Understanding Error in Hadamard Transform for Convolution

In signal processing, both dyadic and circular convolutions can be computed efficiently in O(N log N) time using the Hadamard transform and the Discrete Fourier Transform (DFT), respectively. The Hadamard transform is often preferred due to its real-valued sign flips, which can simplify computations. However, using the Hadamard transform as a substitute for the DFT introduces an algebraic error. This research provides a detailed characterization of this substitution error. It identifies that certain input and output positions universally result in zero error, and this error cannot be eliminated by reordering the output. Furthermore, the error operator is found to be nearly full rank, with its null space having only a logarithmic dimension. The study also reveals that the expected error is primarily governed by a single alignment scalar, for which a closed-form expression is derived by averaging over random filters. In general, this substitution error can asymptotically double the output energy, except for specific filters within the universal zero-error subspace, which incur no error. These findings collectively demonstrate that the substitution error is structured, predictable, and dependent on alignment.