Yeah, i Know but i want to know it in practically how change in time produce exponential multiplication??
Hi again,
Ok, i took a look at your PM and i think i know what you are asking here now. You probably want to see how the factor e^-ajw really affects the function as in F(w)*e^-ajw.
I'll see if i can come up with a good example of this to illustrate a little better.
The simplest way to understand it is using Laplace instead of Fourier. If we multiply F(s) times e^-sT, in the time domain we get f(t) delayed by time T. This of course means that if we had a time function like sin(wt) and multiplied its transform Laplace(sin(wt)) times e^-sT, we would end up with sin(wt) but delayed by T, which would mean the time function does not start (remains at 0) until we reach the time t=T, and then it starts and appears as it normally would).
For example, if we set the frequency to 1Hz in sin(wt) we would of course get a sine wave that starts out at t=0 and rises up as time progresses and then peaks with amplitude 1 and then later -1 and then back to zero, because it's just a sine wave. But in Laplace(sin(wt))*e^-sT with T=1, we would get a wave that is zero at t=0 and STAYS at zero until we reach the point t=1 and then and only then would the sine wave start rising and proceed in the normal way (it would again reach zero at t=2 because freq=1).
It should be pointed out however that this is not the same as sin(wt+TH), where TH is a phase shift, because that allows the sine to start at t=0 (even though it is now phase shifted) and that is not the same as using a delay of e^-sT.
The interesting thing about this is that we can have a sine wave that starts at just about any time we want just by setting T=time delay. If we say set T=0.5 in the above we would get a wave that starts out at zero and stays at zero until t=0.5 seconds and then and only then the sine wave would start to rise and take on its normal shape. This would look like a sine wave shifted by 180 degrees with the exception that it is zero for the first 0.5 seconds.
The interesting consequence of being able to use a delay like this is that we can generate other kinds of waves too where we simply add delayed (and possibly inverted) waves together.
I'll try to work up a better example of this at some point.