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The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant 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; the latter being easier to solve because of its algebraic form.
For example, convolution of digit sequences is the kernel operation in multiplication of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C; von zur Gathen & Gerhard 2003, §8.2).
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform , the Mellin transform , the Z-transform and the ordinary or one-sided Laplace transform .
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
For example, for a system described by transfer function = +, the impulse response ... the Laplace transform of the unit step response is = + and so the step response ...
In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.. An important example of the unilateral Z-transform is the probability-generating function, where the component [] is the probability that a discrete random variable takes the value.
An example of a weakly nonlinear circuit. The inverse multidimensional Laplace transform can be applied to simulate nonlinear circuits. This is done so by formulating a circuit as a state-space and expanding the Inverse Laplace Transform based on Laguerre function expansion.
Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K ( t , s ) = ( t 2 − s 2 ) − α {\displaystyle K(t,s)=(t^{2}-s^{2})^{-\alpha }} with 0 < α < 1 {\displaystyle 0<\alpha <1} , Volterra integral equation of the first kind can conveniently be transformed into a ...