formed by the main diagonals of $\boldsymbol{X}$. $\|\cdot\|$ denotes the $l_2$ norm and $\left\|\cdot\right\|_F$ denotes the Frobenius norm. $\otimes$ denotes the Kronecker product and $*$ denotes the linear convolution. The distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean $\mu$ and variance $\sigma^2$ is denoted by $\mathcal{CN}(\mu,\sigma^2)$; and $\sim$ stands for “distributed as". $\mathbb{C}^{x\times y}$ denotes the space of $x\times y$ complex matrices. $\mathbb{R}$ denotes the space of real numbers. $\max(x,y)$ denotes the maximum between two real numbers $x$ and $y$. ${\mathbb E}\left\{ \cdot \right\}$ stands for the expectation operation and $\mathcal{O}(\cdot)$ stands for the standard big-O notation.

  --------------------------- --------------------------------------------------------------------------
  $\boldsymbol{h}_d$          BS-user (direct) channel
  $\boldsymbol{h}_r$          BS-IRS-user (reflected) channel
  $\tilde{\boldsymbol{h}}$    Overall impulse response for combined BS-user and BS-IRS-user channels
  $\boldsymbol{v}$            Overall frequency response for combined BS-user and BS-IRS-user channels
  $\boldsymbol{h}_m$          BS-IRS channel for the $m$-th IRS element
  $\boldsymbol{g}_m$          IRS-user channel for the $m$-th IRS element
  $\boldsymbol{\nu}_m$        BS-IRS and IRS-user composite channel for the $m$-th IRS element
  $\boldsymbol{\nu}'_k$       BS-IRS and IRS-user composite channel for the $k$-th IRS group
  $\boldsymbol{V}$            BS-IRS and IRS-user composite channel for all IRS elements
  $\boldsymbol{V}'$           BS-IRS and IRS-user composite channel for all IRS groups
  $\boldsymbol{\phi}$         IRS reflection coefficients
  $\bar{\boldsymbol{\phi}}$   IRS group reflection coefficients
  --------------------------- --------------------------------------------------------------------------

  