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I'm reading the paper Model-Driven Deep Learning for Joint MIMO Channel Estimation and Signal Detection by Hengtao He, Chao-Kai Wen, Shi Jin, and Geoffrey Ye Li on Orthogonal Approximate Message Passing (OAMP) detectors.

There is an step that it is getting me crazy, in Equation $(42)$ in Appendix B. It is the following:

enter image description here

At first, I thought that it was a consequence of a Hypothesis, but then, to derive the next expression in Equation $(44)$, it is used again this formula.

enter image description here

Is there some kind of property which I might be missing?

Regards.

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I am not really sure which part you didn't understand. I'm assuming it's the step that goes from the rightmost expression at the top to the one at the bottom. Assuming $\mathbf{q} = \mathbf{x} - \hat{\mathbf{x}}$ is i.i.d. Let me give it a try...

$ \begin{aligned} \textrm{E}[ (\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} \mathbf{H}^{\mathrm{H}}\mathbf{H} (\mathbf{x} - \hat{\mathbf{x}}) ] & = \textrm{tr} \{ \textrm{E}[ (\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} \mathbf{H}^{\mathrm{H}}\mathbf{H} (\mathbf{x} - \hat{\mathbf{x}}) ] \} \\ & = \textrm{E}[ \textrm{tr} \{(\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} \mathbf{H}^{\mathrm{H}}\mathbf{H} (\mathbf{x} - \hat{\mathbf{x}}) \} ] \\ & = \textrm{E}[ \textrm{tr} \{\mathbf{H}^{\mathrm{H}}\mathbf{H} (\mathbf{x} - \hat{\mathbf{x}}) (\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} \} ] \\ & = \textrm{tr} \{\mathbf{H}^{\mathrm{H}}\mathbf{H} ~\textrm{E}[ (\mathbf{x} - \hat{\mathbf{x}}) (\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} ] \} \\ & = \textrm{tr} \{\mathbf{H}^{\mathrm{H}}\mathbf{H} ~\textrm{E}[ (\mathbf{x} - \hat{\mathbf{x}}) (\mathbf{x} - \hat{\mathbf{x}})^{\mathrm{H}} ] \} \\ & = \textrm{tr} \{\mathbf{H}^{\mathrm{H}}\mathbf{H} ~\sigma_{q}\mathbf{I}] \} \\ & = \sigma_{q} \textrm{tr} \{\mathbf{H}^{\mathrm{H}}\mathbf{H}] \}. \end{aligned}$

where I am assuming $\sigma_{q} = \textrm{E}[ \|(\mathbf{x} - \hat{\mathbf{x}}) \|^2 ] = \frac{\textrm{E}[ \|\mathbf{q} \|^2 ]}{N_t} $.

As you can see, I am making a lot of assumptions here. I don't know if they hold since I couldn't access the paper. But I think it can shed some light on your problem (if your question is strictly about the trace operations).

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    $\begingroup$ Yes, you are right. I was so focus in the techincality and formulas that I was missing the i.i.d hypothesis. Thanks! $\endgroup$
    – Cobretti
    Commented Jan 9, 2021 at 15:19

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