A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain


During this paper, we tend to design a new iterative low-complexity algorithm for computing the Walsh–Hadamard rework (WHT) of an $N$ dimensional signal with a $K$ -sparse WHT. We tend to suppose that $N$ may be a power of 2 and $K=O(N^alpha)$ , scales sublinearly in $N$ for a few $alpha in (0,one)$ . Assuming a random support model for the nonzero transform-domain parts, our algorithm reconstructs the WHT of the signal with a sample complexity $O(K log _2(N/K))$ and a computational complexity $O(Klog _2(K) log _2(N/K))$ . Moreover, the algorithm succeeds with a high likelihood approaching one for massive dimension $N$ . Our approach is especially based mostly on the subsampling (aliasing) property of the WHT, where by a rigorously designed subsampling of the time-domain signal, a suitable aliasing pattern is induced within the transform domain. We have a tendency to treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We have a tendency to analyze the properties of the ensuing bipartite graphs and borrow ideas from codes outlined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a particular sparse-graph code transmitted over a binary erasure channel. This allows us to use tools from coding theory (belief-propagation analysis) to characterize the- asymptotic performance of our algorithm within the very sparse ( $alpha in (zero, 1/3]$ ) and therefore the less sparse ( $alpha in (1/3, 1)$ ) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm.

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