IISM Noise Models

• IISM Noise Models are defined by a Poisson field and shot-noise framework that captures non-Gaussian impulsive interference.
• They employ α-stable limits along with analytic expressions for characteristic functions and power spectral density, enabling precise calibration.
• Calibration involves discrete-time simulation, pulse shape fitting, and parameter estimation to replicate heavy-tailed, colored noise in practical settings.

IISM (Impulsive Interferer Shot-noise Model) noise models constitute a framework for describing the aggregate noise processes arising from random, impulsive interference sources, especially in environments such as electrical substations or wireless channels subject to transient events. Unlike purely Gaussian or stationary disturbance models, IISM explicitly models the non-Gaussian, impulsive nature of interference using statistical shot-noise theory, stochastic geometry, and α-stable distributions. These models are grounded in the superposition of physical impulsive waveforms generated at random times/locations by a Poisson field of interferers, allowing for both analytic tractability and direct calibration to empirical data.

1. Poisson Field Construction and Composite Noise Process

IISM models the occurrence of impulsive noise as a space–time Poisson point process Ψ of intensity λ per unit space-time volume, each event ψᵢ = (rᵢ, tᵢ) producing at the receiver an elementary transient h(t–tᵢ), scaled by a random amplitude Kᵢ. The aggregate process is: $X(t) = \sum_{ψᵢ∈Ψ} Kᵢ\,h(t-tᵢ) + n(t)$ where n(t) is an optional additive Gaussian background. The number of impulses in a measurable set B follows a Poisson law with mean λ·Vol(B); amplitudes and times {Kᵢ, tᵢ} are i.i.d. This formulation is agnostic to the specific details of the underlying impulsive sources, making it robust for modeling interference in practical environments.

2. Statistical Properties: Characteristic Function and α-Stable Limit

The moment-generating function (MGF) for IISM aggregates (by Campbell's theorem) to: $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$ and characteristic function: $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$ If K and/or h(u) produces heavy power-law tails such that $P(|K h(u)| > x) \sim C x^{-(α+1)}$ for $0 < α < 2$, $X(t)$ converges in law (via the generalized central limit theorem) to an α-stable random variable with characteristic exponent α, scale γ, location δ, and skew β determined in closed form:

• $\gamma = [\lambda E[|K|^α] \int |h(u)|^α du]^{1/α}$ (scale)
• $\delta = \lambda E[K] \int h(u) du$ (location)
• $$\beta = \frac{\int h(u)^α \mathrm{sign}[h(u)] du}{\int |h(u)|^α du}$$ (skewness)

This result generalizes IISM to describe environments where extreme interference dominates the noise statistics and precludes convergence to Gaussianity.

3. First- and Second-Order Statistics; Power Spectra

The mean and autocorrelation are:

• $E[X(t)] = \lambda E[K] \int h(u) du$
• $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$0

The power spectral density (PSD) for $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$1 is: $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$2 with $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$3. If $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$4 is a double exponential or AR(2) process, $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$5 can exhibit power-law decay $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$6 at high frequencies, with $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$7 a resonant frequency and $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$8 determined by the pulse shape and pole structure.

4. Discrete-Time Simulation Procedure

Discrete-time simulation of IISM noise is achieved via the following sequence:

1. Draw N ~ Poisson(λT) impulsive event times $$M_X(s) = \exp\left\{ \lambda \int_{-\infty}^\infty [E_K(e^{-sK h(u)}) - 1] du \right\} \cdot \exp\left\{ \frac{1}{2} \sigma_n^2 s^2 \right\}$$9 over $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$0.
2. For each event:
   • Sample amplitude $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$1 from a specified law (exponential/heavy-tail).
   • Generate a discrete waveform $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$2, typically as a closed-form pulse or AR(2) realization with coefficients $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$3.
   • Inject the scaled pulse $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$4 into the output sequence $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$5 at position $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$6.
3. Add Gaussian noise $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$7.
4. Ensure the time resolution $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$8 resolves the finest pulse feature adequately.

Parameter choice (λ, pulse shape, amplitude law) is directly motivated by calibration to empirical noise waveforms.

5. Model Calibration and Practical Data Fitting

• λ (impulse rate): Empirically determined via thresholding to detect count rate.
• $$\phi_X(\xi) = \exp\left\{ -\lambda \int [1 - E_K(e^{j\xi K h(u)})] du - \frac{1}{2} \sigma_n^2 \xi^2 \right\}$$9 (impulse shape): Extracted by aligning and averaging isolated detected impulses; parametric fitting applied (e.g., double exponential or AR(2)).
• α-stable parameters (α, β, γ, δ): Estimated via quantiles or characteristic function fitting to the empirical PDF tail of X(t).
• PSD exponent and resonance: Fitted by periodogram analysis in log–log coordinates to yield the slope (–k) and resonant frequency $P(|K h(u)| > x) \sim C x^{-(α+1)}$0.
• $P(|K h(u)| > x) \sim C x^{-(α+1)}$1: Inferred from the spectrum floor.

This structured calibration enables a single IISM parameter set to capture real-world impulsive/interferer-rich noise statistics.

Parameter	Physical Meaning	Calibration Method
λ	Impulse rate	Thresholded event count per time
Pulse shape $P(|K h(u)| > x) \sim C x^{-(α+1)}$2	Single impulse	Alignment + nonlinear least squares
α	Stability exponent	PDF tail fit / quantile estimator
k, $P(|K h(u)| > x) \sim C x^{-(α+1)}$3	PSD slope, resonance	Periodogram, log–log fit
$P(|K h(u)| > x) \sim C x^{-(α+1)}$4	Gaussian floor	Spectrum floor

6. Context, Applicability, and Limitations

IISM noise models are particularly relevant to environments where transient noise bursts dominate, such as substations, highly industrial wireless environments, or urban communication networks impacted by impulsive electromagnetic events. The generality of the Poisson field plus physical pulse allows adaptation to broad classes of noise statistics observed in practice, capturing both non-Gaussian tails and colored spectral content. The regime where IISM models are most essential is when the impulsive/interferer power spectrum is power-law or heavy-tailed—contexts where conventional AWGN models fail dramatically. Limitations include the breakdown of the independent-pulse assumption in the presence of correlated impulsers or highly structured periodic interference, and the need for careful calibration in multi-modal or highly heterogeneous environments.

7. Relationship to Other Noise Models and Theoretical Significance

Unlike classical stationary noise models (AWGN, colored noise), IISM explicitly describes non-Gaussian, impulsive, and nonstationary phenomena through its shot-noise basis and α-stable limits (Au et al., 2015). The model is analytically tractable—MGF, characteristic function, and moments can be computed in closed form for diverse pulse shapes and amplitude distributions. This universality and direct physical linkage to empirical data distinguish IISM from purely phenomenological statistical models. A plausible implication is that IISM provides a foundational structure for modeling environments where risk, rarity, and energy concentration of interference cannot be neglected, and it offers a pathway for the synthesis, simulation, and mitigation of real-world impulsive noise in systems analysis and design.