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JOURNAL OF COMPUTERS, VOL. 7, NO. 2, FEBRUARY 2012
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A Robust Signal Selective TDOA Estimation Algorithm for Cyclostationary Signals Source Location Yang Liu
Dalian University of Technology, Dalian, 116024, China Inner Mongolia University, Huhhot, 010021, China Email: [email protected]
TianShuang Qiu
Dalian University of Technology, Dalian, 116024, China Email: [email protected]
Abstract—For the problem of estimation time-difference-ofarrival (TDOA) for cyclostationary signals in the presence of interference and impulsive noise, a new robust multicycle signal-selective algorithm is introduced. By fusing second-order cyclic moments and fractional lower-order statistics, a novel cyclic fractional lower-order statistics is developed. The proposed algorithm combines the benefits of cyclostationarity based method and fractional lower-order statistics based estimator, by exploiting cyclostationarity property of signals with cyclic fractional lower-order statistics. Simulation results indicate that the new method is highly tolerant to interference and impulsive noise and gives higher estimation accuracy than conventional TDOA estimation methods. Index Terms—cyclostationarity, impulsive noise, fractional lower-order statistics, time-difference-of-arrival (TDOA)
I. INTRODUCTION The problem of locating a Mobile Station (MS) has drawn considerable interest in recent years. A variety of wireless location schemes have been extensively investigated in [1], [2]. One typical method used to estimate the mobile location is time-difference-of-arrival (TDOA) which does not require knowledge of the transmit time of the received signal from the transmitter and has better accuracy than angle of arrival (AOA) [3], [4]. Although several techniques are used to reduce the effects of interference and noise in communication systems [3], it is necessary to develop effective TDOA algorithms. Most man-made signals encountered in radar, sonar, and communication systems are appropriately modeled as cyclostationary time series [5]. A class of signal-selective TDOA methods for passive location is introduced by Gardner et al. [6], [7]. These methods that exploit inherent cyclostationarity of signals are highly tolerant to both interference and Gaussian noise, which are neither cyclostationary signals, nor exhibiting the same cycle frequency of the source signal. Since almost man-made © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.2.393-398
signals in communication systems have more than one cycle frequency [5], some modified multi-cycle algorithms which exploit more than one cycle frequency are developed in [8-10]. The multi-cycle methods can achieve better performance than single-cycle estimators which utilize only one cycle frequency [10]. The primary single-cycle and multi-cycle methods focus on the case where the environment noise is assumed to follow the Gaussian distribution model. However, many types of noises encountered in practice such as atmospheric noise, multiuser interference and some man-made noise in urban region are heavy tailed non-Gaussian impulsive processes [11], [12]. Studies and experimental measurements have shown that alpha-stable distribution is more suitable for modeling noise of impulsive nature than Gaussian distribution in communication, telemetry, radar, and sonar systems [13], [14]. The alpha-stable model is of a statistical-physical nature, arising under very general assumptions satisfies the stability and the Generalized Center Limit Theorem [16], [17]. It can be described conveniently by four parameters, in which the characteristic exponent α (0 < α ≤ 2) determines the heaviness of its tail. A small positive value of α indicates severe impulsiveness, while a value of α close to 2 indicates a more Gaussian type behavior. When α = 2 , the stable distribution reduced to the Gaussian distribution. So the alpha-stable model is more suitable for modeling noise than Gaussian distribution in real signal processing applications. Several TDOA methods take account the impulsive noise using fractional lower order statistics (FLOS) have been proposed in literature [15]. As stable distribution does not have finite second-order moments (except for α = 2 ), conventional signalselective methods based on the second-order cyclostationarity will be considerably weakened in symmetric alpha-stable ( Sα S ) noise environments. Although the fractional lower-order statistics (FLOS) based methods are robust to both Gaussian and non-
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Gaussian noise [15], the interfering signals which occupy the same spectral band as the source signal can severely degrade the performance of these methods. To extend the signal-selective and FLOS based methods and achieve high accuracy of TDOA estimation, a robust TDOA algorithm which combines multi-cycle estimator with fractional lower-order statistics is proposed in this paper. The new method takes advantages of both multi-cycle estimator and FLOS method can exploit cyclostationarity in the presence of interference and Sα S impulsive noise. Simulation results demonstrate that the proposed algorithm can acquire high accuracy of TDOA estimation. The rest of this paper is organized as follows: The modeling of the signal measurement is briefly described in Section Ⅱ. In Section Ⅲ, the cyclic fractional lowerorder moment and fractional lower-order multi-cycle TDOA algorithm are proposed. The performance of the new algorithm via Monte Carlo simulations is presented in Section Ⅳ. Finally, conclusions are given in Section Ⅴ. II. SIGNAL MODEL In general, for a signal radiating from a remote source through a channel with interference and noise, the model for time difference of arrival estimation between received signals at two base stations is given by x(t ) = s (t ) + n(t ) y (t ) = rs (t − D) + m(t )
(1) (2)
where s (t ) is the signal of interest (SOI), n(t ) and m(t ) are the signals not of interest (SNOI), including interference and noise, D is the TDOA to be estimated, and parameter r represents the magnitude mismatch between two receivers. To simplify the problem, it is assumed that s (t ) is statistically independent of SNOI, n(t ) and m(t ) do not share the same cycle frequency as that of the SOI. However, since n(t ) and m(t ) may contain the same interfering signals (with different times of arrival), they can be statistically dependent [6]. III. THE PROPOSED FRACTIONAL LOWER ORDER MULTI-CYCLE TDOA ESTIMATION METHOD Some single-cycle signal-selective TDOA estimation algorithms based on the second-order cyclostationarity are proposed in [6]. Since the cycle frequency for most communication signals are not unique, the single-cycle estimators cannot achieve the best performance. It was introduced in [8] and [10] that multiple single-cycle estimators can be fused together to form a multi-cycle estimator, g ε (τ )
∑ wi ∫ Ryxεi (τ + u )( Rxεi (u ))* du
(3)
εi ∈A
where ε i is one element in the cycle frequency set A of s (t ) , wi is the weight and wi = 1 . According to the © 2012 ACADEMY PUBLISHER
definition in [6], the cyclic autocorrelation function of s (t ) at cycle frequency ε is defined as s (t + τ / 2) s ∗ (t − τ / 2)e − j 2πε t
Rsε (τ )
where
⋅ = lim(1/ T ) ∫
T /2
−T / 2
T →∞
(⋅)dt
(4)
is the time-averaging
operation, and “ ∗ ” denotes the conjugation. The multi-cycle estimator (3) makes better use of the cyclostationarity features and acquires more accurate estimates of TDOA. However, Due to the thick tails, stable distributions do not have finite second-order moments, except for the limiting case α = 2 . If the received noises contain stable processes, the second-order cyclic moments such as cyclic autocorrelation function in (4) will not be applicable. Thus, the conventional multicycle estimator (3) based on the cyclic correlations will degrade severally in impulsive noise environments. In order to outperform the drawbacks of the conventional multi-cycle algorithm, we define a new cyclic fractional lower-order statistics and propose a new signal-selective estimator by utilizing cyclic fractional lower-order statistics instead of second-order cyclostationarity. Considering the random process s (t ) , the cyclic fractional lower-order autocorrelation function at cycle frequency ε is defined as Rsε , p (τ )
1 T /2 * [ s (t + τ / 2)]〈 p 〉 [ s ∗ (t − τ / 2)]e − j 2πε t dt T ∫−T / 2 (5) [ s* (t + τ / 2)]〈 p 〉 [ s∗ (t − τ / 2)]e − j 2πε t
lim
T →∞
where 1 ≤ p < α and z 〈 k 〉 = z convention z 〈 k 〉 = z
k −1
k −1
z ∗ . In equation (5), the
z ∗ can also be rewritten in a polar
form as z 〈 k 〉 = r k −1 z ∗ = r k e − jθ
(6)
where z = re jθ . Equation (6) demonstrates that convention z 〈 k 〉 contains fractional restrain magnitude but maintain period of random variable z . Thus, cyclic fractional lower-order autocorrelation function (5) and conventional cyclic correlation (4) share the same cycle frequency. In other words, z 〈 k 〉 attenuates only the magnitude value of z , but it maintains the periods of z . Both of them exhibit the same cyclostationarity property. Note that the cyclic autocorrelation Rsε (τ ) could be expressed as the conventional cross correlation for the pair of time series u (t ) and v(t ) [6], Rsε (τ )
s (t + τ / 2) s ∗ (t − τ / 2)e− j 2πε t = u (t + τ / 2)v∗ (t − τ / 2)
(7)
=Ruv0 (τ )
where u (t )
s (t )e − jπε t
(8)
v(t )
s(t )e jπε t
(9)
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are obtained by frequency shifting s (t ) . By substituting (8) and (9) into (5), with the properties of z 〈 k 〉 e± jπε t = ( ze∓ jπε t )〈 k 〉 and ( z ∗ )〈 k 〉 = ( z 〈 k 〉 )∗ , the cyclic fractional lower-order autocorrelation Rsε , p (τ ) can be reexpressed as Rsε , p (τ ) = [u ∗ (t + τ / 2)]〈 p 〉 [v(t − τ / 2)] = Ruvp (τ ).
(10)
and v (t ) are reduced to fractional powers. Unlike cyclic ε
autocorrelation Rs (τ ) , even if the received noise in n(t ) and m(t ) are Sα S distribution impulsive processes, the cyclic fractional lower-order autocorrelation Rsε , F (τ ) is also applicable. Thus, the cyclic fractional lower-order autocorrelation function can be expected to exploit spectral correlation in the presence of impulsive noise. In particular, for ε = 0 , cyclic fractional lower-order autocorrelation Rsε , p (τ ) becomes conventional fractional
lower-order autocorrelation R (τ ) . And for p = 2 , it reduces to the conventional second-order cyclic autocorrelation Rsε (τ ) . Substitute (1) and (2) into (5) and assume that r is a real number, the cyclic fractional lower-order autocorrelation and cross correlation functions are given by p s
Rxε , p (τ ) = Rsε , p (τ )
(11)
ε,p Ryx (τ ) = r 〈 p 〉 Rsε , p (τ − D)e − jπε D .
(12)
To overcome the drawbacks of conventional signalselective methods, we propose a new algorithm based on cyclic fractional lower-order correlations. As an alternative, the new estimator w ∫ Rε ∑ ε i
i,p
yx
(τ + u )( Rxε i , p (u ))* du (13)
i ∈A
is called fractional lower-order statistics multi-cycle estimator (FLOS-MCE). By substituting the idealized measurements (11) and (12) into (13), we obtain the ideal function of the FLOS-MCE
∑r ε
∑r
〈 p〉
εi ∈A
≤ r
p
wi e − jπε i D ∫ Rsε i , p (τ − D + u )( Rsε i , p (u ))∗ du
∑∫R
εi , p
s
εi ∈A
2
(15)
(u ) du .
It is noted that only if τ = D can the equality in (15) hold. When wi = e jπε i D and τ = D , we have 2
Equation (10) reveals that the cyclic fractional lowerorder autocorrelation can be expressed as conventional fractional lower-order cross-correlation of the frequencyshifted versions u (t ) and v (t ) of s (t ) . It should be noted that by using signed-power nonlinearity z 〈 k 〉 , impulsive noise components in u (t )
g Mε , p (τ )
g Mε , p (τ ) =
g Mε , p ( D) = r 〈 p 〉 ∑ ∫ Rsε i , p (u ) du .
(16)
εi ∈A
Assume r is a real number, then r g Mε , p ( D) = r
p
∑ ∫ Rε ε s
i,p
p
= r , p
2
(u ) du = max g Mε , p (τ ) . (17)
i ∈A
To see that g Mε , p (τ ) does indeed peak at τ = D , the solution of FLOS-MCE is given by D = arg max{ g Mε , p (τ ) }, wi = e j 2πε i D . τ
(18)
Since the true value of D is unknown, wi can not be computed directly in real applications. In order to obtain accurate estimate of D , it should let wi = 1 get the initial gˆ ε , p (τ ) and Dˆ , then estimate w through iteration [9]. M
i
Note that when p = 2 the FLOS-MCE reduces to the conventional multi-cycle estimator (MCE). VI. SIMULATION RESULTS In this section, we present the results of a simulation study designed to compare the performance of the above algorithms for TDOA estimation. Simulation results are provided to demonstrate the effectiveness and robustness of the proposed method. We consider a real simulated BPSK signal [7], [9] with uncorrelated Sα S noise added to the two received signals. The carrier frequency of the BPSK signal is f c = 0.25 / Ts , keying rate of ε = 0.0625 / Ts with chip width To = 16Ts and sampling increment Ts . The TDOA between two received signals is fixed at D = 48Ts . The length of the received signals is 1024 × 32 baud. As the stable distribution makes the standard SNR meaningless, then a new generalized SNR (GSNR) is defined as GSNR = 10 log σ s2 γ n , where σ s2 is the variance of the signal, γ n is the dispersion parameter of the Sα S noise.
wi e − jπε i D ∫ Rsε i , p (τ − D + u )( Rsε i , p (u ))∗ du .(14) A. The Robustness of New Multi-cycle Algorithm The TDOA function of the conventional single-cycle i ∈A estimator and multi-cycle estimator (MCE) in the It follows from (14) that presence of impulsive noise is shown in Fig. 1, and Fig. 2, respectively. The GSNR= 0 dB, characteristic exponent of the Sα S noise is α = 1.5 , and p = 1.2 . It can be seen from Fig. 1 and Fig. 2 that although the conventional single-cycle and multi-cycle signal-selective TDOA methods are immune to Gaussian noise, both of them fail g Mε , p (τ ) =
〈 p〉
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to give correct estimation in this case where alpha-stable impulsive noise is present. The dominant peak of the single-cycle estimator is at τ = 56Ts . And the peak of interest is only one of many peaks, any one of which might be taken as the TDOA estimate. Similarly, the highest peak of the multi-cycle estimator is also at the wrong TDOA value τ = 52Ts .
1 0.9 0.8 0.7 Amplitude
396
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1
0.2 0.9
0.1 0.8
0 0.7
0
50
100
150
200
τ /Ts
Amplitude
0.6
Figure 4. TDOA estimation function of the FLOS-MCE with cycle frequencies ε , 2ε and 0.
0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
140
160
180
200
τ /Ts
Figure 1. TDOA estimation function of the single-cycle estimator with cycle frequency ε = 0.0625 / Ts . 1 0.9 0.8 0.7
Amplitude
0.6
To illustrate the robustness of the new multi-cycle algorithm to the impulsive noise, the performances of the FLOS-MCE, MCE and FLOS [15] methods are given in Fig. 5 when GSNR is -5 dB. The cycle frequencies exploited by the algorithms are keying rate ε = 0.0625 / Ts , 2ε , and 0. Fig. 5 shows that the FLOS-MCE and FLOS methods greatly outperform the MCE method in impulsive noise condition. The MCE outperforms FLOS-MCE and FLOS only when the additive noise is Gaussian ( α = 2 ). This is due to the fact that the effect of impulsive noise is diminished by fractional lower-order statistics.
0.5
10
4
0.4
10
0.3 0.2
10
3
2
0.1
10 0
20
40
60
80
100 τ /Ts
120
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200
Figure 2. TDOA estimation function of the conventional MCE with cycle frequencies ε , 2ε , and 2 f c .
In contrast to this, the TDOA estimation functions of the proposed double-cycle and triple-cycle FLOS-MCE algorithms are shown in Fig. 3 and Fig. 4. It is clear that both the double-cycle and triple-cycle FLOS-MCE estimators are highly robust to the impulsive noise. The peak of interest is only one, and the highest peak is at the correct TDOA value τ = D = 48Ts . Furthermore, the TDOA function of the triple-cycle FLOS-MCE has sharper peaks than that of the double-cycle FLOS-MCE. 1 0.9 0.8 0.7
Amplitude
0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
140
160
180
200
τ /Ts
Figure 3. TDOA estimation function of the FLOS-MCE with cycle frequencies ε and 2ε .
© 2012 ACADEMY PUBLISHER
MSE
0
10 10 10 10
1
0
-1
-2
-3
1
FLOS
ε=0,1/Tc,2/Tc;MCE ε=2/Tc;FLOS-MCE ε=0,1/Tc;FLOS-MCE ε=0,1/Tc,2/Tc;FLOS-MCE
1.2
1.4
1.6
1.8
2
α Value
Figure 5. Performance of the proposed and conventional TDOA estimation algorithms when GSNR= -5 dB.
The estimation accuracy of the robust double-cycle ( ε , 2ε ) and triple-cycle ( 0, ε , 2ε ) FLOS-MCE estimators are given in Fig. 6 and Fig. 7 in the presence of impulsive noise as the characteristic exponent α changes. Simulation results show that double-cycle estimator is inferior to triple-cycle estimator, and the performance of both double-cycle and triple-cycle estimators degrade as α changes from 1.8 to 1.2. It can be understood from the fact characteristic exponent α controls the tails heaviness of Sα S distribution. As the value of α changes from 1.8 to 1.2, the impulsiveness of the impulsive noise becomes more severe. So performance of the FLOS-MCE is degraded.
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MSE
10
10
10
10
Figure 8. TDOA estimation accuracy interference and Sα S impulsive noise environment.
4
2
0
-2
α=1.2 α=1.4 α=1.6 α=1.8
-4
-12
-9
-6 -3 GSNR (dB)
0
3
Figure 6. TDOA estimation accuracy of the proposed double-cycle FLOS-MCE in the presence of impulsive noise 10
MSE
10
10
10
10
397
multiple
α = 1.6 . From definition (5) we find that the value of p has an impact on the performance of FLOS-MCE. In Fig. 8, the performance of the FLOS-MCE and FLOS methods [15] is evaluated in the presence of multiple interference and impulsive noise as p changing from 1 to 1.7. It can be seen that FLOS-MCE is substantial tolerant to the multiple interferences and impulsive noise as p changing from 1 to 1.4. The estimation accuracy using FLOS has the worst performance among these methods. It is clear that in order to acquire reliable TDOA estimate, the parameter p of the FLOS-MCE should be set to be p < 1.3 .
4
C. Narrowband Interference The interfering signal [5] is a BPSK signal with carrier frequency of f1 = 0.2 / Ts , keying rate of ε1 = 0.025 / Ts , and TDOA of τ 1 = 58 / Ts . The characteristic exponent of Sα S noise is α = 1.8 . The cycle frequency used by the algorithms are ε = 0.0625 / Ts , 2ε and 2 f c . Even though the interference is the same type of SOI, it does not exhibit spectral correlation at ε , 2ε and 2 f c .
2
0
-2
α=1.2 α=1.4 α=1.6 α=1.8
-4
-12
-9
-6 -3 GSNR (dB)
0
3
10
Figure 7. TDOA estimation accuracy of the proposed triple-cycle FLOS-MCE in the presence of impulsive noise.
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10
10
10
10
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4
3
2
ε=1/To;FLOS-MCE ε=1/To,2/To;FLOS-MCE ε=1/To,2/To,2fc;FLOS-MCE
0
-1
-2
1
1.1
10
10
10
4
2
0
-2
-4
-15
ε=1/To;MCE ε=1/To,2/To;MCE ε=1/To,2/To,2fc;MCE ε=1/To;FLOS-MCE ε=1/To,2/To;FLOS-MCE ε=1/To,2/To,2fc;FLOS-MCE
-12
-9
-6
-3
0
3
6
9
GSNR (dB)
Figure 9. TDOA estimation accuracy for narrowband interference and impulsive noise environment.
The TDOA estimation accuracy results are shown in Fig. 9. It can be seen that the accuracy of conventional MCE approximates very close to the new FLOS-MCE when GSNR < -12 dB. Whereas GSNR > -12 dB, the MCE are inferior to the new FLOS-MCE. Furthermore, it is clear that the double-cycle FLOS-MCE performs better than single-cycle estimator, but is inferior to the triplecycle one. V. CONCLUSION
FLOS
1
10
MSE
B. Multiple Interference In this case, the interference [4] consists of five AM signals with carrier frequencies of f1 = 0.156 / Ts , f 2 = 0.203 / Ts , f3 = 0.266 / Ts , f 4 = 0.313 / Ts , and f 5 = 0.375 / Ts , with bandwidths of B1 = 0.04 / Ts , B2 = 0.05 / Ts , B3 = 0.045 / Ts , B4 = 0.04 / Ts , and B5 = 0.08 / Ts , and corresponding TDOAs of τ 1 = 28Ts , τ 2 = 68Ts , τ 3 = 78Ts , τ 4 = 38Ts , and τ 5 = 58Ts . The signal-to-interference ratio (SIR) of each AM signal is 0 dB and GSNR is 0 dB, which yields a total signal-tonoise ratio (SNR) of -8 dB. The cycle frequencies exploited by the algorithms are ε = 0.0625 / Ts , 2ε and 2 f c . The characteristic exponent of Sα S noise is
MSE
in
1.2
1.3
1.4 p
© 2012 ACADEMY PUBLISHER
1.5
1.6
1.7
Conventional second-order cyclostationarity based and FLOS based TDOA estimation methods perform poorly in non-Gaussian impulsive noise and interference environments. A robust multi-cycle signal selective TDOA algorithm based on cyclic fractional lower-order correlations for cyclostationary signals is presented. The new method makes better use of the cyclostationarity property and fractional lower-order statistics, and is tolerant to both interference and impulsive noise. The
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proposed algorithm is a class of methods parameterized by p , include the conventional multi-cycle estimator (3) as a special case ( p = 2 ). Simulation results illustrate that the new FLOS-MCE estimator provides better accuracy for TDOA estimation compared with conventional single-cycle and multi-cycle signal-selective estimators and FLOS method. And the FLOS-MCE algorithm exhibits robustness in a wide range of interference and noise environments. ACKNOWLEDGMENT This work was supported by National Science Foundation of China under Grant 60872122. REFERENCES [1] T. S. Rappaport, J. H. Reed, B. D. Woerner, “Position location using wireless communications on highways of the future”, IEEE Communication Magazine, vol.34, No.10, 1996, pp. 33-41. [2] A. H. Sayed, A. Tarighat, N. Khajehnouri, “Network-based wireless location: Challenges faced in developing techniques for accurate wireless location information”, IEEE Signal Processing Magazine, vol.22, No.4, 2005, pp. 22-40. [3] L. Cong, W. Zhuang, “Hybrid TDOA/AOA mobile user location for wideband CDMA cellular systems”, IEEE Transactions on Wireless Communications, vol.1, No.3, 2002, pp. 439-447. [4] K. Yang, J. An, X. Bu, G. sun, “Constrained total leastsquares location algorithm using time-difference-of-arrival measurements”, IEEE Transactions on Vehicular Technology, vol.59, No.3, 2010, pp. 1558-1562. [5] W. A. Gardner, A. Napolitano, L. Paura, “Cyclostationarity: Half a century of research”, Signal Processing, vol.86, No.4, 2006, pp. 639-697. [6] W. A. Gardner, C. K. Chen, “Signal–selective timedifference-of-arrival estimation for passive location of man-made signal sources in highly corruptive environments, Part 1: Theory and method”, IEEE Transactions on Signal Processing, vol.40, No.5, 1992, pp. 1168-1184. [7] C. K. Chen, W. A .Gardner, “Signal–selective timedifference-of-arrival estimation for passive location of man-made signal sources in highly corruptive environments, Part 2: Algorithms and performance”, IEEE Transactions on Signal Processing, vol.40, No.5, 1992, pp. 1185-1197. [8] W. A. Gardner, C. M. Spoooner, “Detection and source location of weak cyclostationary signals: Simplifications of the maximum-likelihood receiver”, IEEE Transactions on Communications, vol.41, No.6, 1993, pp. 905-916. [9] Z. Huang, Y. Zhou, “Multi-cycle estimator for timedifference-of-arrival (TDOA) and its performance”, IEE Proceedings Radar Sonar and Navigation, vol.153, No.5, 2006, pp. 381-388.
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[10] Z. Huang, Y. Zhou, W. Jiang, “TDOA and Doppler estimation for cyclostationary signals based on multi-cycle frequencies”, IEEE Transactions on Aerospace Electronic System, vol.44, No.4, 2008, pp. 1251-1264. [11] K. Gulati, A. Chopra, B. L. Evans, K. R. Tinsley, “Statistical modeling of co-channel interference”, Proceeding of Global Telecommunication Conference, Holulu, HI, vol.1, 2009, pp. 1-6. [12] C. L. Nikias, M. Shao, Signal processing with AlphaStable Distributions and Applications, Wiley, New York, 1995. [13] R. F. Brcich, D. R. Iskander, A. M. Zoubir, “The stability test for symmetric alpha-stable distributions”, IEEE Transactions on Signal Processing, vol.53, No.3, 2005, pp. 977-986. [14] J. G. Gonzalez, J. L. Paredes, G. R. Arce, “Zero-order statistics: A mathematical framework for the processing and characterization of very impulsive signals”, IEEE Transactions on Signal Processing, vol.54, No.10, 2006, pp. 3839-3851. [15] X. Ma, C. L. Nikias, “Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics”, IEEE Transactions on Signal Processing, vol.44, No.11, 1996, pp. 2269-2687. [16] G. A. Tsihrintzis, C. L. Nikias, “Evaluation of the FLOSbased detection algorithm on real sea-clutter data”, IEE Proceedings Radar, Sonar, and Navigation, vol.144, No.1, 1997, pp. 29-38. [17] P. G. Georgious, P. Tsakalides, “Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise”, IEEE Transactions on Multimedia, vol.1, No.3, 1999, pp. 291-301. Yang Liu Liaoning Province, China. Birthdate: Feb, 1981. is Communication Engineering M.S. graduated from Dept. Electronic Engineering Inner Mongolia University. He is currently working toward the PhD degree in Electronic Engineering at faculty of Electronic Information and Electrical Engineering of Dalian University of Technology. He is a lecturer of Dept. Electronic Engineering Inner Mongolia University. TianShuang Qiu was born in 1954, Jiangsu Province, China. He received the PhD degree from Southeastern University, Nanjing, China, in 1996. He worked as a post doctoral fellow in the Department of Electrical Engineering at Northern Illinois University, USA, from 1996 to 2000. He is currently a professor and doctor advisor in Dalian University of Technology. His research interests include of biomedical computer programs, adaptive signal processing.
393
A Robust Signal Selective TDOA Estimation Algorithm for Cyclostationary Signals Source Location Yang Liu
Dalian University of Technology, Dalian, 116024, China Inner Mongolia University, Huhhot, 010021, China Email: [email protected]
TianShuang Qiu
Dalian University of Technology, Dalian, 116024, China Email: [email protected]
Abstract—For the problem of estimation time-difference-ofarrival (TDOA) for cyclostationary signals in the presence of interference and impulsive noise, a new robust multicycle signal-selective algorithm is introduced. By fusing second-order cyclic moments and fractional lower-order statistics, a novel cyclic fractional lower-order statistics is developed. The proposed algorithm combines the benefits of cyclostationarity based method and fractional lower-order statistics based estimator, by exploiting cyclostationarity property of signals with cyclic fractional lower-order statistics. Simulation results indicate that the new method is highly tolerant to interference and impulsive noise and gives higher estimation accuracy than conventional TDOA estimation methods. Index Terms—cyclostationarity, impulsive noise, fractional lower-order statistics, time-difference-of-arrival (TDOA)
I. INTRODUCTION The problem of locating a Mobile Station (MS) has drawn considerable interest in recent years. A variety of wireless location schemes have been extensively investigated in [1], [2]. One typical method used to estimate the mobile location is time-difference-of-arrival (TDOA) which does not require knowledge of the transmit time of the received signal from the transmitter and has better accuracy than angle of arrival (AOA) [3], [4]. Although several techniques are used to reduce the effects of interference and noise in communication systems [3], it is necessary to develop effective TDOA algorithms. Most man-made signals encountered in radar, sonar, and communication systems are appropriately modeled as cyclostationary time series [5]. A class of signal-selective TDOA methods for passive location is introduced by Gardner et al. [6], [7]. These methods that exploit inherent cyclostationarity of signals are highly tolerant to both interference and Gaussian noise, which are neither cyclostationary signals, nor exhibiting the same cycle frequency of the source signal. Since almost man-made © 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.2.393-398
signals in communication systems have more than one cycle frequency [5], some modified multi-cycle algorithms which exploit more than one cycle frequency are developed in [8-10]. The multi-cycle methods can achieve better performance than single-cycle estimators which utilize only one cycle frequency [10]. The primary single-cycle and multi-cycle methods focus on the case where the environment noise is assumed to follow the Gaussian distribution model. However, many types of noises encountered in practice such as atmospheric noise, multiuser interference and some man-made noise in urban region are heavy tailed non-Gaussian impulsive processes [11], [12]. Studies and experimental measurements have shown that alpha-stable distribution is more suitable for modeling noise of impulsive nature than Gaussian distribution in communication, telemetry, radar, and sonar systems [13], [14]. The alpha-stable model is of a statistical-physical nature, arising under very general assumptions satisfies the stability and the Generalized Center Limit Theorem [16], [17]. It can be described conveniently by four parameters, in which the characteristic exponent α (0 < α ≤ 2) determines the heaviness of its tail. A small positive value of α indicates severe impulsiveness, while a value of α close to 2 indicates a more Gaussian type behavior. When α = 2 , the stable distribution reduced to the Gaussian distribution. So the alpha-stable model is more suitable for modeling noise than Gaussian distribution in real signal processing applications. Several TDOA methods take account the impulsive noise using fractional lower order statistics (FLOS) have been proposed in literature [15]. As stable distribution does not have finite second-order moments (except for α = 2 ), conventional signalselective methods based on the second-order cyclostationarity will be considerably weakened in symmetric alpha-stable ( Sα S ) noise environments. Although the fractional lower-order statistics (FLOS) based methods are robust to both Gaussian and non-
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Gaussian noise [15], the interfering signals which occupy the same spectral band as the source signal can severely degrade the performance of these methods. To extend the signal-selective and FLOS based methods and achieve high accuracy of TDOA estimation, a robust TDOA algorithm which combines multi-cycle estimator with fractional lower-order statistics is proposed in this paper. The new method takes advantages of both multi-cycle estimator and FLOS method can exploit cyclostationarity in the presence of interference and Sα S impulsive noise. Simulation results demonstrate that the proposed algorithm can acquire high accuracy of TDOA estimation. The rest of this paper is organized as follows: The modeling of the signal measurement is briefly described in Section Ⅱ. In Section Ⅲ, the cyclic fractional lowerorder moment and fractional lower-order multi-cycle TDOA algorithm are proposed. The performance of the new algorithm via Monte Carlo simulations is presented in Section Ⅳ. Finally, conclusions are given in Section Ⅴ. II. SIGNAL MODEL In general, for a signal radiating from a remote source through a channel with interference and noise, the model for time difference of arrival estimation between received signals at two base stations is given by x(t ) = s (t ) + n(t ) y (t ) = rs (t − D) + m(t )
(1) (2)
where s (t ) is the signal of interest (SOI), n(t ) and m(t ) are the signals not of interest (SNOI), including interference and noise, D is the TDOA to be estimated, and parameter r represents the magnitude mismatch between two receivers. To simplify the problem, it is assumed that s (t ) is statistically independent of SNOI, n(t ) and m(t ) do not share the same cycle frequency as that of the SOI. However, since n(t ) and m(t ) may contain the same interfering signals (with different times of arrival), they can be statistically dependent [6]. III. THE PROPOSED FRACTIONAL LOWER ORDER MULTI-CYCLE TDOA ESTIMATION METHOD Some single-cycle signal-selective TDOA estimation algorithms based on the second-order cyclostationarity are proposed in [6]. Since the cycle frequency for most communication signals are not unique, the single-cycle estimators cannot achieve the best performance. It was introduced in [8] and [10] that multiple single-cycle estimators can be fused together to form a multi-cycle estimator, g ε (τ )
∑ wi ∫ Ryxεi (τ + u )( Rxεi (u ))* du
(3)
εi ∈A
where ε i is one element in the cycle frequency set A of s (t ) , wi is the weight and wi = 1 . According to the © 2012 ACADEMY PUBLISHER
definition in [6], the cyclic autocorrelation function of s (t ) at cycle frequency ε is defined as s (t + τ / 2) s ∗ (t − τ / 2)e − j 2πε t
Rsε (τ )
where
⋅ = lim(1/ T ) ∫
T /2
−T / 2
T →∞
(⋅)dt
(4)
is the time-averaging
operation, and “ ∗ ” denotes the conjugation. The multi-cycle estimator (3) makes better use of the cyclostationarity features and acquires more accurate estimates of TDOA. However, Due to the thick tails, stable distributions do not have finite second-order moments, except for the limiting case α = 2 . If the received noises contain stable processes, the second-order cyclic moments such as cyclic autocorrelation function in (4) will not be applicable. Thus, the conventional multicycle estimator (3) based on the cyclic correlations will degrade severally in impulsive noise environments. In order to outperform the drawbacks of the conventional multi-cycle algorithm, we define a new cyclic fractional lower-order statistics and propose a new signal-selective estimator by utilizing cyclic fractional lower-order statistics instead of second-order cyclostationarity. Considering the random process s (t ) , the cyclic fractional lower-order autocorrelation function at cycle frequency ε is defined as Rsε , p (τ )
1 T /2 * [ s (t + τ / 2)]〈 p 〉 [ s ∗ (t − τ / 2)]e − j 2πε t dt T ∫−T / 2 (5) [ s* (t + τ / 2)]〈 p 〉 [ s∗ (t − τ / 2)]e − j 2πε t
lim
T →∞
where 1 ≤ p < α and z 〈 k 〉 = z convention z 〈 k 〉 = z
k −1
k −1
z ∗ . In equation (5), the
z ∗ can also be rewritten in a polar
form as z 〈 k 〉 = r k −1 z ∗ = r k e − jθ
(6)
where z = re jθ . Equation (6) demonstrates that convention z 〈 k 〉 contains fractional restrain magnitude but maintain period of random variable z . Thus, cyclic fractional lower-order autocorrelation function (5) and conventional cyclic correlation (4) share the same cycle frequency. In other words, z 〈 k 〉 attenuates only the magnitude value of z , but it maintains the periods of z . Both of them exhibit the same cyclostationarity property. Note that the cyclic autocorrelation Rsε (τ ) could be expressed as the conventional cross correlation for the pair of time series u (t ) and v(t ) [6], Rsε (τ )
s (t + τ / 2) s ∗ (t − τ / 2)e− j 2πε t = u (t + τ / 2)v∗ (t − τ / 2)
(7)
=Ruv0 (τ )
where u (t )
s (t )e − jπε t
(8)
v(t )
s(t )e jπε t
(9)
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are obtained by frequency shifting s (t ) . By substituting (8) and (9) into (5), with the properties of z 〈 k 〉 e± jπε t = ( ze∓ jπε t )〈 k 〉 and ( z ∗ )〈 k 〉 = ( z 〈 k 〉 )∗ , the cyclic fractional lower-order autocorrelation Rsε , p (τ ) can be reexpressed as Rsε , p (τ ) = [u ∗ (t + τ / 2)]〈 p 〉 [v(t − τ / 2)] = Ruvp (τ ).
(10)
and v (t ) are reduced to fractional powers. Unlike cyclic ε
autocorrelation Rs (τ ) , even if the received noise in n(t ) and m(t ) are Sα S distribution impulsive processes, the cyclic fractional lower-order autocorrelation Rsε , F (τ ) is also applicable. Thus, the cyclic fractional lower-order autocorrelation function can be expected to exploit spectral correlation in the presence of impulsive noise. In particular, for ε = 0 , cyclic fractional lower-order autocorrelation Rsε , p (τ ) becomes conventional fractional
lower-order autocorrelation R (τ ) . And for p = 2 , it reduces to the conventional second-order cyclic autocorrelation Rsε (τ ) . Substitute (1) and (2) into (5) and assume that r is a real number, the cyclic fractional lower-order autocorrelation and cross correlation functions are given by p s
Rxε , p (τ ) = Rsε , p (τ )
(11)
ε,p Ryx (τ ) = r 〈 p 〉 Rsε , p (τ − D)e − jπε D .
(12)
To overcome the drawbacks of conventional signalselective methods, we propose a new algorithm based on cyclic fractional lower-order correlations. As an alternative, the new estimator w ∫ Rε ∑ ε i
i,p
yx
(τ + u )( Rxε i , p (u ))* du (13)
i ∈A
is called fractional lower-order statistics multi-cycle estimator (FLOS-MCE). By substituting the idealized measurements (11) and (12) into (13), we obtain the ideal function of the FLOS-MCE
∑r ε
∑r
〈 p〉
εi ∈A
≤ r
p
wi e − jπε i D ∫ Rsε i , p (τ − D + u )( Rsε i , p (u ))∗ du
∑∫R
εi , p
s
εi ∈A
2
(15)
(u ) du .
It is noted that only if τ = D can the equality in (15) hold. When wi = e jπε i D and τ = D , we have 2
Equation (10) reveals that the cyclic fractional lowerorder autocorrelation can be expressed as conventional fractional lower-order cross-correlation of the frequencyshifted versions u (t ) and v (t ) of s (t ) . It should be noted that by using signed-power nonlinearity z 〈 k 〉 , impulsive noise components in u (t )
g Mε , p (τ )
g Mε , p (τ ) =
g Mε , p ( D) = r 〈 p 〉 ∑ ∫ Rsε i , p (u ) du .
(16)
εi ∈A
Assume r is a real number, then r g Mε , p ( D) = r
p
∑ ∫ Rε ε s
i,p
p
= r , p
2
(u ) du = max g Mε , p (τ ) . (17)
i ∈A
To see that g Mε , p (τ ) does indeed peak at τ = D , the solution of FLOS-MCE is given by D = arg max{ g Mε , p (τ ) }, wi = e j 2πε i D . τ
(18)
Since the true value of D is unknown, wi can not be computed directly in real applications. In order to obtain accurate estimate of D , it should let wi = 1 get the initial gˆ ε , p (τ ) and Dˆ , then estimate w through iteration [9]. M
i
Note that when p = 2 the FLOS-MCE reduces to the conventional multi-cycle estimator (MCE). VI. SIMULATION RESULTS In this section, we present the results of a simulation study designed to compare the performance of the above algorithms for TDOA estimation. Simulation results are provided to demonstrate the effectiveness and robustness of the proposed method. We consider a real simulated BPSK signal [7], [9] with uncorrelated Sα S noise added to the two received signals. The carrier frequency of the BPSK signal is f c = 0.25 / Ts , keying rate of ε = 0.0625 / Ts with chip width To = 16Ts and sampling increment Ts . The TDOA between two received signals is fixed at D = 48Ts . The length of the received signals is 1024 × 32 baud. As the stable distribution makes the standard SNR meaningless, then a new generalized SNR (GSNR) is defined as GSNR = 10 log σ s2 γ n , where σ s2 is the variance of the signal, γ n is the dispersion parameter of the Sα S noise.
wi e − jπε i D ∫ Rsε i , p (τ − D + u )( Rsε i , p (u ))∗ du .(14) A. The Robustness of New Multi-cycle Algorithm The TDOA function of the conventional single-cycle i ∈A estimator and multi-cycle estimator (MCE) in the It follows from (14) that presence of impulsive noise is shown in Fig. 1, and Fig. 2, respectively. The GSNR= 0 dB, characteristic exponent of the Sα S noise is α = 1.5 , and p = 1.2 . It can be seen from Fig. 1 and Fig. 2 that although the conventional single-cycle and multi-cycle signal-selective TDOA methods are immune to Gaussian noise, both of them fail g Mε , p (τ ) =
〈 p〉
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to give correct estimation in this case where alpha-stable impulsive noise is present. The dominant peak of the single-cycle estimator is at τ = 56Ts . And the peak of interest is only one of many peaks, any one of which might be taken as the TDOA estimate. Similarly, the highest peak of the multi-cycle estimator is also at the wrong TDOA value τ = 52Ts .
1 0.9 0.8 0.7 Amplitude
396
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1
0.2 0.9
0.1 0.8
0 0.7
0
50
100
150
200
τ /Ts
Amplitude
0.6
Figure 4. TDOA estimation function of the FLOS-MCE with cycle frequencies ε , 2ε and 0.
0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
140
160
180
200
τ /Ts
Figure 1. TDOA estimation function of the single-cycle estimator with cycle frequency ε = 0.0625 / Ts . 1 0.9 0.8 0.7
Amplitude
0.6
To illustrate the robustness of the new multi-cycle algorithm to the impulsive noise, the performances of the FLOS-MCE, MCE and FLOS [15] methods are given in Fig. 5 when GSNR is -5 dB. The cycle frequencies exploited by the algorithms are keying rate ε = 0.0625 / Ts , 2ε , and 0. Fig. 5 shows that the FLOS-MCE and FLOS methods greatly outperform the MCE method in impulsive noise condition. The MCE outperforms FLOS-MCE and FLOS only when the additive noise is Gaussian ( α = 2 ). This is due to the fact that the effect of impulsive noise is diminished by fractional lower-order statistics.
0.5
10
4
0.4
10
0.3 0.2
10
3
2
0.1
10 0
20
40
60
80
100 τ /Ts
120
140
160
180
200
Figure 2. TDOA estimation function of the conventional MCE with cycle frequencies ε , 2ε , and 2 f c .
In contrast to this, the TDOA estimation functions of the proposed double-cycle and triple-cycle FLOS-MCE algorithms are shown in Fig. 3 and Fig. 4. It is clear that both the double-cycle and triple-cycle FLOS-MCE estimators are highly robust to the impulsive noise. The peak of interest is only one, and the highest peak is at the correct TDOA value τ = D = 48Ts . Furthermore, the TDOA function of the triple-cycle FLOS-MCE has sharper peaks than that of the double-cycle FLOS-MCE. 1 0.9 0.8 0.7
Amplitude
0.6 0.5 0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
120
140
160
180
200
τ /Ts
Figure 3. TDOA estimation function of the FLOS-MCE with cycle frequencies ε and 2ε .
© 2012 ACADEMY PUBLISHER
MSE
0
10 10 10 10
1
0
-1
-2
-3
1
FLOS
ε=0,1/Tc,2/Tc;MCE ε=2/Tc;FLOS-MCE ε=0,1/Tc;FLOS-MCE ε=0,1/Tc,2/Tc;FLOS-MCE
1.2
1.4
1.6
1.8
2
α Value
Figure 5. Performance of the proposed and conventional TDOA estimation algorithms when GSNR= -5 dB.
The estimation accuracy of the robust double-cycle ( ε , 2ε ) and triple-cycle ( 0, ε , 2ε ) FLOS-MCE estimators are given in Fig. 6 and Fig. 7 in the presence of impulsive noise as the characteristic exponent α changes. Simulation results show that double-cycle estimator is inferior to triple-cycle estimator, and the performance of both double-cycle and triple-cycle estimators degrade as α changes from 1.8 to 1.2. It can be understood from the fact characteristic exponent α controls the tails heaviness of Sα S distribution. As the value of α changes from 1.8 to 1.2, the impulsiveness of the impulsive noise becomes more severe. So performance of the FLOS-MCE is degraded.
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MSE
10
10
10
10
Figure 8. TDOA estimation accuracy interference and Sα S impulsive noise environment.
4
2
0
-2
α=1.2 α=1.4 α=1.6 α=1.8
-4
-12
-9
-6 -3 GSNR (dB)
0
3
Figure 6. TDOA estimation accuracy of the proposed double-cycle FLOS-MCE in the presence of impulsive noise 10
MSE
10
10
10
10
397
multiple
α = 1.6 . From definition (5) we find that the value of p has an impact on the performance of FLOS-MCE. In Fig. 8, the performance of the FLOS-MCE and FLOS methods [15] is evaluated in the presence of multiple interference and impulsive noise as p changing from 1 to 1.7. It can be seen that FLOS-MCE is substantial tolerant to the multiple interferences and impulsive noise as p changing from 1 to 1.4. The estimation accuracy using FLOS has the worst performance among these methods. It is clear that in order to acquire reliable TDOA estimate, the parameter p of the FLOS-MCE should be set to be p < 1.3 .
4
C. Narrowband Interference The interfering signal [5] is a BPSK signal with carrier frequency of f1 = 0.2 / Ts , keying rate of ε1 = 0.025 / Ts , and TDOA of τ 1 = 58 / Ts . The characteristic exponent of Sα S noise is α = 1.8 . The cycle frequency used by the algorithms are ε = 0.0625 / Ts , 2ε and 2 f c . Even though the interference is the same type of SOI, it does not exhibit spectral correlation at ε , 2ε and 2 f c .
2
0
-2
α=1.2 α=1.4 α=1.6 α=1.8
-4
-12
-9
-6 -3 GSNR (dB)
0
3
10
Figure 7. TDOA estimation accuracy of the proposed triple-cycle FLOS-MCE in the presence of impulsive noise.
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10
10
10
10
10
10
4
3
2
ε=1/To;FLOS-MCE ε=1/To,2/To;FLOS-MCE ε=1/To,2/To,2fc;FLOS-MCE
0
-1
-2
1
1.1
10
10
10
4
2
0
-2
-4
-15
ε=1/To;MCE ε=1/To,2/To;MCE ε=1/To,2/To,2fc;MCE ε=1/To;FLOS-MCE ε=1/To,2/To;FLOS-MCE ε=1/To,2/To,2fc;FLOS-MCE
-12
-9
-6
-3
0
3
6
9
GSNR (dB)
Figure 9. TDOA estimation accuracy for narrowband interference and impulsive noise environment.
The TDOA estimation accuracy results are shown in Fig. 9. It can be seen that the accuracy of conventional MCE approximates very close to the new FLOS-MCE when GSNR < -12 dB. Whereas GSNR > -12 dB, the MCE are inferior to the new FLOS-MCE. Furthermore, it is clear that the double-cycle FLOS-MCE performs better than single-cycle estimator, but is inferior to the triplecycle one. V. CONCLUSION
FLOS
1
10
MSE
B. Multiple Interference In this case, the interference [4] consists of five AM signals with carrier frequencies of f1 = 0.156 / Ts , f 2 = 0.203 / Ts , f3 = 0.266 / Ts , f 4 = 0.313 / Ts , and f 5 = 0.375 / Ts , with bandwidths of B1 = 0.04 / Ts , B2 = 0.05 / Ts , B3 = 0.045 / Ts , B4 = 0.04 / Ts , and B5 = 0.08 / Ts , and corresponding TDOAs of τ 1 = 28Ts , τ 2 = 68Ts , τ 3 = 78Ts , τ 4 = 38Ts , and τ 5 = 58Ts . The signal-to-interference ratio (SIR) of each AM signal is 0 dB and GSNR is 0 dB, which yields a total signal-tonoise ratio (SNR) of -8 dB. The cycle frequencies exploited by the algorithms are ε = 0.0625 / Ts , 2ε and 2 f c . The characteristic exponent of Sα S noise is
MSE
in
1.2
1.3
1.4 p
© 2012 ACADEMY PUBLISHER
1.5
1.6
1.7
Conventional second-order cyclostationarity based and FLOS based TDOA estimation methods perform poorly in non-Gaussian impulsive noise and interference environments. A robust multi-cycle signal selective TDOA algorithm based on cyclic fractional lower-order correlations for cyclostationary signals is presented. The new method makes better use of the cyclostationarity property and fractional lower-order statistics, and is tolerant to both interference and impulsive noise. The
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proposed algorithm is a class of methods parameterized by p , include the conventional multi-cycle estimator (3) as a special case ( p = 2 ). Simulation results illustrate that the new FLOS-MCE estimator provides better accuracy for TDOA estimation compared with conventional single-cycle and multi-cycle signal-selective estimators and FLOS method. And the FLOS-MCE algorithm exhibits robustness in a wide range of interference and noise environments. ACKNOWLEDGMENT This work was supported by National Science Foundation of China under Grant 60872122. REFERENCES [1] T. S. Rappaport, J. H. Reed, B. D. Woerner, “Position location using wireless communications on highways of the future”, IEEE Communication Magazine, vol.34, No.10, 1996, pp. 33-41. [2] A. H. Sayed, A. Tarighat, N. Khajehnouri, “Network-based wireless location: Challenges faced in developing techniques for accurate wireless location information”, IEEE Signal Processing Magazine, vol.22, No.4, 2005, pp. 22-40. [3] L. Cong, W. Zhuang, “Hybrid TDOA/AOA mobile user location for wideband CDMA cellular systems”, IEEE Transactions on Wireless Communications, vol.1, No.3, 2002, pp. 439-447. [4] K. Yang, J. An, X. Bu, G. sun, “Constrained total leastsquares location algorithm using time-difference-of-arrival measurements”, IEEE Transactions on Vehicular Technology, vol.59, No.3, 2010, pp. 1558-1562. [5] W. A. Gardner, A. Napolitano, L. Paura, “Cyclostationarity: Half a century of research”, Signal Processing, vol.86, No.4, 2006, pp. 639-697. [6] W. A. Gardner, C. K. Chen, “Signal–selective timedifference-of-arrival estimation for passive location of man-made signal sources in highly corruptive environments, Part 1: Theory and method”, IEEE Transactions on Signal Processing, vol.40, No.5, 1992, pp. 1168-1184. [7] C. K. Chen, W. A .Gardner, “Signal–selective timedifference-of-arrival estimation for passive location of man-made signal sources in highly corruptive environments, Part 2: Algorithms and performance”, IEEE Transactions on Signal Processing, vol.40, No.5, 1992, pp. 1185-1197. [8] W. A. Gardner, C. M. Spoooner, “Detection and source location of weak cyclostationary signals: Simplifications of the maximum-likelihood receiver”, IEEE Transactions on Communications, vol.41, No.6, 1993, pp. 905-916. [9] Z. Huang, Y. Zhou, “Multi-cycle estimator for timedifference-of-arrival (TDOA) and its performance”, IEE Proceedings Radar Sonar and Navigation, vol.153, No.5, 2006, pp. 381-388.
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[10] Z. Huang, Y. Zhou, W. Jiang, “TDOA and Doppler estimation for cyclostationary signals based on multi-cycle frequencies”, IEEE Transactions on Aerospace Electronic System, vol.44, No.4, 2008, pp. 1251-1264. [11] K. Gulati, A. Chopra, B. L. Evans, K. R. Tinsley, “Statistical modeling of co-channel interference”, Proceeding of Global Telecommunication Conference, Holulu, HI, vol.1, 2009, pp. 1-6. [12] C. L. Nikias, M. Shao, Signal processing with AlphaStable Distributions and Applications, Wiley, New York, 1995. [13] R. F. Brcich, D. R. Iskander, A. M. Zoubir, “The stability test for symmetric alpha-stable distributions”, IEEE Transactions on Signal Processing, vol.53, No.3, 2005, pp. 977-986. [14] J. G. Gonzalez, J. L. Paredes, G. R. Arce, “Zero-order statistics: A mathematical framework for the processing and characterization of very impulsive signals”, IEEE Transactions on Signal Processing, vol.54, No.10, 2006, pp. 3839-3851. [15] X. Ma, C. L. Nikias, “Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics”, IEEE Transactions on Signal Processing, vol.44, No.11, 1996, pp. 2269-2687. [16] G. A. Tsihrintzis, C. L. Nikias, “Evaluation of the FLOSbased detection algorithm on real sea-clutter data”, IEE Proceedings Radar, Sonar, and Navigation, vol.144, No.1, 1997, pp. 29-38. [17] P. G. Georgious, P. Tsakalides, “Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise”, IEEE Transactions on Multimedia, vol.1, No.3, 1999, pp. 291-301. Yang Liu Liaoning Province, China. Birthdate: Feb, 1981. is Communication Engineering M.S. graduated from Dept. Electronic Engineering Inner Mongolia University. He is currently working toward the PhD degree in Electronic Engineering at faculty of Electronic Information and Electrical Engineering of Dalian University of Technology. He is a lecturer of Dept. Electronic Engineering Inner Mongolia University. TianShuang Qiu was born in 1954, Jiangsu Province, China. He received the PhD degree from Southeastern University, Nanjing, China, in 1996. He worked as a post doctoral fellow in the Department of Electrical Engineering at Northern Illinois University, USA, from 1996 to 2000. He is currently a professor and doctor advisor in Dalian University of Technology. His research interests include of biomedical computer programs, adaptive signal processing.
