I have been trying to solve fourier transform of f(t)^n...where f(t)
is any random function.I require an answer in terms of fourier
transform of f(t)... Thanks a lot.
F[f^n] = F[f*f^(n-1)]
the fourier transform of a product is a convolution
F[f] * F[f^(n-1)]
Hence you end up with something like F[f] * .... * F[f]
which is in terms of f only(where f is not the original f but the
transform of f).
also
I[n] = int(f^n*e^(iwx))
can be reduced by parts
u = f^n
du = n*f^(n-1)*f'dx
dv = e^(iwx)dx
v = 1/iw*e^(iwx)
so
f^n*e^(iwx)/iw - n/iw*int(f^(n-1)*f'*e^(iwx))
the last integral, by parts again
u = f'
du = f''dx
dv = f^(n-1)*e^(iwx)dx
v = I[n-1]
so
I[n-1]*f' - int(I[n-1]*f')
or
I[n] = f^n*e^(iwx)/iw - n/iw*int(f^(n-1)*f'*e^(iwx)) = f^n*e^(iwx)/iw
- n/iw*(I[n-1]*f' - int(I[n-1]*f''))
or something similar.
The point here is that you can find a recursive formula for computing
it. You can then potentially reduce it even farther to a close
formula. At least for polynomials you can easily do this since the
formula will definitely terminate after a finite number of steps. Also
with sinusoids the formula should simplify to a closed form after a
few steps.
I'll leave you to do the rest of the work since it's difficult to show
mathematical derivations in ascii.
)