Imagine we have two signals in the frequncy domain X(f) and H(f), for one of the signal H(f) I have the complex pair and the frequncy at which that complex pair exists. The second could be any random signal with a single frequency in MATLAB with its time domain signal represented in MATLAB as
x = A*sin(2*pi*fc*t + phi)
now we if we want to multiply both these signals for a known frequency lets suppose 100Hz then multiplication simply becomes.
H(jw) * X(jw) = H(j*(2*pi*100)) * X(j*(2*pi*100))
where w = 2pif , its clear that w is same in both these frequencies Now if signal has multiple frequency components
x = A*sin(2*pi*fc1*t + phi) + A*sin(2*pi*fc2*t + phi) + A*sin(2*pi*fc3*t + phi)
and I have complex pairs of the signal for fc1 , fc2, fc3 , how does the multiplication exactly occur? like how does the multiplication makes sure same frequency components are multiplied in each signal , this is bit confusing , maybe I am missing something. My second question is I can reconstruct the signal of H(f) by taking the frequncy phase shift and the magnitude calculated through the complex pair add them and up in time domain and do a convolution in time domain by doing.
g = conv(x,h)
Is there any other way ?