Connection between laplace transform and fourier transform pairs

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Yet these systems exist and we can stabilize them in control loops if we could only characterize them properly Laplace can do this! Intuitive Laplace That said, like Fourier, the Laplace Transform is a form of "correlation" in that it is an integration of complex conjugate products see linked post above. The Fourier Transform results in a one dimensional plot of the magnitude and phase of this correlation versus the frequency variable for which we are correlating over.

Each sample actually has a magnitude and phase which we could show as separate plots, but important to note the results are complex quantities. With Laplace we add the ability for these phasors to grow or decay with time, and thus we determine all the base components in an arbitrary waveform often the "waveform" is the impulse response of a system as introduced above as spinning phasors that are allowed to grow or decay in time.

The result of the Laplace transform when plotted is a surface plot since we vary both parameter rate of spin or frequency and rate of decay. We typically simply plot the singularities where this surface goes to infinity as poles and where it goes to zero as zeros since every other point on that surface is uniquely determined from those poles and zeros thus it is all we need to show to completely represent it. Very cool! More Detail Laplace Transform The Laplace transform is a mathematical tool which is used to convert the differential equations representing a linear time invariant system in time domain into algebraic equations in the frequency domain.

Fourier Transform Fourier transform is a transformation technique which transforms signals from continuous-time domain to the corresponding frequency domain and viceversa. 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.

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