An acoustic vector sensor provides measurements of both the pressure and particle velocity of a sound field in which it is placed. These measurements are vectorial in nature and can be used for the purpose of source localization. A straightforward approach towards determining the direction of arrival (DOA) utilizes the acoustic intensity vector, which is the product of pressure and particle velocity. The accuracy of an intensity vector based DOA estimator in the presence of noise has been analyzed previously. In this paper, the effects of reverberation upon the accuracy of such a DOA estimator are examined. It is shown that particular realizations of reverberation differ from an ideal isotropically diffuse field, and induce an estimation bias which is dependant upon the room impulse responses (RIRs). The limited knowledge available pertaining the RIRs is expressed statistically by employing the diffuse qualities of reverberation to extend Polack’s statistical RIR model. Expressions for evaluating the typical bias magnitude as well as its probability distribution are derived.
REFERENCES
1.
Y.
Huang
, J.
Benesty
, and J.
Chen
, “Time delay estimation and source localization
,” in Springer Handbook of Speech Processing
, edited by J.
Benesty
, M. M.
Sondhi
, and Y.
Huang
(Springer
, New York
, 2007
), pp. 1043
–1063
.2.
P.
Stoica
, Z.
Wang
, and J.
Li
, “Robust capon beamforming
,” IEEE Signal Process. Lett.
10
, 172
–175
(2003
).3.
G.
D’Spain
, W.
Hodgkiss
, and G.
Edmonds
, “The simultaneous measurement of infrasonic acoustic particle velocity and acoustic pressure in the ocean by freely drifting Swallow floats
,” IEEE J. Ocean. Eng.
16
, 195
–207
(1991
).4.
B. A.
Cray
and A. H.
Nuttall
, “A comparison of vector-sensing and scalar-sensing linear arrays
,” Naval Undersea Warfare Center Division Report No. 10632, Newport, RI, 1997
.5.
H. -E.
de Bree
, P.
Leussink
, I. T.
Korthorst
, D. H.
Jansen
, D. T.
Lammerink
, and P. M.
Elwenspoek
, “The microflown, a novel device measuring acoustical flows
,” in 8th International Conference on Solid-state Sensors and Actuators, and Eurosensors IX
, Stockholm, Sweden (1995
), Vol. 1
, pp. 536
–539
.6.
A.
Nehorai
and E.
Paldi
, “Acoustic vector-sensor array processing
,” IEEE Trans. Signal Process.
42
, 2481
–2491
(1994
).7.
M.
Hawkes
and A.
Nehorai
, “Acoustic vector-sensor correlations in ambient noise
,” IEEE J. Ocean. Eng.
26
, 337
–347
(2001
).8.
F.
Jacobsen
and T.
Roisin
, “The coherence of reverberant sound fields
,” J. Acoust. Soc. Am.
108
, 204
–210
(2000
).9.
J. -D.
Polack
, “Transmission of sound energy in concert halls
,” Ph. D. thesis, Universit’e du Maine
, Le Mans (1988
).10.
J.
Jot
, L.
Cerveau
, and O.
Warusfel
, “Analysis and synthesis of room reverberation based on a statistical time-frequency model
,” in 103th Audio Engineering Society Convention
, New York (1997
), Preprint No. 4629.11.
A.
Nehorai
and M.
Hawkes
, “Performance bounds for estimating vector systems
,” IEEE Trans. Signal Process.
48
, 1737
–1749
(2000
).12.
A. D.
Pierce
, Acoustics—An Introduction to Its Physical Principles and Applications
(Acoustical Society of America
, Melville, NY
, 1991
), Chap. 1, p. 39
.13.
Obviously this can not be literally accurate since the direct arrival would defy causality. Rather, it should be viewed as a time shifted version of the RIR such that the reflections start at . The benefits of this notation are twofold. Firstly, reverberation starting at time is less cumbersome to manipulate mathematically. Furthermore, any problems arising from the propagation delay of the direct arrival being a fraction of the sampling rate are circumvented.
14.
M. R.
Schroeder
, “New method of measuring reverberation time
,” J. Acoust. Soc. Am.
37
, 409
–412
(1965
).15.
M. R.
Schroeder
, “Statistical parameters of the frequency response curves of large rooms
,” J. Audio Eng. Soc.
35
, 299
–306
(1987
);M. R.
Schroeder
, translated from “Eigen-frequenzstatistik und anregungsstatistik in Raümen
,” Acustica
4
, 456
–468
(1954
).16.
H.
Kuttruff
, Room Acoustics
, 4th ed. (Spon
, London, UK
, 2000
), pp. 97
–100
, pp. 115
–119
.17.
P. M.
Morse
and K. U.
Ingard
, Theoretical Acoustics
(McGraw-Hill
, New York
, 1968
), Chap. 9, pp. 576
–579
.18.
J. B.
Allen
and D. A.
Berkley
, “Image method for efficiently simulating small-room acoustics
,” J. Acoust. Soc. Am.
65
, 943
–950
(1979
).19.
E. A. P.
Habets
and S.
Gannot
, “Generating sensor signals in isotropic noise fields
,” J. Acoust. Soc. Am.
122
, 3464
–3470
(2007
).20.
It is instructive to note an analogy to a better known example from the field of communication. Wireless channels are often modeled as consisting of multiple reflections arriving with varying phases and magnitudes. A similar analysis indicates that the net effect is a symmetrical Gaussian distribution—the phase is distributed uniformly, whereas the magnitude is a Rayleigh variable (Chi distribution of second order). This communication model is known as the Rayleigh channel (Ref. 26) and can be considered a two dimensional analog to our discussion. In fact, the aforementioned distribution is named after Lord Rayleigh who proposed its use to model the summation of sound waves with uniformly distributed phases (Ref. 27).
21.
E. A. P.
Habets
, “Room impulse response (RIR) generator
,” http://home.tiscali.nl/ehabets/rir_generator.html (Last viewed 4/30/2010).22.
The choice of room shape was dictated by the peculiarities of the image source method. Since the room used is rectangular and the reflections specular, permitted directions of a sound ray will be limited by its initial direction and they will not properly (Ref. 28). This results in sound decay rate being highly dependant upon direction. For a room containing boundaries with identical decay coefficients, reverberation propagating parallel to the longest room dimension will eventually dominate the RIR tail. Choosing a cube shaped room mitigates (but does not eliminate) this nonisotropic phenomenon.
23.
The actual distance was adjusted by a fraction of a millimeter so that the propagation delay would correspond to an integer multiple of the sampling period.
24.
P. J.
Brockwell
and R. A.
Davis
, Time Series: Theory and Methods
, 2nd ed. (Springer
, New York
, 1991
), Chap. 7, pp. 218
–219
.25.
The convergence demonstrated herein is . Since , it follows by definition that also converges in the sense of . It is possible to prove that and converge in this sense by way of an argument utilizing the fact that the range of values these r.v.s may take on is bounded [see (2) and (10b)].
26.
D.
Tse
and P.
Viswanath
, Fundamentals of Wireless Communication
(Cambridge University Press
, Cambridge, UK
, 2005
), Chap. 2, p. 36
.27.
J. W. S.
Rayleigh
, The Theory of Sound
, 2nd ed. (Dover
, New York
, 1945
), Vol. I
, Sec. 2-42, pp. 35
–42
.28.
J. -D.
Polack
, “Playing billiards in the concert hall: The mathematical foundations of geometrical room acoustics
,” Appl. Acoust.
38
, 235
–244
(1993
).© 2010 Acoustical Society of America.
2010
Acoustical Society of America
You do not currently have access to this content.