Time-difference-of-arrival estimation algorithms for cyclostationary signals in impulsive noise
Highlights
► The TDOA estimation for cyclostationary signals in impulsive noise environment is considered. ► Two new types of cyclic statistics are developed to exploit cyclostationarity property. ► Two new classes of signal-selective TDOA algorithms for cyclostationary signals are presented. ► The new cyclic statistics are used in the TDOA methods to suppress interference and noise. ► Simulation results show effectiveness and robustness of the approaches in various scenarios.
Introduction
Time-difference-of-arrival (TDOA) estimation between received signals at spatially separated sensors in the presence of interference and noise is an important problem in several fields, including target localization using radar and sonar, and communication systems. Many man-made signals arising in communications, telemetry, radar, and sonar applications exhibit cyclostationarity [1], [2], [3], [4]. A class of signal-selective TDOA estimation methods based on the second-order cyclic statistics was introduced by Gardner et al. [5], [6]. By exploiting the unique cyclostationarity property of the signal of interest (SOI), the signal-selective methods can obtain substantial tolerance to interference and Gaussian noise, which do not exhibit the same cyclostationarity as the SOI. A performance comparison of stationary and cyclostationary TDOA estimators was presented in [6], [7], [8].
Many modulated signals in radar, sonar, and communication systems have more than one cycle frequency. The periods of cyclostationarity correspond to the carrier frequency, pulse rate, time-division multiplexing rates, etc. [1], [3]. Because the cycle frequency for most cyclostationary signals is not unique, the performance of single-cycle estimators is subject to the selected cycle frequency. To make better use of the cyclostationarity property, some multi-cycle signal-selective algorithms that exploit more than one cycle frequency were developed in [9], [10]. Performance analysis and simulation results indicated that the multi-cycle estimator can achieve better performance than single-cycle estimators [10]. An analytical comparative performance study on cyclic-correlation-based TDOA methods was introduced in [11]. Furthermore, an asymptotic cyclic-correlation-based Cramér-Rao bound (CRBCRB) and an approximate maximum likelihood estimator in the AWGN environment were presented. The CRBCRB can serve as a lower bound on the performance attainable by using an unbiased estimator that is based solely on the estimated correlations [11].
Most of the conventional cyclostationary TDOA methods assume that the noise present is additive Gaussian noise. However, the assumption of Gaussian noise is often unrealistic. Studies and experimental measurements have shown that a broad and increasingly important class of noise such as underwater acoustic noise, atmospheric noise, multiuser interference, and radar clutters in real world applications are non-Gaussian processes, primarily owing to impulsive phenomena [12], [13], [14], [15], [16]. It has been shown that a class of α-stable distributions is more appropriate for modeling impulsive noise than Gaussian distribution in signal processing applications [17], [18], [19]. The main difference between the non-Gaussian stable distribution and the Gaussian distribution is that the tails of the stable density are heavier than those of the Gaussian density. The smaller the characteristic exponent (α∈[0,2]), the heavier the tails of the density function (the case α=2 correspondence to the Gaussian distribution). The positive-valued scalar γ is the dispersion of the stable distribution. The dispersion plays a role analogous to that of the variance for second-order process. This type of α-stable distribution provides attractive theoretical and practical tools for many fields including communication, radar, sonar, etc. [20], [21], [22], [23]. Since the stable distribution does not have finite second-order moments (1≤α<2), or even first-order moment (α<1) due to the heavy tails [17], [19], the performance of the existing signal-selective TDOA methods based on second-order cyclostationarity will degrade severely. From the viewpoint of real world applications, we are interested in developing TDOA algorithms accounting for interference, and Gaussian and non-Gaussian random processes.
This paper addresses the problem of TDOA estimation for cyclostationary signals in the presence of interference and impulsive noise. The problem of TDOA estimation in heavy-tailed noise arising from a symmetric α-stable (SαS) distribution has been studied in [18], [19]. It is known that fractional lower-order statistics (FLOS) algorithms can be robust TDOA estimation techniques in impulsive noise environments [19]. Although the FLOS-based algorithms can be robust against impulsive noise, it has been demonstrated that they are unable to generate separate unbiased TDOA estimates when multiple signals are spectrally overlapped. To overcome the limitations of conventional signal-selective methods and FLOS-based methods, it is necessary to exploit additional signal properties that enable the signal, interference and impulsive noise to be separated. By using the phased fractional lower-order moment (PFLOM) operation, we first define two types of fractional lower-order cyclic representations for revealing the cyclostationarity property of signals. Then, we generalize the conventional single-cycle signal-selective algorithms by using fractional lower-order cyclic statistics instead of second-order cyclostationarity. The proposed algorithms take advantages of signal-selective methods and FLOS estimators, are tolerant to interference, and are robust to both Gaussian and non-Gaussian impulsive noise. The paper is organized as follows. In Section 2, conventional second-order cyclostationarity-based TDOA estimation approaches are reviewed. The fractional lower-order cyclic statistics and robust signal-selective TDOA estimation algorithms are described in Section 3. In Section 4, the variance and consistency of the proposed algorithms are analyzed. In Section 5, the performance of the proposed algorithms is demonstrated via simulations. Finally, conclusions are drawn in Section 6.
Section snippets
Preliminaries
In general, signals radiating from a remote source, impinging on a pair of antenna elements, together with interfering signals and received noises, can be modeled aswhere s(t) is the SOI, n(t) and m(t) are signals not of interest (SNOI), which include interfering signals and independent receiver noises. D is the TDOA parameter of interest, and r1 and r2 represent the attenuation. To simplify the problem, it is assumed that s(t) is statistically independent of
Robust signal-selective TDOA estimation methods
The performance of conventional signal-selective TDOA algorithms degrades in impulsive noise, because of the unboundedness of both the cyclic correlation function and the cyclic spectrum. In this section, we propose new algorithms for estimating the TDOA of cyclostationary signals in the presence of interference and impulsive noise.
Performance analysis
One interpretation of the cyclic correlation is that it is the cross-correlation of frequency-shifted versions of the original processes,where
Substitution of (37), (38) into (12), by utilizing the properties of z〈k〉e±jπεt=(ze±jπεt)〈k〉 and (z⁎)〈k〉=(z〈k〉)⁎, we obtain
Furthermore, can also be expressed aswhere
Simulation results
In this section, we present various simulation results with which we describe the performance of the proposed algorithms and compare them with conventional FLOS-based and signal-selective methods. In our simulations, we consider a BPSK signal. The carrier frequency of the BPSK signal is fc=0.25/Ts, keying rate εk=0.0625/Ts, and the TDOA between two received signals is D=48Ts. The discrete time sampling increment is Ts=10−7s, with the integration time T=NTs and N=32,768. Since the α-stable
Conclusions
In this paper, new algorithms for the TDOA estimation of cyclostationary signals in the presence of interfering signals and heavy-tailed α-stable impulsive noise are introduced. The methods are based on pth-order statistics and fractional lower-order cyclic statistics, respectively. It is shown that the cyclostationarity property of signals can be exploited from these statistics. The performance of the proposed methods is examined via simulations. Simulation results demonstrate the
Acknowledgments
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions, which helped improve the manuscript. This work is partly supported by the National Science Foundation of China, under Grants 61172108 and 61139001.
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