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Discrete-time models and performance of phase noise channels

Amina Piemontese, Giulio Colavolpe, Thomas Eriksson

TL;DR

This work addresses phase noise as a fundamental limit in communication systems by developing a measurement-based discrete-time PN channel model that ties PN effects to oscillator PSD parameters. It derives closed-form performance metrics, including a SIR expression as a function of the relative PN bandwidth $\rho$, and introduces a Wiener-filter–based phase-tracking framework that yields a closed-form residual PN variance and an optimal symbol rate $R_s$. The analysis encompasses both free-running and PLL-locked oscillators and accounts for PN-induced power loss and ISI, with bounded approximation errors. Numerical results using a practical Kalman smoother confirm the theory and illustrate PN-aware receiver design and performance prediction from oscillator measurements.

Abstract

This paper deals with the phase noise affecting communication systems, where local oscillators are employed to obtain reference signals for carrier and timing synchronizations. The most common discrete-time phase noise channel model is analyzed, with the aim to fill the gap between measurements and analytical models. In particular, the power loss and the intersymbol interference due to the presence of phase noise is evaluated with reference to the measurements parameters and to the system bandwidth. Moreover, the impact on the communication systems' performance of the phase noise originating from the oscillator non idealities is considered, in case of free-running and phase-locked oscillators. The proposed analysis allows to extrapolate useful information about the performance of practical systems by investigating the power spectral density of the oscillator phase noise. An expression for the variance of the residual phase error after tracking, which depends on the main parameters of practical oscillators, is derived, and used to study the dependence of the performance on the symbol rate.

Discrete-time models and performance of phase noise channels

TL;DR

This work addresses phase noise as a fundamental limit in communication systems by developing a measurement-based discrete-time PN channel model that ties PN effects to oscillator PSD parameters. It derives closed-form performance metrics, including a SIR expression as a function of the relative PN bandwidth , and introduces a Wiener-filter–based phase-tracking framework that yields a closed-form residual PN variance and an optimal symbol rate . The analysis encompasses both free-running and PLL-locked oscillators and accounts for PN-induced power loss and ISI, with bounded approximation errors. Numerical results using a practical Kalman smoother confirm the theory and illustrate PN-aware receiver design and performance prediction from oscillator measurements.

Abstract

This paper deals with the phase noise affecting communication systems, where local oscillators are employed to obtain reference signals for carrier and timing synchronizations. The most common discrete-time phase noise channel model is analyzed, with the aim to fill the gap between measurements and analytical models. In particular, the power loss and the intersymbol interference due to the presence of phase noise is evaluated with reference to the measurements parameters and to the system bandwidth. Moreover, the impact on the communication systems' performance of the phase noise originating from the oscillator non idealities is considered, in case of free-running and phase-locked oscillators. The proposed analysis allows to extrapolate useful information about the performance of practical systems by investigating the power spectral density of the oscillator phase noise. An expression for the variance of the residual phase error after tracking, which depends on the main parameters of practical oscillators, is derived, and used to study the dependence of the performance on the symbol rate.

Paper Structure

This paper contains 13 sections, 2 theorems, 82 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Statistical model of the phase noise
  3. Power spectral density of the phasor
  4. Application to communications
  5. Error analysis of the discrete-time channel
  6. Optimal phase tracking theory
  7. Auxiliary channel models
  8. Low PN approximation
  9. High SNR approximation
  10. Residual error after phase tracking
  11. Performance in the case of free-running oscillators
  12. RESULTS
  13. CONCLUSION

Key Result

Lemma 1

Consider the system model described in (e:r), where the phasor $e^{j\theta(t)}$ has PSD given in (e:free_phasor_spectrum). Let $e^{j \theta_k}$ be the sample $e^{j \theta(k T_s)}$. The MSE in (e:eta1) between $z_k$ defined in (e:discretetime) and $y_k$ defined in (e:rec) is given by where and the terms $g_{n,m}$ are given in (e:g).

Figures (8)

  • Figure 1: Flat part of the phasor power spectral density given in (\ref{['e:flat']}), for $B_\theta=10^9$, $\ell^2_\infty=-130$ dBc/Hz and $\ell^2_{100}=-90$ dBc/Hz.
  • Figure 2: Block diagram of the system model.
  • Figure 3: Normalized variance of the approximation error for the auxiliary channel models on which the design of the phase estimator is based.
  • Figure 4: Power spectral densities of the phase noise and of the residual phase error after tracking in the system bandwidth.
  • Figure 5: SIR after the matched filter at the receiver as a function of the relative bandwidth parameter $\rho$.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1