The inverse Laplace transform is a powerful tool for converting functions from the s-domain back to the time domain. It's crucial for analyzing system behavior after performing operations in the s-domain, like working with transfer functions or convolution.

Partial fraction expansion and Laplace transform pairs are key techniques for finding inverse transforms. These methods allow us to break down complex s-domain functions into simpler terms, making it easier to convert back to the time domain and understand system responses.

Inverse Laplace Transform

Inverse Laplace transform definition

• Converts a function from the s-domain (Laplace domain) back to the time domain (t-domain)
• Denoted as $\mathcal{L}^{-1}{F(s)} = f(t)$, where $F(s)$ is the Laplace transform of the time-domain function $f(t)$
• Recovers the original time-domain function from its Laplace transform representation
• Essential for analyzing the behavior of systems in the time domain after performing operations in the s-domain (transfer functions, convolution)

Partial fraction expansion for inversion

• Decomposes rational Laplace transforms into a sum of simpler terms
• Rational Laplace transform is a ratio of two polynomials in the s-domain, $F(s) = \frac{P(s)}{Q(s)}$
• Factors the denominator polynomial $Q(s)$ and expresses the rational function as a sum of partial fractions
  • Each partial fraction has a denominator that is a linear or quadratic factor of $Q(s)$ (first-order poles, second-order poles)
• Resulting partial fractions are easier to inverse Laplace transform individually using Laplace transform pairs or tables
• Time-domain function $f(t)$ is obtained by summing the inverse Laplace transforms of each partial fraction term

Laplace transform pairs and tables

• Set of correspondences between time-domain functions and their respective Laplace transforms
• Derived from the properties of the Laplace transform and well-established for common functions (exponential, sinusoidal)
• Compilation of these pairs used as a reference for finding the inverse Laplace transform of a given function
  • Tables list the time-domain function $f(t)$ and its corresponding Laplace transform $F(s)$
• Locate the entry in the table that matches the given Laplace transform and read off the corresponding time-domain function
• Common Laplace transform pairs:
  • $\mathcal{L}{1} = \frac{1}{s}$
  • $\mathcal{L}{t^n} = \frac{n!}{s^{n+1}}$, where $n$ is a non-negative integer
  • $\mathcal{L}{e^{at}} = \frac{1}{s-a}$
  • $\mathcal{L}{\sin(at)} = \frac{a}{s^2+a^2}$
  • $\mathcal{L}{\cos(at)} = \frac{s}{s^2+a^2}$

System Response in Time Domain

Time-domain response from Laplace transforms

• Determined by finding the inverse Laplace transform of the system's transfer function
• Transfer function $H(s)$ is the ratio of the Laplace transform of the output $Y(s)$ to the Laplace transform of the input $X(s)$, $H(s) = \frac{Y(s)}{X(s)}$
• Given $H(s)$ and $X(s)$, the Laplace transform of the output is calculated as $Y(s) = H(s) \cdot X(s)$
• To find the time-domain output $y(t)$:
  1. Take the inverse Laplace transform of $Y(s)$ using partial fraction expansion (if necessary)
  2. Use Laplace transform pairs or tables
  • $y(t) = \mathcal{L}^{-1}{Y(s)} = \mathcal{L}^{-1}{H(s) \cdot X(s)}$
• Resulting time-domain function $y(t)$ represents the system's response to the given input in the time domain (step response, impulse response)