Alternatives to Spherical Microphone arrays: Hybrid Geometries

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• Alternatives to Spherical Microphone arrays
  Hybrid Geometries

Aastha Gupta Prof. Thushara Abhayapala Applied
Signal Processing CECS
To be presented at ICASSP, 20-24 April 2009,
Taipei, Taiwan
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Outline

• Spherical harmonic analysis of wavefields
• Spherical microphone arrays and limitations
• Theory of Non-spherical (Hybrid) arrays
• Combination of Circular Arrays
• Conclusions

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Spherical Coordinates

• Elevation
• Azimuth
• r Radial Distance

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Spherical Harmonics
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Wave Propagation

• Wavefields/ soundfields are governed by the wave
  equation.
• Homogeneous fields. They could be due to
  scattering, diffraction, and refraction.
• Basic solution can act as a set of building
  blocks

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Modal Analysis
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Arbitrary Soundfield General Solution
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Spherical Microphone arrays

• Spherical microphone arrays capture soundfield on
  a surface of a sphere.
• Natural choice for harmonic decomposition.
• Open Sphere Abhayapala Ward ICASSP 02
• Rigid Sphere Meyer Elko, ICASSP 02.
• Bessel zeros are a problem in open spheres.
• Rigid spheres are less practical for low
  frequencies.
• Strict orthogonality condition on sensor
  locations.

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Problem

• Spherical harmonic decomposition of
  wavefields/soundfield is a great way to solve
  difficult array signal processing problems.
• How can we estimate spherical harmonics from an
  array of sensors?
• What are the alternatives to spherical arrays?

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Circular Microphone Arrays

• Let be the soundfield
  on a circle at

Spherical Harmonics
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Hybrid Arrays

• We multiply
  by and integrate
  with respect to over to get

where
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Circular harmonic Decomposition

• Left hand side of this equation is a weighted
  sum of soundfield coefficients for a
  given .
• It can be evaluated for
  where the truncation number is dependent on
  the radius of the circle.
• We show how to extract from a
  number of carefully placed circular arrays

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Sampling of Circles

• In practice, we can not obtain soundfield at
  every point on these circles. Thus, needs
  sampling
• According to Shannon's sampling theorem for
  periodic functions, can be
  reconstructed by its samples over
  with at least samples . We
  approximate the integral in by a summation
  
• are the number of sampling
  points on the circle .

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Least Squares

• Suppose our goal is to design a Nth order
  microphone array to estimate (N 1)2
  spherical harmonic coefficients. By placing Q
  (N 1) circles of microphones on planes given by
  (rq, ?q), q 1, . . . ,Q, for a specific m, we
  have

where
The harmonic coefficients can be calculated by
solving the simultaneous system of equations or
evaluating a valid Moore-Penrose inverse of the
matrix
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Legendre Properties
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Bessel Properties

• Infinite summation can be truncated by using
  properties of Bessel functions

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Number of Coefficients
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Combination of Circles

• Consider two circles placed at and
  where 0 .
• That is one circle above the x-y plane and the
  second
• circle below the x-y plane but equal distance
  rq from the origin
• The circular harmonics of the soundfield on the
  circle on or above the
• x-y plane are given by

Above xy plane
Below xy plane
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Circular Harmonic combination

• Right hand side is a weighted sum of coefficients
  for a specific
• For l0 the sum only consists of a weighted sum
  of with n is even.
• For l1 the sum only consists of a weighted sum
  of with n is odd.

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Findings so far..

• Thus, we can separate odd and even spherical
• harmonics from the measurement of soundfield on
  two circles
• placed on equal distance above and below the x-y
  plane.
• This is a powerful result, which we can use
• to extract spherical harmonics from soundfield
  measurements
• on carefully placed pairs of circles.

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Odd Coefficients
Guidelines to choose
systematically such that is always non
singular

• There are specific patterns of the normalized
  associated Legendre function when n-m 1, 3,
  5... There are number of different range of
  elevation angles we can choose for ?q. Note that
  ?q could be same for all q or a group of values.
• For a Nth order system, there are N(N 1)/2 odd
  spherical harmonic coefficients from total of (N
  1)2 coefficients. We use N (for N odd) or N - 1
  (for N even) pairs of of circular microphone
  arrays. We choose the radii of these circles as

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Findings

• With this choice, the soundfield at frequency k
  on a circle
• with rq is order limited to
  due to
• the properties of Bessel functions. This property
  limits the
• higher order components of the soundfield present
  at a particular
• radius rq. Also, the lower order components are
  guaranteed
• to be present due to the choice of radii in (14)
  which
• avoids the Bessel zeros.
• Thus, selecting rq and ?q from the legendre
  and Bessel plots, we can guarantee that is
  non singular.

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Normalised Legendre function-odd
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Even Coefficients

• Suppose, we have selected Q pairs of
  such that
• when
  is even.

We have following guidelines to choose
systematically such that is always non
singular
As in the case of odd coefficients, we can choose
range of values for ?q, which plots
for even. Note that on
the x-y plane (? p/2), all even associate
Legendre functions are non zero. Thus, placing
circles on the x-y plane seems to be an obvious
choice to estimate even coefficients, where we do
not need pairs of circles.
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Normalised Legendre Function-even
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Findings

• Depending on our choice, we can design different
  array configurations, which will be capable of
  estimating spherical harmonic coefficients.
• For a Nth order system, we place N/2 (N even) or
  (N1)/2 (N odd) circles on the x-y plane. We
  choose the radii of these circles based on the
  bessel plots

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Simulations-5th Order System

• We first place four circular arrays (two pairs)
  with 11, 11, 7 and 7 microphones at (4/ko, p/3),
  (4/ko, p - p/3), (5/ko, p/6), and (4/ko, p -
  p/6). Then we place a pair of microphones at
  (5/ko, 0) and (5/ko, p).
• This sub array consists of 38 microphones are
  designed to calculate all odd spherical harmonics
  up to the 5th order (total of 15 coefficients).
• We place three circular arrays on the x-y plane
  together with a single microphone at the origin
  to complete the design. We have 7, 11, and 13
  microphones in three arrays on x-y plane at
  radial distances 2/ko, 4/ko, and 5/ko,
  respectively.

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Simulations

• Test Octave - 3000Hz to 6000Hz (kl 55.44)
• 40dB signal to noise ratio (SNR) at each sensor,
  where the noise is additive white Gaussian
  (AWGN).
• Estimate all 36 spherical harmonic coefficients
  for a plane wave sweeping over the
  entire 3D space and for all frequencies within
  the desired octave.
• We plot the real and imaginary parts of
  against the azimuth and elevation of the
  sweeping plane wave for lower, mid,
• and upper end of the frequency band.

Real part of the estimated harmonic coefficient
a54 for a plane wave sweeping over entire 3D
space (a) Theoretical pattern (b), (c), (d) are
at frequencies 3000, 4500 and 6000Hz,
respectively, and all at SNR 40dB
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Conclusions

• Spherical harmonic decomposition is a useful tool
  to analyse 3D soundfields.
• Spherical arrays have inherent limitations that
  make them unfeasible for practical
  implementation.
• Circular microphone arrays and hybrid arrays
  need carefully designing based on underlying wave
  propagation and theory.
• Combining circular arrays enables us to calculate
  odd and even harmonics independently, providing
  cleaner more accurate results.

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Thanks Questions/Feedback?