A Remark on Adaptive Decomposition for Nonlinear Time-frequency Analysis

LIU XU AND WANG HAI-NA

（School of Applied Mathematics，Jilin University of Finance and Economics，Changchun，130117）

A Remark on Adaptive Decomposition for Nonlinear Time-frequency Analysis

LIU XU AND WANG HAI-NA*

（School of Applied Mathematics，Jilin University of Finance and Economics，Changchun，130117）

In recent study the bank of real square integrable functions that have nonlinear phases and admit a well-behaved Hilbert transform has been constructed for adaptive representation of nonlinear signals.We first show in this paper that the available basic functions are adequate for establishing an ideal adaptive decomposition algorithm.However，we also point out that the best approximation algorithm，which is a common strategy in decomposing a function into a sum of functions in a prescribed class of basis functions，should not be considered as a candidate for the ideal algorithm.

Hilbert transform，empirical mode decomposition，adaptive decomposition algorithm，best approximation

2000 MR subject classification:46E2O，46A35

Document code:A

Article ID:1674-5647（2O16）O4-O319-O6

1　Introduction

An important objective for the time-frequency analysis of a signal is to obtain a timefrequency-energy distribution representing at each time its frequencies and the energy corresponding to each frequency.Instantaneous amplitude and frequency are basic concepts in the implementation of this objective.

A classical way of defining without ambiguity the instantaneous amplitude and frequency of a real signal f∈L2（R）is through the Hilbert transform，which is defined for each function g∈Lp（R），1≤p＜∞，at x∈R as

if the Cauchy principal value（p.v.for short）of the above singular integral exists（see［1］）. To see this，we form the analytic signal Af of f by letting

and then write Af as

where ρ≥O.If θ′≥O，then we define ρ（t），θ′（t）as the instantaneous amplitude and frequency of the signal f at time t，respectively.The quantity ρ（t）is further viewed as the energy of f at time t.To make the above method for defining instantaneous amplitude and frequency applicable，we need an adaptive algorithm A to decompose an arbitrary real signal into a monotone function and a sum of signals in the following class

We also require that the summand in M for each decomposition decay fast so that the algorithm A is useful in practice.

In engineering literature，the empirical mode decomposition（EMD）was proposed in［2］to decompose a real signal into a monotone function and a finite sum of functions called intrinsic mode functions（IMFs）.An IMF is defined in［2］as a real function ψ with the following two properties:

（a）ψ has exactly one zero between any two consecutive local extrema；

（b）the local mean of ψ is zero.

A basic assumption in the establishment of EMD and the Hilbert-Huang transform（HHT）in［2］is that properties（a）and（b）are an empirical sufficient condition for a signal ψ to belong to M.Along this direction of using the Hilbert transform to define instantaneous amplitude and frequency，the task of building the mathematical foundation for EMD consists of two stages（see［3］）.The first is to construct a large bank of functions in M with explicit expression.The second is to establish the ideal algorithm A described above.

Results on the construction of functions in M with explicit expression can be found in［4］and［5］.This study aims at a better understanding of the ideal algorithm A.As a first step，we shall prove in Section 2 that spanM is dense in the space of all the real functions in L2（R）and show in Section 3 that the best approximation，which is a common strategy in decomposing a function into a sum of functions in a prescribed class of basis functions，should not be considered as a candidate for the algorithm A.

2　Completeness of span

Set C+:=｛z∈C:Im（z）＞O｝.We let H（C+）be the space of all the holomorphic functions on C+and introduce the Hardy space H2（C+）by setting

Each f∈H2（C+）has a non-tangential boundary limit in L2（R），which we still denote as f.The Hardy space H2（C+）is a Hilbert space with the inner product〈.，.〉defined for each f，g∈H2（C+）as

The mapping by corresponding each f∈H2（C+）to its boundary limit is an isomorphism from H2（C+）to the subspace of all the functions in L2（R）whose Fourier transform is supported on R+:=［O，∞）.

We make use of an important property of outer functions in H2（C+）.A function f∈H2（C+）is called an outer function if for each z:=x+iy∈C+，x，y∈R，

It was proved in［8］that for each outer function f∈H2（C+）there holds

Theorem 2.1The space spanM is dense in the space of all the real functions in L2（R）.

For each real function f∈L2（R），the Fourier transform of f+iHf has support contained in R+（see［9］）.By the isomorphism mentioned above，we get that

Note that for each ξ∈R+，

As a matter of completeness of discussion，we remark that since

3　Limitations of the Best Approximation

Let H be a separable Hilbert space with the inner product denoted by〈.，.〉and the norm‖.‖defined at each h∈H as

A common method for decomposing an element h∈H into a sum of elements in ℇ is by the best approximation.To explain the method，we set h0:=h.The method works by successively finding for each j∈N，ij∈I and cj∈C such that

and then letting

provided that the infimum（3.1）is attainable at each iteration.In this way，we can successively extract modes in ℇ from h，that is，eio，ei1，....

If h is a finite linear combination of elements in ℇ having the form

where Nn:=｛1，...，n｝，｛ij:j∈Nn｝⊆I，｛cj:j∈Nn｝⊆C，we expect that the best approximation has the maximum-energy-extraction property with respect to ℇ，that is，the first mode eioextracted from h using the method described above satisfies that i0∈｛ij:j∈Nn｝and

This property says that the first mode in ℇ extracted from h should be the one whose corresponding summand in h has the largest norm.If ℇ is orthogonal，namely，

then it is clear that the best approximation has the maximum-energy-extraction property with respect to ℇ.We prove the converse.

Theorem 3.1If the best approximation has the maximum-energy-extraction property with respect to ℇ，then ℇ is orthogonal.

Proof.Suppose that the best approximation has the maximum-energy-extraction property with respect to ℇ.

We first show that for each i∈I there holds

We process by contradiction.Assume that there exists an i0∈I such that

Then there exists some finite set｛jk:k∈Nn｝⊆N+｛i0｝such that

∑

and

We introduce for each ε∈（O，1）an element hε∈H as

Note that for each f，g∈H，

By the maximum-energy-extraction property of the best approximation with respect to ℇ，

we have

Since ℇ is linear independent，the above inequality implies for each ε∈（O，1）that

However，we compute for each ε∈（O，1）that

We observe by（3.3）that there must exists some ε∈（O，1）such that

This contradiction implies that（3.2）is valid for each i∈I.

Suppose that there exists a pair of distinct elements i1，i2∈I such that

By（3.2），we have

We define，for each ε∈（O，1）and j∈N+｛i1，i2｝，an element hε，j∈H as

The maximum-energy-extraction property of the best approximation with respect to ℇ implies for each ε∈（O，1）and j∈N+｛i1，i2｝that

It follows from the equation above that

We let j tend to infinity and observe by（3.4）that for each ε∈（O，1），

The above equation holds for each ε∈（O，1）if and only if

This contradiction proves the theorem.

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1O.13447/j.1674-5647.2O16.O4.O3

date:June 27，2014.

The NSF（11426113）of China，the Science and Technology Research Project（2014161，2014163）of Jilin Provincial Department of Education of China，and the Project（20160520110JH）of Science and Technology Development Plan for Jilin Provincial.

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Communicated by Ma Fu-ming