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What we do when we multiply a signal for fourier transform?

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A.Anand Srinivasan

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actually for taking fourier transform of a signal we integrate the signal after multiplying it by exp(-st).... actually what are we doing when we multiply the signal by exp(-st).....
 

Re: fourier transform

the e^(..) is an orthogonal basis. when we multiply we split the signal to its projection on the axces. same like:

(3,5,7) = 3•i + 5•k + 7•j


where i,j,k are the regular cartezian basis.
 

Re: fourier transform

the exp(-jwt) or exp(-st) as you told is a projection on to a basis function. the basis function here is usually a sine wave of a particular frequency say w1. so it tells how much the given signal is projected on the frequency w1. liike wise when we integrate it between infinite limits, we actually see the projection of our signal on the various frequencies. thus a fourier transform gives the amplitude of the signals at different frequencies(actually the projections).
 

Re: fourier transform

one more thing. as amriths04 said e^(..) = sin( ) +cos( ) and the transforn is actualy a projection on them. This is why it is called the frequancy domain because now we know how much of the signal is in every frequancy.

for examle 5Cos(3t) means we got a 5 on the frequancy 3 ( 5 is the part of the signal projected on the axces cos(3t) ).
 
Re: fourier transform

Let me explain the thing in a different way!!!

The FT of a signal \[x(t)\] is given by the formula

\[X(j\omega)=\int\limits_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\]

By closely looking at the equation we can see that we are actually finding the cross-correlation,, at origin, between the signal \[x(t)\], which may consist of infinitely many frequencies, and the signal \[e^{-j\omega t}\] which has only one frequency \[\omega\] of course for a given \[\omega\]. Now let us see quantitatively for various values of \[\omega\] on \[x(t)\].

If \[\omega \] is small and the signal \[x(t)\] varies slowly then we can intuitively see that the integral value is large which says that there is high correlation between \[x(t)\] and \[e^{-j\omega t} \] at \[\omega \]. How much correlation i.e., the measure of the correlation is given by the number \[X(j\omega)\]. So for slowly varying signals this value will be large for low values of \[\omega \] and small for large values \[\omega \]. Or in other words we can say that \[X(j\omega)\] gives the measure of frequency content in the given signal \[x(t)\].

Hope I think I have tried to explain the FT relation in an intutive way.
 

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