By Pawel Strumillo
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3o. 2 demonstrated that the directivity index and the beamwidth can be improved by adding an additional pole. Figure 4 illustrates the directivity index and the beamwidth for the case of two equal roots or poles in the denominator of the discriminating function. As a means of comparison, it is instructive to consider the dyadic sensor which has a polynomial of the second degree as its discriminating function. 54 dB and the associated beamwidth is 65o. The directivity index in Fig. 0 dB at Q = 10 .
A generalized directional acoustic sensor of order ν can be defined as one whose beamformer output is given by ν po ( t ) = ∑ anLn ⎣⎡ p ( t , r ) ⎦⎤ = gu L (ψ ) p ( t , r ) n=0 ν gu L (ψ ) = ∑ an cosn ψ n=0 (19) (20) The discriminating function is a polynomial in cosψ of degree ν. The optimum directivity index is 20 log ( 1 + ν ) (Schmidlin, 2007). It is a very difficult matter to implement the operations Ln ⎡⎣ p ( t , r ) ⎤⎦ for n ≥ 2 . This accounts for the sparsity of work on higher-order directional acoustic receivers.
Ignoring the self-noise at each velocity sensor and letting a j = − aυ for j = 1, 2, 3 simplifies Eq. (1) to po ( t ) = a0 p ( t ) − ρ0caυ v ( t ) ⋅ u L (3) The particle velocity at time t and position r is related to the pressure as follows (Ziomek, 1995) v (t , r ) = − p (t + u ⋅ r c ) ρ 0c u (4) Setting r to 0 and placing the result into Eq. (3) results in po ( t ) = ( a0 + aυ u ⋅ u L ) p ( t ) (5) The unit-vector u points in the direction of the arriving plane wave and the unit-vector u L points in the look direction.
Advances in Sound Localization by Pawel Strumillo