On Computing the Discrete Hirschman Transform

The Discrete Hirschman Transform (DHT) is a generalization of the entropy-based Hirschman Transform that canonically represents digital signals with sparse basis functions. The DHT is computationally attractive in providing the flexibility in hardware implementations since it has a sparse structure that is similar to that of the Discrete Fourier Transform (DFT) basis. This Hirschman-based algorithm has been widely applied in many digital signal processing algorithms such as denoising, filtering, and linear convolution. In this paper, we have developed two fast algorithms: the radix-2 and radix-4 DHT algorithms. These efficient algorithms significantly reduce the number of nontrivial real computations when compared to the original DHT algorithm and other popular Fast Fourier Transforms (FFTs). We show that it is feasible to realize our new algorithms in hardware through the use of a repetitive application of butterfly computations in a structure that will be familiar to those already working with the similar FFT algorithms. Our proposed algorithms reduce the number of nontrivial real computations while efficiently using available hardware resources.