pith. sign in

arxiv: 2606.26020 · v1 · pith:SCNMQ6T5new · submitted 2026-06-24 · 📡 eess.SP

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

Cameron M. Pike , Animesh Yadav This is my paper

Pith reviewed 2026-06-25 19:27 UTC · model grok-4.3

classification 📡 eess.SP
keywords memoryless nonlinearityFourier seriescharacteristic functionnon-Gaussian signalspower amplifier distortionstochastic analysisRF impairmentsmmWave systems
0
0 comments X

The pith

Fourier series representation of memoryless nonlinearities converts output correlation integrals into summations for non-Gaussian inputs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Rice's characteristic-function approach to analyze signals and noise passing through memoryless nonlinear devices such as power amplifiers. It replaces the usual Fourier-transform representation of the nonlinearity with a Fourier-series expansion, so that output correlation functions are obtained from finite sums instead of multiple improper integrals. The resulting expressions remain valid for arbitrary input distributions, including multiple sinusoids plus non-Gaussian noise. Numerical checks are shown for a GaN HEMT transconductance curve driven by a sinusoid and Gaussian noise. The method supplies a practical route to quantify distortion effects that appear in mmWave vehicular links.

Core claim

By writing the memoryless nonlinearity as a Fourier series whose coefficients discretely parameterize the generalized characteristic function, the evaluation of output correlation functions reduces from computationally heavy double or triple improper integrals to ordinary summations that preserve full generality for non-Gaussian signal ensembles.

What carries the argument

Fourier-series expansion of the memoryless nonlinearity, which yields a discrete parameterization of the generalized characteristic function and converts correlation integrals into summations.

If this is right

  • Output statistics become computable for any combination of sinusoidal carriers and non-Gaussian noise without numerical quadrature.
  • The framework applies unchanged to one or more deterministic signals plus arbitrary noise processes.
  • Distortion-induced beam-pattern deviations and array-gain loss in mmWave systems can be evaluated with far lower computational cost.
  • The same formulation supplies closed-form expressions for higher-order moments when the input distributions are known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The summation form may be inserted directly into link-level simulators that already track characteristic functions.
  • Extension to mildly memory-dependent amplifiers would require only an additional convolution step outside the series.
  • The discrete parameterization invites pre-computation of coefficient tables for common amplifier curves, enabling real-time impairment prediction.

Load-bearing premise

The nonlinearity is memoryless and can be represented to sufficient accuracy by a Fourier series.

What would settle it

Direct Monte-Carlo simulation of the nonlinear device with a non-Gaussian input ensemble produces output autocorrelation values that differ from those obtained by the proposed summations.

Figures

Figures reproduced from arXiv: 2606.26020 by Animesh Yadav, Cameron M. Pike.

Figure 1
Figure 1. Figure 1: Spectra of the noise contributions in (15) for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nonlinear transfer function of GaN HEMT, adapted [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Expansion coefficient σ 2h 2 1,1 (16) for Vbias =β =−3.0,−2.5,−2.0V, SNR=40 dB. computationally intensive multidimensional improper integrals into tractable summations. The resulting framework enabled efficient characterization of nonlinear distortion and noise propagation while retaining applicability to arbitrary signal and noise distributions. Numerical results based on a GaN HEMT demonstrated the abili… view at source ↗
read the original abstract

Beam tracking in vehicular communication systems is inherently challenging due to high mobility and the use of narrow millimeter-wave (mmWave) beams. These challenges are further exacerbated by power amplifier (PA) nonlinearities, which introduce distortion-induced beam pattern deviations, array-gain loss, and non-Gaussian signal distortions. Motivated by the need for analytical tools capable of characterizing such effects, this paper extends Rice characteristic-function (ch. f.) method for the stochastic analysis of signals and noise in memoryless nonlinear systems. The proposed approach represents the nonlinearity using a Fourier series rather than a Fourier transform, transforming the evaluation of output correlation functions from computationally intensive double or triple improper integrals into tractable summations. The resulting framework preserves the generality of the original method, supporting one or more sinusoidal signals and noise processes that are not restricted to Gaussian distributions. A new fundamental ch. f.-based formulation is derived in terms of Fourier-series coefficients and a discrete parameterization of the generalized characteristic function. Numerical results are presented for a nonlinear GaN HEMT transconductance characteristic driven by a sinusoidal signal and Gaussian noise, demonstrating the applicability of the proposed method. The framework provides a computationally efficient tool for analyzing nonlinear RF front-end impairments and their impact on future wireless and vehicular communication systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends Rice's characteristic-function method for the stochastic analysis of signals and noise through memoryless nonlinear systems. It replaces the Fourier-transform representation of the nonlinearity with a Fourier-series expansion, converting the evaluation of output correlation functions from double or triple improper integrals into finite summations. The resulting formulation retains generality for one or more sinusoids plus non-Gaussian noise via a discrete parameterization of the generalized characteristic function; a concrete numerical demonstration is given for a GaN HEMT transconductance driven by a sinusoid plus Gaussian noise.

Significance. If the claimed integral-to-summation reduction holds with controllable truncation error, the method supplies a computationally lighter tool for characterizing nonlinear RF impairments (beam-pattern deviation, array-gain loss) in mmWave vehicular systems while preserving the original method's applicability to non-Gaussian ensembles. The explicit construction in terms of Fourier-series coefficients and the discrete ch.f. parameterization is a clear strength.

minor comments (3)
  1. [Abstract / §1] The abstract and introduction should explicitly state the truncation criterion or error bound used for the Fourier-series representation of the nonlinearity (e.g., number of harmonics retained and residual error metric).
  2. [§2] Notation for the discrete parameterization of the generalized characteristic function should be introduced once, with a clear mapping to the continuous Rice ch.f., to avoid reader confusion between the original and new formulations.
  3. [§4] Figure captions for the numerical GaN HEMT example should report the number of retained Fourier coefficients, the sampling grid for the ch.f., and a direct comparison of run-time against the classical integral formulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript, as well as the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation extends the external Rice ch.f. method by substituting a Fourier-series representation of the memoryless nonlinearity for the Fourier transform. This substitution converts the output-correlation integrals into summations via the stated discrete parameterization of the generalized characteristic function. No equation reduces to a fitted parameter renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the ansatz is introduced explicitly in the present work rather than smuggled via prior author citation. The numerical GaN HEMT example is presented as verification, not as the source of the claimed reduction. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the domain assumption that the PA characteristic is memoryless and admits a Fourier series representation; no free parameters, invented entities, or additional axioms are specified in the abstract.

axioms (1)
  • domain assumption The nonlinearity is memoryless
    Explicitly stated in abstract as applying to memoryless nonlinear systems.

pith-pipeline@v0.9.1-grok · 5754 in / 1131 out tokens · 26085 ms · 2026-06-25T19:27:35.603533+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references

  1. [1]

    On the power amplifier nonlinearity in MIMO transmit beamforming systems,

    J. Qi and S. Aissa, “On the power amplifier nonlinearity in MIMO transmit beamforming systems,”IEEE Trans. Commun., vol. 60, no. 3, pp. 876–887, Mar. 2012

    2012

  2. [2]

    Fast channel estimation and beam tracking for millimeter wave vehic- ular communications,

    S. Shaham, M. Ding, M. Kokshoorn, Z. Lin, S. Dang, and R. Abbas, “Fast channel estimation and beam tracking for millimeter wave vehic- ular communications,”IEEE Access, vol. 7, pp. 141 104–141 118, 2019

    2019

  3. [3]

    Extended Kalman filter beam tracking for millimeter wave vehicular communications,

    S. Shaham, M. Kokshoorn, M. Ding, Z. Lin, and M. Shirvanimoghad- dam, “Extended Kalman filter beam tracking for millimeter wave vehicular communications,”IEEE Commun. Lett., vol. 24, no. 4, pp. 839–842, Apr. 2020

    2020

  4. [4]

    A compact, low-power, low-NF, millimeter-wave cascode LNA with magnetic coupling feedback in 22-nm FD-SOI CMOS for 5G applications,

    E. Kobal, T. Siriburanon, R. B. Staszewski, and A. Zhu, “A compact, low-power, low-NF, millimeter-wave cascode LNA with magnetic coupling feedback in 22-nm FD-SOI CMOS for 5G applications,”IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 70, no. 4, pp. 1331–1335, 2023

    2023

  5. [5]

    A wideband 2–5 GHz noise canceling subthreshold low noise amplifier,

    A. R. A. Kumar, B. D. Sahoo, and A. Dutta, “A wideband 2–5 GHz noise canceling subthreshold low noise amplifier,”IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 65, no. 7, pp. 834–838, 2018

    2018

  6. [6]

    A Gm-boosting technique for millimeter-wave low-noise amplifiers in 28-nm triple-well bulk CMOS using floating resistor in body biasing,

    E. Kobal, T. Siriburanon, X. Chen, H. M. Nguyen, R. B. Staszewski, and A. Zhu, “A Gm-boosting technique for millimeter-wave low-noise amplifiers in 28-nm triple-well bulk CMOS using floating resistor in body biasing,”IEEE Trans. Circuits Syst. I, Regul. Pap., vol. 69, no. 12, pp. 5007–5017, Dec. 2022

    2022

  7. [7]

    Channel estimation and performance analysis of one-bit massive MIMO systems,

    Y . Li, C. Tao, G. Seco-Granados, A. Mezghani, A. L. Swindlehurst, and L. Liu, “Channel estimation and performance analysis of one-bit massive MIMO systems,”IEEE Trans. Signal Process., vol. 65, no. 15, pp. 4075–4089, Aug. 2017

    2017

  8. [8]

    Hardware distortion corre- lation has negligible impact on UL massive MIMO spectral efficiency,

    E. Bj ¨ornson, L. Sanguinetti, and J. Hoydis, “Hardware distortion corre- lation has negligible impact on UL massive MIMO spectral efficiency,” IEEE Trans. Commun., vol. 67, no. 2, pp. 1085–1098, Feb. 2018

    2018

  9. [9]

    On optimal MMSE channel estimation for one-bit quantized MIMO systems,

    M. Ding, I. Atzeni, A. T ¨olli, and A. L. Swindlehurst, “On optimal MMSE channel estimation for one-bit quantized MIMO systems,”IEEE Trans. Signal Process., vol. 73, pp. 617–632, Mar. 2025

    2025

  10. [10]

    Distortion-aware linear precoding for massive MIMO downlink systems with nonlinear power amplifiers,

    S. R. Aghdam, S. Jacobsson, U. Gustavsson, G. Durisi, C. Studer, and T. Eriksson, “Distortion-aware linear precoding for massive MIMO downlink systems with nonlinear power amplifiers,”IEEE Trans. Signal Process., vol. 69, pp. 3699–3713, 2021

    2021

  11. [11]

    Mathematical analysis of random noise,

    S. O. Rice, “Mathematical analysis of random noise,”Bell Syst. Tech. J., vol. 23, no. 3, pp. 282–332, Jul. 1944

    1944

  12. [12]

    Mathematical analysis of random noise,

    ——, “Mathematical analysis of random noise,”Bell Syst. Tech. J., vol. 24, no. 1, pp. 46–156, Jan. 1945

    1945

  13. [13]

    N. M. Blachman,Noise and its Effect on Communication, 2nd ed. Malabar, FL: Robert E. Krieger Publ. Co., 1982

    1982

  14. [14]

    K. W. Cattermole,Mathematical Topics in Telecommunications, V ol. 2: Problems of Randomness in Communication Engineering. John Wiley, 1984

    1984

  15. [15]

    Interference suppression by biased nonlinearities,

    P. C. Jain, N. M. Blachman, and P. M. Chapell, “Interference suppression by biased nonlinearities,”IEEE Trans. Inf. Theory, vol. 41, no. 2, pp. 496–507, Mar. 1995

    1995

  16. [16]

    Intermodulation performance of biased- nonlinearities,

    M. T. Abuelma’atti, “Intermodulation performance of biased- nonlinearities,”Appl. Math. Comput., vol. 107, no. 2–3, pp. 211–224, Jan. 2000

    2000

  17. [17]

    Performance analysis and comparison of multi-level hard limiters,

    M. T. Abuelma’atti and A. M. T. Abuelma’atti, “Performance analysis and comparison of multi-level hard limiters,”Int. J. Electron., vol. 97, no. 9, p. 1021, 2010

    2010

  18. [18]

    Analytical and experimental evaluation of SSB phase noise reduction in self-injection locked oscillators using optical delay loops,

    L. Zhang, A. Poddar, U. Rohde, and A. Daryoush, “Analytical and experimental evaluation of SSB phase noise reduction in self-injection locked oscillators using optical delay loops,”IEEE Photon. J., vol. 5, no. 6, p. 6602217, Dec. 2013

    2013

  19. [19]

    The Bussgang decomposition of nonlinear systems: Basic theory and MIMO extensions [Lecture Notes],

    O. T. Demir and E. Bj ¨ornson, “The Bussgang decomposition of nonlinear systems: Basic theory and MIMO extensions [Lecture Notes],”IEEE Signal Process. Mag., vol. 38, no. 1, pp. 131–136, Jan. 2020

    2020

  20. [20]

    A class of nonlinear signal-processing schemes for bandwidth-efficient OFDM transmission with low envelope fluctuation,

    R. Dinis and A. Gusmao, “A class of nonlinear signal-processing schemes for bandwidth-efficient OFDM transmission with low envelope fluctuation,”IEEE Trans. Commun., vol. 52, no. 11, pp. 2009–2018, Nov. 2004

    2009

  21. [21]

    Optimum performance analysis and receiver design for OFDM-based frequency-splitting SWIPT with strong nonlinear effects,

    A. Rajaram, J. Guerreiro, R. Dinis, D. N. K. Jayakody, and M. Beko, “Optimum performance analysis and receiver design for OFDM-based frequency-splitting SWIPT with strong nonlinear effects,”IEEE Internet Things J., vol. 10, no. 23, pp. 20 928–20 940, Dec. 2023

    2023

  22. [22]

    Crosscorrelation functions of amplitude-distorted gaussian signals,

    J. J. Bussgang, “Crosscorrelation functions of amplitude-distorted gaussian signals,” Res. Lab. Electron., MIT, Cambridge, MA, Tech. Rep. 216, Mar. 1952

    1952

  23. [23]

    A useful theorem for nonlinear devices having gaussian inputs,

    R. Price, “A useful theorem for nonlinear devices having gaussian inputs,”IRE Trans. Inf. Theory, vol. 4, no. 2, pp. 69–72, Jun. 1958

    1958

  24. [24]

    R. V . Churchill and J. W. Brown,F ourier Series and Boundary V alue Problems, 3rd ed. McGraw-Hill, 1978

    1978

  25. [25]

    Cascaded dual drive MZM and dual parallel MZM architecture for optical linearization of MZM nonlinearity,

    S. Duraikannan, S. B. B. A. Anas, B. M. Ali, Z. B. Zan, S. Duraikannan, and V . Thiruchelvam, “Cascaded dual drive MZM and dual parallel MZM architecture for optical linearization of MZM nonlinearity,” in Proc. IEEE Int. Conf. Photonics (ICP), Langkawi, Malaysia, 2018, pp. 1–4

    2018

  26. [26]

    J. P. Z. Peebles,Probability, Random V ariables, and Random Signal Principles, 2nd ed. McGraw-Hill, 1987

    1987

  27. [27]

    Watson,A Treatise on the Theory of Bessel Functions

    G. Watson,A Treatise on the Theory of Bessel Functions. Cambridge, U.K.: Cambridge Univ. Press, 1922

    1922

  28. [28]

    Impact of source resistance on linearity of AlGaN/GaN HEMTs at Ka-band,

    P.-H. Lee, Y .-C. Lin, C.-C. Chiang, K.-H. Lai, H.-T. Hsu, P. Su, and E. Y . Chang, “Impact of source resistance on linearity of AlGaN/GaN HEMTs at Ka-band,”ECS J. Solid State Sci. Technol., vol. 12, 2023

    2023