a system identification problem from lord rayleigh

Boston University--Harvard University--University of Illinois--University of Maryland

A System Identification Problem from Lord Rayleigh

P. S. Krishnaprasad University of Maryland, College Park Department of Electrical and Computer Engineering & Institute for Systems Research Center for Communicating Networked Control Systems ------------ New Directions & Applications in Control Theory Texas Tech University, Lubbock November 14-15, 2003


Recollections of Case Western Reserve University


Lord Rayleigh

About 10 years prior to the Michaelson- Morley experiment, Rayleigh was creating the theory of sound

1842-1919

“The problem of the whispering gallery”,Philosophical Magazine, pp 1001-1004, 1910


Sound following


Sound Source Localization

Outline

Model-free approachesA model and an identification problemAmbiguity and determining elevationReferences

Demonstration

This is joint work with Amir Handzel, Sean Andersson, and Martha Gebremichael. Also thanks to Shihab Shamma for inspiration. Vinay Shah did recent measurements and demos.


Barn Owl and RobotBarn Owl and RobotCan we capture the barn owl’s auditory acuity in a binaural robot? 2 degrees ~ microseconds resolution


Sound Localization in Nature

• Localization: spatial aspect of auditory sense• Sensory organ arrangement: Vision -- spatial “topographic” Audition -- tonotopic, transduction to sound pressure in frequency bands special computation required, performed in dedicated brainstem circuits and cortex


Acoustic Cues for Localization

• Binaural/Inter-aural

Level/Intensity Difference (ILD) Time/Phase Difference (IPD) On-set difference/precedence effect

• Monaural: spectral-directional filtering by Pinna, mostly for elevation


Place Theory (L. Jeffress)Place Theory (L. Jeffress)J. Comp. Physiol. & Psychol., (1948) 41:35-39J. Comp. Physiol. & Psychol., (1948) 41:35-39

Jeffress model and schematic of brainstem auditory circuits for detection of interaural time (ITD) differences; from Carr & Amagai (1996)


Stereausis (S. Shamma et. al.)Stereausis (S. Shamma et. al.)J. Acoust. Soc. Am. (1989) 86:989-1006J. Acoust. Soc. Am. (1989) 86:989-1006

Ipsi-lateral cochleaCharacteristic frequency

SoundC

hara

cter

istic

fre

quen

cyContara-lateralcochlea

AVCN

AVC

N

Ipsi- center contra-

lateral lateral

C kk +1

C kk

C kk -1

2jiij yxC jiij yxgC )0,max()(gwhere

Yj

CijXi

or


-45 deg (left)

Stereausis shifts from the main diagonal according to the source location.

45 deg (right)0 deg center

Incoming sound: a pure tone

Stereausis scheme (courtesy Shihab Shamma, UMd)


Initial Motivation

The above approaches are static, and do not take account of motion. But psychophysical experiments show active horizontal head rotations improve localization, break inter-aural symmetry, and thus provide information on elevation (Perret & Noble 1997, Wightman & Kistler 1999). Formulating this as an identification problem provides insight. We will show how movement can be helpful is resolving ambiguities. Applications arise in guiding a robot towards an acoustic source.

First, for a model, we turn to Rayleigh.


Lord Rayleigh and Binaural Perception

• ILD and ITD both needed for azimuth. (What about elevation?)

See section 385 of The Theory of Sound1945Edition

• Rayleigh set up the problem of sound propagation around an acoustically hard sphere. He introduced the head-related transfer function (HRTF).

• HRTF computed from sound pressure field generated at a point on the sphere by a point source located at (θ, φ).

• For a sphere, one has to solve the Helmholtz equation.


Coordinate Systems

zimuthal PolarElevationzimuth

Microphones at poles on horizontal plane


Static Solution

• Pressure field proportional to

• Does not depend on azimuthal angle (• Head Related Transfer Function (HRTF)• Numerical (e.g. FMP), and empirical methods for non-spherical heads


A representation of HRTF information


Blauert, J. (1997). Spatial Hearing (Revised Edition) (MIT Press, Cambridge, MA).

C. Phillip Brown, “Modeling the Elevation Characteristics of the Head-Related Impulse Response”, Dept. of Elec. Engr., San Jose State Univ. M. S. Thesis, (May, 1996).

Duda, R. O. (1995). “Estimating azimuth and elevation from the interaural head related transfer function,” in Binaural and Spatial Hearing, R. Gilkey and T. Anderson, Eds. (Lawrence Erlbaum Associates, Hillsdale, N.J.)

Some Related Work on HRTFs

R. O. Duda


Feature Plane (cylinder) and Signatures

• ILD & IPD constitute an intermediate computational space for localization.

• At each frequency a source gives rise to a point in the ILD-IPD plane (cylinder).

• A (broadband) point source imprints a signature curve on this feature plane (cylinder) according to its location.


Symmetry of Static Localization

• Sound pressure and resulting inter-aural functions depend only on polar angle;

azimuth invariant -- SO(2) symmetry• Sources on same circle of directions have

identical signatures. Hence the localization confusion.

• Introduce distance measures.


Symmetry and Rotations

zimuthal PolarElevationzimuth


Breaking the Symmetry

• Azimuthal invariance, but polar rotations do change the localization functions.

• Key mathematical step: infinitesimal rotations act as derivative operator -- generate vector fields on signatures.

• Derivatives ‘modulated’ by Cos(-- thus elevation extracted from horizontal rotation!

(Head movement helps)


Experimental Results

Broad band source - sum of pure tones 43 Hz – 11 KHz in steps of 43Hz. Passed through anti-aliasing filter and sampled at 22KHz. Knowles FG-3329 microphones used on head of 22.6 cm maximum diameter. To determine ILD and IPD, each 512 point segment (23 ms) of data was passed through an FFT. Measured IPD and ILD were smoothed by a nine-point moving average. This yields empirically determined (discrete) signature curves on ILD-IPD space. Localization computations based on minimizing distance functions. Implementation of this step on mobile robot achieved as a table lookup.


Pumpkin head side-view (left) and top view (right). Minimumdiameter 19 cm and maximum diameter 22.6 cm.


Plot on left displays smoothed ILD against theoretical ILD forsource at 17.5 degrees in horizontal plane. Plot on right showssmoothed IPD against theoretical IPD for same source.


Plot on left shows distance functions for source at 15 deg and17.5 deg. Plot on right shows distance functions for source at72.5 deg and 75 deg.


Performance plots for IPD-ILD algorithm (left) and traditional ITD algorithm (right)


Implications and Applications

• Psychophysics: auditory displays, auditory component of virtual environments and hearing aids.

• Bio-mimetic active robot head

References: A. A. Handzel and P. S. Krishnaprasad. “Bio-mimetic sound source localization”, IEEE Sensors

Journal, 2(6), 607-617, 2002

A. A. Handzel, S. B. Andersson, M. Gebremichael, and P. S. Krishnaprasad. “A bio-mimetic apparatus for sound source localization”, Proc. 42nd IEEE Conf. on Decision and Control, Dec. 2003 (in press).

S. B. Andersson, A.A. Handzel, V. Shah and P. S. Krishnaprasad. “Robot phonotaxis with dynamicsound source localization”, submitted (November 2003).


Front Back Demo

Without front-back distinction With front-back distinction


Happy BirthdayClyde

a system identification problem from lord rayleigh

Documents