# Relation between fourier and laplace transforms for dummies

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Fourier transformation sometimes has physical interpretation, for example for some mechanical models where we have quasi-periodic solutions usually because of symmetry of the system Fourier transformations gives you normal modes of oscillations. Sometimes even for nonlinear system, couplings between such oscillations are weak so nonlinearity may be approximated by power series in Fourier space. Many systems has discrete spatial symmetry crystals then solutions of equations has to be periodic so FT is quite natural for example in Quantum mechanics.

With any of normal modes you may tie finite energy, sometimes momentum etc. So during evolution, for linear system, such modes do not couple each other, and system in one of this state leaves in it forever. Every linear physical system has its spectrum of normal modes, and if coupled with some external random source of energy white noise , its evolution runs through such states from the lowest possible energy to the greatest.

It depends on initial conditions and boundary values and restrictions but for finite systems and linear equations Fourier Transform gives you transformation from linear differential equation to matrix one which is nearly always soluble and has clear theory and meaning whilst Laplace Transform from DE to algebraic one with all advantages and disadvantages of it.

Laplace transform gives you solution in terms of decaying exponents so it is quite useful in relaxation processes, but it has no physical interpretation, usually no invariants are connected to any "vectors" of such representation, there is no discrete version of such transform with physical meaning.

It is used in various engineering problems such that electrical circuits, queue theory etc. Definitely it would be easier to advice you what method of solution to use if you would describe what is the process you are trying to describe. References: try to Google such words: energy spectrum, normal modes, eigenstates, eigenvectors in context of linear differential equations - solving DE by means of integral transforms in practical way is usually described in books on Mathematical Methods in Physics, and is connected to response functions, distribution theory, Hilbert and Banach functional spaces etc.

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. The Fourier transform is rarely used for solving the differential equations since the Fourier transform does not exists for many signals.

The Laplace transform has a convergence factor and hence it is more general. The Fourier transform does not have any convergence factor. The Fourier transform is equivalent to the Laplace transform evaluated along the imaginary axis of the s-plane.

### URBAN FOREX INDICATORS

This was soon shown to be wrong. The problem was that if you watch the planets carefully, sometimes they move backwards in the sky. So Ptolemy came up with a new idea - the planets move around in one big circle, but then move around a little circle at the same time. Think of holding out a long stick and spinning around, and at the same time on the end of the stick there's a wheel that's spinning.

The planet moves like a point on the edge of the wheel. Well, once they started watching really closely, they realized that even this didn't work, so they put circles on circles on circles Eventually, they had a map of the solar system that looked like this: This "epicycles" idea turns out to be a bad theory.

One reason it's bad is that we know now that planets orbit in ellipses around the sun. The ellipses are not perfect because they're perturbed by the influence of other gravitating bodies, and by relativistic effects. But it's wrong for an even worse reason that that, as illustrated in this wonderful youtube video. In the video, by adding up enough circles, they made a planet trace out Homer Simpson's face.

It turns out we can make any orbit at all by adding up enough circles, as long as we get to vary their size and speeds. So the epicycle theory of planetary orbits is a bad one not because it's wrong, but because it doesn't say anything at all about orbits.

Claiming "planets move around in epicycles" is mathematically equivalent to saying "planets move around in two dimensions". Well, that's not saying nothing, but it's not saying much, either! A simple mathematical way to represent "moving around in a circle" is to say that positions in a plane are represented by complex numbers, so a point moving in the plane is represented by a complex function of time.

If you start by tracing any time-dependent path you want through two-dimensions, your path can be perfectly-emulated by infinitely many circles of different frequencies, all added up, and the radii of those circles is the Fourier transform of your path. Caveat: we must allow the circles to have complex radii.

This isn't weird, though. Mapping between phase and frequency on the unit circle Relationship between Laplace transform and Z-transform We employ the Laplace transform in DSP in analyzing continuous-time systems. Conversely, the z-transform is used to analyze discrete-time systems. 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.

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