ICSV 14 Cairns • Australia 9-12 July , 2007 PROPAGATION PHASE REPRESENTATION IN 3 D SPACE USING POLES AND ZEROS IN COMPLEX FREQUENCY PLANE
This paper describes the transfer function, including the coherent sound field, as the increased distance between source and observation points, from the point of view of the number of zeros in the complex frequency plane. The authors estimated the propagation phase obtained by linear regression analysis of the narrow-frequency-band phase characteristics of the minimumphase component of the transfer function. The direct sound field estimated from the propagation phase of the minimum-phase characteristics consequently seems to be the coherent field within the critical distance. We determined that the zeros drift based on complex-integration analysis. When the sound source distance (SSD) increases, the zeros move toward a higher frequency; thus, an accumulated phase is produced following the propagation phase. When the SSD is long, even in the coherent field, we could confirm a pole-line around the theoretical value.
Range and frequency dependence of transfer function phase
The phase trend in multi‐degree‐of‐freedom systems can be determined by an algorithm that tracks the relative signs of adjacent resonances in a modal expansion of transfer functions. This algorithm is expressed in analytical form for two‐dimensional systems. The analytical expression shows the transition from input function phase to the reverberant phase limit. The phase trend for two‐dimensional systems greatly exceeds that of the average response of such systems and the reason for this difference is analyzed.
The Invariance of the Vacuum is the Invariance of the World
The following theorem is demonstrated: If the vacuum is invariant under the group generated by the space integral of the time component of a local vector current, then the Hamiltonian is invariant also. A similar theorem holds for the group generated by the space integral of a space component of a local axial-vector current.