虚轴上具有重极点的拉普拉斯变换与傅里叶变换相互计算的方法

Mutual Calculation of Laplace Transform and Fourier Transform with Multiple Poles on Imaginary Axis

提出了一种虚轴上具有重极点的拉普拉斯变换(LT)与傅里叶变换(CTFT)相互计算的方法。对虚轴上具有重极点的LT,通过部分分式展开,将LT在虚轴上的极点分离出来,使其成为虚轴上含极点和不含极点两部分之和,针对虚轴上含极点的LT部分,分区左极点和区右极点两种情况,导出了利用LT计算CTFT的方法;将信号的CTFT分解成解析部分与不解析部分之和,针对CTFT的不解析部分,分因果信号和反因果信号两种情况,导出了利用CTFT计算LT的方法。

A mutual calculation method of LT(Laplace transform)and CTFT(continuous-time Fourier transform)with multiple poles on imaginary axis is proposed.Firstly,the LT with multiple poles on imaginary axis is separated into the sum of the two parts including the poles on imaginary axis and the poles not on imaginary axis through partial-fraction expansion.Aiming at the LT part with the poles on imaginary axis and via distinguishing the two cases of the left pole and the right pole,the methods for calculating CTFT by using LT are derived;then,based on decomposing CTFT of the signal into the sum of the analytical part and the non-analytical part,aiming at the non-analytical part and via distinguishing two cases of the causal signal and anti-causal signal,the methods for calculating LT by using CTFT also derived.