Wideband Direction-of-Arrival (DOA) Estimation Methods for Unattended Acoustic Sensors

Transcription

1 Wideband Direction-of-Arrival (DOA) Estimation Methods for Unattended Acoustic Sensors Nicholas Roseveare Department of Electrical and Computer Engineering Colorado State University Thesis Defense, September 28, 27 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 1 / 73

2 Outline of Presentation Introduction Outline 1 Introduction Outline Previous Work Research Objectives 2 Wideband DOA Estimation Signal Model Review of Wideband DOA Estimation Algorithms 3 General Source Error Models Non-Ideal Source Models General Source Error Coherence 4 Robust Wideband DOA Estimation Methods 5 Conclusions and Future Work Conclusions and Summary Suggestions for Future Work (5 min) (1 min) (1 min) (1 min) (5 min) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 2 / 73

3 Introduction Previous Work Background on DOA Estimation using UGS Unattended ground sensors (UGS) have application in battlefield surveillance and situation awareness: They are rugged, reliable, and can be left in the field for a long time after deployment They can be used to capture acoustic signatures of a variety of sources in different types of terrain (MOUT, etc.) The acoustic information may then be used to spatially locate and track sources such as ground vehicles, airborne targets, or even personnel Generally, high performance DOA estimation can separate multiple closely spaced sources. Complications to this arise in acoustic arrays due to: Variability and nonstationarity of source acoustic signatures Signal attenuation and fading effects as a function of range and Doppler Coherence loss due to environmental conditions and wind effects Near-field, multipath, or other non-plane wave effects (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 3 / 73

4 Review of Literature Introduction Previous Work Direction-of-Arrival (DOA) estimation Wideband DOA estimation and tracking with acoustic arrays [Pham] Benchmark of wideband DOA estimation algorithms [1] [Azimi] Localization of multiple wideband sources and the use of distributed arrays to combat sensor location and non-uniform fading error [2, 3] [Damarla],[Hohil] Tracking, counting, and classifying vehicles [4, 5] Narrowband DOA estimation algorithms with application to acoustic arrays [Capon] Minimum Power Distortionless Response (MPDR) [6] [Schmidt] MUltiple SIgnal Classification (MUSIC) method [7] [Viberg] Weighted Subspace Fitting (WSF) method [8] (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 4 / 73

5 Review of Literature Introduction Previous Work Wideband frequency combining methods [Krolik] Coherent focusing with Steered Covariance Matrices [Pham],[Azimi] Wideband incoherent averaging methods [Kaveh],[Di Claudio] Coherent MUSIC and WSF [12, 13] [11, 2] [9, 1] Non-idealized array source models for DOA estimation [Swindlehurst] Models for array geometry and calibration errors [14] [Asztély] Multipath model for local scattering [15, 16] [Valaee] Models for spatially coherent or incoherent sources [17] [Meng],[Scharf] Array source model for partial incoherence [18, 19, 2] (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 5 / 73

6 Introduction Research Objectives Objectives of this Research To benchmark and illustrate deficiencies in existing wideband DOA estimation algorithms To develop a better understanding of non-ideal array signal models This understanding will help appropriately modify the computationally simple Capon to make it robust to the errors in our acoustic data sets To benchmark and show the performance improvements of the developed algorithms (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 6 / 73

7 Outline Wideband DOA Estimation 1 Introduction Outline Previous Work Research Objectives 2 Wideband DOA Estimation Signal Model Review of Wideband DOA Estimation Algorithms 3 General Source Error Models Non-Ideal Source Models General Source Error Coherence 4 Robust Wideband DOA Estimation Methods 5 Conclusions and Future Work Conclusions and Summary Suggestions for Future Work (5 min) (1 min) (1 min) (1 min) (5 min) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 7 / 73

8 Wideband DOA Estimation Wideband Signal Model Signal Model Consider d far-field sources observed by L sensors in an arbitrary noise wavefield for frequency bin f j and sample k d x(f j, k) = A(f j,φ)s(f j, k) + n(f j, k) = a(f j,φ i )s i (f j, k) + n(f j, k) (1) A(f j,φ) = [a(f j,φ 1 ),..., a(f j,φ d )] array manifold of steering vectors s(f j, k) = [s 1 (f, k),..., s d (f, k)] T vector of sources φ = [φ 1,...,φ d ] vector of source directions n(f j, k) noise vector The sample covariance matrix for frequency f j is given as K R xx (f j ) = x(f j, k)x H (f j, k) = A(f j,θ)r s (f j )A H (f j,θ) + R n (f j ) k=1 i=1 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 8 / 73

9 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms Basic DOA Estimation: Beamforming The most basic DOA estimation method is the beamformer which is given by the inner product of the array output vector and a weight vector as y(f j, k,θ) = w H (f j,θ)x(f j, k), (2) where the weight vector w(f j,θ) steers the beam response of the array to observation angle, θ. The quadratic power spectrum becomes p(f j,θ) = w H (f j,θ)r xx (f j )w(f j,θ), (3) from which the peaks are used as DOA estimates. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 9 / 73

10 Wideband DOA Estimation Capon Beamforming Review of Wideband DOA Estimation Algorithms Minimizes the overall received power while requiring the signal of interest (SOI) to be received at unit power, i.e. min w(f j,θ) wh (f j,θ)r xx w(f j,θ) s.t. w H (f j,θ)a(f, θ) = 1 (4) This results in the optimal beamformer weights w (f j,θ) = this beamformer yields the power spectrum p Capon (f j,θ) = R 1 xx (f j )a(f j,θ) a H (f j,θ)r 1 xx (f j )a(f j,θ), (5) 1 a H (f j,θ)r 1 xx (f j )a(f j,θ). (6) incoherent averaging across frequency using the geometric mean is J J 1 P G (θ) = p Capon (f j,θ) = a H (f j,θ)r 1 xx (f j )a(f j,θ). j=1 j=1 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 1 / 73

11 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms Review MUSIC, WSF, and STCM MUSIC Idea Exploits orthogonality between signal and noise subspaces, DOA estimate is min distance between steering vector and noise eigenvectors Drawbacks Sensitive to choice of noise subspace and requires SVD WSF STCM Fits data to a search array manifold matrix in least-squares sense, DOA estimates found from minimum of error between fitted and actual array response Uses unitary transforms to focuses the frequency spectra from multiple narrowband bins; applied in tandem with a narrowband algorithm Requires SVD and an intense multi-dimensional search over sets of angles Focusing reduces the resolution capabilities of whichever algorithm it is used with (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

12 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms Results on Baseline and Distributed Array Data Sets The baseline array data: Textron R five-element wagon-wheel ADAS type array A variety of military vehicle types provided acoustic sources Samples were collected at 124Hz for the uncalibrated time series Phase/gain calibration for 5 25Hz and 5% overlap sliding Hamming window produced 248 samples per observation period for the calibrated data The distributed array data: Fifteen wireless Crossbow Telos R sensor nodes Two types of mid-sized moving trucks were the acoustic sources Samples were collected at 124Hz; only 876 samples per observation period No calibration; additional error due to sensor position uncertainty (up to.2m) due to GPS measurement error (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

13 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms DOA Estimation Results - Baseline Run Arithmetic Capon Geometric Capon Harmonic Capon STCM Geometric MUSIC WSF The markers,, and or, correspond to the DOAs obtained from the first, second, and third peaks of the spectrum. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

14 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms DOA Estimation Results - Baseline Run Arithmetic Capon Geometric Capon Harmonic Capon STCM Geometric MUSIC WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

15 Wideband DOA Estimation Error Statistics for Run 4 Review of Wideband DOA Estimation Algorithms Distribution of Error No. Errors in Normalized Bins Geometric Capon Arithmetic Capon Harmonic Capon Geometric MUSIC STCM WSF Error (degrees) Geo. Capon Ari. Capon STCM Har. Capon Geo. MUSIC WSF µ e σ 2 e (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

16 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms DOA Estimation Results - Distributed Run Arithmetic Capon Geometric Capon Harmonic Capon STCM Geometric MUSIC WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

17 Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms DOA Estimation Results - Distributed Run DOA Estimates using geometric Capon for Run 2 with the failed node (node 2) removed. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

18 Outline General Source Error Models 1 Introduction Outline Previous Work Research Objectives 2 Wideband DOA Estimation Signal Model Review of Wideband DOA Estimation Algorithms 3 General Source Error Models Non-Ideal Source Models General Source Error Coherence 4 Robust Wideband DOA Estimation Methods 5 Conclusions and Future Work Conclusions and Summary Suggestions for Future Work (5 min) (1 min) (1 min) (1 min) (5 min) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

19 General Source Error Models The Need for a General Error Model The reviewed models have been reduced to an understanding of the type of error coherence the source has and how this affects the covariance matrix. This is important because It is useful to have a general model for different errors which you can tailor to the type error present in a particular scenario The basis for developing a particular robust algorithm is expressly dependent on the coherence of the error or mismatch assumed present in the data The error coherence distinguishes the rank of the source in the covariance matrix, and therefore determines how to develop an algorithm to match to this source. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

20 General Source Error Models Error Models in terms of Source Coherence Coherence types Complete spatial coherence. This response models sources which are spatially coherent and temporally persistent within the observation period. Sensor position error, phase/gain or other array miscalibration Near-field effects and local scattering multipath effects Complete spatial incoherence. For this case there is no spatial coherence between source samples during the observation period. Uncorrelated reflections of a source off tropospheric scatterers (radiowave source) Partial spatial incoherence. The spatial coherence of the source in this scenario persists infrequently throughout the observation period. Multipath for non-local scattering When the array geometry is flexible and alters within the observation period (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 2 / 73

21 General Source Error Models Array and Calibration Error Model Non-Ideal Source Models This type of error includes: Sensor position error, array miscalibration, quantization errors, gain/phase errors, other errors without structure. The array signal model for this type of error can be expressed as the nominal response, A(f j,θ), plus an error matrix, Ã(f j,θ). The perturbed sample covariance can be written as R xx (f j ) = (A(f j,θ)+ã(f j,θ))r s (f j )(A(f j,θ) + Ã(f j,θ)) H + R n (f j ). (7) The structure of the array manifold has changed, but the perturbed signal covariance is still rank-one. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

22 Multipath Model General Source Error Models Non-Ideal Source Models This multipath model considers local scattering. The array response is the sum of multiple coherent planewaves arriving at nearby angles. Thus the multipath spatial response for the i th source is N i v i (f j, φ i ) = α ik (f j )a(f j, φ i + φ ik ). (8) k=1 The local scattering enables a first order derivative approximation and results in the following spatial covariance R xx (f j ) = [A(f j, φ) + D(f j, φ)]γ(f j )R s (f j )Γ H (f j )[A(f j, φ) + D(f j, φ)] + R n (f j ). (9) where Γ(f j ) is a diagonal fading matrix and D(f j, φ) is the first derivative of the array manifold (by vector). As is evident, this is still a rank-one matrix. This would be similar for near-field effects. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

23 General Source Error Models Non-Ideal Source Models Model for Coherent and Incoherent Sources Consider the noise-free covariance matrix for the i th source (uncorrelated from other sources) R i (f j, ψ) = a(f j, φ)p i (f j, φ, φ ; ψ i )a H (f j, φ )dφdφ (1) φ Φ φ Φ where, for the i th source, ψ i spreading parameter vector p i (f j, φ, φ ; ψ i ) = E[s i (f j, φ; ψ i )si (f j, φ ; ψ i )] is the angular auto-correlation The coherent source signal density can be written as s i (f j, φ; ψ i ) = γ i (f j )g(f j, φ; ψ i ) (11) which results in the angular auto correlation function (ACF) p(f j, φ, φ ; ψ) = η(f j, )g(f j, φ; ψ)g (f j, φ ; ψ). (12) The incoherent source angular ACF is p(f j, φ, φ ; ψ) = p(f j, φ; ψ)δ(φ φ ). (13) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

24 General Source Error Models General Source Error Coherence Conclusions on Source Error Coherence Cohrence types Complete spatial coherence. This response models sources which are spatially coherent that persist temporally within the observation period. Rank one source covariance of unknown structure. This corresponds to the type of error coherence in the data sets of this study. Complete spatial incoherence. For this case there is no coherence between source samples during the observation period. Full rank source covariance. Partial spatial incoherence. The coherence in this scenario persists infrequently throughout the observation period. Multi-rank source covariance, but typically not full rank. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

25 Outline Robust Wideband DOA Estimation Methods 1 Introduction Outline Previous Work Research Objectives 2 Wideband DOA Estimation Signal Model Review of Wideband DOA Estimation Algorithms 3 General Source Error Models Non-Ideal Source Models General Source Error Coherence 4 Robust Wideband DOA Estimation Methods 5 Conclusions and Future Work Conclusions and Summary Suggestions for Future Work (5 min) (1 min) (1 min) (1 min) (5 min) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

26 Robust Wideband DOA Estimation Methods Review of Wideband Robust Capon Algorithm The robust Capon is designed to be robust against rank-one errors in the steering vector, i.e. sensor position error and near-field effects. It belongs to the class of diagonal loading approaches for robust estimation and is introduced by Li and Stoica [21]. It operates based on an ellipsoidal uncertainty constraint for the steering vector and is formulated as min a a H (f j, θ)r 1 xx (f j )a(f j, θ) s.t. a(f j, θ) ā(f j, θ) 2 ǫ. (14) The optimal beamformer weight vector is w(f j, θ) = ā(f j, θ) U(f j )(I + λ(f j )Σ(f j )) 1 U H (f j )ā(f j, θ). (15) The wideband robust Capon power spectrum becomes P GRobust (θ) = J j=1 1 ā H (f j,θ)u(f j )Σ(f j )[λ 2 (f j )+2λ 1 (f j )Σ(f j )+Σ 2 (f j )] 1 U H (f j )ā(f j,θ). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

27 Robust Wideband DOA Estimation Methods Review of Wideband Beamspace Capon The beamspace method is a preprocessing method that performs spatial filtering by focusing on a region of interest [6]. Initially was implemented to counteract the effects of low SNR sources and wind noise. The L M beamspace matrix B bs is a matrix in C L which projects the input from the element space to the beamspace, in C M, where L M. A general form of a non-orthogonalized beamspace matrix is B no (f j,θ) = [ b(f j,φ P + θ) b(f j,φ + θ) b(f j,φ P + θ) ], (16) where for an arbitrary array geometry and c the speed of sound e j2πf j/c(α cos(φ p)+β sin(φ p)) e j2πf j/c(α 1 cos(φ p)+β 1 sin(φ p)) b(f j,φ p ) =.. (17) e j2πf j/c(α L 1 cos(φ p)+β L 1 sin(φ p)) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

28 Robust Wideband DOA Estimation Methods Review of Wideband Beamspace Capon To ensure orthogonality of the beamspace matrix, we perform B bs = B no [B H no B no] 1 2 (18) Clearly, this whitening yields orthogonality in B H bs B bs = I M. The beamspace Capon method results in the weight vector w bs (f j, θ) = R 1 vv (f j, θ)a bs (f j, θ) a H bs (f j, θ)r 1 vv (f j, θ)a bs (f j, θ), (19) where R vv (f j, θ) = B H bs (f j, θ)r xx (f j )B bs (f j, θ) and a bs (f j, θ) = B H bs (f j, θ)a(f j, θ) are the transformed sample covariance and steering vector, respectively. The wideband geometric mean beamspace Capon power spectrum output then becomes P Gbs (θ) = J j=1 1 a H bs (f j, θ)r 1 vv (f j, θ)a bs (f j, θ). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

29 Robust Wideband DOA Estimation Methods DOA Estimation Results - Baseline Run Beamspace Geometric Capon Standard Geometric Capon Geometric MUSIC WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

30 Robust Wideband DOA Estimation Methods DOA Estimation Results - Baseline Run Robust Geometric Capon Standard Geometric Capon Robust Geometric Capon detail Standard Geometric Capon detail (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 3 / 73

31 Robust Wideband DOA Estimation Methods DOA Estimation Results - Baseline Run Beamspace Geometric Capon Standard Geometric Capon Geometric MUSIC WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

32 Robust Wideband DOA Estimation Methods Error Statistics for Run 4 Distribution of Error.5 Geometric Capon Geometric MUSIC WSF Beamspace Geo. Capon No. Errors in Normalized Bins Error (degrees) Geo. Capon Bmspc Capon Geo. MUSIC WSF µ e σ 2 e (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, 2 Sep / 73

33 Robust Wideband DOA Estimation Methods DOA Estimation Results - Baseline Run Robust Geometric Capon Standard Geometric Capon (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

34 Robust Wideband DOA Estimation Methods Error Statistics for Run 4.3 Distribution of Error.25 Geometric Capon Robust Capon No. Errors in Normalized Bins Error (degrees) Geo. Capon Robust Capon µ e σ 2 e (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

35 Robust Wideband DOA Estimation Methods DOA Estimation Results - Distributed Run Beamspace Geometric Capon Standard Geometric Capon Robust Geometric Capon WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

36 Outline Conclusions and Future Work 1 Introduction Outline Previous Work Research Objectives 2 Wideband DOA Estimation Signal Model Review of Wideband DOA Estimation Algorithms 3 General Source Error Models Non-Ideal Source Models General Source Error Coherence 4 Robust Wideband DOA Estimation Methods 5 Conclusions and Future Work Conclusions and Summary Suggestions for Future Work (5 min) (1 min) (1 min) (1 min) (5 min) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

37 Conclusions and Future Work Conclusions and Summary Conclusions and Summary The tested benchmark algorithms provided good results however, with some drawbacks and deficiencies. The error coherence source model enabled the appropriate choice of rank-one robust DOA estimation algorithms. The wideband robust Capon provided robust DOA estimates in the presence of sensor location uncertainties and near-field effects. Both the wideband beamspace Capon and the robust Capon provided good DOA estimation performance even when there was data loss or corruption. Additionally, the beamspace Capon method provided better performance in wind noise and in locating far range sources. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

38 Conclusions and Future Work Suggestions for Future Work Suggestions for Future Work Algorithm for detecting level of coherence and performing DOA estimation with the appropriate matching to whichever error coherence is present. Expand the application of these algorithms in this research to other wideband data. Explore better ways of finding algorithm parameters, namely, the number of beams and estimated error in the beamspace and robust Capon methods, respectively. Using DOA estimation algorithms for providing input to data fusion and tracking methods. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

39 Conclusions and Future Work Suggestions for Future Work Questions and Thanks Questions? My most sincere an deep thanks to Dr. Azimi for his ideas, editing my writing, and encouragement through this long process. Gratitude is also due Dr. s Scharf and Breidt for their willingness to be members in my committee. Also to the members of the signal and image processing laboratory: Bryan, Gordon, Jered, Amanda, Jaime, Makoto, Neil, Derek, Mike and Tim; you have made grad school an interesting and great learning experience. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

40 References I Appendix Bibliography T. Pham and B. M. Sadler, Wideband array processing algorithms for acoustic tracking of ground vehicles, tech. rep., Army Research Laboratories, Adelphi, MD, M. R. Azimi-Sadjadi, A. Pezeshki, L. Scharf, and M. Hohil, Wideband DOA estimation algorithms for multiple target detection and tracking using unattended acoustic sensors, in Proc. of SPIE 4 Defense and Security Symposium - Unattended Ground Sensors VI, vol. 5417, pp. 1 11, Apr. 24. M. R. Azimi-Sadjadi, Y. Jiang, and G. Wichern, Properties of randomly distributed sparse acoustic sensors for ground vehicle tracking and localization, in Proc. of SPIE 6 Defense and Security Symposium, vol. 621, Apr. 26. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 4 / 73

41 References II Appendix Bibliography T. R. Damarla, J. Chang, and A. Rotolo, Tracking a convoy of multiple targets using acoustic sensor data, in Proc. of SPIE 3 Defense and Security Symposium - Acquisition, Tracking, and Pointing XVII, vol. 582, pp , Aug. 23. M. E. Hohil, J. R. Heberley, J. Chang, and A. Rotolo, Vehicle counting and classification algorithms for unattended ground sensors, in Proc. of SPIE 3 Defense and Security Symposium - Unattended Ground Sensor Technologies and Applications V, pp , Sept. 23. H. L. V. Trees, Optimum Array Processing. Wiley Interscience, 22. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

42 References III Appendix Bibliography R. O. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. on Antennas and Propagation, vol. 34, pp , Mar M. Viberg and B. Ottersten, Sensor array processing based on subspace fitting, IEEE Trans. on Signal Processing, vol. 39, pp , May J. Krolik and D. Swingler, Multiple broad-band source location using steered covariance matrices, IEEE Trans. on Aoustics, Speech, and Signal Processing, vol. 37, pp , Oct (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

43 References IV Appendix Bibliography J. Krolik, Focused wideband Array Processing for Spatial Spectral Estimation, ch. 6. in Advances in Spectrum Analysis and Array Processing, Vol. II, Prentice-Hall, T. Pham and M. Fong, Real-time implementation of MUSIC for wideband acoustic detection and tracking, in Proc. of SPIE AeroSense 97 - Automatic Target Recognition VII, vol. 369, pp , Apr H. Wang and M. Kaveh, Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources, IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. 33, pp , Aug (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

44 References V Appendix Bibliography E. D. D. Claudio and R. Parisi, WAVES: Weighted average of signal subspaces for robust wideband direction finding, IEEE Trans. on Signal Processing, vol. 49, pp , Oct. 21. A. L. Swindlehurst and M. Viberg, Bayesian approaches for robust array signal processing. research supported by NSF grant MIP , D. Asztély, Spatial models for narrowband signal estimation with antenna arrays, tech. lic. thesis, Royal Institute of Technology, Stockholm, Sweden, Nov D. Asztély, B. Ottersten, and A. L. Swindlehurst, Generalised array manifold model for wireless communication channels with local scattering, IEE Proc.-Radar, Sonar Navig., vol. 145, pp , Feb (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

45 References VI Appendix Bibliography S. Valaee, B. Champagne, and P. Kabal, Parametric localization of distributed sources, IEEE Trans. on Signal Processing, vol. 43, pp , Sept A. Pezeshki, B. D. V. Veen, L. L. Scharf, H. Cox, and M. Lundberg, Eigenvalue beamforming using a multi-rank MVDR beamformer and subspace selection, to appear in IEEE Trans. on Signal Processing, submitted Sept. 26. L. L. Scharf, A. Pezeshki, and M. Lundberg, Multi-rank adaptive beamforming, in Proc. IEEE 13th Workshop Statistical Signal Processing, (Bordeaux, France), July 25. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

46 References VII Appendix Bibliography Y. Meng, P. Stoica, and K. M. Wong, Estimation of the directions of arrival of spatially dispersed signals in array processing, in Proceedings of IEE Conf. on Radar, Sonar, and Navig., vol. 143, Feb J. Li, P. Stoica, and Z. Wang, On robust capon beamforming and diagonal loading, IEEE Trans. on Signal Processing, vol. 51, pp , July 23. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

47 Appendix Back Up Slides Other Wideband Algorithm Power Spectrums The wideband arithmetic mean Capon power spectrum is P A (θ) = J p(f j, θ) = j=1 J j=1 1 a H (f j, θ)r 1 xx (f j )a(f j, θ) (2) The wideband harmonic mean Capon power spectrum is P H (θ) = 1 J j=1 wh (f j, θ)r xx (f j )w(f j, θ) (21) The WSF power spectrum is P WSF (θ) = 1 tr{ J j=1 P A 1 (f j, θ)u s (f j )W(f j )W H (f j )U H s (f j )P A 1 (f j, θ)}, (22) where P A1 (f j, θ) = a(f j, θ)a (f j, θ) is the rank-one projection matrix. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

48 Appendix Back Up Slides Derivation of Arithmetic Mean Wideband Capon I The wideband arithmetic Capon problem can be cast as min w(fj,θ) P A (θ) = J j=1 K k=1 wh (f j, θ)x(f j, k)x H (f j, k)w(f j, θ) = J j=1 wh (f j, θ)r xx (f j )w(f j, θ) under the constraints (23) w H (f j, θ)a(f j, θ) = 1, j [1, J] (24) It is assumed that w(f j, θ) is independent of k within the observation period T. This leads to a constrained minimization problem min w(f j,θ) J w H (f j, θ)r xx (f j )w(f j, θ) + j=1 j=1 J λ(f j )(w H (f j, θ)a(f j, θ) 1) (25) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

49 Appendix Back Up Slides Derivation of Arithmetic Mean Wideband Capon II where λ(f j ) s are frequency dependent Lagrange multipliers. This minimization problem leads the optimal beamformer, but here this optimization produces J narrowband rank-one Capon beamformers for w(f j, θ) s, R 1 xx (f j )a(f j, θ) w(f j, θ) = a H (f j, θ)r 1, j [1, J] (26) xx (f j )a(f j, θ) and the wideband Capon spectrum, P A (θ) = J p(f j, θ) = j=1 J j=1 1 a H (f j, θ)r 1 xx (f j )a(f j, θ) (27) (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

50 Appendix Back Up Slides MUltiple SIgnal Classification (MUSIC) The MUSIC algorithm is a type of subspace-based algorithm which uses the decomposition of the orthogonal (or unitary for this complex case) signal and noise subspaces some eigendecomposition R xx (f j ) = U s (f j )Σ s (f j )U H s (f j ) + U n (f j )Σ n (f j )U H n (f j ). (28) Thus, the squared Euclidean distance between the steering vector a(f j, θ) and the noise subspace, 2 = a H (f j, θ)u n (f j )U H n (f j )a(f j, θ), (29) will be minimum in the direction of a source. The incoherent wideband geometric mean MUSIC algorithm is formulated by the inverse of this distance as P MUSICG (θ) = J p MUSIC (f j, θ) = j=1 J j=1 1 a H (f j, θ)u n (f j )U H n (f j )a(f j, θ). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 5 / 73

51 Appendix Back Up Slides Weighted Subspace Fitting (WSF) Narrowband subspace fitting attempts to fit the data to a search array manifold as the minimization problem the signal estimate is min θ x(f j, k) A(f j, θ)s(f j, k) 2 F. (3) ŝ(f j, k) = A (f j, θ)x(f j, k), (31) The error in this LS estimation of the signal estimate is given by where e(f j, k) = x(f j, k) A(f j, θ)ŝ(f j, k) = P A (f j, θ)x(f j, k) (32) P A (f j, θ) = I P A (f j, θ) = I A(f j, θ)[a H (f j, θ)a(f j, θ)] 1 A H (f j, θ) (33) is the orthogonal projection complement operator onto the subspace spanned by the columns of the array response matrix A(f j, θ). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

52 Appendix Back Up Slides Weighted Subspace Fitting (WSF) The algorithm is a multi-dimensional search through θ for the minimum of the squared error between this estimate and the actual signal, J θ = arg min tr{ P A (f j,θ)r xx (f j )P A (f j,θ)}. (34) θ j=1 The decomposition of R xx (f j ) allows it to be written approx. as R xx (f j ) U s (f j )Σ s (f j )U H s (f j ). (35) The algorithm generalizes this decomposition as R xx (f j ) U s (f j )W(f j )W H (f j )U H s (f j ), (36) where the weighting matrix, W(f j ) = (Σ s (f j ) σn 2I)Σ_s 1/2 (f j ), is commonly used. The weighted subspace replaces R xx (f j ) and the multi-dimensional search becomes θ = arg min θ tr{ J P A (f j,θ)u s (f j )W(f j )W H (f j )U H s (f j)p A (f j,θ)}. j=1 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

53 Appendix Back Up Slides STeered Covariance Matrix (STCM) Method The desired effect of the STCM algorithm is to generate a single coherent signal subspace by focusing to a reference frequency those subspaces at other frequencies. Focusing matrices T(f j,θ), j = 1, 2,..., J, exist so that T(f j,θ)a(f j,θ) = A(f,θ), s.t. T(f j,θ)t H (f j,θ) = I. (37) The steered or focused spatial covariance matrix may therefore be defined as R(θ) = J R(f j,θ) = j=1 J T(f j,θ)r xx (f j )T H (f j,θ) (38) j=1 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

54 Appendix Back Up Slides STeered Covariance Matrix (STCM) Method Using the array signal model, we can rewrite R(θ) as R(θ) = J A(f,θ)R s (f j )A H (f,θ) + R nθ. (39) j=1 R nθ is the focused noise covariance. The STCM focusing method can be applied in tandem with any other narrowband DOA estimation algorithm, as its results in a single covariance matrix. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

55 Array Geometries Appendix Back Up Slides North North South (meters) East West (meters) Baseline array Distributed array (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

56 Appendix Back Up Slides Examples of Vehicle Movement and Acoustic Data: Baseline Array 1 5 N North South (m) end:6 end:5 5 end:2 end:3 end:4 1 node 1 node 3 node 2 start:1 start:2 start:4 start:3 start:5 start:6 15 end: East West (m) Movement path baseline Run 1 Spectrogram of mic, baseline Run 1 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

57 Appendix Back Up Slides Examples of Vehicle Movement and Acoustic Data: Distributed Array North South (m) 6 Array Site N East West (m) Data collection site Spectrogram of mic, distributed Run 2 (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

58 Bad Time-series Data Appendix Back Up Slides Amplitude 25 2 Amplitude Time (sec) Working node Time (sec) Bad mic amplifier node Amplitude 25 2 Amplitude Time (sec) Missing data node Time (sec) Detail of missing data node (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

59 Appendix Back Up Slides Bearing Response Analysis - Baseline Array BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Arithmetic Capon Bearing Response, 5 25 Hz BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Geometric Capon Bearing Response, 5 25 Hz BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Harmonic Capon Bearing Response, 5 25 Hz Arithmetic Capon Geometric Capon Harmonic Capon BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o STCM Bearing Response, 5 25 Hz BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Geometric MUSIC Bearing Response, 5 25 Hz BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o WSF Bearing Response, 5 25 Hz STCM Geometric MUSIC WSF 5-element circular array with two sources at separations of 2, 23, and 26. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

60 Appendix Back Up Slides New Alg. Bearing Response Analysis - Baseline Array BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Beamspace Capon Bearing Response, 5 25 Hz Beamspace Geometric Capon BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Geometric Capon Bearing Response, 5 25 Hz Standard Geometric Capon 1.5 BR for sources at 17 o and 19 o BR for sources at 169 o and 192 o BR for sources at 167 o and 193 o Robust Capon Bearing Response, 5 25 Hz Robust Geometric Capon 5-element circular array with two sources at separations of 2, 23, and 26. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 6 / 73

61 Appendix Back Up Slides Bearing Response Analysis - Random Array BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Arithmetic Capon Bearing Response, 5 25 Hz BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Geometric Capon Bearing Response, 5 25 Hz BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Harmonic Capon Bearing Response, 5 25 Hz Arithmetic Capon Geometric Capon Harmonic Capon BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o STCM Bearing Response, 5 25 Hz BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Geometric MUSIC Bearing Response, 5 25 Hz BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o WSF Bearing Response, 5 25 Hz STCM Geometric MUSIC WSF 15-element randomly distributed array with two sources at separations of 1, 3, and 4. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

62 Appendix Back Up Slides New Alg. Bearing Response Analysis - Random Array BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Beamspace Capon Bearing Response, 5 25 Hz Beamspace Geometric Capon BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Geometric Capon Bearing Response, 5 25 Hz Standard Geometric Capon BR for sources at 18 o and 181 o BR for sources at 179 o and 182 o BR for sources at 178 o and 183 o Robust Capon Bearing Response, 5 25 Hz Robust Geometric Capon 15-element randomly distributed array with two sources at separations of 1, 3, and 4. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

63 Appendix Spatially Coherent Signals Back Up Slides Coherent sources includes slow fading multipath, near-field effects, and even the arrays errors discussed previously. In this case the spatial dependent signal density is given by which results in the angular auto correlation s i (f j, φ; ψ i ) = γ i (f j )g(f j, φ; ψ i ) (4) p(f j, φ, φ ; ψ) = η(f j, )g(f j, φ; ψ)g (f j, φ ; ψ) (41) with η(f j ) = E{γ(f j )γ (f j )}. The integral is separable as R i (f j, ψ i ) = a(f j, φ)p i (f j, φ, φ ; ψ i )a H (f j, φ )dφdφ φ Φ φ Φ = η i a(f j, φ)g(f j, φ; ψ i )dφ g (f j, φ ; ψ i )a H (f j, φ )dφ. φ Φ φ Φ and demonstrates that the coherent signal is rank-one. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

64 Appendix Illustration of Coherent Signal Back Up Slides g(φ; ψ i ) δ φ φ γ g(φ ; ψ i ) Array Illustration of distributed source with spatial coherence. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

65 Appendix Spatially Incoherent Sources Back Up Slides An incoherent signal exists when the signal rays arriving from different directions can be assumed uncorrelated. The angular auto-correlation is written as p(f j, φ, φ ; ψ) = p(f j, φ; ψ)δ(φ φ ) (42) The noise-free array correlation matrix for this signal is R i (f j, ψ) = a(f j, φ)p i (f j, φ; ψ i )a H (f j, φ)dφ. (43) φ Φ This incoherent source covariance R i (f j, ψ i ) is always full rank. Using an approximation to the full rank representation is practical for many distributions (and spreading widths). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

66 Appendix Partially Incoherent Sources Back Up Slides As a simple example consider the uniform distribution on [ ε 2, ε 2 ] which results in the covariance R s (ψ i ) = p i (φ;ψ i )a(φ)a H (φ)dφ = 1 ε φ Φ ε/2 ε/2 a(φ)a H (φ)dφ σ 2 s U sλu H s, (44) where rank(r s ) ε 2π L = p. The matrix U s is the L p basis for the p dimensional subspace U s and Λ I, i.e. signal power is even across all distributed components. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

67 Appendix Back Up Slides Partially Incoherent Sources So the generation of a input sample, x, for an partially incoherent noise-free source looks like x = U s bs E[bb H ] = Λ s E[ss ] = σ 2 s R s = E[xx H ] = σ 2 su s Λ s U H s, (45) where every sample within the observation period is formed from a random linear combination of the p signal subspace basis vectors. The coherent source is a similar formulation, except that the random combining vector b is fixed for the observation period. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

68 Appendix Back Up Slides DOA Estimation Results - Baseline Run Beamspace Geometric Capon Standard Geometric Capon Geometric MUSIC DOA estimation is the presence of high wind noise WSF (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

69 Appendix Back Up Slides DOA Estimation Results - Distributed Run Arithmetic Capon Geometric Capon Harmonic Capon STCM Geometric MUSIC WSF DOA estimation for a single extremely near-field source. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

70 Appendix Back Up Slides DOA Estimation Results - Distributed Run Beamspace Geometric Capon Standard Geometric Capon Robust Geometric Capon WSF DOA estimation for a single extremely near-field source. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 27 7 / 73

71 Appendix Back Up Slides DOA Estimation Results - Distributed Run DOA Estimates using geometric robust Capon for Run 3 with estimated error of 1 (instead of.7). (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

72 Appendix Back Up Slides Additional DOA Estimation Results - Distributed Run Beamspace Geometric Capon Standard Geometric Capon Robust Geometric Capon WSF DOA Estimates on two source distributed array run 4. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73

73 Appendix Back Up Slides Additional DOA Estimation Results - Distributed Run Beamspace Geometric Capon Standard Geometric Capon Robust Geometric Capon WSF DOA Estimates for case with two sensor nodes that have missing data. (Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep / 73