Structure of the Circular-Dyadic Convolution Error

Short summary

This paper characterizes the algebraic error introduced when substituting the Hadamard transform for the DFT in circular-dyadic convolution. The authors identify exact error cancellation at specific input/output positions, show the error operator is nearly full rank with logarithmic null space, and derive a closed-form expression for expected error via a single alignment scalar. The substitution error asymptotically doubles output energy except for filters in a universal zero-error subspace.

• •Hadamard substitution for DFT introduces structured, predictable convolution error
• •Error operator is nearly full rank; null space has only logarithmic dimension
• •Expected error governed by a single alignment scalar with closed-form expression

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