Convolution in the time domain can be represented as a Hadamard (pointwise) product in the Frequency domain. Using the instructions specified at https://in.mathworks.com/matlabcentral/answers/38066-difference-between-conv-ifft-fft-when-doing-convolution I am able to obtain a convolution between two vectors both in the time and frequency domain. Now, I would like to do the reverse operation. Suppose (Using Matlab/Octave notation)
$a=[1, 2, 3, 4]$ and $b=[7, 5, 3, 1]$; Then
$c = a.*b = [7, 10, 6, 4]$. If I take a Fourier transform of this vector, I obtain
$c_{fft} = [34 + 0i, -2 - 2i, -2 + 0i, -2 + 2i]$.
Now, If I apply the duality property of convolution,
$c_{fft} = conv(FFT(a),FFT(b))$ should be true (as Hadamard multiplication in time domain is equivalent to convolution in Fourier domain). However, when I implement the same, I get the following:
$c_{fft} = [224 + 0i, 24 + 40i, -24 + 32i, -8 + 8i, -88 + 0i, -32 - 48i, 16 - 32i]$
Am I doing something wrong here? Help from domain specialists would be greatly appreciated.