based on (\[direct\_est\]) and (\[reflect\_est\]), the average mean square error (MSE) of channel estimation on each SC can be derived as $$\begin{aligned}
\varepsilon &=\frac{1}{N}  \cdot {\mathbb E}\left\{  \left\|
\boldsymbol{h}_d +{\boldsymbol{V}}'{\bar{\boldsymbol{\phi}}}-\hat{\boldsymbol{h}}_d -\hat{\boldsymbol{V}}' {\bar{\boldsymbol{\phi}}}
\right\|^{2}
\right\}\nonumber\\
&=\frac{1}{N}  \cdot {\mathbb E}\left\{  \left\|
\breve{\boldsymbol{n}}_0
+\left[\breve{\boldsymbol{n}}_1-\breve{\boldsymbol{n}}_0,\ldots, \breve{\boldsymbol{n}}_K-\breve{\boldsymbol{n}}_0 \right] {\bar{\boldsymbol{\phi}}}
\right\|^{2}
\right\}\nonumber\\
&\le \frac{\sigma^2(K+1)L+\sigma^2KL_0 }{P_{t}},\label{MSE}\end{aligned}$$ where $P_t$ denotes the total transmit power during channel training. It can be observed from that the MSE is inversely proportional to the training SNR and proportional to the number of groups $K$, i.e., we can reduce the channel estimation error by increasing the transmit power during the training phase and/or the group size (i.e., smaller $K$). Meanwhile, it is worth noting that, with the proposed IRS elements grouping, the pilot training period $T_p$ is reduced to $K+1$ OFDM symbols as compared to $M+1$ without the grouping, if $K<M$.[^6]

Processing and Feedback
-----------------------

In the second phase, based on the estimated CSI $\hat{\boldsymbol{h}}_d$ and $\hat{\boldsymbol{V}}'$ at the BS, the transmit power allocation and IRS group reflection coefficients are optimized at the BS to maximize the achievable rate, which is given by $$\begin{aligned}
r(\boldsymbol{p},\bar{\boldsymbol{\phi}}|\hat{\boldsymbol{h}}_d, \hat{\boldsymbol{V}}')=\left(1-\frac{T_{\mathrm{p}}+\tau_{D}}{T_{c}}\right)\frac{1}{N+N_{\mathrm{CP}}} \sum_{n=0}^{N-1}\log_2 \left(1+\frac{\left|\boldsymbol{f}_n^H \hat{\boldsymbol{h}}_d+\boldsymbol{f}_n^H \hat{\boldsymbol{V}}'{\bar{\boldsymbol{\phi}}}
    \right|^2 p_n }{\Gamma\sigma^2}\right),\label{eqn:r2}\end{aligned}$$ where $\tau_D$ denotes the processing and feedback delay. For simplicity, 