## Abstract

The intrinsic mode functions (IMFs) arise as basic modes from the application of the empirical mode decomposition (EMD) to functions or signals. In this procedure, instantaneous frequencies are subsequently extracted from the IMFs by the simple application of the Hilbert transform, thereby providing a multiscale analysis of the signal's nonlinear phases. The beauty of this redundant representation method is in its simplicity and extraordinary effectiveness in many important and diverse settings. A fundamental issue in the field is to better understand these demonstrated qualities of the EMD procedures and the elementary modes they produce. For example, it is easily observed that when an EMD procedure is applied to the sum of two arbitrary IMFs, the original modes are rarely reproduced in the generated collection of IMFs. An interesting question from a representation point of view may be stated as follows: for any given sufficiently smooth function and fixed *n*≥2, when is it possible to represent the function as a sum of (at most) *n* intrinsic modes? A more interesting question is whether such a decomposition is possible when the extracted modes are constructed from a common formulation of the intrinsic properties of the function being analysed.

We provide an answer to these questions for a relaxed version of IMFs, called *weak IMFs*, which has been shown to be characterized in terms of eigenfunctions of Sturm–Liouville operators. The objective of this study is to further extend that analogy to the relationship between sums of weak IMFs and coupled Sturm–Liouville *systems*. The construction of this decomposition also provides a guide to an alternate characterization of the instantaneous frequency and bandwidth.

## 1. Introduction

The empirical mode decomposition (EMD) method was developed by Huang *et al*. (1998) to decompose functions into a superposition of natural modes, each of which could be easily analysed for their instantaneous frequencies and bandwidths. These natural modes, which were termed intrinsic mode functions (IMFs), were generated at each scale, going from fine to coarse, by an iterative procedure to locally isolate the modal behaviour. Application of EMD to real signals *f*(*t*) has the purpose of representing these signals as sums of simpler modes *ψ*, i.e.(1.1)where each *ψ*_{j} (see Cohen 1995) comes with a specific polar representation of the form(1.2)These modes generalize the standard real and imaginary parts of Fourier components, in which it is required that each be constant and *θ*_{j}(*t*)=*jt* be linear, but are more suitable for the study of non-stationary phase. A richer analysis of signals is provided when the amplitudes are allowed to vary and the phases are permitted nonlinear, or non-stationary, behaviour as in the representation (1.2). We wish to emphasize that even if one is provided a decomposition of a signal *f* in the form (1.1), for each given *ψ*_{j} the particular polar representation (1.2) that should be used is ambiguous with many possible selections of reasonable pairs of amplitudes and phases. The objective of EMD is to extract the decompositions from among such highly redundant representations so that the amplitudes *r*(*t*) and the corresponding phases *θ*(*t*) at each scale are both physically and mathematically meaningful. In the case that the signal *ψ* is *causal*, the polar representation may be recovered by the application of an *appropriate* (cf. Sharpley & Vatchev 2006) Hilbert transform. In this case each mode *ψ*, after a phase shift of *π*/2, is represented as the real part of a complex signal *Ψ*(1.3)Obviously, the choice of amplitude–phase (*r*, *θ*) in the representation (1.3) is equivalent to the selection of an imaginary part *ϕ* since(1.4)once some care is taken to handle the branch cut.

An alternative to the Hilbert transform method for extracting instantaneous frequencies is the class of *quadrature* methods (see Cohen 1995). Whichever method is to be used should be governed by the context of the phenomena under study and the analysing procedure should produce, for each signal *ψ* within the signal population, a properly chosen companion *ϕ* for the imaginary part. This companion function *ϕ* should be unambiguously defined and should properly encode information about the component signal, which is in this case the IMF. Therefore, the collection of instantaneous phases present at a given instant (i.e. *t*=*t*_{0}) for a signal *f*(*t*) is heavily dependent upon both the decomposition (1.1) and the selection of representations (1.2) for each monocomponent. A discussion of the application of Hilbert transforms and alternative quadrature methods to IMFs to obtain instantaneous frequencies are provided in Sharpley & Vatchev (2006).

A partial sum of IMFs in the representation (1.1) can be considered as an approximation of a function *f* by IMFs. The stoppage criteria used in applying EMDs are numerous, but they could be considered as control of approximation errors in various metrics.

According to Huang *et al*. (1999, p. 423), the EMD method was motivated ‘from the simple assumption that any data consists of different simple intrinsic mode oscillations’. Three methods of estimating the time scales of *f* at which these oscillations occur have been proposed are

the time between successive zero-crossings;

the time between successive extrema; and

the time between successive curvature extrema.

The use of a particular time scale is application dependent. Following the development in Huang *et al*. (1999), we define a specific class of signals with special properties that make them very well suited for phase and wave band analysis.

A function *ψ*(*t*) is defined to be an *intrinsic mode function*, or more briefly an IMF, of a real variable *t*, if it satisfies the following two characteristic properties:

*ψ*has exactly one zero between any two consecutive local extrema and*ψ*has zero ‘local mean’.

A function that is required to satisfy only condition (i) will be called a *weak IMF* and will be used to designate all such functions. Note that *ψ*∈ precisely when the number of zeros and the number of extrema of *ψ* on *I* differ at most by 1.

In general, the *local mean* in condition (ii) in the EMD procedure is typically the pointwise average of the ‘upper’ and ‘lower’ envelopes as determined by the spline fits of knots consisting of, respectively, the local maxima and the local minima of *ψ*.

The concept of decomposing functions into oscillating modes is the essence of the Sturm–Liouville theory. In Sharpley & Vatchev (2006) we related the concept of monocomponents from phase analysis to the Sturm–Liouville theory by characterizing the weak IMFs as solutions of self-adjoint differential equations.

*Let ψ*∈*C*^{2}(*I*) *be a weak IMF with simple zeros and extrema, then there exist positive continuously differentiable functions P and Q such that ψ is the solution of the initial-value problem*(1.5)*for some τ*∈*I*.

From this result, it follows that an instantaneous frequency and bandwidth pair can be defined implicitly through the Prüfer substitution given by(1.6)

The goal of this paper is to further extend the relationship between weak IMFs and eigenfunctions of self-adjoint differential operators by the representation of very general functions by superposition of weak IMFs from a common framework. Therefore, a natural question arises which must be answered if an EMD procedure, such as Hilbert–Huang transform (Huang *et al*. 1999), is to be realized in terms of some nonlinear minimization procedure in the context of structured dictionaries (e.g. the expository article of DeVore 1998). The question then becomesUnder what conditions can functions be decomposed into a superposition of at most

*n* IMFs that are inherently extracted from the function?

Here, *n* is a fixed number that is set in advance. We prove that any smooth function can be decomposed into two (or fewer) weak IMFs (see theorem 3.1). This is not surprising since the two IMFs are unrelated to *f* other than the one which should be a sinusoidal with amplitude larger than the uniform norm of *f* and frequency higher than any local frequency of *f*. In this case, there are no requirements coupling the two corresponding Sturm–Liouville operators for the *ψ*_{j}. However, with more care, weak IMF pairs *ψ*_{1}, *ψ*_{2} can be constructed from *f* so that they are solutions corresponding to different eigenvalues of a Sturm–Liouville operator (SLO; with coefficients *P, Q*) and *f*=*ψ*_{1}+*ψ*_{2}. This is the content of theorem 3.2. The analogy between a decomposition such as EMD and coupled Sturm–Liouville systems with the same primary coefficients *P* and *Q* is provided in theorem 2.1. We prove that for any sufficiently smooth function *f*, there exists a coupled linear system of two differential equations, such that *f* represents one of the components of the solution. These coupled equations can be formulated in terms of a mechanical system of a pair of variable masses and springs and *f* then represents the displacement from equilibrium of the terminal mass of the system. This formulation gives an indication of a possible deeper relationship between decompositions of signals into superpositions of IMFs and concepts from system identification.

We begin by defining a general SLO that acts on sufficiently smooth functions *f* and is defined by(1.7)where *P* and *Q* are positive continuous functions on some interval *I*=[*a*,*b*]. The spectrum of *L* consists of all *ω* for which the equation *Lψ*=*ωψ* has a solution on *I* that satisfies certain boundary conditions. In a more general situation the Sturm–Liouville (SL) system is defined as the solutions of the boundary-value problem (BVP)(1.8)for positive functions *P* and *Q* and real constants *ω*, *α* and *β*.

It is well known (see Birkhoff & Rota 1989) that (1.8) has a solution for a discrete set of eigenvalues, such that 0<*ω*_{1}<*ω*_{2}<⋯ and . The corresponding eigenfunctions form an orthogonal basis in the weighted Hilbert space with inner product , and hence any function *f*∈*L*_{2}(*I*)(*Q*) has a decomposition(1.9)with convergence in *L*_{2}(*I*)(*Q*). For a fixed SL system, there exists a unique representation of any *f*∈*L*_{2} in the form of (1.9) but, in general, requiring infinitely many terms. This should be contrasted to theorem 3.2.

## 2. Coupled differential systems for smooth functions

In this section, we prove the main result of this paper for representing a function as a component of a solution of a linear system of differential equations. In the proofs of the theorems, we require two technical lemmas. In order not to disturb the main flow of ideas, we refer the reader to appendix A for their proofs. Throughout this section assume *f* is defined on a finite interval *I*=[*a*,*b*] and has only simple inflections and set has *M*<∞ elements.

*For a function f*∈*C*^{2}(*I*) *with simple extrema and for any positive numbers k*_{1} *and k*_{2}*, there exist continuous and positive functions P and Q, depending upon f, and twice differentiable function x such that f and x satisfy the following system of differential equations*:(2.1)

For each point *ξ*_{j}∈*Ξ*, suppose *I*_{j−1}=[*b*_{j−1},*a*_{j}] contains *ξ*_{j} and the initial-value problem (IVP) considered in lemma A.1 has one solution. Let *a*_{0}=*a*, *b*_{M}=*b* and *J*_{j}=[*a*_{j},*b*_{j}]. First assume that *a*_{0}∈*J*_{0}. The existence of *P* and *Q* is shown by inductively constructing them on *I*_{0} and *J*_{0} and repeating the same procedure on *I*_{j}, *J*_{j} for *j*>0.

For fixed *k*_{1}, *k*_{2} positive, set(2.2)and(2.3)Since *f*, *f*′ and *f*″ are bounded on *I*, then for any given real number *u*, it follows that and are finite constants. Pick . (We note that these are the primary constants and parameters in lemma A.2 of the appendix. Although somewhat redundant, introducing these here aids in motivating the proof of alternating the successive application of the two lemmas.)

On the interval *J*_{0} the function *f* does not have an inflection, and hence . By applying lemma A.2 with and , we obtain a twice differentiable solution *Φ* that does not change sign on *J*_{0}. Furthermore, since *f*″ does not change its sign on *J*_{0} and owing to the choice of the initial condition for *Φ*(*a*_{0}), it follows that the function is positive and continuous on *J*_{0} for any fixed function *P*_{0}>0. Fix *P*_{0} a positive constant and define *x*=*f*−(1/*k*_{1})*Φ*, then it follows that(2.4)Twice differentiating the integral equation in lemma A.2, we get and after substituting in , we obtain the following differential equation for *x*:(2.5)By combining (2.4) and (2.5), we have constructed the desired system (2.1) for *J*_{0}.

Next we construct the functions *P* and *Q* on the interval *I*_{0}, but here the function *f* has an inflection point and lemma A.2 cannot be applied. Instead, on this interval this will be the role of lemma A.1. Note that the system (2.1) is equivalent to the system(2.6)where *p*≔(*P*′/*P*) and *q*≔(*Q*/*P*). Let *q* be the linear function connecting the points and . Since on *I*_{0}, then the second equation can be solved for *p*(2.7)and upon substituting in the first equation of (2.6), we obtain the differential equation in lemma A.2 for *x*. The initial conditions naturally come from the already determined values . On *J*_{0} we have that and including the right endpoint *b*_{0}, and hence the estimate holds true. From the definition of *q* it follows that and since *q* is linear on *I*_{0} we have that the last estimate holds on over all of *I*_{0}. All the conditions in lemma A.1 are met, and hence there exists a function *x* that is a solution to the differential equation (A2). Substituting *x* in (2.7), we get a continuous function *p* on *J*_{0}∪*I*_{0}. The functions *P*_{1} and *Q*_{1} are determined from the systemand have solutions , where *P*_{0}=*P*_{1}(*b*_{0}) and *Q*_{0}(*b*_{0})=*Q*_{1}(*b*_{0}). We have now constructed the continuous functions *P* and *Q* on the interval *J*_{0}∪*I*_{0}. Inductively, we can continue the construction on *J*_{1} with the same constants *λ*_{1} and *λ*_{2} as on *J*_{0} and initial conditions , *u* arbitrary. Note that and hence lemma A.1 provides continuous positive *P*_{2} and *Q*_{2} on *J*_{1}. Repeating the same procedure on *I*_{j} and *J*_{j}, *j*≥1, we construct continuous functions *P*, *Q* and *x*, which satisfy the system (2.1). This completes the proof if the partition of *I*=[*a*,*b*] begins with *J*_{0}.

In the case *a*_{0}∈*I*_{0}, we then start by instead using lemma A.1 with appropriate choice for the initial conditions and the function *q*. ▪

In the previous theorem, we showed the existence of continuous positive coefficients *P* and *Q* assuming that the function *f* has a continuous second derivative. If *f* has fourth-order continuous derivatives, i.e. *f*∈*C*^{4}(*I*), the next result shows that we can modify our construction to guarantee in addition that the coefficient *P* is continuously differentiable, so that the equations (2.1) hold in the classical sense.

*Let f be a real-valued function in C*^{4}(*I*)*. If f has only simple extrema, then for any two positive numbers k*_{1} *and k*_{2} *there exist positive functions P and Q such that P, P′ and Q are continuous on I, and f is a solution to* (2.1).

Following the proof of theorem 2.1, it is clear that the only points where *P*′ could have breaks are the points *a*_{j}, *j*=1, 2, …, *M*. We show that around these points the construction of *P* can be adjusted in a way that provides continuous *P*′.

Let *k*_{1} and *k*_{2} be fixed and *λ*_{1}, *λ*_{2} and be as in theorem 2.1, and(2.8)where *m*=(*P*/*Q*) and *s*=(*P*′/*Q*). The system (2.1) is equivalent to the fourth-order equation(2.9)where *L*^{2}*f*=*L*(*Lf*). After substituting (2.8) into (2.9) and isolating *s*″ in the resulting equation we obtain, for the given *f*, the second-order linear equation for *s*(2.10)whereIn the proof of theorem 2.1, the functions *P* and *Q* were constructed continuous and bounded on *I*, and hence the corresponding *p* and *q* are continuous and bounded on *I*. Clearly *P*′=(*s*/*m*)*P*, and hence *P*, *P*′, and *Q* are continuous if and only if *s* and *m* are continuous. On the other hand, *q*=(1/*m*) and *p*=(*s*/*m*). By using the existence theorem for a solution of BVP (appendix lemma A.1), we alter *p* and *q* in one-sided neighbourhoods of the points *a*_{j} in a way that *p* and *q* are continuous on the entire interval *I*.

For a fixed *a*_{j}, there exists a neighbourhood in which the function *f* does not have extremal points and inflections, and hence the function (*f*″/*f*′) has a constant sign on that neighbourhood, say . We modify *q*, obtained from theorem 2.1, on the interval *I*_{j} preserving all necessary conditions required for the proof of theorem 2.1. Let and *ρ*=(*e*/4*R*), where *R*>0, then on the interval *I*_{j,ρ}=[*a*_{j}−*ρ*,*a*_{j}] the straight line is strictly positive and . If necessary, *R* might be increased in order that *I*_{j,ρ} be contained entirely in *I*_{j}. By using the previously constructed *P* and *Q* from theorem 2.1, the new *q* is modified only on the interval *I*_{j}. Namely, on *I*_{j−1}*\I*_{j,ρ} we define *q* as the linear function that connects the points and and on *I*_{j,ρ} we define *q*(*t*)=1/*m*(*t*). Similar to the construction in theorem 2.1, we construct the function *p* on *I*_{j−1}\*I*_{j,ρ} and let . By using the already defined *p* on *J*_{j}, let . We show that on *I*_{j,ρ}, the BVP (2.10) with boundary conditions has a solution , and hence the newly defined *q* and *p* are continuous on .

Without loss of generality, we can consider . Let *R*>0 be large enough such that . Since is linear with respect to *s*′ on *E*(*ρ*, *R*) (see appendix A1), then for a fixed *R* there exists constants *γ* and *C*>0 depending only on *f* and *R* such that *γR*<1 and on *E*(*ρ*, *R*). Next we show that the conditions for hold in limit for *R*→∞. From the choice of *ρ*, it follows that independent of *R* on *I*_{ρ}. Let *μ* be a constant with a value either 1 or −1. Since *f* and its derivatives do not depend on *R* and *m*″=0 on *I*_{ρ}, it follows thatholds uniformly on *I*_{ρ}. Hence we can pick large enough *R* for which and . With that choice of *R*, all the necessary conditions in lemma A.1 are met and the solution to (2.10) with provides a solution *p*.

By repeating the same considerations for all of the *a*_{j}'s, we can construct positive coefficients *P* and *Q* for the system (2.1) such that *P*, *P*′ and *Q* are continuous functions on the entire *I*.

In the previous discussion, we proved that for an appropriate function *f* there exists companion function *x* such that the pair (*f*, *x*) is a solution of a system of type (2.1). The function *x* was constructed first on neighbourhoods around the inflections of *f* and then, starting from the beginning of the interval *I*, the pieces were smoothly connected. In order to construct a particular function *x*, we need to specify the initial values *x*(*a*) and *x*′(*a*). On the other hand, we can start from any interior point with initial values and and construct the function on the left and on the right from .

Furthermore, for functions that satisfy the conditions in theorem 2.2, we considered a method to smoothly connect two disjoint segments of *x*. Summing up, we can construct the function *x* to satisfy boundary–initial conditions (BVP) by initializing the ‘left’ branch of *x* at *a* and the ‘right’ branch of *x* at *b*, followed by connecting smoothly the two branches on an appropriate subinterval of *I*.

In the next section, the relation between the EMD and the representation of *f* as a solution of system of differential equations is discussed.

## 3. Decomposition into pairs of weak IMFs

In the current section, we prove that any function *f*∈*C*^{2}(*I*) with simple zeros and extrema can be decomposed into a sum of two or fewer weak IMFs. The decomposition is constructive and it is not obtained by applying EMD. Two types of decompositions are considered. The first one is a more general result about the existence of only two weak IMFs. In general, the two weak IMFs are not strongly related to physical properties of the function. The second method for decomposition can be associated with the displacement from equilibrium of one of the two masses in a simple mechanical system with time-dependent masses and forces. Details are given later in this section. In either case, the decompositions are non-unique. We denote the interval by *I*=[*a*,*b*].

*Let f*∈*C*^{1}(*I*)*, then there exist two weak IMFs ψ*_{1} *and ψ*_{2}*, such that f=ψ*_{1}*+ψ*_{2} *on I*.

Let *ψ*_{1} be any weak IMF such that is also a weak IMF and the extrema of *ψ*_{1} do not coincide with the extrema of *f*. For example, *ψ*_{1}(*t*) could be sin (*μt*+*ζ*) for appropriate choices of *μ* and *ζ*. Define for *α*∈** R**. Since

*f*is bounded on

*I*, then for any point

*t*∈

*I*we have that and . Since

*f*,

*f*′,

*ψ*

_{1}and are continuous on

*I*, there exists large enough

*α*

_{1}for which is a weak IMF and on

*I*. Since

*α*

_{1}

*ψ*

_{1}is a weak IMF the proof is complete. ▪

From the proof, it is clear that *ψ*_{1} and *ψ*_{2} may not be related in general to the local properties of the function and further analysis based on that representation will not reveal any useful local characteristics of *f*. Another type of decomposition into weak IMFs, which does admit physical interpretation for a sufficiently smooth function *f*, can be obtained from theorem 2.1. That is the content of the next result.

*Let f*∈*C*^{2}(*I*), *then there exist weak IMFs ψ*_{1} *and ψ*_{2}*, such that* *on I, and ψ*_{1} *and ψ*_{2} *are solutions of the self-adjoint differential equations* *, for some positive continuous functions P and Q and positive* .

Fix *k*_{1}, *k*_{2}>0. By applying theorem 2.1 to *f*, it follows that there exist functions *P*, *Q* and *x* such that the vectoris a solution of the system of linear differential equations , where andDirect calculations show that the numbers −*ω*_{1} and −*ω*_{2}, where and are the two eigenvalues of the matrix *K*, and hence there exists an invertible matrix of real numbers *A* such that , whereDefine the vectorThe functions *ψ*_{1} and *ψ*_{2} are two weak IMFs such that for some real *α*'s. Indeed, we have(3.1)From (3.1) it follows that the functions *ψ*_{1} and *ψ*_{2} are solutions to the differential equations for *ω*=*ω*_{1} and *ω*=*ω*_{2}, respectively. Hence they are weak IMFs generated by one and the same self-adjoint operator. Adding the initial conditions from *f*, and , where are determined from the proof of theorem 2.1, we can determine constants *α*_{1} and *α*_{2} such that . ▪

In general, the functions *ψ*_{1} and *ψ*_{2} do not satisfy the same boundary conditions, and hence they are not part of one and the same Sturm–Liouville system. In the case *f*∈*C*^{4}(*I*), we can use remark 2.3 to construct *ψ*_{1} and *ψ*_{2} as two eigenfunctions of one and the same Sturm–Liouville system.

*Let f*∈*C*^{4}(*I*) *and have simple zeros, extrema and inflections. If ω*_{1} *and ω*_{2} *are as in* *theorem* 3.2*, then there exists a Sturm–Liouville system of type* (1.8)*, such that f is a linear combination of two, or fewer, of its eigenfunctions*.

We modify *P* and *Q* in such a way that the functions *ψ*_{1} and *ψ*_{2} constructed in theorem 3.2 satisfy the same boundary condition of type (1.8). From the proof of theorem 3.2, it follows thatwhere *A*^{−1} is a non-singular matrix, and hence *ψ*_{1} and *ψ*_{2} satisfy the same boundary conditions if and only if *f* and *x* satisfy the same boundary conditions. Since the function *f* is given, the parameters in the boundary condition(3.2)(3.3)should be chosen appropriately. The function *P* is coupled with the function *x* but *α* and *β* could be arbitrary.

Let *x*_{l} denote a function *x* that is constructed in theorem 2.2 by starting from the end point *a*. We show that *x*_{l} can be constructed to satisfy (3.2). Depending on the values of *f* and *f*′ at *a*, we consider the following three possible cases.

*Case*(*i*). The function*f*does not have both a zero and an extrema at*a*. Then, (3.2) is equivalent to , and so*x*must satisfy .In the case

*a*∈*J*_{0}, from theorem 2.1, it follows that*x*_{l}constructed as satisfies , if*Φ*is the solution of the initial-value integral equation (A11) with initial condition*Φ*(*a*)=*T*and .In the case

*a*∈*I*_{0}, from theorem 2.1, it follows that the function*x*_{l}, constructed as the solution of the IVP (A2) with initial conditions*x*_{l}(*a*)=*f*(*a*) and =*f*′(*a*), satisfies (3.2).*Case*(*ii*). The function*f*does not have zero at*a*, but*f*′(*a*)=0. By choosing*α*=*π*/2 in (3.2), the initial condition for*f*becomes*f*′(*a*)=0 and the same considerations as in case (i) lead to a construction of*x*_{l}with =0 and any choice of*x*_{l}(*a*).*Case*(*iii*). The function*f*does not have extrema at*a*, but*f*(*a*)=0. In that case, we construct*P*with zero at*a*and positive elsewhere. Choosing*α*=*π*/2 in (3.2), we have a singular Sturm–Liouville system. Since*f*has simple zeros and extrema, it follows that*f*′(*a*)≠0 and*a*∈*I*_{0}. Furthermore, by repeating the same construction of*p*and*q*as in theorem 2.2 with boundary values*s*(*a*)=0 and*s*(*ρ*)=*v*, for an appropriate*ρ*and*v*, we obtain a continuously differentiable non-negative function*P*with its only zero value at the point*a*. The corresponding*Q*is a continuous positive function.

In a similar way, starting from *b*, a function *x*_{r} can be constructed that satisfies the boundary conditions (3.3). Finally, *x*_{l} and *x*_{r} can be connected smoothly on an appropriate interval as in theorem 2.2 to construct a function *x* that satisfies (3.1) and (3.2). The resulting weak IMFs *ψ*_{1} and *ψ*_{2} are eigenfunctions of a Sturm–Liouville system of type (1.8) with corresponding eigenvalues *ω*_{1} and *ω*_{2}. One immediate consequence is that *ψ*_{1} and *ψ*_{2} are orthogonal on *I* with a weight function *Q*. ▪

The main purpose of the EMD method is to decompose a function into a sequence of IMFs in order to extract important physical characteristics from each component by the application of the Hilbert transform. Next we discuss possible physical interpretations of the decompositions considered in theorems 3.1 and 3.2.

Let *SL*(*P*, *Q*) denote the set of all the eigenfunctions of Sturm–Liouville operators defined in (1.7). It is clear that *SL*(*P*, *Q*) contains the eigenfunctions of any Sturm–Liouville system. For any fixed boundary conditions, the system of corresponding eigenfunctions is complete in *L*_{2}(*I*, *Q*) and it follows that the set of functions *SL*(*P*, *Q*) is complete and redundant in *L*_{2}(*I*, *Q*). Furthermore, since the weight function *Q* is strictly positive, continuous on the finite interval *I*, then the space *L*_{2}(*I*, *Q*) is norm equivalent to the space *L*_{2}(*I*). Varying *P* and *Q* over all continuous positive functions on *I*, we observe from theorem 1.2 that .

Self-adjoint differential equations describe many physical processes and have embedded physical characteristics. A typical example is the periodic BVPthat defines the trigonometric system on the interval [0,2*π*). This differential equation models the frictionless displacement from equilibrium of a unit mass attached to a spring with a spring constant *ω*^{2}. If the ‘periodic’ boundary condition is replaced with an initial condition, then the SL operator *Lf*=−*f*″ generates trigonometric functions cos *ωt*, sin *ωt* that still obey the physical relation with mass–spring system, but is a redundant system. One method to decompose a function into a linear combination of elements from a redundant system is by using ‘greedy’-type algorithms (for reference see DeVore 1998 and Temlyakov 1999). Generally, the resulting representation consists of infinitely many terms.

In analogy with the trigonometric system, the characteristic equation for a weak IMF can be interpreted as the physical model according to which a variable mass *P*(*t*) is attached to a spring with a variable spring constant *Q*(*t*) and vibrates frictionlessly around its equilibrium position *ψ*(0). The solution *ψ*(*t*) is the displacement of the mass at the instant *t*. An equivalent form of the characteristic differential equation is(3.4)where *q*>0 and *p* is an arbitrary function. In that form the mass–spring interpretation could be that *ψ*(*t*) is the displacement of a unit mass attached to a spring with a variable spring constant *q* if the motion is subjected to a frictional force −*pψ*′. In both interpretations, or is considered the frequency of the motion. In the second case, *p* can be considered as the instantaneous bandwidth of *ψ*. The instantaneous quantities of *ψ* can be defined by using the Prüfer substitution (1.6).

Summarizing the results of this section, theorems 3.1 and 3.2 not only show the existence of the decomposition of a function *f* into finitely many weak IMFs, but they also provide an additional analytical resource to the Hilbert transform method in order to define instantaneous frequency and bandwidth by using the physical interpretation from a model system of differential equations.

## Acknowledgments

This work was supported in part by the AFRL and AFOSR MURI grant no. F49620-03-1-0381 and by the ARO grant no. W911NF-05-1-0227.

## Footnotes

- Received November 20, 2007.
- Accepted March 18, 2008.

- © 2008 The Royal Society