Optimal Loop-Shaping in the Cepstral Domain for Active Noise Canceling Headsets

Simple active noise-canceling systems (such as noise canceling headphones) use a feedback mechanism whereby the signal recorded by microphone placed near the loudspeaker is phase-inverted and sent back through the loudspeaker via a feedback filter. Such systems can be implemented using analog circuitry, are fairly inexpensive, and achieve reasonably good performance at low frequencies (below 500 Hz). The feedback filter is designed to achieve a high gain at low-frequencies (for best noise reduction) while maintaining closed-loop stability under various conditions. To design the feedback filter, fairly ad-hoc techniques were traditionally used under the broad denomination of "loop-shaping" until the advent in the 80\´s of optimal H_infin algorithms. The design technique outlined in this paper shares the intuitive approach of traditional loop-shaping methods, but removes most of the guess-work while producing optimal filters that guarantee closed-loop stability. The main idea is to represent the filter in the cepstral domain, where gain and phase constraints take a convenient linear form, and use linear programming to design the optimal feedback filter