The equivalent baseband received signal in the frequency domain after CP removal is then given by $$\begin{aligned}
\label{direct_pilot}
\boldsymbol{s}_0 = \boldsymbol{X}_p\boldsymbol{F}_N\boldsymbol{h}_d +\boldsymbol{n}_0,\end{aligned}$$ where $\boldsymbol{n}_0$ denotes the receiver noise with each element modelled by an independent CSCG random variable with mean zero and variance $\sigma^2$. Then, at the $(k+1)$-th pilot symbol, $k\in\mathcal{K}$, only the elements in the $k$-th group are switched on for full-reflection, while the remaining elements in other groups are switched off, i.e., $\bar{\boldsymbol{\phi}}=\boldsymbol{e}_k$, $k\in\mathcal{K}$. The equivalent baseband received signal in the frequency domain after CP removal is therefore given by $$\begin{aligned}
\label{IRS_pilot}
\boldsymbol{s}_k = \boldsymbol{X}_p\boldsymbol{F}_N(\boldsymbol{h}_d +\boldsymbol{\nu}'_k) +\boldsymbol{n}_k,
\quad\quad k\in\mathcal{K},\end{aligned}$$ where $\boldsymbol{n}_k$ denotes the receiver noise at the ($k+1$)-th pilot symbol duration, with $\boldsymbol{n}_k\!\sim\!\mathcal{CN}\left(0,\sigma^2\boldsymbol{I}_N\right)$. Then, based on (\[direct\_pilot\]) and (\[IRS\_pilot\]), the BS-user channel and the composite reflecting channel can be obtained at the receiver by employing the least square (LS) estimation, which are expressed as [@lsest; @lsest2] $$\begin{aligned}
\hat{\boldsymbol{h}}_d &=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \boldsymbol{s}_0\right]^T_{1:L},\; \boldsymbol{0}_{1\times (N-L)}\right]^T=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \left(\boldsymbol{X}_p\boldsymbol{F}_N\boldsymbol{h}_d +\boldsymbol{n}_0\right) \right]^T_{1:L},\; \boldsymbol{0}_{1\times (N-L)}\right]^T\notag\\
&={\boldsymbol{h}}_d+\breve{\boldsymbol{n}}_0
,\label{direct_est}\\ 
\hat{\boldsymbol{\nu}}'_k &=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \boldsymbol{s}_k-\hat{\boldsymbol{h}}_d\right]^T_{1:L_0},\; \boldsymbol{0}_{1\times (N-L_0)}\right]^T\notag\\
&=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \left( \boldsymbol{X}_p\boldsymbol{F}_N(\boldsymbol{h}_d +\boldsymbol{\nu}'_k) +\boldsymbol{n}_k \right)-{\boldsymbol{h}}_d-\breve{\boldsymbol{n}}_0\right]^T_{1:L_0},\; \boldsymbol{0}_{1\times (N-L_0)}\right]^T\notag\\
&={\boldsymbol{\nu}}'_k +\breve{\boldsymbol{n}}_k-\breve{\boldsymbol{n}}_0
, \quad k\in\mathcal{K},\label{reflect_est}\end{aligned}$$ where $\breve{\boldsymbol{n}}_0=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \boldsymbol{n}_0 \right]^T_{1:L},\; \boldsymbol{0}_{1\times (N-L)}\right]^T$ and $\breve{\boldsymbol{n}}_k=\left[\left[\frac{1}{N}\boldsymbol{F}^H_N \boldsymbol{X}_p^{-1} \boldsymbol{n}_k \right]^T_{1:L_0},\; \boldsymbol{0}_{1\times (N-L_0)}\right]^T$ for $k\in\mathcal{K}$.[^5] Moreover, 