Product in frequency domain of discrete data.

Question

I have discrete data x[n]=[1 2 2] and h[n]=[3 4 1]. I can find their frequency counterparts using the fourier series x[iw]=[5 -1 -1] & h[iw]=[8 (1/2-3*sqrt(3)*i/2) (1/2+3*sqrt(3)*i/2)]. How can I multiply them in the frequency domain so that I can then convert them back into time domain (which I can do)? Convolution in time, but I am trying to multiply in freq. domain and using the freq. data.

Answer 1

Componentwise. That is, the array describing the product in the frequency domain has three entries, each of which is the product of the corresponding entries in $x [iw]$ and $h [iw]$.

Answer 2

If you want linear convolution, then you should first zero-pad the signals to the length $N_1+N_2-1$, where $N_1$ and $N_2$ are the length of the two signals. Then compute FFT of the zero-padded signals, and multiply the spectra component by component. After that, the IFFT is the linear convolution result. Without zero-padding, the spectrum multiplication will give circulant convolution.