---
title: Generalized Quadratic Noise Modulation
url: https://www.emergentmind.com/topics/generalized-quadratic-noise-modulation-gqnm
type: topic
---

# Generalized Quadratic Noise Modulation

Searching arXiv for papers on Generalized Quadratic Noise Modulation and related work.
Generalized Quadratic Noise Modulation (GQNM) is a noise-modulation framework in which each transmitted symbol is encoded jointly in two statistical dimensions of the waveform: its mean and its variance. In the formulation introduced for noise-modulated links, a symbol carries a “mean” bit and a “variance” bit, so that a single block realizes a 4-ary constellation; in a later extension, the superposition of two independent GQNM outputs yields a 16-ary composite modulator with four distinct means and four distinct variances [2509.11378; 2510.01776]. The framework has been presented for Gaussian and non-Gaussian noise families, including Generalized Gaussian (GG), Laplacian, and Gaussian Mixture of Two Gaussians (GMoTG or MoG), with closed-form and simulation-based Bit Error Probability (BEP) analyses emphasizing threshold detection from sample mean and sample variance statistics [2509.11378].

## 1. Formal definition and signaling alphabet

In GQNM, symbols are transmitted in blocks of \(N\) samples, and each block carries two bits. The first bit, \(b_0\in\{0,1\}\), selects one of two bias levels, written as \(v_1\equiv m_L\) and \(v_2\equiv m_H\). The second bit, \(b_1\in\{0,1\}\), selects one of two noise-variance modes, described as “low-spread” and “high-spread” [2509.11378].

The transmitted sample has the form
\[
X_n = v_{b_0+1} + N_n,
\]
where \(N_n\) is an i.i.d. zero-mean, even noise sample whose parameters depend on \(b_1\). In the Gaussian specialization used in the composite construction, the modulator output over symbol interval \(n\) is
\[
x_n = m_n + v_n,
\]
with
\[
m_n =
\begin{cases}
m_L,& b_0=0,\\
m_H,& b_0=1,
\end{cases}
\qquad
v_n\sim
\begin{cases}
\mathcal N(0,\sigma_0^2),& b_1=0,\\
\mathcal N(0,\sigma_1^2),& b_1=1.
\end{cases}
\]
This realizes a 4-ary constellation in the two dimensions \((\text{mean},\text{variance})\) [2510.01776].

The four transmitted hypotheses can therefore be written as follows: \((b_0,b_1)=(0,0)\) gives mean \(m_L\) and variance \(\sigma_0^2\); \((0,1)\) gives mean \(m_L\) and variance \(\sigma_1^2\); \((1,0)\) gives mean \(m_H\) and variance \(\sigma_0^2\); and \((1,1)\) gives mean \(m_H\) and variance \(\sigma_1^2\) [2509.11378]. Classical noise modulation uses only the variance degree of freedom to send 1 bit, whereas GQNM uses both the mean and the variance dimensions, so each symbol carries 2 bits and doubles the data rate of variance-only schemes [2509.11378].

## 2. Receiver statistics, threshold detection, and BEP analysis

The receiver observes
\[
r_n = X_n + w_n,
\qquad
w_n\sim\mathcal N(0,\sigma_w^2),
\]
and computes two statistics over the \(N\)-sample block:
\[
\hat m=\frac{1}{N}\sum_{n=1}^N r_n,
\qquad
\hat\sigma^2=\frac{1}{N}\sum_{n=1}^N (r_n-\hat m)^2.
\]
Because the decision variables are separated into mean and variance statistics, the detector applies independent thresholds,
\[
\hat b_0 =
\begin{cases}
0,& \hat m<\mathrm{Th}_m,\\
1,& \hat m>\mathrm{Th}_m,
\end{cases}
\qquad
\hat b_1 =
\begin{cases}
0,& \hat\sigma^2<\mathrm{Th}_v,\\
1,& \hat\sigma^2>\mathrm{Th}_v,
\end{cases}
\]
with midpoint choices
\[
\mathrm{Th}_m=\frac{m_L+m_H}{2},
\qquad
\mathrm{Th}_v=\frac{\sigma_{\min}^2+\sigma_{\max}^2}{2}.
\]
The same midpoint construction appears in the Gaussian single-GQNM exposition, where \(\hat m\) is compared against \((m_L+m_H)/2\) and \(\hat\sigma^2\) against \((\sigma_0^2+\sigma_1^2)/2\) [2509.11378; 2510.01776].

For large \(N\), the sample mean is approximately Gaussian:
\[
\hat m\approx \mathcal N\!\Bigl(m_{b_0,b_1},\,\frac{1}{N}(\sigma_{b_0,b_1}^2+\sigma_w^2)\Bigr).
\]
By the Delta-method or CLT, \(\hat\sigma^2\) is also approximately Gaussian in the purely Gaussian case, with mean
\[
m_{\sigma^2}=m_{b_0,b_1}^2+\sigma_{b_0,b_1}^2+\sigma_w^2
\]
and variance
\[
\frac{1}{N}\Bigl(2(\sigma_{b_0,b_1}^2+\sigma_w^2)^2 +4\,m_{b_0,b_1}^2(\sigma_{b_0,b_1}^2+\sigma_w^2)\Bigr).
\]
Similar moment calculations are stated to apply for GG and GMoTG via the known fourth moments [2509.11378].

The BEP decomposition is written as
\[
p_{b_0}=\Pr\{\hat b_0\neq b_0\},
\qquad
p_{b_1}=\Pr\{\hat b_1\neq b_1\},
\qquad
p_b=\tfrac12(p_{b_0}+p_{b_1}).
\]
Closed-form expressions are given for the Gaussian case and for the GMoTG case; in the Gaussian case, the resulting formulas are sums of \(Q\)-functions parameterized by the mean separation, the variance ratio, receive-noise variance, and the number of samples \(N\) [2509.11378].

## 3. Non-Gaussian generalization

A central generalization of GQNM is the replacement of Gaussian excitation by zero-mean, even, non-Gaussian noise distributions, while retaining the same two-bit signaling structure. The letter introducing this broader framework considers three noise families: Generalized Gaussian (GG), Laplacian, and Gaussian Mixture of Two Gaussians (GMoTG or MoG) [2509.11378].

For the Generalized Gaussian family with shape parameter \(\beta>0\),
\[
f_{N}(n;\alpha,\beta)
=
\frac{\beta}{2\,\alpha\,\Gamma(1/\beta)}
\exp\!\bigl(-(\lvert n\rvert/\alpha)^\beta\bigr),
\]
and two spread choices \(\alpha=\alpha_0\) and \(\alpha=\alpha_1\) produce variances
\[
\sigma_i^2=\alpha_i^2\frac{\Gamma(3/\beta)}{\Gamma(1/\beta)},
\qquad i=0,1.
\]
The Laplacian distribution is identified as the special case \(\beta=1\), equivalently
\[
f_N(n;b)=\frac{1}{2b}\exp(-|n|/b),
\qquad
\mathrm{Var}\{N\}=2b^2.
\]
For GMoTG,
\[
f_N(n)=p\,\mathcal N(0,\sigma_0^2)+(1-p)\,\mathcal N(0,\sigma_1^2),
\]
with distinct low-spread and high-spread parameter sets, and the resulting noise-mode variance is
\[
\sigma_{\mathrm{MoG}}^2
=
p\,\sigma_{0\bullet}^2+(1-p)\,\sigma_{1\bullet}^2.
\]
These constructions preserve the mean/variance decoding logic while altering the estimator statistics through the underlying noise law [2509.11378].

The reported simulations indicate that all three schemes achieve roughly the same total BEP, which is presented as confirmation of the Gaussian-CLT approximations. The behavior of the two sub-bits, however, is distribution-dependent: GMoTG gives a slight edge in mean-bit error \(p_{b_0}\), because the narrow mode in the mixture tightens the distribution of \(\hat m\), whereas Laplacian yields the lowest variance-bit error \(p_{b_1}\), since the sample-variance estimator is more decisive under heavy tails in the high-spread state [2509.11378]. A common misunderstanding is therefore that non-Gaussian noise is uniformly superior; the stated results support a narrower conclusion, namely that non-Gaussian shaping can improve one sub-bit at the cost of a slightly worse other sub-bit [2509.11378].

The same study also states that increasing \(N\) reduces both sub-bit BEPs as \(O(1/\sqrt{N})\), confirming the CLT scaling [2509.11378]. This suggests that the estimation window length is itself a primary modulation-design parameter, not merely a receiver implementation detail.

## 4. Composite 16-ary GQNM by signal addition

A higher-order extension is obtained by simply adding the outputs of two independent GQNM modulators:
\[
X_{\rm comp}=X_0+X_1=(m_{n,0}+m_{n,1})+(v_{n,0}+v_{n,1})=m^{(n)}+v^{(n)}.
\]
Each sub-channel \(i\in\{0,1\}\) carries its own pair \((b_{0,i},b_{1,i})\), so the composite symbol encodes the 4-bit vector
\[
\mathbf b_n=[b_{0,0},\,b_{1,0},\,b_{0,1},\,b_{1,1}]^\top,
\qquad
\mathbf b_n\in\{0,1\}^4,
\]
yielding a \(2^4=16\)-ary constellation [2510.01776].

The construction is parameterized by
\[
m_{L_1}=\beta\,m_{L_0},\quad
m_{H_i}=\alpha\,m_{L_i},\quad
\sigma^2_{1i}=\eta\,\sigma^2_{0i},\quad
\sigma^2_{i1}=\gamma\,\sigma^2_{i0},
\qquad
\alpha,\beta,\gamma,\eta>1.
\]
These choices generate four distinct means,
\[
m_1=(1+\beta)m_{L_0},\qquad
m_2=(\alpha+\beta)m_{L_0},
\]
\[
m_3=(1+\alpha\beta)m_{L_0},\qquad
m_4=\alpha(1+\beta)m_{L_0},
\]
and four variances,
\[
\sigma_1^2=(1+\gamma)\sigma_{00}^2,\qquad
\sigma_2^2=(\eta+\gamma)\sigma_{00}^2,
\]
\[
\sigma_3^2=(1+\gamma\eta)\sigma_{00}^2,\qquad
\sigma_4^2=\gamma(1+\eta)\sigma_{00}^2.
\]
Each of the 16 bit patterns maps to one \((m_j,\sigma_k^2)\) pair [2510.01776].

Detection proceeds in two stages. In the mean dimension,
\[
(\hat b_{0,0},\hat b_{0,1})=
\begin{cases}
(0,0),&\hat m < \tfrac{m_1+m_2}{2},\\
(1,0),&\tfrac{m_1+m_2}{2}<\hat m<\tfrac{m_2+m_3}{2},\\
(0,1),&\tfrac{m_2+m_3}{2}<\hat m<\tfrac{m_3+m_4}{2},\\
(1,1),&\hat m>\tfrac{m_3+m_4}{2},
\end{cases}
\]
and in the variance dimension,
\[
(\hat b_{1,0},\hat b_{1,1})=
\begin{cases}
(0,0),&\hat\sigma^2<\tfrac{\sigma_1^2+\sigma_2^2}{2},\\
(1,0),&\tfrac{\sigma_1^2+\sigma_2^2}{2}<\hat\sigma^2<\tfrac{\sigma_2^2+\sigma_3^2}{2},\\
(0,1),&\tfrac{\sigma_2^2+\sigma_3^2}{2}<\hat\sigma^2<\tfrac{\sigma_3^2+\sigma_4^2}{2},\\
(1,1),&\hat\sigma^2>\tfrac{\sigma_3^2+\sigma_4^2}{2}.
\end{cases}
\]
The authors describe the resulting 16-ary structure as resembling QAM modulators in classical communication, but with modulation taking place on four different means and four different variances rather than on conventional in-phase and quadrature amplitudes [2510.01776].

## 5. Distinguishability conditions and parameter design

The composite 16-ary construction is accompanied by sufficient “6-\(\sigma\)” separation bounds intended to guarantee that the 16 Gaussian components can be sharply distinguished by simple thresholding [2510.01776]. In the mean dimension, the sufficient condition is
\[
m_r + 3\,\sigma_4 \ll m_{r+1} - 3\,\sigma_4,
\qquad r=1,2,3.
\]
Because \((m_2-m_1)\) is the tightest gap, this is reduced to
\[
m_2-m_1 \gg 6\,\sigma_4
\quad\Longleftrightarrow\quad
\alpha\,m_{L_0} \gg 6\,\sqrt{\gamma(1+\eta)}\,\sigma_{00}.
\]
The dependence on \(\sigma_4\), the largest variance term, makes explicit that the narrowest mean separation must be judged against the worst-case spread [2510.01776].

In the variance dimension, \(\hat\sigma^2\) is approximated as Gaussian by the CLT, with
\[
\mathbb E\{\hat\sigma^2\mid f\} = \sigma_f^2,
\]
\[
\mathrm{Var}(\hat\sigma^2\mid f)
=
16\,N^{-4}\,\sigma_f^2\,\frac{\Gamma\!\bigl(\tfrac{N+7}{2}\bigr)}{\Gamma\!\bigl(\tfrac{N-1}{2}\bigr)}
-\sigma_f^4.
\]
A sufficient separation condition is then
\[
\sigma_f^2 + 3\,\sqrt{\mathrm{Var}(\hat\sigma^2\mid f)}
\ll
\sigma_{f+1}^2 - 3\,\sqrt{\mathrm{Var}(\hat\sigma^2\mid f+1)},
\qquad f=1,2,3.
\]
Again the tightest gap occurs at \(f=1\), so
\[
\sigma_2^2-\sigma_1^2
\gg
3\,\sqrt{\mathrm{Var}(\hat\sigma^2\mid 1)}
+
3\,\sqrt{\mathrm{Var}(\hat\sigma^2\mid 2)}.
\]
The paper states that by freely choosing the six design parameters \(\{m_{L_0},\alpha,\beta,\sigma_{00}^2,\eta,\gamma\}\) and/or increasing \(N\), these inequalities can be met with a large margin [2510.01776].

In the broader GQNM framework, analogous design guidance is given for the single-modulator case. Mean separation \(\Delta m=m_H-m_L\) should satisfy a target-error condition based on the standard deviation of \(\hat m\), and variance separation \(\Delta\sigma^2=\sigma_1^2-\sigma_0^2\) should similarly exceed a target-error condition based on \(\mathrm{Var}(r^2)\) [2509.11378]. This suggests that the single-stream and composite constructions share the same fundamental design logic: distinguishability is governed by separations between statistical states relative to estimator dispersion.

## 6. Reported performance, trade-offs, and extensions

The non-Gaussian GQNM study reports simulations with parameter choices chosen to equalize average transmit power: means \(m_L=1\times10^{-3}\), \(m_H=1\times10^{-2}\); GG parameters \(\sigma_0=1\times10^{-3}\), \(\sigma_1=20\times10^{-3}\); MoTG parameters \(\sigma_{0L}=0.5\times10^{-3}\), \(\sigma_{1L}=1\times10^{-3}\), \(\sigma_{0H}=5\times10^{-3}\), \(\sigma_{1H}=21\times10^{-3}\); and Laplace parameters \(b_0=0.1\times10^{-3}\), \(b_1=14.2\times10^{-3}\). The receive-noise standard deviation \(\sigma_w\) is varied from \(6.6\times10^{-6}\) to \(10^{-4}\), with \(N=10\) and additional sweeps over \(N=5\) to \(40\) [2509.11378].

For the composite 16-ary scheme, Monte-Carlo simulation is performed under additive white Gaussian channel noise \(w\sim\mathcal N(0,\sigma_w^2)\), comparing three modulators: a KLJN modulator, a single GQNM (4-ary), and the composite GQNM (16-ary). The stated simulation parameters are: GQNM-0 with \(\sigma_{00}=1\times10^{-5}\), \(\gamma=20\Rightarrow\sigma_{10}=1.4142\times10^{-5}\), \(m_{L_0}=10^{-3}\), \(\alpha=20\Rightarrow m_{H_0}=20\times10^{-3}\); GQNM-1 with \(\eta=5\Rightarrow\sigma_{01}=2.2361\times10^{-5}\), \(\gamma\eta=100\Rightarrow\sigma_{11}=10^{-4}\), \(\beta=5\Rightarrow m_{L_1}=5\times10^{-3}\), \(m_{H_1}=0.1\); \(N=100\); and \(\sigma_w=2\times10^{-5}\) [2510.01776].

Two experiments are highlighted for the composite scheme. In Experiment 1, BEP is plotted against \(N\in[40,100]\), and the composite GQNM is reported to show the lowest BEP and to improve as \(N\) grows. In Experiment 2, BEP is plotted against \(\sigma_w\in[10^{-5},5\times10^{-5}]\), and the composite GQNM again outperforms single-GQNM and KLJN across the entire noise range. The paper further states that the plots demonstrate up to orders-of-magnitude BEP reduction by the 16-ary composite scheme at moderate \(N\) and channel SNRs [2510.01776].

The performance gains are explicitly qualified by a complexity trade-off. The better result in terms of smaller BEP is achieved by increasing the complexity in the modulator, the transmitter, and the detectors in the receiver [2510.01776]. This is the main practical counterpoint to any interpretation of higher-order GQNM as a uniformly dominant replacement for simpler schemes.

The same superposition principle extends formally to \(s>2\) independent GQNM streams, immediately yielding a \(2^{2s}\)-ary noise modulator carrying \(2s\) bits per symbol [2510.01776]. The authors note, however, that a unified closed-form BEP analysis and practical implementation of threshold detectors for arbitrary \(s\) are subjects of ongoing and future work [2510.01776]. A plausible implication is that the central open issue is no longer only symbol construction, but the joint scaling of detector complexity, separability conditions, and error analysis as the number of superposed mean/variance channels increases.

## 7. Position within noise-modulated communication

Within the family of noise-modulated communication methods, GQNM is defined by its use of both mean and variance as information-bearing degrees of freedom. In the single-stream setting, this doubles the data rate relative to variance-only noise modulation; in the composite setting, simple output addition converts two independent 4-ary modulators into a 16-ary modulator carrying four bits per symbol [2509.11378; 2510.01776].

The framework is presented as hardware-friendly in settings where resistor-based noise sources are available and mixing a DC bias is easy, and applications listed in the literature include ultra-low-power IoT links, covert or secure communications leveraging KLJN-style hardware, and differential sensing systems or key-generation schemes that can exploit the extra bit in a single symbol period without extra bandwidth [2509.11378]. These application statements are prospective rather than deployment reports; they locate GQNM as a modulation concept whose practical value depends on the joint control of estimator statistics, threshold separability, and implementation complexity.

Taken together, the available papers describe GQNM as a statistical-constellation approach in which the transmitted symbol is identified not by deterministic waveform points but by the pair \((\text{mean},\text{variance})\), with later work showing that superposition enlarges the constellation to higher-order forms resembling classical multilevel modulations while retaining threshold detection on empirical mean and empirical variance [2509.11378; 2510.01776].

Source: https://www.emergentmind.com/topics/generalized-quadratic-noise-modulation-gqnm