Inverse Laplace Transform¶

Introduction¶

Inversion of the Laplace transform to find the signal x(t) from its Laplace transform X(s) is called the inverse Laplace transform. It is symbolically denoted as :

$$x(t)=\mathcal{L}^{-1}\left\{X(s)\right\}$$

Inversion formula¶

There is a procedure that is applicable to all classes of transform functions that involves the evaluation of a contour (closed-line) integral in complex s-plane; that is,

$$x(t)=\cfrac{1}{2\pi{}j}\int_{c-j\infty{}}^{c+j\infty{}}X(s)e^{st}ds$$

In this integral, the real c is to be selected such that if the ROC of X(s) is a₁ < Re(s) < a₂, then a₁ < c < a₂. The evaluation of this inverse Laplace transform integral requires understanding of complex variable theory. This is quite complicated and not common.

Use of Tables of Laplace Transform Pairs¶

In the second method for the inversion of X(s), we attempt to express X(s) as a sum :

$$X(s)=X_1(s)+\dots{}+X_n(s)$$

where $X_1(s)$, ... , $X_n(s)$ are functions with known inverse transforms $x_1(t)$, ... , $x_n(t)$.

Partial-Fraction expansion¶

If $X(s)$ is a rational function, that is, of the form

$$X(s)=\cfrac{N(s)}{D(s)}=k\cfrac{(s-z_1)\dots{}(s-z_m)}{(s-p_1)\dots{}(s-p_n)}$$

a simple technique based on partial-fraction expansion can be used for the inversion of X(s).

Case 1: Poles having single occurance¶

If all poles of X(s) are distinct (power of unity), then X(s) can be written as:

$$X(s)=\cfrac{c_1}{s-p_1}+\dots{}+\cfrac{c_n}{s-p_n}$$

Where coefficients c_k are given by

$$c_k=(s-p_k)X(s)|_{s=p_k}$$

Case 2: Poles having multiple occurances¶

If some poles of X(s) are having form of $(s-p_i)^r$, then the expansion of X(s) will consist of terms of the form

$$\cfrac{\lambda{}_1}{s-p_i}+\cfrac{\lambda{}_2}{(s-p_i^2)}+\dots{}+\cfrac{\lambda{}_r}{(s-p_i)^r}$$

Where,

$$\lambda_{r-k}=\cfrac{1}{k!}\cfrac{d^k}{ds^k}[(s-p_i)^rX(s)]\Big|_{s=p_i}$$

Case 3: Improper rational function¶

When X(s) is an improper rational function, that is, when m > n: $$X(s)=\cfrac{N(s)}{D(s)}=Q(s)+\cfrac{R(s)}{D(s)}$$ where N(s) and D(s) are the numerator and denominator polynomials in s, respectively, of X(s), the quotient Q(s) is a polynomial in s with degree m - n, and the remainder R(s) is a polynomial in s with degree strictly less than n. The inverse Laplace transform of Q(s) can be computed by using the transform pair:

$$\cfrac{d^k\delta{}(t)}{dt^k}\leftrightarrow{}s^k$$

Numerical techniques to find inverse Laplace transform¶

Fixed Talbot algorithm
Gaver-Stehfest algorithm
de Hoog, Knight and Stokes algorithm
Cohen acceleration algorithm