SS4 - Spatial array processing

Transcription

1 Doctoral Program on Electrical Engineering and Communications Sérgio M. Jesus Universidade do Algarve, PT Faro, Portugal 24 February 2010 thanks to Paulo Santos for figures and examples.

2 Outline of SS4 Optimal detector of one signal in white noise (1) the generalized matched-filter (2) plane waves vs. full-field Plane wave model: beamforming (1) Delay-sum beamformer (2) Horizontal line array (3) Examples (4) Frequency beamformer 3D spatial array processing (1) Horizontal line array: left/right ambiguity (2) Vertical line array (3) Horizontal/vertical planar array (4) Cubic (volumetric) array Vector sensor array (1) simulation (2) real data during MakaiEx experiment (Hawai, USA)

3 Optimal detector of one signal in white noise Data model, in space (K sensors) and at time snapshot n { p(θo ) = H(θ o )s y n (θ o ) = p(θ o ) + w n with usual assumptions, best detector is the GMF (with K sensors, N time snapshots) T [y(θ)] = N 1 n=0 y t n(θ o )p(θ) 2 thus an estimate ˆθ o of θ o is ˆθ o = arg{max θ p t (θ) ˆR yy p(θ)} ˆR yy is the sample covariance matrix. Signal replicas p(θ) depend on point-to-point / time-to-time impulse response matrix H(θ).

4 Signal model(s) Plane wave model: one source, frequency ω 0, direction κ 0 p(t, θ) = s 0 [1,..., e j(ω 0t κ 0 r k ),..., e j(ω 0t κ 0 r K ) ] t with K sensors at positions { r 0, r 1,..., r K }. Full field model: normal modes, parabolic equation, ray tracing,... p k (t, θ) = j 2π M m=1 Ψ m (θ s )Ψ m (θ k ) e j(k mθ r π/4) α m θ r ) e jωt dω km θ r

5 Plane wave model: beamforming Assuming the number of sources unknown, try all θ and look for a maxima, b(t, θ) = y t (t, θ o )p (t, θ) 2 = K y k (t, θ o )e j(ω 0t κ r k ) 2 where s 0 was assumed = 1. For a linear equispaced array κ r k = (kd sin θ)/c, d=array spacing, c = sound speed. k=1

6 Delay-sum beamformer: time domain So, assuming a constant sound speed, the noise free beam pattern b(t, θ) = [ ] Ld sin 2c ω 0(sin θ 0 sin θ) [ ] d sin 2c ω 2 0(sin θ 0 sin θ) In presence of noise, the delay-sum beamformer output is averaged over a certain snapshot span N. Optimum for detecting a single source in temporal and spatially white noise. ˆθ o = arg{max θ N 1 n=0 b n (t, θ)}

7 Horizontal line array (1) f 0 =500 Hz, L=10 sensors, d =1 m, c = 1500 m/s, θ o =40

8 Horizontal line array (2) f 0 =500 Hz, L=10,30,50 sensors, d =1 m, c = 1500 m/s, θ o =0, 40, 90

9 Horizontal line array (3) f 0 =0.5, 3, 6 khz, L=10 sensors, d =1 m, c = 1500 m/s, θ o =40

10 Frequency beamformer (1) In frequency domain time delay phase shift. B(f, θ) = K k=1 j2πfkd sin θ/c Y k (f, θ o )e efficiently implemented by an FFT from space to wavenumber by making fdm sin θ m =. Since 1 sin θ 1, the number of actual beam numbers is c limited for each frequency Md λ m Md λ where m should take integer values.

11 Frequency beamformer (2) Horizontal line array: efficient implementation with a double FFT

12 3D spatial array processing κ 0 = ω 0 c [ sin θ 0 cos θ 0 ; cos θ 0 cos φ 0 ; sin φ 0 ]

13 Horizontal line array: ambiguity

14 Horizontal deformed array: ambiguity less 1 SACLANTCEN Towed array: 160 m, 40 6 compasses, 6 tiltmeters, 6 accelerometers, 2 pressures gauges [MAST 2 sea trial, Strait of Sicily, March 1992] 1 from P. Felisberto and S.M. Jesus, Towed array beamforming during ship s maneuvering, IEE Proc. Radar, Sonar and Navigation, vol. 143, No. 3, pp , 1996.

15 Vertical line array: ambiguity

16 Horizontal planar array: ambiguity

17 Vertical planar array: ambiguity

18 Cubic (volumetric) array ambiguity

19 Vector sensors: omni versus 3D (1) 1 omnidirectional hydrophone and 3 particle velocity sensors (x y z).

20 Vector sensors: omni versus 3D (2)

21 Vector Sensor Array during Makai Ex (Hawaii, 2005)2 2 P.J. Santos, P. Felisberto and P. Hursky, Source localization with a Vector Sensor Array during the Makai Experiment, in Proc. Underwater Acoustic Measurements, pp , Heraklion, Greece, June 2007.

22 Bibliography 1. P. Felisberto and S.M. Jesus, Towed array beamforming during ship s maneuvering, IEE Proc. Radar, Sonar and Navigation, vol. 143, No. 3, pp , P.J. Santos, P. Felisberto and P. Hursky, Source localization with a Vector Sensor Array during the Makai Experiment, in Proc. Underwater Acoustic Measurements, pp , Heraklion, Greece, June S.M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Vol.2, Prentice-Hall, New Jersey (USA), S.M. Jesus, C. Martins and F. Zabel, Maritime Rapid Environmental Assessment / Blue Planet 07: Acoustic Oceanographic Buoys Data Report, Rep. 03/07, SiPLAB Report, University of Algarve, August F.B. Jensen, W.A. Kuperman, M.B. Porter and H. Schmidt, Computational Ocean Acoustics, AIP, New York (USA), 1993.