Properties and Theorems
The Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, by s (similarly to logarithms changing multiplication of numbers to addition of their logarithms). Because of this property, the Laplace variable s is also known as operator variable in the L domain: either derivative operator or (for s−1) integration operator. The transform turns integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the time domain.
Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s):
the following table is a list of properties of unilateral Laplace transform:
| Time domain | 's' domain | Comment | |
|---|---|---|---|
| Linearity | Can be proved using basic rules of integration. | ||
| Frequency differentiation | F′ is the first derivative of F. | ||
| Frequency differentiation | More general form, nth derivative of F(s). | ||
| Differentiation | f is assumed to be a differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts | ||
| Second Differentiation | f is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to f′(t). | ||
| General Differentiation | f is assumed to be n-times differentiable, with nth derivative of exponential type. Follow by mathematical induction. | ||
| Frequency integration | This is deduced using the nature of frequency differentiation and conditional convergence. | ||
| Integration | u(t) is the Heaviside step function. Note (u * f)(t) is the convolution of u(t) and f(t). | ||
| Time scaling | |||
| Frequency shifting | |||
| Time shifting | u(t) is the Heaviside step function | ||
| Multiplication | the integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F. | ||
| Convolution | f(t) and g(t) are extended by zero for t < 0 in the definition of the convolution. | ||
| Complex conjugation | |||
| Cross-correlation | |||
| Periodic Function | f(t) is a periodic function of period T so that f(t) = f(t + T), for all t ≥ 0. This is the result of the time shifting property and the geometric series. |
- Initial value theorem:
- Final value theorem:
- , if all poles of sF(s) are in the left half-plane.
- The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If a function's poles are in the right-hand plane (e.g. or sin(t)) the behaviour of this formula is undefined.
Read more about this topic: Laplace Transform
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)