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.
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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.
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 $h [iw]$.