平均爲2個噪聲信號

Question

之間的相移我有兩個有噪聲的信號：

vv = 

130.004701157424 0.0324551387935370 -0.00451469350442727 
130.011647685172 0.0293209698881577 -0.00443672007848742 
130.018593981487 0.0174427299727758 -0.00604150583046056 
130.025540509235 0.00932950167238858 -0.00674349719282885 
130.032484953732 0.00662683499867775 -0.00652548843970368 
130.039431597197 0.00963962170597527 -0.00271561211198806 
130.046378009228 0.00916636267883020 -0.00458483912768415 
130.053323495409 0.00848153573266126 -0.00542464163811399 
130.060270138874 0.0104189382416956 -0.00464131395425218 
130.067215277788 0.00953497010962494 -0.00598548927183946 
130.074161689819 0.0125977970865675 -0.00236431037166667 
130.081106944449 0.0157313089275160 -0.00124953996090587 
130.088053587915 0.0103333882964656 -0.00417575754540960 
130.095000231496 0.0108738998659066 -0.00134972601854648 
130.101946874961 0.0137694481434158 0.000439871274351066 
130.108893402826 0.0106008069840205 0.00186444703878395 
130.115839930575 0.00601970125521576 0.000359414960086779 
130.122786342606 0.0108009060625473 0.000440760947416785 
130.129732986071 0.00503317203604985 -0.00189482334703281 
130.136679629653 0.00348671530646311 -0.00597670527650866 
130.143626273118 0.00300096809365072 -0.00400691470698012 
130.150572569433 0.0142183275779423 -0.00804978952801293 
130.157519328731 0.0230525240949860 -0.00521843258757276 
130.164466319463 0.0280086287368616 -0.00668926653224850 
130.171412962929 0.0214224391615698 -0.00787607758546161 
130.178359490793 0.0168846248864780 -0.0145337319344219 
130.185306134259 0.0239192800093970 -0.0181421164380445 
130.192253009274 0.0210498390508919 -0.0184306704321636 
130.199198611055 0.0179971812563166 -0.0218520121706652 
130.206145486070 0.0103460380242347 -0.0159440755718102 
130.213092361085 0.00807329993777163 -0.0159555444729480 
130.220039467651 0.00912526344030146 -0.0127874433100412 
130.226986342550 0.00987789917673930 -0.0133068804083725 
130.233932291714 0.0146335693682593 -0.0146516075525065 
130.240879166613 0.0197736097409092 -0.0166339708382938 
130.247826041628 0.0237299835348641 -0.0125980372608054 
130.254772337943 0.0159187859188923 -0.0143153263534319 
130.261718518540 0.0108969610019023 -0.0176225456453581 
130.268663425904 0.0113948628098992 -0.0154783872650768 
130.275610416636 0.00550151729857086 -0.0162109325490846 
130.282555902817 0.0109876298475660 -0.0153331746399511 
130.289503124982 0.00549384751028331 -0.0166060194162694 
130.296449999998 0.00578884116574096 -0.0159344196006499 
130.303394675927 0.00790183949292702 -0.0153635837219086 
130.310341550943 0.00618938542567369 -0.0150381094970244 
130.317288078691 0.00265099365518004 -0.0171663309551115 
130.324235300973 0.00273994060860974 -0.0145340255449315 
130.331182060155 -0.00151750931024957 -0.0143517639050771 
130.338128935171 -0.00358804234355178 -0.0198922665378706 
130.345076620320 -0.00382232746918610 -0.0176678440329436 
130.352024305586 -0.000693949984437276 -0.0169212697877281 
130.358971064768 0.00650620967722442 -0.0170218577527335 
130.365918055500 0.0201962605199323 -0.0140533168589787 
130.372864236098 0.00731475005500598 -0.0150833328404646 
130.379811921248 0.0147341961991546 -0.0129639299478812 
130.386759143530 0.00921475919358350 -0.0161000768186993 
130.393706365695 0.00593047075608162 -0.0133553645712844 
130.400653240711 0.00516910535303972 -0.0173593220662748 
130.407599305592 0.00458812986283153 -0.0159890785453408 
130.414546759308 0.-0.0185143837967815 
130.421493750040 0.00831229147094524 -0.0153532081457558 
130.428440740772 0.00875956790369065 -0.0163487971058285 
130.435387499954 0.00900962706783174 -0.0185691489107274 
130.442333796269 0.-0.0182074899466804 
130.449281365701 0.0152967329894126 -0.0184572729268484 
130.456228472267 0.0165450555725402 -0.0164807811464941 
130.463175000015 0.0260987363550796 -0.0140089856402594 
130.470122338014 0.0321476361526084 -0.0141177661243760 
130.477070254597 0.0316615905504380 -0.0116512722757364 
130.484017245329 0.0198831104884075 -0.0119145477910928 
130.490964930595 0.0266420661754116 -0.0100765551384381 
130.497912037044 0.0247996470494997 -0.00841304990074580 
130.504859259210 0.0263440804991336 -0.00568623791784806 
130.511805439808 0.0311188115790501 -0.00903519920180561 
130.518751736122 0.89672 -0.00538382519970312 
130.525697800887 0.0235173617390519 -0.00261262367193117 
130.532644328752 0.0312466106165505 -0.00251050736973128 
130.539590277800 0.0297431011841171 -0.00751586454533614 
130.546537962949 0.0286264834035095 -0.00575901110708388 
130.553483449130 0.0270058241397517 -0.00251086749982294 
130.560426620391 0.0269740082953162 0.00194420717998295 
130.567370370354 0.0269780887795340 -0.00105897340483095 
130.574313194465 0.0247531916317612 -0.000658590614742004 
130.581258564812 0.0235006810769230 0.00175385001759932 
130.588203356459 0.0184218003234307 0.00594495497455634 
130.595147337971 0.0145457172743063 0.00130263829493817 
130.602092592602 0.0140290282748285 0.00636268308868845 
130.609038888917 0.0112095778161458 0.00405890169473626 
130.615984606531 0..00667915590023187 
130.622929976878 0.0173522535329642 0.00610129390593689 
130.629873495407 0.0243085279171429 0.00778791439018682 
130.636818634204 0.0204177173633550 0.00727095690422537 
130.643763425967 0.0136653938804994 0.00804683674033229 
130.650709375015 0.0124415695556724 0.00786145469637907 
130.657653240694 0.0138946573618692 0.00858570669557563 
130.664598611158 0.0166125649093933 0.00692805363164324 
130.671541550869 0.0170153994249751 0.00376972837524243 
130.678487615750 0.0259979013326192 0.00613225792836833 
130.685432291680 0.0292872930450090 0.00680183673070099 
130.692378125011 0.0241836040209334 0.00341128173237620 
130.699322569391 0.0211658785218161 0.00525108568057152 
130.706269097256 0.0174065996629161 0.0102735112806192 
130.713215162020 0.0244029939810822 0.00617649608165125 
130.720160648201 0.0308685919870497 0.0102049760766938 
130.727105902831 0.0318680007000358 0.0151940055609854 
130.734052430580 0.0335329704999579 0.0130576282886414 
130.740998263911 0.0317550665189883 0.
130.747943981525 0.0338530590274587 0.00800534362963763 
130.754889351872 0.0355948634475302 0.00803021790432679 
130.761835532353 0.0379717809064523 0.0118224061440078 
130.768780324117 0.0344709868173581 0.0144617653568690 
130.775726041614 0.0347918293905032 0.0110525719816612 
130.782670601827 0.0367645195321477 0.0120799232062315 
130.789616203725 0.0334616641122058 0.0122765083970491 
130.796561458355 0.0342625020064189 0.00921761682666889 
130.803506365719 0.0285523281234837 0.0134444207226606 
130.810451157365 0.0331506945581515 0.0191899527295108 
130.817396296279 0.0397013227991109 0.0170379501667732 
130.824339004673 0.0407669203934199 0.0200727547353931 
130.831283333336 0.0478871637734401 0.0244247582307331 
130.838228240700 0.0408618839785955 0.0243414762641529 
130.845170949094 0.0415804166925540 0.0250367157898041 
130.852113078672 0.0516414302749366 0.0278602293650065 
130.859058333302 0.0534956297050274 0.0256323115805525 
130.866004861167 0.0587947836338708 0.0268043272107924 
130.872949305547 0.0555185884955733 0.0243414626671708 
130.879896874979 0.0575501025180937 0.0232977173937327 
130.886843518561 0.0628553903575701 0.0245039237817853 
130.893791319427 0.0662032501163601 0.0192134916738424 
130.900737847176 0.0635672161301594 0.0244152928076735 
130.907684375040 0.0631971340972655 0.0218225356874009 
130.914629282404 0.0601956351175908 0.0220323768257514 
130.921575810152 0.0615947603717025 0.0216393032715897 
130.928521296333 0.0633700871608101 0.0211651378529622 
130.935468865765 0.0604455389157448 0.0180482921102740 
130.942414004679 0.0598902794889758 0.0225695023361722 
130.949360648144 0.0609750166589351 0.0198710505657090 
130.956305787058 0.0601066368353604 0.0221214388738460 
130.963251041688 0.0650144837082769 0.0188895816667075 
130.970198726838 0.0612597200752886 0.0157961054554685 
130.977145138869 0.0640914309476881 0.0136365447754670 
130.984092708302 0.0642525972057988 0.0149770898087904 
130.991039004617 0.0641777756342788 0.0154976461227700 
130.997986921316 0.0689766791171059 0.0105116066241767 
131.004932407380 0.0670590977086140 0.0106867556420315 
131.011877546294 0.0616023731596080 0.00995940011167283 
131.018823842609 0.0642233148019742 0.00610075043484476 
131.025770138923 0.0618348820108556 0.00499665297278329 
131.032717245372 0.0628526022483347 0.00617459925891050 
131.039663425970 0.0595922198501596 0.0103103830130933 
131.046609143468 0.0553992117880178 0.00660506573306566 
131.053555324066 0.0591929936021142 0.00625505819393851 
131.060501273114 0.0597653633182459 0.00986437021675432 
131.067447337995 0.0577586417733864 0.00569030714546696 
131.074393518476 0.0572022829298018 0.00552767005897976 
131.081339699074 0.0521482842922390 0.00300737088905171 
131.088285879672 0.0493907530001598 0.00402396921038598 
131.095232060153 0.0543100574278050 0.00211340817585024 
131.102177430501 0.0532726315849213 -0.000996186333010725 
131.109123611098 0.0435924023275021 -0.000732867398863349 
131.116068171311 0.0393832279694316 -0.00194512568853067 
131.123014120359 0.0401122489975245 -0.000548658148349468 
131.129960532391 0.0385925348998968 -4.06833992466838e-05 
131.136905671304 0.0367309617348125 -0.00207215411154727 
131.143851620352 0.0349056157508192 -0.00205058210954806 
131.150799074068 0.0328696183679943 -0.00133161915021032 
131.157742013896 0.0347941977690749 -0.00524369632099903 
131.164687037093 0.0384693528516317 -0.00447981034266742 
131.171632638874 0.0388991133955227 -0.00610384757612499 
131.178577893504 0.0392851833874640 -0.00493109497479251 
131.185522800984 0.0367626741897185 -0.00410241565669029 
131.192468865749 0.0383606171007927 -0.00593377345607378 
131.199413078695 0.0362952056937326 -0.00784902599550285 
131.206357870342 0.0332167791201225 -0.00372184256451437 
131.213304398206 0.0331134762296300 -0.00501735693009854 
131.220250925955 0.0412988060560842 -0.00759454495853217 
131.227197453703 0.0397244899760068 -0.00818996641892559 
131.234143750018 0.0446725914614798 -0.00612847296947744 
131.241090046332 0.0420463586455278 -0.00583859004196371 
131.248036226814 0.0437993288470138 -0.00856079080564080 
131.254982523129 0.0450911819989674 -0.00635831140902296 
131.261928703726 0.0443881954439863 -0.00520235738468686 
131.268874189791 0.0491447266115222 -0.00704099197742497 
131.275820833372 0.0454428394558733 -0.00485611415531471 
131.282765856453 0.0486849065468990 -0.00707865868309159 
131.289712152767 0.0504940094988358 -0.00157654006424823 
131.296658680516 0.0530386674799791 -0.00299290807107003 
131.303604745397 0.0524715115891931 -0.00612116946824107 
131.310551273145 0.0458682347702377 -0.00662768338605840 
131.317497569460 0.0422240054249507 -0.00618994355653073 
131.324444097208 0.0440449530840628 -0.00752876790101361 
131.331390624956 0.0449314555713677 -0.00789688984671315 
131.338337152821 0.0426203808125579 -0.00468593894550849 
131.345283564853 0.0435575017230648 -0.00784818858984622 
131.352230324035 0.0462593958946013 -0.0102682137161973 
131.359176967616 0.0419842136605030 -0.00820027661676955 
131.366123958349 0.0359594511173479 -0.00828472695248002 
131.373070717647 0.0429619362473594 -0.0126610969699444 
131.380017824096 0.0430590735887860 -0.00647807171819450 
131.386964699111 0.0455767105043226 -0.00711765350493441 
131.393911458293 0.0454022131609273 -0.00580341785603082 
131.400858217592 0.0519837468779914 -0.00649023302327943 
131.407803819489 0.0426887633990038 -0.00765415296896267 
131.414750347237 0.0529420425058456 -0.00240406198943850 
131.421697106536 0.0552825937677333 -0.00646959165440068 
131.428644212985 0.0531969080546970 -0.0116452901767395 
131.435591203717 0.0524321541185253 -0.00941207581419048 
131.442537963015 0.0489587867794195 -0.00804619299670356 
131.449484722197 0.0427053575910049 -0.00856783969084956 
131.456431828646 0.0429849815306349 -0.0120665959836715 
131.463378472228 0.0550113252037017 -0.0100722989698091 
131.470324189842 0.0579340574183199 -0.0120806064304246 
131.477270717590 0.0619338602951362 -0.0109737177241927 
131.484217129622 0.0577345418840181 -0.00793030529414225 
131.491164236071 0.0446941325653314 -0.00593256657854394 
131.498111458353 0.0421887453324615 -0.00589779401034811 
131.505058449085 0.0372948048064539 -0.00900849544825724 
131.512005208384 0.0390507438427083 -0.00487194785307266 
131.518951620415 0.0381645326113884 -0.00762705271160442 
131.525898611147 0.0424783441620848 -0.00905173841441040 
131.532845370329 0.0374813583196310 -0.0101765263988325 
131.539792245344 0.0337188491616810 -0.00475201255835527 
131.546739004643 0.0259299793832763 -0.00178519956054124 
131.553685879658 0.0321498819202716 0.00238691770363199 
131.560632870390 0.0300569314111337 0.00120781582813816 
131.567579861122 0.0307351421660412 0.00300503480598206 
131.574526967571 0.0292989773626216 0.00866006998614298 
131.581474074020 0.0318885513737449 0.00791080395207525 
131.588421064869 0.0258069503673378 0.00587880681072778 
131.595368287035 0.0270707938909797 0.00470772377918112 
131.602315393483 0.0325353331692663 0.00858558282908448 
131.609262731483 0.0398784442635936 0.00943622753906860 
131.616209374948 0.0384161518960841 0.00683768394739982 
131.623155208279 0.0468400462968921 0.00208007792448528 
131.630102083294 0.0586917777411348 0.00605114983907562 
131.637049189859 0.0589534186872856 0.00547178474305857 
131.643996527742 0.0603768516875771 0.00752685551571799 
131.650942361073 0.0526976485337436 0.00734070651683581 
131.657889351805 0.0520469031455501 0.00567018319498953 
131.664836805521 0.0540518383496729 0.
131.671782754685 0.0562401102247814 0.0113474809775515 
131.678729745417 0.0568184246162380 0.00725399693866520 
131.685675578748 0.0616967276210481 0.00969433247042134 
131.692622106496 0.0603025377084500 0.00906003694074906 
131.699567824111 0.0600305457975510 0.0117322333977469 
131.706514236142 0.0635171905367912 0.0116720570118878 
131.713460185190 0.0640325233774840 0.0144966303282292 
131.720407407356 0.0655702112941154 0.0141653570142126 
131.727354398204 0.0664223501395319 0.0153506023772672 
131.734301736087 0.0568297701213287 0.0112170862209991 
131.741248379668 0.0587528093575650 0.0130772212166544 
131.748195486117 0.0574672497096236 0.0138703639635315 
131.755142476849 0.0607343424970708 0.0141523627277801 
131.762089699132 0.0619089300233533 0.0116567212317497 
131.769036805581 0.0617794897319985 0.0108939077743208 
131.775981481510 0.0612434185962576 0.0112365970668752 
131.782926273157 0.0547896090646844 0.00972175041832612 
131.789872685189 0.0575534173959433 0.00809440304988744 
131.796818518520 0.0641312328569411 0.00913708366471404 
131.803765972235 0.0610186291333401 0.00877283777068790 
131.810713425951 0.0663360444319159 0.00902461178030641 
131.817660416683 0.0676411654974640 0.00932510752681921 
131.824607291699 0.0619242157724937 0.00785619694365744 
131.831554050881 0.0628359586714490 0.0104562939388846 
131.838500578745 0.0664714844478518 0.00923287592114393 
131.845447569478 0.0702064114647339 0.00948997468914856 
131.852395023193 0.0696941076249289 0.00897967106407636 
131.859340856524 0.0687255189747678 0.0108746015666911 
131.866288310240 0.0695042997873630 0.00915533160361407 
131.873235648123 0.0700710161567484 0.00982633437035050 
131.880181365737 0.0811208383681068 0.00811718880538073 
131.887128703736 0.0736566824440302 0.00804455472113599 
131.894075810185 0.0762650532682489 0.00517845027886387 
131.901021643516 0.0713812210339911 0.00822540517635032 
131.907969328665 0.0652518387026985 0.00370735893536933 
131.914915161996 0.0645016193955100 0.00271099761681276 
131.921860995327 0.0656205496183650 0.00567917887363935 
131.928804976840 0.0626818284978336 0.000282063156230429 
131.935752430556 0.0635464647057962 0.000580274100755748 
131.942699537005 0.0566494388730308 0.00431660946708819 
131.949646643570 0.0583525411330150 0.00277525764449119 
131.956592013917 0.0596529719736909 0.00493360735378987 
131.963538888932 0.0610801327966719 0.00343154478076365 
131.970485763831 0.0631608078188618 0.00207255618093954 
131.977432754680 0.0599516632575944 0.00172334574260520 
131.984380092562 0.0598524038736555 0.00429462031922425 
131.991327199128 0.0567873735033396 0.0
131.998274189860 0.0538338332361625 0.00198994909017887 
132.005221064759 0.0550579551158673 -0.00249959734021306 
132.012167361073 0.0513248026494203 -0.00122223260656512 
132.019111921286 0.0491895154110539 -0.00180010672543284 
132.026056828650 0.0476692357284076 -0.00476450974438637 
132.032998495360 0.0524899258214720 -0.00475888809953603 
132.039943287007 0.0500982073998819 -0.00555833539140166 
132.046888078653 0.0522093648368018 -0.00660130469926532 
132.053832175883 0.0499191108638875 -0.00771434016901260 
132.060776736096 0.0535031956712866 -0.00733786776070131 
132.067719444490 0.0555732983026725 -0.00822099892651191 
132.074665277821 0.0560626959851276 -0.00611423395938753 
132.081607175965 0.0522720025083649 -0.00949864547015161 
132.088552199045 0.0460141727021084 -0.0111864004085499 
132.095497800943 0.0456659392245141 -0.00871128395314326 
132.102443055573 0.0433286927215380 -0.0107623349828017 
132.109386921252 0.0419849745397525 -0.00801739990674093 
132.116329629647 0.0387390188634245 -0.0107573515089166 
132.123275347261 0.0330727751424722 -0.0150569553020259 
132.130220717608 0.0318127443263756 -0.0127272723348182 
132.137166203672 0.0364567670450558 -0.0128709276252399 
132.144109375051 0.0342287121069688 -0.0119896103757337 
132.151054629681 0.0297918019223798 -0.00897914169021081 
132.157999884311 0.0296256037886811 -0.0126487854216226 
132.164945717642 0.0360290821093720 -0.00801020601982271 
132.171890740723 0.0403938174748625 -0.00705915996178831 
132.178836342620 0.0353021767827033 -0.00735206089326397 
132.185779629599 0.0331795996056022 -0.0100213235631181 
132.192725115779 0.0331569058696126 -0.00769504149961907 
132.199670254576 0.0414409707000433 -0.00979003314090525 
132.206615740757 0.0419021773926508 -0.00670772042685770 
132.213561458371 0.0375111015969745 -0.00575890297225611 
132.220507060152 0.0337352238165512 -0.00751052061730450 
132.227453356466 0.0285397639189766 -0.00771402657163913 
132.234397222262 0.0325982663895901 -0.00732173094862477 
132.241343518460 0.0277308387294906 -0.0119691716795284 
132.248288425940 0.0218910146898687 -0.00696870032603571 
132.255234837972 0.0211966534869462 -0.00817345784534812 
132.262181018570 0.0247869588376762 -0.00984313802684687 
132.269127430601 0.0270024807721633 -0.00778388333983148 
132.276073263932 0.0256695309425160 -0.
132.283018287013 0.0263989215368988 -0.00785086317462576 
132.289964930594 0.0259620568447301 -0.0113905764832590 
132.296910185225 0.0209857975644172 -0.00700874702672643 
132.303855208305 0.0215751228686248 -0.00432029366027902 
132.310800231502 0.0223669196947528 -0.00815668089673350 
132.317746759250 0.0228319289448673 -0.00912167858220261 
132.324692245314 0.0216307542625950 -0.00713917029470587 
132.331638657462 0.0135028453915764 -0.00277651187195760 
132.338585416644 0.00813390381829617 -0.00339156231002729 
132.345532175943 0.0135096298509657 -0.00555915723450343 
132.352478819408 0.0148582193704977 -0.00532521352475677 
132.359423148187 0.0185929605425803 -0.00498162030696373 
132.366368634251 0.0191984097086291 -0.00267056555763903 
132.373315277833 0.0179767799398566 -0.000709246696449680 
132.380261805581 0.0182765396118518 -0.00175696059383416 
132.387206828661 0.0195270150964356 -0.00760521397608591 
132.394151851858 0.0190134494656363 -0.00423139300074821 
132.401097800906 0.0179654813687979 -0.000831554941118412 
132.408044560204 0.0119242568063669 -0.00256134253015786 
132.414990972236 0.0164268827087314 -0.00292530923495597 
132.421937268551 0.0180304514957133 -0.00839390541281580 
132.428883912016 0.0184064573515405 -0.00664066030639073 

其中第一列是時間十進制天，和在第二和第三列中的兩個信號。該數據的曲線圖如下所示：

情節清楚地表明一個信號滯後於其他。因此，我想計算信號之間的平均相移。要做到這一點，我已經嘗試以下操作：

Nfft = 512; 
n = 50; 
Fs = 6; 
[Pxy,~] = cpsd(vv(:,2),vv(:,3),hamming(n),n/2,Nfft,Fs); 
coP = real(Pxy); 
quadP = imag(Pxy); 
phase = atan2(coP,quadP); 
Sphase = nanmean(phase); 
Sphase = Sphase * 180/pi; 

這給

Sphase = 18.40 

難道我已經示範意義的方法，並且是正確的呢？

Answer 1

正如評論中所述，一種選擇是用低通濾波器對信號進行預處理。這可以方便地在傅立葉域中完成。我在an image registration application中用互相關函數乘以不同的截斷點。

適應於您的問題，它看起來像（從this excellent answer複製的代碼的部分）：

N = size(vv, 1); 
nfft = 2^nextpow2(2*N-1); 
F=fft(vv(:,2), nfft); 
M=fft(vv(:,3), nfft); 

R=F.*conj(M); 

風靡交叉譜和逆FFT的截止過濾：

r = ones(nfft,1); 

for k = 0.1:0.05:0.9 
    r = ifft(R(1:end-ceil(k*nfft))); 
    r = r.*r; 
end 

估計的翻譯：

rmax = max(abs(r)); 
x = find(abs(r) == rmax); 

現在，讓我們來看看相關函數N：

很明顯，最大不相關峯的中心。 在高峯附近擬合Gaussian會更合適。這是用於亞像素註冊的典型方法。

讓我們嘗試在10％附近：

f = fit((1:2*ceil(0.05*N)).', abs(r(x-ceil(0.05*N):x+ceil(0.05*N)-1)),'gauss1'); 

高斯的中心發現24.94樣品，這與你估計多少有些一致。你的方法似乎在某種程度上起作用，我的猜測是這種差異是由於信號的平滑。

校正曲線：