Upper Bound on the Capacity of DiscreteTime Wiener Phase Noise Channels
Abstract
A discretetime Wiener phase noise channel with an integrateanddump multisample receiver is studied. An upper bound to the capacity with an average input power constraint is derived, and a high signaltonoise ratio (SNR) analysis is performed. If the oversampling factor grows as for , then the capacity prelog is at most at high SNR.
I Introduction
Instabilities of the oscillators used for up and downconversion of signals in communication systems give rise to the phenomenon known as phase noise [1]. The impairment on the system performance can be severe even for highquality oscillators, if the continuoustime waveform is processed by long filters at the receiver side. This is the case, for example, when the symbol time is very long, as happens when using orthogonal frequency division multiplexing. A study of the signaltonoise ratio (SNR) penalty induced by filtering of a white phase noise process has been recently done in [2], where it is shown that the best projection receiver suffers an SNR loss that depends on the phase noise statistics.
Typically, the phase noise generated by oscillators is a random process with memory, and this makes the analysis of the capacity challenging. The phase noise is usually modeled as a Wiener process, as it turns out to be accurate in describing the phase noise statistics of certain lasers used in fiberoptic communications [3], and of freerunning microwave oscillators [1]. Tight numerical bounds on the information rate of discretetime phase noise channels with memory are given in [4, 5, 6, 7], while analytical results on singleuser Wiener phase noise channels are given in [8, 9, 10, 11, 12] where it is shown that even weak phase noise becomes the limiting factor at high SNR.
In [11] an achievable rate region for the discretetime Wiener phase noise channel with an integrateanddump oversampling receiver was derived. For the same channel and receive filter, in this paper we develop an upper bound to the capacity and characterize the prelog^{1}^{1}1The factor in the capacity highSNR expansion in front of . at high SNR.
The paper is organized as follows. The system model for the continuoustime channel is described in Sec. II, along with a simplification that leads to the discretetime model under consideration. The upper bound to the capacity is derived in Sec. III, and the results are discussed in Sec. IV. Conclusions are drawn is Sec. V.
Notation: Capital letters denote random variables or random processes. The notation with is used for random vectors. With we denote the probability distribution of a real Gaussian random variable with zero mean and variance . The symbol means equality in distribution.
The symbol denotes the reduction of modulo , and the binary operator denotes summation modulo .
Given a complex random variable , we use the notation and to denote the amplitude and the phase of , respectively.
The operators , , and denote expectation, differential entropy, and mutual information, respectively.
The function denotes the natural logarithm of .
Ii System model
In this Section we describe how to obtain a discretetime version of the continuoustime channel, and we point out the main assumption that leads to the simplified model analyzed in Sec. III.
The output of a continuoustime phase noise channel can be written as
(1) 
where , is the databearing input waveform, and is a circularly symmetric complex white Gaussian noise. The phase process is given by
(2) 
where is a standard Wiener process, i.e., a process characterized by the following properties:

,

for any , is independent of the sigma algebra generated by ,

has continuous sample paths almost surely.
One can think of the Wiener phase process as an accumulation of white noise:
(3) 
where is a standard white Gaussian noise process.
Iia Signals and Signal Space
Suppose is in the set of finiteenergy signals in the interval . Let be an orthonormal basis of . We may write
(4) 
where
(5) 
is the complex conjugate of , and the are independent and identically distributed (iid), complexvalued, circularly symmetric, Gaussian random variables with zero mean and unit variance.
The projection of the received signal onto the th basis function is
(6)  
(7)  
(8) 
The set of equations given by (8) for can be interpreted as the output of an infinitedimensional multipleinput multipleoutput channel, whose fading channel matrix is .
IiB Receivers with Finite Time Resolution
Consider a receiver whose time resolution is limited to seconds, in the sense that every projection must include at least a second interval. More precisely, we set , where is the number of independent symbols transmitted in and is the oversampling factor, i.e., the number of samples per symbol. The integrateanddump receiver with resolution time uses the basis functions
(9) 
for . With the choice (9), the fading channel matrix is diagonal and the channel’s output for is
(10)  
(11) 
where we have used the notation and . In (10) we have used (2), the property , the substitution
(12) 
and the property . Finally, in step we have used the substitution .
Since the oversampling factor is , and the basis functions are square in time domain, we have for , and we can write the model (11) as
(13) 
for .
The vectors , , and are independent of each other. The variables are chosen as identically distributed with zero mean and variance , and the average power constraint is
(14) 
Since we set the power spectral density of to 1, the power is also the SNR, i.e., .
Using (3), the variables follow a discretetime Wiener process:
(15) 
where the ’s are iid Gaussian variables with zero mean and variance . The fading variables ’s are complexvalued and iid, and is independent of . In other words, is correlated only to , and is independent of the vector .
Note that for any finite , or equivalently for any finite oversampling factor , the vector does not represent a sufficient statistic for the detection of given in the model (1). In other words, the finite time resolution receiver is generally suboptimal.
In this paper we study a simplified model, where the fading variables are all one, i.e., we have
(16) 
This is a commonlystudied model, e.g., see [11, 13], and it is referred to as the discretetime Wiener phase noise channel. The complete model (13) is harder to analyze than the model (16), because in the former the dependency between and must be addressed. On the other hand, if the oversampling factor grows unbounded, then each random variable converges to ; this suggests that the analysis of the model (16) can give insights into the analysis of model (13) for receivers with high time resolution.
Iii Upper bound on capacity
We compute an upper bound to the capacity of the discretetime Wiener phase noise channel (16). For notational convenience, we use the following indexing for and :
(17) 
and we group the output samples associated with in the vector .
The capacity is defined as
(18) 
where the supremum is taken among the distributions of such that the average power constraint (14) is satisfied.
The mutual information rate can be upperbounded as follows:
(19) 
where step holds by a data processing inequality and because is independent of , because is conditionally independent of given , follows by stationarity of the processes, and by polar decomposition of and the chain rule.
For the amplitude channel, i.e., the first term in the righthand side (RHS) of (19), we have
(20) 
where holds by a data processing inequality and because is independent of , holds due to the circular symmetry of the ’s, because is independent of any other quantity, because the processed variable is a sufficient statistic for the detection of , and is an upper bound to the capacity of a noncoherent channel under an average power constraint [8, Eq. (16)] where represents a function independent of that vanishes for .
For the phase channel, i.e., the second term on the RHS of (19), we have
(21)  
(22) 
where in step we bound by the information extracted by a genieaided receiver that knows the additive noise , is obtained by deleting the amplitude contribution of , holds because forms a Markov chain, is obtained by applying reversible transformations, holds because the random variables are independent of any other quantity, holds by choosing a uniform distribution in for , and the last inequality is derived in the Appendix with
(23) 
Iv Discussion
As a byproduct of (25), an upper bound to the capacity prelog is
(26) 
As shown in the previous Section, a prelog of comes from the amplitude channel, while a contribution of comes from the phase channel. For example, if no oversampling is used (), one can let go to zero and obtain just the degrees of freedom provided by the amplitude channel, i.e., . This means that, without oversampling, the Wiener phase noise channel has the same degrees of freedom of the noncoherent channel. This is in accordance with the result given in [8].
If the oversampling factor grows as , for , then a prelog higher than can not be achieved. Indeed, a prelog as high as can be achieved with the processing described in [11]: the amplitude channel contributes with prelog by using the statistic
(27) 
to detect , and the phase channel contributes with prelog by using the processing
(28) 
to detect .
Figure 1 plots the known upper and lower bounds to the capacity prelog at high SNR. The upper bound is the result of this paper, expressed in (26). The lower bound is based on results derived in [11]. More specifically:

the lower bound for the amplitude channel, shown as the dashed black line, was derived independent of the growth rate of the oversampling factor;

for the phase channel it was shown how to achieve prelog for , hence the same prelog can be achieved for . It is not difficult to use the results of [11] to extend the lower bound in the range . It turns out that the achievable prelog linearly increases from to .
From the Figure, the capacity prelog is exactly known in the range , where the upper and lower bounds agree. The upper bound derived in this paper does not rule out the possibility to achieve prelog if the oversampling factor grows faster than .
Consider the case of receivers without oversampling ( or ). The analysis of Sec. III shows that the simplified model (16) has capacity prelog , while the general discretetime model that accounts also for the amplitude fading (13) has a behavior at high SNR [14], i.e., prelog . This means that, at least in the case , the simplified model is not a good approximation of the complete model.
V Conclusions
We have derived an upper bound to the capacity of discretetime Wiener phase noise channels. As a byproduct, we have obtained an upper bound to the capacity prelog at high SNR that depends on the growth rate of the oversampling factor used at the receiver. If the oversampling factor grows proportionally to , then a capacity prelog higher than can not be achieved.
Previous results on a lower bound to the capacity prelog allow to state that the capacity at high SNR is exactly for .
A lower bound to
The probability density function of is
(29) 
for and zero elsewhere, and can be upperbounded as follows for :
(30) 
where step follows by using for the terms with and for the terms with . Inequality holds because for . The differential entropy of can be lowerbounded as follows:
(31) 
where inequality is due to (30) and the monotonicity of the logarithm, and the last inequality is due to . A lower bound to the second moment of is:
(32) 
where
(33) 
is the error function. Since all the terms of the summation are positive, the inequality follows by considering only the term for . An upper bound to the last expectation on the RHS of (31) is
(34) 
The bound (31) with inequalities (32) and (34) give
(35) 
Also, note that , so we have that the bound (35) is tight for small , i.e.,
(36) 
Acknowledgment
L. Barletta was supported by the Technische Universität München â Institute for Advanced Study, funded by the German Excellence Initiative. G. Kramer was supported by an Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research.
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