Adaptive cancellation of selected harmonics from a signal

The problem of extracting a cosine of unknown frequency in the presence of cosines with known frequencies is presented. Two methods are compared: the constant coefficient digital notch filter and the adaptive subtraction method where the known frequencies are input along with a guess for the unknown frequency. In the latter method, the amplitudes and phases of the known components are estimated using a Kalman filter and this information is used to subtract out these components from the signal to leave the cosine of interest. Two implementations of the Kalman filter are considered: an `optimal´ method, where all the elements of the estimated error covariance matrix are kept and a `suboptimal´ method, where the off-diagonal elements of this matrix are put to zero. Simulated and experimental data are analysed. The high-order Yule-Walker method is applied to determine the frequency content of the filtered signal. It is shown that, if the assumed frequency of the harmonic of interest is in error, then the suboptimal method can have a better performance than the optimal method. The reasons for this are explained by using a theoretical analysis of the estimation equations. The notch filter has by far the worst performance, as the frequencies to be subtracted out do not, in general, correspond to the zeros of the notch filter