Wiener Phase Noise Channel Research Articles

End-to-End Optimization of Constellation Shaping for Wiener Phase Noise Channels With a Differentiable Blind Phase Search

As the demand for higher data throughput in coherent optical communication systems increases, we need to find ways to increase capacity in existing and future optical communication links. To address the demand for higher spectral efficiencies, we apply end-to-end optimization for joint geometric and probabilistic constellation shaping in the presence of Wiener phase noise and carrier phase estimation. Our approach follows state-of-the-art bitwise auto-encoders, which require a differentiable implementation of all operations between transmitter and receiver, including the DSP algorithms. In this work, we show how to modify the ubiquitous blind phase search (BPS) algorithm, a popular carrier phase estimation algorithm, to make it differentiable and include it in the end-to-end constellation shaping. By leveraging joint geometric and probabilistic constellation shaping, we are able to obtain a robust and pilot-free modulation scheme improving the performance of 64-ary communication systems by at least 0.1 bit/symbol compared to square QAM constellations with neural demappers and by 0.05 bit/symbol compared to previously presented approaches applying only geometric constellation shaping.

Models and Information Rates for Wiener Phase Noise Channels

A waveform channel is considered where the transmitted signal is corrupted by Wiener phase noise and additive white Gaussian noise. A discrete-time channel model that takes into account the effect of filtering on the phase noise is developed. The model is based on a multi-sample receiver, i.e., an integrate-and-dump filter whose output is sampled at a rate higher than the signaling rate. It is shown that, at high Signal-to-Noise Ratio (SNR), the multi-sample receiver achieves a rate that grows logarithmically with the SNR if the number of samples per symbol (oversampling factor) grows with the cubic root of the SNR. Moreover, the pre-log factor is at least 1/2 and can be achieved by amplitude modulation. For an approximate discrete-time model of the multi-sample receiver, the capacity pre-log at high SNR is shown to be at least 3/4 if the number of samples per symbol grows with the square root of the SNR. The analysis shows that phase modulation achieves a pre-log of at least 1/4 while amplitude modulation still achieves a pre-log of 1/2. This is strictly greater than the capacity pre-log of the (approximate) discrete-time Wiener phase noise channel with only one sample per symbol, which is 1/2. Numerical simulations are used to compute lower bounds on the information rates achieved by the multi-sample receiver. The simulations show that oversampling is beneficial for both strong and weak phase noise at high SNR. In fact, the information rates are sometimes substantially larger than when using commonly-used approximate discrete-time models.

Computing the information rate of discrete-time Wiener phase noise channels by parametric Bayesian tracking.

A new upper bound (UB) on the information rate (IR) transferred through the additive white Gaussian noise channel affected by Wiener's laser phase noise is proposed in the paper. The bound is based on Bayesian tracking of the noisy phase. Specifically, the predictive and posterior densities involved in the tracking are expressed in parametric form, therefore tracking is made on parameters. This make the method less computationally demanding than known non-parametric methods, e.g. methods based on phase quantization and trellis representation of phase memory. Simulation results show that the UB is so close to the lower bound that we can claim of having virtually computed the actual IR.

Low Complexity Message Passing-Based Receiver Design for Wiener Phase-Noise Channels

This letter concerns low complexity receiver design for Wiener phase-noise channels, where the variance of the additive white Gaussian noise (AWGN) is unknown. By representing the system with a factor graph, both the variational message passing and the sum-product message passing are used to achieve efficient joint decoding, phase noise estimation, and AWGN precision (the reciprocal of AWGN variance) estimation. Compared with the state-of-the-art receiver, the proposed receiver is able to achieve the same performance with much lower complexity.

On discrete-time modeling of the filtered and symbol-rate sampled continuous-time signal affected by Wiener phase noise

This paper investigates the differences between the symbol-spaced discrete-time Wiener phase noise channel model, commonly assumed in the literature to represent the effect of phase noise, and that obtained by symbol-rate sampling the filtered continuous-time received signal affected by continuous-time Wiener phase-noise. In particular, for comparison, we consider some statistical tests to check temporal and distributional properties of the two models. We show that the fit between the two models is very good even for quite strong values of phase noise. The main result is that when the standard deviation of the discrete-time Wiener phase noise increment σPN is below a threshold of approximately σ¯PN≃0.1rad, the discrete-time Wiener model provides a good approximation to the actual symbol-spaced sampled filtered signal affected by continuous-time Wiener phase noise. We show that when σPN is below σ¯PN the ratio between the power of the signal and the power of the model mismatch is greater than 20dB. Simulation results are also presented to compare bit error rates of the two models in case of QPSK and 16-QAM transmission and to compare the power spectral densities of their associated complex exponential phase noises. Our results suggest that the discrete-time Wiener phase noise model can be adopted for many real-world systems, where, according to experimental results available in the literature, σPN in the order of 0.1rad is rarely found even when the nonlinearity of the optical channel is deeply stressed.

A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery

A new lower bound below the information rate transferred through the Additive White Gaussian Noise (AWGN) channel affected by discrete-time multiplicative Wiener's phase noise is proposed in the paper. The proposed lower bound is based on the Kalman approach to data-aided carrier phase recovery, and is less computationally demanding than known methods based on phase quantization and trellis representation of phase's memory. Simulation results show that the lower bound is close to the actual channel capacity, especially at low-to-intermediate signal-to-noise ratio.

Performance Analysis of OFDM with Wiener Phase Noise and Frequency Selective Fading Channel

We analyze the effect of Wiener phase noise on the capacity and signal-to-interference-plus-noise (SINR) ratio. Our analysis includes phase noise at the transmitter and receiver end of an OFDM communication link. We see that the capacity and SINR are random variables whose distribution depends on the phase noise and the fading channel. Using a Taylor series approximation, we show that the random variable, characterizing the phase noise, in the performance metrics can be expressed as a sum of correlated gamma variables with rank-deficient square-root normalized covariance matrix. The approximation holds well when the ratio between the subcarrier spacing and the 3dB bandwidth of the oscillator power spectral density is at least one order of magnitude, which for most practical oscillators is the case. In earlier literature, the probability density function of a sum of correlated gamma variables with full-rank square-root normalized covariance matrix was derived. We extend these results to the rank-deficient case and apply them to the random variable of our case. With the probability density functions characterizing the phase noise and the fading channel at hand, we proceed to obtain closed-form statistical measures of capacity and SINR. The simulations show the good agreement with our analytical expressions.