So I'm given a pass band filter with specific transfer function $H_p(f)$, I want to implement this via baseband processing. I already know how to take the input signal $u(t)$ and process it such that I get the I/Q components $u_I(t), u_Q(t)$. I also have figured out that the following convolution relation holds for the filter output $y$'s I/Q components $y_I=(u_I * h_I) - (u_Q*h_Q)$ and $y_Q = (u_I*h_Q) + (u_Q*h_I)$ where $h_Q$ and $h_I$ are the filters I/Q components.
I have run into an issue where I can't figure out a simple way to determine $h_Q$ and $h_I$. I know for the complex envelope of the filter I have: $h(t) = h_I(t) + j\cdot h_Q(t)$. I can take the fourier transform of this to obtain $H(f) = H_I(f) + j\cdot H_Q(f)$.
So essentially my issue is with how to obtain $H_I$ and $H_Q$. My book basically gives the answer but I must be missing something since I can't figure out the reasoning behind it. They state that $H_I(f) = (H(f) + H^*(-f))/2$ and $j\cdot H_Q(f) = (H(f)-H^*(-f))/2$.
I'm guessing its some property to do with the fourier that I've forgotten but I'm hoping someone can explain the reasoning for it. Thanks!