Antennas and Propagation. Chapter 5c: Array Signal Processing and Parametric Estimation Techniques

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1 Antennas and Propagation : Array Signal Processing and Parametric Estimation Techniques

2 Introduction Time-domain Signal Processing Fourier spectral analysis Identify important frequency-content of signal Matched/Wiener Filter Optimize signal to noise ratio of output (known signal / noise cov.) Array Signal Processing Exploit spatial dimension similar to time-domain SP This Lecture Classical methods: direct extensions of time-domain SP Parametric (superresolution) methods Main Resource: Krim / Viberg paper Antennas and Propagation Slide 2

3 Array Signal Processing Direction of Arrival Estimation Waves arriving from different directions Induce different phase shifts across array Fourier-type analysis: Identify different spatial frequencies Optimal (Linear) Beamforming Wiener / Matched-filtering in spatial domain Limitations of Linear Methods Performance limited by size of aperture (regardless of SNR / number of samples) Nonlinear (superresolution) methods Antennas and Propagation Slide 3

4 Signal Model Narrowband signal: Signal on the array: Signal received at origin Narrowband Assumption Changes in s(t) appear simultaneously on array Antennas and Propagation Slide 4

5 Signal Model (2) Restrict attention to xy plane Signal Ant. Coords Collect signals from L antennas Steering Vector Antennas and Propagation Slide 5

6 Signal Model (3) Steering vector for a ULA Multiple signals are baseband waveforms More compact form Presence of additive noise Steering matrix Vector of signal waveforms Antennas and Propagation Slide 6

7 Assumptions Exploit spatial dimension: Spatial covariance matrix Source covariance Noise covariance Assuming noise is white or uncorrelated from one sensor to the next Assume P is non-singular matrix (e.g. uncorrelated signals) Antennas and Propagation Slide 7

8 Signal / Noise Subspaces Suppose that L > M (more antennas than signals) Can partition R according to Signal Subspace Note: Noise Subspace Columns of U s span range space of A Columns of U n span its orthogonal complement (null space) Projection Operators Antennas and Propagation Slide 8

9 Problem Statement Estimating DOAs Find θ m for each of the incoming signals Given a finite set of observations {x(t)} Note: In practice have only estimates Assumption: Know M or how many signals present Estimating Signals Recover signals s(t) once DOAs known Antennas and Propagation Slide 9

10 Summary of Estimators Definitions Coherent signals Signals that are scaled/delayed versions of each other Consistency Estimate converges to true value for infinite data Statistical efficiency Asymptotically attains CRB (lower bound on covariance matrix of any unbiased estimator) Antennas and Propagation Slide 10

11 Summary of Estimators (2) Antennas and Propagation Slide 11

12 Spectral-Based vs. Parametric Spectral Form a function of parameter of interest (DOA) Sweep that function with respect to some parameter Identify peaks Typically a 1D search. Find DOAs independently Parametric Simultaneous search of all parameters Higher accuracy Increased complexity Antennas and Propagation Slide 12

13 Spectral-Based Methods Beamforming Steer a beam and measure output power Peaks give DOA estimates θ 2 Sources θ 1 Linear beamformer θ 0 = steering angle Power θ 1 θ 2 θ 0 Antennas and Propagation Slide 13

14 Bartlett Beamformer Same as uniform excitation we saw before Maximize power collected from look angle θ For a ULA Resolution approximately 100º/L Antennas and Propagation Slide 14

15 Bartlett Beamformer (2) Example L=10 Elements, λ/2 spacing Resolution of standard ULA approximately 100º/L = 10º (Obtain from HPBW expression) Antennas and Propagation Slide 15

16 Bartlett Beamformer (3) Advantages Simple Robust Disadvantages Resolution is limited Interference of close-by arrivals Strong side lobes Antennas and Propagation Slide 16

17 Capon s Beamformer Revised problem Minimize total power collected Maintain gain in look direction θ to be 1 What does this mean? Like a sharp spatial bandpass filter Reduce interference from directions other than θ when we are looking in direction θ Antennas and Propagation Slide 17

18 Capon s Beamformer (2) Solution Antennas and Propagation Slide 18

19 Capon s Beamformer (3) Advantage Provides much narrower main beam. How? Nulls directions that are near look direction Disadvantages Sacrifice some noise performance Also, can be unstable (consider inverse) Resolution still depends on aperture size and SNR Antennas and Propagation Slide 19

20 Subspace-Based Methods MUSIC (Multiple Signal Classification) Introduced by R. Schmidt in 1980 Breakthrough in DOA Estimation Exploit structure of signal/noise subspaces Resolution no longer depend on array size Antennas and Propagation Slide 20

21 MUSIC Decompose covariance with EVD Assume P to be full rank, A (LxM) is tall (L>M) U s and A span same (column) subspace U n spans the orthogonal complement of U s Each vector in A is orthogonal to U n Idea: Sweep θ and see where this goes to 0. Music spectrum: Exhibits peaks when θ is a DOA. Antennas and Propagation Slide 21

22 Comparison: Spectral-based Methods Parameters: L = 10 d = λ/2 M = 200 samples Antennas and Propagation Slide 22

23 Coherent Signals Problem Signals are correlated with each other P is no longer full rank MUSIC spectrum will not exhibit peaks Example? Multipath Techniques to Decorrelate signals ULA Forward-backward averaging Spatial smoothing Antennas and Propagation Slide 23

24 Forward-Backward Averaging Reverse signals in x vector (reverse antennas) followed by complex conjugate Introduces a unique phase shift for each steering vector (or source) Can treat as another sample of the same signal But phase shift introduces decorrelation Antennas and Propagation Slide 24

25 Forward-Backward Averaging (2) Including backward signals in our covariance estimate Consider: pairs of sources are correlated New effective source covariance not correlated Antennas and Propagation Slide 25

26 Spatial Smoothing Idea Related to FB averaging Form multiple looks of sources by shifting the array This shifts each steering vector (source) by a different phase Relative phases in each steering vector are preserved (shift invariance) Spatial smooth by factor N to decorrelate N sources Antennas and Propagation Slide 26

27 Parametric Methods Drawback of Spectral Methods May be inaccurate (e.g. correlated signals) Parametric Methods Fully exploit the underlying data model Powerful, but in general require multi-dimensional search Exception: For ULA can exploit model without search Variants ML (deterministic or stochastic) Subspace fitting Root MUSIC ESPRIT Antennas and Propagation Slide 27

28 Deterministic Maximum Likelihood Assume Zero-Mean, White Gaussian Noise pdf of observed signal (complex Gaussian) Form Likelihood Function If noise is uncorrelated between samples Likelihood of observing x(t) = As(t) + n(t) given noise, DOAs, signals Idea Find DOAs / signals that make observed x(t) as likely as possible Antennas and Propagation Slide 28

29 Deterministic Maximum Likelihood (2) (Negative) Log-Likelihood Function Minima satisfy Projection onto nullspace of A Sample covariance Pseudoinverse of A Substituting into Log-Likelihood Minimum: Make σ as small as possible Interpretation? When we remove DOAs exactly, resulting power is minimal Antennas and Propagation Slide 29

30 Deterministic Maximum Likelihood (3) How do we minimize? Requires a multidimensional search (numerical) Becomes very complicated for large M Acceleration method Find an initial guess with spectral method Followed by local optimizer Antennas and Propagation Slide 30

31 Parametric Methods for ULAs Uniform Linear Arrays Steering matrix has Vandermonde structure Can exploit this strcuture Allows close to ML estimate to be found without searching ESPRIT Estimation of Signal Parameters by Rotation Invariant Techniques Uses the shift-invariance property of A Antennas and Propagation Slide 31

32 ESPRIT Recall the EVD of R Steering matrix for ULA Vandermonde Matrix Antennas and Propagation Slide 32

33 ESPRIT (2) Shift property of A A Can find a direct method to get Φ P is full rank, span of U s and A same, which means For some invertible matrix T Antennas and Propagation Slide 33

34 ESPRIT (3) Consider relationship of Ψ and Φ Similar matricies Have same eigenvalues Antennas and Propagation Slide 34

35 ESPRIT (4) Tasks Solve for Ψ How do we solve this? Compute eigenvalues to get Φ φ 1, φ 2,... Compute DOAs using Antennas and Propagation Slide 35

36 Total Least Squares (TLS) Want to find A that solves (X,Y tall, A is N x N) Form N dimensional orthogonal basis that best spans both X and Y In the N-dimensional subspace, can now equate Antennas and Propagation Slide 36

37 Summary Array Signal Processing Like filtering, but in spatial dimension Can enhance signals Estimate locations of sources Spectral-based Methods Beamforming (Bartlett, Capon) Subpace-based Method (MUSIC) Parametric Methods Directly exploit underlying signal model DML ESPRIT Antennas and Propagation Slide 37

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