Synchroextracting chirplet transform for accurate IF estimate and perfect signal reconstruction

Digital Signal Processing 93 (2019) 172–186

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Digital Signal Processing www.elsevier.com/locate/dsp

Synchroextracting chirplet transform for accurate IF estimate and perfect signal reconstruction Xiangxiang Zhu a , Zhuosheng Zhang a,∗ , Jinghuai Gao b , Bei Li a , Zhen Li b , Xin Huang c , Guangrui Wen c a b c

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China National Engineering Laboratory for Offshore Oil Exploration, Xi’an 710049, China School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Available online 9 August 2019 Keywords: Time-frequency analysis Synchroextracting transform IF estimation Signal reconstruction

a b s t r a c t Synchroextracting transform (SET) is a recently developed time-frequency analysis (TFA) method aiming to achieve a highly concentrated TF representation. However, SET suffers from two drawbacks. The one is that SET is based upon the assumption of constant amplitude and linear frequency modulation signals, therefore it is unsatisfactory for strongly amplitude-modulated and frequency-modulated (AMFM) signals. The other is that SET does not allow for perfect signal reconstruction, which leads to large reconstruction errors when addressing fast-varying signals. To tackle these problems, in this paper, we first present some theoretical analysis for the SET method, including the existence of the fixed squeeze frequency, the performances of the instantaneous frequency (IF) estimator and the SET reconstruction. Then, a new TFA method, named synchroextracting chirplet transform (SECT), is proposed, which sharpens the TF representation by extracting the TF points satisfying IF equation, and retains an excellent signal reconstruction ability. Numerical experiments on simulated and real signals demonstrate the effectiveness of the SECT method. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Time-frequency (TF) analysis (TFA) method is an effective tool to analyze nonstationary signals and has been successfully applied in astronomical [1], radar and sonar [2], biomedicine [3,4] and mechanical engineering [5,6], etc. Traditional TFA methods can be roughly classified into linear TF and quadratic TF methods, but they have respective shortcomings. The TF concentration of linear TF methods, such as short-time Fourier transform (STFT) and continuous wavelet transform (CWT), is limited by the Heisenberg uncertainty principle [7]. As for quadratic TF methods represented by Wigner-Ville distribution (WVD), interference terms are introduced for multicomponent signals, which reduces the readability of the TF representation (TFR) [7,8]. To improve the performance of conventional TFA methods, many effective methods have been presented. Auger and Flandrin [9] proposed a TF post-processing method called reassignment (RM) to improve the energy concentration, which transfers the TF

*

Corresponding author. E-mail address: [email protected] (Z. Zhang).

https://doi.org/10.1016/j.dsp.2019.07.015 1051-2004/© 2019 Elsevier Inc. All rights reserved.

coefficients from the original position to the center of gravity of signal’s energy distribution, both along the time axis and the frequency axis. However, this method loses its ability in reconstructing original signals due to that RM is based on the spectrogram [10]. By squeezing the TF coefficients into the instantaneous frequency (IF) trajectory only in frequency direction, synchrosqueezing transform (SST) [11] can recover the interested components with an improved TF concentration. In spite of this, when addressing strongly AM-FM signals, SST still suffers from a low TF resolution [12,13]. In order to overcome the drawbacks of the SST method and obtain a better TF resolution for strong FM signals, various TF methods have been developed in the past few decades. Li et al. [13,14] proposed a generalized SST by demodulating the time-varying signal into a purely harmonic version. In [15], an iterative generalized SST was introduced to address multicomponent signals with distinct FM laws. In [16,17], a matching demodulated transform based on the extended polynomial or the Fourier mathematical model was proposed, which tries to match the time-varying FM law in a short time progressively. For the demodulated processing methods above, the precise time-varying FM law of the signal is needed in advance. However, due to the complexity and diversity of practical

X. Zhu et al. / Digital Signal Processing 93 (2019) 172–186

cases, it is difficult to determine the precise demodulated parameters, especially for dealing with multicomponent AM-FM signals [18,19]. Rather than using the demodulated TF methods, a popular way is to use the high-order information of signal, such as chirp rate (CR), to enhance the TF energy concentration for strong modulation signals, which involves second-order SST [12], high-order SST [20]. When using the high-order information of signal, greater computational burden will be introduced with the increasing SST order. Moreover, such high-order SST methods are easy to be disturbed by noise because they are computed by the high-order derivative of the STFT/CWT. Recently, a new TF method, named multi-SST (MSST) [21], is presented, which employs an iterative reassignment procedure to concentrate the blurry TF energy in a step-wise manner. For MSST, in fact, it squeezes the TF coefficients into the ridge, i.e., the curve at the TF plane along which the signal energy is locally maximum. For the TF ridge, however, it gives a biased estimate for the true IF when dealing with the signal which contains strong modulations [22,23], therefore it is difficult for MSST to obtain an ideal TFR in fast-varying signals processing. Besides, a popular TFA method, named synchroextracting transform (SET) [24], has attracted considerable interest recently. Differing from the squeezing manner of SST, SET method retains only the TF information related to the IFs of the signal and removes most smeared TF energy. Although SET can enhance the energy concentration, but it easily produces the TF trajectories deviating from the true IFs in addressing strong modulation modes due to that SET lies on the assumption of constant amplitude and linear frequency modulation signals. Moreover, SET does not allow for perfect signal reconstruction, which leads to the reconstruction error proportional to the strengths of the amplitude and frequency modulations of the signal (see Section 2 for theoretical analysis). Considering the limitations of the standard SET method, in this paper, we first give some theoretical analysis for the SET method based on STFT framework. On this basis, a novel synchroextracting chirplet transform (SECT) is proposed, which can effectively improve the accuracy of IF estimation, and allowing for perfect signal reconstruction. The remainder of the paper is organized as follows. In Section 2, we provide the detailed theoretical analysis of SET. In Section 3, a new SECT method and an ameliorated reconstruction technology are introduced. The simulation test is carried out in Section 4. In Section 5, a bat signal and a vibration signal are utilized to show the effectiveness of the proposed method. Finally, the conclusions are drawn in Section 6.

Consider a monocomponent AM-FM signal as j φ(t )

f (t ) = A (t )e , (1) √ where −1 = j, A (t ) > 0 and φ(t ) are the instantaneous amplitude and instantaneous phase, respectively. φ  (t ) > 0 is referred to as the IF of f (t ), which describes the time-varying behavior of the signal. Let g (t ) be a real and even window function in the Schwartz class (i.e., the space of smooth functions with fast decaying derivatives of any order), the STFT of signal f (t ) ∈ L 2 ( R ) with respect to window g (t ) is defined by g

S f (t , ω) =

S f (t , ω) = A (t )e j φ(t ) gˆ (ω − φ  (t )) + r (t , ω), g

where gˆ (ω) is the Fourier transform of the window g (t ), and r (t , ω) ∼ O (| A  (t ) |, | φ  (t ) |). If there is a small constant ε such that | A  (t ) |< ε , | φ  (t ) |< ε , then (3) admits the following firstorder approximation [25]:

S f (t , ω) ≈ A (t )e j φ(t ) gˆ (ω − φ  (t )). g

(4)

From the STFT result (4), and neglecting the terms on the order O (| A  (t ) |) and O (| φ  (t ) |) resulting from the assumption of | A  (t ) |< ε and | φ  (t ) |< ε , a two-dimensional IF estimate ωˆ (t , ω) can be obtained as follows [11,26]:

ωˆ (t , ω) = {

∂ g ∂ t S f (t , ω) g

j S f (t , ω)

},

(5)

g

here | S f (t , ω) |> 0, and {·} is the real part of complex number. Synchroextracting transform (SET) [24] presents a concentred TFR only using the TF coefficients in the IF trajectory, which can be represented as g

ˆ (t , ω)), T (t , ω) = S f (t , ω)δ(ω − ω

(6)

ˆ (t , ω)) is defined by where the synchroextracting operator δ(ω − ω



δ(ω − ωˆ (t , ω)) =

1

ω = ωˆ (t , ω),

0

otherwise.

(7)

According to approximation (4), the SET-based mode reconstruction is achieved from:

f (t ) ≈

T (t , φ  (t )) gˆ (0)

(8)

.

For the SET method described above, three questions need to be taken into account: (i) whether there exists TF points (t , ω) satisfyˆ (t , ω); (ii) if exist, whether or not there TF points (t , ω) ing ω = ω give a good estimate for the true IF; (iii) how about the reconstruction formula (8)? The answer to these three questions is given in the following sections. 2.1. Existence of the fixed frequency In this paper, we suppose that the window g (t ) is taken as −

t2

Definition 1 ([22]). TF pairs (t , ω) is called the ridge points of STFT if satisfying the following two conditions:

∂2 ∂ g g ln | S f (t , ω) |= 0, ln | S f (t , ω) |< 0. ∂ω ∂ ω2

Proof. Since the STFT of signal f (t ) can be expressed as g

(2)

−∞

The general form for the STFT of signal f (t ) can be written in the following [25]:

(9)

Theorem 1. For a AM-FM signal f (t ) = A (t )e j φ(t ) , the ridge point (t , ω) ˆ (t , ω). of STFT satisfies ω = ω

+∞

S f (t , ω) = f (μ) g (μ − t )e − j ω(μ−t ) dμ.

(3)

Gaussian function g (t ) = e 2σ 2 . The below definition and theorem give a definite answer to the first question.

2. Theoretical analysis for SET method

+∞

173

f (μ) g (μ − t )e − j ω(μ−t ) dμ

−∞

(10)

= M (t , ω)e j (t ,ω) , where M (t , ω) and (t , ω) respectively denote the magnitude and phase of the STFT. Differentiating with respect to t for (10), and

174

X. Zhu et al. / Digital Signal Processing 93 (2019) 172–186 g

dividing both sides by S f (t , ω) (here consider the TF points (t , ω) satisfying |

g S f (t ,

g

∂ g ∂ t S f (t , ω) g S f (t ,

ω) |> 0), we have

=−

ω)

S f (t , ω) g S f (t ,

ω)

+ jω =

∂ ∂ ln M (t , ω) + j (t , ω), ∂t ∂t (11)

g S f (t ,

where ω) denotes the STFT of f (t ) with the window g  (t ). Similarly, differentiating with respect to ω , we obtain ∂ g ∂ ω S f (t , ω) g

S f (t , ω)

tg

=−j

S f (t , ω) g

S f (t , ω)

=

amplitude and linear frequency signals due to that SET extracts ˆ (t , ω). However, for the TF points that satisfy IF equation ω = ω the strongly AM-FM signals, like the broadband signals, how does the SET perform? To illustrate this question, let us consider a more intuitive description of the behavior of the signal by relaxing the hypothesis made on the amplitude and frequency modulations, which will make the calculus more relevant for rapidly vary signals. Assume that around μ = t the phase of the signal f (t ) = A (t )e j φ(t ) can be well approximated by its second-order local expansion:

1

φ(μ + t ) ≈ φ(t ) + φ  (t )μ + φ  (t )μ2 .

∂ ∂ ln M (t , ω) + j (t , ω), ∂ω ∂ω (12)

In the same way, we consider an expansion of the amplitude modulation law around μ = t, that is,

tg

where S f (t , ω) denotes the STFT of f (t ) with the window t g (t ).

ˆ (t , ω) = { According to (5) and (11), we have ω g

− {

S f (t ,ω) g

S f (t ,ω)

∂ g ∂ t S f (t ,ω) g

j S f (t ,ω)

}=

} + ω (where {·} is the imaginary part of complex

number), and then g

ω = ωˆ (t , ω) ⇔ {

S f (t , ω) g

S f (t , ω)

} = 0,

ω) = −

1

σ

tg S (t , 2 f

ω).

(14)

Combining (12)–(14), we have

∂ ln M (t , ω) = 0. (15) ∂ω ˆ (t , ω), which finishes That is, the ridge point (t , ω) satisfies ω = ω

ω = ωˆ (t , ω) ⇔ the proof.

2

Theorem 1 proves the existence of the fixed frequency, i.e., ˆ (t , ω). Without considering for each ridge point, it satisfies ω = ω some noise points in TF plane, SET sharpens the TF representation by extracting the ridge points of the STFT, which is crucial to have a better understanding of SET method.

In this section, we will discuss whether the fixed frequency approximation for the true IF of the signal. First, we consider a particular case of constant amplitude and linear frequency signal, i.e., f (t ) = Ae j φ(t ) , where A is a positive constant and φ(t ) = a + bt + 2c t 2 . In this case, the STFT under the Gaussian window

g S f (t ,

t2 2σ 2

where Q is a positive integer, and A (k) (t ) is the k-order derivative of A (t ). Under expressions (18) and (19), the STFT of f (t ) is a simple gaussian integral and is given by [27]

 g S f (t ,

ω) = e

j φ(t )

σ

Q  jk (k) ˆ (k) A (t )h (ω − φ  (t )), 2  k! 1 − j σ φ (t )

2π

k =0

(20) where hˆ (ω) = exp(− σ2

2

ω

2

1− j σ 2 φ  (t )

). Strictly, expression (20) should

be a approximate equality, but for convenience and illustrative purpose, an exact equality is employed. ˆ (t , ω) is derived: From (20), the IF estimate ω

 Q −1

jk (k+1) A (t )hˆ (k) ( k! k j A (k) (t )hˆ (k) ( k =0 k ! g ∂ ∂ ω S f (t , ) }. g S f (t , )

ω − φ  (t ))

k =0 ωˆ (t , ω) = φ  (t ) + {  Q

− φ  (t ) {

ω − φ  (t ))

ω

ω

}

(21)

In particular, we consider the simple case of Q = 1. In this case, it is easy to find that the set of points ω = φ  (t ) satisfies ∂ g ∂ ω S f (t , ω) g

S f (t , ω)

} = ω.

(22)

This result shows that SET, which retains the TF points of ω = ωˆ (t , ω), inevitably introduces a error in the extraction of the IF of the strongly AM-FM signals. A simple example is the non-constant c 2

amplitude linear chirp signal f (t ) = ate j (bt + 2 t ) defined in a finite time interval, whose IF law satisfies equation (22), not the relation ω = ωˆ (t , ω).

is given by



σ 2 (ω − φ  (t ))2 exp(− ω) = f (t )σ ), 2 2 1 − iσ c 1 − jσ c 2π

2.3. Error analysis for SET reconstruction

(16)

where φ  (t ) = b + ct. By simple calculation, it is easy to get:

ωˆ (t , ω) = {

(19)

k =0

ωˆ (t , ω) + φ (t ) {

ω = ωˆ (t , ω) implemented by the operator (7) provides a good

−

Q  A (k) (t ) k μ, k!



2.2. Performance of the SET-based IF estimation

g (t ) = e

A (μ + t ) ≈

(13)

where the symbol “⇔” denotes the equivalence relation. Since g  (t ) = − σ12 t g (t ) for the Gaussian window g (t ), we obtain g S f (t ,

(18)

2

∂ g ∂ t S f (t , ω) g j S f (t ,

ω)

} = φ  (t ) +

σ 4 c 2 (ω − φ  (t )) . 1 + σ 4c2

(17)

Equation (17) shows that the set of points ω = φ  (t ) satisfies ω = ωˆ (t , ω). That is, the IF estimate by SET is exact for the constant

As presented in (4), only when the amplitude and the phase of the signal satisfy | A  (t ) |< ε , | φ  (t ) |< ε (i.e., f (μ + t ) ≈  A (t )e j (φ(t )+φ (t )μ) ), can the SET reconstruction formula (8) work well. However, this constraint condition is too restrictive as many real signals, such as vibration signals [28], audio signals [29], are made up of very strongly modulated modes. Only using the firstorder derivative information to recover such strongly AM-FM signals, like SET reconstruction does, inevitably results in a serious error.

X. Zhu et al. / Digital Signal Processing 93 (2019) 172–186

175

Fig. 1. Reconstructed results of the SET method. (a) Reconstruction SNRs with the variation of parameter a, the parameter b being set to 0.1. (b) Reconstruction SNRs with the variation of parameter b, the parameter a being set to 0.

Indeed, evaluating the STFT in (20) along the IF φ  (t ), we have



S f (t , φ  (t )) = f (t )σ

2π

g

1 − j σ 2 φ  (t )



+ e j φ(t ) σ

hˆ (0)

Q  jk (k) ˆ (k) A (t )h (0). k! 1 − j σ 2 φ  (t )

2π

k =1

(23)

√

2 2 2π , and hˆ (ω) = exp(− σ2 1− j σω2 φ  (t ) ), which satisfies hˆ (0) = 1, and hˆ (2k−1) (0) = 0 for k = 1, 2, · · · , Q 2 = Q2 , (here · is to take its integer portion), so (23) can be further expressed as

Since gˆ (0) = σ

T (t , φ  (t )) gˆ (0)

S f (t , φ  (t )) g

=

gˆ (0)

+

information of the amplitude is considered. In the following, a concise and general IF equation is derived by chirplet transform (CT) [30,31]. CT is an outstanding TFA method and generalizes the STFT by using an extra CR parameter, which is defined as

=

f (t )

g C f (t ,

Q2  j 2k (2k) ˆ (2k) A (t )h (0). 1 − j σ 2 φ  (t ) k=1 (2k)!

ω, β) =

(25)

g

+∞ Q  j (φ  (t )−β)μ2 1 (k)  2 A (t ) μk g (μ)e e − j (ω−φ (t ))μ dμ. k! k =0

−∞

(26) If β ≡ φ  (t ), then (26) can be written as a more simple form: g

C f (t , ω) = e j φ(t )

Q  jk (k) A (t ) gˆ (k) (ω − φ  (t )). k!

(27)

k =0

g

The transform C f (t , ω) is called matching or adaptive CT because the fact that β = φ  (t ). From (27), one can then show that the set of points ω = φ  (t ) satisfies

{

∂ g ∂ t C f (t , ω) g

C f (t , ω)

+ φ  (t )

g

where | C f (t , w ) |>

g ∂ ∂ ω C f (t , ω) g

C f (t , ω)

} = ω,

(28)

γ (the parameter γ > 0 is a hard threshold

g C f (t , w )

g

on | | to overcome the shortcoming that | C f (t , w ) |≈ 0). Expression (28), indeed, is the desired IF equation, and this result is different from the high-order IF estimators presented in [12,20] because (28) is an equation relationship which the IF satisfies, not the IF estimation. Improving the TF resolution by IF equation, such as equation (28), is a new way, and further investigations are worthy and needed. According to IF equation (28), we can define a new synchroextracting operator that is written as



3.1. SECT The STFT of a AM-FM signal f (t ) = A (t )e j φ(t ) under the hypotheses (18) and (19) is expressed by (20). Directly from (20), it is difficult to obtain an accurate IF equation because the high-order

e − j ω(μ−t ) dμ,

C f (t , ω, β)

(24)

3. Synchroextracting chirplet transform

j β(μ−t )2 2

−∞

= e j φ(t )

From (24), we can observe that: when the amplitude of signal slowly varies (e.g., | A  (t ) |< ε ), a smaller window width can effectively eliminate the reconstruction error. However, a small window width leads to a coarse frequency resolution, so this way is unworkable in practice. Another, the chirp rate φ  (t ) and the high-order derivatives of amplitude (i.e., A (2k) (t )) degrade the performance of the SET reconstruction when the window length is fixed. Generally, the larger the strengths of the φ  (t ) and A (2k) (t ) are, the greater the error is. To validate this point experimentally, a AM-FM signal, that is f (t ) = (1 + at 3 ) cos(2π × (20t + bt 3 )) (t ∈ [0 8]), is employed. This test signal contains two parameters a and b. In our experiments, we make one parameter vary in a certain range and another as a fixed value. Fig. 1 shows the reconstructed results of SET, where a Gaussian white noise with SNR = 13 dB is introduced. From the results, we can see that with increase of parameters a, b, the reconstruction SNRs decrease, which means that the stronger the signal modulation, the worse the reconstruction performance of SET is. In general, the reconstruction error is proportional to the strengths of the amplitude and frequency modulations of the signal.

f (μ) g (μ − t )e −

where β is the CR parameter. Based on CT framework, the CT result of f (t ) under the hypotheses (18) and (19) is obtained, which is given by

1 − j σ 2 φ  (t )

e j φ(t )

+∞

δ(ω − ω˜ (t , ω)) =

1

ω = ω˜ (t , ω),

0

otherwise,

˜ (t , ω) is defined as where ω

(29)

176

X. Zhu et al. / Digital Signal Processing 93 (2019) 172–186

∂ g ∂ t C f (t , ω)

ω˜ (t , ω) = {

g

C f (t , ω)

g ∂ ∂ ω C f (t , ω)

+ φ  (t )

g

C f (t , ω)

g

}.

(30)

Therefore, a novel TF representation called the synchroextracting chirplet transform (SECT) is proposed, that is,

T c (t , ω) =

g C f (t ,

ω)δ(ω − ω˜ (t , ω)).

(31)

Obviously, in order to implement SECT in practice, the CR φ  (t ) of the signal is required to well match. Indeed, one can directly use the existing CR estimation methods in TF domain, such as the methods proposed in [12,32,33]. Beyond this, in the following section, we will introduce an effective alternative which well integrates match step with extraction step.

where C˜ f (t , ω, α ∗ ) and C˜ f (t , ω, α ∗ ) denote the CT of f (t ) obtg

tained by using the window function g  (t ) and t g (t ) respectively. g Similarly, for ∂∂ω C˜ f (t , ω, α ∗ ), we also have

∂ ˜g tg C (t , ω, α ∗ ) = − j C˜ f (t , ω, α ∗ ). ∂ω f Combining (37)–(39), we have g

ω˜ (t , ω) = ω − {

C˜ f (t , ω, α ∗ )

Fs

β=

2T s

tan(α ),

π π α ∈ (− , ), 2

(32)

2

where T s denotes the sampling time, and F s the sampling frequency. Suppose that the parameter α has N α values, as

αk = −

π

kπ

+

}.

(40)

δ(ω − ω˜ (t , ω)) =

⎧ ⎪ ⎨

C˜

g

(t ,ω,α ∗ )

1 { ˜ fg } = 0, C f (t ,ω,α ∗ )

⎪ ⎩0 otherwise.

(41)

Considering the calculation error, it is suggested that (41) be rewritten as

δ(ω − ω˜ (t , ω)) =

⎧ ⎪ ⎨

C˜

g

(t ,ω,α ∗ )

1 | { ˜ fg } |< ω/2, C f (t ,ω,α ∗ )

⎪ ⎩0 otherwise,

(42)

where ω is the discrete frequency interval.

(33)

C˜ f (t , ω, αk ) g

According to (27), we can deduce the following approximate expression (i.e., Q = 1)

C f (t , ω) ≈ A (t )e j φ(t ) gˆ (ω − φ  (t )) + j A  (t )e j φ(t ) gˆ  (ω − φ  (t )). g

(43)

+∞ f (μ) g (μ − t )e

Fs − j 2T tan(αk )(μ−t )2 /2 − j ω(μ−t ) s

e

dμ.

(34)

−∞

Evaluating the SECT along the φ  (t ), we obtain that

T c (t , φ  (t )) = C f (t , φ  (t )) ≈ A (t )e j φ(t ) gˆ (0) + j A  (t )e j φ(t ) gˆ  (0). g

Due to a well match leads to the TFR around its IF having high g energy concentration and the amplitude | C˜ f (t , ω, α ) | reaching the maximum among all values. Thus, we can derive the best value of α from the amplitude of | C˜ gf (t , ω, αk ) |, i.e.,

α ∗ = arg max | C˜ gf (t , ω, αk ) | .

(35)

k

(44) A more concise expression of (44) can be derived when the window g (t ) is a real even function because in this case gˆ  (0) equals zero. Therefore, original signal f (t ) can be effectively estimated by its SECT on the ridge, i.e.,

g

Then, the matching CT C f (t , ω) (27) can be obtained by



g C f (t ,

g

3.3. Reconstruction

, k = 1, 2, · · · , N α .

2 Nα + 1 Based on this discretization, the chirplet transform (25) with regard to parameter αk can be rewritten as

=

C˜ f (t , ω, α ∗ )

Then, our proposed synchroextracting operator (29) can be rewritten as

3.2. Algorithm implementation For CT, its amplitude in the IF will achieve the maximum if the parameter β is consistent with the CR of the signal [34,35]. Based on this fact, we can utilize a series of discrete β to get optimized parameter β such that β ≈ φ  (t ). Similar to the discretization method used in [34], we introduce a rotating parameter α , that is,

(39)

ω) =

C˜ f (t , ω, α ∗ ) if | C˜ f (t , ω, α ∗ ) |> γ , g

f (t ) ≈

g

0

otherwise.

(36)

˜ (t , ω) is derived by Based on equation (36), the quantity ω

ω˜ (t , ω) = {

∂ ˜g ∗ ∂ t C f (t , ω, α ) g C˜ (t , ω, α ∗ )

+ β∗

f

∂ ˜g ∗ ∂ ω C f (t , ω, α ) }, g C˜ (t , ω, α ∗ )

(37)

f

where | C˜ f (t , ω, α ∗ ) |> γ and β ∗ = g

Fs 2T s

tan(α ∗ ).

For more precise parameter estimation, ∂∂t C˜ f (t , ω, α ∗ ) can also be calculated by g

∂ ˜g C (t , ω, α ∗ ) ∂t f +∞ ∂ ∗ 2 = ( f (μ) g (μ − t )e − j β (μ−t ) /2 e − j ω(μ−t ) dμ) ∂t −∞ g

= −C˜ f (t , ω, α ∗ ) + j β ∗ C˜ f (t , ω, α ∗ ) + j ω C˜ f (t , ω, α ∗ ), tg

g

(38)

T c (t , φ  (t )) gˆ (0)

.

(45)

Compared with the SET reconstruction (8), the SECT reconstruction (45) is more general and accurate because it relies on a second-order local expansion of phase function, which can properly eliminate the reconstruction error generated by CR of the signal. Moreover, SECT is based on a more precise IF estimation operator, therefore SECT reconstruction can effectively reduce the error caused by the inaccuracy of IF estimate at some level. 4. Simulation test In this section, we use two numerical examples to illustrate the performance of SECT comparing with other advanced TFA methods, including STFT, SST [11], RM [9], Second-SST [12] and SET [24]. Note that, the parameters γ and N α of SECT should be chosen appropriately. In our experiments, similar to hard threshold methods used in [11,36], the parameter γ is simply selected as: γ = 0.3  f 2 , where  · 2 denotes l2 -norm. The selection criteria for N α in general is that the faster the IF varies, the larger the

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Fig. 2. TF representations obtained by (a) STFT, (b) SST, (c) RM, (d) Second-SST.

value of increase eter N α function

N α is. However due to a large N α (e.g., N α > 100) will the computational cost of SECT, the range of the paramis recommended as [6 50] in experiments. The window used in these methods is unified as the Gaussian window.

[20] for specific formula) of these TF results and list them in Table 1, where a lower Rényi entropy value denotes a more energyconcentrated TFR [20,24]. It can be seen that, the SECT result has the lowest Rényi entropy, which shows that SECT can generate the most energy-concentrated TFR among all TF analysis methods.

4.1. Monocomponent signal test The following monocomponent signal is used to test the performance of the proposed approach.

f (t ) = (1 + 0.1 sin(20π t )) cos(80π t + 5 sin(2π t )),

(46)

whose IF is φ  (t ) = 40 + 5 cos(2π t ). The sampling frequency of this signal is 128 Hz, and the time duration is [0 4]. 4.1.1. Performance comparison of different TFA methods The TFRs of the test signal (46) obtained by STFT, SST, RM, Second-SST, SET and SECT (N α = 15) are shown in Fig. 2 and Fig. 3, where we add the Gaussian white noise to this signal with SNR = 15 dB, and the window width is taken as 85 points. For the result of STFT shown in Fig. 2(a), it suffers from a poor TF resolution due to the limitation of Heisenberg uncertainty principle. For the SST, the energy of TFR is higher in the harmonic part than that in the modulated part, and it cannot achieve a high TF resolution for such modulated signal since SST squeezes the TF coefficients only in the frequency direction. As extension and improvement of SST, RM and Second-SST methods provide more energy-concentrated TF results than the two before, as shown in Fig. 2(c-d). From Fig. 3, it is clear that our proposed SECT method generates the TF result with narrower energy distribution and larger TF amplitude than SET method. Compared with the TF results of STFT, SST, RM and Second-SST, SECT achieves the best TF performance, leading to the result almost identical to the ideal TF spectrogram. Furthermore, to evaluate the energy concentration of different TFA methods quantitatively, we calculate the Rényi entropies (see

4.1.2. IF estimation comparison In this subsection, we test the ability of SECT on detecting the IF feature of the signal to validate the performance of SECT against noise. The STFT and SET are selected as the comparison reference. We add a Gaussian noise with different SNR into the signal (46). The IF can be detected by the peak data from the TFR, and the detected result can be evaluated by the mean relative error (MRE), which is calculated by

MRE =

1 NI



IF − IF IF

1 ,

(47)

where N I is the discrete length of the IF,  · 1 denotes l1 -norm, IF is the original clean IF, and IF represents the estimated IF. Fig. 4 shows the calculated errors of estimated IF by the three methods. It can be seen that the IF estimated by SECT is more accurate than by STFT and SET, and SECT result has a stronger noise robustness. Compared with SET method, STFT achieves a more accurate IF estimation in low SNR (e.g. ≤4), and has a similar performance in high SNR. This also tells us that although both of the estimated IFs by STFT and SET correspond to the same STFT ridge, but strong noise lowers the performance of SET. Furthermore, to illustrate the detected procedure more clearly, we list the TFRs and the estimated IFs under SNR = 0 dB. As shown in Fig. 5, the IF detected by STFT and SET occurs a large deviation with the true IF in low SNR environment. For the estimated IF by SECT, it shows satisfactory accuracy with the true IF.

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Fig. 3. TF results of the SET and SECT methods. (a) SET result, (b) zoom of the SET, (c) SECT result, (d) zoom of the SECT.

Table 1 Rényi entropy comparison of the six TF analysis methods. TF methods

STFT

SST

RM

Second-SST

SET

SECT

Rényi entropy

14.8124

12.5589

11.1513

10.6816

9.9264

9.2631

where f 0 is the original clean signal, and ˜f represents the reconstruction. Fig. 6 displays the performance of ISTFT (ds =45, where ds denotes the reconstruction bandwidth), SST (ds=15), SET and SECT methods over different noise levels. From the results we can see that SECT clearly outperforms SET, especially in high SNR environment. SECT obtains higher reconstruction SNRs than ISTFT and SST in low SNR, which means that SECT has the advantage of noise robustness. In addition, Fig. 7 shows the reconstruction signals and the reconstruction errors at 0-1 s when the input SNR is fixed as 6 dB. Obviously, SECT provides the best reconstruction among the selected methods. That is due to the reconstructed region in TF plane obtained by SECT is smaller, the noise introduced into the reconstructed signal will be less. Moreover, SECT takes advantages of the second-order derivative information of the phase function, which performs better reconstruction for strong modulation signals.

Fig. 4. The errors of detected IF by STFT, SET and SECT.

4.1.3. Signal reconstruction comparison In the following experiments, we test the ability of SECT reconstructing the signal of signal (46) compared to three commonly used methods, i.e., inverse STFT (ISTFT), SST and SET reconstruction (8). The reconstruction signal-to-noise ratio (SNR) will be used to measure the quality of a reconstructed signal, which is defined as

SNR = 10 log10 (

 f 0 22 ),  f 0 − ˜f 22

4.2. Multicomponent signal test In this section, SECT method is applied to multicomponent signal analysis. We first consider a two-component nonstationary signal, including a strongly time-varying mode and a weakly timevarying mode, that is,

(48) f (t ) = f 1 (t ) + f 2 (t ) + n(t ),

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Fig. 5. When SNR = 0 dB, TF result and the detected IF by STFT (top), TF result and the detected IF by SET (middle), TF result and the detected IF by SECT (bottom), where the original IF (black - -), the estimated IF (red -). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

f 1 (t ) = (1 + 0.2 cos(10π t ))

× cos(2π (340t − 2 exp(−2t + 0.4) sin(14π (t − 0.2)))), f 2 (t ) = (1 − 0.3 exp(−0.5t ))

× sin(2π × (100t + 10t 2 + 120(t − 0.4)3 )), (49) where n(t ) is a Gaussian noise with the SNR = 10 dB. The sampling frequency is 1024 Hz, and time duration is [0 1]. Fig. 8 presents the TFRs of the signal (49) generated by STFT, SST, RM and Second-SST. Obviously, STFT spreads out in a large area around IF, which results in a low resolution. SST and SecondSST can provide an energy-concentrated result for mode f 2 (t ). However, for mode f 1 (t ), the results smear heavily. RM describes TF content properly for f 1 (t ) as well as f 2 (t ), which denotes that RM is more suitable for characterizing the fast time-varying features than SST. Besides, the TF results of SET and SECT (N α = 25) are displayed in Fig. 9. Obviously, the proposed SECT approach generates a more accurate and concentrated TF result, and has obvious advantage in addressing strong modulation signals. In addition, Fig. 10 shows the detected IF trajectories from SET and SECT results. The comparative results again prove that SECT

Fig. 6. The SNRs of the reconstructed signal by four methods.

has higher accuracy and better noise robustness than SET. The errors between the original components and the reconstructions obtained by SET and SECT methods are shown in Fig. 11, and the reconstruction SNRs are calculated as 5.6347 dB ( f 1 (t ), SET), 14.7425 dB ( f 2 (t ), SET), 15.8503 dB ( f 1 (t ), SECT), 16.7960 dB ( f 2 (t ), SECT), respectively. It can be seen from Fig. 11 that SET leads to a few

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Fig. 7. Reconstruction results by (a) ISTFT, (b) SST, (c) SET, (d) SECT, where the top row displays the original signal (black - -), the reconstructed signal (red -), the bottom row shows the reconstruction errors.

Fig. 8. TF representations obtained by (a) STFT, (b) SST, (c) RM, (d) Second-SST.

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Fig. 9. TF results of the SET and SECT methods. (a) SET result, (b) zoom of the SET, (c) SECT result, (d) zoom of the SECT.

Fig. 10. IF estimation from the SET and SECT results. (a) SET result, (b) SECT result, where the true IF (black - -), the detected IF (red -), Error1/Error2 denotes the MRE of mode f 1 (t )/ f 2 (t ).

sharp errors due to some gaps are presented in the TFR of SET. Compared to SET method, SECT obtains more accurate reconstructions. So SECT is preferable in reality due to better decomposition accuracy. Next, we consider to address an overlapped multicomponent signal, that is,

f (t ) = f 1 (t ) + f 2 (t ) + f 3 (t ) + n(t )

= cos(2π × (0.35 + 45t − 1.2t 2 )) + (1 + 0.2 cos(0.6π t )) cos(10t + 10t 2 )

(50)

STFT of this test signal and the detected IF ridges from STFT result are illustrated in Fig. 12. Fig. 13 displays the TF result and local feature of the overlapped signal obtained by SECT (N α = 11) method. It can be observed that, in the TF plane around the cross point, SECT fails to characterize the true IF trajectories. Therefore, the SET-based technique is more suitable for addressing the well-separated signals. Future we should explore effective ways to obtain a clear TF representation for overlapped multicomponent signals. 5. Application

+ cos(2π × 200t + 200 sin(2π × 0.1π t )) + n(t ), where n(t ) is a Gaussian white noise with the SNR = 10 dB. The sampling frequency is 128 Hz, and the time duration is [0 16]. The

In this section, two real signals, i.e., an echolocation signal emitted by brown bat [37] and a vibration signal from a test rig, are utilized to validate the capability of SECT approach.

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Fig. 11. The reconstruction results of (a) mode f 1 (t ) obtained by SET, (b) mode f 2 (t ) obtained by SET, (c) mode f 1 (t ) obtained by SECT, (d) mode f 2 (t ) obtained by SECT, where the top row displays the original signal (black - -), the reconstructed signal (red -), the bottom row shows the reconstruction errors.

Fig. 12. Overlapped multicomponent signal. (a) STFT of the signal, (b) the detected IFs.

Fig. 13. TF representation of SECT for overlapped signal. (a) SECT result, (b) zoom of the SECT.

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5.1. Echolocation signal

5.2. Vibration signal

By means of the echolocation signal, the bats can identify the object successfully in the complex environment. Therefore, analyzing the echolocation signal becomes very important in practical application. The bat signal is sampled at 1100 points and its sampling frequency is 140 kHz. The wave form of the signal is displayed in Fig. 14. Fig. 15 and Fig. 16 show the corresponding TFRs associated with the six different methods. It can be seen that, STFT and SST can generate the TFRs for the purely harmonic part (about 2-5 ms) with high energy concentration, but fail to characterize the TF features in the modulated part (about 0.5-1.5 ms). Although RM, Second-SST and SET improve the TF result, but they still suffer from smear effect. For SECT (N α = 15), obviously, it provides us with an intuitive and clear TF result with higher energy concentration. Utilizing the IF trajectory estimated by SECT, the extracted IF ridges are depicted in Fig. 17(a). In the case of signal reconstruction, the comparison between the original bat signal and the reconstructed signal are shown in Fig. 17(b). It can be seen that, the reconstruction error is small compared to the original signal, which again proves that the proposed SECT method has an excellent reconstruction ability.

The instantaneous rotating speed is an important indicator to evaluate the operating condition of rotary machinery, so the estimation of rotating frequency (RF) through vibration signal has been a hot topic in engineering applications [38,39]. In this subsection, experimental data measured from a real test rig (see Fig. 18) is utilized to validate the capability of SECT. The vibration signal is captured by accelerometers under speed fluctuation conditions from 3200 rpm to 3900 rpm. Simultaneously, the rotating speed of shaft is measured by a key phasor. The signal is recorded with a sampling rate of 1024 Hz and about 19 seconds data is used for analysis. The waveform of the vibration signal and its STFT are shown in Fig. 19. The TFR generated by SECT (N α = 15) is shown in Fig. 20(a). Obviously, the TF result is concentrated along the ridge. The extracted IF ridge from SECT result is depicted in Fig. 20(b). From the results, it can be observed that instantaneous RF measured by SECT result almost perfectly matches the estimated result of the key phasor. Furthermore, for comparisons, we calculate the errors between the estimated RF by SET and SECT methods and the result by the key phasor, that is, MRE = 0.0497 (SET), MRE = 0.0326 (SECT). From the results, it clearly interprets that SECT can more accurately reflect RF fluctuation of the vibration signal. Therefore, SECT has a competitive advantage to characterize the features of the AM-FM vibration signals. 6. Conclusion

Fig. 14. Bat echolocation signal.

In this paper, synchroextracting chirplet transform (SECT) was introduced to circumvent the drawbacks of the standard SET method. We analyzed the SET method mathematically, including the existence of the fixed squeeze frequency, and the performances of the IF estimate operator and the SET reconstruction formula. We emphasized that SET would lead to large errors for IF estimation and mode reconstruction when addressing fast-varying

Fig. 15. TF representations obtained by (a) STFT, (b) SST, (c) RM, (d) Second-SST.

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Fig. 16. TF results of the SET and SECT methods. (a) SET result, (b) zoom of the SET, (c) SECT result, (d) zoom of the SECT.

Fig. 17. SECT results. (a) The estimated IF trajectories, (b) summation of the two reconstructed modes and the error.

Fig. 18. The sketch of test rig.

signals. In the SECT, a more precise IF estimate operator was proposed and meanwhile remained a perfect reconstruction ability. The advantages of SECT were demonstrated through simulated and real data comparable to some advanced TF methods. The experimental results indicated that SECT achieves a better performance in concerning TF energy, obtaining a more accurate signal estimate.

Since the accuracy of the estimated CR parameter will influence the performance of the SECT, we should explore a better way to diminish the parameter effect. In addition, it is significant to provide theoretical analysis of the SECT method, and to study the influence of noise on the synchroextracting operator. There are the extension and development of this work.

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Fig. 19. Vibration signal. (a) The waveform of the vibration signal, (b) its STFT.

Fig. 20. Analysis results for the vibration signal. (a) The TF representation by SECT, (b) the detected IF.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the Major Research Plan of the National Natural Science Foundation of China under grant No. 91730306, the National Key R and D Program of the Ministry of Science and Technology of China under Grant no. 2018YFC0603501, and the National Natural Science Foundation of China under Grant no. 51775409. The authors thank J.J. Wang and the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this paper. References [1] S. Liu, T. Shan, Y.D. Zhang, Detection of weak astronomical signals with frequency-hopping interference suppression, Digit. Signal Process. 72 (2018) 1–8. ´ S. Stankovic, ´ T. Thayaparan, L. Stankovic, ´ Multiwindow S-method for [2] I. Orovic, instantaneous frequency estimation and its application in radar signal analysis, IET Signal Process. 4 (4) (2010) 363–370. [3] B. Boashash, A. Aïssa-El-Bey, Robust multisensor time-frequency signal processing: a tutorial review with illustrations of performance enhancement in selected application areas, Digit. Signal Process. 77 (2018) 153–186. [4] C. Park, D. Looney, P. Kidmose, M. Ungstrup, D.P. Mandic, Time-frequency analysis of EEG asymmetry using bivariate empirical mode decomposition, IEEE Trans. Neural Syst. Rehabil. Eng. 19 (4) (2011) 366–373.

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Xiangxiang Zhu received the B.S. degree from Shandong University of Technology, Zibo, China, in 2014. He is currently pursuing the Ph.D. degree in applied mathematics with Xi’an Jiaotong University, Xi’an. He will be a Visiting PhD. student with National University of Singapore, Singapore, from 2019 to 2020. His research interests include digital signal processing, time-frequency analysis, nonnegative matrix factorization and adaptive learning. Zhuosheng Zhang received the M.S. degree in applied mathematics from Xi’an Jiaotong University, Xi’an, China, in 1989. He received Ph.D. degree in information and communication engineering from Xi’an Jiaotong University in 2001. From 2003 to 2006, he was a Research Scientist A in Wavelet and Information Processing Center, Temasek Laboratory of National University of Singapore. He is currently a Professor with the School of Mathematics and Statistics,

Xi’an Jiaotong University. His research interests include wavelet theory, adaptive high-resolution time-frequency analysis, sparsity method and seismic exploration signal processing. Jinghuai Gao received the M.S. degree in applied geophysics from Chang’an University, Xi’an, China, in 1991, and the Ph.D. degree in electromagnetic field and microwave technology from Xi’an Jiaotong University, Xi’an, in 1997. From 1997 to 2000, he was a Post-Doctoral Researcher with the Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China. In 1999, he was a Visiting Scientist with the Modeling and Imaging Laboratory, University of California at Santa Cruz, Santa Cruz, CA, USA. He is also an Associate Director with the National Engineering Laboratory for Offshore Oil Exploration, Xi’an Jiaotong University. He is the Project Leader of the Fundamental Theory and Method for Geophysical Exploration and Development of Unconventional Oil and Gas, Xi’an Jiaotong University, which is a major program of the National Natural Science Foundation of China under Grant 41390450. He is currently a Professor with the School of Electronic and Information Engineering and the School of Mathematics and Statistics, Xi’an Jiaotong University. His research interests include seismicwave propagation and imaging theory, seismic reservoir and fluid identification, and seismic inverse problem theory and method. Dr. Gao was a recipient of the Chen Zongqi Geophysical Best Paper Awardin 2013. He is an Editorial Board Member of the Journal of Chinese Journal of Geophysics and Applied Geophysics and Chinese Science Bulletin. Bei Li received the B.S. degree in Mathematics and Applied Mathematics from University of Jinan, Jinan, China, in 2014. She is currently working toward the Ph.D. degree in information science at School of Mathematics and Statistics, Xi’an Jiaotong University. Her research interests include nonstationary signal analysis, time-frequency signal processing. Zhen Li received the B.S. degree in department of applied mathematics from North University of China, Tai Yuan, China, in 2013. He is currently working toward the Ph.D. degree in the School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include seismic reservoir and fluid representation, time-frequency analysis and signal processing. Xin Huang received the B.S. and M.S. degrees in mechanical engineering from Xinjiang University, Urumqi, China, in 2013 and 2016, respectively. He is currently pursuing the Ph.D. degree in mechanical engineering with Xi’an Jiaotong University. His research interests include mechanical signal processing, mechanical system fault diagnosis and prognosis. Guangrui Wen received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Xi’an Jiaotong University, Xi’an, China, in 1998, 2001, and 2006, respectively. He was a Post-doctoral Fellow with the Xi’an Shaangu Power Company Ltd., Xi’an, from 2008 to 2010. Since 2012, he has been a Professor with the School of Mechanical Engineering, Xi’an Jiaotong University, and also with the School of Mechanical Engineering, Xinjiang University. He is a member of Chinese Mechanical Engineering Society (CIME) and Chinese Society for Vibration Engineering (CSVE). He has published two books and more than 80 articles and held more than 20 items of patents. His research interests include mechanical system fault diagnosis and prognosis, mechanical equipment life cycle health monitoring and intelligent maintenance.