A fast Hadamard transform for signals with sub-linear sparsity

A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT, where N is a power of two and K = O(N^α), scales sublinearly in N for some 0 <; α <; 1. Assuming a random support model for the nonzero transform domain components, the algorithm reconstructs the WHT of the signal with a sample complexity O(K log₂(N/K)), a computational complexity O(K log₂(K) log₂(N/K)) and with a very high probability asymptotically tending to 1. The approach is based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time domain signal, one can induce a suitable aliasing pattern in the transform domain. By treating the aliasing patterns as parity-check constraints and borrowing ideas from erasure correcting sparse-graph codes, the recovery of the nonzero spectral values has been formulated as a belief propagation (BP) algorithm (peeling decoding) over an sparse-graph code for the binary erasure channel (BEC). Tools from coding theory are used to analyze the asymptotic performance of the algorithm in the “very sparse” (α ∈ (0, 1/3]) and the “less sparse” regime (α ∈ (1/3, 1)).