On the Uniqueness of Sparse Time-Frequency Representation of Multiscale Data

0.24 MB·English

by  Chunguang Liu

On the Uniqueness of Sparse Time-Frequency Representation of
Multiscale Data
Chunguang Liu ∗ Zuoqiang Shi † Thomas Y. Hou ‡
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1 May 8, 2015
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2

y
a
M Abstract

7 In this paper, we analyze the uniqueness of the sparse time frequency decomposition and

investigate the efficiency of the nonlinear matching pursuit method. Under the assumption
]
T of scale separation, we show that the sparse time frequency decomposition is unique up to
I
. an error that is determined by the scale separation property of the signal. We further show
s
c that the unique decomposition can be obtained approximately by the sparse time frequency
[
decomposition using nonlinear matching pursuit.

2 Keywords. sparse time frequency decomposition; scale separation; nonlinear matching
v
pursuit.
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0
7
4 1 Introduction
0
.
1 Nowadays, data play more and more important role in our life. It has become increasingly
0
important to develop effective data analysis tools to extract useful information from massive
5
1 amount of data. Frequency is one of the most important features for oscillatory data. In
:
v many physical problems, frequencies encode important information of the underlying physical
i
X mechanism. Manytime-frequencyanalysismethodshavebeendevelopedtoextractinformation
r offrequenciesandthecorrespondingamplitudesfromthemeasurementofsignals. Theseinclude
a
the windowed Fourier transform, the wavelet transform [5, 12], the Wigner-Ville distribution
[10],andtheEmpiricalModeDecomposition(EMD)method[9,17]. Amongthesedifferenttime-
frequency analysis methods, the EMD method provides an efficient adaptive method to extract
frequency information from nonlinear and nonstationary data and it has been successfully used
in many applications. However, due to its empirical nature, the EMD method still lacks a
rigorous mathematical foundation. Recently, a number of attempts have been made to provide
a mathematicalfoundation for this method, see e.g. the synchrosqueezedwavelettransform[6],
the Empirical wavelet transform [8], the variational mode decomposition [11].
In the last few years, inspired by the EMD method and compressive sensing [1, 2, 7], Hou
and Shi proposed a novel time frequency analysis method based on the sparsest representation
∗Department of Mathematics, Jinan University,Guangzhou, China, 510632. Email: [email protected].
†MathematicalSciencesCenter,TsinghuaUniversity,Beijing,China,100084. Email: [email protected].
‡Applied and Comput. Math, MC 9-94, Caltech, Pasadena, CA 91125. Email: [email protected].
1
ofmultiscaledata[14]. Inthismethod,thesignalisdecomposedintoafinitenumberofintrinsic
mode functions with a small residual:
M
f(t)= a (t)cosθ (t)+r(t), t R, (1.1)
j j
∈
j=1
X
where a (t), θ (t) > 0, j = 1, ,M and r(t) is a small residual. We assume that a (t) and
j j′ ··· j
θ are less oscillatory than cosθ (t). What we mean by “less oscillatory” will be made precise
j′ j
below by the definition of scale separation property of the signal. Borrowing the terminology
from the EMD method, we call a (t)cosθ (t) the Intrinsic Mode Function (IMF) [9]. After the
j j
decomposition is obtained, the instantaneous frequencies ω (t) are defined as
j
ω (t)=θ (t), (1.2)
j j′
and the amplitude is given by a (t).
j
One main difficulty in computing the decomposition (1.1) is that the decomposition is not
unique. To pick up the ”best” decomposition among all feasible ones, Hou and Shi proposed
to decompose the signal by looking for the sparsest decomposition by solving the following
nonlinear optimization problem:
Minimize M
(ak)1≤k≤M,(θk)1≤k≤M
(1.3)
M
Subject to: f akcosθk L2 ǫ, akcosθk ,
k − k ≤ ∈D
k=1
X
where ǫ is the noise level and is the dictionary consist of all IMFs (see [14] for its precise
D
definition).
The idea of looking for the sparsest representation over the time frequency dictionary has
beenexploitedextensivelyinthesignalprocessingcommunity,see,e.g. [13,3]. Comparingwith
other existing methods, the novelty of the method proposed by Hou and Shi is that the time
frequency dictionary being used is much larger. This method has the advantage of being fully
adaptive to the signal.
The optimization problem (1.3) is nonlinear and nonconvex. It is challenging to solve this
nonlinear optimization problem efficiently. To overcome this difficulty, Hou and Shi proposed
an efficient algorithm based on nonlinear matching pursuit and Fast Fourier transform to solve
the above nonlinear optimization problem. In a subsequent paper [16], the authors proved the
convergenceoftheir nonlinear matching pursuit algorithmfor periodic data that satisfy certain
scale separationproperty. Further, they have demonstrated the effectiveness of this method by
decomposing realistic signals arising from various applications. However, from the theoretical
point of view, one important question remains open. That is the uniqueness of the solution of
the optimization problem (1.3). This is precisely the main focus of this paper.
In this paper, we will show that under the assumption of scale separation, the solution of
optimizationproblem(1.3)isuniqueuptoanerrordeterminedbythescaleseparationproperty.
First, we give a precise definition of scale separation of a signal as follows:
Definition 1.1 (scale-separation). One function f(t) = a(t)cosθ(t) is said to satisfy a scale-
separation property with a separation factor ǫ > 0, if a(t) and θ(t) satisfy the following condi-
2
tions:
a(t) C1(R), θ C2(R), inf θ′(t)>0,
∈ ∈ t R
∈
Inthe abovedseiunfipfntt∈i∈tRiRoθθn′′,((ttt))he=fiMrst′t<hr+ee∞a,ssum(cid:12)(cid:12)(cid:12)(cid:12)aθp′′((ttti))o(cid:12)(cid:12)(cid:12)(cid:12)ns≤aǫr,eo(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)n(θθt′′h(′(tet)))r2e(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)gu≤laǫr,ity∀otf∈thRe.envelopea(t)
andthe instantaneousfrequencyθ (t). This regularityis relativelymild andcanbe relaxedtoa
′
piecewise smooth function. The key assumptions on the scale separation property are the last
two assumptions. The assumption a′(t) ǫ quantifies what we mean by the envelope is less
θ′(t) ≤
oscillatory than the normalized sign(cid:12)(cid:12)(cid:12)al co(cid:12)(cid:12)(cid:12)s(θ(t)). The last assumption (θθ′′(′t()t))2 ≤ ǫ essentially
saysthat the frequency oscillationis relativelyweakcomparedwith the(cid:12)square(cid:12)ofthe frequency
(cid:12) (cid:12)
itself. Inthe previoustheoreticalstudy oftime-frequency analysis,a mu(cid:12)chstron(cid:12)gerassumption
on the frequency oscillation is made, i.e. θ′′ ǫ, see e.g. [6, 4]. In the problems that we
θ′ ≤
consider, the instantaneous frequency θ′(t)(cid:12)is t(cid:12)ypically quite large. Therefore, the assumption
θ′′(t) ǫ is much weaker than the typica(cid:12)(cid:12)l a(cid:12)(cid:12)ssumption, θ′′ ǫ.
(θ′(t))2 ≤ θ′ ≤
(cid:12) With(cid:12) the definition of scale separation, we construct t(cid:12)he(cid:12)dictionary ǫ by putting all the
(cid:12) (cid:12) (cid:12) (cid:12) D
f(cid:12)unction(cid:12)s satisfying scale separation together. (cid:12) (cid:12)
:= a()cosθ():(a,θ) U , (1.4)
ǫ ǫ
D { · · ∈ }
where
Uǫ :=(cid:26)(a,θ):a>0,θ′ >0; (cid:12)aθ′′(cid:12)≤ǫ, (cid:12)[θθ′′]′2(cid:12)≤ǫ, siunpftt∈∈RRθθ′′((tt)) =M′ <+∞(cid:27). (1.5)
(cid:12) (cid:12) (cid:12) (cid:12)
is the dictionary that we will (cid:12)use(cid:12)to rep(cid:12)resen(cid:12)t the signal. For any given signal f(t), we
Dǫ (cid:12) (cid:12) (cid:12) (cid:12)
decompose f over the dictionary by looking for the sparsest representation:
ǫ
D
Minimize M
M
subject to f(t) a (t)cosθ (t) ǫ , (P )
k k 0 0
| − |≤
k=1
X
a (t)cosθ (t) , k =1,2, ,M.
k k ǫ
∈D ···
Here ǫ is a giventhreshold of the accuracyof the decomposition. Typically, ǫ is set according
0 0
to the amplitude of noise.
To get the uniqueness, we need to assume that the signal f(t) is well separated which is
defined in Definition 1.2.
Definition 1.2 (Well-separated signal). A signal f : R R is said to be well-separated with
→
separation factor ǫ and frequency ratio d if it can be written as
M
f(t)= a (t)cosθ (t)+r(t) (1.6)
k k
k=1
X
where all f (t) = a (t)cosθ (t) satisfies the scale-separation property with separation factor ǫ,
k k k
r(t)=O(ǫ ) and their phase functions θ satisfy
0 k
θ (t) dθ (t), t R, (1.7)
k′ ≥ k′−1 ∀ ∈
and d>1, d 1=O(1).
−
3
In the above definition, the key measurement of separation among different components is
thevalued,whichmeasuresthefrequencygapamongdifferentcomponents. Forawellseparated
signal,we canprovethatthe solutionofthe optimizationproblem(P )is unique upto anerror
0
O(ǫ). The main result is summarized in Theorem 1.1.
Theorem 1.1. Let f(t) be well separated with separation factor ǫ 1 and frequency ratio d as
≪
definedinDefinition1.2. Then(a ,θ ) isanoptimalsolutionoftheoptimization problem
k k 1 k M
≤ ≤
(P ) and it is unique up to the error ǫ, i.e. if a˜ ,θ˜ is another optimal solution of
0 k k
1 k M˜
(P ), then M˜ =M and (cid:16) (cid:17) ≤ ≤
0
θ (t) θ˜ (t)
k k
a (t) a˜ (t) =O(ǫ), | − | =O(ǫ), t, k =1, ,M. (1.8)
k k
| − | θ (t) ∀ ···
k′
ThistheoremisprovedbycarefullystudyingthewavelettransformofeachIMF.Thedetails
of the proof can be found in Section 2.
Weremarkthattherehasbeensomeveryniceprogressindevelopingamathematicalframe-
work for an EMD like method using synchrosqueezed wavelet transforms [6] and windowed
Fouriertransform[4]. Thesemethods arebasedoncontinuouswavelettransformandwindowed
Fourier transform respectively and do not look for the sparsest decomposition directly. So the
question of uniqueness is not the same as that we consider here in this paper.
The rest of the paper is organized as follows. The uniqueness of the sparse time-frequency
decompositionisanalyzedinSection2. InSection3,weanalyzetheperformanceofanalgorithm
based on matching pursuit. Some concluding remarks are made in Section 4. We defer a few
technical lemmas to the appendices.
2 Uniqueness of P for well-separated signals
0
In this section, we assume that the signal, f(t), satisfies the scale-separation property with
separation factor ǫ and frequency ratio d as defined in Definition 1.2.
For this kind of signals, the existence of the solution of (P ) is obvious. Since f(t) already
0
has a representation,
M
f(t)= a (t)cosθ (t)+r(t), (2.1)
k k
k=1
X
this gives a feasible decomposition. Each feasible decomposition gives a positive integer, by
collectingallthesepositiveintegerstogether,wegetasetA. ThenweknowthatAisnonempty
and has a lower bound. Let M =infA. Since A consists of positive integers, then M =infA
0 0
can be achieved. Then the existence of the solution of (P ) is proved.
0
But the uniqueness is much more complicated. In the next subsection, we will prove that if
f is a well-separated signal with separation factor ǫ and frequency ratio d>1 and ǫ 1, then
≪
the solution of (P ) is unique up to ǫ and ǫ .
0 0
To simplify the notation, in the rest of this paper, we assume ǫ and ǫ have the same order
0
and denote both of them by ǫ.
We prove Theorem 1.1 by carefully studying the wavelet transform of each IMF. As we
show in Lemma 2.1, the continuous wavelet transform (CWT) of each IMF is confined in a
4
narrow band by properly choosing an appropriate wavelet function. Then the uniqueness can
be obtained by comparing the continuous wavelet transform of different decompositions.
In order to complete the proof, first, we need to estimate the width of CWT of each IMF.
This estimation is given in the following lemma.
Lemma 2.1. let ψ is a wavelet function such that
I = ψ(τ)dτ <+ , I = τψ (τ)dτ <+ , I = τ2ψ (τ)dτ <+ .
1 2 ′ 3 ′′
R| | ∞ R| | ∞ R| | ∞
Z Z Z
Suppose (a,θ) U , i.e.
ǫ
∈
a′ ǫ, θ′′ ǫ, supt∈Rθ′(t) =M′ <+
(cid:12)θ′(cid:12)≤ (cid:12)[θ′]2(cid:12)≤ inft∈Rθ′(t) ∞
(cid:12) (cid:12) (cid:12) (cid:12)
then we have (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1 τ t
(ae−iθ)(t,ω)= a(τ)e−iθ(τ)ψ − dτ =√ωa(t)e−iθ(t)ψ(ωθ′(t))+C√ωǫ
W √ω R ω
Z (cid:18) (cid:19)
b
where C =(A+4|a(t)|+1)I1+[M′+(M′+1)|a(t)|]I2+M′|a(t)|I3 and A=supt∈R|a(t)|.
The proof of this lemma can be found in Appendix A.
Remark 2.1. One sufficient condition to make sure that I ,I ,I < + is that the Fourier
1 2 3
∞
transform of the wavelet, ψ C4(R) and has compact support. In the proof of Theorem 1.1,
∈
we use the wavelet whose Fourier transform is a fifth order B-spline function such that above
b
sufficient condition is satisfied.
2.1 Proof of Theorem 1.1
Now, we are ready to give the proof of Theorem 1.1.
Proof. of Theorem 1.1
The details of the proof may be a bit tedious but the idea is very clear. First, we assume
there are two decompositions:
M M˜
f(t)= a (t)cosθ (t)+O(ǫ)= a˜ (t)cosθ˜ (t)+O(ǫ). (2.2)
k k k k
k=1 k=1
X X
Using Lemma 2.1, we have
M M˜
ω 1/2 (f)(t,ω)= a (t)e iθk(t)ψ(ωθ (t))+O(ǫ)= a˜ (t)e iθ˜k(t)ψ(ωθ˜ (t))+O(ǫ).(2.3)
− W k − k′ k − k′
k=1 k=1
X X
b b
By analyzing the support of (f)(t,ω), we can get M =M˜ and
W
ak(t)e−iθk(t)ψ(ωθk′(t))=a˜k(t)e−iθ˜k(t)ψ(ωθ˜k′(t))+O(ǫ), k =1,··· ,M. (2.4)
Using these equalities,bwe can show that (1.8) ibs true.
First,wepickupaspecificwaveletfunctionψ. WerequirethatitsFouriertransformψ C4
∈
has support in [1 ∆,1+∆] with 0<∆< √d 1 and ψ(1)=1 is the maximum of ψ . In this
− √d−+1 | | b
proof, we choose ψ to be a fifth order B-spline function after proper scaling and translation.
b b
b
5
For any θ C1, define
∈
U = (t,ω) R2 : ψ(ωθ (t)) >ǫ , U (t)= ω R: ψ(ωθ (t)) >ǫ .
θ ′ θ ′
∈ | | ∈ | |
n o n o
Now, fix a time t=t [0b,1], for any l 1, ,M , let ω =b 1 . Using (2.3), we have
0 ∈ ∈{ ··· } l,0 θl′(t0)
M
ω 1/2 (f)(t ,ω ) = a (t )e iθk(t0)ψ(θ (t )/θ (t ))+O(ǫ)
− W 0 l,0 k 0 − k′ 0 l′ 0
k=1
X
M˜ b
= a˜k(t0)e−iθ˜k(t0)ψ(θ˜k′(t0)/θl′(t0))+O(ǫ). (2.5)
k=1
X
b
Since θ (t )/θ (t ) 1/d < 1 ∆ or θ (t )/θ (t ) d > 1+∆ for any k = l, in the first
k′ 0 l′ 0 ≤ − k′ 0 l′ 0 ≥ 6
summation of the above equation, we have only one term left,
M˜
a (t )e iθl(t0)ψ(1)= a˜ (t )e iθ˜k(t0)ψ(θ˜ (t )/θ (t ))+O(ǫ). (2.6)
l 0 − k 0 − k′ 0 l′ 0
k=1
X
b b
Then, there exists at least a I(l,t ) 1, ,M˜ , such that
0
∈{ ··· }
ψ(θ˜ (t )/θ (t )) >0, (2.7)
I′(l,t0) 0 l′ 0
(cid:12) (cid:12)
which means that 1−∆ < θ˜I′(l,(cid:12)(cid:12)t0b)(t0)/θl′(t0) < 1+(cid:12)(cid:12)∆. Using the assumption that f is well-
separated with frequency ratio d and 0<∆< √d 1, for any k =l, we obtain
√d−+1 6
θ (t ) θ (t ) 1
k′ 0 d, or k′ 0 . (2.8)
θ (t ) ≥ θ (t ) ≤ d
l′ 0 l′ 0
This gives that
θ˜I′(l,t0)(t0) = θ˜I′(l,t0)(t0) θl′(t0) d(1 ∆)>1+∆, (2.9)
θ (t ) θ (t ) · θ (t ) ≥ −
k′ 0 l′ 0 k′ 0
or
θ˜I′(l,t0)(t0) = θ˜I′(l,t0)(t0) θl′(t0) 1+∆ <1 ∆. (2.10)
θ (t ) θ (t ) · θ (t ) ≤ d −
k′ 0 l′ 0 k′ 0
Then, for any k =l, we have
6
ψ(θ˜ (t )/θ (t )) =0. (2.11)
I′(l,t0) 0 k′ 0
(cid:12) (cid:12)
This implies that I(k,t )=I(l,t(cid:12)b), k =l. Then we g(cid:12)et that
0 (cid:12)0 (cid:12)
6 6
M˜ M. (2.12)
≥
Since a˜ ,θ˜ is a solution of (P ), we also have M˜ M. This implies that
k k 0
1 k M˜ ≤
(cid:16) (cid:17) ≤ ≤
M˜ =M, (2.13)
and for any t [0,1], I(,t) : 1, ,M 1, ,M is a one to one map. Then, we can
∈ · { ··· } → { ··· }
define its inverse map I 1(,t): 1, ,M 1, ,M .
−
· { ··· }→{ ··· }
6
For any k = 1, ,M, we study the function I 1(k, ) : [0,1] 1, ,M . Using the
−
··· · → { ··· }
condition that θ˜′′ ǫ and the signal f is well-separated, it is easy to see that I 1(k, ) is a
(θ˜′)2 ≤ − ·
constant over [0,1], i.e.
I 1(k,t)=I 1(k,0), t [0,1], k =1, ,M. (2.14)
− −
∀ ∈ ···
Otherwise, suppose there exists t [0,1], such that I 1(k,t ) = I 1(k,0). Let A = 0 t
0 − 0 −
∈ 6 { ≤ ≤
t : I 1(k,t ) = I 1(k,0) and ξ = supA. Then for any η > 0, there exist t ,t [0,1], such
0 − 0 − 1 2
} ∈
that
t <ξ <t , t t <η, I 1(k,0)=I 1(k,t )=I 1(k,t ).
1 2 2 1 − − 1 − 2
| − | 6
Denote
I−1(k,t1)=k1, I−1(k,t2)=k2, k1 =k2. (2.15)
6
Then, by the definition of the map I(k,t), we have
θ˜ (t ) θ˜ (t )
1 ∆< k′ 1 <1+∆, 1 ∆< k′ 2 <1+∆.
− θ (t ) − θ (t )
k′1 1 k′2 2
Without loss of generality, we assume that θ >θ . Then, we have
k′2 k′1
θ˜ (t )<(1+∆)θ (t ), θ˜ (t )>(1 ∆)θ (t ).
k′ 1 k′1 1 k′ 2 − k′2 1
Now, let η 0, have t ,t ξ, which gives
1 2
→ →
θ˜k′(ξ)≤(1+∆)θk′1(ξ), θ˜k′(ξ)≥(1−∆)θk′2(ξ).
On the other hand, since f is well-separated, we know that
θ (t)
k′1 1/d.
θ (t) ≤
k′2
Then, we have
θ˜ (ξ) (1+∆)θ (ξ), θ˜ (ξ) d(1 ∆)θ (ξ)>(1+∆)θ (ξ),
k′ ≤ k′1 k′ ≥ − k′1 k′1
which is a contradiction. This means that I 1(k, ) is a constant over [0,1] and we can assume
−
·
I 1(k,t)=k, t [0,1], k=1, ,M. (2.16)
−
∀ ∈ ···
Now we have that
θ˜ (t)
1 ∆< k′ <1+∆, t [0,1], k=1, ,M. (2.17)
− θ (t) ∀ ∈ ···
k′
Using the assumption that the signal f is well-separated with ratio d and the choice of ψ, we
have that for any k, l =1, ,M, k =l
··· 6
ψ(ωθl′(t))=ψ ωθ˜l′(t) =0, ∀(t,ω)∈Uθk, (2.18)
(cid:16) (cid:17)
ψb(ωθl′(t))=ψb ωθ˜l′(t) =0, ∀(t,ω)∈Uθ˜k. (2.19)
(cid:16) (cid:17)
b b
7
On the other hand, using Lemma 2.1 we have
M M
ω 1/2 (f)(t,ω)= a (t)e iθk(t)ψ(ωθ (t))+O(ǫ)= a˜ (t)e iθ˜k(t)ψ(ωθ˜ (t))+O(ǫ).(2.20)
− W k − k′ k − k′
k=1 k=1
X X
b b
Using the three relations (2.18),(2.19) and (2.20), we have
ak(t)e−iθk(t)ψ(ωθk′(t))−a˜k(t)e−iθ˜k(t)ψ(ωθ˜k′(t)) =O(ǫ), ∀(t,ω)∈Uθk Uθ˜k, (2.21)
(cid:12) (cid:12) [
which(cid:12)implies thatb b (cid:12)
(cid:12) (cid:12)
a (t)ψ(ωθ (t)) a˜ (t)ψ(ωθ˜ (t)) =O(ǫ), (t,ω) U U . (2.22)
k k′ − k k′ ∀ ∈ θk θ˜k
(cid:12) (cid:12) (cid:12) (cid:12) [
Next, we wil(cid:12)(cid:12)l provebthat |ak(cid:12)(cid:12)(t)−(cid:12)(cid:12) a˜k(t)b|=O(ǫ)(cid:12)(cid:12)and |θk′(t)−θ˜k′(t)|=O(ǫ) from (2.22). First, we
consider the envelopes a and a˜ .
k k
If a (t)>a˜ (t), we choose ω =1/θ (t) to get
k k k′
a (t) ψ(1) a˜ (t) ψ(θ˜ (t)/θ (t)) =O(ǫ), (2.23)
k − k k′ k′
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)b (cid:12) (cid:12)b (cid:12)
where we have used the fact(cid:12)that(cid:12)ψ(ξ) ac(cid:12)hieves its max(cid:12)imum at ξ = 1. Since a (t) > a˜ (t),
k k
we have (cid:12) (cid:12)
(cid:12)b (cid:12)
(cid:12) (cid:12)
0 ψ(1) (a (t) a˜ (t)) a (t) ψ(1) a˜ (t) ψ(θ˜ (t)/θ (t)) =O(ǫ). (2.24)
≤ k − k ≤ k − k k′ k′
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
This proves th(cid:12)abt (cid:12) (cid:12)b (cid:12) (cid:12)b (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
a (t) a˜ (t)=O(ǫ). (2.25)
k k
−
If a (t)<a˜ (t), we take ω =1/θ˜ (t). By following a similar argument, we can prove
k k k′
a˜ (t) a (t)=O(ǫ). (2.26)
k k
−
Combining these two cases, we obtain
a (t) a˜ (t) =O(ǫ). (2.27)
k k
| − |
Substituting the above relation to (2.22), we get
ψ(ωθ (t)) ψ(ωθ˜ (t)) =O(ǫ), (t,ω) U U . (2.28)
k′ − k′ ∀ ∈ θk θ˜k
(cid:12) (cid:12) (cid:12) (cid:12) [
For any t∈[0,1],(cid:12)(cid:12)lbet ω =(1(cid:12)(cid:12)−∆(cid:12)(cid:12)b/2)/θk′(t)(cid:12)(cid:12), then we have
ψ(1−∆/2)) − ψ (1−∆/2)θ˜k′(t)/θk′(t) =O(ǫ). (2.29)
(cid:12) (cid:12) (cid:12) h i(cid:12)
Since ψ isafifthord(cid:12)(cid:12)ebrB-Splinefu(cid:12)(cid:12)nct(cid:12)(cid:12)ibonandǫ 1,itiseasyt(cid:12)(cid:12)o seethatthereexistsaconstant
≪
C >0, such that
b
(1 ∆/2) θ˜ (t)/θ (t) 1 Cǫ. (2.30)
− k′ k′ − ≤
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
Since ψ(ω) is a fifth order B(cid:12)-Spline function, ψ(ω) is sy(cid:12)mmetric with respect to ω = 1. Thus
we have ψ(1 ∆/2)=ψ(1+∆/2). Then, there is also another possibility:
b − b
b b(1−∆/2)θ˜k′(t)/θk′(t)−(1+∆/2) ≤Cǫ. (2.31)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 8 (cid:12)
If (2.30) holds, then we get
θ (t) θ˜ (t)
| k′ − k′ | =O(ǫ). (2.32)
θ (t)
k′
If (2.31) holds, then we have
θ˜ (t) 1+∆/2 Cǫ
k′ − 1+∆, (2.33)
θ (t) ≥ 1 ∆/2 ≥
k′ −
where we have used the assumption that ǫ 1.
≪
Then, let ω =1/θ (t) in (2.28), we have
k′
1= ψ(1) ψ θ˜ (t)/θ (t) =O(ǫ). (2.34)
− k′ k′
(cid:12) (cid:12) (cid:12) (cid:16) (cid:17)(cid:12)
This argument shows that (2.3(cid:12)1b) ca(cid:12)nno(cid:12)tbbe true. This c(cid:12)ompletes the proof.
(cid:12) (cid:12) (cid:12) (cid:12)
2.2 Discussion on Signals with close frequencies
In this section, we will give a brief discussion on the signal with close frequencies, i.e. the
frequency ratio d 1 in the definition of well-separated signal.
→
IntheproofofTheorem1.1,thefrequencyratiodseemstobearbitrary,aslongasitislarger
than 1. Actually, the distance between d and1 can notbe too small. The gapis determined by
the separationfactorǫ. First,itiseasyto showthatthe integralsI ,I ,I inLemma2.1satisfy
1 2 3
I1 =O(∆−1), I2 =O(∆−2), I3 =O(∆−3). (2.35)
Whendapproaches1,∆= √d 1 becomessmaller,thenI ,I ,I becomelarger. WhenI ,I ,I
√d−+1 1 2 3 1 2 3
is as large as 1/ǫ, Lemma 2.1 breaks down, which in turn leads to the break down of the proof
of Theorem 1.1.
Ontheotherhand,thefrequencyratioisdefinedpointwisely. Aslongasthereexistsapoint,
suchthatthefrequencyratioatthispointisawayfrom1comparingwithǫ,theargumentbefore
(2.13)oftheproofofTheorem1.1stillapplies,whichmeansthatthedecompositiongivenin(1.6)
is still an optimal solution of the optimization problem (P ). But in this case, the uniqueness
0
is not guaranteed. To illustrate this point, we construct an example such that the solution of
(P ) is not unique.
0
f(t)=cosθ (t)+cosθ (t), t [0,1], (2.36)
1 2
∈
θ (t)=6πkt+kπ, θ (t)=8πkt+ksin2πt, (2.37)
1 2
where k is a positive integer. We can choose k large enough, such that cosθ ,cosθ U . At
1 2 ǫ
∈
t = 0, θ (0)/θ (0) = 4/3. From the discussion above, we know that (2.36) gives a solution of
2′ 1′
(P ). On the other hand, it is easy to check that the following decomposition is also a solution
0
of (P ).
0
f(t)=cosφ (t)+cosφ (t), t [0,1], (2.38)
1 2
∈
6πkt+kπ, t [0,1/2], 8πkt+ksin2πt, t [0,1/2],
φ (t)= ∈ φ (t)= ∈
1 2
( 8πkt+ksin2πt, t (1/2,1], ( 6πkt+kπ, t (1/2,1].
∈ ∈
9
Thisexampleshowsthatwhenthefrequenciesintersect,thesolutionof (P )maynotbeunique.
0
Inthiscase,weneedtoimposesomeextraconstraintstoobtainuniquenessofthedecomposition.
One natural idea is to pick up the solution according to the regularity of a(t) and θ (t). The
′
solutionwithsmootheramplitude andfrequency isfavorable. One methodbasedonthis idea is
proposed in [15] to decompose signals that do not have well-separated IMFs. In that method,
the regularity is related with the sparsity over Fourier (or Wavelet) dictionary. In the above
example, this method would prefer the decomposition (2.36), since in this decomposition, both
of the amplitude and the frequencies are very sparse over the Fourier dictionary.
3 Optimal Solutions to the P Problem
2
Using the result in the previous section, we know that for signals that are well-separated, the
solution of (P ) is unique up to the separation factor ǫ. One natural question is that how to
0
find this unique solution.
Basedonmatchingpursuit, weproposethe followingalgorithmtosolve(P )approximately.
0
r =f, k=1.
0
•
Step 1: Solve the following nonlinear least-square problem:
(a ,θ ) Argmin r acosθ 2
k k ∈ a,θ k k−1− kl2 (3.1)
Subject to: a,θ U .
ǫ
∈
Step 2: Update the residual
k
r =f a cosθ . (3.2)
k j j
−
j=1
X
Step 3: If rk l2 <ǫ0, stop. Otherwise, set k =k+1 and go to Step 1.
k k
Intheabovealgorithmbasedonmatchingpursuit,eachIMFisgivenbysolvingthefollowing
nonlinear least-square problem:
Minimize p(a,θ):= f(t) a(t)cosθ(t) 2 .
k − kL2 (P )
2
subject to (a,θ) U .
ǫ
∈
We wouldliketoknowthatunderwhatconditionsoneachIMFa (t)cosθ (t)off(t)theabove
k k
nonlinearleast-squareproblemcouldprovidealocal(approximate)optimizertotheP problem.
0
In the computations, we always deal with signals with finite time span. Without loss of
generality, in this section we assume the time span of the signal is [0,1]. For the signal with
finitelength,itisunavoidabletointroducelargererrorsneartheendpointsofthetimeinterval.
This is also known as the “end effect”. To simplify the analysis, we assume that the signal is
periodic over [0,1]. For those signals that are not periodic, we first multiply a cutoff function
to make the signalvanish near the boundary and then treat it as a periodic signal. This means
that the analysis in this section is valid only in the interior region away from the boundary.
For a periodic signal, we prove that each IMF is a local minimizer of (P ) as long as the
2
signal satisfies some assumptions. This result is stated in Theorem 3.1.
10