Laplace Transform

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The Laplace transform is an important concept from the branch of mathematics called Functional Analysis . In actual physical systems the Laplace transform is often interpreted as a transformation from the '' Time-domain '' point of view, in which inputs and outputs are understood as functions of time, to the '' Frequency-domain '' point of view, where the same inputs and outputs are seen as functions of Complex Angular Frequency , or Radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system. The Laplace transform has many important applications in Physics , Optics , Electrical Engineering , Control Engineering , Signal Processing , and Probability Theory . The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace , who used the transform in his work on Probability Theory . The transform was discovered originally by Leonhard Euler , the prolific eighteenth-century Swiss mathematician. FORMAL DEFINITION The Laplace transform of a Function ''f''(''t''), defined for all Real Number s ''t'' ≥ 0, is the function ''F''(''s''), defined by: : $F(s)\; =\; \backslash mathcal\{L\}\; \backslash left\backslash \{f(t)\; ight\backslash \}\; =\backslash int\_\{0^-\}^\backslash infty\; e^\{-st\}\; f(t)\backslash ,dt.$ The lower limit of $0^-$ is short notation to mean $\backslash lim\_\{\backslash epsilon\; ightarrow\; +0\}\; -\backslash epsilon\; \backslash$ and assures the inclusion of the entire Dirac Delta function $\backslash delta\; (t)\; \backslash$ at 0 if there is such an impulse in ''f''(''t'') at 0. The parameter ''s'' is in general Complex : :$s\; =\; \backslash sigma\; +\; i\; \backslash omega.\; \backslash ,$ This Integral Transform has a number of properties that make it useful for analysing linear Dynamical System s. The most significant advantage is that Differentiation and Integration become multiplication and division, respectively, with $s$. (This is similar to the way that Logarithm s change an operation of multiplication of numbers to addition of their logarithms.) This changes Integral Equation s and Differential Equation s to Polynomial Equation s, which are much easier to solve. Inverse Laplace transform The inverse Laplace transform is the Bromwich Integral , which is a Complex integral given by: : $f(t)\; =\; \backslash mathcal\{L\}^\{-1\}\; \backslash left\backslash \{F(s)\; ight\backslash \}$ = rac{1}{2 \pi \imath} \int_{ \gamma - \imath \infty}^{ \gamma + \imath \infty} e^{st} F(s)\,ds, where $\backslash gamma\; \backslash$ is a real number so that the contour path of integration is in the '' Region Of Convergence '' of $F(s)\; \backslash$ normally requiring $\backslash gamma\; >\; \backslash operatorname\{Re\}(s\_p)\; \backslash$ for every Singularity $s\_p\; \backslash$ of $F(s)\; \backslash$ and $\backslash imath=\backslash sqrt\{-1\}$. If all singularities are in the left half-plane, that is for every $s\_p\; \backslash$, then $\backslash gamma\; \backslash$ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier Transform . Bilateral Laplace transform When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally intended. The Laplace transform can be alternatively defined as the ''bilateral Laplace transform'' or Two-sided Laplace Transform by extending the limits of integration to be the entire real axis. If that is done the common unilateral transform simply becomes a special case of the bilateral transform where the definition of the function being transformed is multiplied by the Heaviside Step Function . The bilateral Laplace transform is defined as follows: : $F(s)\; =\; \backslash mathcal\{L\}\backslash left\backslash \{f(t)\; ight\backslash \}\; =\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; e^\{-st\}\; f(t)\backslash ,dt.$ 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.'' 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. PROPERTIES AND THEOREMS Given the functions ''f''(''t'') and ''g''(''t''), and their respective Laplace transforms ''F''(''s'') and ''G''(''s''): :$f(t)\; =\; \backslash mathcal\{L\}^\{-1\}\; \backslash \{\; F(s)\; \backslash \}$ :$g(t)\; =\; \backslash mathcal\{L\}^\{-1\}\; \backslash \{\; G(s)\; \backslash \}$ the following is a list of properties of the Laplace transform: Linearity : $\backslash mathcal\{L\}\backslash left\backslash \{a\; f(t)\; +\; b\; g(t)\; ight\backslash \}\; =\; a\; F(s)\; +\; b\; G(s)$ Differentiation : $\backslash mathcal\{L\}\backslash \{f\text{'}\backslash \}\; =\; s\; \backslash mathcal\{L\}\backslash \{f\backslash \}\; -\; f(0)$ : $\backslash mathcal\{L\}\backslash \{f\text{'}\text{'}\backslash \}\; =\; s^2\; \backslash mathcal\{L\}\backslash \{f\backslash \}\; -\; s\; f(0)\; -\; f\text{'}(0)$ : $\backslash mathcal\{L\}\backslash left\backslash \{\; f^\{(n)\}\; ight\backslash \}\; =\; s^n\; \backslash mathcal\{L\}\backslash \{f\backslash \}\; -\; s^\{n\; -\; 1\}\; f(0)\; -\; \backslash cdots\; -\; f^\{(n\; -\; 1)\}(0)$ Frequency Division : $\backslash mathcal\{L\}\backslash \{\; t\; f(t)\backslash \}\; =\; -F\text{'}(s)$ : $\backslash mathcal\{L\}\backslash \{\; t^\{n\}\; f(t)\backslash \}\; =\; (-1)^\{n\}\; F^\{(n)\}(s)$ Frequency Integration : Integration f(t) ight\} = {1 \over s} F(s) Scaling : $\backslash mathcal\{L\}\; \backslash left\backslash \{\; f(at)\; ight\backslash \}\; =\; \{1\; \backslash over\; a\}\; F\; \backslash left\; (\; \{s\; \backslash over\; a\}\; ight\; )$ Initial value theorem : $f(0^+)=\backslash lim\_\{s\; o\; \backslash infty\}\{sF(s)\}$ Final value theorem : $f(\backslash infty)=\backslash lim\_\{s\; o\; 0\}\{sF(s)\}$, all poles in left-hand 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 functions poles are in the right hand plane (e.g. $e^t$ or $\backslash sin(t)$) the behaviour of this formula is undefined. Frequency shifting : $\backslash mathcal\{L\}\backslash left\backslash \{\; e^\{at\}\; f(t)\; ight\backslash \}\; =\; F(s\; -\; a)$ : $\backslash mathcal\{L\}^\{-1\}\; \backslash left\backslash \{\; F(s\; -\; a)\; ight\backslash \}\; =\; e^\{at\}\; f(t)$ Time shifting : $\backslash mathcal\{L\}\backslash left\backslash \{\; f(t\; -\; a)\; u(t\; -\; a)\; ight\backslash \}\; =\; e^\{-as\}\; F(s)$ : $\backslash mathcal\{L\}^\{-1\}\; \backslash left\backslash \{\; e^\{-as\}\; F(s)\; ight\backslash \}\; =\; f(t\; -\; a)\; u(t\; -\; a)$ :Note: $u(t)$ is the Heaviside Step Function . $n$th-power shifting : $\backslash mathcal\{L\}\backslash \{\backslash ,t^nf(t)\backslash \}\; =\; (-1)^nD\_s^n${Link without Title} Convolution g(t)\} = F(s) \cdot G(s) Periodic Function Period $T$ : $\backslash mathcal\{L\}\backslash \{\; f\; \backslash \}\; =\; \{1\; \backslash over\; 1\; -\; e^\{-Ts\}\}\; \backslash int\_0^T\; e^\{-st\}\; f(t)\backslash ,dt$ Laplace transform of a function's derivative It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. For the unilateral case, this approach becomes: : $\backslash mathcal\{L\}\backslash left\backslash \{\; \{\; df\; \backslash over\; dt\; \}\; ight\backslash \}$ = s \int_{0^-}^{+\infty} e^{-st} f(t)\,dt - f(0) = s \cdot \mathcal{L} \{ f(t) \} - f(0) And in the bilateral case, we have : $\backslash mathcal\{L\}\backslash left\backslash \{\; \{\; df\; \backslash over\; dt\; \}\; ight\backslash \}$ = s \int_{-\infty}^{+\infty} e^{-st} f(t)\,dt = s \cdot \mathcal{L} \{ f(t) \} Relationship to other transforms Fourier transform The Continuous Fourier Transform is equivalent to evaluating the bilateral Laplace transform with complex argument $s\; =\; i\backslash omega$: ::$F(\backslash omega)\; =\; \backslash mathcal\{F\}\backslash left\backslash \{f(t)\; ight\backslash \}$

This relationship between the Laplace and Fourier transforms is often used to determine the Frequency Spectrum of a Signal or Dynamical System .


Mellin transform

The Mellin Transform and its inverse are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform
:
we set $heta\; =\; \backslash exp(-t)$ we get a two-sided Laplace
transform.


Z-transform

The Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of

:$z\; \backslash equiv\; e^\{s\; T\}\; \backslash$

: where $T\; =\; 1/f\_s\; \backslash$ is the Sampling period (in units of time e.g. seconds) and $f\_s\; \backslash$ is the Sampling Rate (in Samples Per Second or Hertz )

Let
:$q(t)\; \backslash equiv\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash delta(t\; -\; n\; T)$

be a sampling impulse train (also called a Dirac Comb ) and
:$x\_q(t)\; \backslash equiv\; x(t)\; q(t)\; =\; x(t)\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash delta(t\; -\; n\; T)$
:: $=\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x(n\; T)\; \backslash delta(t\; -\; n\; T)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x${Link without Title} \delta(t - n T)

be the continuous-time representation of the sampled $x(t)\; \backslash$.
: $x${Link without Title} \equiv x(nT) \ are the discrete samples of $x(t)\; \backslash$.

The Laplace transform of the sampled signal $x\_q(t)\; \backslash$ is
:$X\_q(s)\; =\; \backslash int\_\{0^-\}^\{\backslash infty\}\; x\_q(t)\; e^\{-s\; t\}\; \backslash ,dt$
::$\backslash \; =\; \backslash int\_\{0^-\}^\{\backslash infty\}\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x${Link without Title} \delta(t - n T) e^{-s t} \, dt
::$\backslash \; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x${Link without Title} \int_{0^-}^{\infty} \delta(t - n T) e^{-s t} \, dt
::$\backslash \; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x${Link without Title} e^{-n s T}.

This is precisely the definition of the Z-transform of the discrete function $x${Link without Title} \
:$X(z)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x${Link without Title} z^{-n}

with the substitution of $z\; \backslash leftarrow\; e^\{s\; T\}\; \backslash$.

Comparing the last two equations, we find the relationship between the Z-transform and the Laplace transform of the sampled signal:
  { Border "1" cellspacing="0" cellpadding="6"
  - Align "center"
  12 ''n''th Root <math> \sqrt "n" class="copylinks" target="_blank">{Link without Title} {t} \cdot u(t) </math> <math> s^{-(n+1)/n} \cdot \Gamma\left(1+rac{1}{n} ight)</math> <math> s > 0 \, </math>
  - Align "center"
  13 "http://wwwseattleluxurycom/encyclopedia/entry/natural_logarithm" class="copylinks">Natural Logarithm <math> \ln \left ( { t \over t_0 } ight ) \cdot u(t) </math> <math> - { t_0 \over s} \ \ \ln(t_0 s)+\gamma \ </math> <math> s > 0 \, </math>
  - Align "center"


  - Align "center"


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  16 "http://wwwseattleluxurycom/encyclopedia/entry/Bessel_function" class="copylinks">Bessel Function <br /> of the second kind, <br /> of order 0 <math> Y_0(\alpha t) \cdot u(t)</math> &nbsp &nbsp
  - Align "center"
  17 Modified "http://wwwseattleluxurycom/encyclopedia/entry/Bessel_function" class="copylinks">Bessel Function <br /> of the second kind, <br /> of order 0 <math> K_0(\alpha t) \cdot u(t)</math> &nbsp &nbsp
  - Align "center"
  18 "http://wwwseattleluxurycom/encyclopedia/entry/Error_function" class="copylinks">Error Function <math> \mathrm{erf}(t) \cdot u(t) </math> <math> {e^{s^2/4} \operatorname{erfc} \left(s/2 ight) \over s}</math> <math> s > 0 \, </math>
  colspan 5'''Explanatory notes:'''

:<math> V O \ \ v(t)_{t=0} </math>   :<math>Z(s) { V(s) \over I(s) } \bigg_{V_o = 0}</math>   :<math>P \left{1 \over (s+\beta)} ight_{s=-\alpha} = {1 \over (\beta - \alpha)} </math>   :<math>R \left{1 \over (s+\alpha)} ight_{s=-\beta} = {1 \over (\alpha - \beta)} = {-1 \over (\beta - \alpha)} = - P </math>

Starting with the Laplace transform,

:

we find the inverse transform by first adding and subtracting the same constant α to the numerator:

:

By the shift-in-frequency property, we have

:$x(t)\; =\; e^\{-\backslash alpha\; t\}\; \backslash mathcal\{L\}\; \backslash left\backslash \{\; \{s\; \backslash over\; s^2\; +\; \backslash omega^2\}\; +\; \{\; \backslash beta\; -\; \backslash alpha\; \backslash over\; s^2\; +\; \backslash omega^2\; \}\; ight\backslash \}$

:::$=\; e^\{-\backslash alpha\; t\}\; \backslash mathcal\{L\}\; \backslash left\backslash \{\; \{s\; \backslash over\; s^2\; +\; \backslash omega^2\}\; +\; \backslash left(\; \{\; \backslash beta\; -\; \backslash alpha\; \backslash over\; \backslash omega\; \}\; ight)\; \backslash left(\; \{\; \backslash omega\; \backslash over\; s^2\; +\; \backslash omega^2\; \}\; ight)\; ight\backslash \}$

:::$=\; e^\{-\backslash alpha\; t\}\; \backslash left[\; \backslash mathcal\{L\}\; \backslash left\backslash \{\; \{s\; \backslash over\; s^2\; +\; \backslash omega^2\}\; ight\backslash \}\; +\; \backslash left(\; \{\; \backslash beta\; -\; \backslash alpha\; \backslash over\; \backslash omega\; \}\; ight)\; \backslash mathcal\{L\}\; \backslash left\backslash \{\; \{\; \backslash omega\; \backslash over\; s^2\; +\; \backslash omega^2\; \}\; ight\backslash \}\; ight]$

Finally, using the Laplace transforms for sine and cosine (see the table, above), we have

:


Example #6: Phase delay

Time function	Laplace transform
$\backslash sin\{(\backslash omega\; t+\backslash phi)\}\; \backslash$
$\backslash cos\{(\backslash omega\; t+\backslash phi)\}\; \backslash$

REFERENCES

• A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4


• William McC. Siebert, ''Circuits, Signals, and Systems'', MIT Press, Cambridge, Massachusetts, 1986. ISBN 0-262-19229-2

SEE ALSO

• Differential Equation s


• Linear Time-invariant Systems


• Fourier Transform


• Pierre Simon Laplace


• Leonhard Euler


• Exponential Function


• Complex Number

EXTERNAL LINKS

• Online Computation of the transform or inverse transform, wims.unice.fr


• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.




• Laplace Transform Table at eFunda: Engineering Fundamentals.