Interference-to-Noise Ratio (INR)

• Interference-to-Noise Ratio (INR) is a metric quantifying interference power relative to thermal noise, crucial in wireless and radar system analysis.
• It categorizes operational regimes—noise-limited, interference-limited, and transitional—thereby guiding MAC/PHY design and spectrum sharing strategies.
• INR underpins capacity bounds and generalized degrees-of-freedom analyses in multi-user systems, influencing decoding strategies and interference management.

Interference-to-Noise Ratio (INR) is a foundational metric in the analysis of wireless communication, radar, and coexistence systems, quantifying the power of interference relative to the thermal noise floor. Its precise definition, operational regimes, and impact on capacity, design, and decoding span theoretical information theory, wireless system design, cognitive coexistence, and physical-layer implementation. The INR establishes thresholds for regime identification, underpins the generalized degrees-of-freedom (GDoF) in multi-user and interference channels, and directly appears in capacity bounds, error exponents, and optimization constraints for spectrum sharing. This article surveys the mathematical formulation, operational significance, and application of INR across core wireless, radar, and coexistence paradigms.

1. Mathematical Definition and Scaling of INR

INR is universally defined as the ratio of total or instantaneous interference power to the noise power over a specified bandwidth. If $P_I$ is the aggregate interference power and $N_0 B$ is the thermal noise (with noise spectral density $N_0$ and bandwidth $B$), then: $\mathrm{INR} = \frac{P_I}{N_0 B}$ This form appears in physical-layer analyses, such as for LoRa and mmWave systems (Rebato et al., 2016, Huang et al., 30 Nov 2025). In decibel scale, $$\mathrm{INR}_{\mathrm{dB}} = 10\log_{10}(P_I) - 10\log_{10}(N_0) - 10\log_{10}(B)$$. For interference-limited MIMO or multi-user channels, INR is parametrized via an exponent $\alpha$ relating the growth of cross-link interference to direct-link SNR: $$\alpha = \frac{\log\,\mathrm{INR}}{\log\,\mathrm{SNR}}, \quad \mathrm{INR} = \mathrm{SNR}^\alpha$$ as standardized in the literature on interference networks and the GDoF framework (Mohajer et al., 2011, Mohanty et al., 2016, Karmakar et al., 2010, Dytso et al., 2015). This exponent $\alpha$ enables a piecewise regime analysis, distinguishing between weak ($\alpha<1$), intermediate ($\alpha=1$), and strong ($\alpha>1$) interference.

For channel models with additive interference, such as communications coexisting with radar, the interference in each symbol is modeled as a deterministic or random signal of power $I$, and noise is normalized to unit variance, yielding INR $= I$ (Brunero et al., 2019). In radar–cellular coexistence, INR is computed per receive element as $|u|^2/\sigma_R^2$, where $u$ is the interference sample and $\sigma_R^2$ is the receiver noise variance (Liu et al., 2017).

2. Operational Regimes and System Behavior

INR partitions the system operating space into noise-limited, interference-limited, and regime-transition zones, which directly inform MAC, PHY, and scheduling design.

• Noise-limited regime (INR $\ll$ 1): Performance is dominated by thermal noise; basic distributed MAC and spatial reuse suffice. For example, in 5G mmWave systems with BS density $\lambda_{BS} \lesssim 30$~BS/km$^2$, $>80\%$ of UEs have $\mathrm{INR}<0$ dB (Rebato et al., 2016).
• Interference-limited regime (INR $\gg$ 1): Aggregate interference dominates; inter-cell coordination becomes critical. For mmWave, this is seen with $\lambda_{BS} \gtrsim 120$~BS/km$^2$ (Rebato et al., 2016).
• Transitional regime: Both effects compete; hybrid MAC or adaptive strategies are needed.

This INR-regime classification is robust across models and scales, from symmetric $K$-user networks and MIMO Z-interference channels to IoT coexistence, radar–cellular environments, and fluid antenna arrays (Mohajer et al., 2011, Dytso et al., 2015, Huang et al., 30 Nov 2025, Ghadi et al., 28 Oct 2024, Liu et al., 2017).

3. INR in Information-Theoretic Capacity and GDoF Analysis

INR is a central parameter in GDoF and capacity region characterizations of multi-user and interference channels. Scaling models employ

$$\mathrm{INR} = \mathrm{SNR}^\alpha, \quad \alpha \geq 0$$

with the regime partitioning:

• Weak interference ($\alpha < 1$): Interference is sub-dominant. Each user achieves nearly full DoF, diminished linearly with $\alpha$. E.g., per-user GDoF in $K$-IC with feedback: $1-\alpha/2$ (Mohajer et al., 2011).
• Intermediate ($\alpha=1$): Direct and interfering links scale identically; this is a singular point where GDoF is not well defined and highly sensitive to scaling details.
• Strong ($\alpha > 1$): Interference may be decodable; per-user GDoF for $K$-IC with feedback: $\alpha/2$ (Mohajer et al., 2011). In MIMO Z-ICs, the sum-GDoF displays a “V-shaped” curve in $\alpha$ with minimum at $\alpha=1$ (Mohanty et al., 2016).

Capacity expressions, DMT analysis, and optimal or GDoF-optimal strategies—such as TIN no Time Sharing (TINnoTS), Han-Kobayashi splitting, or cooperative interference alignment—are parameterized or explicitly depend on INR and $\alpha$ (Karmakar et al., 2010, Dytso et al., 2015, Mohajer et al., 2011).

Regime	$\alpha$ range	Characteristic	GDoF trend
Weak	$0 \leq \alpha < 1$	Interference negligible	GDoF decreases linearly
Intermediate	$\alpha = 1$	Singular/transition	GDoF not well-defined
Strong	$\alpha > 1$	Interf. decodable/align	GDoF increases linearly

4. INR in Wireless/Radar Coexistence and Practical Decoding

In coexistence scenarios (e.g., radar–cellular), INR defines both interference constraints and performance metrics:

• The radar receiver enforces per-antenna INR constraints, $r_m = |u_m|^2/\sigma_R^2$, limiting communications-induced interference (Liu et al., 2017).
• Optimization problems simultaneously constrain INR (to protect radar) and SINR (for downlink users), with the INR bound $R_m$ trading off radar protection versus communication power (Liu et al., 2017).
• In additive radar interference channels, the decoding metric is parameterized by INR: $$p_{Y|X}(y|x) = \frac{1}{\pi}\exp(-|y-\sqrt{S}x|^2 - I) I_0(2\sqrt{I}|y-\sqrt{S}x|)$$ where $I$ is INR (Brunero et al., 2019). Algorithms that incorporate measured or estimated INR into their decoding metrics achieve strong BER improvements over legacy systems that assume INR $= 0$.

Simulation studies confirm that for $I \gg S$, performance is dominated by INR, while for $I \ll S$ (low-INR), AWGN assumptions suffice (Brunero et al., 2019). A plausible implication is that robust coexistence only needs INR-adaptive decoding, and code redesign delivers negligible extra benefit.

5. Physical-Layer Effects and Empirical Thresholds

INR captures the susceptibility of physical-layer modulations and receivers to interference and directly informs system-level design:

• For LoRa under narrowband interference, the INR at fixed SNR at which symbol error rate (SER) crosses $10^{-2}$ varies by interferer type (AWGN, BPSK, GMSK), with non-AWGN interference being less damaging (Huang et al., 30 Nov 2025).
• Analytical fits relate maximum tolerable INR to SNR, exhibiting a transition from rapid INR tolerance degradation near sensitivity (low SNR) to a linear (1:1 dB) relationship at high SNR: $$\mathrm{INR}(\mathrm{SNR}) = \alpha\,\mathrm{SNR} + \beta + \frac{\gamma}{\mathrm{SNR} - (R_T - N_0 - 1)}$$ where $\alpha\approx1$ in the high-SNR regime (Huang et al., 30 Nov 2025).
• For Demod-Remod RFI cancellation, IRR grows $\sim$1 dB per 1 dB INR above threshold, with the regime entering parameter-estimation-limited cancellation only when effective INR exceeds $-5$ dB (Li et al., 20 Dec 2025).
• In fluid antenna systems, INR appears as an exponentially distributed (in Rayleigh) random variable per port, and the statistics of its maxima over spatial ports are critical to evaluating outage, delay, and ergodic capacity (Ghadi et al., 28 Oct 2024).

6. INR-Driven Design Guidelines and Optimization

Operational and design rules in wireless, radar, and IoT networks are anchored in measured or modelled INR:

• mmWave cellular: MAC/PHY approaches change with INR; noise-limited regimes tolerate decentralized scheduling, while interference-limited require coordinated interference suppression (Rebato et al., 2016).
• IoT coexistence: Modulation-specific INR bounds (empirically fit) guide link-budget allocations, carrier-sensing thresholds, and coexistence planning, avoiding “worst-case AWGN” assumptions and overly conservative designs (Huang et al., 30 Nov 2025).
• Coexistence/cognitive radio: Setting per-antenna INR constraints bounds radar performance loss while enabling substantial communication system power savings; robust optimization extensions handle imperfect CSI (Liu et al., 2017).
• Decoding strategies: Methods that explicitly incorporate observed or estimated INR outperform AWGN-based decoders in high-interference environments, with minimal incremental gain from re-optimized codes (Brunero et al., 2019).

Key physical and algorithmic parameters—such as cell density, antenna array patterns, main/side-lobes, and code/distribution choices—mediate the realized INR distribution, strongly impacting the trade space between throughput, robustness, and coexistence (Rebato et al., 2016, Liu et al., 2017, Ghadi et al., 28 Oct 2024).

7. Advanced Topics: Statistical Models, Outage Exponents, and Capacity Approximations

INR and its statistics (PDFs, CDFs, marginals, maxima) underlay outage and reliability analysis in fading and random spatial networks:

• In random spatial networks (e.g., PPP-distributed BSs), INR is a shot-noise sum over random placements, link states (LoS/NLoS/outage), and beam directions, requiring Monte-Carlo or integral simulation for accurate ECDFs (Rebato et al., 2016).
• For FAMA with spatially correlated ports, INR maxima are evaluated via Gaussian copula models, enabling closed-form or tight approximate expressions for outage probability, delay outage rate, and ergodic capacity (Ghadi et al., 28 Oct 2024).
• Outage probability and DMT in MIMO interference channels track thresholds of INR relative to SNR and derive critical values ($\alpha=5/4$) beyond which no-CSIT and CSIT strategies become indistinguishable (Karmakar et al., 2010).
• In cognitive and radar–cellular optimization, INR constraints are enforced via convex relaxations and robustified through worst-case analysis in the presence of channel estimation errors (Liu et al., 2017).

A plausible implication is that advanced system analysis and resource allocation in interference-prone networks fundamentally depend on accurately parameterized and empirically validated INR models, jointly considering physical, spatial, and network-level phenomena.

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References:

(Mohajer et al., 2011, Rebato et al., 2016, Liu et al., 2017, Mohanty et al., 2016, Huang et al., 30 Nov 2025, Dytso et al., 2015, Brunero et al., 2019, Ghadi et al., 28 Oct 2024, Karmakar et al., 2010, Li et al., 20 Dec 2025)