**time-frequency representation**

Cite as: Rev. Sci. Instrum. **90, 064901 (2019);**https://doi.org/10.1063/1.5082776

Submitted: 24 November 2018 . Accepted: 12 May 2019 . Published Online: 03 June 2019 Yongliang Cheng, Jie Shao, Yihe Zhao, Shu Liu, and Reza Malekian

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## An improved separation method

## of multi-components signal for sensing

## based on time-frequency representation

**Cite as: Rev. Sci. Instrum. 90, 064901 (2019);**doi: 10.1063/1.5082776

Submitted: 24 November 2018 • Accepted: 12 May 2019 • Published Online: 3 June 2019

Yongliang Cheng,1,2 _{Jie Shao,}1,2 _{Yihe Zhao,}1,2 _{Shu Liu,}1,2 _{and Reza Malekian}3,a)

AFFILIATIONS

**1**_{Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronauts and Astronauts),}

Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

**2**_{Key Laboratory of Underwater Acoustic Signal Processing, Ministry of Education, Southeast University, Nanjing 210096, China}

**3**_{Department of Computer Science and Media Technology, Malmö University, Malmö 20506, Sweden and Internet of Things}

and People Research Center, Malmö University, Malmö 20506, Sweden

**a)**_{E-mail:}_{reza.malekian@ieee.org}

ABSTRACT

In many situations, it is essential to analyze a nonstationary signal for sensing whose components not only overlapped in time-frequency domain (TFD) but also have different durations. In order to address this issue, an improved separation method based on the time-frequency distribution is proposed in this paper. This method computes the time-frequency representation (TFR) of the signal and extracts the instan-taneous frequency (IF) of components by a two-dimensional peak search in a limited area in which normalized energy is greater than the set threshold value. If there is more than one peak from a TFR, IFs of components can be determined and linked by a method of minimum slope difference. After the IFs are obtained, the improved time-frequency filtering algorithm is used to reconstruct the component of the signal. We continue this until the residual energy in the TFD is smaller than a fraction of the initial TFD energy. Different from previous methods, the improved method can separate the signal whose components overlapped in TFR and have different time durations. Simulation results have shown the effectiveness of the proposed method.

Published under license by AIP Publishing.https://doi.org/10.1063/1.5082776

I. INTRODUCTION

In many research fields, such as radar, sonar, and mechanical engineering, the signals received by sensors which have time-varying frequency contents are always multicomponent nonstationary sig-nals.1–5Therefore, it is necessary to separate components from the signal in order to effectively analyze them, some of which may overlap in the time-frequency domain (TFD).

Time-frequency representation (TFR) is a powerful processing tool for a multicomponent nonstationary signal. When dealing with the multicomponent signal, linear transform TFRs do not produce cross terms and have the advantage of low computation complexity. However, they are not robust to noise and have a low time-frequency resolution. Nonlinear transform TFRs overcome the drawback of

linear transform TFRs, but they suffer from serious cross terms. In order to obtain high time-frequency resolution and cross terms sup-pressed TFRs, a series of new time-frequency analysis methods are proposed in recent years.6–14

When a TFR is obtained, the instantaneous frequencies (IFs) of signal components can be extracted in the time-frequency domain. For a monocomponent signal, its IF is denoted by a thin and energy concentrative curve called “ridge.” By extracting the accurate posi-tion of the ridge, the IF of a signal can be got. However, how to extract IFs from the TFR of the multicomponent signal is still a problem, even if a noise free, cross terms are suppressed and high time-frequency resolution TFR is obtained.

In the past decades, many signal decomposition methods have been proposed to separate the multicomponent signal. Some

representative methods include the empirical mode decomposition and its improvement,15,16the synchrosqueezed wavelet transform,17 the empirical wavelet transform,18the variational mode decompo-sition,19 the nonlinear mode decomposition,20 data-driven time-frequency analysis,21sparse time-frequency decomposition,22 varia-tional nonlinear chirp mode decomposition,23adaptive chirp mode pursuit,24 and so on. However, most of the decomposition meth-ods require that the components of the signal satisfy strict separation conditions in their time-frequency plane, and thus, they cannot sep-arate close and overlapped signal components in the time-frequency plane completely.

Recently, some approaches are proposed in order to separate overlapped signal components. Barkat presents two signal separa-tion algorithms in Ref.25. One of them assumes that all compo-nents exist at all time instants in the time-frequency plane and any component intersection is a cross point. A method based on com-pressive sensing is put forward in Ref.26. It can effectively separate nonstationary and stationary signal components overlapping in the time-frequency plane. The intrinsic chirp component decomposi-tion (ICCD) in Ref.27can separate overlapped signal components by using a joint-optimization technique. A multicomponent signal separation method based on ridge path regrouping (RPRG) and ICCD is proposed in Ref.28. It can extract all IFs of components from TFR and regroups them using ridge path regrouping.

It can be found that separating multicomponent signals with overlapping components and different time durations is still a chal-lenging task for the methods above. To solve the issue, an improved method based on TFR is proposed in this paper. The method com-putes the TFR of the signal and extracts the IF of a component by finding the peaks of each time slice of TFR. Then, the minimum slope difference method is used to determine and link IF of time instants. Next, reconstruct the component by an improved time-frequency filtering method and remove it from the original signal. Finally, based on the improved energy criterion, the above steps are repeated for the residual signal until all components are extracted from the original signal.

This paper is organized as follows: In Sec.II, we briefly describe some IF separation methods of the multicomponent signal. Then, our improved method is proposed in Sec.III. In Sec.IV, simula-tion results and discussion of multicomponent signal separasimula-tion are provided to demonstrate the effectiveness of the proposed method. Finally, we draw a conclusion in Sec.V.

II. METHODS A. Signal model

A multicomponent nonstationary signal considered in this
paper is
S(t) =
K
∑
k=1
Ak(t) exp[ j ⋅ φk(t)] + w(t), (1)
where K is the number of signal components, Ak*(t) and 𝜑k(t) denote*
the instantaneous amplitude (IA) and the instantaneous phase of kth
signal components, respectively, and w(t) is a white Gaussian noise.
B. Time frequency representation

To capture the time-varying signatures of the nonstationary signal, a TFR is needed. As the earliest TFR, short time Fourier

transform (STFT) is one of the most widely used TFR, which has the advantage of cross terms free and low computation complexity. STFT is defined as follows: TFR(t, f ) = STFT(t, f ) =∫ ∞ −∞gσ(τ)S(t + τ)e −j2πf τ dτ, (2)

where gσ(τ) is a Gaussian window with length of σ, S(t) is the sig-nal to be transformed, and TFR(t, f ) is the obtained TFR. When a TFR of the signal is obtained, IFs of the components can be extracted from it.

C. IFs extraction algorithm of multicomponent signal

IFs can be extracted by ridge detection from a TFR. The ridge detection finds a maximum energy point of the TFR in the frequency slice at each time instant and considers the corresponding frequency as the IF of current time instant. It can be denoted as

IF(t) = arg max
**F**

∣TFR(t, F)∣, t = t0, . . . , tN−1, (3)
**where F is the set of frequency bins and N is the number of sampling**
points.

An efficient method sharing the idea of formula(3)proposed in Ref.25is introduced in Algorithm1, where TFR(t, f ) is the TFR of signal S(t), ∆f is the allowable frequency variation between two consecutive time instants, ε is the threshold of TFR energy, and IF(t) is the extracted IF.

This method can extract the IFs from the strongest to weak-est one by one. However, if there are overlapped areas of compo-nents in the time frequency plane, the method will fail to extract them.

In Ref.28, the RPRG first detects the ridges of a multicompo-nent signal from a TFR and then extracts the desired IFs by regroup-ing the ridge curves accordregroup-ing to their variation rates at the intersec-tions. After the IFs are obtained, component separation is achieved by using ICCD.

III. IMPROVED SEPARATION METHOD OF MULTICOMPONENTS SIGNAL

The RPRG + ICCD method can separate the overlapped com-ponents in the time-frequency domain completely. However, it can-not separate the components existing at different time periods well and mixes unexpected frequencies.

With the aid of a method that stops the extraction algorithm in Ref.24, a new adaptive energy criterion based on residual time fre-quency energy is proposed. The flow chart of the improved method is given inFig. 1.

The first step of the algorithm consists in noise thresholding to remove the undesired “low” energy peaks in the time-frequency domain and obtain the original total energy TE1, where ε1 is a properly selected threshold (in our simulations, we used ε1= 0.05). Then, the various components are extracted sequentially. Normally, the overall energy in the TFR becomes smaller and smaller after each extraction and after the last component has been retrieved the remaining energy in the TFR matrix should be a fraction of the original one. It is the energy criterion that stops the extraction algorithm.

**ALGORITHM 1. Components separation procedure proposed in Ref.**25.
**Input**: TFR(t, f ), ∆f, ε

1: Compute total time-frequency energy TE1= N−1 ∑ i=0 M−1 ∑ j=0

∣TFR(ti, fj)∣, where M is the point number of Fourier transform. 2: Find (tmax, fmax) =arg max

t,f

∣TFR(t, f )∣, and set TFR(tmax, F), F ∈ [ fmax− ∆f, fmax+ ∆f ] 3: Initialize the center frequencies of searching to the left and right fL= fR= IF(tmax) = fmax 4: for r = max + 1, ⋯, N−1, and l = max − 1, ⋯, 0

5: IF(tr) =fR= arg max
**F**∈[ fR−∆f ,fR+∆f ]

∣TFR(tr, F)∣, and then set TFR(tr**, F), F ∈ [fR**− ∆f, fR+ ∆f ]
6: IF(tl) =fL= arg max

**F**∈[ fL−∆f ,fL+∆f ]

∣TFR(tl**, F)∣, and then set TFR(t**l**, F), F ∈ [fL**− ∆f, fL+ ∆f ]
7: Compute the remaining time-frequency energy TE2=

N−1 ∑ i=0 M−1 ∑ j=0∣TFR(ti, fj)∣ 8: If TE2>TE1⋅ε

9: Jump to step 2 to start a new component extraction. 10: Else

11: stop component separation.

**Output**: IF(t), t = t0, ⋯, tN−1

A. Improved ridge detection algorithm

IF can be extracted by formula(3)for the monocomponent
sig-nal, but it is not suitable for the multicomponent signal. It can be
**modified to search local maxima also known as the peaks of F rather**

**than the maximum of F. It is denoted as**

**F**i(t) = arg maxiam
**F**

∣TFR(t, F)∣, t = t0, . . . , tN−1, (4) where i is the number of peaks at time instant t.

**FIG. 1. Flow chart of the method proposed (TFR, e.g.,**

**FIG. 2. Illustration of the possible detected ridge points for two overlapped**

components.

The possible detected ridge points for two overlapped compo-nents is illustrated inFig. 2, where IFk(t) and IFm(t) are the IFs of two crossed components and point A is the cross-point of compo-nents. Point B and point C are the peaks at the same time instant after point A, respectively. ∆fkand ∆fmare the frequency variation of points A, B and points A, C, respectively.

After cross-point A, many methods fail to extract IFs of the overlapped components correctly. For Algorithm1, when ∆fk<∆f, ∆fm<∆f and the amplitude of point B is greater than that of point C, the frequency of point B will be considered as the frequency of com-ponent k, so comcom-ponents k and m cannot be accurately identified. The peak frequencies of points B and C can be obtained by formula

(4). We can compute the minimum distance between them and the frequency of time instant of point A. However, if ∆fm<∆fk, the fre-quencies of component m may be mixed into those of component k. To avoid this drawback, we introduce the minimum slope difference method.

Usually, for the continuous frequency modulated signal, its fre-quency variation in a short time can be considered as linear. Based on this assumption, the adjacent points with nearly equal slope should be considered as belong to the same component.

As shown inFig. 2, our method computes slope differences of ED, DB and ED, DC, respectively. They are denoted as follows:

slopdk= ∣k0−k_{k}∣, (5)
slopdm= ∣k0−km∣, (6)
k0=IF(tE) −IF(tD)
tdis , (7)
km=IF(tD
) −IF(tB)
tdis , (8)
kk=IF(tD
) −IF(tC)
tdis , (9)

where tdis is the time interval between point E to point D and is set to 3%–5% of signal length N.

slopdk is obviously smaller than slopdm, so the frequency of point C should be considered as the frequency of component k rather than component m. Finally, the IF of component k can be extracted completely and correctly.

The ridge detection procedure is introduced in Algorithm2, where inds and inde are the indices of the start time and end time of a component, respectively. thres is the threshold of amplitude when finding the peaks in limited area, trb and tlb save the time instants whose number of peaks is zero, tb is the maximum allow-able breakpoint range, and if there is no peak at a set of con-secutive time instants whose length is greater than tb, the com-ponent is considered as an appearing end point or start point, respectively.

B. Component reconstruction based on improved time frequency filtering

A component can be reconstructed by combining the
instan-taneous amplitude (IA) with the instaninstan-taneous phase. The
instanta-neous phase can be achieved by the integral of IF. IF extracted from
a TFR is not accurate because of the influence of the noise, and the
frequency bins of IF are fixed. So, IF should be smoothed to obtain
a more accurate instantaneous phase. A modified second difference
**matrix 𝛀 is used to smooth IF in Ref.**23. IA extraction algorithm can
be utilized by integrating dechirp with a low-pass filter.29The time
frequency filtering algorithm is improved to filter the dechirp signal
ranging from tindsto tinde.

The component reconstruction procedure is introduced in Algorithm3, where IF(t), inds, and inde are the IF, start time index, and end time index of the component obtained by Algorithm2, respectively, and β is the smooth coefficient.

IV. RESULTS AND DISCUSSION

In this section, some numerical results are provided for demon-strating the advantages of the proposed method.

A. Simulation and parameter settings

All the tests are carried out using MATLAB 2016a. The sam-ple frequency fs is 1000 Hz, the signal length N is 1000, and the frequency bins of FFT are 1024. The number of Monte Carlo sim-ulations is 100. The threshold of time frequency energy ε1= 0.05 and ε2= 0.1.

B. Quantitative results

To quantitatively compare the performances of algorithms, the mean absolute error (MAE) and mean squared error (MSE) are introduced.30They are defined as follows:

MAE = 1 N N−1 ∑ n=0 ∣IFe(tn) −IFo(tn)∣, (10) MSE = 1 N N−1 ∑ n=0 ∣IFe(tn) −IFo(tn)∣2, (11) where IFe(t) is the IF extracted and IFo(t) is the true IF.

**ALGORITHM 2. Ridge detection.**
**Input**: TFR(t, f ), ∆f, tdis, thres

1: Find (tm, fm) = arg max t,f

∣TFR(t, f )∣, and let fL= fR= IF(ttm) = fm
2: Normalize TFR(t, f ), TFD(t, f ) =_{max∣TFR(t,f )∣}∣TFR(t,f )∣

3: Initialize inds = 0, inde = N− 1, j = tm + 1, k = tm − 1, trb = tlb = [], tb = 0.3N 4: While j ≤ N− 1 do

5: Fi(tj)← Argument of maxima of TFD(ti, F), F ∈ [fR− ∆f, fR+ ∆f ] whose amplitude is greater than thres. 6: if i == 0 do

7: trb = [trb, j], tx = min{⌊( j − tm)/2⌋, tdis},IF(tj) = fR= IF(tj−tx) + [IF(tj−tx)− IF(tj−2tx)], j = j + 1 8: else if i == 1 do

9: IF(tj) = fR= F1(tj), j = j + 1

10: else

12: tx = min{⌊( j − tm)/2⌋, tdis}, IF(tj) =fR=arg max Fi(tj)

∣Fi(tj)−IF(t_{tx} j−tx)−IF(tj−tdis)−IF(t_{tx} j−2tx)∣, j = j + 1
13:inde ← continues breakpoints analysis of trb

14: While k ≥ 0 do

15: Fi(tk) ← Argument of maxima of TFD(tj, F), F ∈ [fL− ∆f, fL+ ∆f ] whose amplitude is greater than thres. 16: if i == 0 do

17: tlb = [tlb, k], tx = min{⌊(inde − k)/2⌋, tdis}, IF(tk) = fL= IF(tk−tx) + [IF(tk−tx)− IF(tk−2tx)], k = k− 1 18: else if i == 1 do

19: IF(tk) = fL= F1(tk), k = k− 1

20: else

21: tx = min{⌊(inde − k)/2⌋, tdis}, IF(tk) =fL=arg max Fi(tk)

∣Fi(tk)−IF(t_{tx} k+tx)−IF(tk+tx)−IF(t_{tx} k+2tx)∣, k = k− 1
22:inds ← continues breakpoints analysis of tlb

**Output**: IF(t), t = t0, ⋯, tN−1, inds, inde
1. Example 1

A three-component nonstationary signal is described as
fol-lows:
S1(t) = Sc1(t) + Sc2(t) + Sc3(t) + w(t), (12)
Sc1(t) = A1(t) exp[j2π(200t2)], 0 ≤ t ≤ 1s, (13)
Sc2(t) = A2(t) exp[j2π(−200t + 150t3)], 0 ≤ t ≤ 1s, (14)
Sc3(t) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0, 0 ≤ t < 0.1s
A3(t) exp[j2π(400t − 200t2)], 0.1s ≤ t ≤ 1s, (15)
where w(t) is noise, A1(t) = 1, A2(t) = e−0.5t, and A3(t) = 0.8 ⋅ e0.5t.
**ALGORITHM 3. Component reconstruction.**

**Input**: IF(t), inds, inds, β, S(t)

**1: Effective IF IFnew**←(IF(t)−M/2)f_{M} s, tinds≤t ≤ tiinde
2: Compute effective IF length L = inde− inds + 1

3: Initialize modified second difference matrix Ω =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−1 1 0 ⋯ 0
1 −2 1 ⋯ 0
⋮ ⋱ ⋱ ⋱ ⋮
0 ⋯ 1 −2 1
0 ⋯ 0 1 −1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦L×L
**3: Smooth effective IF Fsmooth**= (2_{β}ΩTΩ**+ I)**

−1

**IF**new, where T is the matrix transpose.
4: phase(tinds) = 2πFsmooth(1)

5: for ii = 2 to L

6: phase(tinds+ii−1) =phase(tinds+ii−2)+ 2π**F**smooth**(ii)+F**_{2f}smooth(ii−1)
s
7: end for

8: Dechirp Sd(t) = S(t) ⋅ e−j⋅phase(t ), tinds≤t ≤ tinde

9: IA(t), tinds≤t ≤ tinde←Estimated by low-pass filter of Sd(t)
10: Component reconstruction ˆSc(t) = IA(t) ⋅ e−j⋅phase(t), tinds≤t ≤ t_{inde}

* FIG. 3. Signal S*1

*(t). (a) Waveform in the time domain. (b) TFR based on STFT.*

The waveform in the time domain and TFR based on STFT of signal S1(t) when SNR = 5 dB are exhibited inFig. 3. The IFs of sig-nal S1(t) separated by the proposed method and RPRG + ICCD are displayed inFig. 4. As shown inFig. 4, both methods can extract all components of signal S1(t). The difference is that the proposed method can identify the start-point of component Sc3(t) and set the frequencies as zero between 0 s and 0.1 s, while RPRG + ICCD mixes unexpected frequencies into Sc3(t).

The IAs extracted by RPRG + ICCD and the proposed method are displayed inFig. 5. As shown inFig. 5, the amplitude of IAs extracted by RPRG + ICCD fluctuates comparatively around the true IAs. The amplitude of IAs extracted by the proposed method fits the true IAs well. However, the error of IAs is large near the end point

* FIG. 4. Extracted IFs of signal S*1

*(t). (a) RPRG + ICCD. (b) The proposed method.*

* FIG. 5. Extracted IAs of signal S*1

*(t). (a) RPRG + ICCD. (b) The proposed method.*

because there are the edge effects of the low-pass filter and the time frequency energy of TFR is not concentrated at the end point.

In order to compare the proposed method with the RPRG + ICCD quantitatively, the MAE and MSE of components of signal S1(t) under the different SNR are listed inTable I. For component Sc1(t), the MAE and MSE of a new method are slightly less than those of RPRG + ICCD under the conditions of SNR ranging from 10 dB to 20 dB, and the MAE and MSE of the new method are almost equal to those of RPRG + ICCD when SNR = 5 dB.

The MAE and MSE for component Sc2(t) of the proposed approach when SNR = 5 dB are reduced 27% and 44% than those of RPRG + ICCD. 20% listed inTable Imeans there is a 20% fail-ure rate of RPRG + ICCD when extracting this component (they are excluded when calculating MAE and MSE).

When SNR = 10 dB, the results of component Sc3(t) show that the MAE and MSE of the proposed method are about 35 and 12 400 fewer than those of RPRG + ICCD, respectively. It means that the proposed method can extract component Sc3(t) effectively, while RPRG + ICCD could not. Because the time frequency energy is not concentrated at the end of component Sc3(t), the MAE and MSE of component Sc3(t) obtained by the new method are obviously greater than those of components Sc1(t) and Sc2(t).

2. Example 2

A four-component nonstationary signal is given as follows: S2(t) = Sc1(t) + Sc2(t) + Sc3(t) + Sc4(t) + w(t), (16)

Sc1(t) = A1(t) exp[j2π(200t2)], 0 ≤ t ≤ 1s, (17) Sc2(t) = A2(t) exp{ j2π[140t + 150 sin(2πt)]}, 0 ≤ t ≤ 1s, (18)

Sc3(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ A3(t) exp{ j2π[−140t − 150 sin(2πt)]}, 0 ≤ t ≤ 0.9s 0, 0.9 s < t ≤ 1s, (19)

**TABLE I. The MAE and MSE of IFs under different SNR.**

MAE MSE

SNR (dB) New method RPRG + ICCD New method RPRG + ICCD

Sc1(t) 20 0.7151 0.7483 1.2169 1.4946 15 0.7215 0.7460 1.2343 1.5051 10 0.7320 0.7515 1.2736 1.4922 5 0.7531 0.7523 1.3406 1.5326 Sc2(t) 20 0.5597 0.8330 1.5730 2.5818 15 0.5689 0.8236 1.6012 2.6410 10 0.5796 0.8340 1.5605 2.6731 5 0.6538 0.8957 1.6722 2.9815 Sc1(t) 20 1.1296 36.6118 52.7316 1.2701 × 104 15 1.3715 36.4747 144.3689 1.2586 × 104 10 1.5966 36.6233 223.5380 1.2615 × 104 5 2.1866 36.6913 437.0432 1.2759 × 104 Sc4(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0, 0 ≤ t < 0.1s A4(t) exp[j2π(400t − 200t2)], 0.1s ≤ t ≤ 1s, (20) where A1(t) = 1, A2(t) = e−0.4t, A3(t) = 1 + 0.3 cos(2πt + π/2), and A4(t) = 0.8⋅e0.4t.

The waveform in the time domain and TFR based on STFT of signal S2(t) when SNR = 10 dB are depicted inFig. 6. The IFs of signal S2(t) separated by the proposed method and RPRG + ICCD are illustrated inFig. 7. It can be seen that both methods separate all components of signal S2(t). The proposed method can correctly extract the component Sc4(t) which starts at time 0.1 s and the com-ponent Sc3(t) which ends at time 0.9 s, while RPRG + ICCD could not.

The MAE and MSE of components of signal S2(t) under the condition of different SNR are listed inTable II.

* FIG. 6. Signal S*2

*(t). (a) Waveform in the time domain. (b) TFR based on STFT.*

According toTable II, the superiority of MAE and MSE for components Sc1(t) and Sc2(t) in the proposed approach is not obvi-ous. However, RPRG + ICCD cannot extract components Sc2(t) and Sc3(t) when SNR = 5 dB. When SNR = 10 dB, the MAEs of com-ponents Sc3(t) and Sc4(t) obtained by an improved approach are 3.0763 and 1.9614, and those of Sc3(t) and Sc4(t) by RPRG + ICCD are 19.3987 and 36.2974, respectively. It is demonstrated that the proposed method almost completely extracts components Sc3(t) and Sc4(t), while RPRG + ICCD fails to do it.

3. Example 3

This example is still a four-component signal, which is described as follows:

S3(t) = Sc1(t) + Sc2(t) + Sc3(t) + Sc4(t) + w(t), (21)

**TABLE II. The MAE and MSE of IFs under different SNR.**

MAE MSE

SNR (dB) New method RPRG + ICCD New method RPRG + ICCD

Sc1(t) 20 0.7637 0.7655 1.2376 1.4519 15 0.7723 0.7580 1.2720 1.4467 10 0.7631 0.7606 1.2479 1.4425 5 0.7607 0.7670 1.2383 1.4944 Sc2(t) 20 0.9942 0.9819 1.6337 2.4743 15 1.0232 0.9740 1.8219 1.8292 10 1.0217 1.0135 1.8404 1.7385 5 1.0707 NA 2.1818 NA Sc3(t) 20 2.9389 19.6054 386.0232 3.4958 × 103 15 2.9909 19.3327 393.6796 3.3914 × 103 10 3.0763 19.3987 406.2418 3.4060 × 103 5 3.1884 NA 421.1892 NA Sc4(t) 20 1.6299 36.4080 210.2445 1.2570 × 104 15 1.8229 36.6571 292.6739 1.2763 × 104 10 1.9614 36.2974 353.1569 1.2466 × 104 5 2.2637 36.3920 459.6359 1.2546 × 104 Sc1(t) = A1(t) exp[j2π(200t2)], 0 ≤ t ≤ 1 s, (22) Sc2(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

A2(t) exp{ j2π[140t + 150 sin(2πt)]}, 0 ≤ t ≤ 1 s

0, 0.15 s < t ≤ 1 s, (23) Sc3(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ A3(t) exp{ j2π[−140t − 150 sin(2πt)]}, 0 ≤ t ≤ 0.9 s 0, 0.9 s < t ≤ 1 s, (24)

* FIG. 8. Signal S*3

*(t). (a) Waveform in the time domain. (b) TFR based on STFT.*

Sc4(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0, 0 ≤ t < 0.1 s A4(t) exp[j2π(400t − 200t2)], 0.1 s ≤ t ≤ 1 s, (25) where A1(t) = A3(t) = 1, A2(t) = e−0.4t, and A4(t) = 0.8 ⋅ e0.4t.

The waveform in the time domain and TFR based on STFT of signal S3(t) when SNR = 20 dB are shown in Fig. 8. The IFs of signal S3(t) separated by the proposed approach and RPRG + ICCD are presented in Fig. 9. As shown in Fig. 9, it can be clearly seen that both methods can achieve the extrac-tion of all components of signal S3(t). The proposed method extracts the components with different durations completely, while

**TABLE III. The MAE and MSE of IFs under different SNR.**

MAE MSE

SNR (dB) New method RPRG + ICCD New method RPRG + ICCD

Sc1(t) 20 0.7683 0.7632 1.3328 1.5097 15 0.7710 0.7711 1.3255 1.5467 10 0.7744 0.7743 1.3338 1.5521 5 0.7823 0.7757 1.3395 1.6086 Sc2(t) 20 3.0633 23.7196 326.7530 3.4913 × 103 15 3.0402 24.3193 322.5048 3.6558 × 103 10 3.1028 23.6586 328.3039 3.4614 × 103 5 3.3733 24.6332(20%) 358.2955 3.6633 × 103(20%) Sc3(t) 20 2.6421 19.2998 341.8645 3.3916 × 103 15 2.7015 19.3194 352.6694 3.3987 × 103 10 2.8760 19.2338 381.7343 3.3616 × 103 5 3.2185 19.3058 442.3798 3.3639 × 103 Sc4(t) 20 1.7250 36.9284 249.6887 1.2940 × 104 15 1.6732 36.5844 234.6703 1.2686 × 104 10 1.8922 36.3784 329.0008 1.2537 × 104 5 2.1345 36.6176 422.0073 1.2656 × 104

RPRG + ICCD mixes undesirable frequencies into components Sc2(t), Sc3(t), and Sc4(t).

The MAE and MSE of components of signal S3(t) in the pro-posed method and the RPRG + ICCD are provided in Table III. With the decrease in SNR, the MAE and MSE of component Sc1(t) gradually increase. Due to the existence of start time or end time in components Sc2(t), Sc3(t), and Sc4(t), the MAE and MSE of the pro-posed method are clearly smaller than those of RPRG + ICCD. It is demonstrated inTable IIIthat the proposed method is superior to RPRG + ICCD under the same condition of noise.

V. CONCLUSION

In this paper, an improved separation method of the multicom-ponent signal for sensing based on time-frequency representation is proposed in order to separate the signal whose components over-lapped in the time-frequency domain and have different time dura-tions. This method first computes the TFR of the signal and extracts the instantaneous frequency of a component by a ridge detection procedure based on the minimum slope difference algorithm. Then, reconstruct the component by an improved time-frequency filtering algorithm. By this, a complete component is extracted and then the extracted component is subtracted from the original signal. Finally, based on the improved energy criterion, the above steps are repeated for the residual signal to separate all components one by one.

To show the effectiveness of the proposed method, we ana-lyzed some simulated multicomponent signals. When there are com-ponents of the signal throughout the period of time, the perfor-mance of the proposed method is similar to that of RPRG + ICCD for linear frequency modulation (LFM) signals. However, for the high order frequency modulation signal, such as the quadratic fre-quency modulation (QFM) and sinusoidal frefre-quency modulation (SFM) signal, the improved method performs obviously better than

RPRG + ICCD. When the components of a signal do not exist at all time instants, the MAE and MSE of our approach can be reduced by up to 89% and 90%, respectively, in comparison with the RPRG + ICCD. IFs extraction is improved significantly by our method.

Simulation and experimental results have demonstrated the superiority of the proposed algorithm to separate the signal whose components have different durations and overlapped in the time-frequency domain. After the signal components are effectively separated, the parameters of the components can be estimated accurately.

ACKNOWLEDGMENTS

This work was supported in part by the Open Project Program of the Key Laboratory of Underwater Acoustic Signal Processing (Southeast University), China’s Ministry of Education (Grant No. UASP1604), and the Fundamental Research Funds for the Central Universities (Grant No. 2242016K3001), a project funded by the Pri-ority Academic Program Development of Jiangsu Higher Education Institutions.

The authors declare that they have no competing interests. NOMENCLATURE

TFR time frequency representation IFs instantaneous frequencies

ICCD intrinsic chirp component decomposition RPRG ridge path regrouping

STFT short time Fourier transform

IA instantaneous amplitude

MAE mean absolute error

MSE mean square error

QFM quadratic frequency modulation SFM sinusoidal frequency modulation REFERENCES

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