Design and analysis of a virtual-microphone active noise control system

Classical active noise control structures can provide high low-frequency noise attenuation at the close vicinity of the so-called error microphones. However, the generated zones of quiet are usually of small dimensions and do not reach user's ears. Placing other microphones, even wireless, at the ears is often unacceptable. Therefore, there is a need to design algorithms capable to shift the zones of quiet to desired locations. For the algorithm considered in this paper two assumptions are made. One of them requires the distance between a user ear and a corresponding error (real) microphone to be much less than the smallest wavelength in the acoustic noise. This is satisfied in many applications where local control is considered. Another assumption is that the noise is stationary or changes little, what is rather usual in many industrial environments. The algorithm is composed of two stages. In the tuning stage knowledge of properties of the noise at the position of the user's ears is gained, whereas in the control stage it is used to synthesize a set point for the controlled variable. In this paper a formal analysis of this algorithm is presented. First the optimal solution in the mean-square sense is given and analyzed. It is based on spectral factorization of power spectral density of the disturbance, the inner-outer factorization of a transfer function and causal/non-causal decomposition. Next, conditions for convergence of the adaptive version are derived. This algorithm has also been experimentally verified to control noise in an active headrest system. It has proven to be very efficient in providing high noise attenuation directly at the ears under time-varying conditions due to changes of position of the user's head.