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.

Models and Information Rates for Wiener Phase Noise Channels

Differential phase shift keying (DPSK) in the presence of both additive white Gaussian noise and a phase impairment, modeled by a Brownian motion is considered. A time-diversity scheme is used for mitigating the effects of phase noise. This scheme renders a repetition coding approach where the transmitter sends multiple replicas of each data bit. An upper bound on the bit error probability, relying on a bivariate moment-generating function admitted by certain real functionals of the phase sample-path, is derived. The approach taken yields a trackable analysis, which rigorously adheres the phase noise effects. The impact of an incomplete statistical characterization on the tightness of the resultant bound is addressed. The theory, which is applicable to assess the design and performance of general heterodyned lightwave systems using (delay) differential demodulation (as DPSK and CP-FSK, or continuous phase frequency-shift keying), is exemplified and explicit results for the considered time-diversity DPSK scheme are provided. The optimum design of the diversity level is discussed and it is concluded that power efficient transmission is feasible even at bit rates comparable with the signal linewidth. >

Time diversity in DPSK noisy phase 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 considers 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.

In this paper, the capacity of the additive white Gaussian noise (AWGN) channel, affected by time-varying Wiener phase noise is investigated. Tight upper and lower bounds on the capacity of this channel are developed. The upper bound is obtained by using the duality approach, and by considering a specific distribution over the output of the channel. In order to lower-bound the capacity, first a family of capacity-achieving input distributions is found by solving a functional optimization of the channel mutual information. Then, lower bounds on the capacity are obtained by drawing samples from the proposed distributions through Monte Carlo simulations. The proposed capacity-achieving input distributions are circularly symmetric, non-Gaussian, and the input amplitudes are correlated over time. The evaluated capacity bounds are tight for a wide range of signal-to-noise-ratio (SNR) values, and thus they can be used to quantify the capacity. Specifically, the bounds follow the well-known AWGN capacity curve at low SNR, while at high SNR, they coincide with the high-SNR capacity result available in the literature for the phase-noise channel.

https://publications.lib.chalmers.se/records/fulltext/224139/local_224139.pdf

On the Capacity of the Wiener Phase-Noise Channel: Bounds and Capacity Achieving Distributions

This paper describes the design and performance of an M ary frequency shift keyed (FSK) signaling and demodulation scheme for an optical communication system using semiconductor lasers and heterodyne detection. Frequency or phase noise in semiconductor lasers causes spectral spreading, producing a nonzero linewidth laser signal. This degrades communication performance when compared to a system using an ideal laser with zero linewidth. We present estimates of the bit error rate (BER) performance of M -ary frequency shift keying (FSK) with noncoherent demodulation in the presence of white Gaussian frequency noise and additive channel noise. This is typical of an optical system using semiconductor lasers and heterodyne detection. Estimates use the union-Chernoff bound with a simplified channel model to predict the effects of frequency noise. Two effects of frequency noise are identified: signal attenuation or suppression, and crosstalk. These cause an offset in the BER curve from the BER in the absence of frequency noise, and an error rate floor, respectively. The error rate floor is lower than previously predicted. When performance is not crosstalk limited, M -ary FSK is found to perform better than binary FSK with the same system bandwidth constraints, as would be predicted if ideal lasers are used. Theoretical results are compared with Monte Carlo simulations of the system.

M-ary FSK performance for coherent optical communication systems using semiconductor lasers

The performance of noncoherent detection of orthogonal phase-noise impaired signals in the presence of additive white Gaussian noise is considered. The authors present a novel class of upper and lower bounds on the error probability of a binary hypothesis test, comprising quadratic forms of the additive noise and of the filtered noisy-phase signal plus noise. Filtering the noisy-phase signal gives rise to bounded nonlinear functionals of the Brownian motion sample-path, the exact statistics of which are unknown. The classical theory of Chebyshev systems is utilized to solve the limiting values of the required stochastic expectations, based on the availability of the corresponding generalized moments. The resulting multidimensional moment bounds constitute the tightest possible error bounds for the given set of generalized moments, and require only modest computational efforts. The theory is applicable to assess the design and performance of optical heterodyne systems and is most suitable for coded systems employing hard-decisions, for which the obtained bounds are remarkably tight. >

Performance bounds for noncoherent detection under Brownian phase noise

The distribution law of the Brownian motion exponential functional: /spl epsiv/=|f/sub [0,1]/exp[j/spl radic/(2/spl beta/W/sub t/]dt|/sup 2/), where {W/sub t/,t/spl isin//spl Rscr//sup +/} is a standard Brownian motion, modeling the laser's phase noise, plays a key role in many of the proposed heterodyne lightwave systems. The exact analytic derivation of these statistics appears to be, however, a formidable mathematical task. We present a closed form formula for the kth power moment E/spl epsiv//sup k/ induced by the unknown distribution law. The results are useful in assessing the performance of noisy phase lightwave systems. >

Y.E. Dallal

Papers

Performance bounds for noncoherent detection under Brownian phase noise

Power moment derivation for noisy phase lightwave systems