A CORRELATION-BASED METHOD FOR DIRECTION FINDING OF MULTIPATH SIGNALS IN FREQUENCY HOPPING SYSTEMS PENG NINGKUN NATIONAL UNIVERSITY OF SINGAPORE 2004/2005 A CORRELATION-BASED METHOD FOR DIRECTION FINDING OF MULTIPATH SIGNALS IN FREQUENCY HOPPING SYSTEMS PENG NINGKUN (M Eng, Huazhong University of Science and Technology) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004/2005 Acknowledgement I would like to express my sincere thanks to my supervisors, Professor Ko Chi Chung, Dr Zhi Wanjun and Dr Francois Chin, for their invaluable guidance, support, encouragement, patience, advice and comments throughout my research work and this thesis Special thanks to my parents, my girlfriend and sister, who always encourage, support and care for me throughout my life I also wish to give my thanks to all students and staffs in Communications Lab at Department of Electrical and Computer Engineering for their helpful discussion and friendship I am grateful for research scholarship from the National University of Singapore for giving me the opportunity to carry out my research work i Contents Acknowledgement i Contents ii List of Figures v List of Tables vi Nomenclature vii Summary viii Introduction 1.1 Evolution of Wireless Communication Systems 1.2 Introduction to Smart Antenna Technology .3 1.3 Direction Finding and Source Localization .4 1.4 Source Localization using DOA, TOA and TDOA .5 1.5 Organization of the Thesis Direction Estimation using Antenna Array 2.1 Introduction to DOA Estimation Methods 2.2 Signal Model for DOA Estimation 10 2.3 Signal Model under Multipath Propagation 12 2.4 Introduction to Frequency Hopping Systems 14 2.5 DOA Estimation Methods 15 2.5.1 Beamforming Techniques .18 2.5.2 Subspace Methods 20 ii 2.5.3 Maximum Likelihood (ML) Method 24 2.6 Summary 27 Frequency Hopping Correlation (FHC) Method to Track Multipath Signals for Frequency Hopping System 29 3.1 Introduction to the DOA estimation for frequency hopping system 29 3.2 Signal Model for the FHC method 30 3.3 Formulation of the FHC method .33 3.4 Search Methods .36 3.4.1 Steepest Descent Method .36 3.4.2 Newton’s Method 38 3.4.3 Gauss-Newton Method 39 3.4.4 Alternating Minimization Method 42 3.5 Highly Oscillation problem .43 3.6 Detailed Algorithm of the FHC method 47 3.7 Summary 50 Simulation Study of the FHC method 51 4.1 Simulation Scenario 51 4.2 Simulation of Estimating the Directions of Stationary FH signals 55 4.3 Simulation of Tracking Slow Moving FH signals 59 4.4 FHML (Frequency Hopping Maximum Likelihood) method 63 4.4.1 Introduction to the FHML method 63 4.4.2 Simulation of the FHML method 66 iii 4.5 Comparison between FHML method and FHC method 70 4.6 Summary 72 Conclusions and Future Work 73 5.1 Conclusions .73 5.2 Future Work .74 References 75 iv List of Figures 1.1 Localization via DOA in two-dimension ……………………………………… 2.1 Two-dimensional array geometry ……………………………………………… 10 2.2 ULA array geometry…………………………………………………………… 11 2.3 Block diagram of an FH spread spectrum system ……………………………… 15 2.4 Subarray structure for spatial smoothing technique …………………………… 22 3.1 Receiving array structure ……………………………………………………… 30 3.2 System Structure for one hop ……………………………………………………33 3.3 The objective function w.r.t ∆τ with one hop frequency …………………… 35 3.4 The objective function w.r.t ∆τ with two hop frequencies ………………… 35 3.5 Illustration of steepest descent method ………………………………………….37 3.6 Two-dimension case of Alternating Minimization method …………………… 43 3.7 The highly oscillatory objective function w.r.t time delay …………………… 44 4.1 Waveform of received signals with AWGN ……………………………………52 4.2 Power of the received signals with AWGN and without AWGN ………………53 4.3 Cross correlation of the received signals ……………………………………… 53 4.4 Convergence figures of FHC method for Stationary Case ………………57 4.5 Convergence figures of FHC method for Slow Moving Case …………62 4.6 Convergence behavior of stationary case for FHML method ………………… 67 4.7 Convergence behavior of slow moving case for FHML method ……………… 69 4.8 Variances of converged values of θ vs SNR…………………………………… 70 v List of Tables 4.1 An example of hop sequence…………… ………………………………………51 vi Nomenclature a scalar a column vector A Matrix A+ Pseudo-inverse of Matrix [•] Transpose of matrix or vector [•] Hermitian transpose [•] Matrix inversion ∇ x f (x) The gradient of f ( x ) with respect to x FSK Frequency Shift Keying PSK Phase Shift Keying T H −1 vii Summary Direction finding is of great interest in many applications such as GPS (Global Positioning System), radar, sonar and wireless communication systems In smart antenna systems, the direction of users is an important factor to increase the capacity, and an antenna array is usually used at the base station to estimate and track the direction of users Conventional direction finding methods solve the problem of direction-of-arrival (DOA) estimation for narrowband signals, and usually these methods require that the number of array elements be larger than the number of signal sources In military communications and some short-range wireless communication systems (e.g Bluetooth), frequency hopping technique is widely used equip a large size antenna array In such systems, it is difficult to Therefore, it is necessary to solve the direction finding problem for the frequency hopping systems by using an antenna array with a lesser number of elements In this thesis, a new method is proposed to estimate and track the directions of frequency hopping signals under multipath propagation Only the power of trans- mitted signal is needed to be known in this method With a two-element array at the receiver, the objective function is established by minimizing the differece between the estimated correlations and the measured correlations of the received signals viii The [ where S p = e − jα h τ p e − jα h+1τ p e ( − jα h τ p + β p ) e ( − jα h+1 τ p + β p ],α )T h = ω c + ω h , h is the hop in- dex and e jnω hTs  0 n D = 0 0  0      e jnω h+1Ts   e jnω h+1Ts 0 e jnω hTs 0 (4.7) The parameter vector is η =[τ ,L ,τ P −1 ,β ,L ,β P −1 ] T (4.8) Equation (4.6) can be written in the form of X(n ) = D n S(η)G (4.9) where S(η) = [S ,L, S P −1 ] , G = [g , L , g P −1 ] T In the presence of independent receiver noise, the received signal vector becomes X(n ) = D n S(η)G + V (n ) (4.10) where V (0) ,…, V ( N − 1) are independent noise vectors with zero mean and a covariance vector σ 2I f (η ) = − ( 2π σ The likelihood function of received signals is ) 4N e N −1 ∑ X (n )− DnS (η )G 2σ n=0 (4.11) The unknown parameters can be found by maximizing the likelihood function From (4.11), this is equivalent to minimizing the objective function ψ (η ) = N N −1 ∑ X(n ) − D S(η)G n (4.12) n=0 The gradient of ψ (η) with respect to G is ∇ Gψ (η) = N ∑ S (η)[(D ) X(n) − S(η)G ] N −1 H n H n =0 64 (4.13) Let (4.13) be zero, the likelihood estimator for G is + ˆ G = S (η ) X D (4.14) [ ] −1 where S(η) = S H (η)S(η) S H (η) and X D = + N N −1 ∑ (D ) X(n ) n H n=0 Substituting the estimator of G into (4.12), we have 1 N ψ (η) = trace N −1 ∑ X(n)X (n) − P(η)X H D n =0 H XD   (4.15) where P(η) = S(η)S(η) + The maximum likelihood estimator of η is then given by ˆ η = arg l (η) (4.16) η where l (η) = P ⊥ (η)X D (4.17) P ⊥ (η ) = I − P (η ) (4.18) ˆ The solution for η can be obtained by applying Gauss-Newton method { ˆ ˆ ˆ η(k + 1) = η(k ) − µ ⋅ ∇ η(k ) l [η(k )] ˆ { ˆ where ∇ η( k ) l [η(k )] ˆ } + ˆ } ⋅ l[η(k )] + (4.19) ˆ is Moore-Penrose pseudo-inverse of ∇ η( k ) l [η(k )] ˆ The derivative of the objective function is given by ˆ ∇ η(k ) l [η(k )] = ˆ ∂ l [η(k )] ∂ηi ( H H & H = −2 Re X D S + S ηi P ⊥ X D ) (4.20) where ∂S  ∂S ∂S  & S ηi = = , L , P −1  ∂ηi  ∂ηi ∂ηi  (4.21) 65 4.4.2 Simulation of the FHML method To compare the FHML method with the FHC method, the simulation scenario of the FHML method is completely same as that of the FHC method [ rameters are set at η = 5.05µs 10.05µs 27 o SNR=20dB sult ] T 47 o The initial pa- The step size is µ = 0.5 Each step of the iteration uses two successive hops to generate one re- Fig 4.6 and Fig 4.7 show the algorithm performance in stationary case and slow moving case, respectively -6 5.07 x 10 FHML method True values 5.06 5.05 τ0 (Sec.) 5.04 5.03 5.02 5.01 4.99 100 200 (a) τ0 300 Number of hops 400 vs number of hops 66 500 600 -5 1.005 x 10 FHML method True values 1.004 τ1 (Sec.) 1.003 1.002 1.001 0.999 100 200 (b) τ1 300 Number of hops 400 500 600 vs number of hops 35 30 25 θ0 (Deg.) 20 15 10 -5 FHML method True values 100 200 (c) θ0 300 Number of hops 400 vs number of hops 67 500 600 55 FHML method True values 50 θ1 (Deg.) 45 40 35 30 25 20 100 200 (d) θ1 300 Number of hops 400 500 600 vs number of hops Fig 4.6 Convergence figure of stationary case for FHML method -6 5.025 x 10 FHML method True values 5.02 τ0 (Sec.) 5.015 5.01 5.005 4.995 100 200 (a) τ0 300 Number of hops 400 vs number of hops 68 500 600 -5 1.0025 x 10 FHML method True values 1.002 1.001 1.0005 0.9995 100 200 (b) τ1 300 Number of hops 400 500 600 vs number of hops 35 FHML method True values 30 25 θ0 (Deg.) τ1 (Sec.) 1.0015 20 15 10 100 200 (c) θ0 300 Number of hops 400 vs number of hops 69 500 600 55 50 θ1 (Deg.) 45 40 35 30 FHML method True values 25 100 200 (d) θ1 300 Number of hops 400 500 600 vs number of hops Fig 4.7 Convergence figure of slow moving case for FHML method It is shown that the FHML method performs well both in estimating directions of stationary FH signals and tracking slow moving FH signals We will compare the converged results of the FHML method with the FHC method in next section 4.5 Comparison between FHML method and FHC method To compare the converged results of the FHML method and the FHC method, the average variances of the angle-of-arrivals θ and θ are measured after they have converged in the stationary case As shown in Fig 4.4 and Fig 4.6, it is obvious that θ and θ converge after 300 iterations The variances of converged values are calculated from the 350th iteration to the last iteration Fifty Monte Carlo runs are done and we take the average of the fifty variances for both methods at a given SNR 70 0.25 FHML method FHC method Variance of converged values of θ0 0.2 0.15 0.1 0.05 0 10 SNR (dB) 15 20 (a) Variances of converged values of θ0 at different SNR 0.25 Variance of converged values of θ1 FHML method FHC method 0.2 0.15 0.1 0.05 0 10 SNR (dB) 15 20 (b) Variances of converged values of θ1 at different SNR Fig 4.8 Variances of converged values of θ vs SNR Fig 4.8 shows the results at SNR=0dB, 5dB, 10dB, 15dB and 20dB respectively The asterisk and plus signs denote the average variances of the converged θ at the corresponding SNR We can see that the variances of converged values for the FHC 71 method is much smaller than that of the FHML method when SNR is low while they are comparable at SNR=20dB Thus the converged values of directions of the FHC method are more accurate than that of the FHML method at low SNR Other advantages of the FHC method are that only the power of transmitted signal is needed to be known and the correlations of received signal can be measured by analog devices 4.6 Summary In this chapter, simulation results of the FHC method are presented, and it is compared with the FHML method It is shown that the converged results of the new method are more accurate than that of the FHML method Furthermore, only the power of transmitted signal is needed to be known for the proposed method The FHC method is easy to implement as analog devices can be used to measure the correlations of received signals 72 Chapter Conclusions and Future Work 5.1 Conclusions Direction finding and DOA estimation are used in various applications In mili- tary communications and some short-range wireless communication systems, frequency hopping techniques are widely used Some work has already been done to estimate directions for the frequency hopping signals lem is not considered in these methods However, the multipath prob- Therefore, in communication systems where multipath propagation exists, these methods may be not effective Furthermore, a large size antenna array (the number of array elements is larger than the number of signal sources) is needed at the receiver, which is difficult to install on the mobile devices In this thesis, the Frequency Hopping Correlation (FHC) method is proposed to solve the problem of direction finding for the frequency hopping signals under multipath propagation using a two-element array By minimizing the difference between the estimated correlations and the measured correlations of received signals, the objective function is established Generally, the objective function is highly oscillatory with respect to time delay parameters Thus a pre-processing is used to obtain re- 73 fined initial values of the time delays Then the Gauss-Newton algorithm is used to find the optimum parameters including directions The simulation results show that the FHC method is effective for both stationary and slow moving frequency hopping signals Furthermore, the FHC method is compared with the FHML method When the Gauss-Newton algorithm is applied to both methods, the converged values of direction parameters of the FHC method are more accurate than those of the FHML method The new method has another advantage that the correlations can be meas- ured by analog devices 5.2 Future Work The FHC method needs to be improved in the following aspects First, the number of unknown parameters may be reduced so that the Jacobian matrix in the Gauss-Newton method could have a smaller dimension This is to reduce the com- putation complexity of the FHC method Second, in the slow moving case it is assumed that only the direction parameters change with respect to hops In practice, other parameters (e.g time delay) may change with respect to 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users is an important factor to increase the capacity, and an antenna array is usually used at the base station to estimate and track... hopping signals, and a new method is proposed to track multipath frequency hopping signals In Chapter 2, several popular methods for DOA estimation are addressed including maximum likelihood methods,... using DOA, TOA and TDOA The direction parameters are usually estimated with an antenna array which has a certain geometric shape, and the estimation task is performed by exploiting the data collected