Fourier Transform notes

If a function is not differentiable(not smooth), then it cannot be represented by a fourier transform(sum of sines and cosines) bcoz the sum of sines and cosines are differentiable(smooth).
Functions/signals like delta function, triangle wave, cannot be represented by sum of sines and cosines. Only way to represent is to use sum of sines/cosines from +infinity to -infinity, bcoz it take a large number of frequencies to make sharp corners(or any kind of corner).
Functions can be represented as sum of sines/cosines(trignometric). Sum from 1 to k.
or
sum of complex exponentials. These are special complex exponentials that satisfy symmetry (complex num + its conjugate = real number). Since they are symmetric, the sum will be from -n to +n.
Ref: stanford lec. on fourier transforms on youtube.
To represent general periodic signals we must consider infinite sums. Mathematically infinite, but practically consider terms only till the series converges.
Any non-smooth signal will generate infinitely many fourier coefficients.
Convergence:
When the signal is continuous - yes, converges.
When the singal is smooth - yes, converges.
When the signal is not smooth and has jump discontinuity(square wave) - converges* (if t0 is a point of discontinuity, then series converges to the average value at t0)


Fourier transform of a signal f(t) is F(T) = Integral -infinity to +infinity ( f(t).coswt-isinwt).
Integral -infinity to +infinity is nothing but sum : Sigma -infinity to +infinity.
FT does not recognize the direction of rotation of a vector. So the result will be in both +x and -x.
To find the FT of a signal f(t), just multiply f(t) by (cos wt-isinwt) for w = 1 thru infinity.
isinwt is the imaginary part. Just multiply f(t) by cos wt.
When you multiply f(t) by coswt (at w=1,2,3,4........), the result will be 0 for all frequencies except at freq of f(t).
For example if f(t)= cos 4t+cos9t, then the product of ( f(t)coswt) will be zero for all w except when w=4 and w=9. Those are the frequency components in f(t).
Ref: http://www.cis.rit.edu/htbooks/nmr/inside.htm