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[.]

Special Issue Article Time delay estimation based on variational mode decomposition Advances in Mechanical Engineering 2017, Vol 9(1) 1–6 Ó The Author(s) 2017 DOI: 10.1177/1687814016688587 journals.sagepub.com/home/ade Jing-Yi Lu1,2, Dong Ye1 and Wen-Ping Ma2 Abstract In order to improve the time delay estimation of colored noise signals, this article proposes generalized crosscorrelation 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 functions 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 estimation 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 computation 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, correlation 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 problems in passive acoustic localization based on microphone array, and it has very important theoretical significance and practical application value.1–4 Typical estimation methods are adaptive TDE, weighted generalized 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 completely, 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 School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, China 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 (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 Advances in Mechanical Engineering make the relevant peak more sharper, but this need to master some of the statistical characteristics of the signal Multi-scale decomposition, such as wavelet analysis 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 corresponding 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 variational model in the process of obtaining the decomposition component, so as to realize the effective separation 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 correlation 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 computation and has a certain practicality and robustness GCC TDE The time delay of arrival (TDOA) value can be estimated 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 function, the cross-power spectrum of the two signals can be obtained by the Fourier transform of the correlation function Gxixj (t) = aj Gss (t) + Gninj (t) ð1Þ where Gxixj (t) is the power spectrum of the crosscorrelation 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 autocorrelation function 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, inhibition 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 ð‘ RGCC (t) = 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 transform (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 signal, 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) = u k (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 realize adaptive signal decomposition, center frequency, and bandwidth of each IMF in the iteration variable sub-optimal solution of the model is updated continuously According to the frequency domain characteristic of the actual signal, the adaptive decomposition of the signal frequency domain is completed, and a number of narrow band IMF components are obtained The bandwidth of each model is estimated by the following steps: By means of the Hilbert transform, the corresponding marginal spectrum of each analytic function is calculated The spectrum of each model is transferred to the baseband by modulating the index to the estimate center frequency Lu et al 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 ( 2 )    X   j ∂t d(t) +  uk (t) ejvk t    pt fuk g, fvk g k X uk = f ð4Þ ð5Þ k where uk : = fu1 , u2 , , uk g is the modal function, vk : = fv1 , v2 , , vk g is the center frequency Above constrained variational problem transforms to a non-constrained variational problem by introducing Lagrange multiplier l(t) and penalty factor a The augmented Lagrange expressions are as follows L(fuk g, fvk g, l) : 2    X   j jvk t   =a ∂t d(t) + pt  uk (t) e  k  2 * +   X X   + f (t)  uk (t) + l(t), f (t)  uk (t)   k k ð6Þ The alternating direction method of multipliers (ADMM) is used to solve the variational problems, and the iterative optimization of uk + , vkk + , and lk + is used to obtain the saddle point of the extended Lagrange expression Iteration steps are as follows: Initialize u1 , v1 , l1 , n = 0; n = n + 1, execute the whole cycle; Execute the first cycle of the inner layer and update vk according to vkk + = arg L uk +1 g, funik g, fvni g, ln ); (funi\k k = k + 1, repeat step (3), end the first cycle until k = K; Execute the second cycle of the inner layer and updateP l according to ln + = ln + n+1 t(f  k uk ); Repeat steps (2)–(7), end the whole cycle until the iterative stopping criteria P n+1 n n (jju  u jj =jju jj )\e is met k k k k 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 correlation coefficient of each IMF component with the original signal 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 surface, the signal energy at the position where the microphone is located is as follows P= I 4pd ð7Þ 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 proportional 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 formula (7), calculate 1/4 frame length energy of the reference 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 backward extracting three times the 1/4 frame length starting 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 Advances in Mechanical Engineering Table Comparison of simulation results with different methods SNR (dB) GCC phat_GCC The presented algorithm 21 20.5 20.1 0.1 0.5 2129 216 39 41 42 42 43 45 2112 20 41 49 43 47 46 45 48 46 47 46 46 44 45 45 SNR: signal-to-noise ratio; GCC: generalized cross-correlation Figure Two time delay signals n P (xi  x)(yi  y) i=1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s r= n n P P (xi  x)2 (yi  y)2 i=1 ð8Þ i=1 where x, y are two vectors and x, y are the corresponding 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 = is called completely positive correlation, r =  is called completely negative correlation, and r = 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 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, Figure The IMF of VMD decomposition 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 signals with 45 time delay points The TDE results under different SNR are shown in Table Table 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 dB However, the error of TDE with the presented method is very small, and the error is not significantly changed with the change in SNR Therefore, the presented method has high precision and strong antiinterference ability Actual signal experiment The validity and stability of the algorithm are verified by the simulation results, and then the advantage and Lu et al Figure Two practical signals and GCC results Figure The effective signal and GCC results feasibility of the presented method is proved with the actual signal The microphone array data acquisition system consists of Ni PXIe1082, Ni PXIe8820, Ni PXIe4492, and AWA14604, and a real signal processing experiment was carried out in the laboratory (15 m m 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 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 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 running time of the program is 0.804501 s (Figure 4) Before the two picture in Figure are the effective signals extracted with the presented method, and the length is 1280 sampling points The signal acquired by the reference microphone 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 original signal, the correlation coefficient of the original signal 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 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 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 presented 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 drawbacks of the GCC TDE First, use the inverse square law of sound signal transmission, the microphone array, and sampling frequency to select the most effective 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 of GCC TDE and the presented 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 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 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