Quantization Signal-to-Noise Ratio (QSNR)

• QSNR is the effective signal-to-noise ratio resulting from quantization nonlinearity and additive noise, providing a clear metric for ADC performance.
• Its analysis shows that optimal low-bit quantization, especially via asymmetric techniques, preserves channel capacity even under moderate-to-high SNR conditions.
• QSNR evaluation is crucial for system design in high-bandwidth communications and compressive imaging, ensuring robust performance despite hardware limitations.

Quantization Signal-to-Noise Ratio (QSNR) refers to the effective signal-to-noise ratio arising from the combination of additive noise and non-linear distortion induced by quantization. QSNR is a central metric in evaluating system performance where analog-to-digital conversion is performed with low precision, such as in high-bandwidth communication receivers, lensless compressive imaging architectures, and digitized fronthaul transport for O-RAN systems. Its value depends on quantization resolution, quantizer symmetry, input distribution, and noise statistics, and it directly impacts achievable communication rates and detection accuracy.

1. Quantization Fundamentals and QSNR Definition

Quantization is the process of mapping a continuous-valued signal to a discrete set of output levels, typically in $M = 2^m$ bins for an $m$-bit quantizer. The output quantized signal $y = Q(x)$ can be modeled by the Bussgang decomposition as

$y = a_x x + \eta_x$

where $x$ is the input, $a_x$ is the Bussgang gain, and $\eta_x$ is distortion uncorrelated with $x$.

QSNR is defined for the noiseless case as

$$\mathrm{QSNR} = \mathrm{SDNR} = \frac{a_x^2 \mathbb{E}[x^2]}{\mathbb{E}[\eta_x^2]} = \frac{a_x^2}{\gamma_x - a_x^2}$$

with

$$a_x = \frac{1}{\mathbb{E}[x^2]} \int x Q(x) p_x(x)dx,\quad \gamma_x = \frac{1}{\mathbb{E}[x^2]} \int Q^2(x) p_x(x)dx$$

where $p_x(x)$ is the probability density of the input (Burr et al., 23 Aug 2024). For noisy observations $x = s+n$, the SDNR is generalized to

$\mathrm{SDNR} = \frac{a_s^2}{\gamma_s - a_s^2}$

where all moments are taken with respect to the signal $s$ (Burr et al., 23 Aug 2024).

QSNR quantifies the effective SNR after the quantization nonlinearity, and in communication contexts, it reflects the ratio of useful signal variance to distortion plus noise variance at the quantizer output.

2. Effect of Quantization on AWGN Channel Capacity and QSNR

In the discrete-time AWGN channel model with quantized output

$Y = Q(X + N)$

where $Q(\cdot)$ is a quantizer and $N$ is Gaussian noise, the channel capacity and associated QSNR are contingent upon the number of quantizer levels, input distribution, and threshold placements.

Major findings (0801.1185):

• For $K$ quantization bins, optimal input distributions have at most $K+1$ mass points.
• At low SNR, output quantization incurs negligible capacity loss; at moderate-to-high SNR (e.g., 10–20 dB), modest losses occur (2–3 bit ADCs retain 80–90% of infinite-precision capacity).
• The quantization bottleneck primarily affects the QSNR at high SNR, when quantization noise dominates.
• Standard uniform $K$-PAM input with ML hard-decision quantization provides rates near channel capacity over a wide SNR range.

This relationship between capacity and QSNR implies that hardware-limited systems using few-bit quantizers suffer only moderate QSNR reduction except at the limits of SNR.

SNR (dB)	2-bit QSNR (Capacity)	3-bit QSNR (Capacity)
-5	~90%	~95%
10	~85%	~92%
20	~80%	~85%

3. One-Bit Quantization: Symmetric vs. Asymmetric QSNR Penalty

For a Gaussian channel with one-bit quantization, QSNR and capacity behavior are dictated by quantizer symmetry:

• Symmetric quantizer (threshold at 0): At low SNR, capacity per unit energy is reduced by a factor of $2/\pi$, causing a $\approx 2$ dB QSNR penalty (Koch et al., 2011, Koch et al., 2012):

$\dot{C}_{\mathrm{sym}}(0) = \frac{1}{\pi\sigma^2}$

compared to unquantized

$\dot{C}(0) = \frac{1}{2\sigma^2}$

• Asymmetric quantizer (optimized threshold and input): Full QSNR is recovered at low SNR; no inherent quantization penalty:

$\dot{C}_{\mathrm{asym}}(0) = \frac{1}{2\sigma^2}$

The penalty is not fundamental to quantization but due to suboptimal quantizer signaling.

Notably, asymmetric signaling (e.g., rare, strong pulses—flash signaling) is essential for optimal QSNR recovery. If the input is peak-power limited, even asymmetric quantization cannot avoid the 2 dB QSNR reduction (Koch et al., 2012).

Quantizer Type	QSNR Penalty at Low SNR	Condition for Recovery
Symmetric, 1-bit	2 dB loss	None
Asymmetric, 1-bit	No loss	Asymmetric threshold, PPM
Peak-limited	2 dB loss	Unavoidable

4. Non-Gaussian Signals and Generalized QSNR via Bussgang Formulation

Conventional QSNR formulas are valid only for Gaussian inputs. For non-Gaussian signals, the Bussgang decomposition and QSNR (SDNR) require adjustment (Burr et al., 23 Aug 2024):

• The correct QSNR computation for non-Gaussian signals or mixed signal+noise requires integrating the actual input distribution.
• For binary signals ($s = \pm A$) and mid-rise uniform quantization, the SDNR ($=$ QSNR) can exceed input SNR at specific quantizer steps due to non-linear effects, but this does not directly translate to improved BER unless coded transmission is employed.

Example: 8-level quantizer ($M = 8$), binary input ($s = \pm 1$), noise variance $\sigma_n^2 = 0.5$

• Maximum SDNR is about 5.1 dB—higher than the 3 dB input SNR—at optimal quantizer interval.
• For 4-PAM input, SDNR peaks near 0 dB at optimum.

These nonlinearities underscore the necessity to use correct integrals and not Gaussian QSNR rule-of-thumb for complex signal modulations in practice.

Signal / Modulation	Max QSNR Achievable	Note
Binary (±A)	> Input SNR	Nonlinear saturation effects
4-PAM	≈ Input SNR	Typical quantizer trims amplitude

5. Resolution-Independence of QSNR in Compressive Imaging Architectures

In lensless compressive imaging (LCI), total QSNR is strikingly robust with respect to image resolution (Jiang et al., 2014):

• For $N$-pixel reconstructions using compressive measurements ($N$), the measurement QSNR (SNR) is invariant with respect to $N$, subject to fixed scene brightness and sensor noise:

$$\mathrm{SNR}_{LCI} > \frac{X^0}{\sqrt{2X^0 + 4\sigma^2}}$$

where $X^0$ is the total scene brightness.

• In contrast, pinhole aperture and lens-based imaging architectures, additive noise accumulates per pixel as resolution $N$ increases, inducing a QSNR that degrades with $N$ as

$\mathrm{SNR}_{PAI} = \frac{X^0}{X^0 + N\rho^2}$

• For LCI, quantization noise is not amplified with increasing resolution, permitting arbitrarily high image resolutions with robust QSNR and measurement SNR.

Architecture	SNR vs. Resolution $N$	High $N$ QSNR Behavior
LCI	Invariant (no decrease)	Constant, bounded below
Pinhole/Lens	Decreases with $N$	Approaches zero

For high-resolution imaging, LCI outperforms traditional architectures in terms of total QSNR, especially relevant for scenarios with aggressive sensor quantization.

6. System Design Implications, QSNR Optimization, and Practical Guidelines

Key recommendations arising from current research include:

• For low-SNR, high-bandwidth applications constrained to one-bit ADC, asymmetric quantization and highly asymmetric, sparse signaling (e.g., pulse-position modulation, flash signaling) are essential for attaining full QSNR (Koch et al., 2011, Koch et al., 2012).
• In O-RAN and PHY fronthaul systems with non-Gaussian modulations, QSNR must be calculated via generalized Bussgang formulas (Burr et al., 23 Aug 2024).
• When deploying low-bit ADCs (2–3 bits), capacity and QSNR losses are tolerable for a significant SNR range, enabling cost-effective and power-efficient system architectures (0801.1185).
• In LCI and other compressive imaging applications, fixed bit-depth enables QSNR preservation regardless of spatial resolution—a unique property compared to traditional sensor arrays (Jiang et al., 2014).

Practical system designers must match quantizer thresholds to input distributions, avoid symmetric quantization at low SNR, and utilize the correct SDNR/QSNR expressions for non-Gaussian input regimes to ensure intended performance.

7. Summary of QSNR Quantification Approaches

Scenario	QSNR Formula	Notes
Symmetric 1-bit Quantizer (AWGN)	$2/\pi$ reduction at low SNR	2 dB penalty, avoid for optimality
Asymmetric 1-bit Quantizer	Matches unquantized QSNR at low SNR	Requires flash signaling, optimized threshold
K-bit Quantizer (AWGN)	Empirical: 80–90% capacity at 2–3 bits	Valid for K+1 masspoint input, moderate SNR
Non-Gaussian Input (Bussgang)	$\frac{a_x^2}{\gamma_x - a_x^2}$	Integrate over actual signal distribution
LCI Imaging	$\frac{X^0}{\sqrt{2X^0 + 4\sigma^2}}$	Resolution-independent QSNR

QSNR encapsulates the performance cost—in terms of effective SNR—imposed by signal quantization. Its accurate quantification enables optimally engineered receiver frontends, fronthaul transport, and imaging systems in the presence of practical hardware constraints and diverse signal statistics.