Eigenvalues and Eigenvectors in Array Antennas. Optimization of Array Antennas for High Performance. Self-introduction

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1 Short in MACAO Eigenvalues and Eigenvectors in Array Antennas Optimization of Array Antennas for High Performance Nobuyoshi Kikuma Nagoya Institute of Technology, Japan 1 Self-introduction 2 Nobuyoshi Kikuma Professor Research interests: adaptive and signal processing array multipath propagation analysis mobile and indoor wireless communication wireless power transmission Hobby: MUSIC (karaoke, etc). 1

2 3 On this short course Spatial signal processing technology using array antenna has been one of the important approaches for improving the performance of communications and radars. This short course expresses the optimization of the array antenna for its high performance in various applications. 4 On this short course(cont d) Significance of eigenvalues and eigenvectors of various matrices used in the array antenna is explained. Key words: Gain of the array antenna Optimum weights of the adaptive array Array weights for direction-of-arrival (DOA) estimation (e.g. MUSIC) Ref.: D.K. Cheng, "Optimization techniques for antenna array," Proc. IEEE, vol.59, No.12, pp , Dec

3 Applications of radio waves Communications, Broadcasting, and Sensors 5 Antennas are important components. Radio Propagation in Mobile Communications 6 No line of sight Reflection, diffraction and scattering (multipath) Heavy fading 10 B) Amplitude (d f = 800MHz v = 60km/h Time (seconds) Nagoya Institute of Technology 3

4 Example of antenna pattern for multipath waves (1) 7 Only one dominant wave is received. Example of antenna pattern for multipath waves (2) 8 All multipath waves are combined appropriately. 4

5 Radio Propagation in Cellular Communication Systems 9 Effective utilization of frequency resources Co-channel interference (CCI) BS MT Interference Multipath waves Same frequency Nagoya Institute of Technology Techniques expected in antenna systems Fading recovery Suppressing or Canceling CCI 10 Control of directional pattern using array antenna: Adaptive array signal processing (pattern optimization) Nagoya Institute of Technology 5

6 11 Why array antenna is used? SNR increased by inphase combination of array-element signals High angular resolution with narrow mainlobe Electronic scan of mainlobe/null General configuration of K-element array antenna Transmitting mode 12 I 1 I k r(r,θ,φ) position vector of observation point I K I k Directional pattern of element Excitation current 6

7 Radiation field from antenna at origin (phase center) 13 Maxwell s equation spherical wave Focus on θ component No r-component Approximation of radiation field (e.g., θ-component) 14 #1 phase difference from origin i #2... phase difference from origin 7

8 Approximation of radiation field 15 Radiation field of array antenna (Combined electric field) spherical wave 16 phase difference due to element position Directional function of array antenna 8

9 Radiation field of array antenna (Cont d) If directional functions of all elements are identical, then 17 Principle of pattern multiplication : array factor : element pattern 18 Directivity and its optimization 9

10 Directivity of array antenna definition impedance 19 power density Identical elements Normalized power pattern of antenna element Directivity of array antenna (Cont d) Vector notation : excitation current vector 20 Directivity : array steering vector Inner product of above two vectors Ratio of two Hermitian forms 10

11 Maximization of directivity of array antenna Generalized Rayleigh quotient What is w maximizing G? 21 Generalized eigenvalue problem: Eigenvector corresponding to maximum eigenvalue λ M [W,Lambda]=eig(A,B); Maximization of directivity of array antenna (Cont d) Uniform linear array of isotropic elements with element spacing of d Example 22 In addition, when d = λ/2, ( :Kronecker delta) Uniform excitation Number of elements 11

12 Q factor & mainbeam radiation efficiency of array antenna Q factor 23 Mainbeam radiation efficiency Numerical Examples 6-element uniform linear array (ULA) 24 Broadside array with isotropic elements 12

13 25 Numerical Examples (Cont d) 6-element uniform linear array (ULA) Endfire array with isotropic elements 26 Numerical Examples (Cont d) 6-element uniform linear array with d = λ/4 Directional patterns w M =[1.11, -0.91, 0.80, 0.80, -0.91, 1.11] T v 0 =[1, 1, 1, 1, 1, 1] T w M =[ j, j, j, j, j, 2.68 ] T v 0 =[1, -j, -1, j, 1, -j] T 13

14 27 Adaptive array and its optimization 28 Adaptive array model (receiving mode) Vector notation of inputs and weights K-element array antenna Output is expressed as inner product of two vectors. 14

15 Adaptive array model (Cont d) 29 Array output t power: Covariance matrix: Hermitian form Hermitian matrix Adaptive array model (Cont d) 30 Output SINR: at output Large value means good receiving performance Maximization of SINR *SINR: Signal-to-Interference-plus-Noise Ratio 15

16 Input vector: Adaptive array output (1) (linear array, no interference) S N 31 Array output: Adaptive array output (1) (Cont d) (linear array, no interference) 32 Input SNR Rayleigh quotient Equal to mainbeam radiation efficiency Weight vector maximizing SINR: 16

17 Adaptive array output (2) (linear array, L interferences) 33 Input vector: S I N Array output: Adaptive array output (2) (Cont d) (linear array, L interferences) 34 Generalized Rayleigh quotient P 0 =Ps 17

18 Adaptive array output (2) (Cont d) (linear array, L interferences) 35 Quasi-Normalized SINR Generalized Rayleigh quotient Directivity and Output SINR 36 Same form Desired signal Angular distribution of incident waves 18

19 Typical criteria of adaptive array 37 Maximum Signal-to-Noise Ration: MSN Minimum Mean Square Error: MMSE MSN adaptive array 38 This adaptive array controls weights to maximize the output SNR(SINR). A priori knowledge: DOA of desired signal Cost function: maximized Angular distribution of incident waves 19

20 39 MSN adaptive array (Cont d) Optimum weight vector: MSN adaptive array (Cont d) Optimum weight vector w opt maximizing SINR γ 20

21 41 MMSE adaptive array Minimizing error e(t) which is a difference between reference signal r(t) and array output y(t): Cost function: minimized 42 MMSE adaptive array (Cont d) Optimum weight vector In the case of one desired signal incident This weight is equal to that t of MSN adaptive array. Maximizing output SINR 21

22 Example of 6-element adaptive array (K = 6) Example of 6-element adaptive array (K = 6) (Cont d) Directional patterns Adaptive (MMSE & MSN) SINR = 27.66dB SINR 0 = Uniform SINR = 15.31dB SINR 0 =

23 45 DOA estimation and array optimization Linear array 46 Beam-scan scheme Beamformer Maximization of directivity in θ Angular spectrum Capon Beamformer Constrained minimization Angular spectrum 23

24 Beam-scan scheme (Rayleigh quotient expression) 47 Beamformer Mainbeam radiation efficiency Capon Beamformer SINR 48 Problems of Beamformer Wave 2 is received by sidelobe. Closely spaced waves cannot be separated. 24

25 Advantage of Capon Beamformer 49 Desired signal Interference Interference Desired signal Examples of directional patterns by optimum weights How angular resolution is enhanced furthermore? 50 Nulls are utilized. Null-scan scheme Nulls are adaptively steered in the respective directions of incident waves. Angular spectrum of null-scan scheme: Inverse of power pattern 25

26 Null-scan scheme (Constrained minimization) linearly constrained minimization (Linear prediction) 51 Quadratically constrained minimization Pisarenko s method, Min-Norm method, MUSIC Null-scan scheme (Rayleigh quotient expression) 52 linearly constrained minimization (Linear prediction) Quadratically constrained minimization (Pisarenko s method) 26

27 Eigenvectors of minimum eigenvalues of covariance matrix 53 Eigenvalues: Eigenvectors: Noise power Example of 6-element ULA with element spacing of a half wavelength (L=3) 3 eigenvectos used individually Pisarenko s method Some smoothing required Min-Norm method and MUSIC Min-Norm method Linear combination of eigenvectors 54 Signal subspace Noise subspace MUSIC Combination of power patterns 27

28 55 5 estimators 6-element ULA 3 waves 0, 1 10, 1 60, 0.5 Noise power Summary and conclusions Optimization of array antennas using eigenvalues and eigenvectors Directivity Adaptive arrays DOA estimation Rayleigh quotient expression Link to eigenvalue problem Optimization of array antennas based on Rayleigh quotient expression Unified array antenna theory including adaptive arrays and DOA estimation 28

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