Noise Modulation (NoiseMod) Communication

• NoiseMod is a communication technique that encodes digital information in the variance of noise-like waveforms, offering noncoherent detection and covert transmission capabilities.
• Key implementations include TherMod, which uses inherent Johnson noise from resistors, and External-NoiseMod, which exploits synthesized Gaussian noise to balance reliability and data rate.
• Enhanced strategies like NC-NoiseMod and TD-NoiseMod leverage noncoherent detection and time diversity to improve error performance, particularly in fading channels under trade-offs between bit rate and reliability.

Noise Modulation (NoiseMod) designates a class of physical-layer communication schemes and signal-processing methodologies in which information is encoded into the statistical properties—most commonly, the variance—of noise-like waveforms, as opposed to using deterministic carriers with prescribed amplitude, phase, or frequency. This inversion of the classical paradigm, where noise is traditionally suppressed or marginalized as a transmission impairment, allows both passive (thermal) and actively synthesized (artificial) noise to function as the information-bearing medium, enabling noncoherent detection, tunable trade-offs between reliability and throughput, and channels with inherent covert/low-probability-of-detection qualities. The foundations, system models, analytical characterizations, and main design strategies of NoiseMod are set forth in (Basar, 2023), with further generalizations and practical extensions elaborated in subsequent works (Silva et al., 6 Nov 2025, Zayyani et al., 14 Sep 2025).

1. Foundations and Conceptual Framework

NoiseMod leverages the controlled statistical shaping—primarily the variance—of noise-like carriers for digital communication. Each symbol (or bit) interval corresponds to the emission of a stochastic waveform with a prescribed variance (or, in more advanced schemes, joint mean and variance, or higher-order structure). For binary signaling, a “low-variance” and “high-variance” noise process are mapped to bits 0 and 1, respectively. This general strategy encompasses two main implementations:

• Thermal-NoiseMod (TherMod): Uses physical Johnson noise, e.g., by switching between resistors of different values.
• External-NoiseMod: Employs actively generated Gaussian waveforms (“software-defined noise”) allowing increased range and flexibility.

Unlike standard amplitude/phase/frequency modulation, the information is entirely carried by the second-order (or higher-order) moment statistics rather than deterministic parameters. The receiver infers the transmitted bit by applying a variance estimator to the sampled received waveform.

2. Mathematical and System Model

The canonical baseband model for each received sample is

$s_n = h\, r_n + w_n$

where:

• $r_n \sim \mathcal{CN}(0, \sigma_i^2)$, with $\sigma_i^2 = \sigma_0^2$ (bit 0) or $\sigma_1^2 = \alpha \sigma_0^2$ (bit 1), $\alpha > 1$.
• $w_n \sim \mathcal{CN}(0, \sigma_w^2)$ is additive white Gaussian noise (AWGN).
• $h$ is the channel coefficient: $h = 1$ for AWGN channels, $h \sim \mathcal{CN}(0,1)$ for flat Rayleigh fading.
• $N$ samples are transmitted per bit.

Key parameters:

• $\delta = \sigma_0^2 / \sigma_w^2$: useful-to-noise variance ratio.
• $\alpha = \sigma_1^2 / \sigma_0^2$: modulation ratio.
• $N$: number of samples per bit, controlling the bit-rate/error tradeoff.

The received sample variance (conditioned on $h$) is

$$\sigma_s^2 = \begin{cases} \sigma_w^2 (1 + |h|^2 \delta) & b = 0 \ \sigma_w^2 (1 + |h|^2 \alpha\delta) & b = 1 \end{cases}$$

3. Modulation, Detection, and Decision Rules

At the transmitter, one maps each symbol to the appropriate variance parameter and generates $N$ independent samples of Gaussian noise. For TherMod, selecting between resistors of different values modulates the Johnson noise power.

At the receiver, a noncoherent energy detector computes

$$\hat{\sigma}_s^2 = \frac{1}{N} \sum_{n=1}^N |s_n|^2$$

By the central limit theorem, $\hat{\sigma}_s^2$ is approximately Gaussian:

$$\hat{\sigma}_s^2 \sim \mathcal{N}\left(\sigma_s^2,\, \frac{\sigma_s^4}{N}\right)$$

A single threshold test is performed:

$$\hat{b} = \begin{cases} 0 & \hat{\sigma}_s^2 < \gamma \ 1 & \hat{\sigma}_s^2 \geq \gamma \end{cases}$$

The threshold $\gamma$ can be computed by equalizing the $Q$-function arguments for both bit levels, producing the optimal (equal-error) normalized threshold

$$\chi = \frac{\gamma}{\sigma_w^2} = \frac{2(1+\delta)(1+\alpha\delta)}{2 + \delta(1+\alpha)}$$

and yielding the bit error probability (BEP)

$$P_b = Q\left( \frac{\sqrt{N}\, \delta (\alpha-1)}{2 + \delta(\alpha+1)} \right)$$

which decays exponentially in $N$ under AWGN.

For fading channels:

$$P_b(|h|) = Q\left( \frac{\sqrt{N}|h|^2\,\delta(\alpha-1)}{2+|h|^2\delta(\alpha+1)} \right)$$

$$\bar{P}_b = \int_0^\infty Q\left( \frac{\sqrt{N} u \delta(\alpha-1)}{2+u \delta(\alpha+1)} \right) e^{-u} du$$

where $u = |h|^2$. Under fading, the average BEP decreases only linearly with $N$, not exponentially.

4. Enhanced NoiseMod Strategies: Noncoherence and Diversity

Two advanced NoiseMod variants are explicitly constructed:

• Non-Coherent NoiseMod (NC-NoiseMod): The bit interval is split into two, with bit 0 sending low-variance then high-variance, and bit 1 the reverse. The receiver compares sample variances in each half. The conditional BER is

$$P_b = Q\left( \frac{\sqrt{N/2}|h|^2\,\delta(\alpha-1)}{\sqrt{(1+|h|^2\alpha\delta)^2 + (1+|h|^2\delta)^2}} \right)$$

No threshold calculation or channel knowledge is required.

• Time-Diversity NoiseMod (TD-NoiseMod): The $N$ samples are distributed across $I$ independent time slots, each experiencing its own (block) fading $h_i$. The detector aggregates over all samples. Conditioned on $\{h_i\}$, the BER is

$$P_b = Q\left( \frac{\sqrt{N}\,\delta(\alpha-1)\sum_{i=1}^I |h_i|^2}{2 + \delta(\alpha+1)\sum_{i=1}^I |h_i|^2} \right)$$

Averaging over the (chi-square) distribution for the sum $\sum |h_i|^2$ shows that TD-NoiseMod achieves $I$th order diversity: $\bar{P}_b\propto (\delta^I N)^{-1}$.

5. Analytical Performance and Trade-offs

Theoretical and simulation results from (Basar, 2023) provide the following performance summaries:

Scheme	AWGN: BEP scaling	Fading: BEP scaling	Comments
TherMod	Exponential in $N$	Linear in $N$	Simple variance estimation
NC-NoiseMod	Exponential in $N$	Linear in $N$	No channel estimation needed
TD-NoiseMod	Exponential in $N$	Diversity order $I$ in $N$	Exploits independent fades per time slot

Key observations:

• In AWGN, all variants achieve exponential decay in error with increasing $N$.
• In fading channels, diversity (TD-NoiseMod with $I>1$) is critical for suppressing error floors.
• Noncoherent detection nearly matches coherent performance while discarding the need for channel knowledge.
• For too small $N$, the sample variance distribution deviates from normality, leading to error floors and undermining theoretical scaling—implicating a minimum per-bit sample requirement for reliable operation.
• There is a strict trade-off between the bit-rate (proportional to $1/N$) and BEP: higher reliability demands more samples per bit, reducing throughput.

6. Practical Implementation and System Guidelines

Implementation of NoiseMod schemes (TherMod and External-NoiseMod) is highly hardware-efficient. In TherMod, information is embedded by switching between physical resistors, emitting only the inherent Johnson noise—requiring no oscillators or power amplifiers. For External-NoiseMod, arbitrary waveform generators synthesize Gaussian (or other) noise with programmatically controlled variance.

Key design strategies:

• Maximize the transmitted-to-noise variance ratio $\delta$ and the variance contrast $\alpha$ subject to hardware/covertness constraints.
• Choose $N$ per bit to balance reliability against desired data-rate, ensuring the sample variance estimator approaches normality.
• In fading or time-varying channels, employ TD-NoiseMod with $I>1$ to harness diversity gains.
• For the lowest complexity, NC-NoiseMod provides robust performance without explicit channel estimation or tracking.

Examples of practical benefits and applications include:

• Physically secure and low-probability-of-detection links, as the noise-like transmitted signal is difficult to distinguish from ambient background.
• Suitability for ultra-low-complexity, battery-free, or energy-constrained devices.

7. Open Problems and Future Directions

The channel capacity of variance-modulation—i.e., the mutual information achievable by encoding data in the second (or higher) moment—remains analytically uncharacterized in (Basar, 2023). Determining the ultimate rate limits of NoiseMod schemes will require new approaches that treat the sequence of noise-variance-modulated samples as non-Gaussian and non-Markovian channels, possibly invoking tools from information theory on mixture models or functional statistics.

Further research directions mentioned or implied include:

• Extension to higher-order statistics (beyond variance) and multidimensional modulations (e.g., including mean or mixture structure).
• Investigation of optimal pilot embedding, statistical channel estimation, and resource allocation for complex multi-user networks.
• Detailed experimental evaluation in real-world RF environments, including the effect of nonideal noise sources, hardware nonlinearities, and ambient interference conditions.

NoiseMod, in its core formulations and analyzed extensions, provides a mathematically rigorous and practically validated framework for digital communication systems that depart radically from deterministic-carrier orthodoxy, opening new prospects for physical-layer security, covert wireless networking, and the integration of communication and sensing modalities (Basar, 2023, Silva et al., 6 Nov 2025).