Example of using FFT to convolve two functions:

First show ordinary (non-FFT) convolution:

octave:2> a=[1 2 3]
a =

  1  2  3

octave:3> b=[4 5 9]
b =

  4  5  9

octave:4> c=conv(a,b)
c =

   4  13  31  33  27

Now show how we can use FFT to get same result:

octave:5> A=fft([a 0 0])
A =

 Columns 1 through 3:

   6.00000 + 0.00000i  -0.80902 - 3.66547i   0.30902 + 1.67760i

 Columns 4 and 5:

   0.30902 - 1.67760i  -0.80902 + 3.66547i

octave:6> B=fft([b 0 0])
B =

 Columns 1 through 3:

   18.0000 +  0.0000i   -1.7361 - 10.0453i    2.7361 +  5.6206i

 Columns 4 and 5:

    2.7361 -  5.6206i   -1.7361 + 10.0453i

octave:7> ifft(A.*B)
ans =

   4.0000  13.0000  31.0000  33.0000  27.0000