What question did this study set out to answer?

The research aims to establish a continuous criterion for efficient parallelization of DSP filter chains using commutator norms.

March 29, 2026Open Access

Commutator Norm-Based Dynamic Parallelization of DSP Filter Chains: A Continuous Non-Commutativity Criterion with Formal SNR Guarantees

Key Points

The research aims to establish a continuous criterion for efficient parallelization of DSP filter chains using commutator norms.
Introduced matrix commutator norm as a basis for parallelization.
Developed error bound independent of signal specifications.
Derived SNR-calibrated threshold rules for design.
Validated through experiments with 45 real DSP filter pairs.
Proved a signal-independent output error bound for the DSP filter chains.
Achieved a 32× speedup in processing time for certain filter pairs.
Confirmed the commutator-norm dependency graph is a directed acyclic graph.

Abstract

This preprint introduces the matrix commutator norm d (A, B) = ‖AB−BA‖F as a continuous, algebraically grounded criterion for DSP filter-chain parallelization. Four contributions are made: (1) a formally proved, signal-independent output error bound ‖ABx−BAx‖ ≤ d (A, B) ‖x‖; (2) an SNR-calibrated threshold rule derived via the Böttcher–Wenzel inequality, enabling design-by-specification from a target SNR to a maximum allowable commutator norm; (3) a proof that the commutator-norm dependency graph is a directed acyclic graph (DAG) under order-preserving edge assignment, reducing critical-path length from O (N) toward O (log N) ; and (4) a statistically consistent adaptive threshold via the Glivenko–Cantelli theorem. Experiments on 45 real DSP filter pairs validate the error bound for all tested pairs and demonstrate 32× speedup on a 32-filter near-commutative chain (7. 4 ms → 0. 23 ms). Related to Japanese Patent Application JP 2026-046614 (SlimeDSP), filed 2026-03-20.

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