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Understand the Physical Meaning of Laplace Transform with a 3D Map of Rotating Vectors

Abstract: The Laplace transform is widely used in different fields, but the physical meaning of the Laplace transform is rarely graphically described in signal and systems textbooks. Three-dimensional amplitude spectrum and phase spectrum of the Laplace transform is proposed based on the three-dimensional graph of the rotation vector. The Laplace transform decomposes a signal into an infinite number of linear combinations of generalized variable amplitude sine waves. The experimental results show that the method intuitively reveals the essence and physical meaning behind the complex mathematical formula of Laplace transform.

1. 引言

2. 旋转矢量ejωt物理意义与三维图

(a) (b)

Figure 1. 3D map of rotation vector ejωt

3. 复指数est的物理意义与三维图

(a) σ < 0 (b) σ = 0 (c) σ > 0

Figure 2. 3D plot and projection plot of complex exponential est rotation vector

Figure 3. The waveform corresponding to each point of the complex frequency plane

Figure 4. The projection of the waveform corresponding to each point in the complex plane

4. 拉普拉斯变换的物理意义

$f\left(t\right)=\frac{1}{2\text{πj}}{\int }_{\sigma -j\infty }^{\sigma +j\infty }F\left(s\right){\text{e}}^{st}\text{d}s$ (1)

(a) $|F\left(s\right)|$ 幅度频谱图 (b) $\phi \left(s\right)$ 相频谱图

Figure 5. Spectrogram of Laplace transform

5. 结语

NOTES

*通讯作者。

 [1] 郑君里, 应启珩, 杨为理. 信号与系统[M]. 第二版. 北京: 高等教育出版社, 2000. [2] (美)奥本海姆. 信号与系统[M]. 第二版. 刘树棠, 译, 北京: 电子工业出版社, 2010. [3] Apte, S.D. (2016) Signals and Systems: Principles and Applications. Cambridge University Press, Cambridge, England. https://doi.org/10.1017/CBO9781316536483