# Relation between fourier laplace and z transform table

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Each strip maps onto a different Riemann surface of the z "plane". Mapping of different areas of the s plane onto the Z plane is shown below. IJSER Summary Transforms are used because the time-domain mathematical models of systems are generally complex differential equations.

Transforming these complex differential equations into simpler algebraic expressions makes them much easier to solve. Once the solution to the algebraic expression is found, the inverse transform will give you the time-domain response. Laplace is used for stability studies and Fourier is used for sinusoidal responses of systems. Fourier is used primarily for steady state signal analysis, while Laplace is used for transient signal analysis. Laplace is good at looking for the response to pulses, step functions, delta functions, while Fourier is good for continuous signals.

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.

This is an important condition for accurate transformation. Mapping of the stable poles on the left-hand side of the imaginary s-plane axis into the unit circle on the z-plane.

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