Jone said:Complex numbers aren't really "complex". Imagine that ONLY the rational numbers where invented and your are to solve the equation x^2=2. So in the rational system this equation has no solution. But if we extend our number system and mnemonically write the solution to this equation as sqrt(2), then we say that we have solved the equation. Now imagine only the real numbers existed and we are to solve x^2=-1. Again, we'll have to extend the number system to solve this equation. We mnemonically write the solution of this equation as x = +/-i, where i=sqrt(-1). When we do calculations where sqrt(2) or i are involved these numbers stands out like a sore tumb and we just calculate "around" them. My point here was to explain that the i isn't anything more "complex" than sqrt(2).
Ok, now to the transforms. Recall that the Fourierseries is a projection of a function into an orthogonal basis exp(inwt), where n=-inf to + inf. This basis is orthogonal and has certain other nice features like that the derivative of it is inw*exp(inwt). But the Fourier series accepted only peridic functions and now we also want to obtain a fourier series for a non-periodic function. But in a sense the period of non-periodic function is infinite, so lets use that and extend the fourier series. Doing so shows that Fourier series x(t) = sum(Cn*exp(inwt)) (summing from -inf to +inf, and Cn are the coefficients) will turn in to the fourier integral. You'll find this derivation in some book on this subject. So, that's where the Fourier integral comes from. But why did we choose this basis? Say, we have an LTI system with the frequency respone H(jw) and we apply the input signal x(t) = A*exp(iwt). What will the output signal look like? Because of LTI, convolution will give y(t) = H(jw)*x(t) and exp(iwt) is an eigenfunction. So the output is just the input signal times the frequency respone of the system. But the input signals doesn't usually look like A*exp(iwt)? The thing is, that if we can tranform the input signals to the form A *exp(iwt), then one can easily calculate the output signals. This is why transforms are useful in signal processing. And, the Laplace transform is similiar to the Fourier transform. Actually the Fourier transform is just a special case of the Laplace transform where the real part of s=sigma+iw is set to zero.
Jones
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