Mirror Image Theory and One-Dimensional Systems

This chapter formulates wave traveling in acoustic pipes from a perspective of the discrete linear system theory based on mirror image methods and z-transforms. The z-transforms would be helpful to represent transfer functions by poles and zeros for wave propagation including time delay. Transfer functions are decomposed into products of all-pass and minimum-phase components. The all-pass component is due to the distance between the source and receiving points, and it may include pure delay in one-dimensional pipes. Minimum-phase parts are made by the poles and zeros of transfer functions of one-dimensional pipes. The phase response, what is called the propagation phase in a one-dimensional system, may follow the pure delay from the source within the minimum-phase local variances. Minimum-phase transfer functions yields no phase accumulation; however, phase accumulation due to the propagation phase may be estimated even from the minimum-phase transfer function, in which the difference in the numbers of poles and zeros except at the origin in the z-plane yields the accumulated phase corresponding to the delay by the distance. The propagation phase may be the standard of phase responses for one-dimensional systems.