Laplace transform

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:

s int_{-infty}^{+infty} e^{-st} f(t),dt.

Applications

The Laplace transform is used frequently in engineering and physics; the output of a linear dynamic system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication, which often makes matters easier. For more information, see control theory.

Related Topics:
Engineering - Physics - Impulse response - Convolution - Multiplication - Control theory

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The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.

Related Topics:
Solve differential equations - Electrical engineering

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Relation to other transforms

Fourier transform

The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iomega:

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::F(omega) = mathcal{F}left{f(t) ight}

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::::= mathcal{L}left{f(t) ight}|_{s = i omega} = F(s)|_{s = i omega}

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::::= int_{-infty}^{+infty} e^{-imath omega t} f(t),mathrm{d}t.

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Note that this expression excludes the scaling factor rac{1}{sqrt{2 pi}}, which is often included in definitions of the Fourier transform.

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This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical system.

Related Topics:
Frequency spectrum - Signal - Dynamical system

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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

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:G(s) = mathcal{M}left{g( heta) ight} = int_0^infty heta^s g( heta) rac{d heta}{ heta}

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we set heta = exp(-t) we get a two-sided Laplace

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transform.

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Z-transform

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

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: z equiv e^{s T}

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: where T = 1/f_s is the sampling period (in units of time e.g. seconds) and f_s is the sampling rate (in samples per second or hertz)

Related Topics:
Sampling - Sampling rate - Samples per second - Hertz

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Let

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: q(t) equiv sum_{n=0}^{infty} delta(t - n T)

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be a sampling impulse train (also called a Dirac comb) and

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: x_q(t) equiv x(t) q(t) = x(t) sum_{n=0}^{infty} delta(t - n T)

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:: = sum_{n=0}^{infty} x(n T) delta(t - n T) = sum_{n=0}^{infty} x delta(t - n T)

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be the continuous-time representation of the sampled x(t) .

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: x equiv x(nT) are the discrete samples of x(t) .

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The Laplace transform of the sampled signal x_q(t) is

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:X_q(s) = int_{0^-}^{infty} x_q(t) e^{-s t} ,dt

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:: = int_{0^-}^{infty} sum_{n=0}^{infty} x delta(t - n T) e^{-s t} , dt

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:: = sum_{n=0}^{infty} x int_{0^-}^{infty} delta(t - n T) e^{-s t} , dt

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:: = sum_{n=0}^{infty} x e^{-n s T}.

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This is precisely the definition of the Z-transform of the discrete function x

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: X(z) = sum_{n=0}^{infty} x z^{-n}

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with the substitution of z leftarrow e^{s T} .

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Comparing the last two equations, we find the relationship between the Z-transform and the Laplace transform of the sampled signal:

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:X_q(s) = X(z) Big|_{z=e^{sT}}

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Fundamental relationships

Since an ordinary Laplace transform can be written as a special case

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of a two-sided transform, and since the two-sided transform can be written as

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the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different

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characteristic problems are associated with each of these four major integral transforms.

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Properties and theorems

• Linearity

: mathcal{L}left{a f(t) + b g(t) ight}

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