Logarithms were first calculated to transform multiplication and division into easier addition and subtraction.They seem to have the same properties such as linearity, shifting and scaling associated with them.
I cant seem to put them separately and identify the purpose of each transform. I wish to see them compared on the same page so that I can have some clarity. Thus, the Laplace transform is useful for, among other things, solving linear differential equations. The Z transform maps a sequence fn to a continuous function F(z) of the complex variable z rejOmega. In fact, so long as youre in the Region of Convergence, its fair game to go back and forth between the two just by replacing jomega with s and vice versa. Laplace To Z Transform Calculator Series Is ForThe Fourier series is for periodic functions; the Fourier transform can be thought of as the Fourier series in the limit as the period goes to infinity. Also, since periodic signals are necessarily time-varying signals, I dont get the distinction youre drawing. ![]() It is just a tool used in computers for fast computations (okay, we can use it manually too). You see, on a ROC if the roots of the transfer function lie on the imaginary axis, i.e. Laplace transforms gets reduced to Continuous Time Fourier Transform. To rewind back a little, it would be good to know why Laplace transforms evolved in the first place when we had Fourier Transforms. You see, convergence of the function (signal) is a compulsory condition for a Fourier Transform to exist (absolutely summable), but there are also signals in the physical world where it is not possible to have such convergent signals. But, since analysing them is necessary, we make them converge, by multiplying a monotonously decreasing exponential e to it, which makes them converge by its very nature. This new j is given a new name s, which we often substitute as j for sinusoidal signals response of causal LTI systems. In the s-plane, if the ROC of a Laplace transform covers the imaginary axis, then its Fourier Transform will always exist, since the signal will converge. It is these signals on the imaginary axis which comprise of periodic signals ej cos t j sin t (By Eulers). ![]() So, assume we have a system that is described with a known differential equation, let say for example that we have a common RLC circuit. Also assume that a common switch is used to switch ON or OFF the circuit. Now if we want to study the circuit in the sinusoid steady state we have to use Fourier transform. Otherwise, if our analysis include the switch ON or switch OFF the circuit we have to implement the Laplace transformation for the differential equations. It Includes not only the transient phenomenon from the initial state of the system but also the final sinusoid steady state. Back at the end of the sixteenth century astronomers were starting to do nasty calculations.
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