Optimal Constraint-Based Loop-Shaping in the Cepstral Domain

Simple active noise-canceling systems (such as noise canceling headphones) use a feedback mechanism whereby the signal recorded by a microphone placed near the loudspeaker is phase-inverted and sent back through the loudspeaker via a feedback filter. 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 1980s of optimal H_infin algorithms. The design technique outlined in this letter uses a cepstral domain approach, where gain and phase constraints take a convenient linear form, and linear programming to design the unique optimal feedback filter