A multi-tone pseudo-cascade filtered-x LMS adaptive notch filter

Cancellation path delay limits the bandwidth performance of the FXLMS (filtered-x least-mean-square) notch filter. A cascade of FXLMS notch filters using multiple reference tones is examined in order to alleviate this problem. An analysis of a single-stage notch filter characterizes the response by the cancellation bandwidth and out-of-band overshoot, which are analytically expressed in terms of the loop gain and cancellation path delay. These results are then used to synthesize a multistage cascade with offset reference tones. A pseudocascade configuration is also formulated that only requires one entry point to the primary disturbance plant, which is a prerequisite for many applications. A numerical example of a three-stage FXLMS notch filter shows that for a 3 dB overshoot constraint, the 20 dB cancellation bandwidth is increased by a factor of 2.56