Development and analysis of delayed-X LMS algorithm for active periodic noise control
The conventional adaptive notch filter uses two single tap Finite Impulse Response (FIR) filters and a 90 degree phase shift unit to cancel narrow-band noise. In this thesis, a new adaptive notch filter, which uses a second-order FIR filter and a delay unit, is proposed and analyzed. The novel feature of this delayed- X Least Mean Square (LMS) adaptive notch filter is the introduction of the delay unit in the reference input, resulting in improved numerical behavior of the notch filter and reduced out-of-band overshoot which degrades the performance of notch filters. The system transfer function of the delayed-X LMS algorithm is derived. The derivation is based on interpreting the delayed-X LMS filter as two fixed HR filters. The result of the derivation is verified by computer simulations. The delayed-X LMS system transfer function turns out to be a notch filter, which is capable of canceling narrow band noise. The stability of the system is analyzed based on this transfer function, which shows that the delayed-X LMS adaptive notch filter is always stable for the properly chosen convergence factor. This transfer function also shows that the out-of-band overshoot effect can be reduced by tuning the delay. Simulations show an average 5 dB overshoot reduction by using delayed-X LMS algorithm. For real time applications, it is important to optimize the numerical behavior of the adaptive notch filter for fixed-point implementation. In this thesis, the numerical behavior of the proposed delayed-X LMS notch filter is analyzed. Based on the steady-state weight vector, the optimal delay of the reference signal is derived to achieve the highest coefficient precision for the fixed-point implementation, thus reducing the coefficient quantization noise and eliminating coefficient overflow during the adaptation. Computer simulations show an averge 10 dB coefficient quantization noise reduction of the adaptive notch filter with the delayed-X LMS algorithm. The convergence behavior of the adaptive notch filter in relation to the sampling frequency is also analyzed based on the eigenvalue spread of the autocorrelation matrix. The eigenvalue spread is derived and shows that a faster sampling frequency does not give a shorter convergence time. It is proved that there is a relationship between the convergence rate and the sampling frequency. Optimal sampling frequency exists and is derived to provide the fastest convergence rate. Simulation results show the improvement of convergence speed with the optimum sampling rate. Finally, the delayed-X LMS algorithm is applied to multiple harmonics noise cancellation. A cascade structure is proposed for the delayed-X LMS algorithm to cancel multiple harmonics. Simulation results are given which show that this structure is suitable to effectively cancel multiple harmonics.