Chapter Summary

Chapter Summary

Key Points

• 1.

  Active RIS trades amplifier noise for gain. Each element has a low-power amplifier with gain $|g_n| \leq g_{\max}$. The passive unit-modulus constraint relaxes to $|\psi_n| \leq g_{\max}$. The reflected signal is boosted by $g_n$, and a new noise term $\sigma^2_{\text{RIS}} |g_n|^2$ appears per element.

• 2.

  The passive-active crossover distance is $d^\star \sim \lambda\sqrt{N/\text{NF}}$. Below $d^\star$ passive wins (product path loss is mild enough); above it active wins (passive's $d^2 d^2$ ceiling is broken by amplification). For mmWave with $N = 256$ and NF = 5 dB, $d^\star \sim 0.1\,\text{m}$ — meaning active RIS is essentially always needed at mmWave.

• 3.

  Active RIS is algorithmically easier than passive. The feasible set $\{|\psi_n| \leq g_{\max}\}$ is convex; combined with the convex total-power constraint, the passive subproblem has a convex feasible set. WMMSE + convex solver (SOCP or gradient projection) gives global inner solutions. Compared with the NP-hard unit-modulus passive QCQP, this is a significant simplification.

• 4.

  Active RIS beats AF relay by $2N$ in SNR. Coherent combining across $N$ amplifiers (factor $N$) plus full-duplex operation (factor 2) gives $2N$ SNR advantage over an AF relay with the same total amplifier power. At $N = 256$: $27$ dB. Active RIS is the cleaner architectural choice for link-budget-limited deployments than a comparable AF relay.

• 5.

  Practical active RIS consumes $\sim 1$-$5$ W DC. Each amplifier needs bias power ($\sim 10$ mW). Total for a 256-element panel: several watts — small compared to a full active array (100 W) but significant compared to passive ($\sim 10$ mW). DC supply, thermal management, and stability (amplifier oscillation avoidance) are the real-world engineering concerns.