Instantaneous Frequency Estimation Based on Synchrosqueezing Wavelet Transform

Transcription

1 Instantaneous Frequency Estimation Based on Synchrosqueezing Wavelet Transform Qingtang Jiang and Bruce W. Suter February 216 1st revision in July 216 2nd revision in February 217 Abstract Recently, the synchrosqueezing transform (SST) was developed as an alternative to the empirical mode decomposition scheme to separate a non-stationary signal with time-varying amplitudes and instantaneous frequencies (IFs) into a superposition of frequency components that each have well-defined IFs. The continuous wavelet transform (CWT)-based SST sharpens the time-frequency representation of a non-stationary signal by assigning the scale variable of the signal s CWT to the frequency variable by a reference IF function. Since the SST method is applied to estimate the IFs of all frequency components of a signal based on one single reference IF function, it may yield not very accurate results. In this paper we introduce the instantaneous frequency-embedded synchrosqueezing wavelet transform (IFE-SST). IFE-SST uses a rough estimation of the IF of a targeted component to produce accurate IF estimation. The reference IF function of IFE-SST is associated with the targeted component. Our numerical experiments show that IFE-SST outperforms the CWT-based SST in IF estimation and separation of multicomponent signals. Keywords: Instantaneous frequency, Empirical mode decomposition (EMD), Synchrosqueezing transform (SST), Signal separation 1 Introduction Recently the study of modeling a non-stationary signal as a superposition of Fourier-like oscillatory modes has been an active research area. To model a non-stationary signal x(t) as K x(t) = A (t) + A k (t) cos ( 2πφ k (t) ) (1) k=1 Qingtang Jiang is with the Department of Mathematics and Computer Science, University of Missouri-St. Louis, MO 63141, USA, jiangq@umsl.edu; Bruce W. Suter is with The Air Force Research Laboratory, AFRL/RITB, Rome, NY 13441, USA, bruce.suter@us.af.mil. 1

2 is important to extract information, such as the underlying dynamics, hidden in x(t). The representation of x(t) in (1) with A k (t), φ k (t) > and A k(t), φ k (t) varying more slowly than φ k(t), is called an adaptive harmonic model (AHM) representation of x(t), where A k (t) are called the instantaneous amplitudes (IAs) and φ k (t) the instantaneous frequencies (IFs), which can be used to describe the underlying dynamics. AHM representations of non-stationary signals have been used in many applications including geophysics (seismic wave), atmospheric and climate studies, oceanographic studies, medical data analysis, speech recognition, non-stationary dynamics in financial system, see for example, [1]-[4]. The empirical mode decomposition (EMD) introduced by Huang et al is a popular method to decompose a non-stationary signal as a superposition of intrinsic mode functions (IMFs) [5]. This is an efficient data-driven approach and no basis of functions is used. It has been widely used in many applications, see [1] and the references therein. Many time-frequency methods have been developed to study the time-varying spectral properties of a given signal x(t) [6]. Recently the synchrosqueezing transform (SST), also called the synchrosqueezed wavelet transform, was developed by Daubechies, Lu and Wu [7] to provide mathematical theorems to guarantee the recovery of oscillatory modes from the SST of x(t). SST, which was first introduced by Daubechies and Maes in 1996 for the consideration of speech signal separation [8], is based on the continuous wavelet transform (CWT), which has scale and time variables. SST re-assigns the scale variable to the frequency variable to sharpen the time-frequency representation of a signal as the method of both time and frequency re-assignments studied by Auger and Flandrin in 1995 [9] (see also [1] for the time-frequency and time-scale representations of signals by the re-assignment method). In addition, the original signal can be recovered from its SST. SST provides an alternative to the EMD method and its variants such as the ensemble EMD (EEMD) scheme [11], and it overcomes some limitations of the EMD and EEMD schemes such as mode-mixing and the possible negativeness of the IFs which arise in EMD and EEMD schemes. See [12]-[14] for a comparison between EMD and SST. Generalized SST was introduced in [15] for the time-frequency representation of signals with significantly oscillating IFs. The stability of SST was studied in [16]. The SST with vanishing moment wavelets with stacked knots was introduced in [17] to process signals on bounded or half-infinite time intervals for real-time signal process. [18] introduced the hybrid EMT-SST computational scheme by applying the modified SST to the IMFs of the EMD. [19] provided the AHM representation of oscillatory signals composed of multiple components with fast-varying instantaneous frequency by optimization. [2] proposed a new method to determine the time-frequency content of time-dependent signals consisting of multiple oscillatory components. The SST introduced in [8] and studied in above papers is also referred to the wavelet-based SST. The short-time Fourier transform (STFT)-based SST was studied in [21]-[24]. Also, the S- 2

3 transform-based SST is introduced in [25] and has been applied in seismic spectral decomposition. Other methods to decompose a non-stationary signal as the superposition of adaptive IMFs/subbands include optimization methods [26]-[31], the empirical wavelet transform to produce an adaptive wavelet frame system for the decomposition of a given signal [32], an alternative algorithm to EMD with iterative filtering replacing the sifting process in EMD [33, 34], and a STFT-based signal separation method in [35]. In this paper we will introduce instantaneous frequency-based synchrosqueezing wavelet transform. Our work is motived by [23] on the demodulation transform with STFT. Our approach gives IFs more straightly. This paper is organized as follows. In Section 2, we review the SST. In Section 3, we introduce the instantaneous frequency-embedded wavelet synchrosqueezing transform (IFE-SST) and study its property. In Section 4, we consider the implementation issue. In Section 5, we use IFE- SST for the separation of multicomponent signals. Our experimental results show that IFE-SST outperforms wavelet-based SST in estimation of IFs and signal separation. The conclusion is given in Section 6. 2 Synchrosqueezed wavelet transform The synchrosqueezed wavelet transform (SST) separates Fourier-like oscillatory mode A k (t) cos ( 2πφ k (t) ) from a superposition in (1), with A(t) and φ (t) positive and slow-varying, compared to φ(t), and they satisfying certain conditions. The SST approach in [8, 7] is based on the continuous wavelet transform (CWT). 2.1 Continuous wavelet transform (CWT) A function ψ(t) L 2 (R) is called a continuous (or an admissible) wavelet if it satisfies (see e.g. [36, 37]) the admissible condition: < C ψ = 2 dξ ψ(ξ) ξ <. In this paper the Fourier transform of a function x(t) L 1 (R) is defined by x(ξ) = x(t)e i2πξt dt, which can be extended to functions in L 2 (R). Denote ψ a,b (t) = 1 a ψ( ) t b a. The continuous wavelet transform (CWT) of a signal x(t) L 2 (R) with a continuous wavelet ψ is defined by W x (a, b) = x, ψ a,b = 3 x(t) 1 a ψ( t b) dt. a

4 The variables a and b are called the scale and time variables respectively. The signal x(t) can be recovered by the inverse wavelet transform (see e.g. [36, 37, 38]) x(t) = 1 C ψ W x (a, b)ψ a,b (t)db da a. The Fourier transform and the CWT given above can be extended to x(t) in the class of tempered distributions, denoted by S, which includes the Dirac delta function, sinusoidal functions or polynomials on R. For x(t) S, x is defined as the Fourier transform of x(t) if x statifies x(ξ) y(ξ) dξ = x(ξ) ŷ(ξ) dξ, y(t) S, where S is the Schwartz space or the space of testing functions all of whose derivatives of any order exist and are rapidly decreasing. Refer to [39] for mathematically rigorous definitions of S and S. For x(t) = e i2πct with frequency c, its Fourier transform is δ(ξ c) (see [4]). If the continuous wavelet ψ is in S, then the CWT W x (a, b) of x S with ψ is well defined. For x(t) = A(t)e i2πφ(t), its CWT W x (a, b) is well defined as long as ψ has certain decay as t to assure A(t)ψ(t) L 1 (R). In addition, if ψ(ξ) has enough decay as ξ, we have W x (a, b) = x, ψ a,b = x(ξ) ψ ( aξ ) e i2πbξ dξ. A function x(t) is called an analytic signal if it satisfies x(ξ) = for ξ <. In this paper, we consider analytic continuous wavelets. In addition, we assume ψ also satisfies c ψ = ψ(ξ) dξ ξ <. (2) For an analytic signal x(t) L 2 (R), it can be recovered by another inverse wavelet transform which does not involve the time variable b (refer to [8, 7]): x(b) = 1 c ψ W x (a, b) da a, where c ψ is defined by (2). Furthermore, for a real signal x(t) L 2 (R), it can be recovered by the following formula (see [7]): ( 2 x(b) = Re c ψ W x (a, b) da a Again, the above two formulas hold for x(t) = A(t)e i2πφ(t) as long as ψ has certain decay as t. The bump wavelet defined by ). ψ(ξ) = e σ 2 (ξ µ) 2 χ (µ 1 σ,µ+ 1 ), (3) σ 4

5 and the (scaled) Morlet s wavelet defined by ψ(ξ) = e 2σ2 π 2 (ξ µ) 2 e 2σ2 π 2 (ξ 2 +µ 2), (4) where σ >, µ >, are commonly used continuous wavelets. The parameter σ in (3) and (4) controls the shape of ψ and has the effect on the CWT of a signal. For a multicomponent signal x(t) = K k=1 A k cos(2πφ k t ) with positive constants A k, φ k, < φ k < φ k+1, there are no interference among the components A k cos ( 2πφ k t ) in W x (b, a) with the bump wavelet as long as the parameter σ is large enough. For a superposition (1) of AHMs with φ k satisfying the separation condition, the larger σ does not necessarily provide a better separation of AHMs. See [13] for the detailed discussion on the effect of σ on the CWT of a signal with the bump wavelet. Here we illustrate that for Morlet s wavelet, a larger σ of Morlet s wavelet does not necessarily result in a sharper representation of the CWT in the time-scale plane as the bump wavelet illustrated in [13]. Let y(t) = e i2π(9t+5t2) + e i2π(13t+1t2), t 1, (5) which is sampled uniformly with 128 sample points. The CWT of y(t) with Morlet s wavelet with σ = 1, µ = 1 and σ = 5, µ = 1 are shown Fig.1. Observe that the wavelet with a larger σ results in more blurring representation of y(t) in the time-scale plane. Figure 1: W y (a, b) : CWT of y(t) = e i2π(9t+5t2) + e i2π(13t+1t2), t 1 by using Morelet s wavelet ψ with σ = 1, µ = 1 (Left picture) and with σ = 5, µ = 1 (Middle picture); CWT of s(t) = cos ( 2π(1t) ), t 1 with wavelet ψ given by (6) (Right picture) The bump wavelet is bandlimited, and hence it has a better frequency localization than Morlet s wavelet. On the other hand, Morlet s wavelet ψ(t) is ψ(t) = 1 σ t 2π e ( ) 2 2σ (e i2πµt e 2π2 σ 2 µ 2 ). Thus Morlet s wavelet enjoys nice localization in both the time and frequency domains. bump wavelet is analytic, but Morlet s wavelet is not. Observe that the second term in (4) is very The 5

6 small for σ 1 and µ 1, e.g., with µ = 1, σ = 1, e 2σ2 π 2 (ξ 2 +µ 2) exp( 2π 2 ) = Thus the second term in (4) could be dropped in practice. In addition, the first term of ψ(ξ) in (4) is also very small for any ξ if σ 1, µ 1. Thus in practice one may use ψ defined by ψ(ξ) = { e 2π 2 (ξ 1) 2, for ξ >,, for ξ. (6) ψ defined by (6) is one of the three wavelets used in [16]. In this paper, unless it is specifically stated, we will use this ψ and we also call it Morlet s wavelet. Observe that ψ(ξ) of Morlet s wavelet ψ given in (6) concentrates at ξ = 1. If an input signal x(t) concentrates around ξ = c in the frequency domain, then its CWT concentrates around the line a = 1 c in the scale-time plane. For example, let us consider x(t) = A cos(2πct), where c > is a constant. Then x(ξ) = A 2 (δ(c) + δ( c)). Thus for a >, W x (a, b) = x(ξ) ψ ( aξ ) e i2πbξ dξ = 1 2 A ψ ( ac ) e i2πbc. Therefore, W x (a, b) concentrates around ac = 1, i.e. a = 1/c. See Fig.1 for W s (a, b) with s(t) = cos ( 2π(1t) ). Observe that W s (a, b) does concentrate around a =.1, the reciprocal of the IF=1 of s(t). However W s (a, b) spreads out around a =.1 and what we see in the scale-time plane is a zone, not a sharp line, around a =.1. This property will cause the problem that we cannot separate the IFs from their CWTs when two signals have close IFs, though for the superposition of signals such as A cos(2πct) or Ae i2πct, there is no such an issue as long as we choose the parameter σ of the wavelet to be large enough. For example, as demonstrated in Fig.1, the CWTs of the two components of y(t) given in (5) are mixed. SST re-assigns the scale variable a to the frequency variable so that it sharpens the time-frequency representation of a signal. 2.2 Synchrosqueezed wavelet transform (SST) The idea of SST is to re-assign the scale variable a to the frequency varilable. As in [7], we first look at the CWT of x(t) = A cos(2πct), where c is a positive constant. As shown above, the CWT of x(t) is W x (a, b) = 1 2 A ψ ( ac ) e i2πbc. Observe that the IF of x(t), which is c, can be obtained by b W x(a, b) 2πiW x (a, b) = c. Thus, for a general x(t), at (a, b) for which W x (a, b), a good candidate for the instantaneous frequency (IF) of x(t) is b Wx(a,b) 2πiW x(a,b) ω x (a, b) =. In the following, denote b W x(a, b) 2πiW x (a, b), for W x(a, b). 6

7 ω x (a, b) is called the reference IF function in [18] and the phase transform in [16]. SST is to transform the CWT W x (a, b) of x(t) to a quantity, denoted by T x (ξ, b), on the time-frequency plane: T x (ξ, b) = where ξ is the frequency variable. {a:w x(a,b) } W x (a, b)δ ( ω x (a, b) ξ ) da a, (7) Figure 2: Assignment of a to ξ in SST for x(t) = A cos ( 2π(ct) ) Fig.2 illustrates the definition of SST for the special case with x(t) = A cos ( 2π(ct) ). See Fig.3 for the SSTs of s(t) = cos ( 2π(1t) ), t 1 and y(t) given in (5). Figure 3: Left: SST of s(t) = cos ( 2π(1t) ) ; Right: SST of y(t) = e i2π(9t+5t2) + e i2π(13t+1t2 ) The input signal x(t) can be recovered from its SST as shown in the following theorem. Theorem 1. ([7]) Let c ψ be the constant defined by (2). Then for a real-valued x(t), ( 2 x(b) = Re c ψ ) T x (ξ, b)dξ ; (8) 7

8 for an analytic x(t), x(b) = 1 c ψ T x (ξ, b)dξ. (9) In practice, a, b, ξ are discretized. Suppose a j, b n, ξ k, j, n, k = 1,, are the sampling points of a, b, ξ respectively. Here we assume ξ k+1 ξ k = ξ for all k. Then the SST of x(t) is given by T x (ξ k, b n ) = j: ω x(a j,b n) ξ k ξ/2, W x(a j,b n) γ W x (a j, b n )a 1 j ( a) j, where ( a) j = a j+1 a j, and γ > is a threshold for the condition W x (a, b) >. The recovering formula (8) for a real signal x(t) leads to ( 2 x(b n ) = Re c ψ while for an analytic x(t), we have, from (9), 2.3 Generalized SST k ) T x (ξ k, b n ), n = 1, 2,, x(b n ) = 1 T x (ξ k, b n ), n = 1, 2,. c ψ k Figure 4: Left:z(t); Right: IF of z(t) As shown in the above examples, SST works well for some signals such as those of constant frequency. However, SST does not work well for the signals whose frequencies change significantly with the time. For example, let z(t) be the signal given by z(t) = {.8 cos 16πt, t < 2.5, cos 2π ( 15t + cos(2πt) ), 2.5 t 4. (1) z(t) is a variant of a signal considered in [15]. Here t is uniformly sampled with sample rate z(t) and its IF are shown in Fig.4. The SST of z(t) is presented in Fig.5. Observe that the IF of z(t) is blurring for t >

9 The generalized SST was introduced by Li and Liang in [15]. The idea is to transform a signal x(t) = A(t) cos(2πφ(t)) or x(t) = A(t)exp(2πiφ(t)) to a signal with a constant frequency by x(t) x(t) exp( 2πiφ (t)), where φ (t) is a function such that φ (t) = φ (t) ξ with ξ being the target frequency. If we choose φ (t) = φ(t) ξ t, then x(t) exp( 2πiφ (t)) is a signal with constant frequency ξ. The problem of this approach is that in practice φ (t) is unknown, one needs to estimate φ (t). Figure 5: SST of z(t) given in (1) 3 Instantaneous frequency embedded SST Motivated by the work of S. Wang et al [23] on the demodulation transform with STFT, we define the instantaneous frequency-embedded CWT (IFE-CWT) as follows. Let ϕ(t) be a differentiable function with ϕ (t) >. For x(t) L 2 (R), we define ( ) x ϕ,b,ξ (t) := x(t)e i2π ϕ(t) ϕ(b) ϕ (b)(t b) ξ t, (11) where ξ. Oberve that if x(t) = A(t) exp(i2πφ(t)) for some φ(t) with φ (t) >, then x ϕ,b,ξ (t) with ϕ(t) = φ(t) has IF φ (b) + ξ. Also note that in the definition of generalized SST in [15], the frequency demodulation of x(t) is x(t) exp ( i2π(ϕ(t) ξ t) ). Definition 1. Suppose ϕ(t) is a differentiable function with ϕ (t) >. The IFE-CWT of x(t) L 2 (R) with a continuous wavelet ψ is defined by ( Wx IFE (a, b) = x ϕ,b,ξ, ψ a,b = x(t)e i2π ϕ(t) ϕ(b) ϕ (b)(t b) ξ t) 1 a ψ( t b) dt. (12) a In the above definition, we assume x(t) L 2 (R). Actually, the definition of IFE-CWT can be extended to slowly growing functions x(t). Next, we have the following property about the IFE-CWT. 9

10 Proposition 1. Let Wx IFE (a, b) be the IFE-CWT of x(t) defined by (12). Then Wx IFE (a, b) = e i2πϕ(b) x(ξ) ψ( aξ + aϕ (b) ) e i2πbξ dξ, (13) where x(t) = x(t)e i2πϕ(t)+i2πξ t. (14) Proof. Let ψ 1 (t) = ψ(t)e i2πϕ (b)at. Then the Fourier transform of ψ 1 (t) is: ψ1 (ξ) = ψ ( ξ +aϕ (b) ). With x(t) given by (14), we have as desired. Wx IFE (a, b) = e i2πϕ(b) x(t)e i2πϕ(t)+i2πξt e i2πϕ (b)(t b) 1 a ψ( t b) dt a = e i2πϕ(b) x(t) 1 a ψ (t b) 1 dt = e i2πϕ(b) x(ξ) ψ1 (aξ) e i2πbξ dξ a = e i2πϕ(b) x(ξ) ψ( aξ + aϕ (b) ) e i2πbξ dξ, We note that the proof of (13) is straightforward. However, the formula (13) plays an important role in our discussion and implementation of IFE-SST. Thus, we consider it in a proposition. If x(t) = Ce i2πφ(t) for a constant C and we choose ϕ(t) = φ(t), then x(ξ) = Cδ(ξ ). Thus W IFE x (a, b) = Ce i2πφ(b) ψ( aξ + aφ (b) ) e i2πbξ. Observe that for Morlet s wavelet given in (6), ψ ( aξ + aφ (b) ) (hence Wx IFE (a, b)) concentrates along a ( ξ + φ (b) ) = 1. Thus IFE-CWT gives more straightforward scale-time representation of a signal. Let u(t) be the signal given by u(t) = e i2π(1t+1t2), t 1. (15) u(t) is a chirp considered in [24]. Here we uniformly sample u(t) with 128 sample points. In Fig.6, with ϕ(t) = t t, ϕ (t) = t , which can be estimated from CWT of u(t), the IFE-CWT of u(t)with ξ = 2 is shown. Observe that the IF of u(t) is 1+2t, < t < 1, a line segment. The IFE-CWT of u(t) displayed in Fig.6 is indeed a zone concentrating a line segment, while the CWT does not give a clear picture for the IF of u(t). The picture of IFE-CWT of u(t) with a different ξ is similar to that with ξ = 2. In the following experiments, we simply set ξ =. Next, we show that x(t) can be recovered back from its IFE-CWT. 1

11 Figure 6: CWT (Left picture) and IFE-CWT (Right picture) with ξ = 2, ϕ(t) = t t, ϕ (t) = t of u(t) = e i2π(1t+1t2), t 1 Theorem 2. Let x(t) be a function in L 2 (R). Then where c ψ is defined by (2). x(b) = 1 c ψ exp( i2πξ b) Wx IFE (a, b) da a, (16) When x(t) satisfies certain condition, x(t) can be recovered from its IFE-CWT with the scale variable a restricted to a >. Theorem 3. Let x(t) be a function in L 2 (R). Suppose there is ϕ with ϕ (t) > such that y(t) defined by y(t) = x(t) exp( i2πϕ(t)) satisfies ŷ(ξ) =, ξ A for some constant A. Let W IFE x (a, b) be the IFE-CWT of x(t) defined by (12) with ϕ(t) and ξ > A. Then where c ψ is defined by (2). x(b) = 1 c ψ exp( i2πξ b) W IFE x (a, b) da a, (17) The proof of Theorem 2 is similar to that of Theorem 3. Here we give the proof of Theorem 3. Proof of Theorem 3 Let x(t) be the function defined by (14). Observe that x(t) = y(t)e i2πξt. Thus x(ξ) = ŷ(ξ ξ ). Hence, by (13) in Proposition 1, we have Wx IFE = = e i2πϕ(b) (a, b) da a = e i2πϕ(b) ( x(ξ) ψ a ( ξ + ϕ (b) )) e i2πbξ dξ da a e i2πϕ(b) ŷ(ξ ξ ) ψ ( a ( ξ + ϕ (b) )) e i2πbξ dξ da a ŷ(ξ) ψ ( a ( ξ + ξ + ϕ (b) )) e i2πb(ξ+ξ) dξ da a = e i2πϕ(b)+i2πξb ( ŷ(ξ) ψ a ( ξ + ξ + ϕ (b) )) da a ei2πbξ dξ 11

12 This shows (17). = e i2πϕ(b)+i2πξb A ŷ(ξ) = e i2πϕ(b)+i2πξb ŷ(ξ) A ( ψ a ( ξ + ξ + ϕ (b) )) da a ei2πbξ dξ ψ(a) da a ei2πbξ dξ (since ξ + ξ + ϕ (b) > when ξ > A) = c ψ e i2πϕ(b)+i2πξb ŷ(ξ)e i2πbξ dξ = c ψ e i2πϕ(b)+i2πξb ŷ(ξ)e i2πbξ dξ = c ψ e i2πϕ(b)+i2πξb y(b) = c ψ e i2πξb x(b). A Remark 1. If the condition ŷ(ξ) =, ξ A is not satisfied, then for large ξ, we have x(b) 1 c ψ exp( i2πξ b) W IFE x (a, b) da a. (18) This can be obtained as follows. Following the proof of Theorem 3 and noting that ŷ(ξ) ψ ( a ( ξ + ξ + ϕ (b) )) = if ξ + ξ + ϕ (b), we have Wx IFE (a, b) da a = ei2πϕ(b)+i2πξb = e i2πϕ(b)+i2πξb = e i2πϕ(b)+i2πξb ξ ϕ (b) ξ ϕ (b) = c ψ e i2πϕ(b)+i2πξb ξ ϕ (b) ŷ(ξ) ŷ(ξ) ( ψ ŷ(ξ)e i2πbξ dξ = c ψ e i2πϕ(b)+i2πξb( ŷ(ξ)e i2πbξ dξ ( ŷ(ξ) ψ a ( ξ + ξ + ϕ (b) )) da a ei2πbξ dξ a ( ξ + ξ + ϕ (b) )) da a ei2πbξ dξ ψ(a) da a ei2πbξ dξ ξ ϕ (b) = c ψ e i2πϕ(b)+i2πξb y(b) c ψ e i2πϕ(b)+i2πξb ξ ϕ c ψ e i2πξb x(b). ) ŷ(ξ)e i2πbξ dξ (b) ŷ(ξ)e i2πbξ dξ Thus (18) holds with error c ψ ξ ϕ (b) ŷ(ξ) dξ. For x(t), at (a, b) for which Wx IFE (a, b), we need to define the reference IF function ωx IFE (a, b). Following the definition of ω x (a, b), b W IFE x (a, b) 2πiW IFE x (a, b) (19) may be a good candidate for the reference IF function. First we look at b W IFE x (a, b). From (13), 12

13 we have b W x IFE (a, b) = i2πϕ (b)e i2πϕ(b) x(ξ) ψ( aξ + aϕ (b) ) e i2πbξ dξ +e i2πϕ(b) x(ξ) ψ( aξ + aϕ (b) ) e i2πbξ i2πξdξ +e i2πϕ(b) x(ξ) ( ψ aξ + aϕ (b) ) aϕ (a)e i2πbξ dξ =: I 1 + I 2 + I 3. (2) where x is given by (14). Clearly, I 1 = i2πϕ (b)wx IFE (a, b). Consider the case x(t) = Ce i2πφ(t) again. As shown above, with ϕ(t) = φ(t), we have x(t) = C exp(i2πξ ), and hence x(ξ) = Cδ(ξ ), and Note that in this case W IFE x (a, b) = Ce i2πϕ(b) ψ( aξ + aφ (b) ) e i2πbξ. I 1 + I 2 = i2πφ (b)w IFE x (a, b) + Ce i2πφ(b) ψ( aξ + aφ (b) ) e i2πbξ i2πξ = i2πφ (b)wx IFE (a, b) + i2πξ Wx IFE (a, b) = i2π ( φ ) (b) + ξ W IFE x (a, b). Therefore, I 1 + I 2 2πiWx IFE (a, b) = φ (b) + ξ is the IF of x(t) plus the target frequency ξ. Hence, for a general x(t), we define the reference IF function ωx IFE (a, b) of the IFE-CWT of x(t) to be ω IFE x (a, b) := I 1 + I 2 2πiW IFE x (a, b) = ϕ (b) + I 2 2πiW IFE x (a, b), (21) where I 1 and I 2 are defined by (2). Observe that the reference IF function ωx IFE (a, b), defined by (21), is not the quantity defined by (19). Instead, it is Clearly, ω IFE x (a, b) depends on ϕ. b W IFE x (a, b) I 3 2πiWx IFE. (a, b) Definition 2. The instantaneous frequency-embedded wavelet synchrosqueezing transform (IFE- SST) of a signal x(t) with ϕ and ξ is defined by Tx IFE (ξ, b) = {a:wx IFE (a,b) } W IFE x (a, b)δ ( ωx IFE (a, b) ξ ) da a, where W IFE x is IFE-CWT of x(t) defined by (12) and ω IFE x (a, b) is defined by (21). 13

14 By Theorem 3, we know the input signal x(t) can be recovered from its IFE-SST as shown in the following theorem. The author refers to Theorem 3.3 in [7] for the recovery of components of a signal from the orginal SST. Theorem 4. Let x(t) be a function in Theorem 3 with A =. Then x(b) = 1 c ψ exp( i2πξ b) T IFE x (ξ, b)dξ. (22) In practice, a, b, ξ are discretized. Suppose a j, b n, ξ k, j, n, k = 1,, are the sampling points of a, b, ξ respectively. Again, we assume ξ k+1 ξ k = ξ for all k. Then the IFE-SST of x(t) is given by T IFE x (ξ k, b n ) = j: ω IFE x (a j,b n) ξ k ξ/2, Wx IFE (a j,b n) >γ Wx IFE (a j, b n )a 1 j ( a) j, where ( a) j = a j+1 a j, and γ > is parameter to set the condition W x (a, b). The recovering formula (22) for x(t) implies x(b n ) = 1 c ψ exp( i2πξ b n ) k T IFE x (ξ k, b n ), n = 1, 2,. (23) We will discuss more about the discretization and the implementation of IFE-SST in the next section. Another issue we need to consider about IFE-CWT and IFE-SST is that, in practice, to estimate IFs of x(t) from its IFE-SST, we need ϕ(t) and ϕ (t) which should be close to φ(t) and φ (t) respectively. We will first use (regular) CWT/STFT to have a rough estimate of φ, φ to be used as the input ϕ, ϕ for IFE-CWT and IFE-SST. Then we use IFE-CWT or IFE-SST to get more accurate estimate of φ, φ. See the next section for more details. 4 Implementation For the implementation of the IFE-SST, one may modify the procedures in [16]. Suppose x(t) is discretized uniformly at points t n = t + n t, n =, 1,, N 1. Let b n = n t, n =, 1,, N 1. Let x C N denote the discretization of x in (14): x = [ x ] T, x 1,, x N 1, where T denotes the transpose of a vector/matrix, and x n = x(t n ) = x(t n )e i2πϕ(tn)+i2πξ t n. 14

15 Let η = 1 N t and η k = { k η, for k [ N 2 ], (k N) η, for [ N 2 ] + 1 k N 1, be the sampling points for the frequency variable η. Let x k = x(η k ), k N 1 denote the discretization of the Fourier transform x of x. One may obtain x = [ x, x 1,, x N 1 ] T by applying the FFT to x: x = t FFT x. The scale variable can be discretized as a j = ν j t, ν j = 2 j/nν, j = 1, 2,, n ν ([ log2 N ] 1), where n ν is a parameter which user can choose. One may choose n ν = 32 or n ν = 64 as suggested in [16]. Then we have x(η) ψ( aη + aϕ (b) ) e i2πbη dη N 1 k= = 1 N 1 (FFT x)(k) N ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη k. k= Thus the IFE-CWT of x(t) can be discretized as η x(η k ) ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη k Wx IFE (a j, b n ) = e i2πϕ(bn) 1 N 1 (FFT x)(k) N ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη k. (24) k= Similarly, the integral I 2 in (2) can be discretized as I 2 (a j, b n ) = i2πe i2πϕ(bn) 1 N 1 (FFT x)(k) N ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη k η k. k= Therefore, ωx IFE (a, b) defined by (21) can be approximated by, for Wx IFE (a j, b n ), N 1 ωx IFE (a j, b n ) = ϕ k= (b n ) + (FFT x)(k) ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη kη k N 1 k= (FFT x)(k) ψ ( a j η k + a j ϕ (b n ) ). (25) e i2πbnη k The frequency variable ξ > of the IFE-SST can be discretized as follows. Let ξ be the frequency resolution parameter (one may set ξ = 1 2(log 2 N 1) t ). Partition the time-frequency [ ] region {(ξ, b) : < ξ 1 2 t, b } into K 1 := 2 t ξ nonoverlap zones: Ω k := n ν 1 { (ξ, b) : ξ k ξ 2 < ξ ξ k + ξ 2, b }, k = 1, 2,, K, where ξ k = k ξ. Let γ > be parameter to set the condition W x (a, b). One may choose γ to be a number between 1 8 and 1 4. Then we obtain the IFE-SST of x(t): T IFE x (ξ k, b n ) = j: ξ k ξ 2 <ωife x (a j,b n) ξ k + ξ 2 15, W IFE x (a j,b n) >γ W IFE x (a j, b n ) log 2 n ν, (26)

16 where we have used the fact a 1 j ( a) j log 2 n ν. Finally, x(b) can be recovered by (23). In the following we summarize our calculation of IFE-SST as Algorithm 1. Algorithm 1. (Calculation of IFE-SST of monocomponent signal) Input: x(t n ), ϕ(t n ), ϕ (t n ), n N 1, ξ, and γ >. Step 1. Calculate W IFE x (a j, b n ) by (24). Step 2. Calculate ωx IFE (a j, b n ) by (25) for all j, n with Wx IFE (a j, b n ) > γ. Step 3. Calculate T IFE x (ξ j, b n ) by (26). Next we give the procedures to estimate IF of a monocomponent signal with IFE-SST. Algorithm 2. (IFE-SST-based IF estimation of monocomponent signal) Input: of x(t n ). x(t n ), ξ, γ >, and initial ϕ(t n ), ϕ (t n ), n N 1 estimated from CWT/SST Step 1. Calculate IFE-SST by Algorithm 1. Step 2. Estimate IF φ (t n ) from IFE-SST and set φ(t n ) = n k= φ (t k ) t. Step 3. Repeat Step1 (with φ(t n ), φ (t n ) obtained in Step 2 as the initial ϕ(t n ), ϕ (t n )) and Step 2 till the error criterion is reached. Figure 7: Left: SST of u(t) = e i2π(1t+1t2), t 1; Middle: IFE-SST of u(t) with ξ =, ϕ(t) = 9.835t t, ϕ (t) = t ; Right: IFE-SST of u(t) with ξ =, ϕ(t) = 1.61t t, ϕ (t) = 2.122t In Algorithms 1 and 2, we need initial estimate ϕ(t n ), ϕ (t n ) for the phase function φ and IF φ. As mentioned above, we may use CWT or SST to obtain a rough estimation of them. For 16

17 a monocomponent signal x(t n ), n N 1, one may give an estimation of its IF from its CWT/SST as follows. Let M n = max{ω x (a j, b n )}, n N 1, j (M n will be max j { W x (a j, b n ) } if we use CWT for the estimation). Then a curve obtained by approximating (b n, M n ), n N 1 gives the SST-based estimation of IF of x(t n ). For the monocomponent signal, one may simply use polynomial fitting least-squares polynomial fitting to obtain the IF estimation. The IF estimation from IFE-SST in Step 2 of Algorithm 2 can be carried out similarly. More precisely, denote, M IFE n = max{ωx IFE (a j, b n )}, n N 1. j Then a curve obtained by approximating (b n, Mn IFE ), n N 1 gives the estimated IF of x(t n ) with IFE-SST. Next let us look at u(t) given in (15) as an example about how to obtain its IF by IFE-SST. Throughout this paper, we set γ = 1 5 and n ν = 1 32 for SST and IFE-SST. From the SST of u(t), which is shown in the left picture of Fig.7, we obtain M n, n 127. Then we obtain linear least square approximation with (b n, M n ), 4 n 123: ϕ (t) = t (27) One could use higher order polynomial least square approximation. Here we use the linear least square approximation for the purpose to compare the estimation to the true IF of u(t): φ (t) = 2t + 1. Also we consider n from 4 to 123 to reduce the boundary effect. Then we use ϕ (t) in (27) and its integral as the initial input IF and phase function to calculate IFE-SST of u(t). From the IFE-SST, we then obtain Mn IFE and IF estimation, denoted by φ (t), by linear least square approximation with (b n, Mn IFE ), 4 n 123. We continue this procedure iteratively as in Algorithm 2. In Table 1, we list the estimated φ (t). Observe that the best estimation we can get is φ (t) = 2.122t which is quite close to the true IF φ (t). IFE-SST of u(t) with ϕ (t) = t and that with ϕ (t) = 2.122t are shown in Fig.7. Note that IFE-SST of u(t) with ϕ (t) = t already gives a sharper representation of IF than SST. Here we also provide the differences M n φ (t n ) and M IFE n φ (t n ) to show the performance of SST and IFE-SST, where φ (t) is true IF of u(t). The comparison between M n φ (t n ), n = 5,, 124 and M IFE n φ (t n ) with ξ =, ϕ (t) = t and that with ξ =, ϕ (t) = 2.122t are shown in the left and middle pictures of Fig.8. 17

18 Iteration φ (t) t t t t t t t t Table 1: Estimated IF of u(t) = e i2π(1t+1t2) by IFE-SST with Algorithm 2 Figure 8: Left: IF estimation errors with SST and with IFE-SST for u(t) = e i2π(1t+1t2), t 1 with ξ =, ϕ (t) = t ; Middle: IF estimation errors with SST and with IFE-SST for u(t) = e i2π(1t+1t2), t 1 with ξ =, ϕ (t) = 2.122t ; Right: IF estimation errors with SST and with IFE-SST for x(t) = e i2π(1t+1t2), t 1 Our IFE-SST-based IF estimation works well with chirps of high frequency. As an example, we show in Fig.8, IF estimation errors M n φ (t n ) and M IFE n φ (t n ) for n = 13,, 5 for x(t) = e i2π(1t+1t2), t 1, which is uniformly sampled with 512 sample points, where φ (t) = 1 + 2t is the true IF. 5 IFE-SST based signal separation We will apply the IFE-SST for signal separation. (AHM) after the trend removal process: We consider the adaptive harmonic model K x(t) = x k (t) + ɛ(t), x k (t) = A k (t) cos ( 2πφ k (t) ), k=1 18

19 where ɛ(t) is the noise. We may separate the components of x(t) as follows. Use CWT/STFT to identify the highest frequency component, say x 1 (t), and estimate initial φ 1 (t) and φ 1(t). Then we use Algorithm 2 to have accurate estimate of φ 1 (t), φ 1(t) and recover x 1 (t). After that we remove x 1 (t) from x(t) and repeat the same procedures to the new signal to recover the component of the 2nd highest frequency and other components. Our method is different from that in [13], where optimization method is used to reconstruct the components of a multicomponent signal simultaneously. Next we modify the definition of IFE-CWT and IFE-SST for the purpose to estimate the IF of a particular component of a multicomponent signal. Consider the case x k (t) = A k e i2πφ k(t) and ξ =. We assume the IFs of different x k (t) lie nonoverlap different zones in the timefrequency plane. Suppose we want to estimate the IF of the lth component x l (t). We should choose( ϕ (t) close) to φ l (t). Assume it happens that ϕ(t) = φ l(t). Then x k (t) = x k (t)e i2πϕ(t) = A k e i2π φ k (t) φ l (t). Thus x l (ξ) = A l δ(ξ), and for k l, the IF of x k (t) lies in a zone away from the line ξ =, and we have x k (ξ) for ξ ɛ, where ɛ is a small positive number. Therefore, if we define the modifies IFE-CWT then W IFE x W x IFE (a, b) := e i2πφ l(b) W IFE x (a, b) = e i2πφ l(b) e i2πφ l(b) ξ ɛ = A l e i2πφ l(b) ψ( aφ l (b) ). (a, b) by K k=1 ξ ɛ ξ ɛ ( x(ξ) ψ a ( ξ + φ l (b))) e i2πbξ dξ, xk (ξ) ψ ( a ( ξ + φ l (b))) e i2πbξ dξ A l δ(ξ) ψ ( a ( ξ + φ l (b))) e i2πbξ dξ Hence, W IFE x (a, b) concentrates along aφ l (b) = 1, the IF of x l(t) in the time-scale plane. Numerically, we consider and W IFE x (a j, b n ) = ei2πϕ(bn) ( U + N ω IFE x (a j, b n ) = ϕ (b n ) + k= N 1 k=n L ( U k= + N 1 ) (FFT x)(k) ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη k, (28) ) k=n L (FFT x)(k) ψ ( a j η k + a j ϕ (b n ) ) e i2πbnη kη k ( U k= + ) N 1 k=n L (FFT x)(k) ψ ( a j η k + a j ϕ (b n ) ) (29) e i2πbnη k instead of Wx IFE (a j, b n ) and ωx IFE (a j, b n ) defined by (24) and (25) resp. Then we define the modified IFE-SST of x(t): T IFE x (ξ k, b n ) = j: ξ k ξ 2 < ωife x (a j,b n) ξ k + ξ 2 19, W IFE x (a j,b n) >γ W IFE x (a j, b n ) log 2 n ν. (3)

20 Here U and L are some nonnegative integers, which are not large. In addition, if φ l to be estimated is the IF of the component with the highest frequency, one may choose L =, while if φ l is the IF of the component with the lowest frequency, one may choose U =. Figure 9: Right: IFE-SST of z(t) with Algorithm 3 and U = L = 2; Middle: v(t) = e i2π(1t+1t2) + e i2π(9t+5t2), t 1 (real part); Right: SST of v(t) In the following, we describe the procedures to estimate the IF of a particular component and separate components of multicomponent signals. Algorithm 3. (IFE-SST-based IF estimation of lth component of multicomponent signal) Input: x(t n ), γ >, ξ, and initial ϕ l (t n ), ϕ l (t n) for lth component estimated from CWT/SST of x(t n ). Choose integers U, L Step 1. Calculate the modified IFE-SST T IFE x,l (ξ j, b n ) of lth component by (3). Step 2. Estimate IF φ l (t n) from IFE-SST T IFE x,l (ξ j, b n ) and set φ l (t n ) = n k= φ l (t k) t. Step 3. Repeat Step 1 (with φ l (t n ), φ l (t n) obtained in Step 2 as the initial ϕ l (t n ), ϕ l (t n)) and Step 2 till the error criterion is reached. Algorithm 4. (IFE-SST based signal separation) Step 1. Use CWT/STFT/SST to identify the number K of frequency components. Choose a (targeted) frequency component, say x l (t). Use CWT/STFT/SST to obtain initial estimation φ l, φ l of φ l, φ l. Step 2. With ϕ = φ l, ϕ = φ l, use Algorithm 3 to have accurate estimation of φ l (t), φ l(t) and obtain x l (t), recovered x l (t). Step 3. Remove x l (t) from x(t) and repeat Steps 1-2 to recover the second targeted component. Step 4. Do Step 3 for other components. 2

21 Step 5. If time permits, set y l (t) = x(t) k K,k l x k(t). Apply Algorithm 3 to y l (t) to have more accurate estimation of φ l (t), φ l(t) and x l (t); and do the same to other components. Figure 1: Left: IFE-SST of estimated 1st component with estimated IF: φ 1(t) = t , t 1; Right: IFE-SST of estimated 2nd component with estimated IF: φ 2(t) = 1.337t , t 1 Before we consider multicomponent signals, we remark that we can also use Algorithm 3 to estimate the IF of a monocomponent signal. For example, for the signal z(t) with significantly changing frequency given by (1), we show in Fig.9 its IF estimation obtained by Algorithm 3 with U = L = 2. Next we consider a signal consisting of two chirps: v(t) = v 1 (t) + v 2 (t), v 1 (t) = e i2π(1t+1t2), v 2 (t) = e i2π(9t+5t2), t 1. (31) v(t) is uniformly sampled with 128 sample points. The real part of this signal v(t) is shown in the middle picture of Fig.9 and the SST of v(t) is shown in the right picture of Fig.9. Fig.1 shows the IFE-SSTs of the estimated 1st and 2nd components by Algorithm 4. From the FE-SSTs, we can recover v 1 and v 2 by recovering formula (22) in Theorem 4. The error between v 1 and recovered v 1 and that between v 2 and recovered v 2 are shown in Fig.11. Figure 11: Left: Error between v 1 (t) = e i2π(1t+1t2), t 1 and recovered v 1 (t); Right: Error between v 2 (t) = e i2π(9t+5t2), t 1 and recovered v 2 (t) 21

22 Figure 12: Left: v(t) and noised v(t) with noise (1dB) (real part); Right: SST of noised v(t) We also consider signal separation in noisy environment. The signal to noise ratio (SNR) of a noised signal x = x + ɛ with noise ɛ is defined by where x is the mean of x. SNR = 2 log x x 2 (db), ɛ 2 We consider the case ṽ(t) = v(t) + ɛ(t) with v(t) defined by (31) and ɛ(t) a Gaussian white noise with 1dB. ṽ(t) is shown in Fig.12, where its SST is also provided. With Algorithm 4 (we choose L = 1, U = for v 2 (t), the component with lower frequency, and choose L =, U = 1 for v 1 (t), the component with higher frequency, and we also do Step 5), estimated IFE-SSTs of v 1 (t) and v 2 (t) in the noise environment are shown in the top row of Fig.13. We also provide the recovered v 1 (t) and v 2 (t) in Fig.13. Next example considered is w(t) = w 1 (t) + w 2 (t) with w 1 (t) = log(2 + t/2) cos ( 2π(5t +.1t 2 ) ), w 2 (t) = exp(.1t) cos ( 2π(4t +.5 cos t) ), (32) for t 8. We sample w(t) uniformly with 124 sample points. We discuss the IF estimation of w(t) in noisy environment. Let w(t) = w(t) + ɛ(t), where ɛ(t) a Gaussian white noise with 1dB. w(t) and its SST are shown in Fig.14. With Algorithm 4, the estimated IFE-SSTs of w 1 (t) and w 2 (t) in the noise environment are shown in the top row of Fig.15, and the recovered w 1 (t) and w 2 (t) are shown in the bottom row of Fig.15. From the SST and IFE-SST provided in Figs.13 and 15, we see IFE-SST gives a better IF representation of a signal than SST. These examples also show that our IFE-SST works well in the noise environment. Our last example is to use IFE-SST to separate the components of a bat echolocation signal. Fig.16 shows an echolocation pulse emitted by the Large Brown Bat (Eptesicus Fuscus). The data can be downloaded from the website of DSP at Rich University: There are 4 samples; the sampling step is 7 microseconds. The IF representation of this bat signal has studied in [23] and 22

23 Figure 13: Top row: IFs of v 1 (t) and v 2 (t) recovered by Algorithm 4; Bottom row: Recovered v 1 (t) and v 2 (t) in noise environment by Algorithm 4 [24] by matching-modulation-transform-sst (MDT-SST) and the second-order SST respectively. Here we remark that since the IFE-SST is based on CWT, we only compare our method with CWT-based SST in this paper, not with the SST based on STFT, including MDT-SST and the second order SST. For the bat signal, we just show that using Algorithm 4, we can get the IF estimation and separate the components. The SST of the bat signal is shown in Fig.16, where Morlet s wavelet with ψ(ξ) = e 2π2 (1 ξ) 2 χ (, ) (ξ) is used. The IFE-SSTs of the four main components obtained by Algorithm 4 are shown in Fig. 17, and the recovered components are provided in Fig Conclusion In this paper we introduce the instantaneous frequency-embedded continuous wavelet transform (IFE-CWT). We establish that the original signal can be reconstructed from its IFT-CWT. Then based on IFE-CWT, we introduce the instantaneous frequency-embedded synchrosqueezing transform (IFE-SST). IFE-SST can preserve the IF of monocomponent signal. For each component of a multicomponent signal, IFE-SST uses a reference IF function associated with that component. Our numerical experiments show that IFE-SST has a better performance than the CWT-based SST in the separation of multiple components of non-stationary signals. The experimental results also show that IFE-SST works well in the noise environment. 23

24 Figure 14: Left: w(t) and noised w(t) with noise (1dB); Right: SST of noised w(t) Figure 15: Top row: IFs of w 1 (t) and w 2 (t) recovered by Algorithm 4; Bottom row: w 1 (t) and w 2 (t) recovered in noise environment by Algorithm 4 ACKNOWLEDGMENT OF SUPPORT AND DISCLAIMER: (a) Contractor acknowledges Government s support in the publication of this paper. This material is based upon work funded by AFRL, under AFRL Contract No. FA and FA (b) Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of AFRL. The authors thank the anonymous reviewers for their valuable comments. The authors wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data in Fig.16 and for permission to use it in this paper. 24

25 Figure 16: Left: Bat echolocation chirp; Right: SST of bat signal Figure 17: IFE-SSTs of four main components of bat signal References [1] N. E. Huang and Z. Wu, A review on Hilbert-Huang transform: Method and its applications to geophysical studies, Rev. Geophys., vol. 46, no. 2, June 28. [2] J. B. Tary, R. H. Herrera, J. J. Han, and M. van der Baan, Spectral estimation What is new? What is next?, Review of Geophys., vol. 52, no. 4, pp , Dec [3] H.-T. Wu, Y.-H. Chan, Y.-T. Lin, and Y.-H. Yeh, Using synchrosqueezing transform to discover breathing dynamics from ECG signals, Appl. Comput. Harmon. Anal., vol. 36, no. 2, pp , Mar

26 Figure 18: Four main components of bat signal obtained by Algorithm 4 [4] S. K. Guharaya, G. S. Thakura, F. J. Goodmana, S. L. Rosena, and D. Houser, Analysis of non-stationary dynamics in the financial system, Economics Letters, vol. 121, no. 3, pp , Dec [5] N. E. Huang, Z. Shen, S. R. Long, M. L. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis, Proc. Roy. Soc. London A, vol. 454, no. 1971, pp , Mar [6] L. Cohen, Time-frequency Analysis, Prentice Hall, New Jersey, [7] I. Daubechies, J. Lu, and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Appl. Computat. Harmon. Anal., vol. 3, no. 2, pp , Mar [8] I. Daubechies and S. Maes, A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models, in A. Aldroubi, M. Unser Eds. Wavelets in Medicine and Biology, CRC Press, 1996, pp [9] F. Auger and P. Flandrin, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Proc., vol. 43, no. 5, pp ,

27 [1] E. Chassande-Mottin, F. Auger, and P. Flandrin, Time-frequency/time-scale reassignment, in Wavelets and Signal Processing, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 23, pp [11] Z. Wu and N. E. Huang, Ensemble empirical mode decomposition: A noise-assisted data analysis method, Adv. Adapt. Data Anal., vol. 1, no. 1, pp. 1 41, Jan. 29. [12] H.-T. Wu, P. Flandrin, and I. Daubechies, One or two frequencies? The synchrosqueezing answers, Adv. Adapt. Data Anal., vol. 3, no. 1 2, pp , Apr [13] S. Meignen, T. Oberlin, and S. McLaughlin, A new algorithm for multicomponent signals analysis based on synchrosqueezing: With an application to signal sampling and denoising, IEEE Trans. Signal Proc., vol. 6, no. 11, pp , Nov [14] F. Auger, P. Flandrin, Y. Lin, S.McLaughlin, S. Meignen, T. Oberlin, and H.-T. Wu, Timefrequency reassignment and synchrosqueezing: An overview, IEEE Signal Process.Mag., vol. 3, no. 6, pp , 213. [15] C. Li and M. Liang, A generalized synchrosqueezing transform for enhancing signal timefrequency representation, Signal Proc., vol. 92, no. 9, pp , 212. [16] G. Thakur, E. Brevdo, N. Fučkar, and H.-T. Wu, The synchrosqueezing algorithm for timevarying spectral analysis: Robustness properties and new paleoclimate applications, Signal Proc., vol. 93, no. 5, pp , 213. [17] C. K. Chui, Y.-T. Lin, and H.-T. Wu, Real-time dynamics acquisition from irregular samples with application to anesthesia evaluation, Anal. Appl., vol. 14, no. 4, pp , Jul [18] C. K. Chui and M. D. van der Walt, Signal analysis via instantaneous frequency estimation of signal components, Int l J Geomath, vol. 6, no. 1, pp. 1 42, Apr [19] M. Kowalskia, A. Meynarda, and H.-T. Wu, Convex optimization approach to signals with fast varying instantaneous frequency, Appl. Comput. Harmon. Anal., in press. [2] I. Daubechies, Y. Wang, H.-T. Wu, ConceFT: Concentration of frequency and time via a multitapered synchrosqueezed transform, Phil. Trans. Royal Soc. A, vol. 374, no. 265, Apr [21] H.-T. Wu, Adaptive analysis of complex data sets, Ph.D. dissertation, Princeton Univ., Princeton, NJ,

28 [22] T. Oberlin, S. Meignen, and V. Perrier, The Fourier-based synchrosqueezing transform, in Proc. 39th Int. Conf. Acoust., Speech, Signal Process. (ICASSP), 214, pp [23] S. Wang, X. Chen, G. Cai, B. Chen, X. Li, and Z. He, Matching demodulation transform and synchrosqueezing in time-frequency analysis, IEEE Trans. Signal Proc., vol. 62, no. 1, pp , 214. [24] T. Oberlin, S. Meignen, and V. Perrier, Second-order synchrosqueezing transform or invertible reassignment? towards ideal time-frequency representations, IEEE Trans. Signal Proc., vol. 63, no. 5, p , Mar [25] Z.-L. Huang, J. Z. Zhang, T. H. Zhao, and Y. B. Sun, Synchrosqueezing S-transform and its application in seismic spectral decomposition, IEEE Trans. Geosci. Remote Sensing, vol. 54, no. 2, pp , Feb [26] S. Meignen and V. Perrier, A new formulation for empirical mode decomposition based on constrained optimization, IEEE Signal Proc. Letters, vol. 14, no. 12, pp , Dec. 27. [27] T. Y. Hou and Z. Shi, Adaptive data analysis via sparse time-frequency representation, Adv. Adapt. Data Anal., vol. 3, no. 1, pp. 1 28, Apr [28] T. Oberlin, S. Meignen, and V. Perrier, An alternative formulation for the empirical mode decomposition, IEEE Trans. Signal Proc., vol. 6, no. 5, pp , May 212. [29] T. Y. Hou and Z. Shi, Data-driven time-frequency analysis, Appl. Comput. Harmon. Anal., vol. 35, no. 2, pp , Sep [3] K. Dragomiretskiy and D. Zosso, Variational mode decomposition, IEEE Trans. Signal Proc., vol. 62, no. 3, pp , Feb [31] N. Pustelnik, P. Borgnat, and P. Flandrin, Empirical mode decomposition revisited by multicomponent non-smooth convex optimization, Signal Proc., vol. 12, pp , Sep [32] J. Gilles, Empirical wavelet transform, IEEE Trans. Signal Proc., vol. 61, no. 16, pp , Aug [33] L. Lin, Y. Wang, and H. M. Zhou, Iterative filtering as an alternative algorithm for empirical mode decomposition, Advances in Adaptive Data Analysis, vol. 1, no. 4, pp , Oct

29 [34] A. Cicone, J. F. Liu, and H. M. Zhou, Adaptive local iterative filtering for signal decomposition and instantaneous frequency analysis, Appl. Comput. Harmon. Anal., vol. 41, no. 2, pp , Sep [35] C. K. Chui and H. N. Mhaskar, Signal decomposition and analysis via extraction of frequencies, Appl. Comput. Harmon. Anal., vol. 4, no. 1, pp , 216. [36] Y. Meyer, Wavelets and Operators Volume 1, Cambridge University Press, [37] I. Daubechies, Ten Lectures on Wavelets, SIAM, CBMS-NSF Regional Conf. Series in Appl. Math, [38] C.K. Chui and Q.T. Jiang, Applied Mathematics Data Compression, Spectral Methods, Fourier Analysis, Wavelets and Applications, Amsterdam: Atlantis Press, 213. [39] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, [4] R. Allen and D. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley-IEEE Press, Piscataway, NJ,