TOA Estimation Algorithm Based on Shift-Invariant Technique for Multi-band Signals Ying-chun Li, Seungryeol Go, Hyung-ryeol Yoo, Sang-deok Kim, and Jong-Wha Chong Hanyang University, Seoul, Korea
[email protected]
Abstract. This paper proposes a new TOA estimation algorithm based on shift invariant structure for multi-band signals. The performance of the proposed method is compared with Cramer-Rao lower bound (CRLB) and the conventional algorithms such as MUSIC, matrix pencil, TLS-ESPRIT in additive white Gaussian noise (AWGN) and multipath channel.
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Introduction
Wireless location has drawn considerable research and provided numbers of application, such as target finding, mine rescuing and navigating. The main technique utilized in wireless location is parameter estimation include time-of-arrival (TOA) estimation, time difference of arrival (TDOA) estimation, received signal strength interference (RSSI) estimation and angle-of-arrival (AOA) estimation. TOA estimation is an important signal processing problem with applications in many areas, such as radar, sonar, and wireless communications. How is the TOA estimation organized? Two kinds of approaches exist for TOA estimation. The first one is the maximum-likelihood (ML) method, which investigates the correlation function of a received signal as in [1]-[2]. Therefore, the resolution is limited by the inverse of the signal bandwidth, and it is always used with a narrow band signal. The second one is the shift invariant method, which transforms the received signal to a sinusoidal signal and handles the received signal in the frequency domain. In previous work, parameter estimation techniques can be divided into two types: onedimensional parameter estimation and two-dimensional parameter estimation. First kind of parameter estimation techniques, which are one dimensional, such as MUltiple Signal Classification (MUSIC) [3], Matrix Pencil (MP) [4] and Total Least Square ESPRIT (TLS-ESPRIT) [5], were applied to the transformed signals. Second kind of parameter estimation techniques, such as 2-dimensional MUSIC, and joint angle and frequency estimation, have only been used for the estimation of TOA parameters in uniform or non-uniform arrays. Although the performances of the conventional algorithms can approximate the CRLB, higher computational complexities and support of different auxiliary information are in demand [5]. Therefore, more efficient and robust TOA estimation techniques still require further research. In this paper, a new TOA estimation by using shift-invariant technique was proposed. I. Zelinka et al. (eds.), AETA 2013: Recent Advances in Electrical Engineering and Related Sciences, Lecture Notes in Electrical Engineering 282, DOI: 10.1007/978-3-642-41968-3_25, © Springer-Verlag Berlin Heidelberg 2014
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Proposed Method
2.1
Signal Model
The received signals can be changed into a form of sinusoids for shift invariant TOA estimation. Our algorithm transforms the received samples by a DFT and a deconvolution, as in [3-4] and [6]. We assume that K multiple symbols are transmitted at different frequency bands, and the discrete time model for the k-th received signals yk(t) is given by yk[n]=yk(nTs) for n=0,1,…,N 1 where Ts=1/fs is the sampling period, such that Tsym=NTs. The sampling period Ts is determined in such a way that the Nyquist criterion is met. For a given sequence of yk[0], yk[1],…, yk[N 1] from the k-th symbol, a sequence of the data for the estimated channel frequency response(CFR) by N-point DFT and deconvolution is modeled from [4] by
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xk [l ] =
M −1
a
m =0
m
exp ( j 2π ( f k + l Δf )τ m ) for l = 0,..., N − 1,
(1)
where am and τm denote the complex amplitude and TOA of the m-th path, respectively; M is the number of the received paths; the frequency spacing Δf is determined by fs/N and fk denotes the center frequency of the k-th symbol. For convenience without loss of generality, we assume that fk=(k 1)B where B denotes the bandwidth for a symbol. By the transformation, a sequence of the received samples yk[n],n=0,…,N 1 in the time domain is changed into a sequence of the samples xk[l],l=0,…,N 1 in the frequency domain. The transformation using the Npoint DFT is based on the shifting property of the Fourier transform. Since the transformed samples are composed of M sinusoids whose frequency is a function of τm, it is also possible to estimate the TOA of the received paths by frequency estimation of xk[l], l=0,…,N 1.
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2.2
Shift-Invariant Technique
In this section, the small-scaled phase shift is defined by ϕm of (2). Using the transformed samples of the k-th symbol, xk[l] for l=0,…,N 1,the Hankel snapshot matrix is defined as
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xk [1] xk [0] x [1] xk [2] Xk = k x [ L − 1] x [ k Lc ] k c
xk [ Lr − 1] xk [ Lr ] , xk [ N − 1]
(2)
where Lr and Lc=N-Lr+1 are selection patameters, referred to as pencil parameter in [7], satisfying the condition Lr≥M and Lc≥M. Let us define the steering matrices P and
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Q, which contain the corresponding M sinusoids in its columns and rows, respectively, such that,
1 φ P= 0 Lc φ0
1
φ1
φ1L
c
1 1 φ0 φM −1 1 φ1 and Q = Lc φM −1 1 φM −1
φ0Lr −1 φ1Lr −1 , φMLr−−11
(3)
where ϕm=exp(j2πΔfτm) form=0,…,M-1. Let us define the diagonal matrix Rk whose elements on its diagonal represents the complex amplitude of the m-th path multiplied with the TOA-induced phase shift φm,k = exp(j2πfkτm), such that
Rk = diag a0ϕ 0, k , a1ϕ1,k ,..., aM −1ϕ M −1,k ,
(4)
and diag[·] denotes the diagonal matrix. As proven in [7], Xk is factorized in terms of P, Q and Rk by
X k = PRk Q for k =1,...,K − 1.
(5)
Let us define two matrices Xk,0 and Xk,1, which are the partitioned sub-matrix of Xk with the last row deleted and the first row deleted, respectively, such that
X k ,0 = J 0 X k and X k ,1 = J1 X k
(6)
where J0=[IN-Lr,0N-Lr×1], J1=[0N-Lr×1, IN-Lr], IN−Lr denotes the identity matrix of N−Lr by N−Lr and 0N-1×1 denotes the zero matrix of N−1by1. Substituting Xk with PRkQ in (5), the sub-matrices Xk,0 and Xk,1 are represented by
X k ,1 = J 0 PRk Q = P0 Rk Q ,
(7)
where P0 =J0P and
X k ,1 = J 1 PRk Q =P1 Rk Q = P0ΦRk Q ,
(8)
where P1=P1Φ and Φ = diag[ϕ0,ϕ1,…,ϕM-1]. The factorizations in (7)-(8) are quite similar to the factorization used in [7]. Based on the factorizations in (7)-(8), the generalized EVD on Xk,0 and Xk,1 can be derived by
(X
k ,1
− λ X k ,0 ) β = 0 ⇔ ( P0ΦRk Q − λ P0 Rk Q ) β = 0
⇔ ( P0 [Φ − λ I M ] Rk Q ) β = 0
(9)
where λ and β denote the eigenvalue and the corresponding eigenvector, respectively. In (9), one can demonstrate that, in general, the rank of (Xk,1-λXk,0) will be M. However, if λ=ϕm, m=0,1,…,M − 1, the i-th row of (Xk,1-λXk,0) equal to
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(P0[Φ-λIM]RkQ) becomes zero and the rank of this matrix is M − 1. Through the generalized EVD on Xk,0 and Xk,1, this rank reducing numbers are given by the eigenvalues, i.e., λm=ϕm for m=0,…,M−1. Based on the estimated ϕm, we can achieve the estimated TOA for the m-th path such that
τˆm = 3
1 arg (φm ) 2πΔf
(10)
Simulation Results
The simulation results of Monte Carlo averaged over 10,000 estimates for the AWGN and multipath channel (two-path, herein). In the simulation results, the TOA estimation performance of our proposed algorithm is compared with that of the conventional algorithms such as the ML-based method, MUSIC, MP and TLSESPRIT. The ML-based TOA estimation method developed for multicarrier signals in [2] is used here. Since the TOA of the first arrival path is meaningful in wireless localization, only the RMSE for the first arrival path is considered. The received signal spassed through the multipath channel are normalized to have unit power in average, and AWGN signals with the specified variance is added for Monte-Carlo simulations. -6
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Fig. 1. Comparsion of RMSEs with CRLB in AWGN channel
In Fig. 1, the performance of the proposed algorithm is compared with that of the conventional shift invariant algorithms and CRLB in the AWGN channel. Among the conventional algorithms, the proposed one exhibits the most accurate TOA estimation performance, since it exploits the dual shift invariant structure of the multi-band signals while the others are constrained to the single shift invariant structure. The
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ML-based estimation algorithm shows the worst performance in the AWGN channel due to its low resolution. By the two-path channel simulation, the decomposition capability of the algorithms can be compared to each other. The proposed method shows the most superior performance among the algorithms as shown in Fig. 2. For the case of ML-based method, it shows the worst performance in two-path channel, since its bandwidth is too small to decompose the received paths.
Fig. 2. Comparsion of RMSEs in two-path channel
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Conclusion
We proposed an accurate shift invariant TOA estimation for multiple reference signals in multipath channels. Performance improvement is observed in the TOA estimation results based on the proposed transformation. Acknowledgement. This work was supported by the IT R&D program of MKE/KEIT. [10035570, Development of self-powered smart sensor node platform for smart&green building]. This work was sponsored by ETRI SW-SoC R&BD Center, Human Resource Development Project.
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[2] Yasir, K., Davide, D., Antonio, A., Umberto, M.: Range Estimation in Multicarrier Systems in the Presence of Interference: Performance Limits and Optimal Signal Design. IEEE Trans. Wireless Commun. 10(10), 3321–3331 (2011) [3] Sarkar, T.K., Pereira, O.: Using the matrix pencil method to estimate the parameters of a sum of complex exponentials. IEEE Antennas Propag. Mag. 37(1), 48–55 (1995) [4] Li, F., Vaccaro, R.J., Tufts, D.W.: Performance Analysis of the State-Space Realization (TAM) and ESPRIT Algorithms for DOA Estimation. IEEE Transactions on Antennas and Propagation 39, 418–423 (1991) [5] Guvenc, I., Chong, C.: A Survey on TOA Based Wireless Localization and NLOS Mitigation Techniques. IEEE Communications Surveys & Tutorials 11(3), 107–124, 3rd Quarter (2009) [6] Li, X., Pahlavan, K.: Super-resolution TOA estimation with diversity for indoor geolocation. IEEE Trans. Wireless Commun. 3(1), 224–234 (2004) [7] Boumard, S., Mammela, A.: Robust and Accurate Frequency and Timing Synchronization Using Chirp Signals. IEEE Transactions on Broadcasting 55, 115–123 (2009)
