Time-Delay Estimation Based on Multilayer Correlation
Time-Delay Estimation Based on Multilayer Correlation Hua Yan, Yang Zhang, and GuanNan Chen School of Information Science and Engineering, Shenyang University of Technology, Shenyang 110870, China [email protected], [email protected]
Abstract. Time-delay estimation technique is a key to acoustic temperature field measurement. In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer auto-correlation (MC method for short) is proposed. Crosscorrelation method and second correlation method are described briefly for comparison. Theory analysis and simulation research in MATLAB prove that MC method can obtain highest estimation precision among these three methods when the signal to noise ratio is low. Keywords: time-delay estimation, multilayer cross-correlation, multilayer autocorrelation, low SNR.
1 Introduction Temperature field measurement based on acoustic time-of-flight (TOF) tomography [1] has been used in atmospheric monitoring and heat management. It has many advantages such as nondestructive, noncontact sensing and quick in response. Timedelay estimation technique is a key to acoustic temperature field measurement. Time-delay estimation is an important signal processing problem and has received significant amount of attentions during past decades in various applications, including radar, sonar, radio navigation, wireless communication, acoustic tomography, etc [1]. The signals received at two spatially separated microphones in the presence of noise can be modeled by
r1 (t ) = s (t ) + n1 (t ), r2 (t ) = s (t − D ) + n2 (t ), (0 ≤ t ≤ T )
(1)
where r1(t) and r2(t) are the outputs of the two microphones, s(t) is the source signal, n1(t) and n2(t)represent the additive noises, T denotes the observation interval, and D is the time-delay between the two received signals. Time-delay estimation is not an easy task because of various noises and the short observation interval. There are many algorithms to estimate the time-delay D. The cross-correlation (CC) is one of the basic algorithms. Many methods, such as second correlation (SC) method [2] and generalized correlation method [3] develop based on this algorithm. There are two forms of correlations: auto- and cross-correlations. The crosscorrelation function is a measure of the similarities or shared properties between two signals. It can be used to detect/recover signals buried in noise, for example the S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 562–567, 2011. © Springer-Verlag Berlin Heidelberg 2011
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detection of radar return signals and delay measurements. The autocorrelation function involves only one signal; it is a special form of cross-correlation function and is used in similar applications. In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer auto-correlation (MC method for short) is proposed in this paper, and compared with cross-correlation method and second correlation method.
2 The Principle of Time-Delay Estimation Based on Correlation 2.1 The Cross-Correlation (CC) Method
The CC method cross-correlates the microphone outputs r1(t) and r2(t), and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The CC method can be modeled by:
DCC = arg max[ Rc (τ )] τ
Rc (τ ) = E[ r1 (t ) r2 (t + τ )] = E[ s (t ) + n1 (t )][ s(t − D + τ ) + n2 (t + τ )]
(2)
= Rss (τ − D) + Rn s (τ − D) + Rsn2 (τ ) + Rn1n (τ ) 1
2
The signal and the noise are assumed to be uncorrelated. Thus Rn1 s (τ − D ) ,
Rsn2 (τ ) are zero. Also, the additive noises are assumed uncorrelated, thus Rn1n2 (τ ) is zero. However, Rn1n2 (τ ) is usually not be neglected in practice due to the existing correlation between the two noises. So we have
Rc (τ ) = Rss (τ − D) + Rn1n2 (τ ) = RS1 (τ − D) + N1 (τ )
(3)
where Rss (τ ) or RS1 (τ ) is the auto-correlation of the source signal s(t), RS1 (τ − D) reaches maximum when τ=D, Rn1n2 (τ ) or N1 (τ ) is the cross-correlation of noise n1(t) and n2(t). RS1 (τ ) and N1 (τ ) can be thought of as the signal component and noise component of Rc (τ ) , respectively. 2.2 Second Correlation (SC) Method
The auto-correlation of r2(t) can be expressed as
R1 (τ ) = E[r2 (t ) r2 (t + τ )] = E[( s (t -D) + n2 (t ))( s (t − D + τ ) + n2 (t + τ ))] =Rss (τ ) + Rn2 s (τ − D) + Rsn2 (τ − D) + Rn2 n2 (τ )
(4)
Assuming the signal and the noise to be uncorrelated, we have
R1 (τ ) = Rss (τ ) + Rn2n2 (τ ) = RS1 (τ ) + N1' (τ )
(5)
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where Rn2 n2 (τ ) or N1' (τ ) is the auto-correlation of noise n2(t). RS 1 (τ ) and N1' (τ ) can be thought of as the signal component and noise component of R1 (τ ) , respectively. The SC method cross-correlates R1 (t ) and Rc (t ) , and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The SC method can be modeled by:
DSC = arg max[ Rs (τ )] τ
Rs (τ ) = E[ R1 (t ) ⋅ Rc (t + τ )] = E ⎡⎣[ RS 1 (t ) + N1' (t )][ RS1 (t − D + τ ) + N1 (t + τ )]⎤⎦
(6)
Assuming the signal and the noise to be uncorrelated, we have
Rs (τ ) = E[ RS1 (t ) ⋅ RS1 (t − D + τ )] + E[ N1' (t ) ⋅ N1 (t + τ )] = RS 2 (τ − D) + N 2 (τ )
(7)
where RS 2 (τ ) is the auto-correlation of the signal RS 1 (τ )
, RS 2 (τ − D ) reaches maximum when τ=D, N 2 (τ ) is the cross-correlation of noise N1' (t ) and N1 (t ) . RS 2 (τ ) , N 2 (τ ) can be thought of as the signal component and noise component of Rs (τ ) , respectively. 2.3 Multi-Correlation (MC) Method
Correlation operation is an effective means of de-noising and increasing the signal to noise ratio. Therefore we try to estimate time-delay in low SNR by multiplayer correlation. The auto-correlation of R1 (t ) can be expressed as
R2 (τ ) = E[ R1 (t ) ⋅ R1 (t + τ )] = E ⎡⎣[ RS1 (t ) + N1' (t )][ RS 1 (t + τ ) + N1' (t + τ )]⎤⎦
(8)
Assuming the signal and the noise to be uncorrelated, we have
R2 (τ ) = E[ RS1 (t ) ⋅ RS1 (t + τ )] + E[ N1' (t ) ⋅ N1' (t + τ )] = RS 2 (τ ) + N 2' (τ ) where RS 2 (τ ) is the auto-correlation of the signal RS1 (τ )
(9)
, N 2' (τ ) is the auto-
correlation of noise N1' (t ) . RS 2 (τ ) , N 2 (τ ) can be thought of as the signal component and noise component of R2 (τ ) , respectively. The MC method cross-correlates R2 (t ) and Rs (t ) , and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The MC method can be modeled by:
DMC = arg max[ Rm (τ )] τ
Rm (τ ) = E[ R2 (t ) ⋅ Rs (t + τ )] = E ⎡⎣[ RS 2 (t ) + N 2' (t )][ RS 2 (t − D + τ ) + N 2 (t + τ )]⎤⎦
(10)
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Assuming the signal and the noise to be uncorrelated, we have
Rm (τ ) = E[ RS 2 (t ) ⋅ RS 2 (t − D + τ )] + E[ N 2' (t ) ⋅ N 2 (t + τ )] = RS 3 (τ − D ) + N 3 (τ ) (11) where RS 3 (τ ) is the auto-correlation of the signal RS 2 (τ )
, RS 3 (τ − D ) reaches maximum when τ=D, N 3 (τ ) is the auto-correlation of noise N 2' (t ) and N 2 (t ) . RS 3 (τ ) , N 3 (τ ) can be thought of as the signal component and noise component of Rm (τ ) , respectively.
3 Fast Implementation of CC, SC, and MC Methods Above correlation operations are implemented in discrete time. Sample r1(t) and r2(t) simultaneously, we have two data sequences r1(n) and r2(n), each containing N data. The cross-correlation R12 ( j ) between r1(n) and r2(n) can be expressed as
R12 ( j ) =
1 N
N
∑ r (n)r (n + j) 1
2
(12)
n=0
For longer data sequences, correlation operations can be speeded up by using the correlation theorem and fast Fourier transform as follows [4]. R12 ( j ) = IFFT ⎡⎣[ FFT (r1 (n)]* ⋅ FFT (r2 (n)]⎤⎦
(13)
where FFT and IFFT denote the inverse fast Fourier transform and fast Fourier transform, respectively. * is conjugate operator. The fast implementation of CC, SC and MC methods is given in Fig.1. In Fig.1, REAL means obtaining the real part and MAX means obtaining the peak position, respectively.
Fig. 1. The fast implementation of CC, SC and MC methods
4 Simulation Research In order to avoid the influence of acoustic travel-distance measuring error and so on, the acoustic travel-time estimation method based on multilayer correlation is verified
566
H. Yan, Y. Zhang, and G.N. Chen
using MATLAB simulation data. The acoustic source signal s(t) is a linear sweptfrequency cosine signal generated by chirp(t,f0,t1,f1) and the acoustic signal at receiving point can be written as s(t-τ). In this paper, f0=200Hz, f1=850Hz, t1=T/2=N/2/fs, N=25000 or 50000, fs=250kHz, τ= 0.014412 s. N is the number of samples, fs is the sample frequency. Simulation research shows that the acoustic travel-time estimation value is stable and exact if the noise is weak. But if the noise isn’t weak, the acoustic travel-time estimation value will fluctuate slightly. The standard deviation (std) and the relative root-mean-square error (R_RMSE) are used to assess the stability and accuracy of the travel-time estimation values. They are defined as follows. n
n
∑ (τ i − τ )
std =
i =1
n −1
,
τ =
∑τ i i =1
n
R _ RMSE =
1 n 2 (τ i − τ ) ∑ n i =1
τ
× 100%
(14)
where τ is the actual acoustic travel-time; τi is the ith estimation value of the acoustic travel-time; n is the number of measurement (estimation), in this paper, n=100. The estimation results when Gaussian white noises and colored noises are added are given in Table1 and Table 2, respectively. The colored noise is obtained by feeding the white noise through a band-rejection filter. The system function of the filter is
H (z) =
1− 2z
−1
2 − 1 .2 2 7 z − 2 − 0 .6 1 9 2 z − 3
(15)
Following can be found from Table 1~Table 4. 1) The stability and accuracy of time-delay estimation will decrease with the decreasing of SNR, or the decreasing of samples. 2) Among CC method, SC method and MC method, MC method has best stability and accuracy of time-delay estimation. Table 1. Estimation results when Gaussian white noise is added
SNR -5 -10 25000 -15 samples -18 -19 -5 -10 50000 -15 samples -18 -19
CC method std R-RMSE(%) 1.78e-5 0.1233 3.14e-5 0.2180 4.88e-5 0.3370 7.24e-5 0.5026 2.38e-4 1.6632 1.41e-5 0.0980 2.89e-5 0.2001 4.26e-5 0.2950 6.32e-5 0.4476 6.34e-5 0.4390
SC method std R-RMSE(%) 5.46e-6 0.0382 1.22e-5 0.0846 3.52e-5 0.2437 5.31e-5 0.3677 7.33e-5 0.5086 4.28e-6 0.0296 1.13e-5 0.0785 3.48e-5 0.2411 4.87e-5 0.3407 5.26e-5 0.3650
MC method std R-RMSE(%) 3.08e-6 0.0215 7.49e-6 0.0518 2.71e-5 0.1875 5.69e-5 0.3948 7.22e-5 0.5001 2.63e-6 0.0182 6.46e-6 0.0447 2.34e-5 0.1622 4.67e-5 0.3231 4.96e-5 0.3434
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Table 2. Estimation results when colored noise is added
SNR -5 -10 25000 -15 samples -18 -19 -5 -10 50000 -15 samples -18 -19
std 2.08e-5 2.04e-5 2.30e-5 2.10e-5 2.30e-5 1.60e-5 1.79e-5 1.77e-5 1.88e-5 1.87e-5
CC method R-RMSE(%) 0.1442 0.1411 0.1591 0.1454 0.1622 0.1106 0.1247 0.1231 0.1318 0.1304
std 1.32e-5 1.13e-5 1.06e-5 1.16e-5 1.27e-5 8.13e-6 8.84e-6 8.87e-6 8.48e-6 1.01e-5
SC method R-RMSE(%) 0.0928 0.0786 0.0736 0.0816 0.0879 0.0562 0.0611 0.0618 0.0585 0.0701
MC method std R-RMSE(%) 8.04e-6 0.0556 8.03e-6 0.0560 7.39e-6 0.0512 8.61e-6 0.0609 8.96e-6 0.0623 5.89e-6 0.0409 5.69e-6 0.0394 5.96e-6 0.0412 6.44e-6 0.0456 5.78e-6 0.0399
5 Conclusion In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer autocorrelation (MC method) is proposed and compared with cross-correlation (CC) method and second correlation (SC) method. Theory analysis and simulation research in MATLAB prove that MC method can obtain highest estimation precision these three methods when the signal to noise ratio is low. Acknowledgments. The work is supported by Natural Science Foundation of China (60772054), Specialized Research Fund for the Doctoral Program of Higher Education (20102102110003) and Shenyang Science and Technology Plan (F10213100).
References 1. Ostashev, V.E., Voronovich, A.G.: An Overview of Acoustic Travel-Time Tomography in the Atmosphere and its Potential Applications. Acta Acust. United Ac. 87, 721–730 (2001) 2. Gedalyahu, K., Eldar, Y.C.: Time-delay Estimation from Low-rate Samples: A Union of Subspaces Approach. IEEE T. Signal Proces. 58, 3017–3031 (2010) 3. Tang, J., Xing, H.Y.: Time Delay Estimation Based on Second Correlation. Computer Engineering 33, 265–269 (2007) (in Chinese) 4. Knapp, C.H., Carter, C.G.: The Generalized Correlation Method for Estimation of Time Delay. IEEE T. Acoust., Speech, Signal Processing ASSP21, 320–327 (1976) 5. Ifeachor, E.C., Jervis, B.W.: Digital Signal Processing: A Practical Approach, 2nd edn. Publishing House of Electronics Industry, Beijing (2003)
Abstract. Time-delay estimation technique is a key to acoustic temperature field measurement. In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer auto-correlation (MC method for short) is proposed. Crosscorrelation method and second correlation method are described briefly for comparison. Theory analysis and simulation research in MATLAB prove that MC method can obtain highest estimation precision among these three methods when the signal to noise ratio is low. Keywords: time-delay estimation, multilayer cross-correlation, multilayer autocorrelation, low SNR.
1 Introduction Temperature field measurement based on acoustic time-of-flight (TOF) tomography [1] has been used in atmospheric monitoring and heat management. It has many advantages such as nondestructive, noncontact sensing and quick in response. Timedelay estimation technique is a key to acoustic temperature field measurement. Time-delay estimation is an important signal processing problem and has received significant amount of attentions during past decades in various applications, including radar, sonar, radio navigation, wireless communication, acoustic tomography, etc [1]. The signals received at two spatially separated microphones in the presence of noise can be modeled by
r1 (t ) = s (t ) + n1 (t ), r2 (t ) = s (t − D ) + n2 (t ), (0 ≤ t ≤ T )
(1)
where r1(t) and r2(t) are the outputs of the two microphones, s(t) is the source signal, n1(t) and n2(t)represent the additive noises, T denotes the observation interval, and D is the time-delay between the two received signals. Time-delay estimation is not an easy task because of various noises and the short observation interval. There are many algorithms to estimate the time-delay D. The cross-correlation (CC) is one of the basic algorithms. Many methods, such as second correlation (SC) method [2] and generalized correlation method [3] develop based on this algorithm. There are two forms of correlations: auto- and cross-correlations. The crosscorrelation function is a measure of the similarities or shared properties between two signals. It can be used to detect/recover signals buried in noise, for example the S. Lin and X. Huang (Eds.): CSEE 2011, Part I, CCIS 214, pp. 562–567, 2011. © Springer-Verlag Berlin Heidelberg 2011
Time-Delay Estimation Based on Multilayer Correlation
563
detection of radar return signals and delay measurements. The autocorrelation function involves only one signal; it is a special form of cross-correlation function and is used in similar applications. In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer auto-correlation (MC method for short) is proposed in this paper, and compared with cross-correlation method and second correlation method.
2 The Principle of Time-Delay Estimation Based on Correlation 2.1 The Cross-Correlation (CC) Method
The CC method cross-correlates the microphone outputs r1(t) and r2(t), and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The CC method can be modeled by:
DCC = arg max[ Rc (τ )] τ
Rc (τ ) = E[ r1 (t ) r2 (t + τ )] = E[ s (t ) + n1 (t )][ s(t − D + τ ) + n2 (t + τ )]
(2)
= Rss (τ − D) + Rn s (τ − D) + Rsn2 (τ ) + Rn1n (τ ) 1
2
The signal and the noise are assumed to be uncorrelated. Thus Rn1 s (τ − D ) ,
Rsn2 (τ ) are zero. Also, the additive noises are assumed uncorrelated, thus Rn1n2 (τ ) is zero. However, Rn1n2 (τ ) is usually not be neglected in practice due to the existing correlation between the two noises. So we have
Rc (τ ) = Rss (τ − D) + Rn1n2 (τ ) = RS1 (τ − D) + N1 (τ )
(3)
where Rss (τ ) or RS1 (τ ) is the auto-correlation of the source signal s(t), RS1 (τ − D) reaches maximum when τ=D, Rn1n2 (τ ) or N1 (τ ) is the cross-correlation of noise n1(t) and n2(t). RS1 (τ ) and N1 (τ ) can be thought of as the signal component and noise component of Rc (τ ) , respectively. 2.2 Second Correlation (SC) Method
The auto-correlation of r2(t) can be expressed as
R1 (τ ) = E[r2 (t ) r2 (t + τ )] = E[( s (t -D) + n2 (t ))( s (t − D + τ ) + n2 (t + τ ))] =Rss (τ ) + Rn2 s (τ − D) + Rsn2 (τ − D) + Rn2 n2 (τ )
(4)
Assuming the signal and the noise to be uncorrelated, we have
R1 (τ ) = Rss (τ ) + Rn2n2 (τ ) = RS1 (τ ) + N1' (τ )
(5)
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where Rn2 n2 (τ ) or N1' (τ ) is the auto-correlation of noise n2(t). RS 1 (τ ) and N1' (τ ) can be thought of as the signal component and noise component of R1 (τ ) , respectively. The SC method cross-correlates R1 (t ) and Rc (t ) , and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The SC method can be modeled by:
DSC = arg max[ Rs (τ )] τ
Rs (τ ) = E[ R1 (t ) ⋅ Rc (t + τ )] = E ⎡⎣[ RS 1 (t ) + N1' (t )][ RS1 (t − D + τ ) + N1 (t + τ )]⎤⎦
(6)
Assuming the signal and the noise to be uncorrelated, we have
Rs (τ ) = E[ RS1 (t ) ⋅ RS1 (t − D + τ )] + E[ N1' (t ) ⋅ N1 (t + τ )] = RS 2 (τ − D) + N 2 (τ )
(7)
where RS 2 (τ ) is the auto-correlation of the signal RS 1 (τ )
, RS 2 (τ − D ) reaches maximum when τ=D, N 2 (τ ) is the cross-correlation of noise N1' (t ) and N1 (t ) . RS 2 (τ ) , N 2 (τ ) can be thought of as the signal component and noise component of Rs (τ ) , respectively. 2.3 Multi-Correlation (MC) Method
Correlation operation is an effective means of de-noising and increasing the signal to noise ratio. Therefore we try to estimate time-delay in low SNR by multiplayer correlation. The auto-correlation of R1 (t ) can be expressed as
R2 (τ ) = E[ R1 (t ) ⋅ R1 (t + τ )] = E ⎡⎣[ RS1 (t ) + N1' (t )][ RS 1 (t + τ ) + N1' (t + τ )]⎤⎦
(8)
Assuming the signal and the noise to be uncorrelated, we have
R2 (τ ) = E[ RS1 (t ) ⋅ RS1 (t + τ )] + E[ N1' (t ) ⋅ N1' (t + τ )] = RS 2 (τ ) + N 2' (τ ) where RS 2 (τ ) is the auto-correlation of the signal RS1 (τ )
(9)
, N 2' (τ ) is the auto-
correlation of noise N1' (t ) . RS 2 (τ ) , N 2 (τ ) can be thought of as the signal component and noise component of R2 (τ ) , respectively. The MC method cross-correlates R2 (t ) and Rs (t ) , and considers the time argument that corresponds to the maximum peak in the output as the estimated time- delay. The MC method can be modeled by:
DMC = arg max[ Rm (τ )] τ
Rm (τ ) = E[ R2 (t ) ⋅ Rs (t + τ )] = E ⎡⎣[ RS 2 (t ) + N 2' (t )][ RS 2 (t − D + τ ) + N 2 (t + τ )]⎤⎦
(10)
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Assuming the signal and the noise to be uncorrelated, we have
Rm (τ ) = E[ RS 2 (t ) ⋅ RS 2 (t − D + τ )] + E[ N 2' (t ) ⋅ N 2 (t + τ )] = RS 3 (τ − D ) + N 3 (τ ) (11) where RS 3 (τ ) is the auto-correlation of the signal RS 2 (τ )
, RS 3 (τ − D ) reaches maximum when τ=D, N 3 (τ ) is the auto-correlation of noise N 2' (t ) and N 2 (t ) . RS 3 (τ ) , N 3 (τ ) can be thought of as the signal component and noise component of Rm (τ ) , respectively.
3 Fast Implementation of CC, SC, and MC Methods Above correlation operations are implemented in discrete time. Sample r1(t) and r2(t) simultaneously, we have two data sequences r1(n) and r2(n), each containing N data. The cross-correlation R12 ( j ) between r1(n) and r2(n) can be expressed as
R12 ( j ) =
1 N
N
∑ r (n)r (n + j) 1
2
(12)
n=0
For longer data sequences, correlation operations can be speeded up by using the correlation theorem and fast Fourier transform as follows [4]. R12 ( j ) = IFFT ⎡⎣[ FFT (r1 (n)]* ⋅ FFT (r2 (n)]⎤⎦
(13)
where FFT and IFFT denote the inverse fast Fourier transform and fast Fourier transform, respectively. * is conjugate operator. The fast implementation of CC, SC and MC methods is given in Fig.1. In Fig.1, REAL means obtaining the real part and MAX means obtaining the peak position, respectively.
Fig. 1. The fast implementation of CC, SC and MC methods
4 Simulation Research In order to avoid the influence of acoustic travel-distance measuring error and so on, the acoustic travel-time estimation method based on multilayer correlation is verified
566
H. Yan, Y. Zhang, and G.N. Chen
using MATLAB simulation data. The acoustic source signal s(t) is a linear sweptfrequency cosine signal generated by chirp(t,f0,t1,f1) and the acoustic signal at receiving point can be written as s(t-τ). In this paper, f0=200Hz, f1=850Hz, t1=T/2=N/2/fs, N=25000 or 50000, fs=250kHz, τ= 0.014412 s. N is the number of samples, fs is the sample frequency. Simulation research shows that the acoustic travel-time estimation value is stable and exact if the noise is weak. But if the noise isn’t weak, the acoustic travel-time estimation value will fluctuate slightly. The standard deviation (std) and the relative root-mean-square error (R_RMSE) are used to assess the stability and accuracy of the travel-time estimation values. They are defined as follows. n
n
∑ (τ i − τ )
std =
i =1
n −1
,
τ =
∑τ i i =1
n
R _ RMSE =
1 n 2 (τ i − τ ) ∑ n i =1
τ
× 100%
(14)
where τ is the actual acoustic travel-time; τi is the ith estimation value of the acoustic travel-time; n is the number of measurement (estimation), in this paper, n=100. The estimation results when Gaussian white noises and colored noises are added are given in Table1 and Table 2, respectively. The colored noise is obtained by feeding the white noise through a band-rejection filter. The system function of the filter is
H (z) =
1− 2z
−1
2 − 1 .2 2 7 z − 2 − 0 .6 1 9 2 z − 3
(15)
Following can be found from Table 1~Table 4. 1) The stability and accuracy of time-delay estimation will decrease with the decreasing of SNR, or the decreasing of samples. 2) Among CC method, SC method and MC method, MC method has best stability and accuracy of time-delay estimation. Table 1. Estimation results when Gaussian white noise is added
SNR -5 -10 25000 -15 samples -18 -19 -5 -10 50000 -15 samples -18 -19
CC method std R-RMSE(%) 1.78e-5 0.1233 3.14e-5 0.2180 4.88e-5 0.3370 7.24e-5 0.5026 2.38e-4 1.6632 1.41e-5 0.0980 2.89e-5 0.2001 4.26e-5 0.2950 6.32e-5 0.4476 6.34e-5 0.4390
SC method std R-RMSE(%) 5.46e-6 0.0382 1.22e-5 0.0846 3.52e-5 0.2437 5.31e-5 0.3677 7.33e-5 0.5086 4.28e-6 0.0296 1.13e-5 0.0785 3.48e-5 0.2411 4.87e-5 0.3407 5.26e-5 0.3650
MC method std R-RMSE(%) 3.08e-6 0.0215 7.49e-6 0.0518 2.71e-5 0.1875 5.69e-5 0.3948 7.22e-5 0.5001 2.63e-6 0.0182 6.46e-6 0.0447 2.34e-5 0.1622 4.67e-5 0.3231 4.96e-5 0.3434
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Table 2. Estimation results when colored noise is added
SNR -5 -10 25000 -15 samples -18 -19 -5 -10 50000 -15 samples -18 -19
std 2.08e-5 2.04e-5 2.30e-5 2.10e-5 2.30e-5 1.60e-5 1.79e-5 1.77e-5 1.88e-5 1.87e-5
CC method R-RMSE(%) 0.1442 0.1411 0.1591 0.1454 0.1622 0.1106 0.1247 0.1231 0.1318 0.1304
std 1.32e-5 1.13e-5 1.06e-5 1.16e-5 1.27e-5 8.13e-6 8.84e-6 8.87e-6 8.48e-6 1.01e-5
SC method R-RMSE(%) 0.0928 0.0786 0.0736 0.0816 0.0879 0.0562 0.0611 0.0618 0.0585 0.0701
MC method std R-RMSE(%) 8.04e-6 0.0556 8.03e-6 0.0560 7.39e-6 0.0512 8.61e-6 0.0609 8.96e-6 0.0623 5.89e-6 0.0409 5.69e-6 0.0394 5.96e-6 0.0412 6.44e-6 0.0456 5.78e-6 0.0399
5 Conclusion In order to acquire a stable acoustic time-of-flight in low SNR, a time-delay estimation method combined multilayer cross-correlation and multilayer autocorrelation (MC method) is proposed and compared with cross-correlation (CC) method and second correlation (SC) method. Theory analysis and simulation research in MATLAB prove that MC method can obtain highest estimation precision these three methods when the signal to noise ratio is low. Acknowledgments. The work is supported by Natural Science Foundation of China (60772054), Specialized Research Fund for the Doctoral Program of Higher Education (20102102110003) and Shenyang Science and Technology Plan (F10213100).
References 1. Ostashev, V.E., Voronovich, A.G.: An Overview of Acoustic Travel-Time Tomography in the Atmosphere and its Potential Applications. Acta Acust. United Ac. 87, 721–730 (2001) 2. Gedalyahu, K., Eldar, Y.C.: Time-delay Estimation from Low-rate Samples: A Union of Subspaces Approach. IEEE T. Signal Proces. 58, 3017–3031 (2010) 3. Tang, J., Xing, H.Y.: Time Delay Estimation Based on Second Correlation. Computer Engineering 33, 265–269 (2007) (in Chinese) 4. Knapp, C.H., Carter, C.G.: The Generalized Correlation Method for Estimation of Time Delay. IEEE T. Acoust., Speech, Signal Processing ASSP21, 320–327 (1976) 5. Ifeachor, E.C., Jervis, B.W.: Digital Signal Processing: A Practical Approach, 2nd edn. Publishing House of Electronics Industry, Beijing (2003)
