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What is the difference between Laplace Transform and fourier Transform???

MIT 6.002 Circuits and Electronics, Spring 2007
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LAPLACE TRANSFORM is used to get directly the final responce of any system having transient behaviour just by knowing the initial conditions of system..initial conditions can b obtained by putting independent variable =0 in the EQ. representing ur system.FOURIER TRANSFORM is used for breakup of any varying signal in to its sine & cosine components..
Good question. Here are several different answers: 1) Laplace has 's' in it, Fourier has 'w' or 'f'. 2) To use Laplace to find an output given a system and input, you find the Laplace of the input X(s), and the system H(s), multiply them together to find the output Y(s), then inverse transform that to find y(t). Use Fourier Transforms you split the input signal X(s) into many pure sinusoids each of a known amplitude, phase, and frequency w, then directly find the associated output sinusoid for each of the inputs (same as the input but with a gain of |H(jw)| and with an added phase of angle(H(jw))). 3) Laplace is used in control system theory, Fourier is for communication theory 4) Laplace assumes zero initial conditions* (e.g. an input that first becomes non-zero at t=0) and Fourier assumes steady-state initial conditions* (e.g. a squarewave input that has been on since t=-infinity). (The asters mean think of it that way, but there are exceptions. You can define bilateral Laplace transforms that can accommodate signals that aren't multiplied by unit steps, and by introducing generalized functions like impulses you can take Fourier Transforms of almost anything*. Yeah, more asters.) 5) There's no difference if the input is zero for t<=0...just sub jw in for the s's in the Laplace and you get Fourier.
Good Question. Basically the laplace transform is used to shift the system transfer function from time domain to the frequecy domain. in fourier transform we get the frequency spectrum of the signal (i.e) the variuos frequencies in the given composite signal and their relative amplitude or gain. in laplace the 't' is replaced by (sigma+j*w) The real part help us to plot the poles and zeros of a given system. In fourier we are mostly interested with the variations of the gain of variuos frequency components in signal. so 'w' takes care of this. Graphically in fourier transform we are interested for the area lying under the graph of signal multiplied by exp(j*w). (i.e) X(W)=int(x(t)*exp(-(j*w))) dt) between the period limits of signal(-pi/2 to pi/2, etc;). And more over the fourier transform is used mainly for the finite energy signals. Fourier series is used for continous random power signals, and helps to break up signal into the corresponding harmonics of the fundemental frequencies. this comes from the basic fact that systems responds with same frequency sinusoidal outputs for a given sinusoidal input frequency.

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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. invariants of motion. 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 as electrical circuits, queue theory etc. many equations in diffusion processes has easy Laplace transform solutions.

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