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| − | == definition == | + | == definition of laplace transform == |
The Laplace transform of a function ''f''(''t''), defined for all real numbers ''t'' ≥ 0, is the function ''F''(''s''), defined by: | The Laplace transform of a function ''f''(''t''), defined for all real numbers ''t'' ≥ 0, is the function ''F''(''s''), defined by: | ||
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The inverse Laplace transform is given by the following complex integral | The inverse Laplace transform is given by the following complex integral | ||
: <math>f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds,</math> | : <math>f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds,</math> | ||
| + | |||
| + | == Region of convergence == | ||
| + | The Laplace transform ''F''(''s'') typically exists for all complex numbers such that Re{''s''} > ''a'', where ''a'' is a real constant which depends on the growth behavior of ''f''(''t''), whereas the two-sided transform is defined in a range | ||
| + | ''a'' < Re{''s''} < ''b''. The subset of values of ''s'' for which the Laplace transform exists is called the ''region of convergence'' (ROC) or the ''domain of convergence''. In the two-sided case, it is sometimes called the ''strip of convergence.'' | ||
| + | |||
| + | The integral defining the Laplace transform of a function may fail to exist for various reasons. For example, when the function has infinite discontinuities in the interval of integration, or when it increases so rapidly that exp(-pt) cannot damp it sufficiently for convergence on the interval to take place. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken. | ||
| + | |||
| + | == relationship to fourier transform == | ||
| + | The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument ''s'' = ''i''ω or ''s = 2πfi'': | ||
| + | |||
| + | :<math> | ||
| + | \begin{array}{rcl} | ||
| + | F(\omega) & = & \mathcal{F}\left\{f(t)\right\} \\[1em] | ||
| + | & = & \mathcal{L}\left\{f(t)\right\}|_{s = i \omega} = F(s)|_{s = i \omega}\\[1em] | ||
| + | & = & \int_{-\infty}^{+\infty} e^{-\imath \omega t} f(t)\,\mathrm{d}t.\\ | ||
| + | \end{array} | ||
| + | </math> | ||
| + | |||
| + | Note that this expression excludes the scaling factor <math>\frac{1}{\sqrt{2 \pi}}</math>, which is often included in definitions of the Fourier transform. | ||
| + | |||
| + | This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamic system. | ||
Latest revision as of 15:26, 24 November 2008
Contents
definition of laplace transform
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:
- $ F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{0^-}^{\infty} e^{-st} f(t) \,dt. $
The lower limit of 0− is short notation to mean
- $ \lim_{\varepsilon\to 0+}\int_{-\varepsilon}^\infty $
and assures the inclusion of the entire Dirac delta function δ(t) at 0 if there is such an impulse in f(t) at 0.
The parameter s is in general complex number:
- $ s = \sigma + i \omega \, $
inverse laplace transform
The inverse Laplace transform is given by the following complex integral
- $ f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds, $
Region of convergence
The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on the growth behavior of f(t), whereas the two-sided transform is defined in a range a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence. In the two-sided case, it is sometimes called the strip of convergence.
The integral defining the Laplace transform of a function may fail to exist for various reasons. For example, when the function has infinite discontinuities in the interval of integration, or when it increases so rapidly that exp(-pt) cannot damp it sufficiently for convergence on the interval to take place. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.
relationship to fourier transform
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iω or s = 2πfi:
- $ \begin{array}{rcl} F(\omega) & = & \mathcal{F}\left\{f(t)\right\} \\[1em] & = & \mathcal{L}\left\{f(t)\right\}|_{s = i \omega} = F(s)|_{s = i \omega}\\[1em] & = & \int_{-\infty}^{+\infty} e^{-\imath \omega t} f(t)\,\mathrm{d}t.\\ \end{array} $
Note that this expression excludes the scaling factor $ \frac{1}{\sqrt{2 \pi}} $, which is often included in definitions of the Fourier transform.
This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamic system.
