Random Sampling and Signal Reconstruction Based on Compressed Sensing

Transcription

1 Sesors & rasducers, Vol 70, Issue 5, May 04, pp Sesors & rasducers 04 by IFSA Publshg, S L Radom Samplg ad Sgal Recostructo Based o Compressed Sesg Cayu Huag School of Electrcal & Iformato Egeerg, Hua Iteratoal Ecoomcs Uversty, Chagsha, 4005, Cha el: E-mal: matlab_wjf@6com Receved: 5 March 04 /Accepted: 30 Aprl 04 /Publshed: 3 May 04 Abstract: Compressed sesg (CS) samplg s a samplg method whch s based o the sgal sparse Much formato ca be extracted as lttle as possble of the data by applyg CS ad ths method s the dea of great theoretcal ad appled prospects I the framework of compressed sesg theory, the samplg rate s o loger decded the badwdth of the sgal, but t depeds o the structure ad cotet of the formato the sgal I ths paper, the sgal s the sparse the Fourer trasform ad radom sparse samplg s advaced by programmg radom observato matrx for peak radom base he sgal s successfully restored by the use of Bregma algorthm he sgal s descrbed the trasform space, ad a theoretcal framework s establshed wth a ew sgal descrpto ad processg By makg the case to esure that the formato loss, sgal s sampled at much lower tha the yqust samplg theorem requrg rate, but also the sgal s completely restored hgh probablty he radom samplg has followg advatages: alas-free, samplg frequecy eed ot obey the yqust lmt, ad hgher frequecy resoluto So the radom samplg ca measure the sgals whch ther frequeces compoet are close, ad ca mplemet the hgher frequeces measuremet wth lower samplg frequecy Copyrght 04 IFSA Publshg, S L Keywords: Compressed sesg, Radom samplg, ouformly samplg, Sparse samplg, Sgal recostructo Itroducto Compressed sesg (CS) looks lterally lke a data compresso meas, ad s deed for etrely dfferet cosderatos Classcal data compresso techology, whether t s audo compresso (eg mp3), mage compresso (eg jpeg), vdeo compresso (mpeg), or geeral ecodg compresso (zp), are both from the characterstcs of the data tself, to fd ad remove data mplct redudacy, so as to acheve the purpose of compresso Such compresso has two characterstcs: frst, t occurs after the data whch has bee collected to complete; secod, whch oe tself requres complex algorthms to complete I cotrast, the calculato s relatvely smple the decodg process he asymmetry of compresso ad decompresso s just the opposte wth the demad of people I most cases, the data acqusto ad processg equpmet ted to be cheap, save electrcty, less computg power portable devces, such as fool camera, or recorder, or remote motor ad so o Resposble for processg (uzp) 48 Artcle umber P_03

2 Sesors & rasducers, Vol 70, Issue 5, May 04, pp formato s ofte coducted o large computers stead, t has hgher computg power, ad ofte does ot portable ad save eergy requremets hat s to say, we are usg cheap, savg eergy equpmet to deal wth complex computg tasks, ad large effcet equpmet for processg s relatvely smple computg tasks hs cotradcto some cases eve are more acute, such as the equpmet the feld operato or mltary operatos, collectg data s ofte exposed the atural evromet At ay tme, eergy supply may lose or ts performace s partal loss, eve ths case, the tradtoal data acqusto - compresso - trasfer - decompresso mode s bascally faled Compressed sesg s to solve such coflcts After collectg the data, you wat to compress the redudacy, ad ths compresso process s relatvely dffcult Why do ot we take drectly data after compresso? Such acqusto task s much lghter, but ths also elmates the trouble of compresso hs s called "compressed sesg", that s the drect percepto of compressed formato he tradtoal yqust samplg theorem requres that the samplg rate s ot less tha twce the hghest frequecy of the sgal Wth the developmet of sgal processg techology ad the surge the amout of data processed, ths samplg method has bee far from the requremets to keep up wth the hgh-speed sgal processg I 006, Dooho proposed compressed sesg (CS) theory [-3], the sgal has a sparse ature, ts sparse features ca bee used the sgal samplg pots whch are less orgal sgal, ad these ca be approxmated to restore the orgal sgal hs theory greatly promoted the process of the sgal processg theory, ad has broad applcato prospects Curretly, compressed sesg theory have a very good applcato compressed mage, covertg aalog formato, bo-sesg, sgal detecto ad classfcato, wreless sesor etworks, data commucatos ad geophyscal data aalyss ad other felds [4] If you wat to collect a few data ad expect to gve a lot of formato a small amout of data from these "decompresso", you eed to esure that: Frst: the amout of the collected data cota the global formato of the orgal sgal, secod: there s a kd of algorthm, by whch the orgal sgal ca bee restored from such a small amout of data formato I ths paper, we have researched a method of radom samplg of compressed sesg ad sgal hgh-resoluto recostructo I the method, there are the advatages of o-uform samplg wthout samplg frequecy lmt, hgh frequecy resoluto ad at-alasg [6] At low samplg frequecy, sgal spectrum s based o o-uform samplg Fourer trasform sgal spectrum, ad the sgal s recostructed by separate Bregma method [7-8], ad the spectrum ose s reduced, whch s caused by o-uform samplg Radom Samplg ad Aalyss Uform Samplg Drawbacks Uformly sampled tme fucto s the stadard lear fucto, the samplg tme s some terval dstrbuto he defto of the sampled sgal s x (t) he samplg terval s Δ t, the samplg tme fuctos s = Δt, ad the samplg t frequecy f s = s satsfy of the samplg Δt theorem, they s greater tha two tmes the maxmum sgal frequecy value For fte-legth sgal samplg dscrete x[ ] = x([: ] Δt), s the umber of samplg pots, the samplg durato = Δt By Fourer trasform, to aalyze the sampled sgal x( t) = s(π ft),( f = 85Hz, = 56, f s = 56Hz) Aalyss of the sgal spectrum s show Fg Fg Sgal spectrum aalyss of uformly samplg (f s =56 Hz) As ca be see from Fg, sce the samplg frequecy s less tha two tmes of the samplg sgal frequecy, ad t appears the alasg frequecy 7 Hz Because the real sgal spectrum s equal wth the spectrum of alasg sgal, the true sgal ca t be dstgushed Sce ths embodmet frequecy resoluto s Hz, the sgal frequecy s a teger multple of the frequecy resoluto, t s possble to accurately measure the frequecy value Of the cases, the other parameters s costat, the chaged samplg frequecy s f s = 5Hz, whch s meet to the lmt samplg theorem Aalyss of the sgal spectrum s show Fg Fg Sgal spectrum aalyss of uformly samplg (f s =5 Hz) It s showed Fg that there are o alasg sgals the (O,) bad, but due to the chage of the samplg frequecy, the frequecy resoluto becomes Hz, the true frequecy s 85 Hz the sgal, ad t s ot a teger multple of the 49

3 Sesors & rasducers, Vol 70, Issue 5, May 04, pp frequecy resoluto, ad thus these cause leakage of the spectrum ad feces pheomeo, so the measured frequecy s f = 88 Hz, ad t s devated from the correct value As ca be see from the above aalyss, uform samplg restrctos are lmted by samplg frequecy; the alasg frequecy s produced; frequecy resoluto s ot hgh; there are the pheomeo of spectral leakage ad feces ad other ssues Radom Samplg ad ts Fourer rasform Radom samplg s ot restrcted by samplg theorem, the detecto rage of frequeces ca be acheved creasgly, ad a frequecy of the hgher order s detected a short legth of the data ad the low samplg frequecy, whch ca quckly meet the requremets of real-tme specfc occasos Most mportatly, due to radom samplg, so that the ouform samplg to elmate the problem of alasg frequecy whch s caused by uform samplg; there s also the advatage of hgh frequecy resoluto, the spectral leakage s reduced, ad the pheomeo of the fece s elmated, etc I the above example, the other parameters are costat, radom samplg s chaged, t = rad ( 0,) = g ( ), x( ) = x( t ), where rad (0,) s radom umber betwee (0,), =,,,, g () s a olear fucto of Fourer trasform: X ( ω ) = x( )exp( jω ) () = t 3 Compressed Sesg Prcple 3 Compressed Sesg Statemets Compressve Sesg (CS) theory ma dea s: Suppose that there are sparse coeffcets of the sgal x of legth (e oly a few o-zero coeffcets) of orthogoal bass or o a tght frame Ψ If the coeffcets are projected to aother observatos base Φ : M, M << whch s urelated wth the trasform base Ψ, ad a coservatve collecto y:m s obtaed he the sgal x ca be recorded precsely by solvg a optmzato problem ad relyg o these observatos CS theory s a ew theory framework wth samplg whle achevg the purpose of compresso, ts compressve samplg procedure s show Fg 4 Frst, f sgal x R s compressble o a quadrature group or a tght frame Ψ, the trasform coeffcets α = Ψ x are obtaed, α s the sparse represetato wth equvalet to x ad approxmato he secod step, a smooth measuremet matrx Φ wth M dmesos s desged, whch s ot assocated wth the trasform base Ψ, After x s observed, x wll be projected to M-dmesoal space to obta observatos set y = Φx, he process s the compressg ad samplg process, e samples s take [5] Fally, the x exact or approxmate approxmato soluto xˆ s take by the optmzg problem he Fourer trasform spectral aalyss of Radom samples ( = 56) s show Fg 3 Fg 4 Compresso samplg process Fg 3 Sgal spectrum aalyss of radom samplg (Average f s =56 Hz) he samplg terval s creased by the radom samplg tme to mprove the frequecy resoluto, ad to elmate the pheomeo of the fece Due to radom samplg, alasg sgal wll ot cocetrate o a specfc pot whch s related to the samplg frequecy, but t s all form uformly dstrbuted to the sgal frequecy bad I addto, spectral leakage ca also cause spectral ose he ose spectrum may be reduced wth creasg umber of samplg pots or Wheξ s observed wth a ose, y = Φx + ξ () I order to be trasformed: m Ψ x s t y Φx < ε x (3) xˆ = arg m y Φx + λ Ψ x (4) x 50

4 Sesors & rasducers, Vol 70, Issue 5, May 04, pp Separable Bregma Iteratve Algorthm Sgal Recovery Problem (4) soluto s coverted frst to sparse vector (5) solvg, whe A = ΦΨ, so λ ˆ α = arg m A α y α + α (5) Bregma algorthm [7-0] s the followg steps: ) Calculatg B = ( λ A A + I ), I s -order ut matrx, F = λa y ; b 0,d 0 are -dmesoal zero vector; ) Gve, λ (= 0), terato termato codto δ (= 000), teratos = ; 3) Calculatg α B F + d b ), d = ( = sg( + b ) max( α + b,0) α, b = b + α d ; 4) If α α δ, = +, go to Step 3); Otherwse, stop the terato, ˆ α = α 5) x ˆ = Ψαˆ 4 Sparse Radom Samplg Desg 4 Sgal Sparse Represetato Adaptve sgal trasformato group ψ, the expresso the sgal at the base s sparse he trasformato coeffcets vector of sgal X s α = ψ X uder trasform base ψ, f 0<P< ad R>0, these Coeffcets are satsfy α = ( α p p ) p R, coeffcet vector s sparse If the potetal the support doma { : α 0} of trasform coeffcets α =< X, ψ > s less tha or equal K, α R s oly K o-zero etres he heret freedom s reflected by the umber K of o-zero etres of the sgal Or the sgal sparsty costtutes of a measurg scale of the umber of o-zero compoets of the coeffcets he represetato of the sgal sparse ca be usually measured by the vector zero-orm A vector zero - orm s o-zero elemets of ths vector umber Fourer trasform s what we commoly used 4 Irrelevat ad Isometrc ature he adaptve ad observatve matrx Φ wth M dmesos s desged whch s relevat wth trasform base ψ he desgg objectve of the observato matrx A s to restore the orgal sgal few observatos I a partcular desg, the eed s cosder the followg two relatos: ) he relatoshp betwee the observato matrx Φ ad base matrx ψ ; ) he relatoshp betwee the matrx A = Φψ ad K-sparse sgal α ; Frst, there s rrelevace (Icoherece) betwee the observato matrx φ ad the base matrx ψ he coherece μ betwee the observatve matrx φ ad the base matrx ψ s defed as: μ ( Φ, ψ ) = max k m < φ k, ψ j > j he coherece μ s the maxmum degree (6) of coherece of ay two vectors betwee Φ ad ψ Whe the coheret vector s cotaed Φ ad ψ, the degree μ of the coherece s greater I the sgal compressg samples, each observato cotas dfferet formato of the orgal sgal, ad the vectors betwee Φ ad ψ are orthogoal as much as possble, the degree μ of coherece s as small as possble, ad ths s the coheret reaso betwee the bass matrx ad the observato matrx [8, ] As log as satsfyg the followg equato, sgal ca be recostructed a hgh probablty [] M Cμ ( Φ, Ψ) K log (7) Secodly, the relatoshp betwee the matrces A ad K-sparse sgal s lked wth the lmted equdstat ature (Restrcted Isometry Property, RIP) [8-], whch s the matrx "equdstat costat" o arbtrary K = l, the sometrc costat δ K of the matrx A s defg as the mmum value satsfyg the followg formula, where α s the arbtrary K-sparse vector: ( δk ) α A α ( + δk ) α (8) If δ K <, ths s ukow as whch the matrx A satsfes K-RIP, t s esured that the Eucldea dstace of K-sparse sgal α s costat approxmately the matrx A, whch meas that α s ot possble the ull space of matrx A (otherwse α wll have ftely may solutos) Radom matrx s commoly used to be observed practce A commo observato matrx s Gaussa matrx, bary observato matrx, Fourer observato matrx ad rrelevat observato matrx Radom observatos provde a effectve way to acheve compresso samples 5

5 Sesors & rasducers, Vol 70, Issue 5, May 04, pp Low-speed Rate Sgal Samplg I fact, the desg of observed part s the desg of effcet observato matrx, amely the useful formato of sparse sgal ca be captured a effcet observato ( e, samplg) protocol whch s desged, so the sparse sgal s compressed to a small umber of data hese agreemets are oadaptve, ad oly a small amout of the fxed waveform s lked wth the orgal sgal, these fxed waveforms provde cocse ad rrelevat sgal group Moreover, the observatve process s depedet of the sgal tself he recostructed sgal ca be collected the small umber of observatos by usg optmzato methods Samplg terval [0, ], M pots are radomly collected ths terval, t = rad( 0,), =,,, M, rad (0,) s radom pot (0,), x = [ x( t), x( t ),, x( t M )], y = x, the radom measuremet matrx s farly desged radom spkes base φ k ( t) = δ ( t k), k s oe of the M values whch are selected radomly from [,,, ], ψ s Fourer-base,, t belogs / πjt ψ j ( t ) = e, j, =,,, to the tme-doma set whch cotas radom collecto ad recostructve sgal hs desg Φ ad ψ s satsfy rrelevace (6) ad restrcted sometry (8) A are cosstg of M (M> K) row vectors whch are radomly selected from Fourer matrx A group amely that s a partal Fourer trasform [] method of the samplg, the Fourer trasform of the sgal s doe, ad the row vectors are radomly extracted from trasform coeffcets, whch the observatos are made wth radom ad ucorrelated propertes he urelated characterstcs of radom Observato Matrx are a suffcet codto to restore the sgal properly, ad the observato matrx s hghly rrelevat, these ca esure the effectve restorato of the sgal 44 Expermetal estg ad Evaluato Expermetal sgal fucto: f ( t ) = s( 40 π t ) + s( 40 πt ) + s( 300 πt ) he samplg frequecy f s = 56 Hz, hghest sgal frequecy f max = 50Hz, f s = 56Hz < f max = 300Hz he samplg frequecy s less tha two tmes of the hghest sgal frequecy, ad t does ot satsfy the yqust samplg theorem I Fg 5, there are alasg ad leakage the Fourer trasform spectrum, whle radom samplg s take by usg the same samplg frequecy, spectrum alasg ad leakage are avoded the dscrete Fourer trasformato (see Fg 5 below) Radom samplg ad recostructo are mproved by the proposed compresso method, ad the recostructo s very cosstet ether the frequecy doma or the tme doma Fg 5 shows Result the compressed sesg sparse samplg ad the sgal recostructo wth a hgh-resoluto he upper pcture shows the orgal sgal ad the recostructed sgal tme doma, the mddle pcture shows the uform samplg of the Fourer trasform, the lower part are the radom sample of the orgal sgal, the frequecy-doma recovered sgal, the relatve error (Relatve error = =0787) of the recostructed sgal tme doma 5 Cocluso ad Outlook he basc dea of compressed sesg s that much formato [3-] s extracted from lttle as possble the data, ad t s o doubt that there s a great prospect theory ad applcato deas It s a exteso of the tradtoal formato theory, but t has goe beyod tradtoal compresso theory, ad t has formed a ew sub-brach CS theory states that whe the sgal has a sparse feature, the sgal ca be accurately recostruct by a small umber of observatos whch s much smaller tha the legth of the sgal source I CS theory, sgal samplg ad compresso are combed to a sgle step, ad the sgal s ecoded, these have broke the tradtoal lmts of the yqust samplg theorem some extet, ad have reduced the burde o the hardware processg I the framework of compressed sesg theory, the samplg rate s o loger determed by the badwdth of the sgal, but t depeds o the structure ad cotet of formato the sgal he sgal s descrbed by the use of a trasform space, a ew theoretcal framework s bult to descrbe ad process sgal, so that the crcumstaces to esure that formato s ot lost, the samplg rate of the sgal s far less tha s requred the yqust samplg theorem, but also the sgal s recovered fully hgh probablty he samplg rate of samplg system ca be effectvely mproved by usg radom samplg techque as a o-uform samplg method I a radom sample, the o-uform dstrbuto characterstcs of the samplg terval lead to collect o eough samples for sgal recostructo I ths paper, f the sgal has the sparsty the Fourer trasform, the radom observatve matrx s desged the spkes radom group that s the radom sparse samplg he sgal s successfully restored by usg Bregma teratve algorthm he hardware cost s ot creased ths method, ad the orgal sgal s recostructed by the lmted radom sample values Expermets show that sparse sgal samplg the frequecy doma s far below the yqust frequecy sgal samplg rate, ad the orgal sgal ca bee accurately recostruct by compressed sesg sgal recostructo algorthm 5

6 Sesors & rasducers, Vol 70, Issue 5, May 04, pp Fg 5 Compressed sesg sparse samplg ad sgal recostructo Refereces [] E J Cades, J Romberg, Sparsty ad coherece compressve samplg, Iverse Problems, 3, 3, [] D L Dooho, Compressed sesg, IEEE ras Iformato heory, 5, 4, 006, pp , pp [3] Lu Z, He J, Lu Z, Hgh resoluto frequecy [] Barauk R G, Compressve sesg, IEEE Sgal estmato wth compressed sesg, Sgal Processg Magaze, 4, 4, 007, pp 8- [3] Dooho D, sag Y, Extesos of compressed sesg, Sgal Processg, 86, 3, 006, pp [4] Sh G M, Lu D H, Gao D H, Lu Z, L J, Wag L J, Advaces theory ad applcato of compressed sesg, Acta Electroca Sca, 009, 37, 5, pp [5] E J Cade`s, J Romberg, ao, Robust ucertaty prcples: Exact sgal recostructo from hghly complete frequecy formato, IEEE ras Iform heory, 5,, Feb 006, pp [6] E J Cade`s, ao, ear-optmal sgal recovery Processg, 5, 8, 009, pp 5-56 [4] He Y P, L H, Wag K R, Zhu X H, Compressve sesg based hgh resoluto DOA estmato, Joural of Astroautcs, 3, 6, 0, pp [5] J J, Gu Y, Me S L, A Itroducto to compressve samplg ad ts applcatos, Joural of Electrocs & Iformato echology, 3,, 00, pp [6] Shapro H S, Slverma R A, Alas-free samplg of radom ose, Joural of the Socety for Idustral ad Appled Mathematcs, 8,, 960, pp 5-48 from radom projectos: Uversal ecodg [7] S Osher, Y Mao, B Dog, W Y, Fast learzed strateges?, IEEE ras Iform heory, 5,, Dec Bregma terato for compressed sesg ad sparse 006, pp deosg, UCLA CAM Report, 08-37, 008 [7] S Che, D L Dooho, M Sauders, Atomc [8] W Y, S Osher, D Goldfarb, J Darbo, Bregma Decomposto by Bass Pursut, SIAM Joural teratve algorthms for l-mmzato wth o Scetfc Computg, 0,, 999, pp 33-6 applcatos to compressed sesg, SIAM J Imagg [8] Cades E J, ao, Decodg by lear Sc,, 008, pp programmg, IEEE rasactos o Iformato [9] J-F Ca, S Osher, Z She, Learzed Bregma heory, 5,, 005, pp teratos for compressed sesg, Math Comp, 78, [9] Cades E J, he restrcted sometry property ad ts 009, pp mplcatos for compressed sesg, Comptes [0] J-F Ca, S Osher, Z She, Covergece of the Redus del Cademe des Sceces, Sere, 346, 9-0, learzed Bregma terato for l-orm 008, pp mmzato, Math Comp, 78, 009, pp 7-36 [0] Barauk R G, Daveport M A, DeVore R, Wak [] L M Brègma, A relaxato method of fdg a M B, A smple proof of the restrcted sometry commo pot of covex sets ad ts applcato property for radom matrces, Costructve to the soluto of problems covex programmg, Approxmato, 8, 3, 008, pp Z Vyčsl Mat Mat Fz, 7, 967, pp [] Haupt J, owak R, A geeralzed restrcted [] E Cadès, J Romberg, ao, Stable sgal sometry property, echcal Report ECB-07-, recovery from complete ad accurate Uvemty of Wscos Madso, May 007 measuremets, Comm Pure Appl Math, 59, 8, Aug 006, pp Copyrght, Iteratoal Frequecy Sesor Assocato (IFSA) Publshg, S L All rghts reserved ( 53