Acoustic travelling wave separation in the time domain using electronic time delay circuits and leaky recursion

Abstract

Acoustic travelling plane wave separation is implemented in the time domain from the sound pressure signals of two closely spaced microphones using analog electronic circuits based on a leaky recursion formulation. The sound pressure and acoustic particle velocity signals at the midpoint between the microphones are estimated from delayed recursive expressions modified for stability by introducing mild loss recursion. The forward and backward travelling wave components are obtained electronically by simple linear combinations of the sound pressure and particle velocity signals. Two travelling wave separation circuits (TWSC) were designed, built, and evaluated from measurements in an acoustic test tube apparatus in the operational frequency range from 710 to 5,680 Hz. The two designs differ in the use of first- or second-order time delay electronic circuits, with performance of the first-order circuit degrading with increasing frequency. Experimental results for the second-order circuit show a travelling wave separation error of -30.4 dB in the operational frequency range, a travelling wave amplitude accuracy within ±0.1 dB in most of the frequency range, with an implementation that offers operation in real-time with low latency, and resilience to unknown initial conditions or similar errors during operation, with typical transient relaxation times of 2 ms, that can be made lower at the expense of reduced accuracy.