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.

中文翻译：

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基于离散Hirschman变换的降低复杂度的最佳卷积

与离散傅立叶变换（DFT）相比，离散Hirschman变换（DHT）在计算上更具吸引力。基于其推导的线性卷积，我们已经确认，与我们先前的工作中基于常规DFT的卷积滤波器相比，基于DHT的卷积滤波器在有条件地减少计算量方面显示出其优越性。由于基于DHT的卷积具有许多取决于参数选择的配置，因此我们推测应该有一个最佳的情况来最大程度地减少计算量。在本文中，对于基于DHT的卷积，我们表达了实际计算中的要求，并提出了一种确定最佳参数以减少计算量的方法。我们进一步将基于DHT的最佳卷积的计算负荷与其他流行卷积的计算负荷进行比较。而且，还使用数字信号处理器（DSP）TMS320C5545估计了其时钟周期的减少。结果表明，基于DHT的最佳卷积可以减少实际计算（乘以 $ 9.09 \％-50 \％$ 和添加 $ 1.12 \％-51.09 \％$ ）和时钟周期，具体取决于输入长度和滤波器大小，但某些情况下的性能与基于radix-2 FFT的竞争对手相同。