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Method for finding optimal exponential decay coefficient in numerical Laplace transform for application to linear convolution

机译：线性卷积中数值拉普拉斯变换中寻找最佳指数衰减系数的方法

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摘要

In this paper, a method based on the numerical Laplace transform is used for calculating the full linear convolution of real or complex signals. An algorithm for obtaining the last N values of the convolution is presented, along with a method for finding an optimal value for the decay coefficient of the transform. It is shown that the use of the numerical Laplace transform formulation allows the calculation of each half of the linear convolution independently, which has computational benefits. The numerical Laplace transform is expressed as the fast Fourier transform of signals that have been premultiplied by a decreasing exponential window characterized by decay coefficient c. The error of the resulting linear convolution depends on the value of the decay coefficient; undervalue results in the generation of wraparound error whereas overvalue causes amplification of Gibbs phenomenon. In this paper, a formula that optimizes the value of the decay coefficient is developed. A trade-off value for c is obtained and error analysis shows that it outperforms other coefficients proposed in the literature when applied to the calculation of linear convolution. The relative errors obtained are of the order of 10~(-6)% and 10~(-9)% for single and double precisions.
机译：在本文中，基于数值拉普拉斯变换的方法用于计算实信号或复信号的全线性卷积。提出了一种用于获取卷积的最后N个值的算法，以及一种为变换的衰减系数找到最佳值的方法。结果表明，使用数值拉普拉斯变换公式可独立计算线性卷积的每一半，这具有计算上的优势。数值拉普拉斯变换表示为信号的快速傅里叶变换，该信号已预先乘以衰减系数c为特征的递减指数窗口。线性卷积的误差取决于衰减系数的值。低估会导致环绕误差的产生，而高估会导致吉布斯现象的放大。本文提出了一个优化衰减系数值的公式。获得了c的折衷值，误差分析表明，当将其应用于线性卷积计算时，其性能优于文献中提出的其他系数。对于单精度和双精度，获得的相对误差约为10〜（-6）％和10〜（-9）％。

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