ON THE TIME/FREQUENCY SIMULTANEOUS ALIGNMENT OF THE SIGNALS COMPORTMENT


 Amberly Underwood
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1 UPB Sci Bll, Series A, Vol 69, No 4, 7 ISSN 377 ON THE TIME/FEQUENCY SIMULTANEOUS ALIGNMENT OF THE SIGNALS COMPOTMENT Li OTAIU În cestă lcrre, conceptl de tomfrecvenţă (T/F), elbort de J von Nemnn şi D Gbor, este pnctt în termeni mtemtici; de semene, nele proprietăţi le trnsformtei Forier c ferestră snt scose în evidenţă (Proposition şi Proposition 3); în finl este prezenttă o plicţie ndinelor pentr descriere mecnismli fiziologic l rechii mne în timpfrecvenţă In this pper, the tomfreqency (T/F) concept, elborted by J von Nemnn nd D Gbor, is pointed ot in mthemticl terms; lso, some properties of Forier trnsformtion with window re mrked ot (Proposition nd Proposition 3); finlly, we give n ppliction of wvelets for (T/F) hmn er physiologicl mechnism description Keywords: wvelets, ttented sine Introdction If : C is L ( ) clss fnction, dhoc clled signl, then we ssign its Forier trnsformtion ˆ : C, which is continos nd bonded fnction, defined by ˆ ( ω ) = (, () this improper integrl being convergent for ny ω The fnction û is clled the freqency spectrm of the signl, nd A ( = ˆ(, the freqency mplifiction of An insfficiency of the clssic Forier trnsformtion is constitted by the fct tht we hve to know the vles of for entire time is (ccording to ()) if we wnt to clclte the spectrm ( ω ) in only one freqency ω Applying the Forier inversion forml in deqte conditions (for emple, if is continos fnction nd L ( ) L ( ) ), Assist, Bnt s University of Agricltrl Sciences nd Veterinry MedicineTimisor, OMANIA
2 4 Li otri i ( ) ˆ( ω t t = e dω, () π it trns ot tht to determinte the smple ( t ) for time moment t it is necessry to know the spectrm ˆ ( in entire freqency bnd Ths, we consider the pirs ( t ), simltneos tken, nd nlyse the time/freqency comportment of some signls in their neighborhood TimeFreqency Atoms It is considered pln relted to n orthogonl fied point where by bsciss is mentioned the generic time t, nd by ordinte line, the generic freqency ω To represent signl in n orthogonl plne t ω mens to tke simltneos its time drtion nd its freqency rtio (hmn voice cse) We fi pir ( t ) Intitively, n ( T / F) tom rond the point ( t ) is ny signl (from L ( ) L ( ) ) with compct spport, which contins t (so is nll ot of the sppor; moreover, û hs to hve compct spport, which contins ω (so freqencies ˆ ( re insignificnt ot of the sppor According to the indeterminism principle, sch nonzero signl doesn t eist, no mtter how smll re the spports of nd û John von Nemnn clled T / F tom ny fmily of fnctions by the form { e i ω t ( t t )}, with ( t ), where ( is fied fnction (from L ( ) L ( ) clss) Distribting the points ω) niformly in the to ω pln, J von Nemnn hs recommnded, in the Signls Theory, to se n orthonormted bse in the Hilbert spce L ( ) reltive to the dot prodct < f, g >= f ( g( dt, mde of T / F toms sinπt Proposition Let be ( = (for t ) nd ( ) = The T / F πt toms lt ( = e π ( t k) ; k, l Z, (3) (corresponding to the vles t = k, ω = πl ), mke n orthonormted bse for L ( )
3 On the time/freqency simltneos lignment of the signls comportment 5 Proof Using the Prsevl forml, it reslts immeditely tht <, pg >= δ lp δ kg We hve representtion by the form f ( = c ( for ny signl f L ( ) ; the coefficients c re immeditely dedced from c =< f, >, for ny k, l Z Therefore, ny continl (nlogicl) signl f is identified by the seqence c, which is n illstrtion of the deltor phenomenon nmed nlogicl/digitl conversion of the signls For the signl ( from Proposition (clled ttented sine ), we hve, if ω ( π, π ), ˆ( =, in rest nd for ny k, l Z fied, it reslts tht ˆ ( ω ) k, l iωt πilt iωt = ( e dt = e ( t k) e dt; mking chnge of vrible, t k = τ, we obtin ik ( ω πl ) ˆ ( = e ˆ( ω πl ), for ny k, l Z The grphics of the fnctions nd û re indicted in Fig ; ), b) y y = ( t ) y ω 3 3 t π π ) Fig b) The signl ˆ( hs good position in t = nd it is insignificnt ot of the system [, ]; ( t k) is the trnsltion of with k time nits nd it is well loclized in the point k Therefore, ( is well loclized in k nd it is
4 6 Li otri insignificnt ot of the intervl [ k, k + ], for k Z In the sme wy, ˆ ( ω πl ) is nll for ω πl [ π, π ], hence ˆ ( is nll ot of the intervl [ π + πl, π + πl ], which mens tht ˆ ( is well loclized rond the πl freqency Let s consider now rectngles from the to ω pln, hchred like in Fig nd centered in the points ( k,π l), with k, l Z In this wy, the time/freqency T / F pln, identified with the to ω pln, is prried with rectngles, like in Fig Fig These kind of rectngles cn be covered between themselves (so tht they cn t mke pln prtition) The orthonormted bse, l, k Z from the Proposition presents si disdvntges, which re connected with wek convergence in dots prodcts clcls < t, >, necessry for signls digitl representtion Also, the fct tht ll the T / F toms hve the sme drtion is n impediment in some type of pplictions (for emple, in Geophysics or dr) Tht s why we proposed some other kind of orthonormted bses for L ( ) 3 T / F Trnsformtion D Gbor proposed the replcement of the discret vles k, l Z with continos vribles ξ, τ, considering T / F toms by the type
5 On the time/freqency simltneos lignment of the signls comportment 7 iξt wξτ ( = e w( ), where w( L ( ) is signl with the norm w = π (clled window) For ny signl L ( ), fnction with two rel vribles, W, defined by iξt W ( τ, ξ ) =<, wξτ >= ( e w( ) dt, (4) ws clled the T / F trnsformtion of, with the window w fied (or eqivlent of Forier trnsformtion with window) It my be remrked the nlogy with the reltion () D Gbor proved, knowing W (similrly with ()), the recprtion of forml, nmely ( = W ( τ, ξ ) w ( dτdξ ξτ (5) Note: If w (constnt fnction), we hve W ( τ, ξ ) = ˆ( ξ ) (Forier clssic trnsformtion) nd if w = τ (Dirc distribtion), then iξτ W ( τ, ξ ) = ( τ ) e In the two cses, w doesn t belong to the spce L ( ) We fi τ nd > We choose window w : which hs to πn hve its spport contined in the intervl [ τ, τ + ] Then, for ξ =, n Z, we hve (ccording to (4)) πin πin πn t t W ( τ, ) = ( w( ) e dt = ( w( ) e dt Proposition Let be L ( ) nd c n the Forier comple coefficients of the fnction ( w( ) restricted to the intervl [, ] nd then etended to by its periodicity In these conditions, πn W ( π, ) = c n, for ny n Z (6) Proof The demonstrtion reslts directly from definitions Therefore, knowing W, we cn determinte the coefficients c n sing the reltion (6); the signl ( is recovered from its Forier coefficients: ( = c n e n Z π in t In other words, choosing convenble windows, from dt bot the signl (, we cn find ot locl dt bot its T / F trnsformtion, W, nd conversely
6 8 Li otri Now we fi window w For ny fied t, we cn consider the fnction h t,ω : C defined by iωτ ht ( τ ) = w( τ e For ny vector h L ( ), we note with h * the fnctionl defined by * h =<, h > With these nottions, we cn write: Proposition 3 For ny window w, we hve the reltion * h = t, ω ht dωdt I, where I is the identity on the Hilbert spce L ( ) Proof Let s tke L ( ), rbitrry fied We hve to prove tht * ( τ ) = h ( ) ( ) t, ω τ ht dωdt (7) Be it t ( τ ) = ( τ ) w( τ Then W ˆ = t ( τ ), nd, ccording to the reltion i (), it reslts tht τ = ω e τω t ( ) W ) dω If we mltiplicte with w( τ π nd integrte relted with t, we obtin which mens we lso know tht reltion: Bt i ( τ ) ( τ ) = (, ( τ ) τω t w t dt dt W t w t e dω, π W nd the reltion (7) is now proved ( τ ) w = W (, ) ( ) t ω ht τ dtdω ; π w = Tht s why we cn write now the following π τ ) = h ( τ ) W dtdω ω ( t, iωτ w( τ ( τ ) e dτ = ht ( τ ) ( τ ) =< *, h >= t ht ( ), = dτ = Corollry For ny signl L ( ), its energy the forml E ( ) = is given by
7 On the time/freqency simltneos lignment of the signls comportment 9 E( ) = W ω ) dtdω * * Proof We hve W = W W = h ( ) h ( ) ; so, W (, ) = t dtdω ω The size W hs the net phisycl interprettion: it is the energy density of the signl relted to the time nit in T / F pln Note: The T / F trnsformtion works with fied drtion of the window, mening tht we hve to consider only w( t b) trnsltions of the window if we wnt to clclte W ( b, ξ ) ; this cn be n inconvenient in T / F nlysis of some signls ( with high vritions in short intervls of time (like in Geophysics, dr, Hmn voice, etc) This ws one of the resons which determinted s to propose more fleible windows, which cn be trnsltted nd lso, delted (or contrcted); this fct mrked the ppernce of the wvelet concept We fi windowfnction Ψ : so tht Ψ ( nd tψ ( belong to L ( ) ; moreover, Ψ ˆ () = ; the fnction Ψ ws clled wvelet For ny signl L ( ), we define the trnsformtion of by the wvelet Ψ s the net fnction of two rel vribles, b, with > : t b W (, b) = ( Ψ dt (8) Applying the Prsevl clssic reltion, we obtin ibω W (, b) = ˆ( e Ψˆ ( dω = π ibω = e ˆ( e Ψˆ ( dω π This reltion shows the connection with the T / F trnsformtion t, ω t, ω 4 Appliction Let be the loction of sensitive cell in the spirl shell corte of the hmn er We sppose tht the dio signl received in, g (, is the convoltion of n costic signl ( with liner filter which depends by the
8 3 Li otri loction, nd hs the trnsfer fnction H ( Then, in the freqency domin, we hve the reltion gˆ ( = ˆ( H ( (9) Sppose tht by the spirl shell geometry itself we hve dely in freqency, ie there eists fnction G so tht ( = G( ln If we note = e, reslts tht = ln nd we consider wvelet Ψ ( with Ψ ˆ ( = G ln ω () Then, sing (9), reslts gˆ ˆ ( = Ψ ( ˆ( nd from the inversion Forier forml (), we obtin the reltion iωt g ( = e Ψˆ ( ˆ( dω = π iωt τ t = ˆ( Ψˆ ( e dω = ( τ ) Ψ dτ π Using gin the Prsevl reltion, reslts tht the reception t the sensoril cell loclized in is given by / g ( = e W e ) The constrction of the W ( wvelet with the property () is still n open problem H E F F E E N C E S Tood Ogden, Essentil Wvelets for Sttisticl Aplictions nd Dt Anlysis, Boston, Birkhser, 997 O Stănăşilă, Anliz mtemtică semnlelor şi ndinelor, Mtri om, L Prsd nd S S Izengr, Wvelet Anlysis with Applictions to Imge Processing, Chpmn nd Hll/CC Press, Jne 997, 4 Jh issnen, Wvelet on selfsimilr sets nd the strctre of the spce M p ( E, μ) 5 I Dbechies, Ten Lectres on Wvelets, CBMSNSF Series in Appl Mth, SIAM, 99 6 S Mllt nd Z Zhng, Mtching prsit with timefreqency dictionries, IEEE Trns Sig Proc, December 993