Improved time difference of arrival estimation algorithms for cyclostationary signals in α-stable impulsive noise

Digital Signal Processing 76 (2018) 94–105

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Digital Signal Processing www.elsevier.com/locate/dsp

Improved time difference of arrival estimation algorithms for cyclostationary signals in α -stable impulsive noise Yang Liu a,b , Yinghui Zhang a , Tianshuang Qiu c , Jing Gao d , Shun Na a,∗ a

College of Electronic Information Engineering, Inner Mongolia University, Hohhot, 010021, China Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China c Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China d Tianjin Key Laboratory of Wireless Mobile Communications and Power Transmission, Tianjin Normal University, Tianjin, 300387, China b

a r t i c l e

i n f o

Article history: Available online 21 February 2018 Keywords: Cyclostationarity Time difference of arrival (TDOA) α -Stable distribution Fractional lower-order statistics

a b s t r a c t In this study, we introduce two new robust signal-selective algorithms based on the fractional lowerorder cyclostationarity in order to address the problem of estimating time difference of arrival (TDOA) for cyclostationary signals in the presence of interference and α -stable distribution impulsive noise. Conventional signal-selective and fractional lower-order statistics (FLOS) based TDOA methods suffer performance degradation in the presence of non-Gaussian α -stable impulsive noise and corruptive interfering signals. By exploiting fractional lower-order cyclostationarity, we are able to develop a novel multi-cycle method and a generalized fractional lower-order spectral coherence method. The proposed methods restrain the effects of α -stable impulsive noise and make better use of the cyclostationarity property of cyclostationary signals. Simulation results indicate that the new methods are highly tolerant to interference and impulsive noise, and provide higher estimation accuracy than conventional algorithms. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The problem of time difference of arrival (TDOA) estimation of a signal received at two separate receivers has garnered considerable interest in recent years [1,2]. This attention results from the potentially wide range of practical applications for TDOA. One of the most important applications is source location, which is a crucial element in radar, sonar, navigation, communications, and wireless sensor networks. Angle of arrival (AOA), time of arrival (TOA), received signal strength (RSS), and TDOA of the emitted signals are commonly used measurements for source location [3–6]. The TDOA technique is generally superior to the RSS and AOA techniques. In addition, compared to TOA, the TDOA technique does not require knowledge of the transmit time of the transmitter. Therefore, a variety of effective emitter location methods rely on the accurate and robust estimation of TDOA. The generally applicable TDOA estimation method requires a generalized cross correlation (GCC) between the signals collected at one receiver with the signals collected at another receiver. GCC is the foundation of some of the most useful methods for estimating TDOA. In addition, other methods for TDOA estima-

*

Corresponding author. E-mail address: [email protected] (S. Na).

https://doi.org/10.1016/j.dsp.2018.02.010 1051-2004/© 2018 Elsevier Inc. All rights reserved.

tion include linear regression analysis of the phase data, adaptive filtering, and bispectrum estimation, etc. [7]. Many manmade modulated signals are appropriately modeled as cyclostationary time series, and the cyclostationary properties have been widely used to perform signal processing tasks in communication, radar, and sonar systems [8–11]. For TDOA estimation of cyclostationary signals, Gardner et al. introduced a class of signalselective methods for passive location [12,13]. Since the noise and interference are not cyclostationary signals or do not exhibit the same cycle frequency of the source signals, the more recent signal-selective TDOA estimation methods that exploit the inherent cyclostationarity of communication signals are highly tolerant to both noise and interference. One of the most successful algorithms based on cyclostationarity is spectral coherence alignment (SPECCOA). Some improved algorithms are generalizations of the SPECCOA method [14–17]. Most of the cyclostationary signals in communication systems have more than one cycle frequency [11,18], researchers have developed some modified multicycle signal-selective algorithms that exploit more than one cycle frequency [18,20]. For instance, the multi-cycle method performs better than single cycle estimators when the spectral correlation for the cycle frequencies used in the sum of the multi-cycle method is stronger than that for the cycle frequency of single cycle method.

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

Conventional single cycle and multi-cycle signal-selective TDOA estimation methods demonstrate signal selectivity, and are immune to interference and noise. However, most existing signalselective methods focus on scenarios where environmental noises are assumed to follow the Gaussian distribution model. Generally, it can be assumed that noises follow the Gaussian distribution model with finite second-order statistics in communication applications, which leads to closed-form solutions. Whereas some types of noises encountered in practice, such as atmospheric noise, multiuser interference, and some man-made noises in urban regions, are heavy tailed non-Gaussian impulsive processes [21–24]. Studies and experimental measurements have shown that α -stable distribution is more suitable for modeling noises of an impulsive nature (compared to Gaussian distribution models) in communication, telemetry, radar and sonar systems [25,26]. The α -stable distributions do not have finite second-order moments (except for α = 2), or even first-order moments (α < 1) [27]. If the received signals contain stable distributed impulsive noises (α < 1), the second-order moments will become infinite, thus the secondorder statistics based TDOA algorithms that perform well under the Gaussian assumption exhibit various degrees of performance degradation in the presence of impulsive noise. Therefore, conventional signal-selective TDOA estimation methods based on secondorder cyclostationarity weakened in additive α -stable impulsive noise environments. Several TDOA methods have been introduced in [27,28], which take into account impulsive noise using fractional lower-order statistics (FLOS). According to the FLOS theory and the characteristics of α -stable distribution, the phased fractional lower-order moment operation can transfer the α -stable distributed process x(t ) in to second-order moment process (x(t ))a when (0 ≤ a < α /2 ≤ 1), and then the impulsive components of the signals can be suppressed by the FLOS. Therefore, the FLOS based methods are robust in terms of both Gaussian noise and non-Gaussian impulsive noise. Nevertheless, the interfering signals that occupy the same spectral band as the source signal can severely degrade the performance of these methods. In order to extend the signalselective and FLOS based methods and achieve higher accuracy in TDOA estimation, [29,30] introduced a class of TDOA methods that take into account impulsive noise and interference using fractional lower-order cyclostationarity. The fractional lower-order cyclic correlation functions are also successfully used to estimate the direction of arrival for cyclostationary signals in impulsive noise [31]. A generalized fractional lower-order cyclic ambiguity function for jointly estimating TDOA/FDOA was proposed in [32]. The generalized fractional lower-order cyclic ambiguity function can provide TDOA estimate when Doppler is zero in the presence of impulsive noise and interference. Although some techniques are used to reduce the effects of interference and noise in fractional lower-order cyclic algorithms, it is necessary to develop more effective TDOA algorithms by making better use of the fractional lower-order cyclostationarity. Highly accurate TDOA measurements ensure highly accurate location information. In order to obtain more accurate TDOA estimates, we propose two robust signal-selective TDOA estimation algorithms that are highly tolerant to interference, Gaussian noise, and impulsive noise. First, we present a robust fractional lower-order multicycle signal-selective method. The new multi-cycle method exploits more than one cycle frequency and makes better use of fractional lower-order cyclostationarity. Oftentimes, it is difficult to obtain enough accurate prior knowledge about signals’ cycle frequency. To circumvent this limitation, a generalized fractional lower-order spectral coherence method is introduced. The generalized fractional lower-order spectral coherence method utilizes fractional lower-order cyclostationarity at only one cycle frequency. These two new methods take advantage of both cyclostationarity

95

property and fractional lower-order statistics that allow for high accuracy TDOA estimation. This paper first addresses the cyclostationarity and fractional lower-order cyclostationarity of cyclostationary signals. We then introduce two new robust signal-selective TDOA algorithms based on fractional lower-order cyclostationarity. Next, we present the performance results of the new algorithms via Monte Carlo simulations. Finally, we provide conclusions. 2. Cyclostationarity and fractional lower-order cyclostationarity For a signal radiating from a source through a channel with interference and noise, the received signals at two base stations are x(t ) and y (t ). The cyclostationary signal of interest (SOI) is s(t ), n(t ) and m(t ) represent the signals not of interest (SNOI), including interference and noise in x(t ) and y (t ). The signal model is expressed as

x(t ) = r1 s(t ) + n(t )

(1)

y (t ) = r2 s(t − D ) + m(t )

(2)

where r1 and r2 represent the magnitude mismatch between the two receivers, and D is the TDOA to be estimated using measurements x(t ) and y (t ). Without loss generality, we made some simplifying assumptions to the model so that n(t ) and m(t ) do not share the same cycle frequency as that of s(t ), and s(t ), n(t ) and m(t ) do not exhibit joint cyclostationarity. In addition, it is assumed that SOI is statistically independent of the SNOI. According to the definition in [11], the cyclic autocorrelation function of x(t ) at certain cycle frequency ε is defined as





R εx (τ )  x(t + τ /2)x∗ (t − τ /2)e − j2π εt (3) ´ T /2 where · = lim T →∞ (1/ T ) − T /2 (·)dt is the time-averaging operation, and “∗” denotes the conjugation. The spectral correlation function is defined as the Fourier transform of the cyclic autocorrelation function, +∞ ˆ

S εx ( f ) 

R εx (τ )e − j2π f τ dτ

(4)

−∞

Based on (1) and (2), the cyclic cross correlation of y (t ) and x(t ), and the autocorrelation of x(t ) are given by ε (τ ) R εyx (τ ) = r2 r1∗ e − j π ε D R εs (τ − D ) + R mn

(5)

R εx (τ ) = |r1 |2 R εs (τ ) + R nε (τ )

(6)

It is assumed that the SNOI does not exhibit the same cyclostationarity as that of the SOI, and s(t ), n(t ) and m(t ) do not exhibit joint cyclostationarity. Thus, if the SOI contain a cycle frequency ε not shared by any of the SNOI, and s(t ), n(t ) and m(t ) do not exhibit joint cyclostationarity at cycle frequency ε , then ε (τ ) = 0. From this, we obtain R nε (τ ) = R mn

R εyx (τ ) = r2 r1∗ e − j π ε D R εs (τ − D )

(7)

R εx (τ ) = |r1 |2 R εs (τ )

(8)

This is the means by which we obtained signal selectivity in the cyclic measurements [12]. In practical applications, it is frequently observed that many physical phenomena are definitively non-Gaussian distributed that exhibit impulsive behavior, such as underwater acoustic, low frequency atmospheric, multiuser interference, radar clusters, and some man-made noises. Among many non-Gaussian distributions,

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Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

the α -stable distribution is used to model such impulsive processes [21–24]. The α -stable distribution is most conveniently defined by its characteristic function, since a closed form expression for its probability density function is not always available. The characteristic function is given by







ϕ (t ) = exp jat − γ |t |α 1 + j β sign(t )ω(t , α )

(9)

where α , β , γ and a are the characteristic exponent, symmetry parameter (the distribution is symmetrical α -stable distribution (S α S) when β = 0), dispersion, and location, respectively. The ω(t , α ) = tan(απ /2) if α = 1 and ω(t , α ) = (α /π ) log |t | if α = 1. In addition, 0 < α ≤ 2, −1 ≤ β ≤ 1, γ > 0, −∞ < a < +∞. The striking characteristic of α -stable distributions is their algebraic tails, which are thicker than Gaussian distribution. The smaller the value of α , the thicker the tails. Due to the thick tails, α -stable distributions do not have finite second-order moments, except for the limiting case of α = 2. If the received noises contain α -stable distribution processes, the second-order cyclic correlation function and the spectral correlation function are not applicable. Hence, it is necessary to define new cyclic statistics that can be used to exploit the cyclostationarity property of cyclostationary signals in the presence of impulsive noise. A type of fractional lower-order cyclic statistics is defined in [29]. The fractional lowerorder cyclic autocorrelation function is defined as

R εx , F (τ )

ˆT /2

1

 lim

T →∞

T

E



a 

x(t + τ /2)

b 

x(t − τ /2)

e − j2π εt dt

− T /2

a  b − j2π εt   x(t + τ /2) x(t − τ /2) e 

0 ≤ a , b < α /2

(10)

where z p  is the phase fractional lower-order moment and z p  = | z| p −1 z. Compared with conventional cyclic autocorrelation, the fractional lower-order cyclic autocorrelation can exploit the cyclostationarity property of signals in impulsive noise. The fractional lower-order cyclic spectrum is given by

ˆ∞ S εx , F ( f ) =

R εx , F (τ )e − j2π f τ dτ

(11)

−∞

The fractional lower-order cyclostationarity is an alternative but equivalent characterization of the second-order periodicity. When a = b = 1, the fractional lower-order cyclic correlation and fractional lower-order spectral correlation functions become to the second-order cyclic correlation and spectral correlation functions. Since the SOI is statistically independent of the SNOI, by substituting (1) and (2) into (10), the fractional lower-order cyclic cross correlation and autocorrelation functions are given by b a

ε , F (τ ) R εyx, F (τ ) = r1 r2 e − j π ε D R εs , F (τ − D ) + R mn a b R εx , F (τ ) = r1 r1 R εs , F (τ ) + R nε, F (τ )

(12) (13)

According to the assumptions that the noise is not a cyclostationary signal and the interfering signals do not share the same cycle frequency as the SOI, and s(t ), n(t ) and m(t ) do not exhibit ε, F ε, F joint cyclostationarity, thus R mn (τ ) = R n (τ ) = 0. Equations (12) and (13) become to (derivation of (14) and (15) see Appendix) b a

R εyx, F (τ ) = r1 r2 e − j π ε D R εs , F (τ − D ) a b R ε, F (τ ) = r r R ε, F (τ ) x

1

1

s

(14) (15)

Fig. 1. The spectral correlation function and fractional lower-order spectral correlation function for the BPSK signal in S α S impulsive noise (α = 1.6). (a) Spectral correlation function. (b) Fractional lower-order spectral correlation function.

From (14) and (15), it is evident that the fractional lower-order cyclic correlation functions exhibit the same signal selectivity as the second-order cyclic correlation functions. However, compared with the second-order cyclostationarity, the fractional lower-order cyclostationarity is robust in terms of impulsive noise, because the impulsive components of the two received signals are suppressed by the phase fractional lower-order moment operation. Fig. 1 shows the effects of symmetrical α -stable (S α S) impulsive noise (α = 1.6) on the second-order spectral correlation function and the fractional lower-order spectral correlation function. The SOI is a binary phase-shift keyed (BPSK) signal with a carrier frequency of f = 0.1 f s and a keying rate of ε = 0.025 f s , where f s represents the sampling frequency. The fractional lower-order exponents are a = b = 0.5. Significantly, the fractional lower-order spectral correlation function is robust in comparison to the secondorder spectral correlation function in impulsive noise. The secondorder spectral correlation function is useless in this scenario because it is completely masked by the impulsive noise. 3. Proposed fractional lower-order signal-selective TDOA estimation methods In this section, we derive two new fractional lower-order signalselective algorithms in order to solve the degradation issue in existing cyclostationary TDOA estimation methods. The proposed algorithms are developed for a large class of signal-selective methods, including the conventional method as a special element.

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

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3.1. Conventional single cycle and multi-cycle signal-selective methods The problem of estimating the time difference of arrival by exploiting cyclostationarity has been closely studied by Gardner and other researchers [12–19]. A class of signal-selective TDOA algorithm that is highly tolerant to interference and noise was introduced in [12] and [13]. Arguably the most widely accepted signal-selective TDOA method is the spectral coherence alignment (SPECCOA) method [12,19]. In order to improve the performance, a generalized SPECCOA (GSPECCOA) algorithm was proposed in [14]. The SPECCOA function is defined as

ˆ



∗

S εyx ( f ) S εx ( f ) e j2π f τ df

aε (τ ) =

(16)

Typical cycle frequencies for many communications signals include the keying rate and its harmonics, the double carrier frequency, as well as sums and differences of these cycle frequencies. For example, for a BPSK signal or a quaternary phase-shift keying (QPSK) signal, the cycle frequencies are εi = ±2 f c + i εk (i = 0, ±1, ±2, . . .), where f c represents the carrier frequency, εk represents the keying rate. Since most of the cyclostationary signals exhibit more than one cycle frequency, the performance of the signal-selective methods is subject to the selected cycle frequency [18,19]. It was introduced in [19] and [20] that multiple single cycle estimators can be fused together to form a multi-cycle estimator. The multi-cycle method is defined by



c (τ ) 

ˆ

 ε

ε

∗

R yxi (τ + u ) R x i (u ) du ,

wi

w i = e j π εi D

(17)

εi ∈ A

where εi is one element in the cycle frequency set A of s(t ), w i is the weight, and | w i | = 1. The single cycle estimator used in (17) is the correlated cyclic cross-correlation (CCCC) method. From the Wiener–Khintchine and Parseval theorems, we can see that

ˆ



∗

R εyx (τ + u ) R εx (u ) du

ˆ ˆ = ¨ = =

ˆ

S εyx ( f )e j2π f (τ +u ) df

 ˆ



S εx ( f )e j2π f u df

∗ du

∗

Fig. 2. The TDOA estimation functions of the conventional single cycle and multicycle estimators in impulsive noise. (a) Single cycle estimator ε = εk . (b) Multi-cycle estimator with cycle frequencies of εk , 2εk .

S εyx ( f )e j2π f (τ +u ) S εx ( f ) e − j2π f u dudf



∗

S εyx ( f ) S εx ( f ) e j2π f τ df

(18)

From (18), it is evident that the CCCC method is equivalent to the SPECCOA method. It is indicated in (16) and (18) that the single cycle SPECCOA which uses only one cycle frequency is a special case of the multi-cycle estimator. However, the multi-cycle method that does process the cycle frequencies which spectral correlation is stronger than that at cycle frequency of single cycle method makes better use of the cyclostationarity features, and acquires more accurate estimates of TDOA [19,20]. Both the single cycle and multi-cycle methods based on secondorder cyclostationarity have been proposed in the assumption that the noise present is additive Gaussian noise. Nevertheless, it has been shown that second-order cyclostationarity is not applicable in α -stable distribution impulsive noise environments. Therefore, the performance of conventional single cycle and multi-cycle signalselective TDOA methods based on second-order cyclostationarity degrades severely in the presence of impulsive noise. Fig. 2 shows the TDOA estimation functions of the conventional single cycle and multi-cycle estimators in the presence of impulsive noise (α = 1.6). The SOI is a QPSK signal with a carrier frequency of f c = 0.1/ T s , and a keying rate of εk = 0.05/ T s . The TDOA between the two received signals is D = 25T s . It can be seen from

Fig. 2 that the two methods failed in impulsive noise. For instance, Fig. 2 clearly shows many spurious peaks for the single cycle and multi-cycle methods. Although the multi-cycle estimator is superior to the single cycle method, neither one can provide accurate TDOA estimates in impulsive noise environments. 3.2. Formulation of proposed fractional lower-order signal-selective TDOA algorithms The performance of conventional signal-selective TDOA algorithms degrades severely in impulsive noise, because of the unboundedness of both the cyclic correlation function and the spectral correlation function. As such, it is necessary to develop effective TDOA algorithms that are highly tolerant to interference, and are robust to Gaussian noise and non-Gaussian impulsive noise. 3.2.1. The fractional lower-order multi-cycle signal-selective algorithm There are two primary limitations to the conventional multicycle method. First, the second-order cyclostationarity exploited by the method is useless in impulsive noise. Second, from (17) it is evident that the TDOA D is unknown even though it is necessary ˆ

ˆ i = e j π εi D must be adjusted for the estimation. Thus, the weight w

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Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

ˆ When an accurate TDOA estimation cannot be as the estimates D. ˆ i unduly influences the performance of the achieved, the weight w method. It has been shown that the fractional lower-order cyclostationarity is useful for exploiting cyclostationarity in the presence of Gaussian and non-Gaussian impulsive noises. Therefore, the fractional lower-order cyclic statistics based method possesses a better noise suppression capability than the conventional single cycle and multi-cycle methods. To make better use of the cyclostationarity and FLOS methods, we define a new robust multi-cycle method based on fractional lower-order spectral correlation functions,



F

c (τ ) 

ˆ

wi

εi , F

S yx

 ε ,F ∗ ( f ) S x i ( f ) e j2π ( f +εi /2)τ df

(19)

εi ∈ A

where εi is one element in the cycle frequency set A of s(t ), w i is the weight, and | w i | = 1. By taking the Fourier transform of (14) and (15), the fractional lower-order cyclic spectra are given by b a

S εyx, F ( f ) = r1 r2 S εs , F ( f )e − j2π ( f +ε/2) D

(20)

a b S ε, F (τ ) = r r S ε, F ( f )

(21)

x

1

s

1

It follows from (20) and (21), the fractional lower-order multicycle method can be expressed as



c F (τ ) =

ˆ

b a ε , F

r1 r2 S s i ( f )

wi

εi ∈ A

 a b ε , F ∗ × e − j2π ( f +εi /2) D r1 r1 S s i ( f ) e j2π ( f +εi /2)τ df (22)

Note that |r a | = |r |a for any constant r, which leads to

F

c (τ ) = |r1 |a+2b |r2 |a

ˆ

εi , F

2 j2π ( f +ε /2)(τ − D ) i

× wi Ss ( f ) e df

(23)

εi ∈ A

By using the triangle inequality and integral inequality in (23), we obtain

F

c (τ ) = |r1 |a+2b |r2 |a

ˆ

ε ,F

2 ×

w i S s i ( f ) e j2π ( f +εi /2)(τ − D ) df

εi ∈ A

|r2 |a ≤ |r1 |

ˆ ε ,F

2 j2π ( f +ε /2)(τ − D ) i i

× Ss ( f ) e df

wi a+2b

≤ |r1 |

ˆ ε ,F

S s i ( f ) 2 df |r2 | a

(24)

εi ∈ A

which indicates ´ that the maximum of the (23) is  ε ,F |r1 |a+2b |r2 |a εi ∈ A | S s i ( f )|2 df . By substituting τ = D into (22), we get

c F (D) =



ˆ

wi



2 a b b a ε , F r1 r1 r1 r2 S s i ( f ) df

(25)

εi ∈ A

And then substituting w i = 1 into (25), we get

εi ∈ A

wi = 1

  ˆ

 ε ,F ∗ ε ,F = arg max

S yxi ( f ) S x i ( f ) e j2π ( f +εi /2)τ df

(27) τ εi ∈ A

From (27) we can see that the proposed fractional lowerorder multi-cycle method can provide a true value of TDOA when w i = 1. It circumvents the estimation of w i with the conventional multi-cycle method. Furthermore, the fractional lower-order multicycle method is a class of method parameterized by a and b. When a = b = 1, the new method reduces to a second-order cyclostationarity based multi-cycle estimator. However, by using (27), the new method does not need to estimate the weight, which is a necessity in the conventional multi-cycle algorithm. 3.2.2. The generalized fractional lower-order SPECCOA algorithm The multi-cycle method requires the knowledge of the cycle frequency set and the fractional lower-order spectral correlation functions of the SOI. Sometimes, there is not enough accurate prior knowledge about signals’ cycle frequencies for this to be applicable. In addition, the ideal spectral correlation functions are not available in actual practice. Therefore, the performance of the multi-cycle TDOA estimation method is subject to the estimates of spectral correlation functions and cycle frequencies [33–35]. The generalized fractional lower-order cyclic ambiguity function makes better use of the cyclostationarity at one cycle frequency and can estimate TDOA when Doppler is zero [31]. However, the generalized fractional lower-order cyclic ambiguity function is concerned with joint estimation of TDOA and FDOA, it does not contain enough appropriate estimators for estimating TDOA only. In order to obtain a more generalized robust signal-selective TDOA method, the approximations of the fractional lower-order spectral correlation function at one cycle frequency should be thoroughly considered. In this section, we introduce a generalized robust signalselective method that utilizes fractional lower-order spectral correlation functions at one cycle frequency. A signal model can be used to reflect the generality of the TDOA estimation. The equivalent signal model is

i = 1, 2

ε, F

εi ∈ A

(26)

(28)

where xi (t ) is the received signal, s(t ) and ni (t ) are the SOI and the SNOI, respectively. r i is the attenuation, D i is the time delay of each received signal, and D = D 2 − D 1 is the TDOA to be estimated. To make the derivation mathematically tractable, it is assumed that the attenuation of the received signals is negligible (r1 = r2 = 1). In fact, the attenuation of signals can not be negligible in practice, however, it has been shown by [14] that the performance of the TDOA estimation method for both r i = 1 and r i = 1 cases is very similar, and provided that the mismatch of r i = 1 is small. It is assumed that there is negligible mismatch between the two receivers. According to the definition of (10), the fractional lower-order correlation function of the received signals xm (t ) and xk (t ) can be expressed by

R mk (τ ) = R εs , F (τ + D k − D m )e − j π ε( D k + D m )

ˆ

F

ε ,F



2

c ( D ) = |r1 |a+2b |r2 |a S s i ( f ) df

 = max c F (τ )

 

ˆ

 εi , F ∗ j2π ( f +ε /2)τ εi , F i

D = arg max wi S yx ( f ) S x ( f ) e df

, τ

xi (t ) = r i s(t − D i ) + ni (t ),

a+2b

εi ∈ A

Equation (26) indicates that the peak of the new proposed multi-cycle occurs when w i = 1 and τ = D. The value of the TDOA can be determined by

(29)

where m, k ∈ {1, 2}. By substituting (29) into (11), we obtain the fractional lower-order spectral correlation function of xm (t ) and xk (t ),

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

ε, F S mk ( f ) = S εs , F ( f )e − j2π ( f +ε/2) D m e j2π ( f −ε/2) D k ε, F

By solving for S s

(30)

( f ), we obtain

F mk ( f ) = S mk ( f )e j2π ( f +ε/2) D m e − j2π ( f −ε/2) D k ε, F

ε, F

(31)

ε, F

( f ) is only approximate, (32)

In order to estimate the desired parameter TDOA, a leastsquares problem for minimizing the sum of squared differences ε, F ε, F between Fˆ mk ( f ) and Sˆ s ( f ) must be formulated [12,14]. [12] and [29] have shown that the solution to the minimization problem (m = 2, k = 1) yields the SPECCOA (a = b = 1) and the fractional lower-order SPECCOA (FSPECCOA, 0 ≤ a, b < α /2) for the TDOA estimation. As a generalization, a more appropriate example can be ε, F formulated in order to jointly fit all pairs of Fˆ mk ( f ) (m, k ∈ {1, 2} and 0 ≤ a, b < α /2). Thus, the general solution to this simplification is

 ˆ∞

D = arg max Re τ

 ε, F ∗ S mk ( f ) S i j ( f ) e j ϕ (τ , f ) df



ε, F

(33)

−∞

where i , j , m, k ∈ {1, 2}´, and ϕ (τ , f ) depends on i, j, m, and k. By ∞ ε, F ε, F substituting (30) into −∞ S mk ( f )( S i j ( f ))∗ df , we obtain

ˆ∞

2π ( f + ε /2)( D i − D m )

τ = D2 − D1 = D

=

b1 (τ )

 ˆ∞

= Re −∞

 ˆ∞ = Re b2 (τ )

 ˆ∞

= Re

 ε, F ∗ ε, F S mk ( f ) S i j ( f ) e j2π ϕ (τ , f ) df

ε, F

S ( f ) 2 e j2π ( f +ε/2)( D i − D m ) e j2π ( f −ε/2)( D k − D j ) e j ϕ (τ , f ) df s

−∞

(35)

 ˆ∞

(36)

ε, F

 ε, F

∗

−∞



ˆ∞ ≤

max Re −∞

 ε, F ∗ S mk ( f ) S i j ( f ) e j ϕ (τ , f ) df ε, F

(41b)



 ε, F ∗ j2π ( f −ε/2)τ

( f ) S 22 ( f ) e df

ε, F

S ( f ) 2 e j2π ( f −ε/2)(τ − D )df



ϕ (τ , f )=2π ( f −ε /2)τ

(41c)

s

b4 (τ )

 ˆ∞

= Re

ε, F

S 22 −∞

= Re



 ε, F ∗ j2π ( f +ε/2)τ

( f ) S 12 ( f ) e df

ε, F

S ( f ) 2 e j2π ( f +ε/2)(τ − D )df



ϕ (τ , f )=2π ( f +ε /2)τ

(41d)

s

b5 (τ )

ε, F

S ( f ) 2 df s

(37)

 ˆ∞

= Re

−∞

 ˆ∞

ϕ (τ , f )=2π ( f +ε /2)τ

−∞



ˆ∞ =

ε, F

S ( f ) 2 df s

−∞

(38)

ε, F

S 21 −∞  ˆ∞

Thus,





−∞

The expression of (35) then becomes

S mk ( f ) S i j ( f ) e j ϕ (τ , f ) df

ε, F

S 21

 ˆ∞





 ε, F ∗ j2π ( f +ε/2)τ

( f ) S 11 ( f ) e df

ε, F

S ( f ) 2 e j2π ( f +ε/2)(τ − D )df

b3 (τ )

Note that



(41a)

s

= Re

−∞

Re

ε, F

S 21

−∞  ˆ∞

Re e j2π ( f +ε/2)( D i − D m ) e j2π ( f −ε/2)( D k − D j ) e j ϕ (τ , f ) ≤ 1

ϕ (τ , f )=2π ( f −ε /2)τ

s

= Re

Then, the formulated measurement of (33) is

 ˆ∞

ε, F

S ( f ) 2 e j2π ( f −ε/2)(τ − D )df



−∞

(34)

−∞

ˆ∞



 ε, F ∗ j2π ( f −ε/2)τ

( f ) S 12 ( f ) e df

−∞

= Re

s

=

ε, F

S 11

 ˆ∞

ε, F

S ( f ) 2 e j2π ( f +ε/2)( D i − D m ) e j2π ( f −ε/2)( D k − D j ) df

(40)

can the equality hold in (38) when i , j , m, k ∈ {1, 2}, which means the estimator provides the correct TDOA estimate. The solutions that can provide correct TDOA estimates (τ = D) to all the scenarios of i , j , m, k ∈ {1, 2} are given by

−∞

 ε, F ∗ S mk ( f ) S i j ( f ) df

−∞

ˆ∞

Furthermore, for the purpose of estimating D, (39) does equal to zero when ε is a cycle frequency of SOI and when τ = D 2 − D 1 = D. Thus, it follows from (39), that only if

ε, F

ˆ∞

(39)

+ 2π ( f − ε /2)( D k − D j ) + ϕ (τ , f ) = 0

ε, F Fˆ mk ( f ) ∼ = Sˆ εs , F ( f )



+ πε ( D i − D m − D k + D j ) + ϕ (τ , f ) = 0

ε, F

It can be seen from (31) that F mk ( f ) = S s ( f ) for all m, k ∈ {1, 2}. However, the ideal fractional lower-order cyclic spectrum ε, F S mk ( f ) is unavailable in actual practice. The estimates of the fractional lower-order spectral function must be obtained, so the ε, F equivalence between the two estimate functions F mk ( f ) and Ss

It can be seen from (36) and (38) that the formulated measurement of (33) reaches the maximum value only if Re{e j2π ( f +ε/2)( D i − D m ) e j2π ( f −ε/2)( D k − D j ) e j ϕ (τ , f ) } = 1, then

2π f ( D i − D m + D k − D j )

= S εs , F ( f ) ε, F

99

= Re



 ε, F ∗ j4π f τ

( f ) S 12 ( f ) e df

ϕ (τ , f )=4π f τ

ε 2 j4π f (τ − D )

S ( f ) e df s

 (41e)

−∞

It follows from (41) that each of the estimators can obtain its peak at τ = D,

100

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

4. Simulation results

Fig. 3. The TDOA estimation functions of b i (τ ) for the QPSK signal with a = b = 0.5, and D = 25T s .



b i ( D ) = max b i (τ ) = Re

 ˆ∞



ε, F

2

S ( f ) df , s

ε = εk ,

i = 1, 2, 3, 4, 5

−∞

(42) Fig. 3 shows a graph of these estimators for the QPSK signal of Fig. 2 with keying rate ε = εk , a = b = 0.5, and D = 25T s . There is a unique peak at τ = D for each estimator, and the TDOA is determined by





D = arg max b i (τ )

(43)

τ

Since knowledge of the SOI is generally not obtainable and the fractional lower-order spectral functions for estimating the TDOA are approximate estimates, the capability of each estimator based on the same signal model is different. Therefore, insight into the capabilities of the approach can be gained by considering all possible cases. The resultant generalized fractional lower-order spectral coherence alignment (GFSPECCOA) algorithm is

⎡´∞

∗ j2π ( f −ε /2)τ df ˆ ε, F ˆ ε, F −∞ S 11 ( f )( S 12 ( f )) e ´ ∞ ε, F ε, F + −∞ Sˆ 21 ( f )( Sˆ 11 ( f ))∗ e j2π ( f +ε/2)τ df ´ ∞ ε, F ε, F + −∞ Sˆ 21 ( f )( Sˆ 22 ( f ))∗ e j2π ( f −ε/2)τ df ´ ∞ ε, F ε, F + −∞ Sˆ 22 ( f )( Sˆ 12 ( f ))∗ e j2π ( f +ε/2)τ df ´ ∞ ε, F ε, F + −∞ Sˆ 21 ( f )( Sˆ 12 ( f ))∗ e j4π f τ df

⎢ ⎢ ⎢ ⎢ G (τ ) = Re ⎢ ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(44)

The estimate of TDOA can be determined by





D = arg max G (τ ) τ

(45)

The generalized fractional lower-order spectral coherence alignment algorithm shares the signal selectivity and is robust to impulsive noise. Moreover, it makes better use of the fractional lowerorder cyclostationarity features at one cycle frequency. It should be noted that both the proposed fractional lowerorder multi-cycle algorithm and the generalized fractional lowerorder spectral coherence alignment algorithm are developed from the narrow-band assumption. In this condition, the time-scale factor of the complex envelope can be assumed unity [17,36]. The proposed methods can provide accurate estimates even in severe impulsive noise and interference environments under narrow-band condition.

In this section, computer simulations are carried out to evaluate the performance of the proposed TDOA estimation methods in comparison to the conventional cyclostationarity based methods and FLOS based methods. We performed the evaluation of the proposed algorithms by using a BPSK signal [13, 19]. The carrier frequency of the BPSK signal is f c = 0.25/ T s , the keying rate is εk = 0.0625/ T s with a chip width of T o = 16T s . The TDOA between the two received signals is D = 48T s . The sampling increment is T s = 10−8 s with an integration time of T = N T s and N = 32768. The modulated signal is affected by independent interference under Gaussian noise and S α S impulsive noise scenarios. For simplification, we made a = b in all instances. Since second-order moments are not defined for α -stable distributions (except for α = 2), we use generalized signal to noise ratio (GSNR) to evaluate the effects of attenuation of received signals. The GSNR defined as the ratio of the signal power over the impulsive noise dispersion γ [27],



N

σ2 1

s(n) 2 GSNR = 10 log10 s = 10 log10 γ γ × N n =1

 (46)

Based on this choice of GSNR metric, the S α S noise samples are power scaled by the dispersion parameter γ . 4.1. Performance under co-band interference and noise The first simulation considers the presence of an AM signal as an interference located within the spectral band of the signal interest. The interference has the same carrier frequency and bandwidth as that of the SOI. The TDOA of the interfering signal is D = 58T s . The signal to interference ratio (SIR) is −3 dB. We consider the scenarios involving Gaussian noise and impulsive noise in the background. The signal to noise ratio (SNR) is −3 dB. The S α S impulsive noise is α = 1.8 and GSNR = −3 dB. The cycle frequency exploited by the single cycle algorithms is the keying rate of εk = 0.0625/ T s . The cycle frequencies exploited by the multicycle algorithm are εk and 2εk . Fig. 4 and Fig. 5 show the estimation functions of the fractional lower-order correlation (FLOC) method [27], the generalized spectral coherence alignment (GSPECCOA) method [15], the proposed fractional lower-order multi-cycle (FMC) method, and the proposed generalized fractional lower-order spectral coherence alignment (GFSPECCOA) method for 20 trails. From Fig. 4, it is evident that a separate TDOA estimation corresponding to the SOI can be made regardless of the spectrally overlapping interfering signal and Gaussian noise by either the conventional signal-selective method or the proposed signal-selective methods. This is an improvement over the FLOC method that produced biased estimations corresponding to the interfering signal under the Gaussian noise scenario. Although the FLOC method is robust to the Gaussian and impulsive noises, it fails in the scenarios where spectrally overlapping interference is present. However, the behavior of the conventional signal-selective method degrades severely in impulsive noise. From Fig. 5(b) we see that the peak of interest for the GSPECCOA method is only one of many peaks, any one of which might be taken as the TDOA estimate. In contrast to this, Fig. 5(c) and Fig. 5(d) indicate that the proposed methods have better estimation capabilities for TDOA estimation when severe interference and impulsive noise cause conventional methods to become unreliable.

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

101

Fig. 4. The TDOA estimation functions of the methods in Gaussian noise. (a) FLOC (a = b = 0.5). (b) GSPECCOA (ε = εk ). (c) FMC (a = b = 0.5, ε = εk , 2εk ). (d) GFSPECCOA (a = b = 0.5, ε = εk ).

Fig. 5. The TDOA estimation functions of the methods in impulsive noise (α = 1.8). (a) FLOC (a = b = 0.5). (b) GSPECCOA (ε = εk ). (c) FMC (a = b = 0.5, ε = 0, 2εk ). (d) GFSPECCOA (a = b = 0.5, ε = εk ).

4.2. Performance under wide band interference and noise The interfering signal in this environment is a BPSK signal with a carrier frequency of f 1 = 0.21875/ T s , a keying rate of ε1 = 0.10/ T s , and a TDOA of D = 58T s . The signal to interference ratio (SIR) is 0 dB. Fig. 6 shows the performance of the proposed fractional lower-order signal-selective methods and the conventional multi-cycle method (MC) in the presence of wide band interference, and Gaussian noise and impulsive noise (α = 1.8) from 2000 Monte-Carlo trials. The cyclic correlation-based Cramér–Rao bound (CRBCRB) is indicated by [15]. It can be seen from Fig. 6 that the proposed multi-cycle algorithms and the conventional multi-cycle methods that use the same cycle frequencies have similar performance in Gaussian noise. However, the proposed fractional lowerorder multi-cycle and GFSPECCOA algorithms significantly outperform the conventional algorithms under impulsive noise. In addition, the curves related to the GFSPECCOA method and the doublecycle FMC method under the same GSNR and α nearly overlap, which indicates that the GFSPECCOA method closely approximates the proposed double-cycle estimator in this environment. The per-

Fig. 6. The TDOA estimation accuracy of different algorithms in the presence of wide band interference and noise. (a) Gaussian noise. (b) Impulsive noise (α = 1.8).

formance of the proposed GFSPECCOA method and the generalized fractional lower-order cyclic ambiguity function (GFCCA) [33] in the presence of interference, Gaussian noise (α = 2) and impulsive noise (α = 1.8) is shown in Fig. 7. Simulation results show that both the two methods can robust against the Gaussian and impulsive noises, and immune to the interference. Moreover, the GFSPECCOA method performs slightly better than the GFCCA method. Simulation results demonstrate that both the proposed methods are robust to the impulsive noise and Gaussian noise, and achieve better performance than conventional methods especially in impulsive noise. 4.3. Performance under narrow band interference and noise In this environment, we employ a BPSK signal with a carrier frequency of f 2 = 0.02/ T s , a keying rate of ε2 = 0.025/ T s , and a TDOA of D = 36T s as interference [13]. The signal to interference ratio (SIR) is 0 dB. The cycle frequency exploited by the single cycle algorithms is the keying rate εk . The cycle frequencies exploited by the multi-cycle algorithm are 0, εk , and 2 f c . In the simulations, 2000 independent Monte-Carlo trials are performed.

102

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

Fig. 7. The TDOA estimation accuracy of GFSPECCOA and GFCCA in the presence of wide band interference, Gaussian noise (α = 2), and impulsive noise (α = 1.8).

The performance of the conventional multi-cycle (MC) method and the proposed fractional lower-order multi-cycle (FMC) algorithm under impulsive noise with α = 1.5 and α = 1.7 are shown in Fig. 8. The characteristic exponent α controls the heaviness of tails for the α -stable distributions. A small positive value of α indicates severe impulsiveness, while a value of α close to 2 indicates a more Gaussian type of behavior. When α equals to 2, the α -stable distribution becomes to Gaussian distribution. Because the impulsiveness of the S α S distribution with α = 1.5 is more severe than the S α S distribution with α = 1.7, the overall performance of the algorithms greatly improved, as shown in Fig. 8. Furthermore, the conventional multi-cycle methods are vastly inferior to the new fractional lower-order multi-cycle methods in the presence of impulsive noise. Previous research results demonstrated that by making better use of cyclostationarity it is possible to achieve better estimation capabilities for multi-cycle methods. For the conventional and proposed multi-cycle methods, it is clear that the double-cycle estimator is superior to the single-cycle estimator. This can be explained by the fact that cycle frequency zero includes the weakest cyclic feature when the interference is present. However, the cycle frequency 2 f c corresponds to the strongest cyclic feature, the best performance for the proposed multi-cycle method occurs for the triple-cycle estimator. Fig. 9 shows the normalized mean-square errors (NMSE) versus the collection times for the GSPECCOA method and the proposed GFSPECCOA method. The characteristic exponent is α = 1.85, and a = b = 0.5. The Cramér–Rao lower bound (CRLB) is addressed by [13]. Simulation results demonstrated that with increases in collection time, all of the methods perform better. However, the GFSPECCOA method performs significantly better than GSPECCOA, particularly as the length of the collection time increased. The performance of TDOA estimation for GSPECCOA and GFSPECCOA when α varied from 1.1 to 2 is shown in Fig. 10 with a = b = 0.2. A small value of α indicates a more severe impulsiveness. As such, in each scenario the accuracy of TDOA degrades as the characteristic exponent decreases from 2 to 1.1. The proposed GFSPECCOA method achieves better robustness to impulsive noise than the GSPECCOA, even when the behavior of the noise is close to a Gaussian distribution (α = 1.9). As the characteristic exponent equals to 2, the S α S distribution becomes to a Gaussian distribution, all the algorithms perform well. Nevertheless, the GFSPECCOA method performs better than the GSPECCOA under Gaussian noise. The fractional exponents play an important role to the robustness of the FLOS based method. According to the

Fig. 8. The TDOA estimation accuracy of the algorithms in the presence of narrow band interference and impulsive noise. (a) Impulsive noise (α = 1.5). (b) Impulsive noise (α = 1.7).

Fig. 9. The normalized mean-square errors versus the collection time for narrow band interference and impulsive noise (α = 1.85).

Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

103

Appendix. Derivation of Eq. (14) and Eq. (15) Let X and Y be two S α S random variables, the fractional lowerorder covariation of Y and X is defined as [27,37]

   X , Y  p = E X Y  p −1  ,

1< p<α≤2

(47)

The fractional lower-order correlation of y (t ) and x(t ) is given by

R Fyx (τ ) = E



a 

y (t + τ /2)

b 

x(t − τ /2)

(48)

According to (47), the fractional lower-order correlation of y (t ) and x(t ) is equivalent to the fractional lower-order covariation of x(t − τ /2) and ( y (t + τ /2))a

R Fyx (τ ) =



a

y (t + τ /2)

 , x(t − τ /2) b+1

(49)

Substituting (1) into (49), we get

R Fyx (τ ) = Fig. 10. The TDOA estimation performance versus

α.

FLOS theory and the α -stable distribution characteristics, if x(t ) is a α -stable distributed signal then (x(t ))a has finite second-order moments when 0 < a < α /2 ≤ 1. Moreover, it has been shown that the characteristic exponent of impulsive noise in application is always between 1.6 to 1.9 [25]. Although it is difficult to choose the optimum a and b, from simulation results of Fig. 10 and [29] we can see that choosing 0.1 ≤ a ≤ 0.5 the proposed fractional lower-order cyclostationarity based methods exhibit effectiveness and robustness. 5. Conclusions Conventional second-order cyclostationarity and FLOS based TDOA estimation methods perform poorly under non-Gaussian impulsive noise and interfering signal environments. As such, we present two new fractional lower-order signal-selective TDOA algorithms based on the fractional lower-order cyclostationarity for cyclostationary signals. The methods we proposed make better use of cyclostationarity property and fractional lower-order statistics, are tolerant to interference, and are robust to the Gaussian noise and α -stable impulsive noise. Simulation results demonstrate that the proposed algorithms provide better accuracy for TDOA estimation compared with conventional signal-selective and FLOS methods, and exhibit effectiveness and robustness in a wide range of interference and noise environments under the narrow-band condition. Author Contributions: The algorithms proposed in this paper have been conceived by Prof. Yang Liu and Prof. Tianshuang Qiu. Prof. Yang Liu, Dr. Yinghui Zhang and Shun Na designed and performed the experiments. Prof. Yang Liu, Dr. Jing Gao and Shun Na analyzed the data. Prof. Yang Liu and Shun Na wrote and revised the paper. Acknowledgments The authors would like to thank the editors and anonymous reviewers for their constructive comments and suggestions, which helped improve the manuscript. The authors are grateful to the National Science Foundation of China for its support of this research. This work is supported in part by the National Science Foundation of China under Grants 61362027, 61461036, 61761033, and 61501325.



a

y (t + τ /2)

 , r1 s(t − τ /2) + n(t − τ /2) b+1

(50)

Based on the properties of the fractional lower-order statistics [37,38], if Y 1 and Y 2 are independent, then

 X , λY 1 + η Y 2  p = λ p −1  X , Y 1  p + η p −1  X , Y 2  p

(51)

η. Thus a  R Fyx (τ ) = r1 y (t + τ /2) , s(t − τ /2) b+1  a  + y (t + τ /2) , n(t − τ /2) b+1

for any constants λ and b 

(52)

From (48) and (49), we can see that (52) can also be expressed as b 

 , y (t + τ /2) a+1  b  + n(t − τ /2) , y (t + τ /2) a+1

R Fyx (τ ) = r1

b

s(t − τ /2)

(53)

Substituting (2) into (53), we get

R Fyx (

τ)

b 

 , r2 s(t − D + τ /2) + m(t + τ /2) a+1  b  + n(t − τ /2) , r2 s(t − D + τ /2) + m(t + τ /2) a+1 a  b  a b  = r2 r1 E s(t − D + τ /2) s(t − τ /2) a  b  b  + r1 E m(t + τ /2) s(t − τ /2)  a  b  + E m(t + τ /2) n(t − τ /2) a  b  b  + r2 E n(t − τ /2) s(t − D + τ /2) (54)

= r1

b

s(t − τ /2)

According to the assumptions that the signal of interest and SNOI are independent, thus, E {(m(t + τ /2))a (s(t − τ /2))b } = 0 and E {(n(t − τ /2))a (s(t − D + τ /2))b } = 0. a b

R Fyx (τ ) = r2 r1 E



a 

s(t − D + τ /2)

b 

s(t − τ /2)

 a  b  + E m(t + τ /2) n(t − τ /2)

(55)

Substituting (55) into the definition of fractional lower-order cyclic correlation function, a b 

R εyx, F (τ ) = r2 r1

a 

s(t − D + τ /2)

b

s(t − τ /2)

e − j2π εt



 a  b − j2π εt  + m(t + τ /2) n(t − τ /2) e a   b a b   = r2 r1 s t + (τ − D )/2 s t − (τ − D )/2  ε , F (τ ) × e − j2π ε(t + D /2) + R mn a b

ε , F (τ ) = r2 r1 R εs , F (τ − D )e − j π ε D + R mn

(56)

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Y. Liu et al. / Digital Signal Processing 76 (2018) 94–105

It follows from (56), the fractional lower-order cyclic autocorrelation function of x(t ) is given by, a b

R εx , F (τ ) = r1 r1 R εs , F (τ ) + R nε, F (τ )

(57)

According to the assumptions that the SNOI do not share the same cycle frequency as that of the SOI, and s(t ), n(t ) and m(t ) ε, F ε, F do not exhibit joint cyclostationarity, then R n (τ ) = R mn (τ ) = 0. Therefore, equations (56) and (57) become to a b

R εyx, F (τ ) = r2 r1 R εs , F (τ − D )e − j π ε D a b R ε, F (τ ) = r r R ε, F (τ ) x

1

1

s

(58) (59)

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Yang Liu received the B.Eng. degree from the Inner Mongolia University and the Ph.D. degree from the Dalian University of Technology, both in electronic engineering, in 2003 and 2012, respectively. Since 2017, he has been a Senior Research Scholar with Department of Electronic Engineering, Tsinghua University. He is currently a Professor with the Department of Electronic Engineering, Inner Mongolia University, China. His research interests include array signal processing, non-Gaussian signal processing, and wireless communications with a focus on multi-antenna techniques. Yinghui Zhang received the B.Eng. and M.S. degree from Xidian University in 2004 and 2007, Xi’an, China, and Ph.D. degree from Beijing University of Posts and Telecommunications in 2015, Beijing, China. She serves as a member of the Inner Mongolia Communications Association. At present, Dr. Zhang is an associate Professor in the College of Electronic Information Engineering, Inner Mongolia University. Her research interests include 5G Technologies, Millimeter wave communication, Cooperative communication and Relay Network. Tianshuang Qiu was born in China. He obtained the B.Eng. degree from the Tianjin University and the Ph.D. degree from the Southeastern University, both in electronic engineering, in 1983 and 1996, respectively. From 1983 to 1996, he was an Electronic Engineer at the Dalian Institute of Chemical Physics, Chinese Academy of Science, China. During 1996–2000, he worked as a Post-Doctoral Fellow at Northern Illinois University, USA. He is currently a Professor in Dalian University of Technology.

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His research interests include adaptive signal processing, biomedical signal processing, non-Gaussian signal processing, and array signal processing. Jing Gao received B.Eng. and M.S. degree from Changchun University of Technology in 2002 and 2005 respectively, and Dr. from Beijing University of Posts and Telecommunication in 2014. Her main research interest is wireless communications, cognitive radio network and multi-hop routing protocol.

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Shun Na received B.Eng. and M.S. degree from Inner Mongolia University of Technology in 2001 and 2009, respectively. He is currently an assistant Professor in the College of Electronic Information Engineering, Inner Mongolia University. His main research interest is communications signal processing and wireless sensor network.