A Simple Numerical Method for Non-Gaussian Signal Ensembles in Nonlinear Power Amplifiers

Technical Summary: A Simple Numerical Method for Non-Gaussian Signal Ensembles in Nonlinear Power Amplifiers

Problem Statement
Modern wireless communication systems, particularly those utilizing 5G/6G, massive MIMO, millimeter-wave (mmWave), and integrated sensing and communications (ISAC), face significant challenges due to hardware nonlinearities. These nonlinearities, primarily arising from radio frequency (RF) power amplifiers (PAs) operating near saturation and low-resolution data converters, introduce amplitude and phase distortions. In high-mobility vehicular communication systems, these distortions degrade beamforming performance, alter beam patterns, and cause array-gain loss.

Existing analytical tools often rely on assumptions that limit their applicability. For instance, Bussgang's theorem is restricted to Gaussian processes, while Volterra series methods can be computationally burdensome. Furthermore, many current approaches require evaluating computationally intensive double or triple improper integrals to characterize the statistical effects of nonlinear distortion on non-Gaussian signal ensembles. There is a need for a general, computationally efficient framework capable of analyzing memoryless nonlinear systems with arbitrary signal and noise distributions to support the design of robust next-generation wireless systems.

Methodology
The paper proposes a numerical extension of Rice's characteristic function (ch. f.) method for the stochastic analysis of signals and noise in memoryless nonlinear systems. The core innovation lies in representing the nonlinearity using a Fourier series of a periodic extension, rather than a Fourier transform or power series.

The derivation proceeds as follows:

1. System Modeling: The input is modeled as a sum of a sinusoidal signal, a DC bias, and a noise process (which may be non-Gaussian). The system output is defined by memoryless nonlinear functions $f(x)$ and $g(x)$.
2. Fourier Series Representation: The nonlinear transfer functions are expressed as Fourier series. This allows the expectation of the product of outputs to be transformed from an integral problem into a summation problem.
3. Characteristic Function Decomposition: The cross-correlation function of the output is derived by separating the expectation into three components:
   • The ch. f. of the DC bias.
   • The joint ch. f. of the sinusoidal signal (derived using Bessel functions).
   • The joint ch. f. of the noise processes.
4. Series Expansion: For Gaussian noise, the noise ch. f. is expanded as a Maclaurin series polynomial in the correlation coefficient $\rho(\tau)$. For non-Gaussian noise, the specific joint ch. f. of that distribution is substituted.
5. Final Formulation: The output correlation $\Psi(\tau)$ is expressed as a double summation over harmonic indices and noise expansion terms. This results in a series expansion where the output spectrum at each harmonic is a weighted sum of the input spectrum and its autoconvolutions.

Key Contributions

• Generalization of Rice's Method: The paper reformulates Rice's ch. f. method by utilizing Fourier series coefficients to represent nonlinearity. This approach retains the generality of the original method, supporting one or more sinusoidal signals and noise processes that are not restricted to Gaussian distributions.
• Computational Efficiency: By transforming the evaluation of output correlation functions from improper integrals into tractable summations, the method significantly reduces computational complexity.
• New Fundamental Formulation: A new ch. f.-based formulation is derived in terms of Fourier-series coefficients and a discrete parameterization of the generalized characteristic function.
• Applicability to Non-Gaussian Ensembles: The framework is applicable to stochastic processes of any distribution, provided the wide-sense stationary random voltage or current has a characteristic function, which always exists.

Numerical Results
The authors validated the proposed method using a nonlinear GaN HEMT transconductance characteristic driven by a sinusoidal signal and Gaussian noise.

• Setup: The device was modeled in Class-AB mode with a bias near the peak of the transconductance curve. The nonlinearity was approximated by a periodic extension of a polynomial fit to the device data.
• Observations:
  • The method successfully decomposed the output spectrum into a fundamental sinusoid component and noise components weighted by expansion coefficients.
  • As the input sinusoid amplitude increased, the fundamental signal term exhibited expected compression, while higher-order noise terms ($k>0$) rapidly diminished in significance.
  • The analysis revealed that the bias voltage significantly impacts noise power. Deviating the bias from the peak gain point increased noise power and altered the noise spectrum shape, introducing localized nulls.

Significance and Claims
The paper claims that the proposed framework provides a computationally efficient tool for analyzing nonlinear RF front-end impairments without relying on extensive Monte Carlo simulations. It enables the characterization of:

• Signal-to-distortion ratio (SDR).
• Bit error rate (BER).
• Outage probability.
• The statistical effects of hardware nonlinearities on beam tracking and beam management in vehicular and future 6G systems.

The authors emphasize that this method is not intended to replace Volterra-series methods but rather addresses a different class of problems by providing a tractable analysis for memoryless nonlinear systems with stochastic inputs. The framework supports the development of distortion-aware communication algorithms, such as likelihood-based data detection and MAP receivers, which account for non-Gaussian distortion statistics rather than relying on conventional Gaussian assumptions. Future work is identified as extending the framework to modulated signals and investigating non-Gaussian distributions for distortion-aware system design.