Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

Appl. Math. J. Chinese Univ.2013, 28(4): 505-530Mathematical theory of signal analysis vs. complexanalysis method of harmonic analysisQIAN TaolZHANG Li-ming2Abstract. We present recent work of harmonic and signal analysis based on the complex Hardyspace approach.δ1 IntroductionProfessor Gong Sheng shared with the first author his view:“On the unit circle it is harmonicanalysis, and inside the unit circle it is complex analysis" ([16]). This is, in fact, common senseamong analysis. In general, we regard the following as what we mean by complex method ofharmonic analysis. Suppose that one is to study analysis on a closed and finite-dimensionalmanifold. One can then imbed the manifold into a space of one more (or several more) dimension(dimensions) with a complex analysis structure, and, in such way, one treats the manifoldunderstudy as a co-dimension 1 (or co-dimension p) space. By a complex structure it meansthat there exist a Cauchy kernel and a Cauchy formula, and the related complex analysisobjects. With the complex structure one can define Hardy spaces of good complex holomorphicfunctions in the regions enclosed by the manifold. Functions on the manifold then can be splitinto a sum of the boundary limits of the related Hardy space functions. Those boundary limitsconstitute boundary Hardy spaces. Such idea was, in fact, taught by M.-T. Cheng and D.-G.Deng when the first author was in his Ph.D. program in Beijing University around 1980. Itis also hinted by the book of Gorusin translated by Jian-Gong Chen ([17]). The author alsolearned this idea from the works by C. Kenig and, separately, by A. McIntosh on complex Hardyspaces and singular integrals on Lipschitz curves and surfaces. This article gives a survey on theresults that the authors and their collaborators obtained by implementing the complex analysisapproach to signal analysis.It has been a controversial issue to the present time about what is instantaneous frequency(IF), or, in brief, frequency. People tend to believe for a general signal there is a certain frequencyReceived: 2013-10-16.MR Subject Classification: 42A50, 32A30, 32A35, 46J15.Keywords: Mobius transform, Blaschke form, mono- component, Hardy space, adaptive Fourier decomposi-tion, rational approximation, rational orthogonal system, time-frequency distribution, digital signal processing,uncertainty principle, higher dimensional signal analysis in several complex variables and the Clifford algebrasetting.Digital Object Identifier(DO): 10. 10751166-013-3225-4.The work was partially supported by Research Grant of University of Macau MYRG115(Y1-L4)-FST13-QT,Macao Science and Technology Fund FDCT/098/2012/A3.中国煤化工MHCNMH G506Appl. Math. J. Chinese Univ.Vol. 28, No.4at any moment of time. This belief is hinted and supported by sinusoidal signals that possessthe constant frequencies. To justify this idea is to define what is frequency. For a generalsignal a well acceptable and reasonable way to define instantaneous frequency has not beenfound. Our view is that there does not exist a frequency concept for a general signal. Our workproposes a definition under which some signals have well defined instantaneous frequencies, andsome not. Signals possessing IF are called mono-components, and otherwise, multi- components.For multi-components one seeks for mono-component decompositions. We now review the storystarted from Gabor.In 1946 Gabor proposed his analytic signal approach ([15]). Throughout this article werestrict ourselves to only signals with finite energy, or L2-functions. Let s(t) be a real-valuedsignal of finite energy. The associated analytic signal, denoted by s+(t), is defined ass+(t)= (<() + iH()，where H is the Hilbert transformation. We note that analytic signals are non-tangential bound-ary limits (the Plemelj formula) of Hardy space functions in the related domain, the latter con-sisting of holomorphic functions with good control close to the boundary. For the real line casethe related domain is the upper-half complex plane. The Hardy space functions in the domainare given by the Cauchy integral of the signal s(t) on the boundary. We note that through outthe paper, except in the final section for multivariate signals, we use the terminology Hardyspace only for the complex Hardy H2 spaces in either the contexts inside or outside of the unitdisc, or the contexts of the upper- or lower- half complex planes. There are esentially paralleltheories in the four contexts. We will feel free in below to switch from one context to another.From paragraph to paragraph we will make sure that we give clear indication to which con-text we are referring. We will use the notations L2(R), H2(C+), L2(OD), H2(D), etc., whereR, C,D and C+ denote, respectively, the real line, the complex plane, the unit disc and theupper-half- complex plane. When we use H2 and L2 we mean that we refer to all the fourcontexts.On the real line, the Fourier transform of Hs is - isgn(.)8(). As consequence, the Fouriertransform of s+ is supported on [0,∞), and, in particular,s+()=赤[^ e(s()dE.This shows that s+ is a“linear combination”of some terms of non-negative frequencies, viz.,of those eit with ξ≥0. It hence has reason to believe that in a single-term-amplitude-phaserepresentation, viz, s+(t) = p(t)ei6(t), one should have that the phase derivative of s+, oralternatively, the analytic phase derivative of s, satisfies θ'(t)≥0. But, unfortunately, this isnot true. To make clear the terminology, we note that if s= s+, then st+ = s. This amountsthat analytic phase derivatives of boundary values of the Hardy space functions coincide withthe phase derivatives of the functions. It is a fact that phase derivatives of any non- trivialanalytic outer function are negative in a set of positive Lebesgue measure [26]. Such examplescan be simply constructed as follows. Consider a fractional linear transform, f, in the complexplane that maps the unit disc centered at the origin onto a disc that dose not contain the originin its topological closure. Restricted to the unit circle the mapping has an amplitude- phase中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 507representation f(eit) = p(t)ei0(t). One observes that 0(t) is not a monotone function in therange [0,2π] when t traverses from 0 to 2π. But, instead, in an interval the phase derivativestrictly increases and in an adjacent interval strictly decreases. This implies that the phasederivative 0'(t) changes sign in a pair of adjacent open intervals, and, in particular, negativein an open interval. Such functions can be constructed from their corresponding real parts byGabor's analytic signal method.Why positivity of frequency of a signal is important? The primary importance is that thefrequency concept is generated from physical practice: it is an extension of vibrating frequency.Secondly, the positivity has a great significance in signal analysis. For instance, the mean offrequency of a real valued signal is zero if we do not restrict to positive frequencies, that makesthe mean concept to be useless.We note that the Hilbert transformation in various contexts play a crucial role when dealingwith boundary values of holomorphic functions. We adopt the definition of Hilbert transforma-tion from S. Bell (3]). Suppose that we deal with a simply-connected domain 2 in the complexplane. Let f be a holomorphic function in the domain. Assume that f has a non-tangentialboundary limit, while this is always true when f is in the Hardy space of the domain. Denotingthe boundary limit of f by f = u+ in, where u and U are scalar-valued. Then the Hilberttransformation H is defined through H: u→U, or U= Hu. This mapping in some cases shouldmodulo a constant. In different contexts the Hilbert transformation has diferent representa-tion. On the real line it is given by the principal value singular integral with the kernel t,or, in the inverse Fourier transform formulation, given by the Fourier multiplier - isgn(ξ). Inthe unit circle case it is the so called circular Hilbert transformation, or induced by the sameFourier multiplier via Fourier series expansion. The general Hilbert transformation on mani-folds in higher dimensional Euclidean spaces may be defined similarly via the Ciford algebraformulation ([1], [46]). .The first task of the study is to find a pool of the functions that have non-negative analyticphase derivative. Precisely, we are to find signals s(t)∈L2 such that s+(t) = p(t)ei0(t) hasnon-negative phase derivative, viz., 0'(t) ≥0. We call such signals s mono- components orreal-mono- component, and, without ambiguity, call s+ mono- component, too, and sometimescomplex-mono-component [27], [28].Definition 1. (Tao Qian 2006) Let s be a real- or complex- valued signal with finite energy.We call s a mono-component if its analytic signal, or its projection into the Hardy space H2，viz, s+(t)=言(s(t) + iHs()), in its phase- amplitude representations+(t) = p(t)ei9(1) satifies .θ'(t)≥0, where the phase derivative θ'(t) is defined through the non-tangential limit of thesame quantity but inside the region. Precisely, in the unit circle case,日'(t)=. lim_ 0,(t)， s+(rei*)= Pr()i.,()，00.8 is called a normal mono-component if it is mono-component and, further more, there exists1>δ>0suchthat0,(t)≥0,ifforalltand1>r>1-δ,intheunitdisccase;orexists.中国煤化工MHCNMH G508Appl. Math. J. Chinese Univ.Vol. 28, No.4h>0 such that (t)≥0, iffor allt and0 0, a.e. The proof is anapplication of the Julia-Wolff- Caratheodory Theory. We haveTheorem 1. (Tao Qian 2009) Let θ be a Lebesgue measurable function. Then the phasefunction eio is a mono-component if and only if ei0 is the non-tangential boundary limit of aninner function, or, equivalently, if and only if H(ei9)= -ieio.The above theorem is valid in both the unit circle and the real contexts.Instead of listing all the important references in relation to the recent developments ofBedrosian identity studies I list the main relevant authors. An incomplete list includes, in thealphabetical order, Q.H. Chen, T. Qian, L.H. Tan, R. Wang, S.L. Wang, Y.S. Xu, D.Y. Yan,L.X. Yan, L.H. Yang, B. Yu, H. Z. Zhang. However, only a moderate percentage of the relevant中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 509literature devote to solve the mentioned mono-component problem ([59], [45], [50], etc.). Most,in fact, develop their own interests in finding out conditions for the Bedrosian identity. Wehave the following basic result: If the phase function part is from a finite or infinite Blaschkeproduct, then the desired amplitude part has to be in the closure of the linear span of therelated Takenaka-Malmquist (TM) system, that is a backward-shift invariant subspace of H2([54]). Precisely, we haveTheorem 2. Ifei0(t)Ii -apeit-ak1 |ar| 1- Treitthen p(t)ei0(t)∈HP(D),1≤p≤∞, if and only ifρ∈spanp{Bk}R=1,whereBe(z)=V1- |a二”.1-az111- uzNote that a TM system is orthonormal under the iner product<1,9>=六[".f(eit )(i)dt2π J)([56]). It becomes a basis in H2 if and only if the non-separable hyperbolic distance condition之a -lax)=∞holds. A basic function Bk in a TM system, called a weighted Blaschke product, consists oftwo parts of which one is a Blaschke product 1I1=1 名影，being a product of kt - 1 Mobiustransforms that is a mono-component, and the other part is the classical Szego kernel, beingan outer function. Szego kernel is the reproducing kernel of the Hardy H2 space. A detailedanalysis show that if al = 0, then all the basic functions Bt, Bi+1...., Bl+k....are mono-components.The stated result for infinite Blaschke products gives rise to suficient and necessary con-ditions on real and non- negative amplitudes ρ for pei0 being new mono-components ([54]).The relation between weighted Blaschke products and p-starlike function in complex analysisfunctions are indicated in [55].A great variety of mono-components can be identified based on Theorem 1 and the Bedrosianidentity results. The next question is how to express a signal in terms of mono- components.Before giving our answer we first draw the reader's attention to some observations and argu-ments. First, if a function f in the Hardy space H2(D), then we can show, for any givenε > 0,there exist two mono-components, m1 and m2 such that([341)|f- (m1- m2)l2 0. For anyparameters a....,a1 in the unit disc, all Bx,k= 1...,l, are pre-mono-components. Ifak = 0,then Bk, Br+1,are mono-components. If all a1....... are zero, then the correspondingTM system becomes Fourier.Suppose we are given a signal f in the Hardy H2(D) space, that is f(z) = Ei=1cz',. 2i=1|c1|2 <∞. Now we seek for a decomposition into a TM system with adaptively selectedparameters. We will use the collction of the functions√1- Tapa∈D1-azwhich are normalized Szego kernels of the unit disc in which a is a parameter. Set f = f1. Firstwrite the identityf()=a.(2)+ 5(2)-< fr.e.20a()二一-1- az中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 511We note that in this step a1 can be any complex number in the unit disc. The above can befurther written asf(z)=< fi,ea>ea(z)+ f2(z)二11-2with .f2(z) =f1(z)- < ft,ea1 > ea1(z)1-12We call the transformation from f1 to f2 the generalized backward shift via a1 and f2 thegeneralized backward shift transform of f1 vnia a1. The notion is related to the classical backwardshift operators(f)(z)=a1+a2z +...+Ck+1zk +...=f(z)- f(0) .Recognizing that f(0) =< f,eo> eo(z), the operator S is generalized backward shift operatorvis 0.In the decomposition f2 is called a reduced reminder. The purpose now is to extract themaximal energy portion from the term < f1,ea1 > eal(z). The energy of the latter, due to thereproducing kernel property of ea, is given by|< fr,eal > ealI2=(1 - |al|)f(a1)I*.The orthogonality between the two term in the right hand side of (2) and the unimodularproperty of Mobius transform on the circle implyIf| = (1 - |a1|)\f+(a1)2 + l11.This shows that to minimize the remainder If1|2 is to maximize (1 - |a1|)|fi(a1)|2.Fortunately, one can show that there exists a1 in the open disc D such thata1 = argmax{(1 - |al?)!fr(a)|2: a∈D}([36]). The existence of such maximal selection is called Marimal Selection Principle. Undersuch a maximal selection of a1 we call the decomposition (2) a maximum sifting. Selecting sucha1 and repeating the process for f2, and so on, we obtain之- akf(z)=Z Br(2)+ fn+1II1_akk=1where for k= 1....n,ak = arg max{(1 - |alQ){fx(a)|2 : a∈D},and, for k= 2...n+ 1,fr(z)=fk-1(z)-< fk-1,eak-1 > eax-1(z) .中国煤化工MHCNMH G512Appl. Math. J. Chinese Univ.Vol. 28, No.4It can be shown thatlim In+(ex(i)) IIexp(i)-a|=01一Tk exp(i)There holds the following theorem.Theorem 3. For any give function f in the Hardy H2 space, under the maximum siftingprocess we have([36])f(z)=E< fr,eak > Br(z).k=1The following relations are noted:< fk,Cak >=<9r,Br>=< f,Br>，where 9k is the standard reminder:9r(z)=f-> < fi,ea;> B(2).Remark 1. We note that the selected parameters a...., an,... in AFD do not have to satisfythe condition (1), and the induced TM system {Bk} may not be a basis. The decompositionprocess exhibit that one is not interested in whether the resulted system is a basis, but interestedin whether it can expand the given signal f. One is indeed able to do so, and, in fact, achievesfast convergence.Remark 2. If we choose a1 = 0, then all Bk are mono-components, and AFD offers a mono-component decomposition. For arbitrary selections of a........ we arrive a pre-mono-component decomposition, of which after multiplying eit all entries in the infinite sum becomemono-components.Remark 3. Based on the same dictionary of Szego kernels AFD is different from greedyalgorithm or orthogonal greedy algorithm in two aspects. One is that at every step we canget a maximal energy portion but not a time of it as in greedy algorithm situation. Thesecond is that we can repeatedly choose the same parameter if necessary to get best possibleapproximation. Using a cyclic AFD algorithm we can get a conditional solution to the n-bestrational function approximation (see subsection 3).Remark 4. The convergence rate for AFD is 1/√n where n is the order of the approximatingAFD partial sum. One has to note that this is a convergence rate for bad functions, includ-ing those being discontinuous. This convergence estimation, therefore, has a different naturecompared with the traditional convergence theorems: the latter are for smooth functions.Several other mono-component decompositions are based on AFD. To emphasis this funda-mental role we sometimes call the above defined AFD as Core AFD.2. Unwending AFDIn DSP there is the following assertion: If f = hg, where f,g are Hardy H2(D) functions,中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 513Decomposition no.=65Energy difterence=0.039941reconstution).5--12004000600800 1000Figure 1: Original signal and reconstructed signal of order 65 AFD decompositionDecompositin no.=65Energy difference=0.1180632reocstructon1罡一21000TimeFigure 2: Original signal and reconstructed signal of order 65 FD decompositionDecomposition no.-=210Energy ditrence=0.033637reconstruction匣-2Figure 3: Original signal and reconstructed signal of order 210 FD decomposition中国煤化工MHCNMH G514Appl. Math. J. Chinese Univ.Vol. 28, No.4and h is an inner function. Let f and g be expanded into their respective Fourier series, viz.,f(2)=>cnxz",9(z)=Sdxz*.k=1Then one has, for any n,k=n(see, for instance [8], [5]).This amounts to say that after factorizing out an inner function factor the reemaining Hardyspace function series converges faster. This suggests that in the above AFD process if oneincorporates a factorization process then the convergence becomes faster. This is reasonable:when a signal by its nature is of high frequency, one should first “unwending”it but not extractfrom it a maximal portion of lower frequency. We proceed it as follows ([29], [35]). Firstwe do factorization f = f1 = I1O1，where I1 and 01 are, respectively the inner and outerfunction factors of f. The factorization is based on Nevanlinna's factorization theorem. Theouter function has the explicit integral representationO1(z)=e≠f8*去lgI(e"t.In the computation we find the boundary value of the outer function by using thboundaryvalue of fi in which the above integral is taken to be of the principal integral sense. Theimaginary part of the integral reduces to the circular Hilbert transform of log |f(ei). Next,we do a maximum sifting to O1. This givesf(z)= Ir(z)[< O1,Cal > ea(2)+ f2(2)1一]where f2 is the backward shift of O1 via a1 :f2(z)=01(z)-< O1,ea1 > ea(z)By factorizing f2 into its inner and outer factors, f2 = I1O2, we havef(z)= I(z)[< Or,eal > ear(z)+ Ia(z)O2(z)二(1]1-可1zNext, we do a maximum sifting to O2, and so on. In such way we obtain the decompositionTheorem 4. Under the assumptions as in Theorem 2, we have the unwending AFD decompo-sitionf(2)= 2II I() B(z)+ f+(>)I二% Ir(2),-akz工where fr+1 = Ik+1Ok+1 is the backward shift of Ok via ak,k= 1..n, and Ik+1 and Ok+1中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 515are respectively the inner and outer functions of fr+1. Furthermore,f(z)=2I(z) Be(z).k=1l=1Remark 5. In most cases unwending AFD is automatically a mono-component decompositionbecause of the inner function factors generated in the process. As in AFD we can manuallyset a1 = 0 to guarantee that all the terms of the unwending AFD are mono-components.Unwending AFD converges very fast. This is shown through comparison of the performancesof AFD, unwending AFD, double-sequence AFD, as well as Fourier series on singular innerfunctions ([35]).Remark 6. There are other AFD variations that first extract factor signals of high frequencies.Those include a double- sequence unwending AFD ([39]) and one using what we call high-orderSzego kernels ([40]). With a similar eectiveness as unwending AFD the algorithm of double-sequence unwending AFD is, however, more complicated. The high- order Szego kernel methodin [40] can get a maximal energy portion as AFD but with a suitable frequency level. Ithowever, does not have a generalized backward shift structure. .Energy difference=0.00815280.6-reconstruction).4-00.2--0.4--0.6--0.8.200400008010001200Figure 4: Original signal and reconstructed signal of order 5 Unwending AFD decomposition3. Optimal approximation by rational functions of order not larger than nAFD and unwending AFD offer fast decomposition of signals into mono-components. They,however, are not fastest. They are of certain uniqueness only from the algorithm. Due to theseobstacles the question on simultaneous selection of n-parameters 1..... an, in an approximatingn-Blaschke form, viz.,< f,Bk> Br(z),k=1arises. Simultaneous selection of the parameters but not one by one in a sequel certainly offersbetter approximate to the given signal. Simultaneous selection of the parameters in an ap-proximating n- Blaschke form is equivalent with the so called optimal approximation by rationalfunctions of order not larger than n. We phrase the problem as best n-rational approximation.It is a long standing open problem till now, formulated as follows.中国煤化工MYHCNMH G516Appl. Math. J. Chinese Univ.Vol. 28, No.4Let p and q denote polynomials of one complex variable. We say that (p,q) is an n-pair ifpand q are co-prime, both of degrees less than or equal to n. Denote the set of all n-pairs by Rn.If (p,q)∈Rn, then the rational function p/q is said to be a rational function of degree less orequal n. Let f be a function in the Hardy H2 space in the unit disc. To find a best n-rationalapproximation to f is to find an n-pair P1,91) such that|f - p1/qll= min{lf - p/all : (p,q)∈Rn}.By using a so called cyclic AFD algorithm we can get a solution of the above mentionedproblem if there is only one critical point for the objective function (30]). We call such asolution a conditional solution. Besides cyclic AFD, to the author's knowledge, there existsanother algorithm, RARL2, by the French institute INRIA, that also can only get a conditionalsolution [2]. The theory and algorithm of cyclic AFD are both explicit. It directly finds outthe poles of the approximating rational function. The other rational approximation modelsall use the cofficients of p and q as parameters in order to set up and solve the optimizationproblem. Using cofficients of polynomials involves tedious analysis and computation. Theultimate solution of the optimization problem lays on optimal selection of an initial status tostart with. Finding an optimal initial status itself is, however, an NP hard problem.We will call2 ; ckBr(z)k=1an n-Blaschke form where the parameters 01...., an defining Bn are arbitrary complex numbersin D. As shown in the literature, with an abuse of terminology, the parameters a,..an areoften called“poles” although they are, in fact, zeros. An n- Blaschke form is said to be non-degenerate if Cn≠0. It is easy to see that a non-degenerate n-Blaschke form is either ann-rational function or an (n - 1)-rational function, depending on whether 0 is a pole of Bn.This shows a little inconsistence with the notion of n-rational functions. If, instead, we workon the parallel context outside the unit disc, then the set of all n-Blaschke forms coincides withthe set of all n-rational functions. Some researchers, including L. Baratchart, choose to workin the context outside the unit disc. To simplify the writing we ignore the inconsistence andstill work on inside the unit disc.For any given natural number n the objective function for the optimization problem isA(-..an)=. If|122 - E(f, Bx)2.(4k-1Definition 2. An n-tuple (a,..,.n) is said to be a coordinate-minimum point of an objectivefunction A(f;z1,..,zn) if for any chosen hi among 1...n, whenever we fix the rest n - 1variables, being Z1 = a1,...,7k-1 = ak-1,Zk+1 = ak+1,...,zn = an, and select the kth variablezk to minimize the objective function, we haveak = arg max{A(f;a..,ak-1,zk,ak+1...,an) : zk∈D}.中国煤化工MYHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 517In the AFD algorithm we repeat the following procedure: Along with choosing a1,..,ak-1in D, we produce the reduced remainders f...., fn. Then to fk we apply the Maximal SelectionPrinciple to find an ak giving rise to max{|<.fk,ea)|:a∈D}. The Cyclic AFD Algorithm repeatsuch procedure for k = n : For any permutation P of 1...n, whenever ap()....ap(n-1)are fixed from previous steps we accordingly and inductively obtain the reduced remaindersf2.... fn, and, next, use the Maximal Selection Principle to select an optimal ap(n).Denote by LMP a local minimum points, by CMP a coordinate-minimum point, and CPa critical point of an objective function. Denote by LM,CM and C the sets, of, respectively,all the LMPs, CMPs and CPs of an objective function. Then we have the following inclusionrelations.Proposition 5. .L.MCCMC C.(5The proposed cyclic AFD algorithm is contained in the following theorem.Theorem 6. Suppose that f is not an m- Blaschke form for anym< n. Letso= {.{......be any n-tuple of parameters inside D. Fix some n- 1 parameters of So and make an optimalselection of the single remaining parameter according to the Maximal Selection Principle basedon the objective function (4). Denote the obtained new n-tuple of parameters by 81. We repeatthis process and make cyclic optimal selections over the n parameters. We thus obtain a sequenceof n-tuples so, ......... with decreasing objective function values d that tend to a limitd≥0, where, in the notation and frmulation of (4) and (3),du= A....4=. If|1 -二1 - |b?1})f"?(6")2.(6Then, (i) IfS, as an n-tuple, is a limit of a subsequence of {s}2o, then歹is inD; (ii)JisaCMP of A(;-); (ii) If the correspondence betwcen a CMP and the corresponding value ofA(f;--) is one to one, then the sequence {si}2o itself converges to the CMP, being dependentof the initial n-tuple so; (iv) If A(j;..) has only one CMP, then {si}2o converges to a limit5 inD at which A(f;--) attains its global minimum value.For further details including examples on cyclic AFD we refer the reader to [30].Remark 7. Various types of AFD related signal expansions have applications in practices ofdifferent areas. For applications in system identification, for instance, see ([22], [21]).Experiments.The experimental function is in the Hardy H2 space with non- trivial singular inner partgiven byf=(在一1- x22)(x+号)(x+ 2)(x + 3)e(辩+带).With this example we compare performances of Fourier series decompcUnwending AFD and n-Best AFD. We also include comparison between their correspondingtime-frequency distributions (with FD replaced by Short Time Fourier Transform or STFT).It is well known that convergence of Fourier series of Hardy space functions with non-trivialsingular inner parts is very slow. The experiments show that to reach a similar accuracy of中国煤化工MHCNMH G518Appl. Math. J. Chinese Univ.Vol. 28, No.4decomposition no.=35energy diterence=0.0332777reconstructionE0中_1-21000TimeFigure 5: Original signal and reconstructed signal of n-Best AFD decomposition for n=35approximation of 65 iterations of Core AFD one needs to run 210 iterations of FD. To reachalmost the same accuracy one needs to run n-Best AFD for n= 35, and only run 5 iterationsof Unwending AFD (We note that electronic version of the pictures gives considerably betterillustrations of the accuracy comparison due to the easily seen colors). In the time-frequencyaspect we see that all of those give two peaks at almost the same time instants. They, however,have different carrier frequencies. This can be understood.In fact, decomposition is notunique. For instance, a mono-component can have many different decompositions into othermono-components. By considering both aspects we recommend Unwending AFD: If a signal isessentially of a certain frequency (or say to have a certain carrier frequency), then UnwendingAFD can first factorize the corresponding inner function. The approximation accuracy andtime-frequency representation of Unwending AFD can be further improved if computation ofthe Hilbert transform can be improved: Circular Hilbert transform is encountered to work outthe corresponding outer function.83 The mathematical theory of phase derivative and its impacts todigital signal processingSignal analysis practice has been longing to have a frequency theory. Many signal ana-lysts tend to believe that frequency, or instantaneous frequency, should be defined as the phasederivative of a complex signal associated with the given real-valued signal. While this seemsto be reasonable, there, however, exist a number of obstacles associated with this idea. First,how to associate a real-valued signal with a complex- valued signal so that we can well definea frequency concept as the phase derivative of the complex signal. Gabor proposed his ana-lytic phase derivative method through Hilbert transform of the signal. Ever since then, signalanalysts have been justifying the analytic signal approach through enormous experiments andarguments, but till now there has been no major progress with this method. Gabor's approach,in fact, cannot be easily implemented due to at least two problems. One is that given a generalsignal, how to mathematically define the analytic phase derivative. The point is that a signal it-self is usually not smooth, and neither is the associated analytic signal. A general signal should中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 519be assumed to be only a function in the Lebesgue L2 space. L2 functions are not considered asfunctions precisely and pointwisely defined. Functions that defer in their values in a Lebesguenull set are considered to be the same L2 function. This completely rules out smoothness ofL2 functions. The phase derivative approach in such case amounts to get smooth objects fromnon-smooth objects. When dealing with a discrete signal one methodology is to treat the dataas the Fourier cofficients through Z-transformation; and the other methodology is to treatthe data as sampling of a continuous or smooth signal. Both of these methodologies run intothe same smoothness verses non- smoothness problem. The following is a concrete exampleconcerning definition of such phase derivative. In L. Cohen's book [7] he proves the followingformula广w18(w)2dw= ()s(t)|2dt .(7under the assumptions that s(t) = p(t)ei0(t)， and s, p(t) and 0(t) are all smooth in t and|s||| = 1. This formula gives a reason why phase derivative should, in general, be consideredas instantaneous frequency. The question is, for non-smooth signals s， for which the phasederivative 0' may not exist, in what capacity the above relation or a similar one would still holdtrue?The second problem is that although non-negativity of analytic phase derivative is desired,it is not always available. The non-negativity of analytic phase derivative is necessary notonly because the physical meaning of the frequency concept, but also because that analysis ofpositive and negative frequencies together sometimes does not give meaningful results. This canbe seen, for instance, from the mean of frequencies. If one does not require non-negativity, themean of Fourier frequency of any real- valued signal is zero. The task is to construct a coherentanalytic signal theory that solves the mentioned and not mentioned problems. The authorand his collaborators have built up such a theory. In the previous sections we introduced themono-component and mono-component decomposition theory. In this section we give a briefsummary on phase derivative vs. frequency and some related results.From our study there are two cases in which we can define phase derivative. One is forsome classes of functions defined in complex analysis and the other is based on Sobolev spaceconditions.1. Inner and outer functionsIf f isa function in the Hardy H2 space, then f has the following canonical factorizationdecomposition, called Nevanlinna factorization:f = OBS,where O,B and S are, respectively, the outer, the Blaschke product and the singular innerfunction part of f. Such classes of functions and the related factorizations are available forall the concerned contexts: the unit disc and outside the closed unit disc, anthe upper- andlower half complex planes. The function I = BS is called the inner function part of f. For anyinner function Theorem 1 shows that its phase derivative as the limit of the same quantity butfrom inside of the disc always exists, and be positive, if non-trivial. For the outer function part,under certain conditions it exists, and has the zero-mean property. This shows that it should be中国煤化工MYHCNMH G520Appl. Math. J. Chinese Univ.Vol. 28, No.4sometimes positive and sometimes negative. Here “sometimes positive (sometimes negative)”means that it is positive (negative) in a set of Lebesgue positive measure. There exist resultsfor amplitude derivatives, too. For details we refer the reader to [26].The inner and outer function theory given by Theorem 1 gives a great impact to the theoryof ll-pass filters, energy delay and signals of minimum phase [8] It is noticeable that textbooksof DSP claim that inner functions have positive phase derivative (see [5]), but this fact had neverbeen rigorously proved until the publication of Theorem 1 in [26]. .2. Signals in the Sobolev spacesFor a function in the L2 space on the boundary one can proceed the Hardy space decompo-sition, viz.,The functions s+ are holomorphic functions in the respective domains in which they are defined.In the real line case the respective domains are the upper and lower half planes, and we have,as a basic and important property of Hardy space functions,lim s*(x+y)=s+(x)，a.e.In order to make，lim. (s+)'(x + iy)also exist a.e. it sufices that (s#)' belong to the Hardy space H2(C*). Fourier analysis showsthat a suficient and necessary condition for (s+)' belonging to the Hardy space H2(C#) is thats belongs to the Sobolev space[i={8∈L2(R) :∈L*(R)},dtwhere stands for the distributive derivative [11]. Under such condition we have nontangentially, lim. s*(x + iy),lim. (s+)(x + igy)both exist a.e. and the limits are a.e. non-zero. Therefore, by definition,(0*)'(t)=_ lim (哧)()=. lim, Im((s+)'(x + iy)土y→0+土y- +0-s+(x + iy)exist and being non-zero and finite a.e. Under the Sobolev condition the formula (7) is gener-alized to[* ()2=/* (+)()()tu+/* ((0-)()s-()2dt(see [11]).What is amazing is that under the same assumption we are able to define the so calledHardy-Sobolev phase derivative0*()= X{s+8- #o}()Im .((s+)(t)+(s-)'(t)'s+()+s-() )'中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 521where s+ are understood as a.e. determined through non-tangential boundary limits. Underthe Hardy- Sobolev phase derivative notion we can show the ultimate relation (see [8])u18(w)2= [~ 0()5+()dt.It is to a great satisfaction to notice that if s(t) = p(t)ei0(t) and the classical derivativess'(to),p'(to) and θ'(to) all exist, then '(to) = 0*(to). Similar generalizations for higher ordermoments and deviations are available under the notion of Hardy-Sobolev derivatives (11], [8]).We finally note that with various notions of the phase and ampltude derivatives we are ableto prove truly stronger uncertainty principle for the classical setting, the LCT setting, as wellas for the general self-adjoint operator setting ([9], [10]).δ4 Time-frequency distribution based on mono componentsSuppose that m(t) = p(t)ei0(t) is a complex signal, then the ideal time-frequency distributionis one of the Dirac type defined as a function of two variables, viz.,P(t,w) = p(t)8(w- 0()). .Here the Dirac function δ is understood as being with value 1 at the zero point and value zerootherwise. The implementation of this idea, however, brings in great controversies. One is thata practical signal, no matter complex-valued or real-valued, cannot be simply expressed in suchform with well defined phase derivative, let alone the requirement for θ'(t)≥0, a.e. Only inthe latter case such time frequency distribution has properties like a probability distributionforw>0.This formulation is valid and practical only for mono-component signals: Although thishas been desired by many signal analysts for many years, it can never be implemented, fothere were no clearly defined notions of mono-component and instantaneous frequency. Signalanalysts generally admit that a signal is said to be a monocomponent (with a little difference inspelling from what we defined mono-component)，“if for this signal, there is only one frequencyor a narrow range of frequencies varying as a function of time; and, it is a multicomponent if itis not a monocomponent" (Boashash, [4]). The ambiguity of such definition of monocomponentlays on the fact that it is based on the notion of frequency and its narrow range. What theycall“frequency" and“narrow range”，however, are not defined in signal processing knowledgesystem. In such way signal analysis has been established based on intuition with undefinedconcepts but not on rigorous mathematics. This not only restricts signal analysis practice,but also restrict the theoretical and concept development. There has been no agreement amongsignal analysts on what is frequency or instantaneous frequency: it depends on personal discreteunderstanding. Most signal analysts tend to believe that something called frequency objectivelyexist, and what human being can do is just to“estimate" the frequency. Boashash proposed theabove definition in 1990's which adopted the idea of Gabor. Until the present time, however,there has been no progress and the situation stays as the same as that more than a half centuryago.Our way to get out of the frequency paradoxes is to define frequency (instantaneous fre中国煤化工MHCNMH G522Appl. Math. J. Chinese Univ.Vol.28, No.4quency) as a function to be the analytic phase derivative if the latter can be well definedand non-negative almost everywhere in the Lebesgue measure sense. Signals that possess in-stantaneous frequency are mono- components. In such way not every signal has instantaneousfrequency. For general signals that do not have a global instantaneous frequency function, or,not a mono- component, one seeks for appropriate mono-component decompositions. In suchway we have, for a complex- valued signal s in the Hardy space,s(t)= Sma(t),k=1where for each k, mk is a mono-component. In the AFD decomposition case, for instance,we have m:(t) =< f,Bk > Bk = Pr(t)ei0e(t) with %(t)≥0, a.e. If 8 is real-valued in L,we use the relation s = 2Res+一Co, and get mr(t) = pr(t)cos0x(t),0'(t) ≥0, a.e. In bothcases we define P(t,w)= Ek=1 Pre(t,w), P:(t,w)= Zk=1 Ph(t)8(w -时()). Such defined time-frequency distributions enjoy almost all of the commonly desired properties for time-frequencydistributions. For details we refer the reader to paper [60].000.350.0.250.20.150o0.05400 600800 1000TimeFigure 6: Time frequency distribution of order 65 AFD decomposition120010000.:2Figure 7: Time frequency distribution of order 5 Unwending AFD decomposition中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 5233500.35000.500.250.20.150.10.0540060800 1000Figure 8: Time frequency distribution of n- Best AFD decomposition for n=35B04020400600 800 1000meFigure 9: Time frequency distribution of STFT85 Further study on TM systems and backward shift operatorinvariant spacesShift and backward shift operator analysis was developed with the cornerstone BeurlingTheorem and Beurling-Lax Theorem. Many of the studies are related to Russian mathemati-cians. The proposed AFD has a close relationship with backward shift invariant spaces. Forany sequence of complex numbers in the unit disc, a....,an..., there exist two classifications.1. The hyperbolic non-separable condition (1) holdsIn such case the related TM system is dense in the HP spaces for 1≤p≤∞, viz,HP = span{Bx}e=1.On the other hand, if the span of {Bk} is dense in HP for any p∈[1, ∞]，then the complexnumbers in the sequence must satisfy (1).2. The hyperbolic non-separable hyperbolic condition (1) does not holdIn such case a Blaschke product φ with the complex numbers 0a..... an... as zeros (to-中国煤化工MHCNMH G524Appl. Math. J. Chinese Univ.Vol. 28, No.4gether with the multiplicities if there are repeatings) can be defined, andH2(D) = span{Bn}田φH2(D).This is to say that a sequence a1,...n....obtained from an AFD process may not satisfythe non-separable condition, and the span of the related TM system { Bw} may not be dense.In such case for the signal s based on which the AFD process is done there holds the relations∈span{Bn}.For general HP(D) spaces we have [53]Theorem 7. Assume that Zx=1(1- |ax|) <∞, and φ is the Blaschke product defined by theak's in the sequence, then for any p∈(1,∞) there holdsH"(D)∩φH呵) = (φH° (R)) = span{Bn}n21,(8where the closure span is in the LP(OD) topology and (φHr' (D))-= {f∈H'(D)|(f,φg) = .0，V g∈HP' (D)}.In [51] we give pointwise convergence results of TM system corresponding to the classicDini, Dirichlet type results in the Fourier series case. In [6] we give a constructive proof ofBeurling-Lax Theorem. In [31] we prove that a TM system {Bk} is a Schauder basis in theclosure of the span{ Bk }.Restricted to bandlimited functions we satisfactorily characterized the solutions of the bandpreserving, phase and amplitude retrieving problems, as well as solutions of the Bedrosianequation when one of the product function is bandlimited. The band preserving and phaseretrieving problems mainly arise from optics, and the amplitude retrieving problem and theBedrosian equation problems are related to signal analysis. These problems are described asfollows.Let A>0.(i) Knowing suppf C [0, A], characterize all functions g in LP,1≤p≤∞，such that .supp(fg) C [0, A]. We phase this as band preserving problem.(i) Knowing suppf C [0, A], characterize all functions g such that |g| = 1, ae, and supp(fg)c[0, A]. We phase this as phase retrieving problem.(ii) Let f(t) = p(t)ei0(t) be a complex mono-component. Characterize all real-valued func-tion g such that fg is again a complex mono-component. We call this as amplituderetrieving problem.(iv) Characterize all solutions for the Bedrosian equation H(fg) = fHg, where f or g isbandlimited. We call this as bandlimited Bedrosian equation.Particular cases of amplitude retrieving problem (ili) have been addressed in the previous sec-tions. It is obvious that Problem (ii) can be considered as a particular case of Problem (i). Bothproblems have been attacked by researchers. The results that the other researchers obtained arebased on Weierstrass' infinite product formula for entire functions. Our approach to Problem(i) and (i), as well as to problem (iv) are based on backward shift invariant spaces that is moreexplicit and computable.中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 525Denote by FH'[A, B] the space of functions in L9(R) whose distributional Fourier transformis supported in [A, B]; by LH(R) the space of functions in L"(R) whose Hilbert transform isalso in L9(R). The notation 8- 1 f denotes the Laplace transform of f whenever definable. Thebandlimited Bedrosian problem (iv) is treated in the following three theorems ([53]).Theorem8. Letf∈[唱(R) andg∈唱(R), where1≤p,q≤∞and0≤r-1=p-1+q-1≤1.Then the following assertions are equivalent.(1) H(fg)= fHg;(2) H(f-9+)=-if-9+_ andH(f+9-)=if+9-;(3) f-g+∈H"(R) and F+9-∈H"(R);(4) f-9+∈FH(R+) and F+9_∈FH"(R+);(5)F∈HP(R)∩Ig+ (R+) and f+∈H(R)∩Ig-H(R) if9+ and g_ are nonzero functions.(6) 9+∈φ1H9(R),∈φ1H"(R) and二]=号，Is+=号，where f+ andf- arenonzero functions, φ1 and2 is a pair of co-prime inner functions, ψ1 and42 is also a pair ofco-prime inner functions.Theorem 9. Let f∈FH'[A,B] andg∈HP(R) be nonzero functions. If A,B∈suppf andA<0< B. Then H(fg)= fHg if and onlyifg+∈e-iAxH9(R) and g二∈eiBxH9(R).Theorem 10. Letg∈ FH'[A, B], where A,B∈suppg, and f∈L&(R) be nonzero functions.Then H(fg)= fHg if and only iff∈span'{r-X)'λ∈E∪Ez，j= 1,-,m()where E1 is the set of all different zeros ofG+(z):= (0-1g+)(z) in the upper half plane, E2 isthe set of all different zeros ofG_ (z) := (8-1g-)(z) in the lower half plane and m(X) be themultiplicity at入. The above representation is with the convention that if one of 9+ and 9- iszero, then the corresponding set of zeros is the empty set.The next two theorems treat the band preserving problem (i) ([52], [53]).Theorem 11. Suppose that0≠g∈FH'[A,B] andf∈Lg(R). Then fg∈FH"[A,B] if andonly ifF∈H"(R)∩Ig(x)H(R) = FH'[0,a2]田Hispa"{a=x，λ∈E, j=1,-. ,m()}, ._1 .andf+∈H"(R)∩Ig(x)Hp(R) = FHPI0,a]θS_1θispamr {2=x λ∈E, j-=1... ,m()},where Ig(x) and Ig2(x) are, respectively,m()m(X)|\2+ 1|Ig(x):= eiarx I]1In(x)=ei I (\入2+ 1斗二=\)”λ∈E2 \(入2+1二)”中国煤化工MHCNMH G526Appl. Math. J. Chinese Univ.Vol. 28, No.4Theorem 11 gives a characterization of the solutions f∈L'(R),1 < P <∞, in terms ofbackward shift invariant subspaces. It, however, does not cover the cases p = 1 and p=∞due to the failure of the relevant Hardy spaces decomposition. Below we will treat the twoexceptional cases by using an alternative approach.Theorem 12. Suppose that function 0≠g∈FH'[A,B] and f∈LP(R),1≤p≤∞，benonzero functions. Then fg∈FH"[A, B] if and only iff∈gH(R)∩Ig.HP(R) = g HP(R)∩Ig1Ig.HP(R)where Ig\(x) := ei(a1x+b1)Br(x) is the inner function of 91(x) := e-iArg(x) and Ig2(x) :=ei(a2+b2) B2(x) is the inner function of 9g2(x):= eiBxg(x).The next three theorems treat phase retrieving problem (i) ([52]).Theorem 13. Assume that 0≠f∈HP(R) and its Laplace transform f(z) is holomorphicacross R. Then there exists an analytic signal g(x)∈HP(R) whose Laplace transform g(x) isholomorphic across R such that |f(x)|= |g(x)| if and only if02+1x-og(x)= eiax+ibB+(x)f(x),(9(n资+二量)(n|a12 + 1|x- anβ2+1x-7where a and b are some real constants, {a';}%=1 are partial zeros of f(z) in the upper halfplane, {βi}%1 is a complex sequence satisfying 22nep2 <∞, {Bn}n=1 can only have anaccumulation point at∞and {β'}∩{o};}= 0.Theorem 14. Assume that analytic signal0≠f∈HF(R) and its Laplace transform f(z) bean entire function. Then there exists an analytic signal g(x)∈HP(R) whose Laplace transformg(z) is an entire function such that |f(x)|= |l9(x)| if and only ifg(x)= eiax+tbx-2)f(x),(nmi β2+1 x-β|a+1|x-a'nwhere a and b are some real constants, {o'}=1 are partial zeros of f(z) in the upper half plane,{βl}≌1 are partial zeros of f(z) satisfying 2 2H1m(}) <∞o,and {β}∩{o'}=0.台1+13h4The following theorem completely solves the phase retrieving problem for bandlimited sig-nals.Theorem 15. Let nonzero analytic signals f∈HP(R) and Suppf E [0,A]. Then there existsan analytic signal g∈HP(R) with Suppg S [0,A] such that |f(x)|= |g(x)| if and only if+1. x-可回2+1.x-可g(x) = eibtiar'+1|x-B3+1工一限(10)where b and a are some real constants,{a%} are partial zeros of f(z) in the upper half plane,{@%} are partial zeros of g(z) in the lower half plane and {B}∩{a'%}= 0.中国煤化工MHCNMH GQIAN Tao, et al. Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis 527δ6Higher dimensional generalizationsIn higher dimensional spaces there are mainly two types of complex structures of which oneis several complex variables which is esentially of the tensor form and the other is Clifordalgebra that treats a vector variable as a complex variable. Quaternionic algebra is a Ciffordalgebra. It has a particular position not because it is the only commutative or non-commutativefield of a finite dimension apart from the real and the complex number felds (Frobenius)but because that the quaternionic space offers an“non-canonical” imbedding of R3 into thespan of the Clifford algebra generated by two Cifford basis elements e1 and e2 satisfyingei = e险=-1,e1e2 = -e2e1. The canonical imbedding of R3 is the one that identifies R3with the set {x1e1 + x2e2 + xge3} in which e1,e2, e3 are independent Cilifford basic elements.By complex structure we mean a Cauchy type structure, including at least a Cauchy kernel, aCauchy theorem and a Cauchy formula. There may also be a related Hardy space in the context,and correspondingly a Szego kernel as reproducing kernel in the boundary L-space, etc. Inprinciple, one can develop an approximation theory by using linear combinations of sampledSzego kernels. We in below mention some particulars in each of the concerned contexts.With the several complex variable setting we have been studying two contexts, the n-torusand the tubes in the sense of Stein and Weiss [49]. In the n-torus we use the direct productof the TM systems associated with each of the complex variables. Taking n= 2, we show thatfor any function f in the Hardy space of the two-disc, we can choose adaptively two sequencesa= {an} and b = {bn} such that the corresponding direct product of the two TM systems,B*θBb= {Bf田B} offer fast decomposition of the given signal [33]. For the tubes case thereis a parallel theory.In the Cifford algebra setting, for the quaternionic case we can construct a theory verysimilar to the one complex variable case. Although there is no backward shift operator in thecontext we can show a mechanism similar to (3) ([41]). For a general Cifford algebra this hasnot been done. The obstacle is that in the Cliford case in the Gram- Schmidt scalar-valued.This gives rise to a technical dificulty. Instead, in the general Ciford algebra setting we usean improved greedy algorithm to get one of the best parameters at each selection (44]). Inthe multi-dimensional case we also developed a higher order Szego kernel method to extractsuitable basic functions of high frequency with the maximal energy [57]These results offer a theory of rational function approximation in higher dimensions. Pre-cisely, the related reproducing kernels are not necessarily rational functions but with squareroot of polynomials. We also developed compressed sensing with Szego kernels [19], supportingvector machine with Szego kernels [23]), as well as adaptive Aveiro discretization method byusing Szego kernels [20]References[1] A. Axelsson, K.I. Kou, T. Qian. Hilbert transforms and the Cauchy integral in Euclidean space,Studia Math, 2009, 193(2): 161-187.中国煤化工MHCNMH G528Appl. Math. J. 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Backward shift invariant subspaces with applications to band preservingand phase retrieval problems, preprint.[53] L.H. Tan, T. Qian, QH. Chen. New aspects of Beurling Lax shift invariant subspaces, preprint.[54] T. Qian, L.H. Qian. Phase and amplitude of analytic signals, preprint.[55] T. Qian, L.H. Qian. Mono-components and p-starlike functions, preprint.[56] J.L. Walsh. Interpolation and Approrimation by Rational Functions in the Complex Plane,American Mathematical Society, Providence, RI, 1969.[57] J.X. Wang, T. Qian. Approrimation of monogenic functions by higher order Szego kernelson the unit ball and the upper half space, accepted to appear in Sci China Math, DOI:10.1007/s11425-013-4710-1.[58] Y.S. Xu. Private comminication, 2005.[59] B. Yu, H.Z. Zhang. The Bedrosian identity and homogeneous semi-convolution equations, JIntegral Equations Appl, 2008, 20: 527-568.[60] L.M. Zhang, T. Qian, P. Dang, W.X. Mai. Diract type time-. frequency distribution based onmono-component decomposition, preprint.Department of Mathematics, University of Macau, Macao. Email: fsttq@umac.mo2Department of Computer and Information Science, University of Macau, Macao.Email: lmzhang@umac.mo中国煤化工MYHCNMH G