Reduced Complexity Optimal Convolution Based on the Discrete Hirschman Transform

The Discrete Hirschman Transform (DHT) is more computationally attractive than the Discrete Fourier Transform (DFT). Based on its derived linear convolution, we have confirmed that the DHT-based convolution filter shows its superiority in reducing computations conditionally, while compared with the conventional DFT-based convolution filter in our previous work. Since the DHT-based convolution has many configurations depending on parameter choices, we conjecture that there should be an optimal case for the largest reduction in computations. In this paper, for the DHT-based convolution, we express the requirement in real computations and propose an approach of how to determine the optimal parameters to reduce computations. We further compare the computational load of the optimal DHT-based convolution with that of other popular convolutions. Moreover, its reduction in clock cycles has also been estimated using a Digital Signal Processor (DSP) TMS320C5545. Results indicate that the optimal DHT-based convolution can reduce real computations (multiplications by 9.09\%-50\% and additions by 1.12\%-51.09\% ) and clock cycles, according to the input length and filter size, except for some cases with identical performance to the radix-2 FFT-based competitor.