A hyperstable adaptive line enhancer for fast tracking of sinusoidal inputs

An IIR adaptive filter is presented for the application of tracking sinusoidal inputs. The filter uses a hyperstable update to provide very fast convergence, and is capable of tracking multiple sinusoids, with the order of the filter being twice the number of sinusoids. The convergence of the filter depends on a certain transfer function being strictly positive real (SPR), and necessary conditions are derived, for this transfer function to be SPR. These conditions hold for the completely general case, and can be very simply expressed in terms of the input sinusoidal frequencies, because of the specific application that is addressed, and because of the filter topology (a resonator-based filter structure) that is used. Further, these conditions are derived under the assumption that no post-error filter is used (unlike most hyperstable schemes). Further, these conditions are also shown to be sufficient for convergence, for the case of up to two input sinusoids (4th order filter). For a larger number of input sinusoids, though it has not been possible to prove the sufficiency of these conditions, we have conducted several experiments and obtained convergence in all cases, leading to the conjecture that the conditions are sufficient for convergence, for the general case as well. Simulations of the performance of the filter structure are also provided to support the theory and the conjectured condition for convergence. The capability of the adaptive filter to track fast changes in frequency make it a candidate for applications like demodulating MFSK signals. Alternatively, the speed can be traded for complexity because for slow adaptation (low μ), the algorithm reduces to a pseudo-gradient-based update that can be implemented very simply