Digital Signal Processing 29 (2014) 138–146

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Digital Signal Processing

www.elsevier.com/locate/dsp

A novel envelope model based on convex constrained optimization ✩

Lijun Yang a, Zhihua Yang b, Feng Zhou c, Lihua Yang c,∗a School of Mathematics and Information Sciences, Henan University, Chinab Information Science School, Guangdong University of Finance & Economics, Chinac Institute of Computing Science and Computer Application, Guangdong Province Key Laboratory of Computational Science, School of Mathematics andComputational Science, Sun Yat-sen University, China

a r t i c l e i n f o a b s t r a c t

Article history:Available online 7 March 2014

Keywords:Instantaneous frequencyMonocomponent signalEnvelopeUndershootConvex constrained optimization

The concept of envelope has been used widely in signal analysis. However a good mathematical definitionof suitable envelope remains an issue. In this paper, we present a novel model to estimate the envelopeof a signal by using the convex constrained optimization. This model is based on the commonly acceptedknowledge about envelope which makes it coincide with the geometric envelope of the signal. Theenvelope based on the new model is smooth and has no undershoots. Experiments comparing with theexisting typical models of envelope are also implemented and discussed.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Fourier transform cannot represent the time-varying frequencyof non-stationary signals since it transforms signals in the wholetime domain. The concept of instantaneous frequency (IF) was pro-posed to characterize the time-varying frequency of non-stationarysignals. It is still far from well-defined even though it has beenstudied for more than a half century since the work by D. Ga-bor [5], in which the IF is defined as follows: Given a signal s(t),find its Hilbert transform Hs(t) and define the analytic signal (AS)as

z(t) = s(t) + jHs(t) = u(t)e jθ(t).

Then u(t) and θ(t) are respectively called the analytic envelope(AE) and the instantaneous phase, whose derivative θ ′(t) is ac-cordingly defined as the IF of the signal, called Gabor’s IF. Thisconcept gives physically meaningful characterization for narrow-band signals, such as cos(ωt + θ0), where ω and θ0 are realconstants [20]. But for general signals, it may lead to physicalparadoxes due to the multi-component frequencies of the signal[3]. To obtain reasonable frequencies, the signal should be de-composed into the so-called “monocomponent signals”. Both theclassical Fourier expansion and the Empirical Mode Decomposition(EMD), a fully data-driven method for adaptive signal decompo-

✩ This research was partially supported by NSFC (Nos. 11371017, 11071261,91130009), RFDP (No. 20130171110016), Project of DEGP (No. 2012KJCX0055) andthe program for the Guangdong Province Computational Science Innovative Re-search Team.

* Corresponding author.E-mail address: mcsylh@mail.sysu.edu.cn (L. Yang).

http://dx.doi.org/10.1016/j.dsp.2014.02.0171051-2004/© 2014 Elsevier Inc. All rights reserved.

sition [11], are such algorithms. However, it is very difficult tounderstand and give a rigorous and reasonable definition for theconcept “monocomponent” [2,18,9].

From the physical point of view, “monocomponent signal”should have single frequency ω(t) at any instant t . Then its phaseat time t is θ(t) = ∫ t

−∞ ω(τ)dτ . Let its magnitude at instant tbe u(t). Then the monocomponent signal should have the forms(t) = u(t) cos θ(t). Our question is: Given a monocomponent sig-nal s(t), how to find its amplitude u(t) and phase θ(t) such thats(t) = u(t) cos θ(t). This process is called the demodulation of s(t).Intuitively, the amplitude u(t) is the envelope of the signal. Ifit can be solved from s(t), then the phase can be obtained asθ(t) = arccos(s(t)/u(t)).

Thus, the demodulation of signals comes down to compute theenvelope of the signal. The concept of envelope is fundamental andused extensively in signal processing. As is known, envelope esti-mation is a key step in the EMD approach, in which the upper andlower envelopes are computed to produce the mean of the signaliteratively. The model for envelope is not sole and the understand-ing to envelope differs from man to man according to their profes-sional backgrounds and practical applications. In the past decades,researchers have contributed a lot in the theory and algorithms forenvelope. Besides the model of analytic envelope, there are anothertwo kinds of envelope detectors: one is the amplitude modula-tion detection used in communications, which involves squaringthe input signal and sending this signal through a lowpass fil-ter to recover the information-bearing signal and eliminate thecarrier of higher frequency from the modulated signal [15]. Thisrecovered signal is a kind of envelope and called the Squaring andLowpass Filtering Envelope (SLFE) for simplicity in this paper; an-ther kind of envelope uses the spline functions to interpolate the

L. Yang et al. / Digital Signal Processing 29 (2014) 138–146 139

peaks of the signal, a typical model of this kind is the cubic splineenvelope (CSE) presented by Huang et al. [11]. A drawback of theCSE is that the undershoots usually occur near the local extremepoints of the signal, which contradicts the fundamental require-ment u(t) � s(t) [25]. To overcome this drawback, Huang et al.designed the “AM/FM demodulation” to calculate the meaningfulIF [13]. Qin and Zhong presented a segment-power-function-basedenvelope model (SPFE) to eliminate the undershoots [19]. Unfor-tunately, it is pointed out that SPFE can reduce but not eliminatethe undershoots [25]. Based on the monotone piecewise cubic in-terpolant, Yang et al. proposed an improved envelope algorithm(IMCE) which can eliminate the undershoots completely [25]. Re-cently, Niu et al. provided an envelope method based on an empir-ical pursuit of knots and natural splines [14]. Huang et al. proposeda construction of envelopes by solving a convex optimization prob-lem [10].

Different models possess different properties and can be usedin different applications. Nevertheless, there still exist some com-mon essential attributes on the concept of envelope. Firstly, fromthe point of view of frequency spectrum, the frequency spectra ofthe envelope u(t) is lower than that of the input signal (the mod-ulated signal) s(t), which implies that u(t) is much smoother thans(t) (in fact, a function is infinite differentiable if its Fourier trans-form is compactly supported [6], the readers are also referred toliteratures on the Bedrosian Identity for this topic, such as [1,22]).Secondly, from the point of view of geometry, u(t) is usually sit-uated above the signal, that is u(t) � s(t). As discussed above, theAE u(t) = √

s2(t) + [Hs(t)]2 � s(t) always holds. For the CSE, theupper envelope u(t) � s(t) holds almost everywhere except the un-dershoots near the local extreme points of the signal [25]. In thecase of SLFE, if the modulated signal is s(t) = u(t) cos(ωct + θc),where the modulating signal u(t) is of lower frequency than ωc ,the frequency of the carrier wave, then the SLFE is |u(t)|, whichsatisfies the condition |u(t)| � s(t). However, if s(t) is noisy, thenthe SLFE is actually an approximation of the envelope of u(t). Thusthe SLFE inequality |u(t)| � s(t) holds no longer. The envelope fornoisy signals will be studied in detail in Section 5.

The constraint that u(t) � s(t) is very important to the time–frequency analysis of nonstationary signals. For a monocomponentsignal, this constraint is necessary in computing the instantaneousphase by θ(t) = arccos(s(t)/u(t)) since the domain of arccos x mustsatisfy |x| � 1. The smoothness of u(t) depends on the envelopemodels used. For a monocomponent signal, if u(t) > 0 for all t ,θ(t) = arccos(s(t)/u(t)) is as smooth as s(t) and u(t), otherwiseθ(t) may have jumps and thus be not continuous. The source of theproblem is the over-modulation in the amplitude which is assumedto be non-negative. If we relax the non-negative assumption on theamplitude and switch the upper and lower envelopes at the dis-continuities, the smoothness can be restored. From the point viewof mathematics, negative amplitude is caused. The readers are re-ferred to [24,23] for details on this question.

Based on the discussion above, it is understood that the unan-imous knowledge about envelope is: the envelope of a signal shouldbe smooth and contain as little oscillation as possible; it should be situ-ated above (below) the signal except for the points where the envelopeand the signal are tangent, and wrap the signal as tightly as possible.

According to the commonly accepted understanding as de-scribed above, the envelope u(t) of a given signal s(t) should sat-isfy (i) u(t) is smooth and contains as little oscillation as possible;(ii) u(t) > s(t) (u(t) < s(t)) except for the points where the enve-lope and the signal are tangent; and (iii) u(t) wraps the signal astightly as possible. This naturally produces a constrained optimiza-tion problem. Inspired by this, we present a new envelope modelbased on the convex constrained optimization in this paper. Exper-iments show the effectiveness of the model.

The remainder of the paper is organized as follows: In Sec-tion 2, we present an optimization model to estimate the envelope.Performance comparisons with some existing methods are given inSection 3. In Section 4, we discuss the applications of the new en-velope model to EMD and AM/FM demodulation. Section 5 focuseson the discussion of the stability of the envelope models to noise.Finally, the work is concluded in Section 6.

2. The new model of envelope

In this section, we present a new model of envelope throughthe constrained optimization. Without loss of generality we dis-cuss the case of upper envelope only. According to the analysis inlast section, the upper envelope u(t) of a given signal s(t) can beobtained by solving the following optimization problem{Find the smoothest u(t);

Subject to u(t) � s(t), andu(t)wraps the signal as tightly as possible.

(1)

To measure and formulate the smoothness and less oscillationof u(t), we can use its total variation of high order [9]:

TVk−1(u(t)) =

b∫a

∣∣u(k)(t)∣∣dt, (2)

where u(k)(t) stands for the kth derivative of u(t) for some k � 1.To avoid the effect of the so-called stair case, we adopt the 3rd-order total variation, i.e., k = 4 as indicated in [9]. In order to makeu(t) wrap the signal as tightly as possible and locate above the sig-nal, u(t) should be tangent with s(t) at some proper points. Thesetangential points are very important to obtain a reasonable en-velope in geometry and they should be near the local maximumpoints intuitively. This requirement leads to the equality constraint:u(ξ j) = s(ξ j) at the proper tangential points ξ j , j = 1, . . . , M . Thusthe problem (1) can be formulated as the following 3rd-order totalvariation minimization problem{

Minimize TV3(u(t));Subject to u(t) � s(t), u(ξ j) = s(ξ j), j = 1, . . . , M.

(3)

Suppose that the signal s(t) is uniformly sampled at ti , i =1,2, . . . ,n. Then, the optimization problem (3) can be written asan L1-minimization problem. The discrete form of the 3rd-ordertotal variation can be formulated by a 4th-order difference matrixmultiplying the signal vector. Accordingly, the problem (3) is refor-mulated as{

Minimize ‖Φu‖1;Subject to u � s, u(ξ j) = s(ξ j), j = 1,2, . . . , M,

(4)

where Φ is the following matrix of order (n − 4) × n:

Φ =

⎛⎜⎜⎜⎝

1 −4 6 −4 1 0 · · · · · · 00 1 −4 6 −4 1 0 · · · 0· · · · · · · · · · · · · · · · · · · · · · · · · · ·0 · · · 0 1 −4 6 −4 1 00 · · · · · · 0 1 −4 6 −4 1

⎞⎟⎟⎟⎠ .

The key issue of this model is how to find the tangential points.The exact positions are difficult to determine and we have to sub-stitute the approximate ones for them. In this paper, we use theCSE to estimate the positions of these tangential points, which isdescribed in the following.

Let y(t) = s(t)− e(t) be the residual between s(t) and its upperCSE e(t). Then, the inequality y(t) � 0 holds almost everywhereexcept in the undershoot intervals. Find all the nonnegative lo-cal maximum points of y(t), denoted by ξ1, . . . , ξM , and regard

140 L. Yang et al. / Digital Signal Processing 29 (2014) 138–146

them as the approximate tangential points of s(t) and u(t). Usu-ally, the tangential points are equal or near to the correspondingmaximum points. For simplicity, we rewrite the constrained condi-tion u(ξ j) = s(ξ j), j = 1, . . . , M , as Au = b, where A is the positionmatrix of order M × n determined by the tangential points andb = {s(ξ j)}M

j=1. Then the problem (4) can be reformulated equiva-lently as follows:{

Minimizeu∈Rn ‖Φu‖1;Subject to u � s, Au = b.

(5)

The L1-minimization problem (5) is convex and non-differential.We can solve it by means of the split Bregman iteration algorithm[7,16,28], which is an efficient tool for solving such convex con-strain optimization problems. Because of the inequality constraintin problem (5), we modify the split Bregman iteration by imple-menting the following projection:

Proj(u)k = max(uk, sk), ∀u ∈Rn, (6)

so that Proj(u) belongs to the convex set Ω = {u ∈ Rn | u � s}. Theimproved algorithm is summarized as follows.

Algorithm 1 Upper Envelope Computation Algorithm.1. initialize: b0 = c0 = d0 = 0, u0 = s;2. while not converge

• bk+1 = bk + (b − Auk), ck+1 = ck + (Φuk − dk);• uk+1 ← min{ λ

2 ‖Au − bk+1‖2 + μ2 ‖dk − Φu − ck+1‖2};

• uk+1 ← Proj(uk+1);• dk+1 ← min{‖d‖1 + μ

2 ‖d − Φuk+1 − ck+1‖2};3. end while.

Remarks.

1) The lower envelope v(t) based on convex constrained opti-mization can be obtained similarly.

2) In [10], B. Huang and A. Kunoth also proposed an optimiza-tion problem to calculate the envelope. Unlike the model (3)in this paper, they used the ‖u(n)(t)‖2

2 as the objective func-tion and the local maximum points, instead of the tangentialpoints, in the equality constraint. On one hand, the ContrastSensitivity curve shows that L1-metric is more appropriatefor measuring the error of the human visual system than theL2-metric [4]. Hence, our model is reasonable to consider theL1-norm of u(n)(t), even though it increases the difficulty ofsolving the minimization. On the other hand, it is easily knownthat the tangential points are usually not the local extremepoints. Thus, the equality constraint condition of this paper isa considerable improvement to their work. The results from

the two optimization models will be compared in the follow-ing section.

3) The computational cost of Algorithm 1 contains the compu-tation of the position matrix A and those of uk and dk inthe Split Bregman iteration. Let n be the length of the signal.Since A is determined by all the M tangential points of a cubicspline interpolation to the signal, it can be computed by solv-ing a linear system with sparse coefficient matrix of M vari-ables and the computational complexity is O (n) since M < n.The uk can be computed by using the existing optimizationtechnique with computational complexity O (n log n) and dk

can be solved by the soft threshold function, which needsonly the operations of vector product and scalar contraction ofcomplexity O (n). If L iterations are implemented, then the to-tal computational complexity is O (n) + L(O (n log n) + O (n)) =L · O (n log n).

3. Methods comparison for monocomponent signals

This section will give some experiments to test the effectivenessof the new envelope model. For simplicity, we denote by “OPE” and“OPEH ” respectively the envelope based on model (5) and Ref. [10].We still discuss the case of upper envelope only.

3.1. OPE vs. AE, CSE, SPFE

Consider the signal s1(t) = (2+cos 8t) cos 10t , t ∈ [1,7], as plot-ted in Fig. 1(a) (blue curve). It can be easily computed that the AEof s1(t) is u1(t) = 2 + cos(8t), as shown in Fig. 1(a) (green dashcurve). It is easy to see that this curve is totally different from thegeometric envelope of the signal. On the contrary, the other threeenvelopes: the OPE (red), CSE (black dot-dashed) and SPFE (pinkdotted) with β = 2.5 as recommended in [19], have much betterintuitive character than the AE. However, among the three en-velopes, OPE is the only one which does not contain undershoots,as illustrated in the enlarged drawing Fig. 1(b).

To test the performance of the proposed model to real physicalsignals, we consider a modeled damped Duffing wave with chirpfrequency [13,17]. The explicit expression of the model is given by

s(t) = e−t/256 cos

(1

2q(t) + 3

10sin q(t)

),

q(t) = π

32

(t2

512+ 32

), t ∈ [64,1024],

which enables us to validate the methods quantitatively. Fig. 2(a)shows the AE, CSE, SPFE with β = 2.5, and OPE. It is easy to see

Fig. 1. (a) The signal s1(t) (blue solid) and the AE (green dash), the CSE (black dot-dashed), the SPFE (pink dotted) and the OPE (red solid); (b) The drawing of a partialenlargement of (a). (For interpretation of the colours in this figure, the reader is referred to the web version of this article.)

L. Yang et al. / Digital Signal Processing 29 (2014) 138–146 141

Fig. 2. The damped Duffing wave s(t) (blue solid) and its envelopes: the AE (green dashed), CSE (black dot-dashed), SPFE (pink dotted) and OPE (red solid); (b) The drawingof a partial enlargement of (a); (c) The errors between these envelopes and u(t) = e−t/256. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 3. Left: The CSE (blue) and OPE (red) of the EEG signal (green); Right: The drawing of a partial enlargement of the left sub-figure. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

that the last three curves give very good approximation to the ge-ometric envelope intuitively. As illustrated in the enlarged drawingFig. 2(b), among them the OPE is the only one which has no un-dershoots. Fig. 2(c) illustrates the errors u(t) − e(t) between thedamped oscillation u(t) = e−t/256 and each above envelope e(t).

We also test the proposed model and the CSE for a real-worldsignal, a six second segment of a sleep EEG data at the secondsleep stage of a NREM (no-rapid eye movement) phase [26]. Thesignal, its CSE and OPE are shown in Fig. 3. It is seen that the OPEperforms as well as or a little better than the CSE.

The advantage of no undershoot is a main advantage of the OPEover the CSE and other existing models. The existence of under-shoots is an obstacle for the AM/FM demodulation of IMFs sincethe inequality −u(t) � s(t)� u(t) is necessary for the computationof the phase θ = arccos(s(t)/u(t)). To eliminate the undershootscaused by the CSE model so that the above inequality is satis-fied, an iterative normalization process is proposed in [13,27]. Itis easily seen that the OPE can be used directly in the AM/FM de-modulation of IMFs.

3.2. OPE vs. AE, IMCE, OPEH

Among the existing typical envelope models, the AE, IMCE andOPEH do not contain undershoots as OPE do. To compare thesemodels with OPE, let us consider the signal s2(t) = t cos(10t).We observe that u2(t) = t is a straight line containing no oscil-lation, located above the signal: s2(t) � u2(t), except for the pointswhere u2(t) and s2(t) are tangent, and wraps the signal tightly thebest. According to the fundamental conditions on the envelope de-scribed in Section 1, u2(t) = t is the ideal envelope of s2(t). Eventhough none of the envelopes AE, IMCE, OPEH and OPE contains

undershoot, there are different more or less. As we know, a goodenvelope should oscillate as small as possible. Thus, to comparethese models, we employ the oscillation rate (OR) of order n forn = 2,3,4, which is defined by

OR(s) =(∫ ∣∣s(n)(t)

∣∣2dt

) 12

. (7)

Meanwhile, the following root mean square (RMS) is used to mea-sure the error between an envelope u(t) and the ideal envelopeu2(t):

RMS(u, u2) =√√√√1

n

n∑i=1

∣∣u(ti) − u2(ti)∣∣2

. (8)

Generally speaking, the smaller both the OR and RMS, the betterthe envelope is.

The ORs and RMSs of the envelopes AE, IMCE, OPEH and OPE ofs2(t) = t cos(10t) are listed in Table 1. It is seen that the proposedOPE is the best among them in both OR and RMS respects.

To further verify and explain the comparison, the envelopes AE,IMCE, OPEH and OPE of s2(t) = t cos(10t) are plotted in Fig. 4(a)

Table 1OR and RMS comparisons.

Methods OR values RMSs

n = 2 n = 3 n = 4

AE 0.1874 0.3740 0.7465 0.0410IMCE 1.075 × 10−4 0.041 × 10−4 0.022 × 10−4 0.0027OPEH 2.3 × 10−3 0.8 × 10−3 1.1 × 10−3 0.0367OPE 4.5 × 10−6 0.082 × 10−6 0.002 × 10−6 8.5796 × 10−4

142 L. Yang et al. / Digital Signal Processing 29 (2014) 138–146

Fig. 4. (a) The signal s2(t) (black) and the envelopes obtained by four models; (b) The local enlarged drawing. (For interpretation of the references to color in this figure, thereader is referred to the web version of this article.)

Fig. 5. (a) The IF θ ′3(t) = 5 + cos t (blue) of the signal s3(t) = t cos(5t + sin t), and the approximate IF based on OPE (red) and OPEH (green); (b) The difference between θ ′

3(t)and the approximate ones. (For interpretation of the colours in this figure, the reader is referred to the web version of this article.)

and a local enlarged drawing is displayed in Fig. 4(b). It is ob-served that OPEH is horizontally tangent with the signal at eachmaximum point of s2(t), such as that marked by “A” in Fig. 4(b).This constraint produces an inflection and results in a deviation ofthe envelope from u2(t) around. On the contrary, the AE, IMCEand OPE approximate to u2(t) much better than OPEH . More-over, among them OPE is of the least oscillation (with smallestOR of order n = 2,3,4) owing to the model (3), which minimizesTV3(u(t)).

3.3. Instantaneous frequency

The instantaneous frequency (IF) of a monocomponent signalcan be easily calculated if the envelope is given. Better envelopewill provide more precise and reasonable estimation of the IF.For a monocomponent signal s(t) = u(t) cos θ(t) with u(t) as itsenvelope, the IF at the discrete sampling {t j} can be calculated ap-proximately as follows [10]:

θ ′(t j) ≈∣∣∣∣ cos θ(t j+1) − cos θ(t j−1)

(t j+1 − t j−1)

√1 − cos2 θ(t j)

∣∣∣∣≈

∣∣∣∣ s̃(t j+1) − s̃(t j−1)

(t j+1 − t j−1)

√1 − s̃2(t j)

∣∣∣∣ (9)

where s̃(t) = s(t)/u(t) is the pure FM signal. For those points t j

satisfying cos2 θ(t j) = 1 (s̃(t j) = ±1), the right hand of (9) doesnot exist and are processed according to the technique presentedin [10].

In the following we compare the proposed OPE with the OPEHby computing the IF using the above formula. Consider the narrow-band signal s3(t) = t cos(5t + sin t). Similarly to the above discus-sion on s2(t), this signal can be regarded as a monocomponentwhose envelope and IF are u3(t) = t and θ ′

3(t) = 5 + cos t . For agiven envelope, the envelope-based IF can be calculated accordingto (9). Fig. 5(a) illustrates the IF θ ′

3(t) = 5 + cos t (blue), the ap-proximate IFs based on the OPEH (green dashed) and that basedon the OPE (red dash-dotted). It is obviously seen that the OPE-based IF approximates to θ ′

3(t) much more accurately than theothers. Fig. 5(b) is the difference between each approximate IFand θ ′

3(t).

4. Discussion on the applications of OPE to EMD and AM/FMdemodulation

In this section, we will discuss the applications of OPE to EMDand AM/FM demodulation. It will be shown that if the CSE is re-placed by the OPE, the EMD can be improved to some extent andthe computational cost for AM/FM demodulation can be totallysaved. The new EMD for a real world signal is also given in thissection.

4.1. Application to the EMD

As a good approximation to the geometric envelope the OPEcan be used in the EMD as the CSE does. Let us consider the signals4(t) = x1(t)+ x2(t), t = 0 : 0.01 : 16, where x1 = sin θ1(t) is an IMFwith constant amplitude and nonlinear phase θ1(t) = 20t + cos(5t)

L. Yang et al. / Digital Signal Processing 29 (2014) 138–146 143

Fig. 6. The EMDs based on the CSE (left) and the OPE (right).

Fig. 7. Left: The errors imfcse1 − x1 (blue) and imfope

1 − x1 (red); Right: The errors imfcse2 − x2 (blue) and imfope

2 − x2 (red). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

and x2(t) = ρ(t) cos θ2(t) is also an IMF with instantaneous ampli-tude ρ(t) = 3 + 2 sin(t) and nonlinear phase θ2(t) = 4t + cos(2t).We decompose the signal x(t) using the CSE-based EMD and OPE-based EMD respectively, in which the stop criteria is set to be

1

2

n∑j=1

∣∣u(t j) + v(t j)∣∣ � ε

n∑j=1

∣∣x(t j)∣∣, ε = 0.05.

Fig. 6 is the experimental results, in which the first rows arethe original signals and those followed from top to bottom arethe IMFs and the residues. The CSE-based EMD generates 5 IMFs,which are denoted by imfcse

i , i = 1,2, . . . ,5, and the OPE-basedEMD generates two IMFs, which are denoted by imfope

i , i = 1,2.It is easy to see that both imfcse

i and imfcsei are approximation of

xi for i = 1,2. Those imfcsei , i = 3,4,5 are pseudo waves caused

by the model or computational errors. Fig. 7 shows the errorsimfcse

i − xi , i = 1,2. Intuitively, the OPE-based EMD performs betterthan the CSE-based EMD since the latter generates only two IMFslike the actual components of x. It means that the error between xand imfope

1 + imfope2 is negligible.

As an example for a real signal, Fig. 8 illustrates the EMDs ofthe EEG data shown in Fig. 3. It is seen that the OPE-based EMDand the CSE based EMD have similar experimental results. Unlikethe signal shown in Fig. 6, the components of the EEG signal isunknown, which makes it impossible to compare these two resultsquantitatively.

4.2. Application to the AM/FM demodulation

The main advantage of OPE over CSE is that it contains no un-dershoots, which can be used in the AM/FM demodulation of IMFs.To explain it clearly, let us consider the demodulation of x2(t). Ac-cording to the method proposed in [12], the CSE of |x2|, denotedby ecse

0 , needs to be computed. Since ecse0 contains undershoots,

|x2(t)/ecse0 (t)| > 1 for some t , as illustrated in Fig. 9, an iteration

of normalization needs to be contacted to improve ecse0 until the

modified one ecsen satisfies |x2(t)/ecse

n (t)| � 1 for all t [12]. If theCSE is replaced by the OPE here, we do not need the normalizationsince |x2(t)/eope

n (t)| � 1 is always satisfied. Thus the computationalcost for the normalization can be totally saved in the AF/FM de-modulation.

5. The envelope for noisy signals

Noise immunity is an important factor to assess a method forsignal processing. Generally, good methods should be stable tonoise. Let T be an operator for signal processing. If T is a boundedand linear operator, then it is stable to noise due to∥∥T sn(t) − T s(t)

∥∥ � C∥∥sn(t) − s(t)

∥∥,

where, s(t) is a signal, sn(t) is its noisy one, C is a constant and‖sn − s‖ stands for a norm of sn(t)− s(t) which is used to estimatethe error between sn(t) and s(t) [21]. This reveals in mathematics

144 L. Yang et al. / Digital Signal Processing 29 (2014) 138–146

Fig. 8. Left: The EMD based on CSE; Right: the EMD based on OPE.

Fig. 9. (a) The signal |x2| (green), its CSE ecse0 (blue), and its OPE eope

0 (red); (b) a local enlarged drawing of (a); (c) ecse0 /|x2| (blue) and eope

0 /|x2| (red); (d) a local enlargeddrawing of (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

that most operators for signal processing is stable to noise, suchas the Fourier, Hilbert, wavelet transforms and lowpass filtering.But there also exist some nonlinear or unbounded operators. Thederivative operation is a typical unbounded operator, which is verysensitive to noise. However, the nonstability to noise is not alwaysnegative and useless. It is well-known that the derivative operatoris very useful in detecting the singularity of a signal, such as theedge detection in image processing [8]. In this section, we pointout that some envelope detectors are nonlinear operators, whichare not stable to noise.

In this section, we discuss the noise immunity of the four kindsof envelope models: AE, SLFE, CSE and OPE, the proposed one.

Consider the signal s(t) = u(t) cos(10t) in which u(t) = 3 + cos t +sin(t/2). By adding white Gaussian noise of SNR = 10 dB to s(t),a noisy signal sn(t) is obtained. We compute the AEs, SLFEs, CSEsand OPEs of s(t) and sn(t) and depict them in Fig. 10. The firstrow is the envelopes of s(t) and the second row is those of sn(t).From left to right are respectively the AEs, SLFEs, CSEs, and OPEs.It is easy to see that the AE and SLFE are stable to noise but theCSE and OPE are not. This results can be explained theoretically.The stability of AE and SLFE are guaranteed by their linearity andboundedness of the Hilbert transform and the Fourier transform(for lowpass filtering). The instability of CSE and OPE is causedby the nonlinearity of the envelope detectors, which can be ex-

L. Yang et al. / Digital Signal Processing 29 (2014) 138–146 145

Fig. 10. The envelopes of s(t) = [3 + cos t + sin(t/2)] cos(10t) (the first row) and the noisy version with SNR = 10 dB (the second row). From left to right are respectively theAEs, SLFEs, CSEs and OPEs.

Fig. 11. Magnified display of Fig. 10(g) and (h) on the sub-domain [9.2,10.4]. TheCSE and OPE are close to the original signal sn(t) since they are determined by themaxima of sn(t).

plained as follows: In the domain [2,12], s(t) has only 16 maximabut sn(t) has 152 maxima because of the noise. As the cubic splineinterpolation passing these maxima, the CSE of sn(t) is quiet differ-ent from that of s(t) which passing only those 16 maxima. Fig. 11

is the magnified display of the CSEs of s(t) and sn(t) in the sub-domain [9.2,10.4]. Since the CSE is determined by the maxima, itis a nonlinear operator and not stable to noise. Similarly, the OPEis not stable to noise either. Unlike the CSE, the OPE has no under-shoots.

The instability of envelope is not always a negative property. Infact, it is very useful in EMD. If the CSE is replaced by the AE orSLFE to estimate the mean of the signal in the sifting algorithmof EMD, the noise of high frequency cannot be extracted from thesignal. Fig. 12 shows the difference sn(t) − m(t), where m(t) is themean of the upper and lower envelopes of the noisy signal sn(t)and from left to right the envelope detectors are respectively theAE, SLFE, CSE and OPE. Since the upper and lower envelopes of AEor SLFE modes are exactly equal, the difference equals exactly thesignal itself sn(t) − m(t) = sn(t), as shown in Fig. 12(a). Thus thenoise cannot be extracted with the AE/SLFE-based EMD. However,the CSE and OPE works well due to their instability to noise. Onthe other hand, while demodulating a monocomponent signal, theAE and OPE can be used to calculate the instantaneous phase sincethey satisfy u(t) � s(t) and θ(t) = arccos(s(t)/u(t)) can be definedwell.

Fig. 12. The residue sn(t) − m(t), where m(t) is the mean of the upper envelope and the lower envelope. The envelope used are respectively (a) AE and SLFE, (b) CSE, and(c) OPE.

146 L. Yang et al. / Digital Signal Processing 29 (2014) 138–146

6. Conclusion

This paper presents a novel envelope model (OPE) by using theconvex constrained optimization. The OPE is smooth, has no under-shoots and coincides with the human perception of the geometryof the signal. Experiments comparing this model with the otherexisting models show very encouraging results.

References

[1] E. Bedrosian, A product theorem for Hilbert transform, Proc. IEEE 51 (1963)868–869.

[2] B. Boualem, Estimating and interpreting the instantaneous frequency of a sig-nal I – fundamentals, Proc. IEEE (ISSN 0018-9219) 80 (4) (April 1992) 520–538.

[3] L. Cohen, Time–Frequency Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1995.[4] R.A. DeVore, B. Jawerth, B.J. Lucier, Image compression through wavelet trans-

form coding, IEEE Trans. Inf. Theory 38 (2) (March 1992) 719–746.[5] D. Gabor, Theory of communication, J. Inst. Electr. Eng. 93 (1946) 429–457.[6] C. Gasquet, P. Witomski, Fourier Analysis and Applications: Filtering, Numerical

Computation, Wavelets, Springer-Verlag New York, Inc., 1998 (Translated by R.Ryan).

[7] T. Goldstein, S. Osher, The split Bregman method for l1-regularized problems,SIAM J. Imaging Sci. 2 (2) (2009) 323–343.

[8] R.C. Gonzalez, R.E. Woods, Digital Image Processing, Addison–Wesley Publish-ing Company, 1992.

[9] T.Y. Hou, Z.Q. Shi, Adaptive data analysis via sparse time–frequency represen-tation, Adv. Adapt. Data Anal. 3 (1) (2011) 1–28.

[10] B.Q. Huang, A. Kunoth, An optimization based empirical mode decompositionscheme, J. Comput. Appl. Math. 240 (2013) 174–183.

[11] N.E. Huang, Z. Shen, S.R. Long, et al., The empirical mode decomposition andthe Hilbert spectrum for nonlinear and non-stationary time series analysis,Proc. R. Soc. Lond. A 454 (1998) 903–995.

[12] N.E. Huang, Z. Wu, S.R. Long, On instantaneous frequency, in: Workshop onthe Recent Developments of the Hilbert–Huang Transform Methodology andIts Application, Taipei, China, March 15th–17th 2006, 2006.

[13] N.E. Huang, Z. Wu, S.R. Long, K.C. Arnold, X. Chen, K. Blank, On instantaneousfrequency, Adv. Adapt. Data Anal. 1 (2) (2009) 177–229.

[14] K. Niu, L. Peng, S. Peng, A direct evaluation of mono-components by instanta-neous frequency, Int. J. Wavelets Multiresolut. Inf. Process. 10 (3) (2012), http://dx.doi.org/10.1142/S0219691312500282.

[15] A.V. Oppenheim, A.S. Willsky, S.H. Nawab, Signals and Systems, 2nd edition,Pearson Education North Asia Limited and Publishing House of Electronic In-dustry, 2002.

[16] S. Osher, M. Burger, D. Goldfarb, J.X.W. Yin, An iterative regularization methodfor total variation-based image restoration, Multiscale Model. Simul. 4 (2)(2006) 460–489.

[17] D.E. Panayotounakos, E.E. Theotokoglou, M.P. Markakis, Exact analytic solutionsfor the damped Duffing nonlinear oscillator, C. R., Méc. 334 (5) (2006) 311–316.

[18] T. Qian, Analytic signals and harmonic measures, J. Math. Anal. Appl. 314(2006) 526–536.

[19] S.R. Qin, Y.M. Zhong, A new envelope algorithm of Hilbert–Huang transform,Mech. Syst. Signal Process. 20 (2006) 1941–1952.

[20] S.O. Rice, Envelope of narrow-band signal, Proc. IEEE 70 (7) (1982) 692–699.[21] W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, 1991.[22] L. Tan, L. Shen, L. Yang, Rational orthogonal bases satisfying the Bedrosian iden-

tity, Adv. Comput. Math. 33 (3) (2010) 285–303.[23] A. Venkitaraman, C. Seelamantula, On computing amplitude, phase, and fre-

quency modulations using a vector interpretation of the analytic signal, IEEESignal Process. Lett. 20 (12) (2013) 1187–1190.

[24] A. Venkitaraman, C.S. Seelamantula, A technique to compute smooth ampli-tude, phase, and frequency modulations from the analytic signal, IEEE SignalProcess. Lett. 19 (10) (2012) 623–626.

[25] L. Yang, Z. Yang, L. Yang, P. Zhang, An improved envelope algorithm for elimi-nating undershoots, Digit. Signal Process. 23 (2013) 401–411.

[26] Z. Yang, L. Yang, D. Qi, Detection of spindles in sleep EEGs using a novel algo-rithm based on the Hilbert–Huang transform, in: M.I.V. Tao Qian, Y. Xu (Eds.),Wavelet Analysis and Applications, Applied and Numerical Harmonic Analysis,Birkhäuser Verlag, Basel, Switzerland, 2006, pp. 543–559.

[27] Z. Yang, L. Yang, C. Qing, D. Huang, A method to eliminate riding waves ap-pearing in the empirical AM/FM demodulation, Digit. Signal Process. 18 (2008)488–504.

[28] W. Yin, S. Osher, D. Goldfarb, Bregman iterative algorithms for L1-minimizationwith applications to compressed sensing, SIAM J. Imaging Sci. 1 (1) (2008)143–168.

Lijun Yang was born in Henan province, China, in 1979. She receivedthe B.S. degree and M.S. degree from the School of Mathematics and In-formation Sciences, Henan University, in 2002 and 2005, the Ph.D. degreefrom the School of Mathematics and Computing Science, Sun Yat-sen Uni-versity, in 2013. Now she is an Assistant Professor at the School of Math-ematics and Information Sciences, Henan University. Her current researchinterests include wavelet and signal processing and time–frequency analy-sis.

Zhihua Yang received his M.S. degree in automation from Hunan Uni-versity and Ph.D. degree in computer science from Sun Yat-sen Universityrespectively in 1995 and 2005. Now he is a Professor at Information Sci-ence School, Guangdong University of Finance & Economics. His researchinterests include signal analysis, pattern recognition and image processing.

Feng Zhou received the B.S. degree from the Department of Mathemat-ics and Information Science, Minnan Normal University, in 2010. Now heis a Ph.D. candidate at the School of Mathematics and Computing Science,Sun Yat-sen University. His research interests include signal processing,time–frequency analysis and image processing.

Lihua Yang received the B.S. degree in mathematics from Hunan Nor-mal University, China, in 1984, the M.S. degree in mathematics fromBeijing Normal University, China, in 1987, and the Ph.D. degree in com-putational mathematics and its applications from Sun Yat-sen University,China, in 1995. He worked as a postdoctoral fellow in the Institute ofMathematics, Academia Sinica, China from 1996 to 1998. As a visitingscholar he worked in short terms in Department of Computer Science,Hong Kong Baptist University (1997–1999), Center of Pattern Recogni-tion and Machine Intelligence, Concordia University (2001), Departmentof Mathematics, Syracuse University, USA (2006), Laboratoire Jacques-LouisLions, Universite Pierre et Marie Curie (2008), Department of Mathematics,Michigan State University (2010), and Department of Mathematics, CityUniversity of Hong Kong (2009, 2010). Now he is a Professor and the Di-rector of the Institute of Computing Science and Computer Application,School of Mathematics and Computing Science, Sun Yat-sen University,China. His current research interests include time–frequency analysis, sig-nal processing and pattern recognition. He has published more than 80papers, 1 book and 3 translations. He was the supervisor of sixteen Ph.D.candidates.