Prerequisites & Notation

Before You Begin

Channel estimation is where the elegant theory of Chapter 3 meets a hard engineering constraint: the RIS does not transmit. This chapter assumes fluency with standard least-squares / MMSE estimation (FSI Ch. 5–7) and DFT-based pilot design (Telecom Ch. 14). The compressed-sensing material of Section 4.4 builds on sparsity concepts from FSI Ch. 13–14.

Least-squares and MMSE channel estimation: $\hat{\mathbf{h}} = (\mathbf{S}^H\mathbf{S})^{-1}\mathbf{S}^H\mathbf{y}$ (Review ch05)
Self-check: Given $\mathbf{y} = \mathbf{S}\mathbf{h} + \mathbf{w}$ with $\mathbf{S}$ known, can you derive the LS estimate and its MSE?
DFT matrices: orthogonality, unit-modulus entries, $\mathbf{F}^H \mathbf{F} = N\mathbf{I}$ (Review ch14)
Self-check: Can you write the $(n,k)$ -th entry of the $N \times N$ DFT matrix?
Pilot-overhead tradeoff in training-based schemes(Review ch03)
Self-check: How does pilot length $\tau_p$ affect the effective throughput $(1 - \tau_p/T) R$ ?
Sparse signal recovery: $\ell_1$ minimization, RIP(Review ch13)
Self-check: What is the RIP condition on a measurement matrix $\mathbf{A}$ that guarantees unique $s$ -sparse recovery?
Cascaded channel model from Chapter 3(Review ch03)
Self-check: Can you write $\mathbf{h}_{\text{eff}}^H = \mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1$ and identify what needs to be estimated?

Notation for This Chapter

Channel-estimation symbols. Note that in this chapter we often work with the cascaded channel $\mathbf{G} = \text{diag}(\mathbf{h}_2^*)\mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$ which is what the RIS actually sees for beamforming purposes — decomposing it into $\mathbf{H}_1$ and $\mathbf{h}_2$ separately is often unnecessary.

Symbol	Meaning	Introduced
$\tau_p$	Pilot length (number of pilot symbols or, equivalently, RIS configurations used)	s02
$\boldsymbol{\Phi}^{(t)}$	RIS configuration during pilot slot $t$	s02
$\mathbf{G} \in \mathbb{C}^{N \times N_t}$	Cascaded channel $\text{diag}(\mathbf{h}_2^*)\mathbf{H}_1$ (single-UE case)	s02
$\mathbf{s}_t \in \mathbb{C}^{N_t}$	BS pilot signal at slot $t$ , $\\|\mathbf{s}_t\\|^2 = P_t$	s02
$\mathbf{F} \in \mathbb{C}^{N \times N}$	DFT matrix, $[\mathbf{F}]_{n,k} = e^{-j 2\pi nk/N}/\sqrt{N}$	s03
$L$	Number of dominant scattering paths (sparsity level)	s04
$\mathbf{A} \in \mathbb{C}^{\tau_p \times N}$	Sensing matrix in the compressed-sensing formulation	s04
$\epsilon_{\text{CSI}}$	Normalized channel estimation error: $\\|\hat{\mathbf{G}} - \mathbf{G}\\|_F / \\|\mathbf{G}\\|_F$	s05

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