# Property of the trace and expectation

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:

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.

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

Regards.

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