# Relation between fourier and laplace transforms of 1

Published в Investing input grounded | Октябрь 2, 2012

The above diagram explains the function of Fourier series. Relation to Fourier -Transform. Taking limit as in Eqn(1), as is absolutely integrable. The Laplace transform is applied for solving the differential equations that relate the input and output of a system. The Fourier transform is. This operation transforms a given function to a new function in a different independent variable. For example, the Laplace transform of ƒ(t) = cos(3t) is F(s). INCREASING AND DECREASING FUNCTIONS USING DERIVATIVES IN INVESTING

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Relation between Laplace Transform and Fourier Transform

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The Laplace transform converts differential equations into algebraic equations. Whereas the Z-transform converts difference equations discrete versions of differential equations into algebraic equations. The Laplace transform maps a continuous-time function f t to f s which is defined in the s-plane.

In the s-plane, s is a complex variable defined as: Similarly, the Z-transform maps a discrete time function f n to f z that is defined in the z-plane. Here z is a complex variable defined as: Derivation Consider a periodic train of impulses p t with a period T.

Now consider a periodic continuous time signal x nT. Take a product of the above two signals as shown below. Multiplying a continuous time signal with an impulse signal is known as the impulse sampling of a continuous time signal. Taking Laplace transform of the above signal and using the identity Thus, Which can be written as: Compare this equation with that of z-transform Thus we finally get the relation: Derived from the Impulse Invariant method Another representation: Derived from Bilinear Transform method Mapping the s-plane into the z-plane Mapping of poles located at the imaginary axis of the s-plane onto the unit circle of the z-plane.

The Laplace transform is applied for solving the differential equations that relate the input and output of a system. The Fourier transform is also applied for solving the differential equations that relate the input and output of a system.

The Laplace transform can be used to analyse unstable systems. Fourier transform cannot be used to analyse unstable systems. The Laplace transform is widely used for solving differential equations since the Laplace transform exists even for the signals for which the Fourier transform does not exist.

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Relationship between Fourier and Laplace transform

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