By Pawel Strumillo
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1. 105. 3. 1 Interconnection of prototype filters The first-order prototype filter can be used as a fundamental building block for generating filters that have discriminating functions which are rational functions of cosψ . As an example, consider a discriminating function that is a proper rational function and whose denominator polynomial has roots that are real and distinct. Such a discriminating function may be expressed as 28 Advances in Sound Localization μ μ gu L (ψ ) = j ∑ d j cos ψ j =0 =K ν j ∑ c j cos ψ j =0 ( ) ( j =1 ) ∏ b j − cosψ j =1 ν ∏ a j − cosψ (46) where cν = 1 and μ < ν .
The general solution to Eq. , 2003) po ( t ,τ ) = f ( t + aτ ) exp ( − γ t a ) (22) The function f ( ⋅) is arbitrary. The forcing function of interest is the harmonic plane wave function p ( t ,τ ) = exp ( jωt ) exp ( jωτ cosψ ) (23) The response to this input can be found by assuming a solution of the form po ( t ,τ ) = B (ω : ψ ) exp ( jωt ) exp ( jωτ cosψ ) (24) The substitution of Eqs. (23) and (24) into Eq. (21) results in B (ω : ψ ) = K γ + jω ( a − cosψ ) (25) The function B (ω : ψ ) is called the beam pattern of the filter.
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 . 7 o at Q = 10 .
Advances in Sound Localization by Pawel Strumillo