# fft of point wise product

I have a sequence

  a =   [1     2     3     4     5     6     7     8],


and a weighting

   w = [ 2     3     4     5     6     7     1     2],


Now, based on convolution theorem, the Fourier transform of point-wise multiplication of two sequences is equivalent to the convolution of the Fourier transform of the two:

$\mathcal{F}\{w \cdot a\}= \mathcal{F}\{w\}*\mathcal{F}\{a\}$

where $\cdot$ indicates point-wise multiplication, $*$ in dicates cpnvolution, and $\mathcal{F}$ indicates Fourier transform.

However, my problem is when I apply this theorem to above example, it does not seem to match the convolution theorem. I am using MATLAB, to calculate

the Fourirt transform the point-wise product of the two:

 fft(w.*a) = fft([ 2     6    12    20    30    42     7    16])

= 1.0e+02 *[
1.3500 + 0.0000i
-0.5628 + 0.1763i
0.1300 - 0.1200i
0.0028 + 0.2763i
-0.3300 + 0.0000i
0.0028 - 0.2763i
0.1300 + 0.1200i
-0.5628 - 0.1763i]


and the convolution of the Fourier transform of the two:

conv(fft(w),fft(a)) =

1.0e+03 * [
1.0800 + 0.0000i
-0.4422 + 0.2072i
0.0459 - 0.0653i
-0.0239 + 0.1975i
-0.2640 + 0.0127i
-0.0597 - 0.2067i
0.0461 + 0.0187i
-0.4503 - 0.1410i
0.0000 + 0.0000i
-0.0081 - 0.0661i
0.0581 - 0.0307i
0.0261 + 0.0235i
0.0000 - 0.0127i
0.0619 - 0.0143i
0.0579 + 0.0773i]


Obviously,

fft(w.*a) $\neq$ conv(fft(w), fft(a)).

Therefore, from what I tested, it does not seem that the fft of the point-wise multiplication equals the convolution of the fft. I don't find an explanation. So where is the problem? Can anyone help please?

Thanks alot.

 1. fft(w.*a)