Ferkans — Interactive Telecom Tutor

The Simplest Idea That Works

Since the continuous-phase algorithms of Chapters 5–7 are mature, the simplest discrete-phase heuristic is: solve the continuous problem, then project each element's phase to its nearest grid point. This inherits all the continuous machinery and adds only a rounding step. For single-user problems, it is asymptotically optimal: as $N \to \infty$ , the fraction of elements at the nearest grid point converges to one, and the discrete-phase solution approaches the continuous one. For multi-user, it is typically within $0.5$ - $1\text{ dB}$ of the optimal discrete solution. This section formalizes the approach and its near-optimality.

Definition:
Nearest-Level Projection

The nearest-level projection operator $\mathcal{P}_{\Theta_B}: [0, 2\pi) \to \Theta_B$ maps a continuous phase to the nearest discrete level:

$\mathcal{P}_{\Theta_B}(\theta) = \arg\min_{\vartheta \in \Theta_B} |\theta - \vartheta|_{\text{mod } 2\pi} = \Delta\theta \cdot \text{round}(\theta / \Delta\theta) \bmod 2\pi.$

Applied element-wise, $\boldsymbol{\phi}^{\text{disc}} = \big(e^{j\mathcal{P}_{\Theta_B}(\theta_1^\star)}, \ldots, e^{j\mathcal{P}_{\Theta_B}(\theta_N^\star)}\big)$ . The projection is the Euclidean-distance-closest point on $\mathcal{M}_B$ to the continuous optimum.

Project-from-Continuous Discrete-Phase RIS

Complexity:

O(T_{\text{AO}} \cdot C_{\text{continuous}})

+

O(N)

for projection +

O(C_{\text{WMMSE}})

optional refine

Input: channels, bit-depth

B

, tolerance

\epsilon

.

Output: discrete-phase

\boldsymbol{\Phi}^{\text{disc}}

and precoder

\mathbf{W}^{\text{disc}}

.

1. Step 1 — Continuous AO: solve the continuous-phase joint problem via Chapter 5's AO.

Outputs

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

with

|\phi_n^\star| = 1

,

\theta_n^\star \in [0, 2\pi)

.

2. Step 2 — Nearest-level projection:

For each

n

:

\tilde\theta_n \leftarrow \Delta\theta \cdot \text{round}(\theta_n^\star / \Delta\theta)

.

Set

\boldsymbol{\Phi}^{\text{disc}} = \text{diag}(e^{j\tilde\theta_n})

.

3. Step 3 — Refine precoder (optional): with

\boldsymbol{\Phi}^{\text{disc}}

fixed, re-optimize

\mathbf{W}

via WMMSE.

Output

\mathbf{W}^{\text{disc}}

.

4. return

(\mathbf{W}^{\text{disc}}, \boldsymbol{\Phi}^{\text{disc}})

.

Step 3 is optional but often worth it: fixing $\boldsymbol{\Phi}^{\text{disc}}$ and re-running WMMSE recovers a bit of the projection loss by optimally matching the active beamformer to the quantized channel. Adds one extra WMMSE call per coherence block — negligible cost.

Theorem: Projection Loss for Single-User Problems

For the single-user matched-filter problem, the projection achieves SNR

$\text{SNR}^{\text{disc}} = \text{sinc}^2\!\left(\frac{\pi}{2^B}\right) \cdot \text{SNR}^{\text{cont}},$

where $\text{SNR}^{\text{cont}}$ is the continuous-phase coherent optimum. Tabulated:

$B$	$\eta_B$	SNR loss (dB)
1	0.405	3.92
2	0.811	0.91
3	0.949	0.22
4	0.987	0.056

This matches the hardware-only quantization loss from Ch. 2. For single-user, projection is essentially equivalent to quantizing the hardware — no optimization penalty beyond the physical quantization.

For single-user problems, the continuous optimum is the matched- filter phase $\theta_n^\star = -\arg(a_n)$ . Nearest-level projection introduces an error $\epsilon_n = \tilde\theta_n - \theta_n^\star$ uniformly distributed on $[-\Delta\theta/2, \Delta\theta/2]$ . The coherent-sum amplitude scales by $\mathbb{E}[e^{j\epsilon_n}] = \text{sinc}(\pi/L) = \text{sinc}(\pi/2^B)$ ; the SNR scales by its square.

Proof

Continuous optimum

$\theta_n^\star = -\arg(a_n)$ (from Chapter 6, Example 6.1).

Quantization error

$\tilde\theta_n - \theta_n^\star \in [-\Delta\theta/2, \Delta\theta/2]$ , uniformly distributed if $\theta_n^\star$ is generic. Error is i.i.d. across $n$ .

Expected coherent sum

$\mathbb{E}[\sum_n a_n e^{j\tilde\theta_n}] = \sum_n a_n \cdot \text{sinc}(\pi/2^B) e^{j\theta_n^\star + j\arg a_n}$ (using $\mathbb{E}[e^{j\epsilon}] = \text{sinc}(\pi/2^B)$ with uniform $\epsilon$ in $[-\pi/2^B, \pi/2^B]$ ).

Squared amplitude → SNR

Coherent magnitude is scaled by $\text{sinc}(\pi/2^B)$ , SNR by its square. The per-element SNR is multiplied by $\eta_B$ , and so is the coherent $N^2$ factor. $\blacksquare$

Projection Is Nearly Optimal, Not Exactly

The projection-then-refine approach achieves $\sim \eta_B$ of the continuous optimum in single-user cases — which is also what an oracle discrete-phase solver would achieve asymptotically. In this sense, projection is (nearly) optimal for single-user.

For multi-user: the continuous optimum's coupled structure means independent per-element rounding breaks the coupling. A careful joint projection (e.g., randomized rounding with multiple trials and retaining the best) recovers $\sim 0.1$ - $0.5\text{ dB}$ . This is the small edge that direct discrete optimization (Section 8.3) tries to close.

Projection Loss vs. $B$ for Single-User and Multi-User

Sweep $B \in \{1, 2, 3, 4, 5\}$ and plot the SNR loss (dB) of the projection method relative to the continuous-phase optimum for single-user and for multi-user ( $K = 4$ ) scenarios. The single-user curve matches the analytical $\eta_B$ ; the multi-user curve is slightly worse, showing the cost of rounding the coupled phases.

Parameters

RIS elements

N

64

Users

K

2

Monte-Carlo trials50

Example: Projection with Precoder Refinement

A single-user MISO-RIS has $N_t = 4, N = 32, B = 2$ . The continuous-phase solution gives rate $R^\star_{\text{cont}} = 5.0$ bits/s/Hz. Predict the discrete-phase rate with and without Step 3 refinement.

Solution

Without refinement

$\text{SNR}^{\text{disc}} = \eta_2 \cdot \text{SNR}^{\text{cont}} = 0.811 \cdot \text{SNR}^{\text{cont}}$ . Rate loss: $\log_2(\text{SNR}^{\text{cont}}/0.811) \approx 0.3$ bits/s/Hz. Discrete rate $\approx 4.7$ bits/s/Hz.

With refinement

Step 3 re-optimizes $\mathbf{W}$ for the quantized channel. For single-user MRT, this is just matched filtering on $\mathbf{h}_{\text{eff}}(\boldsymbol{\Phi}^{\text{disc}})$ — automatic. Gain: another $\sim 0.05$ - $0.1$ bit/s/Hz typically. For multi-user, more significant.

Compare

Final discrete rate: $\sim 4.75$ - $4.8$ bits/s/Hz. Total loss from continuous: $\sim 0.2$ - $0.25$ bits/s/Hz, matching the $0.91\text{ dB} / (2.3\text{ dB/bit}) \approx 0.4$ estimate. $\blacksquare$

Common Mistake: Don't Skip the Precoder Refine in Multi-User

Mistake:

"Project phases, re-use the pre-projection precoder. It's a small discretization error anyway."

Correction:

For multi-user, skipping Step 3 is a $\sim 0.3$ - $1\text{ dB}$ sum-rate loss at $B = 2$ - $3$ . The active beamformer was matched to the continuous phases; after projection, it's mismatched. Running one more WMMSE iteration on the quantized channels recovers most of this. Cost: one inner WMMSE = $\sim 10\,\mu\text{s}$ . Never skip it.

🔧Engineering Note

Projection in Real-Time Controllers

Projection adds negligible cost to the continuous-phase pipeline:

Step 2 (projection): $O(N)$ comparisons, $< 10\,\mu\text{s}$ at $N = 256$ .
Step 3 (refine): one extra WMMSE call, $< 1\,\text{ms}$ .
Total added latency: $< 1\,\text{ms}$ .

For a $20$ -ms coherence block, this is 5% overhead — well within budget. The big computational cost is the continuous AO in Step 1 ( $\sim 10$ - $100$ ms); projection is free relative to it.

Practical Constraints

•
Step 2 alone: 1 μs at N = 256 on a modern CPU.
•
Step 3 refinement: ~500 μs for $N_t = 16, K = 4$ .
•
Total projection overhead: $\leq 1\%$ of continuous AO time.

Project-from-Continuous Solutions