Convolution and discrete Fourier

In the discrete Fourier setting, the convolution theorem still holds, but with an important modification. The multiplication of discrete Fourier transforms corresponds to a convolution that is circular. 

An alternative could be to simply sum the remaining products when, in some position, the shape sticks out. That would be equivalent to assuming that the signal is zero beyond the available data. In the CSTR example above that was a reasonable assumption, but in general it is not. 

Yet another way of solving the problem of missing data at the edges of the signal is to think of the signal as something that repeats itself. After the end 

In a discrete convolution, either we lose part of the signal, or we deform that part. As long as the signal is long compared to the shape it is being convoluted with, we do not worry too much about the deformation. Under those circumstances, we will loosely use the convolution theorem, as if the circular aspect were not there. 

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