Maximum Likelihood Direction of Arrival Estimation ...

Maximum Likelihood Direction of Arrival Estimation Using Spherical Harmonics Hugh L. Kennedy BAE Systems Australia [email protected]

Workshop on Progress in Radar Research 13 - 14 May 2009

© 2009 BAE Systems Australia Limited. All rights reserved.

Problem overview – COMINT/ELINT ESM application. – Passive array consisting of multiple omni-directional sensing elements (antennas) arranged in an arbitrary geometric configuration. – Intercepting CW signals of short wavelength (relative to the array dimensions). – Want to estimate the Direction of Arrival (DOA) of a single radiation source. – Typically solved using Phase Interferometry (PI) [1].


Mathematical definition of Spherical Harmonics (SH) [2] Ylm (θ , φ ) = with

l≥0

(2l + 1) (l − m )!e imφ P m (cos θ ) l 4π (l + m )! and

−l ≤ m ≤ + l

z

where

Associated Legendre function, of the first kind.

θ

y

φ x


Visual representation of spherical harmonics z

x

y

l = 0, m = 0


z

Visual representation of spherical harmonics (cont.) z

x

y

z

l = 1, m = -1

x

y

l = 1, m = 0

x

y

l = 1, m = +1


z

Visual representation of spherical harmonics (cont.) z

x

z

y

l = 2, m = -2 x

y

z

x

y

l = 2, m = -1

z

l = 2, m = 0 y

x

x

y

l = 2, m = +1 l = 2, m = +2


Visual representation of spherical harmonics (etc …) z

z

x

y

z

l = 3, m = -3 x

y

z

l = 3, m = -2

z x

y

l = 3, m = -1

y

x

z

x

y

l = 3, m = 0 z

l = 3, m = +1

x

y

l = 3, m = +2

x

y

l = 3, m = +3


Properties of spherical harmonics [2] – Form an Orthonormal set. – The Product of two spherical harmonics (with l1 and l2) yields a finite expansion of spherical harmonics (with a maximum l parameter of l1+l2). Use the Wigner 3jm coefficients. – Rotation of a spherical harmonic yields a finite sum of spherical harmonics (with the maximum l parameter unchanged). Use the Wigner D-function. – Convolution of a linear combination of spherical harmonics with another, yields a linear combination of spherical harmonics.


Example applications – – – – – –

Physics and chemistry [2&3]. Computer graphics [4]. Acoustics [5]. Antenna modelling [6&7]. Beamforming [8-10]. Tracking …


Bayesian estimation Posterior pdf.

Likelihood fcn.

p (x z ) = State vector being estimated.

Prior pdf.

p (z x ) p(x )

∫

p (z x ) p(x )dx

Measurement.

xˆ MAP = arg max p (z x ) p(x ) x © 2009 BAE Systems Australia Limited. All rights reserved.

Gaussians in Cartesian Bayesian estimation (the Kalman filter)

20

20

15

15

vely (km/Tdelta)

pos y (km)

2D measurement space; 4D state space.

10 5 0

0

5

10 15 pos x (km)

Use of Gaussians greatly simplifies the Cartesian problem.

20

25

10 5 0

0

5

prior pdf predicted pdf likelihood function posterior pdf

10 15 20 velx (km/Tdelta)

25

Similar benefits in the polar problem when SH are used …


Spherical harmonics in polar Bayesian estimation – – – – – –

Consider a simple hypothetical scenario … Single stationary emitter in the far field. Unobstructed stationary array in free space. 1 central sensor, 4 peripheral sensors, 4 baselines of equal length. 2-channel fast A/D converter. 5 digitized samples are collected in the time it takes for light to travel along the axis of a baseline. – The measured group delay error is assumed to be Gaussian distributed with a mean of zero and a standard deviation of half the sample period. – Accuracy depends on the signal wavelength. – E.g. 1: Wavelength of the intercepted signal is greater than twice the baseline length (λ = 2.5D). Measurements are conically ambiguous. – E.g. 2: Wavelength of the intercepted signal is less than twice the baseline length (λ = 1.5D). Measurements may have multiple conical ambiguities. © 2009 BAE Systems Australia Limited. All rights reserved.

Spherical harmonics in polar Bayesian estimation e.g. 1 Baseline axis (dashed green). Sensor location (green asterisk). 1.5

z (m)

1

z

0.5

y

0 x -0.5

0

True DOA (dashed magenta).

1 2 x (m)

-0.5

0

0.5

1

y (m)


Spherical harmonics in polar Bayesian estimation e.g. 1 Measurement on 1st baseline (on +ve x axis).

0 tht, phi = 0 330

30

θ’ is angle relative to baseline axis

60

300

Likelihood function. 270 Gaussian in lag space. Conical in polar coordinates. LSQ fit SH, with m=0 (real) and up 240 to l=10, to 22 specified “control” points (blue dots). Multiply by complex conjugate to ensure +ve everywhere.

0.05

0.1

0.2

0.15

90

120

True θ’ (red circle). Meas. θ’ (green circle). 210

150 180


Spherical harmonics in polar Bayesian estimation e.g. 1 Prior pdf (red). Likelihood fcn (green). Posterior pdf (blue). All +ve and real.

z

Rotate likelihood function from baseline-centric coordinate system into array-centric coordinate system.

Spherically ambiguous (uniform) prior.

y

x

Posterior pdf equals likelihood function.

The DOA estimate is the maximum of the posterior pdf.


Spherical harmonics in polar Bayesian estimation e.g. 1 Measurement on 2nd baseline (on +ve x axis).

0 tht, phi = 0 330

30

60

300

0.05

0.1

0.2

0.15

270

90

240

120

210

150 180


Spherical harmonics in polar Bayesian estimation e.g. 1 z

Likelihood function is a “fuzzy” cone

y

x

Posterior pdf (the product of the red and green fuzzy cones) has two maxima – above and below the x-y plane.


Spherical harmonics in polar Bayesian estimation e.g. 1 Measurement on 3rd baseline (on +ve z axis).

0 tht, phi = 0 330

30

60

300

0.05

0.1

0.2

0.15

270

90

240

120

210

150 180



Incorporation of this measurement resolves the up/down ambiguity.

y

x


Spherical harmonics in polar Bayesian estimation e.g. 1 Measurement on 4th baseline (midway between +ve x, y and z axes).

0 tht, phi = 0 330

30

60

300

0.05

0.1

0.2

0.15

270

90

240

120

210

150 180



Incorporation of this redundant measurement “sharpens” and shifts the “beam” slightly.

y

x

Prune SH in all functions, with coefficients less than 0.001. 340 non-negligible SH basis functions, with a max l of 20, are required to represent this solution.


Spherical harmonics in polar Bayesian estimation e.g. 2

1.6 1.4 1.2

z (m)

1

z

0.8 0.6 0.4 0.2

y

0 -0.5

1.5

x 0

1 0.5

0.5

1

0 y (m)

x (m)


Spherical harmonics in polar Bayesian estimation e.g. 2 The likelihood function is composed of two fuzzy cones.

0 tht, phi = 0 330

30

300

60

0.15 0.1 0.05 270

90

120

240

210

150 180



y

x


Spherical harmonics in polar Bayesian estimation e.g. 2 0 tht, phi = 0 330

At this frequency, not all measurements are ambiguous.

30

60

300

0.05

0.1

0.2

0.15

270

90

240

120

210

150 180



Fat cone (green), intersects with concave dumbbells (red) to form butterfly wings (blue).

y

x



30

300

60

0.15 0.1 0.05 270

90

120

240

210

150 180



Four feasible DOAs in the prior pdf (red), reduced to two in the posterior pdf (blue).

y

x



30

60

300

0.05

0.1

0.2

0.15

270

90

240

120

210

150 180



y

Final disk-like measurement resolves the ambiguity.

x

Still have an ambiguous lobe of lesser probability density. © 2009 BAE Systems Australia Limited. All rights reserved.

Spherical harmonics in polar Bayesian estimation – Monte Carlo simulation results, RMSE (°). – SH compared with PI (in parentheses). Seed 1

2

3

4

5

6

7

8

9

2.5D

4.22 (5.02)

6.54 (5.88)

6.36 (5.32)

1.62 (1.21)

8.54 (7.87)

4.79 (6.11)

7.06 (7.21)

4.73 (3.32)

4.04a (4.01)

1.5D

4.13 (5.02)

6.58 (5.88)

6.24 (5.32)

153 (153)

8.5 (7.87)

4.82 (6.11)

7.07 (7.21)

4.67b (3.32)

4.36 (4.01)

λ

a e.g. 1 b e.g. 2

Neither method resolves the ambiguity in this case. Both may have been lucky in the other 1.5D cases (50/50 chance of choosing the correct maxima when two are present).


Conclusions – Use of spherical harmonics allows the direction-of-arrival problem to be solved using Bayesian methods. – This provides a framework for … – Refining the estimate over time. – Visualising uncertainty and ambiguity.

– Accuracy is similar to phase interferometry. – Greater complexity and execution time.


References – – – – – – – – – –

[1] L. Dinoi, A. Di Vito and G. Lubello, "Direction Finding of ground based emitters from airborne platforms," Radar Conference, 2008. RADAR '08. IEEE, pp.1-6, 26-30 May 2008. [2] D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, “Quantum Theory of Angular Momentum,” World Scientific Pub Co Inc, 1988. [3] H.L. Kennedy and Y. Zhao, “Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal-coordinate and zeta-function expansions,” International Journal of Quantum Chemistry, vol.71, no.1, pp.1-13, 1999. [4] R. Green, “Spherical harmonic lighting: The gritty details”, Game Developers' Conference Archive, 2003, (available at www.research.scea.com/gdc2003/spherical-harmonic-lighting.pdf). [5] T.D. Abhayapala, "Generalized framework for spherical microphone arrays: Spatial and frequency decomposition," Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on, pp.5268-5271, March 31 2008-April 4 2008. [6] R.J. Allard and D.H. Werner, "The model-based parameter estimation of antenna radiation patterns using windowed interpolation and spherical harmonics," Antennas and Propagation, IEEE Transactions on , vol.51, no.8, pp.1891-1906, Aug. 2003. [7] A.R. Runnalls, "Likelihood function for a simple cardioid sonobuoy," Radar, Sonar and Navigation, IEE Proceedings - , vol.153, no.5, pp.417-426, Oct. 2006. [8] R.A. Kennedy, T.D. Abhayapala and D.B. Ward, "Broadband nearfield beamforming using a radial beampattern transformation," Signal Processing, IEEE Transactions on , vol.46, no.8, pp.2147-2156, Aug 1998. [9] B. Rafaely and M. Kleider, "Spherical Microphone Array Beam Steering Using Wigner-D Weighting," Signal Processing Letters, IEEE , vol.15, pp.417-420, 2008. [10] J. Meyer and G. Elko, "Spherical harmonic modal beamforming for an augmented circular microphone array," Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on, pp.5280-5283, March 31 2008-April 4 2008.
