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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
Outline
  •   Spherical harmonic analysis of wavefields
  •   Spherical microphone arrays and limitations
  •   Theory of Non-spherical (Hybrid) arrays
  •   Combination of Circular Arrays
  •   Conclusions
Spherical Coordinates
  •             : Elevation
  •               : Azimuth
  •   r: Radial Distance
Spherical Harmonics
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
Modal Analysis
Arbitrary Soundfield – General Solution
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.
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?
Circular Microphone Arrays

Let        be the soundfield on a circle at
                                 Spherical
                                 Harmonics
Hybrid Arrays
 We multiply
 by       and integrate with respect to
 over    to get



   where
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
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        .
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, θq), 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
Legendre Properties
Bessel Properties
 Infinite summation can be truncated by using properties of Bessel
functions:
Number of Coefficients
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
Circular Harmonic combination




•   Right hand side is a weighted sum of
    coefficients for a specific
•   For l=0 the sum only consists of a weighted
    sum of         with n is even.
•   For l=1 the sum only consists of a weighted
    sum of         with n is odd.
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.
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 θq. Note that θq 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
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 θq from the legendre and Bessel plots, we
 can guarantee that     is non singular.
Normalised Legendre function-odd
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 θq, which plots               for         even.
       • Note that on the x-y plane (θ = π/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.
Normalised Legendre Function-even
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
    (N+1)/2 (N odd) circles on the x-y plane. We
    choose the radii of these circles based on the
    bessel plots
Simulations-5th          Order System
   We first place four circular arrays (two pairs) with
    11, 11, 7 and 7 microphones at (4/ko, π/3), (4/ko, π −
    π/3), (5/ko, π/6), and (4/ko, π − π/6). Then we place a
    pair of microphones at (5/ko, 0) and (5/ko, π).
   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.
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    Real part of the estimated harmonic coefficient α54 for a
                                       plane wave sweeping over entire 3D space: (a) Theoretical pattern
wave for lower, mid,                   (b), (c), (d) are at frequencies 3000, 4500 and 6000Hz,
and upper end of the frequency         respectively, and all at SNR= 40dB
band.
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.
      Thanks
        &
Questions/Feedback
         ?

								
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