Fourier Transform question

[likephysics]

Why can't Fourier transform distinguish between a clockwise and a counter clockwise rotating vector? Why does it give peaks at both + and -.
If we discard the -ve frequency and use only the +ve frequency, we can just use
$$\int$$ f(t)coswt instead of {f(t)(coswt-isinwt)}

 

[Xezlec]

What do you mean by "rotating vector"? Do you mean a complex-valued function whose phase angle rotates with time, or do you really mean a vector and some kind of multi-dimensional Fourier transform?

Fourier transforms certainly do distinguish between functions with phases that rotate in different directions! I don't know why you think they don't.

I'm also not sure what you mean by "peaks at + and -". Are you just asking why a real-valued signal has both positive and negative frequency components? That's because the positive and negative frequencies represent complex-valued functions that spiral in opposite directions, and they add together to produce a real-valued function.

I feel like I'm not answering your question. Can you try to explain a little more clearly?

 

[Xezlec]

Oh, one other thing: you mention trying to use an integral of cos(wt) f(t). That won't work because it assumes that the phase of every frequency component is zero. Not all signals have this property. For example, no matter how many cosines you add together, the value at time t=0 will just keep getting bigger and bigger, because all cosines have a value of 1 at t=0.

 

[likephysics]

Xezlec, I was reading this FT tutorial here - http://www.cis.rit.edu/htbooks/nmr/inside.htm
By 'rotating vector' , I meant complex-valued function whose phase angle rotates with time.
By + and - peaks, I did mean positive and negative frequencies.
I was trying to calculate the FT of a delta function by multiplying the delta function by (coswt-isinwt) for different w and then adding up.
It works for a function like cos 4t+cos9t (as in the link above). I had never thought of FT in this way. Me so dumb. For this example, it is enough to multiply only by coswt. No need for isinwt.