I am having trouble understand what I think should be a pretty simple concept. Conceptually I can understand that a signal that has been down-converted, and then delayed in time is different than a signal that has been delayed in time and then down-converted.
I am trying to compensate for this difference on an FPGA (I am down-converting the signal, delaying in time, and then up-converting)
The way I see it, the delayed (and attenuated signal) looks like: $A e^{i(\omega_{\text{c}}-\omega_{\text{lo}})(t-\tau_1)} m(t-\tau_1)$
That is different from a down-converted delayed signal, so I need a phase correction constant: $\phi = e^{i\omega_{\text{lo}}\tau_1}$
Which gives: $A e^{i(\omega_{\text{c}}-\omega_{\text{lo}})t - i\omega_{\text{c}}\tau_1} m(t-\tau_1)$
So the correction is a function of the LO and the delay itself, right? I have my sampled signal in I and Q, but I need to multiply it by the correction before up-converting, and that is where I am lost.