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If I have a communications channel where a digital QAM signal is transmitted, I can either model the channel's impulse response as a complex $h(t)$, or as 2 separate impulse responses $p(t)$ and $q(t)$ for the I and Q channels, respectively. Let $s(t)$ be the transmitted signal, with $s_I(t)$ and $s_Q(t)$ denoting its I and Q components.

For the complex case, the filtered signal would be $$[s_I*\operatorname{Re}(h)-s_Q*\operatorname{Im}(h)]+j[s_I*\operatorname{Im}(h)+s_Q*\operatorname{Re}(h)]$$

For the 2 separate case, it would be $$[s_I*p]+j[s_Q*q]$$

Clearly, they look rather different. Is either one a "better" model than the other? Better in the sense of, for example, using an adaptive filter to equalize the channel. We can train for a complex response with the cross coupling of the former, or run the channels separately. If not, can we talk about some sort of mathematical equivalence between them? Is there a mathematical mapping?

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    $\begingroup$ Your second model is wrong. The wireless channel "mixes" the I and Q signal compoments, as seen in your complex model. In other words, you can't model the wireless channels as two independent channels, one for I, and the other for Q. $\endgroup$ – MBaz Apr 23 '18 at 0:38
  • $\begingroup$ In a strictly linear system, you can split the input into I and Q channels. However, after convolution (with an IR that has any finite support), neither sub-channel will remain strictly I or Q. $\endgroup$ – hotpaw2 Apr 23 '18 at 19:52
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After giving it more thought, I see my original confusion. We need to be careful about what we mean by the channel's impulse response. Normally for an equalizer for this application, we are not modeling the channel alone, but rather also including the frequency mixing up and down, before and after, transmission through the channel. When we include the mixers in this way, we get the cross coupling as I mentioned for the first case. So if $h(t)$ is the response of the channel alone, then $g(t)=e^{-j\omega_ct}h(t)$ is the response of the entire system from baseband to baseband. The second case with independent responses would be for applications where the channels are not mixed together at any stage and are transmitted and received independently.

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