Ferkans — Interactive Telecom Tutor

Continuous Phases Are a Fiction

Chapter 1 optimized the received SNR over phases $\theta_n \in [0, 2\pi)$ , a continuous set. Real RIS hardware supports only a finite set of phase states: a varactor driven by a 4-bit DAC realizes $2^4 = 16$ phase levels; a PIN-diode element might realize just 2. How much SNR is lost by rounding the "ideal" $\theta_n^\star$ to the nearest available quantized level? The answer — surprisingly favorable — is derived here.

Definition:
$B$ -Bit Phase Quantization

A $B$ -bit RIS element can realize $L = 2^B$ distinct phase values, typically evenly spaced:

$\Theta_B = \left\{\, 0,\ \frac{2\pi}{L},\ \frac{4\pi}{L},\ \ldots,\ \frac{2\pi (L-1)}{L}\,\right\}.$

The quantized phase is the nearest element of $\Theta_B$ :

$\theta_n^{(B)} = \arg\min_{\vartheta \in \Theta_B} |\vartheta - \theta_n^\star|_{\text{mod } 2\pi}.$

The step size is $\Delta\theta = 2\pi / L = 2\pi / 2^B$ .

Theorem: SNR Loss from $B$ -bit Phase Quantization

Under uniform-quantization with step $\Delta\theta = 2\pi/2^B$ and assuming the unquantized phase errors are uniform on $[-\Delta\theta/2, \Delta\theta/2]$ , the expected received signal power degrades by the factor

$\eta_B = \left(\frac{\sin(\pi/2^B)}{\pi/2^B}\right)^2 = \text{sinc}^2\!\left(\frac{\pi}{2^B}\right).$

The corresponding power penalty in dB is $-10 \log_{10} \eta_B$ . In particular:

Bits $B$	Levels $2^B$	$\\eta_B$	dB loss
1	2	$4/\pi^2 \approx 0.405$	$3.92\text{ dB}$
2	4	$0.811$	$0.91\text{ dB}$
3	8	$0.949$	$0.22\text{ dB}$
4	16	$0.987$	$0.056\text{ dB}$
$\infty$	$\infty$	$1$	$0\text{ dB}$

Quantization introduces a zero-mean phase error uniformly distributed in $[-\Delta\theta/2, \Delta\theta/2]$ . The coherent sum $\sum_n e^{j\epsilon_n}\,r_n$ (where $r_n$ is the aligned-phase amplitude and $\epsilon_n$ the phase error) has expected value $\mathbb{E}[e^{j\epsilon}]\sum_n r_n = \text{sinc}(\Delta\theta/2) \sum_n r_n$ . Thus the received amplitude is scaled by $\text{sinc}(\Delta\theta/2)$ , and the power by $\text{sinc}^2(\Delta\theta/2)$ .

Proof

Quantized sum

The received amplitude is $Z = \sum_n r_n e^{j\epsilon_n}$ , where $r_n = |(\mathbf{h}_2)_n^*(\mathbf{h}_1)_n|$ and $\epsilon_n = \theta_n^{(B)} - \theta_n^\star$ is the quantization error. Under uniform quantization the errors are approximately uniform on $[-\Delta\theta/2, \Delta\theta/2]$ and independent across $n$ (for randomly-distributed optimal phases).

Expected amplitude

$\mathbb{E}[Z] = \sum_n r_n \mathbb{E}[e^{j\epsilon_n}] = \big(\sum_n r_n\big) \mathbb{E}[e^{j\epsilon}]$ . Compute: $\mathbb{E}[e^{j\epsilon}] = \frac{1}{\Delta\theta}\int_{-\Delta\theta/2}^{\Delta\theta/2} e^{j\epsilon} d\epsilon = \frac{2 \sin(\Delta\theta/2)}{\Delta\theta} = \text{sinc}(\Delta\theta/(2\pi)) = \text{sinc}(\pi/2^B) \cdot (1/\pi) \cdot \pi = \frac{\sin(\pi/2^B)}{\pi/2^B}$ .

Expected power

For large $N$ , $|Z|^2 \approx |\mathbb{E}[Z]|^2 + \text{Var}(Z)$ . The first term dominates at large $N$ (coherent sum); the variance term is $\mathcal{O}(N)$ while the deterministic coherent part is $\mathcal{O}(N^2) \cdot \eta_B$ . Hence $\text{SNR}^{(B)} \approx \eta_B \cdot \text{SNR}^\star$ . $\blacksquare$

Key Takeaway

Three bits of phase resolution lose only $0.22\text{ dB}$ of coherent SNR. This is the practical argument for the industry standard of 3-bit RIS panels: the marginal gain from 3 → 4 bits ( $\sim 0.16\text{ dB}$ ) rarely justifies doubling the number of PIN diodes or control lines. Going down to 1 bit costs $3.92\text{ dB}$ — an uncomfortable but survivable penalty for a cheap prototype.

Quantization Loss vs. Bits per Element

Vary the bit resolution $B$ and compare the quantized-phase sum against the continuous-phase ideal. The blue curve shows the theoretical $\text{sinc}^2(\pi / 2^B)$ prediction; the orange markers show a Monte-Carlo empirical loss for a random set of optimal phases.

Parameters

Number of RIS elements

N

256

Max bits on the x-axis6

Rounding Continuous Phases onto a Discrete Grid

Continuous optimal phases (blue dots on the unit circle) are projected onto the nearest

B

-bit level (red). As

B

increases, the rounding residual shrinks and the coherent-sum amplitude approaches its continuous-phase ideal.

Example: A 1-bit RIS Prototype

Tang et al. (2021) built a 1-bit RIS with $N = 256$ elements at 5.8 GHz. Compute the coherent-sum loss relative to a continuous-phase ideal. If the coherent-phase ideal gives $40\text{ dB}$ received SNR, what is the 1-bit SNR?

Solution

Apply the $\eta_B$ formula

$\eta_1 = \text{sinc}^2(\pi/2) = (2/\pi)^2 = 0.405$ , so the power loss is $-10 \log_{10}(0.405) = 3.92\text{ dB}$ .

1-bit SNR

$40\text{ dB} - 3.92\text{ dB} = 36.08\text{ dB}$ . The 1-bit RIS loses almost $4\text{ dB}$ but retains $\sim 90\%$ of the capacity at high SNR (since at high SNR, $\Delta R \approx \log_2(\eta_1) / 1$ per-element scaling).

Practical note

The Tang prototype was a first-of-its-kind demonstration; its goal was to validate the $N^2$ coherent-combining law, not to maximize throughput. For that purpose, 1-bit was sufficient. Commercial deployments target at least 2 bits.

Nearest-Level Projection for $B$ -bit RIS

Complexity:

O(1)

per element,

O(N)

for the full RIS

Input: continuous optimal phase

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

; bit depth

B

Output: quantized phase

\theta^{(B)} \in \Theta_B = \{0, 2\pi/2^B, \ldots\}

1.

L \leftarrow 2^B

2.

\Delta\theta \leftarrow 2\pi / L

3.

k \leftarrow \text{round}(\theta^\star / \Delta\theta) \bmod L

4. return

\theta^{(B)} = k \Delta\theta

Nearest-level projection is the universal starting point, but it is not always optimal when the elements interact (e.g., if the objective is not a simple coherent sum). Chapter 8 revisits quantization as an integer program and compares direct discrete optimization with projection-from-continuous.

Common Mistake: Quantization Loss Is Not Additive

Mistake:

"Each element loses $\text{sinc}^2(\pi/2^B)$ of its power, so the total loss is $N\,\eta_B$ ."

Correction:

The quantization loss factor $\eta_B$ applies to the coherent sum, not to individual elements' power. Individual elements' reflected signal strength is unaffected — only the phase alignment is imperfect. The coherent-sum amplitude is reduced by $\text{sinc}(\Delta\theta/2)$ , and the coherent-sum power by $\eta_B = \text{sinc}^2(\Delta\theta/2)$ , uniformly for all $N$ . So the total coherent SNR becomes $\eta_B \cdot N^2 \alpha^2 \beta^2$ , not $N \cdot \eta_B \cdot \alpha^2 \beta^2$ .

Quick Check

A 4-bit RIS loses approximately how much coherent SNR compared to a continuous-phase RIS?

$\sim 4\text{ dB}$

$\sim 1\text{ dB}$

$\sim 0.06\text{ dB}$

$\sim 0\text{ dB}$ (indistinguishable from continuous)

Correction:

\sim 0.06\text{ dB}

From the table: $\eta_4 = \text{sinc}^2(\pi/16) = 0.987$ , giving $-10\log_{10}(0.987) \approx 0.056\text{ dB}$ .

⚠️Engineering Note

How Many Bits Should a Real RIS Have?

The bit-depth choice is not about asymptotic SNR — it is about hardware complexity vs. calibration burden:

1-bit: two PIN diodes per element. Cheapest. $\sim 4\text{ dB}$ loss. Good for prototypes, rapid beam steering, secrecy applications.
2-bit: four phase levels. Requires two diodes with carefully matched asymmetric loading. $\sim 1\text{ dB}$ loss. Common in first-generation products.
3-bit: eight phase levels. Three diodes per element, or one varactor with a 3-bit DAC. $\sim 0.2\text{ dB}$ loss. Industry default for mmWave RIS.
Continuous (varactor with $\geq 6$ -bit DAC): no bit- quantization loss but significant calibration overhead (the voltage-phase curve is nonlinear and varies with temperature).

The sweet spot depends on the application: for high-rate communication, 3-bit is enough; for radar / ISAC where phase precision is paramount, prefer continuous; for deep-pocket IoT deployments, 1-bit remains the cost optimum.

Practical Constraints

•
3-bit elements typically have 3 PIN diodes and 3 control lines per unit cell.
•
Continuous (varactor) requires per-element DAC calibration for temperature stability.
•
Control overhead scales as $B \cdot N$ bits per configuration update.

Phase Quantization: 1-bit, 2-bit, Continuous