Fast Subband Adaptive Filtering (FSAF)

Theory and applications of Fast Subband Adaptive Filtering, including Near Perfect Reconstruction Open Loop Delayless FSAF.

The set of new techniques, summarily named FSAF, is faster in all respects: converging faster, taking less MIPS, having lower processing latency, etc. FSAF allows nesting / recomposing of the subband architecture, to facilitate efficient fast converging low-latency low MIPS applications for AEC, Dereverberation, Feedback Cancellation, ANC, etc.

The RLS Joint Time Frequency initialization is discussed in details. At this time, it appears to be the MA/FIR recursive generalization of kernel-based regularization a.k.a. kernel methods a.k.a. ReLS (see System Identification Toolbox for more details). Extensive simulations and some converegence eigen-spectrum based analysis are in the Part II. Part IV contains practical examples of real-life adaptive signal processing, with AEC converging within the first utterance, before a word ends.

A new class of sub-optimal adaptive algorithms called Diagonal Least Squares (DLS) is introduced, to be integrated into audio applications. Meta-adaptive algorithms are discussed.

FSAF, besides usual per-subband processing, can be used to solve arbitrary high dimension system identification problems in a divide and conquer style, including Near Perfect Reconstruction Open Loop Delayless FSAF. For any predefined precision δ, FSAF solves M smaller, better-conditioned problems in subbands, using either RLS or diagonal / scalar step-size algorithms like Kaczmarz a.k.a. [N]LMS ((~1/(M + o(1/δp)):LMS; ~1/(M + o(1/δp))^2:RLS less MIPS vs full-band), and converts them back to the original full-band time domain with required precision δ.

For example, Room Impulse Response identification on a arbitrary excitation (15 sec of music, Fs=48kHz, RT60=0.5) take 36 hours on i7-10700K using full band Least Squares (LS) but less than 0.5 sec using FSAF LS, also providing significantly lower residuals and higher precision.

Detailed test and demo scripts are provided to ensure full reproducibility.