Ferkans — Interactive Telecom Tutor

Three Ways to Split the Energy

The passivity constraint $|r_n|^2 + |t_n|^2 \leq 1$ forces the RIS element to decide how to allocate its incident energy between reflection and transmission. Three canonical protocols have emerged in the literature, each corresponding to a different choice of feasibility structure:

Energy Splitting (ES): continuous partition per element.
Mode Switching (MS): each element chooses reflect-only or transmit-only (binary).
Time Switching (TS): all elements operate in one mode at a time, alternating across time.

This section formalizes each protocol, states its feasibility set, and compares their algorithmic and performance properties.

Definition:
Energy-Splitting (ES) Protocol

In the energy-splitting protocol, each element continuously splits its incident energy between reflection and transmission:

$a_n^r, a_n^t \in [0, 1], \quad (a_n^r)^2 + (a_n^t)^2 = 1,$

with phases $\theta_n^r, \theta_n^t$ independently adjustable (or coupled per hardware). The feasible set is the unit complex sphere per element:

$\{(r_n, t_n) \in \mathbb{C}^2 : |r_n|^2 + |t_n|^2 = 1\}.$

Every element contributes to both half-spaces simultaneously. Both user sets $\mathcal{U}^r, \mathcal{U}^t$ are served in every coherence block.

ES is the most flexible protocol — no hard discretization. The $(a_n^r, a_n^t)$ are real variables on the unit circle (parametrized by a single angle $\alpha_n \in [0, \pi/2]$ : $a_n^r = \cos\alpha_n, a_n^t = \sin\alpha_n$ ). This extra real variable per element makes ES the highest-performing but also the most complex.

Definition:
Mode-Switching (MS) Protocol

In the mode-switching protocol, each element commits to one mode:

$(a_n^r, a_n^t) \in \{(1, 0), (0, 1)\},$

i.e., $\beta_n = 1$ (reflect) or $\beta_n = 0$ (transmit), selected per element. The feasible set is the 2-point set per element, or equivalently $\{0, 1\}^N$ for the vector $\boldsymbol{\beta}$ .

MS is a combinatorial problem, closer in spirit to discrete beamforming. It has fewer variables (each element has a binary choice + one phase) and can be solved via element assignment heuristics or integer-relaxation methods.

MS is the simplest protocol to implement in hardware: each element is just a switch (PIN diode) that routes incident energy to the reflection or transmission path. No continuous amplitude control is needed. This simplicity comes at a cost: $\sim 1$ dB worse than ES at typical $N$ because the per-element split is restricted to 0 or 1.

Definition:
Time-Switching (TS) Protocol

In the time-switching protocol, the entire RIS panel operates in one mode at a time, alternating across time:

Fraction $\tau$ of the coherence block: all elements reflect ( $a_n^r = 1, a_n^t = 0$ ).
Fraction $1 - \tau$ : all elements transmit ( $a_n^r = 0, a_n^t = 1$ ).

The phases are independently optimizable in each sub-slot. $\mathcal{U}^r$ is served during the reflect-slot with rate penalty factor $\tau$ ; $\mathcal{U}^t$ during the transmit-slot with factor $1 - \tau$ .

TS is the easiest to analyze: each sub-slot is a standard passive-RIS problem. The only joint variable is $\tau \in [0, 1]$ , optimized separately from the two sub-slot beamformers. TS is strictly suboptimal to ES (because TS is a special case of ES with binary amplitudes over time, while ES has continuous control), but the performance gap is small at large $N$ .

Theorem: Protocol Performance Hierarchy

For any channel realization and optimization objective (sum rate, max-min, etc.):

$R_{\text{ES}}^\star \geq R_{\text{MS}}^\star \quad \text{and} \quad R_{\text{ES}}^\star \geq R_{\text{TS}}^\star.$

$R_{\text{MS}}$ and $R_{\text{TS}}$ are not generally comparable — which wins depends on the specific geometry. However, for large $N$ with symmetric user distributions across the two half-spaces, $R_{\text{MS}} \approx R_{\text{ES}}$ (the combinatorial search approximates the continuous optimum well).

ES is the most general: continuous amplitude split per element per time. MS is ES restricted to $\{0, 1\}$ amplitudes. TS is ES restricted to homogeneous (all-elements-same-mode) amplitudes. Hence ES $\supseteq$ MS and ES $\supseteq$ TS. The rates therefore satisfy $R_{\text{ES}} \geq \max(R_{\text{MS}}, R_{\text{TS}})$ .

Proof

Feasibility inclusions

Every MS configuration is feasible under ES (the binary amplitudes are valid continuous values). Every TS configuration is also feasible under ES (a fraction of time in all-reflect, a fraction in all-transmit). Hence the feasibility sets satisfy $\mathcal{F}_{\text{MS}}, \mathcal{F}_{\text{TS}} \subseteq \mathcal{F}_{\text{ES}}$ .

Optimum ordering

Maximum of an objective over a superset is at least the maximum over a subset. So $R^\star_{\text{ES}} \geq R^\star_{\text{MS}}, R^\star_{\text{TS}}$ . $\blacksquare$

STAR-RIS Protocol Comparison

Property	ES (Energy Splitting)	MS (Mode Switching)	TS (Time Switching)
Per-element amplitude	$a_n^r, a_n^t \in [0,1]$ continuous	$(a_n^r, a_n^t) \in \{(1,0),(0,1)\}$	All elements same mode per slot
Variables per element	3 (amplitude + 2 phases)	2 (mode + 1 phase)	1 (phase); shared $\tau$
Hardware complexity	High (fine amplitude control)	Medium (binary switch)	Low (simple on/off panel)
Optimization difficulty	Non-convex + continuous	Mixed-integer (combinatorial)	Convex in $\tau$ ; passive subproblems in slots
Rate ordering (typical)	Best (upper bound)	Close to ES at large $N$	$\sim 1$ - $2$ dB below ES
Implementation maturity (2024)	Research prototypes	Commercial demos	Straightforward from passive RIS

Time-Switching STAR-RIS Optimization

Complexity:

O(S_\tau \cdot 2 \cdot C_{\text{passive}})

where

S_\tau

is the

\tau

-grid size, typically 20

Input: channels, BS power

P_t

, user sets

\mathcal{U}^r, \mathcal{U}^t

.

Output:

\tau^\star, \boldsymbol{\Phi}^{r,\star}, \boldsymbol{\Phi}^{t,\star}

, precoders.

1. For each $\tau \in [0, 1]$ (bisect or grid-search):

2.

\quad

Sub-slot 1 (reflection): solve passive-RIS problem for

\mathcal{U}^r

→

(\mathbf{W}^{r}, \boldsymbol{\Phi}^{r,\star})

.

3.

\quad

Sub-slot 2 (transmission): solve passive-RIS problem for

\mathcal{U}^t

→

(\mathbf{W}^{t}, \boldsymbol{\Phi}^{t,\star})

.

4.

\quad

Compute total rate

R(\tau) = \tau \sum_{k \in \mathcal{U}^r} R_k^r + (1-\tau) \sum_{k \in \mathcal{U}^t} R_k^t

.

5. Pick

\tau^\star = \arg\max_\tau R(\tau)

.

6. return

\tau^\star

and the corresponding sub-slot solutions.

TS is the easiest protocol to optimize: the $\tau$ dimension is 1D and convex given sub-slot optima, and each sub-slot is a standard passive-RIS problem solvable via Chapters 5-7. In practice, 20 $\tau$ values suffice for fine-grained exploration.

ES vs. MS vs. TS Sum Rate

Compare the achievable sum rate of the three protocols across $N$ , SNR, and user-set ratio $|\mathcal{U}^r|/|\mathcal{U}^t|$ . ES dominates; MS approaches ES at large $N$ ; TS trails by a small margin that widens when user sets are imbalanced.

Parameters

RIS elements

N

64

BS antennas

N_t

8

Users on reflect side3

Users on transmit side3

Transmit SNR (dB)10

Key Takeaway

Choose your protocol based on hardware, not theory. ES is best in simulation; TS is easiest to deploy. MS sits between: it needs only binary switches per element, yet achieves near-ES performance at typical $N$ . For commercial deployment in 2024-2025: MS with appropriate element selection is the most common choice. For academic benchmarks and upper bounds: ES.

Common Mistake: Don't Compare Protocols Without Normalization

Mistake:

"TS has rate $\tau R_r + (1-\tau) R_t$ ; ES has $R_r + R_t$ (both served simultaneously). So ES always beats TS by 2×."

Correction:

This is an apples-to-oranges comparison. TS serves each side only a fraction of the time — its achievable rate per user is reduced. ES serves both simultaneously but with split energy per element, so per-user rate is also reduced. The correct comparison is total sum-rate over the full coherence time: in TS, total is $\tau R_r + (1-\tau) R_t$ ; in ES, total is $R_r^{\text{ES}} + R_t^{\text{ES}}$ . The ES-TS gap is typically $1$ - $3\text{ dB}$ — significant but not the 3 dB naive factor of 2.

The Three Protocols: ES, MS, TS