Difference between delayed down-converted signal and a down-converted delayed signal 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.
Let your original signal be (leaving out any constant multiplicative factor):

$$s(t)=e^{i\omega_ct}m(t)$$

Then the down-converted signal is

$$\tilde{s}(t)=e^{i(\omega_c-\omega_{lo})t}m(t)$$

and its delayed version is

$$\tilde{s}(t-\tau)=e^{i(\omega_c-\omega_{lo})(t-\tau)}m(t-\tau)$$

whereas the delayed and down-converted signal is

$$s(t-\tau)e^{-i\omega_{lo}t}=e^{i\omega_c(t-\tau)}e^{-i\omega_{lo}t}m(t-\tau)=
\tilde{s}(t-\tau)e^{-i\omega_{lo}\tau}$$

So the correction factor $e^{-i\omega_{lo}\tau}$ indeed depends on both $\omega_{lo}$ and the delay $\tau$.