and $\boldsymbol{h}_m$ are not required. Hence, the overall CFR can be rewritten as $\boldsymbol{v}=\boldsymbol{F}_N\tilde{\boldsymbol{h}}=\boldsymbol{F}_N\left(\boldsymbol{h}_{d}+\boldsymbol{V}\boldsymbol{\phi}\right)$, and the CFR at each $n$-th SC is given by $$\label{eqn:vn}
    v_n=\boldsymbol{f}_n^H\boldsymbol{h}_{d}+\boldsymbol{f}_n^H\boldsymbol{V}\boldsymbol{\phi}, \quad n\in\mathcal{N},
$$ where $\boldsymbol{f}_n^H$ denotes the $n$-th row of the DFT matrix $\boldsymbol{F}_N$. Therefore, the maximum achievable rate in bits per second per Hertz (bps/Hz) is given by $$\begin{aligned}
    r(\boldsymbol{p},\boldsymbol{\phi})\!=\!\frac{1}{N\!+\!N_{\mathrm{CP}}}\!\sum_{n=0}^{N-1}\! \log_2\!\left(\!1\!+\!\frac{|\boldsymbol{f}_n^H\boldsymbol{h}_{d}\!+\!\boldsymbol{f}_n^H\boldsymbol{V}\boldsymbol{\phi}|^2p_n}{\Gamma\sigma^2} \!\right)\!,\label{eqn:rk}\end{aligned}$$ where $\Gamma\geq 1$ is the gap from channel capacity owing to a practical modulation and coding scheme (MCS); the receiver noise at each SC is modelled as an independent CSCG random variable with mean zero and variance $\sigma^2$.

Note that (\[eqn:rk\]) represents the theoretical upper-bound of the achievable rate of the considered OFDM system, which is difficult to be achieved in practice. This is because to perform OFDM coherent detection at the receiver as well as to design the transmit power allocation and IRS reflection coefficients, accurate knowledge of the CSI (i.e., $\boldsymbol{h}_d$ and $\boldsymbol{V}$) is required, which needs to be acquired at the cost of channel training and feedback overhead. Moreover, note that the dimension of the composite channel $\boldsymbol{V}$ grows linearly with the number of IRS reflecting elements, $M$, which can be very large in practice (e.g., current metasurfaces 