Chapter Summary

Chapter Summary

Key Points

• 1.

  The RIS is passive, so only the cascaded channel is estimable. The observable quantity is $\mathbf{G} = \text{diag}(\mathbf{h}_2^*)\mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$; $\mathbf{H}_1$ and $\mathbf{h}_2$ are inherently unidentifiable separately, but the BS optimization needs only $\mathbf{G}$ so the ambiguity is harmless.

• 2.

  Identifiability requires $\tau_p \geq N$ pilot slots. Each pilot slot applies a different RIS configuration; the stacked configuration matrix $\boldsymbol{\Phi}^{\text{stack}} \in \mathbb{C}^{N \times \tau_p}$ must have row rank $N$ for the LS estimator to be well-posed.

• 3.

  DFT codebook beats ON/OFF by a factor of $2N$ in MSE. ON/OFF activates one element at a time, wasting $(N-1)/N$ of the RIS aperture per pilot slot. DFT keeps all $N$ elements active with orthogonal phase patterns, improving estimation SNR by $N$ at the same pilot length. ON/OFF is pedagogically useful; DFT is the practical default.

• 4.

  Compressed sensing breaks the $\tau_p = N$ barrier for sparse channels. When the angular-domain cascaded channel has only $L \ll N$ dominant paths (typical of mmWave and sub-THz), CS recovers $\mathbf{G}$ from $\tau_p = \mathcal{O}(L \log N)$ pilots — exponentially fewer than the naive bound. The price is computational: LASSO/OMP replaces a cheap matrix inverse.

• 5.

  Optimal pilot length scales as $\sqrt{T/\text{SNR}}$. Balancing CSI error against pilot overhead gives an interior optimum that grows much more slowly than $N$. For typical coherence budgets, $\tau_p^\star$ is a small fraction of $T$, retaining $> 95\%$ of the coherent SNR gain at $< 10\%$ overhead. The "impossible pilot cost" critique of large-$N$ RIS is overstated.