Root-MUSIC Time Delay Estimation Based on Propagator Method Bin Ba, Yun Long Wang, Na E Zheng & Han Ying Hu

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1 International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 15) Root-MUSIC ime Delay Estimation Based on ropagator Method Bin Ba, Yun Long Wang, Na E Zheng & an Ying u Zhengzhou Institute of Information Science and echnology, Zhengzhou, 45, China Keywords: time delay estimation, Root-MUSIC, propagator method, computational complexity. Abstract. his paper presents a new root multiple signal classification (Root-MUSIC) time delay estimation method based on the concept of well nown propagator method (M). he M does not need the eigenvalue decomposition whose computational complexity is high. his method has great performance of time delay estimation under the condition of high signal to noise ratio (SNR) and need spectral pea searching whose computational complexity is very high. o address those issues, the new method is proposed. he proposed algorithm has high resolution capabilities and has lower computational complexity than conventional M. Its computational complexity significantly reduces by using the method of finding the root of polynomial instead of the spectral pea searching and the eigenvalue decomposition. heoretical analysis and computer simulation results show that the proposed method is effective. Introduction ositioning refers to the geographic information of a target. Various needs from commercial and emergency promote the improvement of high accuracy, low cost positioning system. Location information is very helpful for improving performance of the communication system. he super resolution techniques are firstly used in direction of arrival (DOA) estimation. he DOA estimation has many applications in wireless localization [1-6]. Moreover, the super resolution time delay estimation can be used in high-precision ranging, which is the basis of highprecision positioning system. he multiple signal classification (MUSIC) is used to estimate time delay [7-9]. Smoothing in frequency domain is introduced to MUSIC algorithm for single snapshot. he MUSIC algorithm has great performance in time delay estimation. owever, it needs eigenvalue decomposition and spectral pea searching whose computational complexity is very high. o address this issue, the estimation of signal parameters via rotational invariance technique (ESRI) is provided to time delay estimation [1]. It does not need spectral pea searching and its computational complexity significantly reduces. owever, the performance of ESRI is lower than MUSIC. In order to avoid spectral pea searching and decline for performance, the Root- MUSIC is introduced to time delay estimation. he spectral pea searching is replaced by finding the roots of polynomial [11]. he Root-MUSIC and MUSIC have the same asymptotic performance, and the Root-MUSIC performance with small samples is better than the MUSIC. owever, the Root-MUSIC algorithm needs eigenvalue decomposition, whose computational complexity needs to reduce. he propagator method (M) can estimate noise subspace without eigenvalue decomposition [1]. In this paper, a new Root-MUSIC time delay estimation algorithm based on M (M-Root- MUSIC) is proposed. he algorithm does not need eigenvalue decomposition and spectral pea searching, so its computational complexity significantly reduces. Signal model he impulse response of a multipath wireless channel in time domain can be modelled as L 1 ht ( ) = aδ( t t), (1) = j where L is the number of multipath components, a = a e θ and τ are the complex fading coefficient and propagation time delay for the th path component, respectively. a presents the 15. he authors - ublished by Atlantis ress 31

2 amplitude of a. θ means the phase of a and obeys a uniform distribution U (, ). In consequence, τ denotes the propagation time delay of the first path. his time delay has to be estimated for the purpose of locating a target. he time domain expression of OFDM symbol is K 1 1 j t x( t) = be, t + G, () = where b= [ b, b1,, bk 1] denotes data vector of carrier,, G, and + G are the period of fast Fourier transform (FF) or inverse FF (IFF), the length of cyclic prefix and the length of OFDM symbol. According to the formula (1) and formula (), the received signal can be expressed as K 1 L 1 1 y t b ae e n t j t i j t ( ) = i + ( ) = i=, (3) K 1 L 1 1 j t i j t = b ae i + n e = i= where n represents the additive white Gaussian noise with the mean of zero and variance of σ. he received data at th carrier y can be obtained as L 1 j τ i y = b ae i + n i=. (4) hen, multipath channel frequency response at th carrier can be expressed as L 1 j τ i = i + i= ˆ ae n. (5) he channel frequency response estimation can expressed in vector form ˆ = + n = Va + n, (6) where = 1 K 1 = [ n n1 nk 1] ˆ ˆ ˆ ˆ, (7) n, (8) V = v( ) v( ) v ( ), (9) τ τ1 τk 1 j τi j ( K 1) τi τ i = 1 e e = a a1 al 1 v ( ), (1) a. (11) Root-MUSIC time delay estimation based on propagator method Under the hypothesis that the steering matrix V L is of full ran, rows of V are linearly independent. he other rows can be expressed as a linear combination of these L rows. We assume that the first L rows are linearly independent. ropagator method he definition of the propagator is based on the partition of the steering vectors (9) in accordance with V A } L V=, (1) V B } K L where V A and V B are the matrices of dimension L L and ( K L) L respectively. Owing to non-singular matrix V A, the propagator is the unique linear operator which satisfies with V A =V B. (13) he formula (13) is equivalent to, I K L V=QV= ( K L), (14) L where I K L and ( K L) are the identity matrix and the null matrix respectively. L 313

3 he autocorrelation matrix R ˆ ˆ ˆ ˆ = E of Ĥ can be bloced as R = [ G F ], (15) where G = R (:,1: L ) and F = R (:, L + 1: K) are the columns from 1 to L and the columns from L + 1 to K, respectively. he matrices G, F, and satisfy the following relations G = F. (16) he least square cost function for solution of (16) can be expressed as J ( ) = F G, (17) F where respresents Frobenius norm. F In consequence, the result of propagator estimation is ˆ = G G G F. (18) ( ) 1 Both Q and V are orthogonal each other, so the space expanded by Q belongs to the noise subspace. In order to improve the performance, we can introduce a projection operator onto the noise subspace, and substitute Q by its orthonormalized version ( ) 1 Q QQQ (19) = In order to estimate time delay, the pseudospectral function can be obtained as 1 M ( τ ) =, () v ( τ) QQ v( τ) which is similar to the MUSIC algorithm. Root-MUSIC based on propagator method he computational complexity of the MUSIC algorithm is so high as to be difficult to be realized in project, because it need estimate autocorrelation matrix and eigenvalue decomposition. he M algorithm can obtain the noise subspace by taing advantage of the submatrix of autocorrelation matrix and does not need eigenvalue decomposition for autocorrelation matrix. he spectral pea searching can be made for time delay estimation by the propagator, and the computational complexity of M is lower than MUSIC algorithm. We propose a Root-MUSIC time delay estimation based on propagator method which does not need eigenvalue decomposition and spectral pea searching. he algorithm principle is as follows. We can estimate Q by M algorithm and obtain Q which can be used as noise subspace by orthogonalization. According to the rationale of Root-MUSIC algorithm, a polynomial can be defined as f ( z) = p ( z) QQ p ( z), (1) τ j where z = e. As long as the roots of the polynomial can be calculated, the time delay can be estimated. It is complicated to solve polynomial, because the polynomial exists z. herefore, the polynomial (1) can be modified as K 1 Τ 1 f ( z) = z p ( z ) QQ p ( z). () he time delay estimation can be obtained by the L roots which are closest to unit circle. he steps of the proposed algorithm can be summarized as step. 1 Calculate the autocorrelation matrix R ˆ ˆ ; step. Estimate Q by M algorithm, and calculate Q by orthogonalization; step. 3 Find the roots of polynomial (); step. 4 Obtain the L roots which are closest to unit circle; ˆi = angle z ˆ i, i =,, L 1. step. 5 Estimate τ ( ) Simulation results and analysis of computational complexity In this section, the simulation results will be discussed for confirming the performance of the 314

4 proposed algorithm, and the analysis of computational complexity will be shown for conforming the lower computational complexity of the proposed algorithm than Root-MUSIC alogrithm. Simulation results We will discuss the performance of super resolution time delay estimation based on Root- MUSIC and M-Root-MUSIC in OFDM system. he number of multipath is. he amplitudes of multipath components is 1 and.9 respectively. he snapshots is 64, and the number of Monte Carlo simulations is. We set the parameters of OFDM system listed in able 1. able 1 OFDM system parameter settings No. arameters Values 1 Guard duration G 1.6μs FF eriod FF 3.μs 3 System Bandwidth B Mz 4 he Number of Subcarrier 64 5 Carrier Frequency f c.4gz M-ROO-MUSIC ROO-MUSIC CRB MSE/nσ /σ /db Fig.1: he performance comparison of 1 st time delay between these algorithms M-ROO-MUSIC ROO-MUSIC CRB MSE/nσ /σ /db Fig.: he performance comparison of nd time delay between these algorithms. he performance comparisons between these algorithms are shown in Fig.1 and Fig.. he mean square error of the Root-MUSIC algorithm and the M-Root-MUSIC algorithm decrease with increase of the power of noise. he performance of the M-Root-MUSIC algorithm is close to the Root-MUSIC algorithm, and approaches Cramer-Rao bound (CRB). Analysis of computational complexity he computational complexity of M algorithm, Root-MUSIC algorithm, and the proposed 315

5 algorithm will be compared below. hese algorithms need estimate the the autocorrelation matrix R ˆ ˆ O( MK ) of Ĥ, whose computational complexity is times multiplication of complex number. M is the snapshots. he computational complexity of eigenvalue decomposition for Root-MUSIC 3 O( K ) algorithm times multiplication of complex number. he M algorithm does not need eigenvalue decomposition, and the computational complexity of orthogonalization and spectral pea searching are (( ) O K L ) L O and ( KLm τ ), where m τ is the number of searching. he Root- MUSIC algorithm maes use of finding the roots of polynormial in order to substitute spectral pea searching, so the computational complexity significantly reduces. he proposed algorithm does not need eigenvalue decomposition and spectral pea searching. So the computational complexity of the proposed algorithm significantly reduces. Conclusions he performance of time delay estimation is vital for the evaluation of performance of positioning system. he super resolution has great performance in time delay estimation. owever, the computational complexity of super resolution is very high. o address this issue, a new Root- MUSIC time delay estimation based on propagator method is proposed. he proposed algorithm does not need eigenvalue decomposition and spectral pea searching. he performance of the algorithm is close to the Root-MUSIC and approaches CRB, and the computational complexity is lower than Root-MUSIC. Simulation results and analysis of computational complexity are conducted to confirm the effectiveness of the proposed algorithm. References [1] Song. B., Wang.-G., ong K., & Wang L., A novel source localization scheme based on unitary esprit and city electronic maps in urban environments. rogress In Electromagnetics Research, 94, pp.43-6, 9. [] Liu.-Q., So.-C., Lui K. W. K., & Chan F. K. W., Sensor selection for target tracing in sensor networs. rogress In Electromagnetics Research, 95, pp.67-8, 9. [3] Liu.-Q. & So.-C., arget tracing with line-of-sight identification in sensor networs under unnown measurement noises. rogress In Electromagnetics Research, 97, pp , 9. [4] Mitilineos S. A. & homopoulos S. C. A., ositioning accuracy enhancement using error modeling via a polynomial approximation approach. rogress In Electromagnetics Research, 1, pp.49-64, 1. [5] Zhang W., oorfar A., & Li L., hrough-the-wall target localization with time reversal music method. rogress In Electromagnetics Research, 16, pp.75-89, 1. [6] Mitilineos S. A., Kyriazanos D. M., Segou O. E., Goufas J. N., & homopoulos S. C. A., Indoor localisation with wireless sensor networs. rogress In Electromagnetics Research, 19, pp , 1. [7] Li X., & ahlavan K., Super-resolution OA estimation with diversity for indoor geolocation. IEEE rans. Wireless Commun., 3(1), pp.4-34, 4. [8] Li X., Ma X., Yan S., & ou C., Super-resolution time delay estimation for narrowband signal. IE Radar Sonar & Navigation, 6(8), pp , 1. [9] Zhang X., Feng G., & Xu D., Blind direction of angle and time delay estimation algorithm for uniform linear array employing multi-invariance music. rogress In Electromagnetics Research Letters, 13, pp.11-, 1. [1] Oh D., Kim S., Yoon S.., & Chong J. W., wo-dimensional ESRI-lie shift-invariant OA estimation algorithm using multi-band chirp signals robust to carrier frequency offset. IEEE rans. Wireless Commun., 1(7), pp ,

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