Capacity of Discrete-Time Wiener Phase Noise Channels to Within a Constant Gap

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by  Luca Barletta

Capacity of Discrete-Time Wiener Phase Noise
Channels to Within a Constant Gap
Luca Barletta Stefano Rini
Politecnico di Milano, Italy National Chiao Tung University, Taiwan
[email protected] [email protected]
7 Abstract—Thecapacityofthediscrete-timechannelaffectedby is not Gaussian. The authors of [4] improve on the results
1 both additive Gaussian noiseand Wienerphasenoise is studied. of [3] by showing that the capacity-achieving distribution,
0 Novel inner and outer bounds are presented, which differ of at similarly to the block-memoryless phase noise channel, is
2 most 6.65 bits per channel use for all channel parameters. The
discrete and possesses an infinite number of mass points.
capacityofthismodelcanbesubdividedinthreeregimes:(i)for
n The WPN channel encompasses the block memoryless
largevaluesofthefrequencynoisevariance,thechannelbehaves
a similarly to a channel with circularly uniform iid phase noise; channel and the non-coherent channel as the two limiting
J
(ii) when the frequency noise variance is small, the effects of cases in which the variance of the innovation process tends
8 the additive noise dominate over those of the phase noise, while to zero and infinity, respectively. This model was fist studied
1 (iii) for intermediate values of the frequency noise variance, the
in [5] where the high SNR capacity is derived using duality
transmission rate over the phase modulation channel has to be

] reduced due to the presence of phase noise. arguments.The authorsof [6] propose a numericalmethod of
T
evaluatingtightinformationrate boundsfor thismodel.In [7]
.I I. INTRODUCTION theauthorsderiveanalyticalapproximationstocapacitywhich
s
Inthediscrete-timeWienerphasenoise(WPN)channel,the are shown to be tight through numerical evaluations.
c
[ channelinputisaffectedbybothadditivewhiteGaussiannoise Capacity bounds for more complex models taking into

(AWGN) and multiplicative Wiener phase noise. The Wiener accountoversamplingatthereceiverand/oreffectofimperfect
1
phase process can be used to model a number of random matchedfilteringareproposedin[8]–[11]:heretheboundsare
v
2 phenomena:fromimperfectionsintheoscillatorcircuitsatthe valid only at high SNR.
8 transceivers,toslowfadingeffectsinwirelessenvironmentsor Contributions: In this paper we derive the capacity of the
9 oscillations in the laser frequency in optical communications. discrete-time WPN channel to within 6.65 bits per channel
4
Despite its relevancein manypracticalscenarios,the capacity use (bpcu) for all channel parameters, namely SNR and
0
oftheWPN channelremainsanopenproblemasthepresence frequency noise variance. This result is shown by separately
.
1 of memory in the phase noise process makes the analysis considering three regimes of the frequency noise variance:
0
challenging. In this paper we provide the first approximate small, intermediate, and high variance. The practical insights
7
1 characterizationofcapacityforthediscrete-timeWPNchannel into these regimes are as follows. In the small frequency
: for all channel parameters and provide an input distribution noise variance regime, the effects of the additive noise dom-
v
which performs close to optimal. inates over those of the phase noise: for this reason the
i
X Stateofthe Art:ChannelmodelsencompassingbothAWGN channel behaves essentially as an AWGN channel. In the
r and multiplicative phase noise have been considered in the highfrequencynoisevarianceregime,theinstantaneousphase
a
literatureunderdifferentassumptionsonthedistributionofthe variations dominate over the memory of the process and the
phase noise process. The channel model in which the phase channel resembles a non-coherent phase noise channel. In
noiseisrandomlyselectedatthebeginningoftransmissionand the intermediate frequency noise variance regime, part of the
is kept fixed throughthe transmission block-lengthis referred transmissionrateoverthephasemodulationchannelhasto be
toasblock-memorylessphasenoisechannel.In[1]theauthors sacrificed due to the presence of the phase noise.
prove that the capacity-achieving input distribution for this Organization:Thechannelmodelandknownresultsinlitera-
model exhibits a circular symmetry and that the distribution ture are presented in Sec. II. Some preliminaryresults, useful
oftheamplitudeoftheinputisdiscretewithaninfinitenumber for obtaining tight capacity bounds, are detailed in Sec. III.
of mass points. In [2], the author presents capacity outer and Outer bounds are derived in Sec. IV, while the capacity to
innerboundsthatcapturethefirsttwotermsoftheasymptotic within a constant gap is shown in Sec. V. Conclusions are
expansion of capacity as the signal-to-noise ratio (SNR) goes drawn in Sec. VI.
to infinity. II. CHANNEL MODELAND KNOWNRESULTS
When the phasenoise processis composedof iid circularly
The discrete-time Wiener phase noise (WPN) channel is
uniform samples, the channel is referred to as non-coherent
described by the input-outputrelationship
phasenoisechannel.Thecapacityofthismodelisfirststudied
in[3]whereitisshownthatthecapacity-achievingdistribution Y =X ejΘi +W , i=1,...,N (1)
i i i
where j = √ 1, W (0,2), i.e. W is a circularly- and chi-square RVs.
i i
− ∼ CN
symmetriccomplexGaussian randomvariable(RV) with zero
Theorem III.1. Wrapped Gaussian entropy.The entropyof
mean and variance 2. The channel input X is subject to an
i
the circularly wrapped Gaussian distribution ∆ of variance
average power constraint over the transmission block-length
N, that is N E[X 2] NP, while Θ , i N is a σ∆2 is lower-bounded as
i=1 | i| ≤ { i ∈ }
Wiener procPess defined by the recursive equation h(∆) log(2π)−21−e−e−σ∆2σ∆2 log(e) σ∆2 >2π/e
Θ0 =0, Θi+1 =Θi+∆i, i 0, (2) ≥( 1log(2πeσ2)+g(σ2)log(e) σ2 2π/e
≥ 2 ∆ ∆ ∆ ≤
(7)
where ∆ (0,σ2), and σ2 is the frequency noise
i ∼ N ∆ ∆
variance. Standard definitions of code, achievable rate and where g(σ2) is obtained as
∆
capacity are assumed in the following.
dKtshqeneugoarewmreenosddioRusftlerufsisrbueuletstdsiqo:ounmAawr,seiint.heoo.ftneodtnhi-ence[on5ut]rt,aplcuiottynhdpaiatsrioamnaeednteoornn|-xc|iXe|n2it|raa=nld|ctxwhii|o-, g(σ∆2)= 21erf p2πσ∆2 !−pe−2π2πσσ2∆2∆2 π+ 41(−π+e−σσππ∆2∆22)−21.
Proof:The boundinvolvingthe term g(σ2) is derivedin
∆
Y 2 X = x χ2(x 2). (3) [11]: the following derivation is an alternative bound to
| i| || i| | i|∼ 2 | i|
1
Moreover,in thecapacity-achievingdistribution,theinputhas h(∆) log(2πeσ2)+g(σ2)log(e). (8)
≥ 2 ∆ ∆
iid circularly symmetric phase X , i [1,N].
i
∈
The pdf of a zero-mean wrapped Gaussian can be written,
Lemma II.1. Ergodic phase noise capacity [5, Sec. VI].
using Jacobi’s triple product as
The capacity of the model in (1) when Θ , i N is any
i
{ ∈ } ∞
stationary ergodic process with finite entropy rate is obtained 1
p (x)= (1 qn)(1+qn−1/2e+ix)(1+qn−1/2e−ix)
as ∆ 2π −
n=1
Y
1 P (9)
= log 1+ +log(2π) h( Θ )+o(1), (4)
k
C 2 2 − { }
(cid:18) (cid:19) where q =e−σ∆2 so that
where h( Θ ) is the entropy rate of the phase noise process
{ k} φ(q) ∞ ( 1)j qj(j+1)/2
and o(1) vanishes as P . h(∆)= log +2 − log(e) (10)
→∞ − 2π j 1 qj
The result in Lem. II.1 establishes the high SNR capacity (cid:18) (cid:19) Xj=1 −
ofthephasenoisechannelforalargeclassofphaseprocesses whereφ(x)istheEulerfunction.Notethatφ(q)in(10)isless
but does not apply to the WPN channel, as Wiener process is thanonebydefinition,sothat log(φ(q)) 0.Moreover,the
− ≥
not stationary. function κ(j,q) defined as
Specializing a result of [10] to model (1) we obtain the
1qj(j+1)/2
following lemma. κ(j,q)= (11)
j 1 qj
Lemma II.2. WPN channel capacity inner bound [10, −
is decreasing in j when q (0,1) so that
Sec. III]. The capacity of the model in (1)-(2) is lower- ∈
bounded as ∞ ( 1)j qj(j+1)/2 q
− . (12)
1 P 2b+2ν j 1 qj ≥−1 q
sup log − j=1 − −
C ≥ 2 π2eνρ X
b≥0,ν>0(cid:26) (cid:18) (cid:19) The upper bound in (14) follows by noting that
2+b−1 3b
+ log(e) , (5)
− ν P 2b h(∆)=h( ejZ) h(Z). (13)
(cid:18) − (cid:19) (cid:27) ≤
where ρ=1 (1 E[Xi −2])2e−σ∆2/2.
− − | |
The bound in (7) essentially states that the entropy of the
The inner bound in (5) is obtained by letting the input be
wrapped Gaussian ∆ = ejZ for Z (0,σ2) is well
a truncated exponential distribution, that is ∼ N ∆
approximated as
1 x b
p|Xi|2(x)= P/2 bexp −P/2− b , x≥b. (6) h(∆) min 1log(2πeσ2),log(2π) , (14)
− (cid:18) − (cid:19) ≈ 2 ∆
(cid:26) (cid:27)
III. PRELIMINARY RESULTS
that is, the minimum between the entropy of the Gaussian
Inthis sectionwe presenttwo theoremsthatwillaid, in the RV Z and a uniformly distributed RV with support [0,2π].
following sections, the development of tight inner and outer As shown in Fig. 1, the approximation in (14) is rather tight:
boundsto capacity. The first theorem boundsthe entropy of a this figure plots the entropy of the uniform RV over [0,2π],
wrapped Gaussian RV while the second the entropy of a chi the wrapped Gaussian RV and the Gaussian RV for different
3.5 5
1log(2πeσ2)
2 ∆ 4.5 outer bound
bits] 3 log(2π) bits] 4
y[ 2.5 h(∆) y[ 3.5 h(χ22(λ))
p p
o o 3
r r
nt 2 nt 2.5 inner bound
E E
2
1.5
1.5
1 1
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10
σ2
∆ λ
σ2 = 2π
∆ e Fig. 2: The entropy of a chi-square distribution (blue solid
line), the inner bound (red dashed line) and the outer bound
Fig. 1: The entropy of a wrapped Gaussian RV (blue solid
(black dash-dotted line) in (15) for the non-centrality param-
line), a Gaussian RV (black dash-dotted line), and a uniform
RV in [0,2π] (red dashed line) as in (14) for σ∆2 ∈[0,6]. eter λ∈[0,10].
values of σ2. entropy bound reads as
∆
TheoremIII.2. Non-centralχ22 andχ2 entropy.Theentropy h(T) E 1log(64)+ 1log(λ2) log(√π+1) (23)
of a non-central chi-square distribution with two degrees of ≥ 2 4 −
(cid:20)
freedom and non-central parameter λ is bounded as 1 64 (cid:3)
log λ
1 1 ≥ 2 (√π+1)2
log(8πeλ) log(3) h(χ2(λ)) log(8πe(1+λ)). (cid:18) (cid:19)
2 − ≤ 2 ≤ 2 1 1
log(8πeλ) log(3). (24)
(15) ≥ 2 − 2
Similarly, the entropy of a chi distribution with non-central An upper bound on h(T) can be obtained through the
parameter λ is lower-bounded as “Gaussian maximizes entropy” property as
1 1 1
h(χ (λ)) log(8πe) log(6)+ Ei( λ/2)log(e). (16) h(T) log(2πeVar[T])
2
≥ 2 − 2 − ≤ 2
1
Proof: The pdf of T χ2(λ) is = log(8πe(1+λ)). (25)
∼ 2 2
pT(t)= 1e−t+2λI0(√λt) (17) For the entropyofa chidistribution,by changeof variable,
2
we can write
where
h(√T)=h(T) E[log√T] log(2)
E[T]=2+λ (18a) − −
(15) 1 1
Var[T]=4(1+λ). (18b) log(8πeλ) log(6) E[logT]
≥ 2 − − 2
Using the bound for I0(x) of Corollary A.2, we have (=22) 1log(8πe) log(6)+ 1Ei( λ/2)log(e). (26)
2 − 2 −
h(T)=E[ logp (T)] (19)
T
−
1 2 1 The bounds (15) are plotted in Fig. 2 as a function of the
E 3log(2)+ √T √λ log(e)+ log(λT)
≥ 2 − 4 non-centrality parameter λ.
(cid:20) (cid:16) (cid:17)
log(√π+1) . (20)
− IV. OUTERBOUNDS
Note that 1(√T √λ)2(cid:3)is convexin T forT 0,so thatwe
2 − ≥ This section introduces novel outer bounds to the capacity
have
of phase noise channel which connect this model to the non-
1 2 1 2
E √T √λ E[T] √λ coherentphasenoisechannelandthememorylessphasenoise
2 − ≥ 2 −
(cid:20) (cid:16) (cid:17) (cid:21) 1(cid:16)p (cid:17)2 channel.
= √λ+2 √λ 0. (21) In the non-coherent phase noise channel, the phase noise
2 − ≥ process Θ , i N is an iid sequence of uniformly dis-
(cid:16) (cid:17) { i ∈ }
Furthermore, we have tributedRVsin[0,2π]:forthisreasonreliablecommunication
can take place only over the amplitude modulation channel
E[logT]=log(λ) Ei( λ/2)log(e) log(λ), (22)
− − ≥ p . The rate attainable over this channel is bounded in
|Y|||X|
where Ei() is the exponential integral function, so that the the next theorem.
·
TheoremIV.1. Outerboundoncapacityovernon-coherent formly distributed input phase X in [0,2π) is capacity-
k
AWGNchannel.Thecapacityofthenon-coherentphasenoise achieving. This also implies that
channel is upper-bounded as
I(Θ ;Y Yk−1)
k−1 k
1 |
sup I(X ; X+Z ) log(2πe(P +2)), (27) =I(Θ ; X ej(Θk−1+∆k−1+∠Xk)+W Yk−1)=0,
pX:E|X|2≤P | | | | ≤ 2 k−1 | k| k|
given that X is independent of Θ . Using the polar
where Z ∼CN(0,2). decompositionkX = X ej∠X, and drokp−p1ing the time index
| |
Proof: Using the Gaussian maximum entropy bound for convenience of notation, write
we bound the positive entropy term as h(X + Z )
1/2log(2πe(P +2)). For the conditional entro|py we w|rite≤ I(Xk;Yk|Θk−1)=I X;Xej∆+W (30a)
=I(X ; X+W )+I X;Xei∆+W X (30b)
(cid:0) (cid:1)
h(X +Z X ) maxh(X +Z X , X >x) | | | | | |
| ||| | ≥ x≥0 | ||| | | | I(X ; X+W )+I((cid:0) X; X ∆) (cid:12) (cid:1) (30c)
∞ ≤ 1 | | | | ⊕ (cid:12)
max h(t+Z )dF (t) log(2πe(P +2))+log(2π) h(∆), (30d)
≥ x≥0 Zx | | |X| ≤ 2 −
1 8πe 1 x2 where (30b) follows by circular symmetry of W, (30c) by
maxP(X >x) log + Ei log(e) ,
≥ x≥0 | | 2 62 2 − 2 revealing W to the receiver, and (30d) from Thm. IV.1 and
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
the circular symmetry of X. The symbol in (30c) denotes
where in the last step we used the bound in Thm. III.2 and ⊕
the addition modulo 2π.
the (increasing) monotonicity of the bound in (16). Since
V. MAIN RESULT
maxf(x)=max max f(x),maxf(x) , (28)
x≥0 0≤x<r x≥r Theorem V.1. Capacity to within a constant gap. The
(cid:26) (cid:27)
and by choosing r such that 1log(8πe) log(6) + capacity of the WPN channel in (1)-(2) is upper-boundedas
2 −
1Ei( r2/2)log(e) = 0 (there exists only one value of r 1
2 − log(1+P/2)
with this property, namely r 0.937), we conclude that C ≤ 2
≈
h0(|Xx+<Zr||g|Xive|)smaunsetgbaetivpeosnituivmeb,esri,ncwehtihleetmheaxmimaxizimatiiozantifoonr 12log(4πe)+21−e−e−2eπ2eπ log(e) σ∆2 > 2eπ
fo≤r x r gives a positive number. + 1log 2 +log(2π)+log2(e) P−1 σ2 2π
The≥next outer bound is a refinement of the result in  21log((cid:16)1σ+∆2 (cid:17)P/2) P−1 >≤σ∆2 ≤ e
2 ∆
Lmeomde.lIiIn.1(1to)-(y2i)elvdaliadnaotufitenriteboSuNnRds.toThtheebcoaupnadciitsydoefrivthede  (31)
and the exact capacity is to within bpcu from the outer
by revealing the past phase realization to the receiver, which
G
bound in (31), where
results in a memoryless phase noise channel.
4 σ2 > 2π
Theorem IV.2. Memoryless phase noise channel outer ∆ e
6.65 P−1 σ2 2π (32)
bound. The capacity of the WPN channel in (1)-(2) can be G ≤ ≤ ∆ ≤ e
1.21 P−1 >σ2
upper-bounded as  ∆
Proof: The achievability proof relies on a simple trans-
1 
min log(2πe(P +2))+log(2π) h(∆), missionschemeemployingiidcomplexGaussianinputswhile
C ≤ 2 −
(cid:26) the converse proof is developed from the outer bound in
log(1+P/2) , (29)
Thm. IV.2.
}
where h(∆) is the entropy of a wrapped Gaussian with Achievability:Asin[10,Eq.(18)],theWPN channelcapacity
variance σ2. can be lower-bounded as
∆
N
log(1P+rooPf:/2)Aandtrivisiaolbtcaainpeadcitbyy poruotveirdinbgouthnedphisaseCnois≤e C ≥ N1 I |Xk|;Y1N X1k−1 +I Xk;Y1N X1k−1,|Xk|
k=1
sequence to the receiver. Another capacity outer bound is X (cid:0) (cid:12) (cid:1) (cid:0) (cid:12) (cid:1)
I X 2; Y 2 +I (cid:12) X ;Y1 X , X . (cid:12) (33)
obtained as follows: ≥ | 1| | 1| 1 0 0 | 1|
Ne(cid:0)xt we lower-(cid:1)bound(cid:0)the term(cid:12)I =I (cid:1)X 2; Y 2 and
I(XN;YN) I(XN,Θ ;Y Yk−1) (cid:12) || | 1| | 1|
≤Xk k−1 k| Ia∠mpl=itudIe anXd1p;hYa0s1e mXo0d,|uXla1ti|o,n cwhhaincnhelw, r(cid:0)eespreefcetrivetoly.a(cid:1)s the
= I(XN;Y Θ )+I(Θ ;Y Yk−1) (cid:0) (cid:12) (cid:1)
k| k−1 k−1 k| Amplitude modul(cid:12)ation channel: Let the input distribution
k •
X(cid:0) (cid:1) be X (0,P). Recognizing that Y 2 = (1 + P/2)K
= k I(Xk;Yk|Θk−1)+I(Θk−1;Yk|Yk−1) . where K∼ ∼CNχ22, we obtain | |
X(cid:0) (cid:1) I X 2; Y 2 =log(e(2+P)) h(Y 2 X 2). (34)
Since the additive noise W is circularly symmetric, a uni-
| | | | − | | || |
(cid:0) (cid:1)
Using the outer bound of Thm. III.2, we bound the term yields the outer bound
h(Y 2 X 2) as
| | || | 1log(1+P/2) P 10
h(|Y|2||X|2)≤ 21Elog(8πe(|X|2+1))≤ 12log(8πe(1+P)). C ≤( 221log(2πe(P +2))+21−e−e−2eπ2eπ log(e) P ≤>10
(35) (40)
Finally, combining (34) and (35) yields By comparing the outer bound in (40) with the inner bound
in (39) we obtain the capacity gap
1 P 1 2π 1 1+P/2
I log 1+ log + log .
|| ≥ 2 2 − 2 e 2 1+P 1log(1+P)+ 1log 2π P 10
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) ((cid:19)36) G ≤( 221log 12++PP +221−e−e−2(cid:0)eπ2eπe (cid:1)log(e)+log(4π) P >≤10
Phase modulation channel: The second term in the RHS (cid:16) (cid:17) (41)
•
of (33) can be lower-bounded as follows:
which is smaller than 4 bpcu for any P >0 and σ2 >2π/e.
∆
I∠ =I X1;Y01 X0,|X1| w•eSmcoanllsifdreerqutheenctryivniaolisoeuvtearribaonucned:σ∆2 <loPg(−11:+InPt/h2is).reWgihmene
=I( X(cid:0)1;X0+W(cid:12) 0 X0, X(cid:1)1 ) C ≤
(cid:12) | | | P <2,capacityisnecessarilyless than1bpcu. WhenP 2,
+I X ;X ej∆0 +W X , X ,X +W ≥
1 1 1 0 1 0 0 the gap between the trivial outer bound and the inner bound
| |
=I (cid:0)X1;X1ej∆0 +W1 X(cid:12) 1 (cid:1) in (39) is at most 1.21bpcu.
≥I((cid:0) X1; X1⊕∆0⊕N(cid:12)(cid:12)||(cid:12)|X|1(cid:1)|) T•hIentoeurtmeredbioautnedfrineqTuhe.ncIVy.2noinisethvisarrieagnimcee: cP1an≤beσ∆2rew≤rit2teπen:
=log(2π) h(∆ N X ), (37)
− 0⊕ || 1| as
where (37) follows by considering just the phase of Y1, 1 P +2
asunpdpoNrt =[0,2π|X],1w|+e cWan1.apSpinlyceth∆e0m⊕axNimuims deenfitrnoepdyothveeorrtehme C ≤ 2log(cid:18) σ∆2 (cid:19)+log(2π)−g(σ∆2)log(e)
1 P +2
to upper-boundthe conditionalentropytermh(∆ N X ) log +log(2π)+log2(e), (42)
0⊕ || 1| ≤ 2 σ2
with the entropy of a wrapped Gaussian RV with variance (cid:18) ∆ (cid:19)
σ∆2 +1/|X1|2: where in the last step we note thatg(σ∆2) is decreasingin σ∆2
so that g(σ2) log(e). We next compare the inner bound
h(∆0⊕N||X1|) (38a) in (39) with∆th≥e o−uter bound in (42) and obtain
1 1
≤ 2E(cid:20)log(cid:18)2πe(cid:18)σ∆2 + |X1|2(cid:19)(cid:19)(cid:21) (38b) G = γ2 +log(e) log(e)+ 21log (2π)212++PP 1+σ2σP∆2P
= 12E log 2πe σ∆2|X1|2+1 − 21E log|X1|2 (cid:16)γ +log(e)(cid:17)log(e)+log(2π(cid:18))+ 3log(2) 6.∆65, (cid:19)
1 (cid:2) (cid:0) (cid:0)σ2P +1 (cid:1)(cid:1)(cid:3)γ (cid:2) (cid:3) ≤ 2 2 ≤
log 2πe ∆ + log(e), (38c) (cid:16) (cid:17)
≤ 2 P 2 where, in the last passage, we have considered the worst case
(cid:18) (cid:18) (cid:19)(cid:19) with σ2 =P−1 and P .
where (38b) follows by Thm. III.1, and (38c) from Jensen’s ∆ →∞
Note that the inner bound in Th. V.1 relies on iid complex
inequality for the first term and from the fact that
Gaussian inputs, thus showing that this input distribution
Elog X 2 =log(Pe−γ), where γ is the Euler-Mascheroni
| 1| performs sufficiently close to capacity.
constant.
Some further insights on the result in Th. V.1 emerge by
simplifyingtheouterboundin(31)asinthefollowinglemma.
Putting together the contributions of amplitude and phase
modulation in (36) and (38) respectively, we obtain the inner Lemma V.2. Larger gap, simpler expression. The capacity
bound of the WPN channel in in (1)-(2) is upper-bounded as
1 P 1 1+P/2 Pe−γ 1log(1+P/2)+4 σ2 > 2π
log 1+ + log . (39) 2 ∆ e
C ≥ 2 2 2 1+P σ2P +1 1log(1+P/2) 1logσ2 +5.5 P−1 σ2 2π
(cid:18) (cid:19) (cid:18) ∆ (cid:19) C ≤ 2 − 2 ∆ ≤ ∆ ≤ e
log(1+P/2) P−1 >σ2
Note that, for any fixed σ2, the pre-log at large P given  ∆
∆ (43)
by (39) is 1/2. Also, note that if σ2 1/P, then the pre-log 
at large P is 1. ∆ ≤ and the exact capacity is to within a gap of 7 bpcu from the
outer bound in (43).
Converseandgapfrom capacity:TheouterboundinTh.IV.2
The three regimes in (43) can be intuitively explained as
canbesub-dividedinthethreeregimesofsmall,intermediate, follows. The inequality σ2 ≷ 2π/e in (43) arises from the
and high frequency noise variance. ∆
result in Th. III.1 and whether the phase noise entropy is
High frequency noise variance: σ2 > 2π/e: The outer better approximated by using a uniform RV over [0,2π] or a
• ∆
bound from Th. IV.2 together with the condition σ2 > 2π/e Gaussian RV. When the frequency noise variance is high, the
∆
Proof: By definition
50 5
40 4.5 I (x)= 1 πexcos(θ)dθ
0
P30 4 π Z0
3.5 1 π/2 1 π
= excos(θ)dθ+ excos(θ)dθ. (46)
20
3 π Z0 π Zπ/2
10 2.5 I1 I2
10 20 30 40 50 60 70 80 90 100 Using the infi|nite pro{dzuctform}ula|for the c{ozsine fun}ctionand
1/σ2 a linear lower bound for the cosine, we obtain the bound
∆
Fb[1oi,gu1.n03d0:]Ai.nctohnetopurroopflootfofTthh.eVex.1acftograPpb∈etw[0e,e5n0i]nnanerda1n/dσo∆2ute∈r excos(θ) ≤( eexx((11−−ππ242θθ)2) π0/≤2θ≤≤θ π≤/π2 (47)
so that
cnhoaisnen,eilnbwehhiacvhestrasnimsmiliasrsliyontotaakecshpalnanceelownliythinutnhiefoarmmpplihtuadsee I1 ≤ π1 π/2ex(1−π42θ2)dθ
Z0
modulation channel. When the frequency noise variance is √π ex
= erf(√x) (48)
small, instead, the effect of the multiplicative noise times the 4 √x
channel input has variance Pσ2 1 which is smaller than
≤ while
the variance of the additive noise. The inequality P−1 σ2,
as in Th. IV.2, intuitively arises from the rate attainab≤le o∆n 1 π ex(1−π2θ)dθ
2
the phase modulation channel: the largest rate attainable on I ≤ π
Zπ/2
this channel is close to 1/2log(P +1), when the frequency 1 e−x
noisevarianceisrelativelysmall.Forahigherfrequencynoise = − . (49)
2x
variance, the rate of the phase modulation channel is instead
Combining (48) and (49) we obtain the bound
close to 1/2log(σ−2).
∆
ex 2e−x(1 e−x)
Although the largest gap between inner and outer bound is
I (x) erf(√x)√π+ − (50)
0
boundedby6.65bpcu,thedifferencebetweeninnerandouter ≤ 4√x √x
(cid:18) (cid:19)
boundsforanyparameterregimecanbeeasilyevaluatedfrom
the proof of Th. V.1. A plot of the exact value of in (32) is The result in Lemma A.1 can be further weakened to
G
presented in Fig. 3.
obtain an expression which can be more easily manipulated
analytically.
VI. CONCLUSIONS
Corollary A.2. An upper bound to I (x) is
0
We have derived outer and inner bounds of the discrete- ex √π+1
time Wiener phase noise channel and have shown that they I0(x) . (51)
≤ √x 4
differ of at most 6.65 bpcu at any SNR and frequency
noise variance. Both bounds have rather simple expressions Proof: From Lemma A.1 we have
and suggest that all channels can be roughly divided in three ex 2e−x(1 e−x)
parameter regimes: high, intermediate, and small frequency I0(x) erf(√x)√π+ −
≤ 4√x √x
noise variance. Moreover, the analysis of the inner bound (cid:18) (cid:19)
ex
shows that a complex Gaussian input distribution performs √π+1 (52)
≤ 4√x
rather close to capacity.
(cid:0) (cid:1)
wherethelastequalityfollowsbyerf(x) 1andthefactthat
≤
APPENDIX 2e−x(1−e−x)<√x.
Note that, although simpler, the expression in (51) is not
Lemma A.1. Upper bound on modified Bessel function tight as x 0, where it actually has an asymptote. The
→
The zero-th order modified Bessel function of the first kind expression in (44), although cumbersome, is tight as x 0.
→
can be upper-bounded as
ex 2e−x(1 e−x) REFERENCES
I (x) erf(√x)√π+ − (44)
0
≤ 4√x √x
(cid:18) (cid:19) [1] R. Nuriyev and A. Anastasopoulos, “Capacity and coding for the
for any x 0, where block-independent noncoherent awgn channel,” IEEE Transactions on
≥ Information Theory,vol.51,no.3,pp.866–883,March2005.
erf(x)= 2 xe−t2dt. (45) [2] cGh.anDnuelr,i”siI,EE“OEnCothmemucanpicaactiitoynsoLfetttheers,bvlooclk.-1m6,emnoo.ry8l,epsps.1p1h5as7e–-1n1o6is0e,
√π
Z0 2012.
[3] G.ColavolpeandR.Raheli,“Thecapacityofthenoncoherentchannel,”
Europeantransactions ontelecommunications, vol.12,no.4,pp.289–
296,2001.
[4] M. Katz and S. Shamai, “On the capacity-achieving distribution of
thediscrete-time noncoherent andpartially coherent AWGNchannels,”
IEEE Transactions on Information Theory, vol. 50, no. 10, pp. 2257–
2270,2004.
[5] A. Lapidoth, “On phase noise channels at high SNR,” in Information
TheoryWorkshop, 2002.Proceedings ofthe2002IEEE. IEEE,2002,
pp.1–4.
[6] L. Barletta, M. Magarini, and A. Spalvieri, “The information rate
transferred through thediscrete-time Wiener’s phase noisechannel,” J.
Lightw. Technol.,vol.30,no.10,pp.1480–1486, May2012.
[7] M.R.Khanzadi,R.Krishnan,J.So¨der,andT.Eriksson,“Onthecapacity
of the Wiener phase-noise channel: Bounds and capacity achieving
distributions,” IEEE Trans. Commun., vol. 63, no. 11, pp. 4174–4184,
2015.
[8] H. Ghozlan and G. Kramer, “On Wiener phase noise channels at high
signal-to-noise ratio,” in IEEEInt. Symp. Inf. Theory (ISIT), 2013, pp.
2279–2283.
[9] ——,“Phasemodulationfordiscrete-timeWienerphasenoisechannels
withoversamplingathighSNR,”inProc.IEEEInt.Symp.Inf.Theory,
June2014,pp.1554–1557.
[10] L.BarlettaandG.Kramer,“Lowerboundonthecapacityofcontinuous-
timeWienerphasenoisechannels,”inProc.IEEEInt.Symp.Inf.Theory,
June2015,pp.2326–2330.
[11] ——,“Upperboundonthecapacityofdiscrete-timeWienerphasenoise
channels,” inProc.IEEEInf.TheoryWorkshop,April2015,pp.1–5.