Time delay estimation based on variational mode decomposition

ADE688587 1 6 Special Issue Article Advances in Mechanical Engineering 2017, Vol 9(1) 1–6 � The Author(s) 2017 DOI 10 1177/1687814016688587 journals sagepub com/home/ade Time delay estimation based on[.]

Advances in Mechanical Engineering

2017, Vol 9(1) 1–6

Ó The Author(s) 2017 DOI: 10.1177/1687814016688587 journals.sagepub.com/home/ade

Time delay estimation based on

variational mode decomposition

Abstract

In order to improve the time delay estimation of colored noise signals, this article proposes generalized cross-correlation time delay estimation based on variational mode decomposition First of all, we put forward the signal energy detection criterion to extract the effective signal from the signal, which can reduce the amount of calculation and improve the real-time performance Second, the effective signal is decomposed into a number of intrinsic mode functions using variational mode decomposition The correlation coefficients of each intrinsic mode function and the original signal are calculated The article reconstructed signal with intrinsic mode functions which extract useful intrinsic mode func-tions by defaulting the correlation coefficient threshold Finally, this article uses generalized cross-correlation to estimate time delay of the reconstructed signal Theoretical analysis and simulation results show that the accurate time delay esti-mation can be obtained under the condition of color noise by the proposed method The measurement accuracy of the proposed method is 15 times that of the generalized cross-correlation, and the running time of the proposed method is 4.0601 times faster than that of the generalized cross-correlation algorithm The proposed method can reduce the com-putation and the running time of the system and also improve the measurement accuracy

Keywords

Variational mode decomposition, time delay estimation, generalized cross-correlation, intrinsic mode function, correla-tion coefficients

Date received: 22 August 2016; accepted: 15 December 2016

Academic Editor: Elsa de Sa Caetano

Introduction

Time delay estimation (TDE) is one of the key

prob-lems in passive acoustic localization based on

micro-phone array, and it has very important theoretical

significance and practical application value.1–4Typical

estimation methods are adaptive TDE, weighted

gener-alized phase TDE, least mean square (LMS) TDE, and

generalized cross-correlation (GCC) TDE based on

empirical mode decomposition (EMD), which are

improved on the basis of cross-correlation TDE

Although the GCC TDE has many advantages, its

defects are obvious It not only requires that the signal

is stationary, the signal and noise are independent, but

also requires a priori knowledge of the signal and noise

In the real sound field environment, due to the presence

of reverberation and colored noise, the prior knowledge

of the signal and the noise cannot be known com-pletely, and in some applications, the noise is often associated with the signal

The existing TDE methods have some limitations in different degrees, which limit the field of application and reduce the precision of the TDE To solve the problem, pre-filtering of the received signal is presented

to improve the output signal-to-noise ratio (SNR) to

1

School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, China

2

School of Electricity and Information Engineering, Northeast Petroleum University, Daqing, China

Corresponding author:

Dong Ye, School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China.

Email: yedong@hit.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

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make the relevant peak more sharper, but this need to

master some of the statistical characteristics of the

sig-nal Multi-scale decomposition, such as wavelet

analy-sis and empirical mode decomposition, is a new way to

solve the problem of the TDE It can separate the noise

and signal at different scales and improve the

corre-sponding SNR of the signal.5–10 In 2014,

Dragomiretskiy et al.11 proposed a new multi-scale

decomposition method, namely, variational mode

decomposition (VMD) The central frequency and

bandwidth of each component are determined by an

iterative search for the optimal solution of the

varia-tional model in the process of obtaining the

decomposi-tion component, so as to realize the effective separadecomposi-tion

of signal and noise adaptively Therefore, it is a better

method to pre-process the noise signal

In this article, a TDE algorithm based on VMD and

GCC techniques are proposed The algorithm first uses

the inverse square law and the size of the microphone

array sound signal transmission to extract the effective

signal, thereby reducing the amount of calculation and

improving the real time Second, it decomposes the

effective signal into a plurality of the intrinsic mode

functions (IMF) by VMD and calculates the

correla-tion coefficient of the IMF and the original signal, then

determines the IMF number of reconstructed signal

using the default correlation coefficient threshold

Finally, the time delay value of reconstructed signal of

each channel is estimated by the GCC Theoretical

analysis and simulation results show that this method

can obtain more precise TDE under the condition of

non-steady signal and existing correlation noise among

the various channels of the collected signal Also, the

presented method greatly reduces the amount of

com-putation and has a certain practicality and robustness

GCC TDE

The time delay of arrival (TDOA) value can be

esti-mated by calculating the correlation function between

the two signals received by the microphone, because

there is a certain correlation between the signals from

the same source According to the principle that the

cross-power spectrum density function of two signals is

exactly the Fourier transform of the correlation

func-tion, the cross-power spectrum of the two signals can

be obtained by the Fourier transform of the correlation

function

Gxixj(t) = aiajGss(t) + Gninj(t) ð1Þ

where Gxixj(t) is the power spectrum of the

cross-correlation function of two signals, Gss(t) is the power

spectrum of the autocorrelation function of the sound,

and Gninj(t) is the power spectrum of the

autocorrela-tion funcautocorrela-tion of the noise

GCC TDE method is to compute the cross-power spectrum of two related signals and weight in the power spectrum domain to highlight related signal part, inhi-bition of noise, and reverberation part, so that the peak

of the correlation function is more prominent Then, inverse transformation to the time domain, ultimately find the delay GCC function is as follows

RGCC(t) =

ð

‘

‘

cij(v)Gxixj(t)eivt ð2Þ

where cij(v) is the weighting function, the introduction

of the weighted function is to improve the SNR, so that

RGCC(t) has a sharp peak

Commonly used weighting functions are phase trans-form (PHAT), Roth processor, and smooth coherent transform (SCOT) However, in the real sound field environment, due to the existence of reverberation and noise, the prior knowledge of the signal and noise is not always known, also the noise is correlated with the sig-nal, so the precision of TDE is not high

VMD

Dragomiretskiy proposed a new method of signal decomposition, which decomposes the original signal into k modal functions based on the center frequency

vk, k of which is the default scale

In the VMD algorithm, the IMF is redefined as an amplitude modulation signal

uk(t) = Ak(t) cos (uk(t)) ð3Þ where Ak(t) and vk(t) are the instantaneous amplitude and instantaneous frequency of uk(t), vk(t) = u0k (t) = du(t)=dt

It assumes that each mode function is the limited bandwidth around the center frequency, by searching constraint variational model, optimal solution to real-ize adaptive signal decomposition, center frequency, and bandwidth of each IMF in the iteration variable sub-optimal solution of the model is updated continu-ously According to the frequency domain characteris-tic of the actual signal, the adaptive decomposition of the signal frequency domain is completed, and a num-ber of narrow band IMF components are obtained The bandwidth of each model is estimated by the fol-lowing steps:

1 By means of the Hilbert transform, the corre-sponding marginal spectrum of each analytic function is calculated

2 The spectrum of each model is transferred to the baseband by modulating the index to the esti-mate center frequency

3 The bandwidth is estimated by the Gauss

smoothness and gradient-squared criterion of

the demodulation signal

The constrained variational problem obtained by the

above procedure is as follows

min

fu k g, fv k g

X

k

∂t d(t) + j

pt

 uk(t)

ejvk t



2

ð4Þ X

k

where uk: =fu1, u2, , ukg is the modal function,

vk: =fv1, v2, , vkg is the center frequency

Above constrained variational problem transforms

to a non-constrained variational problem by

introdu-cing Lagrange multiplier l(t) and penalty factor a The

augmented Lagrange expressions are as follows

L(fukg, fvkg, l) :

= aX

k

∂t d(t) + j

pt

 uk(t)

ejvk t



2 + f (t)X

k

uk(t)











 2

2

+ l(t), f (t)X

k

uk(t)

The alternating direction method of multipliers

(ADMM) is used to solve the variational problems,

and the iterative optimization of uk + 1, vk + 1

k , and

lk + 1 is used to obtain the saddle point of the extended

Lagrange expression Iteration steps are as follows:

1 Initialize u1, v1, l1, n = 0;

2 n = n + 1, execute the whole cycle;

3 Execute the first cycle of the inner layer and

update vk according to vk + 1

k = arg min

u k L (fun + 1

i\k g, fun

ikg, fvn

ig, ln);

4 k = k + 1, repeat step (3), end the first cycle

until k = K;

5 Execute the second cycle of the inner layer and

update l according to ln + 1= ln+

t(f P

kun + 1k );

6 Repeat steps (2)–(7), end the whole cycle until

the iterative stopping criteria

P

k(jjun + 1

k  un

kjj22=jjun

kjj22)\e is met

The central frequency and bandwidth limit are

obtained by the iterative search for the optimal solution

of the variational model The effective components of

each center frequency are adaptively decomposed to

obtain an IMF component in the frequency domain

Therefore, VMD has better accuracy and stability in

feature extraction, which can suppress the correlated

noise

GCC TDE based on VMD

In this article, a GCC TDE method based on VMD is proposed First, the starting point to detect the actual signal to select the effective signal segment greatly reduces the amount of computation Then, the correla-tion coefficient of each IMF component with the origi-nal sigorigi-nal is calculated by the use of VMD to each valid segments of mode decomposition and the IMF that is greater than correlation coefficient threshold is extracted to reconstruct the signal And then, estimate the time delay value of the two reconstructed signals using the GCC

Signal starting point detection criteria

According to the inverse square law of propagation of sound signal, the variation in signal energy is used to determine the starting point and the effective length of the signal in the process of calculating the actual signal time delay, which achieves the purpose of reducing the amount of computation In the ideal case, ignore the change of the internal energy of the medium on the propagation path caused by the sound signal and the influence of environmental noise According to the characteristic of energy spread along the spherical sur-face, the signal energy at the position where the micro-phone is located is as follows

P = I

where I is the energy of the sound source signal and d is the distance from the microphone to the sound source From formula (7), it is known that the energy of the sound signal at certain point in space is proportional to the energy of the source signal and is inversely propor-tional to the square of the distance from the sound source

The frame length of the data is set based on the microphone array size and sampling frequency By for-mula (7), calculate 1/4 frame length energy of the refer-ence microphone data acquisition to determine the position of the maximum value And then, use the data points as the starting point to process, which is obtained by forward extracting two times and back-ward extracting three times the 1/4 frame length start-ing point Thereby, the amount of calculation of the actual signal processing is greatly reduced

Correlation coefficient

Pearson Carle proposed a unified indicator of the degree of correlation between the two variables, namely, the correlation coefficient,12 which is used to reflect the degree of correlation between variables The formula of correlation coefficient is as follows

r =

Pn

i = 1

(xi x)(yi y) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pn

i = 1

(xi x)2 Pn

i = 1 (yi y)2

where x, y are two vectors and x, y are the

correspond-ing expectations

The range of correlation coefficient is [21, 1]; r.0

indicates positive correlation, r\0 expresses negative

correlation, and jrj indicates the level of correlation

between variables In particular, r = 1 is called

com-pletely positive correlation, r =  1 is called completely

negative correlation, and r = 0 is called not related

Experiments and analysis

Simulation experiment

To prove the validity of this method, the synthetic input

signal containing noise is chosen as formula (9)

fn(t) = 0:9 cos (40pt) + 0:25 cos (400pt)

+ 0:125 cos (600pt) + s ð9Þ where s is colored noise signal

The two constructed signals with 45 time delay

points are shown in Figure 1

The effectiveness and feasibility of the method are

analyzed with the first signal as an example The mode

components decomposed by VMD are shown in

Figure 2; it can be seen that the IMF1 is highly similar

to the original signal and the similarity of the other

three signals is very small

The correlation coefficient of the original signal and

IMF1 is 0.9610, which is highly correlated And the

correlation coefficient of the original signal with IMF2,

IMF3, and IMF4 are 0.1422, 0.1221, and 0.1185,

respectively, all very low relative to the original The system only extracts IMF1 as the reconstructed signal GCC, phat_GCC, and the presented algorithm in this article are used to estimate the two constructed sig-nals with 45 time delay points The TDE results under different SNR are shown in Table 1

Table 1 is a series of data selected from multiple measurement results under the condition of different SNR It is found that the TDE of the original algorithm has great changes in the experiment The time delay error of the GCC reaches a maximum, and the result of the estimation is very unstable with SNR ranging from

21 to 2 dB However, the error of TDE with the pre-sented method is very small, and the error is not signifi-cantly changed with the change in SNR Therefore, the presented method has high precision and strong anti-interference ability

Actual signal experiment

The validity and stability of the algorithm are verified

by the simulation results, and then the advantage and

Figure 1 Two time delay signals.

Table 1 Comparison of simulation results with different methods.

SNR (dB) GCC phat_GCC The presented

algorithm

21 2129 2112 48 20.5 216 20 46

SNR: signal-to-noise ratio; GCC: generalized cross-correlation.

Figure 2 The IMF of VMD decomposition.

feasibility of the presented method is proved with the

actual signal

The microphone array data acquisition system

con-sists of Ni PXIe1082, Ni PXIe8820, Ni PXIe4492, and

AWA14604, and a real signal processing experiment

was carried out in the laboratory (15 m 3 7 m 3 4 m)

There are two microphones with 2-m distance on the

X-axis of the microphone array, and computer is used

to control the high-fidelity speakers which are placed

on the extension of the X-axis to produce the sound

source for experiments The sampling frequency of the

data acquisition system is 44,100 Hz, the length of the

signal is 200,000 sampling points, respectively, use the

method and GCC to estimate the time delay

In practical signal processing, there is a large part of

the useless signal contained in the actual signal, so the

introduction of starting point criterion to select the

most effective part of a signal can reduce the amount

of computation and improve the accuracy of TDE

The distance between two microphones is 2 m, the

indoor sound speed is 343 m/s, and the signal sampling

frequency is 44,100 Hz; it can be calculated theoretically

that there is a time delay of about 256 sampling points,

a frame data length is set to 256 sampling points when

introducing the signal starting point detection criterion

And the result of TDE is represented by the sampling

point in the course of the experiment

Figure 3 shows the two actual signals of the system

and the results of applying GCC, and the TDE is 241

sampling points, the difference is 15 sampling points

with practical results, and the actual delay error is

0.3401 ms, which is due to the error caused by the noise

and reverberation in the indoor measurement The

run-ning time of the program is 0.804501 s (Figure 4)

Before the two picture in Figure 4 are the effective

signals extracted with the presented method, and the

length is 1280 sampling points The signal acquired by

the reference microphone 1 is decomposed with VMD,

the correlation coefficient of the original signal with IMF1 and IMF2 are 0.7177 and 0.7298, respectively, which means the significant correlation with the origi-nal sigorigi-nal, the correlation coefficient of the origiorigi-nal sig-nal with IMF3 and IMF4 are 0.4099 and 0.2083, respectively, micro related So the system extracts IMF1 and IMF2 whose correlation coefficients are greater than 0.5 as the reconstructed signal The final picture of Figure 4 is the result of applying GCC to the reconstructed signals The TDE result obtained by the presented method is 257 sampling points, the error is 1 sampling points, namely, the actual time delay error is 0.0227 ms And the program running time is 0.198156 s

By comparing the results of GCC TDE and the pre-sented method, it is concluded that the delay error of the former is 15 times that of the latter, and the running time of the program is 4.0601 times that of the latter

Conclusion

In this article, the GCC TDE and the limitations of the technique are studied, and the VMD algorithm, the inverse square law of sound signal transmission, and the correlation coefficient have been researched A novel TDE method is presented to overcome the draw-backs of the GCC TDE First, use the inverse square law of sound signal transmission, the microphone array, and sampling frequency to select the most effec-tive data segment Then, decompose the extracted effective data segment into multiple IMF with VMD and calculate the correlation coefficient of the modal components with the original signal to determine the IMF of the reconstructed signal Finally, the delay value of the reconstructed signal is estimated by GCC Theoretical analysis and simulation results show that this method under the condition of colored noise can obtain more accurate TDE By comparing the results

Figure 3 Two practical signals and GCC results.

Figure 4 The effective signal and GCC results.

of GCC TDE and the presented method, it is

con-cluded that the delay error of the former is 15 times

that of the latter, and the running time of the program

is 4.0601 times that of the latter Therefore, the method

not only can reduce the computation time of the system

but also has a certain practicality and robustness

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of this

article.

Funding

The author(s) received no financial support for the research,

authorship, and/or publication of this article.

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