Prerequisites & Notation

Before You Begin

Active RIS is the natural generalization of passive RIS when the product-path-loss limitation is severe. Each element has a low- power amplifier that can boost the reflected signal, at the cost of adding noise and consuming power. The optimization framework is similar to passive RIS (AO, WMMSE, element-wise updates) but the constraints change: $|\phi_n|$ is now bounded by the amplifier gain rather than fixed at 1.

Passive RIS joint optimization (Chapter 5)(Review ch05)
Self-check: Recall the unit-modulus passive constraint and why it makes the problem non-convex.
AF relay model: amplified signal + relay noise (Telecom Ch. 22)(Review ch22)
Self-check: Write the AF relay received signal with amplifier gain $g$ and relay noise $\sigma_{\text{relay}}^2$ .
Power amplifier basics: small-signal gain, noise figure
Self-check: What does a 3-dB noise figure mean in terms of SNR degradation?
Product path loss (Chapter 1)(Review ch01)
Self-check: Recall the $1/(d_1^2 d_2^2)$ path loss scaling for passive RIS.
RIS vs. relay comparison (Chapter 1)(Review ch01)
Self-check: State the Björnson threshold: when does passive RIS beat an AF relay?

Notation for This Chapter

Active-RIS-specific notation. We reuse passive symbols and add amplifier gains and noise parameters.

Symbol	Meaning	Introduced
$g_n$	Amplifier gain for RIS element $n$	s01
$\|g_n\|^2 \leq g_{\max}^2$	Maximum amplifier gain (per-element power constraint)	s01
$P_{\text{RIS}}$	Total active-RIS transmit power (sum over elements)	s01
$\sigma^2_{\text{RIS}}$	Per-element RIS amplifier noise variance	s01
$\boldsymbol{\Psi}$	Active RIS response matrix: $\boldsymbol{\Psi} = \text{diag}(g_n e^{j\theta_n}) = \text{diag}(\psi_n)$ with $\|\psi_n\| \leq g_{\max}$	s01
$\mathbf{w}_{\text{RIS}}$	Amplified RIS noise: $\mathbf{w}_{\text{RIS}} \sim \mathcal{CN}(\mathbf{0}, \sigma^2_{\text{RIS}} \mathbf{I})$ at the RIS	s01
$\eta_{\text{ampl}}$	Amplifier efficiency (DC-to-RF ratio)	s03

← Ch 8 The Active RIS Signal Model