received signals with an independent reflection coefficient. Specifically, let $\boldsymbol{\phi}=[\phi_1, \dotsc, \phi_M]^T\in\mathbb{C}^{M\times 1}$ denote the IRS reflection coefficients, where each $\phi_m=\beta_m e^{j\theta_m}$ comprises an amplitude coefficient $\beta_m\in[0,1]$ and a phase shift $\theta_m\in[-\pi,\pi)$, i.e., $|\phi_m|\leq 1$. The composite BS-IRS-user channel for the $m$-th reflecting element is thus the concatenation of the BS-IRS channel, IRS reflection, and IRS-user channel, which is given by $\boldsymbol{h}_m*\phi_m*\boldsymbol{g}_m=\phi_m\boldsymbol{h}_m*\boldsymbol{g}_m\in\mathbb{C}^{L_0\times 1}$, where $L_0=L_1+L_2-1$. For ease of exposition, define $\boldsymbol{V}=[\boldsymbol{\nu}_1,\dotsc,\boldsymbol{\nu}_M]\in\mathbb{C}^{N\times M}$ as the zero-padded concatenated BS-IRS and IRS-user channels, where each $\boldsymbol{\nu}_m\triangleq[(\boldsymbol{h}_m*\boldsymbol{g}_m)^T,\boldsymbol{0}_{1\times(N-L_0)}]^T\in\mathbb{C}^{N\times 1}$. The composite BS-IRS-user channel for all IRS reflecting elements, denoted by $\boldsymbol{h}_r\in\mathbb{C}^{N\times 1}$, can thus be expressed as $$\label{eqn:hr1}
    \boldsymbol{h}_r=\boldsymbol{V}\boldsymbol{\phi}.$$ Hence, the superposed channel impulse response (CIR) between the BS and the user by combining the BS-user (direct) channel and the BS-IRS-user (IRS-reflected) channel is given by $$\label{eqn:hall}
    \tilde{\boldsymbol{h}}=\boldsymbol{h}_{d}+\boldsymbol{h}_{r}.
$$

Assume OFDM modulation at the BS with a cyclic prefix (CP) of length $N_{\mathrm{CP}}$, with $N_{\mathrm{CP}}\geq\max(L, L_0)$. The channel frequency response (CFR) $\boldsymbol{v}=[v_{0},\dotsc,v_{N-1}]^T\in\mathbb{C}^{N\times 1}$ of the CIR $\tilde{\boldsymbol{h}}$ is thus given by $$\begin{aligned}
    \boldsymbol{v}=\boldsymbol{F}_N\tilde{\boldsymbol{h}}.\label{eqn:vall}
$$ It is worth noting that with given $\boldsymbol{\phi}$, knowledge of $\boldsymbol{\nu}_m$ suffices to characterize the BS-IRS-user composite channel for the $m$-th reflecting element, thus explicit knowledge of the individual channels $\boldsymbol{g}_m$ 