arXiv:1705.08148v1 [] 23 May 2017

Capacity Outer Bound and Degrees of Freedom of
Wiener Phase Noise Channels with Oversampling
Luca Barletta

Stefano Rini

Politecnico di Milano, Italy
luca.barletta@polimi.it

National Chiao Tung University, Taiwan
stefano@nctu.edu.tw

Abstract—The discrete-time Wiener phase noise channel with
an integrate-and-dump multi-sample receiver is studied. A novel
outer bound on the capacity with an average input power
constraint is derived as a function of the oversampling factor.
This outer bound yields the degrees of freedom for the scenario
in which the oversampling factor grows with the transmit power
P as P α . The result shows, perhaps surprisingly, that the largest
pre-log that can be attained with phase modulation at high signalto-noise ratio is at most 1/4.

I. I NTRODUCTION
In the discrete-time Wiener phase noise (WPN) channel, the
channel input is affected by both additive white Gaussian noise
(AWGN) and multiplicative WPN. The Wiener phase process
can be used to model a number of random phenomena: from
imperfections in the oscillator circuits at the transceivers, to
slow fading effects in wireless environments or laser imperfections in optical communications. For the WPN channel, the
sampled output of the filter matched to the transmit filter does
not always represent a sufficient statistic [1], and oversampling
does help in achieving higher rates over the continuous-time
channel [2], [3]. For this reason, it is of interest to study the
effect of oversampling on the maximum achievable rates [4].
In this paper, we study the discrete-time Wiener phase noise
with oversampling channel, a model obtained by sampling the
output of the continuous-time phase noise channel faster than
the symbol frequency. For this model, we determine a novel
outer bound on capacity and provide the generalized degrees
of freedom (GDoF) for the scenario in which the oversampling
factor grows to infinity as P α where P is the transmit power.
This result shows that, even in the high signal-to-noise ratio
(SNR) regime, it is not possible to attain more than a pre-log
factor of 1/4 through phase modulation.
State of the Art: The literature on channel affected by both
additive noise and phase noise considers three models: (i) the
continuous-time model, (ii) the discrete-time model and (iii)
the discrete-time model with oversampling.
For the continuous-time case, the joint effect of phase noise
and additive white Gaussian noise is first considered in [5].
In [6], the authors investigate white (Gaussian) phase noise
for which they observe a “spectral loss” phenomenon induced
by white phase noise. The continuous-time channel in the
presence of white noise is proposed and discussed in [1]. Here
it is shown that, for linear modulation, the output of the baudsampled filter matched to the shaping waveform represents a

sufficient statistic. Bounds on the SNR penalty for the case of
Wiener phase noise affecting the channel input are developed
in [7].
The discrete-time phase noise channel is obtained by considering a discrete-time WPN process sampled at symbol
frequency.
This model was first studied in [8]: here the high SNR
capacity is derived using duality arguments. The authors of [9]
propose a numerical method of precise evaluation of information rate bounds for this model. In [10] the authors derive
closed-form approximations to capacity which are shown to
be tight through numerical evaluations.
Finally, in the discrete-time model with oversampling multiple samples for every input symbol are obtained in output. This
model was first considered in [11] where it is shown that, if the
number of samples per symbol grows with the square root of
the SNR, the capacity pre-log is at least 3/4. The result in [11]
is extended in [12] to consider all scaling of the oversampling
coefficient of the form P α . Further simulations to compute
lower bounds on the information rates achieved by the multisample receiver have been recently shown in [13], [14].
Contribution: In this paper, we investigate the capacity of
the point-to-point channel corrupted by Wiener phase noise
and additive white Gaussian noise with an integrate-anddump multi-sample receiver, which we refer to as oversampled
Wiener phase noise (OWPN) channel. Our main contributions
are described as follows:
• Sec. IV– Capacity outer bound: Using the I-MMSE
relation [15] and a lower bound on the minimum mean-square
error (MMSE) estimate expressed through a recursive equation
[16], we obtain a novel outer bound on the capacity of the
OWPN channel.
• Sec. V–Degrees of Freedom: The outer bound in Sec. IV
is shown to be tight at high SNR; more specifically we derive
the GDoF for the model in which the transmit power P , and
the oversampling factor L, grow large for L = ⌊P α ⌋.

Paper Organization: The channel model is presented in
Sec. II while the known results in the literature are presented
in Sec. III. Outer bounds are derived in Sec. IV, while the
degrees of freedom analysis is shown in Sec. V. Conclusions
are drawn in Sec. VI.

II. S YSTEM M ODEL

III. K NOWN R ESULTS

We consider the OWPN, that is the point-to-point channel
corrupted by Wiener phase noise and additive white Gaussian
noise with an integrate-and-dump multi-sample receiver. The
main assumptions are as in [12], [17], that is (i) the amplitude
fading component, obtained by low-pass filtering the continuous time Wiener process, is neglected by setting it to one, and
(ii) the complex envelope of the transmitted waveform is constant in each symbol time interval. Under these assumptions,
the channel output for this model is obtained as
YmL+l = Xm ejΘmL+l + WmL+l ,

(1)

for m ∈ [1, . . . , M ] and l ∈ [0, . . . , L − 1] and where Wj ∼
CN (0, 2) is the additive noise, and L ∈ N is the oversampling
factor, that is equal to the reciprocal of the sampling time by
assuming that the symbol time is unitary. The Wiener phase
noise process {Θj } is defined as
ΘL−1 ∼ U([0, 2π])

(2a)

Θk+1 = Θk + Nk

Nk ∼ N (0, σ 2 L−1 ),

(2b)
k ∈ [L...∞),

(2c)

where U(I) indicates the uniform distribution over the set I.
The channel input is subject to the power constraint
E[|Xm |2 ] ≤ L−1 P,

(3)

which correspond to an average power constraint of P for the
transmitted waveform. As a consequence, the SNR is equal
(m+1)L−1
to P/2. Define Ym = YmL
= [YmL , . . . , Y(m+1)L−1 ]
(m+1)L−1
and Wm = WmL
: with this notation, the capacity of
the channel in (1) can be expressed as
C(P, σ 2 , L) = lim

M→∞

1
sup I(YM ; X M )
M

(4)

where the supremum is over all the distributions of X M such
that the power constraint P is satisfied and for an oversampling
factor equal to L.1
We also consider the high-SNR asymptotics of the expression in (4) which are described by the GDoF, defined as
C(P, σ 2 , ⌊P α ⌋)
,
P →∞
log(P )

D(α) = lim

(5)

that is, the capacity pre-log factor when P grows to infinity
while L = ⌊P α ⌋.
Note that, in the above formulation, the additive noise
variance is not affected by the oversampling factor, while the
transmit power of a sample is. The detailed derivation of the
discrete-time model in (1) from the continuous-time one is
presented in [13]. Since P and SNR are directly related, the
degrees of freedom formulation in (5) correctly captures the
asymptotic behaviour of capacity at high SNR.
1 In the following, we indicate the dependency of C to [P, σ 2 , L] only when
necessary.

The OWPN encompasses the classic discrete-time Wiener
phase noise (WPN) channel as the special case in which
L = 1. We have recently derived the capacity of the WPN
channel to within a small additive gap.
Theorem III.1. Capacity bounds on the WPN channel [18,
Th. V.1]. The capacity of the WPN channel is upper-bounded
as
1
C ≤ log(1 + P/2)
2

− 2π
e

σ 2 > 2π
 12 log(4πe) + 2 e − 2π log(e)
e
e
1−e

2
+
1
2
−1
log σ2 + log(2π) + log (e) P ≤ σ 2 ≤ 2π

e
 12
P −1 > σ 2 ,
2 log(1 + P/2)
(6)

and the exact capacity is to within G bpcu from the outer
bound in (6), where

σ 2 > 2π
 4
e ,
7.36 P −1 ≤ σ 2 ≤ 2π
(7)
G≤
e ,

1.8 P −1 > σ 2 .

The result in Th. III.1 is interesting at it shows that the
capacity of the WPN channel can be sub-divided in three
regimes: (i) for large values of the frequency noise variance σ 2 ,
the channel behaves similarly to a channel with circularly
uniform iid phase noise; (ii) when the frequency noise variance
is small, the effect of the additive noise dominates over that
of the phase noise, while (iii) for intermediate values of the
frequency noise variance, the transmission rate over the phase
modulation channel has to be reduced due to the presence of
phase noise.
A lower bound on the GDoF of the OWPN channel for
α = 1/2 is obtained in [4] and is later extended in [12] to
yield an inner bound to the GDoF region.2
Theorem III.2. GDoF lower bound [4], [12]. The function
D(α) in (5) can be lower-bounded as
 1+α
0 ≤ α < 21
2
(8)
D(α) ≥
3/4 α ≥ 12 ,
The inner bound in Th. III.2 is obtained by letting the
channel input have uniformly distributed phase in [0, 2π] while
the amplitude has a shifted exponential distribution. At the
receiver, the statistic used for detecting
|Xk | is ||Yk ||, and the

⋆ 
one used for detecting Xk is ∠ YkL YkL−1 e−j∠Xk−1
.
An outer bound on the capacity of the OWPN channel is
derived in [12] which, together with Th. III.2, yields the exact
GDoF expression for α ∈ [0, 1/2).
Theorem III.3. OWPN channel outer bound [12]. The
capacity of the OWPN channel is upper-bounded as




1
P
2π
1
+ log
+ O(1).
(9)
C ≤ log 1 +
2
2
2
eσ 2 L−1
2 The original result is derived for α ∈ [0, 1] but can be easily extended to
the case of a ∈ R+ .

Combining the results in [4] and [12], for α ∈ [0, 1/2] we
obtain that
1+α
.
(10)
if α ∈ [0, 1/2], then D(α) =
2
In the next section, we derive an outer bound tighter than that
in Th. III.3 which yields the GDoF region for any α ∈ R+ .
IV. O UTER B OUND

= I(X1 , Y1 |ΘL−1 )
= I ( |X1 |; Y1 | ΘL−1 ) + I ( X1 ; Y1 | ΘL−1 , |X1 |)

where the first equality follows from stationarity of the processes, and the last step by polar decomposition of X1 . In the
following, we refer to I ( |X1 |; Y1 | ΘL−1 ) as the rate of the
amplitude channel and I ( X1 ; Y1 | ΘL−1 , |X1 |) as the rate
of the phase channel. Analogously to [12, Eq. (20)], the rate
of the amplitude channel rate can be bounded as

We begin by deriving an outer bound on the capacity which
improves over the result in Th. III.3. Specifically, we provide
a better estimate of the transmission rate that can be attained
through phase modulation of the channel input.
Theorem IV.1. Capacity Outer bound. The capacity of the
OWPN channel is upper-bounded as


P
1
+ log (2π)
(11)
C ≤ log 1 +
2
2
!!
r
1
1
+ log L−1 P
1 + 4 2 −2 − 1
.
2
σ L P
Proof: Let us begin by upper bounding the information
rate in (4) as

I X1M ; Y1M
=

≤
=

M
X

k=1
M
X

k=1
M
X
k=1

I X1M ; Yk Y1k−1



I X1M , ΘkL−1 ; Yk Y1k−1



|Xk |, Wk |

i=0





(12)

ℓ=1


= I ΘkL−1 ; ΘkL−1 ⊕ Xk ⊕ NkL−1 , |Xk |, Wk | Y1k−1
=0
(13)
where (13) follows from the fact that the input is circularly
symmetric, so that |Xk | ⊥ Xk , and independent of the phase
ΘkL−1 . Similarly to [12, Eq. (19)], we note that the term
I ( Xk ; Yk | ΘkL−1 ) can be divided into two contributions: one
from the channel input amplitude and the other from channel
input phase. In fact, using (13), we can write
M

1 X
1
I X1M ; Y1M ≤
I ( Xk ; Yk | ΘkL−1 )
M
M
k=1

≤ I |X1 |; Y1 , Θ2L−1
ΘL−1
L

=I



L−1
√
1 X
WL+ℓ
|X1 | ; LX1 + √
L ℓ=0

!

(15a)

1
log(2πe(P + 2)),
(15b)
2
where (15b) follows from the result in [18, Thm. IV.1]
which bounds the entropy of a non-central chi-square random
variable.
The rate in the phase modulation channel can be written as
≤

I ( X1 ; Y1 | ΘL−1 , |X1 |)
= h ( X1 | ΘL−1 , |X1 |) − h ( X1 | ΘL−1 , |X1 |, Y1 ) (16a)
(16b)
= log(2π) − h ( X1 | ΘL−1 , |X1 |, Y1 ) ,
where (16b) follows from the fact that Xk ∼ U([0, 2π)). The
entropy term in (16b) can be rewritten as
X1 | |X1 |, Y1 e−jΘL−1
= h (ΘL−1 | |X1 |,
(
= −h

where (12) follows from the Markov chain Yk ⊸−
−
Xk , ΘkL−1 ⊸−
− Y1k−1 . Since the noise is circularly symmetric, a circularly distributed input is capacity achieving:
accordingly we have

I ΘkL−1 ; Yk | Y1k−1

)L
(
ℓ−1
X

≤ I ΘkL−1 ; ΘkL−1 ⊕ Xk ⊕
NkL+i−1
Y1k−1

I ( |X1 |; Y1 | ΘL−1 )

− h ( X1 | ΘL−1 , |X1 |, Y1 )

I ( Xk ; Yk | ΘkL−1 ) + I ΘkL−1 ; Yk | Y1k−1

(14)

|X1 | exp j





ΘL−1 +

= −h ΘL−1 | |X1 |, YeL2L−1


≤ −h ΘL−1 | |X1 |, YeL∞ ,



(17a)
k−1
X

i=L−1

Ni

!!

+ Wk

(17b)
)2L−1 
k=L



(17c)
(17d)

where Yek = |X1 |ejΘk + Wk . Here, (17a) follows from the
fact that ΘL−1 ∼ U([0, 2π)) and independent of all other
variables. In (17b) we let X1 = ΘL−1 : this substitution is to
stress the fact that X1 is independent of all other variables
and is thus statistically equivalent to ΘL−1 . In other words,
the entropy of the phase of X1 given the knowledge of ΘL−1
and (YL2L−1 , |X1 |) is equivalent to the entropy of ΘL−1 given
(YeL2L−1 , |X1 |). Finally, (17d) follows from the “conditioning
reduces entropy” property.
From the I-MMSE relationship [15, Eq. (182)], we have

 1
h ΘL−1 | |X1 |, YeL∞ = log(2πeVar [ΘL−1 ])
2
Z 
Var [ΘL−1 ]
1 ∞
−
2 0
1 + ρVar [ΘL−1 ]
i
√
−mmse(ΘL−1 | ρΘL−1 + Z, |X1 |, YeL∞ ) dρ, (18)

where Z ∼ N (0, 1) and independent of any other quantity,
and


(19)
mmse(S|K) , E (S − E [ S | K])2 .

The crucial step in bounding the entropy term in (17d) using
the relationship in (18) is in obtaining a tight lower bound to
the MMSE. To obtain such lower bound we rely on the result
√
in [16, Prop. 1]. To this end, let us rewrite U = ρΘk + Z
and
√
(20)
mmse(ΘL−1 | ρΘL−1 + Z, |X1 |, YeL∞ )
k−1
= lim mmse(Θk |U, |X1 |, Ye
)

Full GDoF
3/4
amplitude modulation
1/2
1/4
phase modulation
0

1/2

1

k→∞

≥

D(α)

lim J −1 ,
k→∞ k

(21)

where we used the time-reversibility of the Wiener process
and the Bayesian Cramer-Rao inequality. Here Jk is defined
as the entry in position (k, k) of the information matrix
associated with the joint distribution of [Θk1 , Ye1k−1 , U ] given
|X1 |. According to [16, Prop. 1], the value Jk can be computed
recursively as

Fig. 1: The degrees of freedom of the OWPN channel in
Lem. V.1.

Proof: As shown in (10), the GDoF in known for α ∈
[0, 1/2]. For α > 1/2 consider the outer bound in Th. IV.1 for
L = ⌊P α ⌋: for the rate in the amplitude modulation channel
in (15), we have

(12)

(22)

Jk = Dk−1 −

(Dk−1 )2
(11)

Jk−1 + Dk−1

,

(22)

lim

L
, n = 1, . . . , k − 1
(23a)
σ2
L
(23b)
Dn(12) = − 2 , n = 1, . . . , k − 1
σ
L
(22)
(23c)
Dk−1 = 2 + ρ,
σ


L
Dn(22) = 2 + E |X1 |2
σ
L
≤ 2 + L−1 P,
n = 1, . . . , k − 2,
(23d)
σ
where (23d) follows from the average power constraint. Using (23) into (22) we can easily find a lower bound on J∞ .
Using this bound in (18) and then in (16b), an upper bound
on the rate of the phase channel is obtained as
 
2π
1
(24)
I ( X1 ; Y1 | ΘL−1 , |X1 |) ≤ log
2
e
!!
r
1
1
L−1 P
+ log
1 + 4 2 −2 − 1
.
2
2
σ L P
Combining (15b) and (24) we obtain an outer bound on
capacity as in (11).
V. D EGREES OF F REEDOM A NALYSIS
The outer bound in Th. IV.1, together with the results in
Th. III.3 and Th. III.2 yields the GDoF of the OWPN channel.
Lemma V.1. The GDoF of the OWPN channel is obtained as
 1+α
0 ≤ α ≤ 21 ,
2
D(α) =
(25)
3
α > 21 .
4

I ( |X1 |; Y1 | ΘL−1 )
log(P )
1 log(2πe(P + 2))
1
≤ lim
= ,
P →∞ 2
log P
2

P →∞

for
Dn(11) =

α

(26)

which holds for any α ∈ R+ . For the rate in the phase
modulation channel in (24) we have
lim

I ( X1 ; Y1 | ΘL−1 , |X1 |)
(27)
log(P )
 −1 q

L P
1 + 4 σ2 L1−2 P − 1
2
1
1 log
= ,
≤ lim
P →∞ 2
log P
4

P →∞

which follows from the fact that, for α > 1/2, we have that
L−2 P = P 1−2α → 0 as P → ∞.
Combining (26) and (27) we obtain the outer bound on
D(α) with matches the inner bound in (8) for the regime
α ∈ [1/2, ∞).
The result in Lem. V.1 is schematically represented in Fig. 1:
the GDoF from amplitude modulation are equal to 1/2 for
all α, while the GDoF from phase modulation are equal to
(1 + α)/2 for α ∈ [0, 1/2] and equal to 1/4 for α > 1/2.
Note that, when α → 0, we obtain the model with L = 1 in
Th. III.1 which has pre-log equal to 1/2.
Discussion: The analysis of the GDoF in Lem. V.1 suggests
that there is a fundamental tension between the AWGN and
the multiplicative WPN, and improving
the resolution of the
√
receive filter beyond L−1 = 1/ P does not improve the
capacity pre-log at large P . From a high level perspective, the
parameter σ 2 is related to the quality of the local oscillators
available at the user: in this sense, then, the result in Lem.
V.1 shows that, regardless of the value σ 2 , the fundamental
tension will eventually reduce the available DoF for a suitably
large P .
From the I-MMSE bound in (18) and (22) used in the

proof of Th. IV.1, it is apparent that the tension between the
AWGN and the WPN is related to the difficulty of predicting a
new sample of a Wiener process when corrupted by AWGN.
The following questions naturally arise: is the limitation of
the available GDoF an artifact of the assumptions used to
derive the model in (1) or is it an inherent limitation of the
physical system? Further, the model in (1) neglects the effect
of amplitude fading for the sake of simplicity. For the model
encompassing both phase and amplitude fading, one wonders
whether it is possible to attain higher DGoF. The model
in (1) is obtained by employing a waveform that allocates
the power uniformly over time. One then naturally wonders
whether it is possible to attain higher performance employing
a waveform that does not allocate energy uniformly in time.
These interesting open questions are to be addressed in future
works.
VI. C ONCLUSION
We have derived an outer bound on the capacity of discretetime Wiener phase noise channels with multi-sample receivers.
In this model, the input of a point-to-point channel is corrupted
by both additive noise and multiplicative phase noise: the additive noise is a white Gaussian process while the phase noise
is a Wiener process. For each symbol in input, the channel
produces L outputs corresponding to the output of an integrateand-dump multi-sample receiver with oversampling factor L.
A novel outer bound is derived using the I-MMSE relationship
and a recursive expression of the minimum mean-square error
though the Fisher information matrix. This novel outer bound
is used to derive the generalized degrees of freedom for the
scenario in which the oversampling factor grows with the
transmit power P as L = ⌊P α ⌋. This latter result shows
that there exists a fundamental tension between the AWGN
and the WPN that limits the available GDoF at 1/4 for phase
modulated signals, regardless of the power of the phase noise.
The degrees of freedom analysis of models encompassing both
multiplicative phase noise and multiplicative amplitude noise
remains an interesting open question.
R EFERENCES
[1] L. Barletta and G. Kramer, “On continuous-time white phase noise
channels,” in Proc. IEEE Int. Symp. Inf. Theory, 2014, pp. 2426–2429.
[2] M. Martalò, C. Tripodi, and R. Raheli, “On the information rate of
phase noise-limited communications,” in Inf. Theory and Applications
Workshop (ITA), 2013.
[3] H. Ghozlan and G. Kramer, “On Wiener phase noise channels at high
signal-to-noise ratio,” in Proc. IEEE Int. Symp. Inf. Theory, 2013, pp.
2279–2283.
[4] ——, “Phase modulation for discrete-time Wiener phase noise channels
with oversampling at high SNR,” in Proc. IEEE Int. Symp. Inf. Theory,
2014, pp. 1554–1557.
[5] G. J. Foschini and G. Vannucci, “Characterizing filtered light waves
corrupted by phase noise,” IEEE Trans. Inf. Theory, vol. 34, no. 6, pp.
1437–1448, 1988.
[6] B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik,
“Calculation of mutual information for partially coherent Gaussian
channels with applications to fiber optics,” IEEE Trans. Inf. Theory,
vol. 57, no. 9, pp. 5720–5736, 2011.
[7] L. Barletta and G. Kramer, “Signal-to-noise ratio penalties for
continuous-time phase noise channels,” in Intern. Conf. on Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM),
2014, pp. 232–235.

[8] A. Lapidoth, “On phase noise channels at high SNR,” in IEEE Inf.
Theory Workshop, 2002, pp. 1–4.
[9] L. Barletta, M. Magarini, and A. Spalvieri, “The information rate
transferred through the discrete-time Wiener’s phase noise channel,”
IEEE/OSA J. Lightw. Technol., vol. 30, no. 10, pp. 1480–1486, May
2012.
[10] M. R. Khanzadi, R. Krishnan, J. Söder, and T. Eriksson, “On the capacity
of the Wiener phase-noise channel: Bounds and capacity achieving
distributions,” IEEE Trans. Commun., vol. 63, no. 11, pp. 4174–4184,
2015.
[11] H. Ghozlan and G. Kramer, “Phase modulation for discrete-time Wiener
phase noise channels with oversampling at high SNR,” in Proc. IEEE
Int. Symp. Inf. Theory, June 2014, pp. 1554–1557.
[12] L. Barletta and G. Kramer, “Upper bound on the capacity of discretetime Wiener phase noise channels,” in Proc. IEEE Inf. Theory Workshop,
April 2015, pp. 1–5.
[13] H. Ghozlan and G. Kramer, “Models and information rates for Wiener
phase noise channels,” IEEE Trans. Inf. Theory, vol. 63, no. 4, pp. 2376–
2393, 2017.
[14] A. Nedelcu and L. Barletta, “Waveform design for Wiener phase noise
channels and multi-sample receivers,” in 21st Intern. ITG Workshop on
Smart Antennas (WSA), March 2017.
[15] D. Guo, S. Shamai, and S. Verdú, “Mutual information and minimum
mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory,
vol. 51, no. 4, pp. 1261–1282, April 2005.
[16] P. Tichavsky, C. H. Muravchik, and A. Nehorai, “Posterior CramerRao bounds for discrete-time nonlinear filtering,” IEEE Trans. Signal
Processing, vol. 46, no. 5, pp. 1386–1396, May 1998.
[17] H. Ghozlan and G. Kramer, “Multi-sample receivers increase information rates for Wiener phase noise channels,” in Global Commun. Conf.
(GLOBECOM), 2013, pp. 1897–1902.
[18] L. Barletta and S. Rini, “Capacity of discrete-time Wiener phase
noise channels to within a constant gap,” in To appear in
Proc. IEEE Int. Symp. Inf. Theory, 2017. [Online]. Available:
http://arxiv.org/pdf/1701.04982v1.pdf

