyindex'）;

ylabel（'Magnitude'）;

title（'WHTCoefficients'）;

Ascanbeseenintheaboveplot,mostofthesignalenergyisconcentratedatlowersequencyvalues.Forinvestigationpurposes,onlythefirst1024coefficientsarestoredandusedtoreconstructtheoriginalsignal.Truncatingthehighersequencycoefficientsalsohelpswithnoisesuppression.Theoriginalandthereproducedsignalsareshownbelow.

y（1025:

length（x））=0;%Zeroingoutthehighercoefficients

xHat=ifwht（y）;%SignalreconstructionusinginverseWHT

figure（'Color','white'）;

plot（x）;

holdon

plot（xHat,'r'）;

xlabel（'Sampleindex'）;

ylabel（'ECGsignalamplitude'）;

legend（'OriginalSignal','ReconstructedSignal'）;

Thereproducedsignalisveryclosetotheoriginalsignal.

Toreconstructtheoriginalsignal,westoredonlythefirst1024coefficientsandtheECGsignallength.Thisrepresentsacompressionratioofapproximately4:

1.

req=[length（x）y（1:

1024）];

whosxreq

NameSizeBytesClassAttributes

req1x10258200double

x1x409632768double

CommunicationusingSpreadSpectrumSpreadspectrum-basedcommunicationtechnologies,likeCDMA,useWalshcodes（derivedfromWalshfunctions）tospreadmessagesignalsandWHTtransformstodespreadthem.SinceWalshcodesareorthogonal,anyWalsh-encodedsignalappearsasrandomnoisetoaterminalunlessthatterminalusesthesamecodeforencoding.Belowwedemonstratetheprocessofspreading,determiningWalshcodesusedforspreading,anddespreadingtorecoverthemessagesignal.

TwoCDMAterminalsspreadtheirrespectivemessagesignalsusingtwodifferentWalshcodes（alsoknownasHadamardcodes）oflength64.ThespreadmessagesignalsarecorruptedbyaadditivewhiteGaussiannoiseofvariance0.1.

Atthereceiver（basestation）,signalprocessingisnon-coherentandthereceivedsequenceoflengthNneedstobecorrelatedwith2^NWalshcodewordstoextracttheWalshcodesusedbytherespectivetransmitters.ThiscanbeeffectivelydonebytransformingthereceivedsignalstosequencydomainusingthefastWalsh-Hadamardtransform.Usingthesequencylocationatawhichapeakoccurs,thecorrespondingWalsh-Hadamardcode（ortheWalshfunction）usedcanbedetermined.TheplotbelowshowsthatWalsh-Hadamardcodeswithsequency（withordering='hadamard'）60and10wereusedinthefirstandthesecondtransmitter,respectively.

loadmess_rcvd_signals.mat

N=length（rcvdSig1）;

y1=fwht（rcvdSig1,N,'hadamard'）;

y2=fwht（rcvdSig2,N,'hadamard'）;

figure（'Color','white'）;

plot（0:

63,y1,0:

63,y2,'r'）;

xlabel（'Sequencyindex'）;

ylabel（'WHToftheReceivedSignals'）;

title（'Walsh-HadamardCodeExtraction'）;

legend（'WHTofTx-1signal','WHTofTx-2signal'）;

Despreading（ordecoding）toextractthemessagesignalcanbecarriedoutinastraightforwardmannerbymultiplyingthereceivedsignalsbytherespectiveWalsh-hadamardcodesgeneratedusingthehadamardfunction.（NotethattheindexinginMATLAB®startsfrom1,henceWalsh-Hadamardcodeswithsequency60and10areobtainedfrombyselectingthecolumns（orrows）61and11intheHadamardmatrix.）

N=64;

hadamardMatrix=hadamard（N）;

codeTx1=hadamardMatrix（:

61）;%Codeusedbytransmitter1

codeTx2=hadamardMatrix（:

11）;%Codeusedbytransmitter2

Thedecodingoperationtorecovertheoriginalmessagesignalis

xHat1=codeTx1.*rcvdSig1;%Decodedsignalatreceiver1

xHat2=codeTx2.*rcvdSig2;%Decodedsignalatreceiver2

Therecoveredmessagesignalsatthereceiversideareshownbelowandsuperimposedwiththeoriginalsignalsforcomparison.

subplot（2,1,1）;

plot（x1）;

holdon

plot（xHat1,'r'）;

legend（'OriginalMessage','ReconstructedMessage','Location','Best'）;

xlabel（'Sampleindex'）;

ylab