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I have a filter whose amplitude and phase response in terms of w(omega) is known to me which i calculated numerically. Hence, the value is known only for discrete values of w(omega).

Now, I want to know the output which this filter will produce on sending a temporal pulse(pulse in time domain) as my input.

I know i need to Fourier transform my pulse( in the form summation(A(w)exp(iwt)) and then write the output as something like summation(A(w)*H(w)*exp(jwt)).

The question is how can i find the Fourier transform (i.e find A(w) at those w where response of filter is known)