JOURNAL, VOL. , NO. , MONTH YEAR

1

Models and Information Rates for
Wiener Phase Noise Channels

arXiv:1503.03130v2 [] 11 Aug 2017

Hassan Ghozlan, Member, IEEE, and Gerhard Kramer, Fellow, IEEE

Abstract—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 integrateand-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 discretetime 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 discretetime models.
Index Terms—Phase noise, Wiener process, waveform channel,
capacity, oversampling.

I. I NTRODUCTION
Phase noise arises due to the instability of oscillators [1] in
communication systems such as satellite communication [2],
microwave links [3] or optical fiber communication [4]. The
statistical characterization of the phase noise depends on the
application. In systems with phase-locked loops (PLL), the
residual phase noise follows a Tikhonov distribution [5]. In
Digital Video Broadcasting DVB-S2, an example of a satellite
communication system, the phase noise process is modeled by
The work of H. Ghozlan was supported by a USC Annenberg Fellowship
and the National Science Foundation (NSF) under Grant CCF-09-05235.
The work of G. Kramer was supported by an Alexander von Humboldt
Professorship endowed by the German Federal Ministry of Education and
Research, as well as by the NSF under Grant CCF-09-05235. Part of the
material in this paper was presented at the IEEE International Symposium on
Information Theory, Istanbul, Turkey, July, 2013, at the Global Communications Conference, Atlanta, GA, December, 2013 and at the IEEE International
Symposium on Information Theory, Honolulu, HI, June/July, 2014.
H. Ghozlan was with the Department of Electrical Engineering, University
of Southern California, Los Angeles, CA 90089, USA. He is now with Intel
Corporation, Hillsboro, OR 97124, USA. (e-mail: hassan.ghozlan@intel.com).
G. Kramer is with the Institute for Communications Engineering,
Technical University of Munich, 80333 Munich, Germany (email: gerhard.kramer@tum.de).
Copyright (c) 2014 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.

the sum of the outputs of two infinite-impulse response filters
driven by the same white Gaussian noise process [2]. In optical
fiber communication, the phase noise in laser oscillators is
modeled by a Wiener process [4].
A commonly studied discrete-time channel model is
Yk = Xk ejΘk + Zk

(1)

where {Yk } are the output symbols, {Xk } are the input
symbols, {Θk } is the phase noise process and {Zk } is additive
white Gaussian noise (AWGN). For example, Katz and Shamai
[6] studied the model (1) when {Θk } is independent and identically distributed (i.i.d.) according to pΘ (·), when Θ is uniformly distributed (called a non-coherent AWGN channel) and
when Θ has a Tikhonov (or von Mises) distribution (called a
partially-coherent AWGN channel). The i.i.d. Tikhonov phase
noise models the residual phase error in systems with phasetracking devices, e.g., phase-locked loops (PLL) and ideal
interleavers/deinterleavers. Katz and Shamai [6] characterized
some properties of the capacity-achieving distribution. Tight
lower bounds on the capacities of memoryless non-coherent
and partially coherent AWGN channels were computed by
solving an optimization problem numerically in [6] and [7],
respectively. Lapidoth studied in [8] a phase noise channel
(1) at high SNR. He considered both memoryless phase noise
and phase noise with memory. He showed that the capacity
grows logarithmically with the SNR with a pre-log factor 1/2
if the differential entropy rate of {Θk } is finite. The prelog 1/2 is achieved by amplitude modulation only, and the
phase modulation contributes only a bounded number of bits.
Dauwels and Loeliger [9] proposed a particle filtering method
to compute information rates for discrete-time continuousstate channels with memory and applied the method to (1)
for Wiener phase noise and auto-regressive-moving-average
(ARMA) phase noise. Barletta, Magarini and Spalvieri [10]
computed lower bounds on information rates for (1) with
Wiener phase noise by using the auxiliary channel technique
proposed in [11] and they computed upper bounds in [12].
They also developed a lower bound based on Kalman filtering
in [13]. Barbieri and Colavolpe [14] computed lower bounds
with an auxiliary channel slightly different from [10]. Durisi
et al. [15], [16] developed capacity bounds for a multi-input
multi-output (MIMO) extension of the discrete-time model (1).
A less-studied model since 1990 is the continuous-time
model for phase noise channels, also called a waveform phase
noise channel, see Fig. 1. The received waveform r(t) is
r(t) = x(t) ejθ(t) + n(t), for t ∈ R

(2)

where x(t) is the transmitted waveform while n(t) and θ(t)
are additive noise and phase noise, respectively. Continuous-

JOURNAL, VOL. , NO. , MONTH YEAR

x(t)
Transmitter

Fig. 1.

2

×

+

ejθ(t)

n(t)

r(t)
Receiver

Waveform phase noise channel.

time white phase noise was recently considered in [17, Section
IV.C] and [18], [19]. We study the waveform channel (2) with
Wiener phase noise. A detailed description of the channel is
given in Sec. II. This model is reasonable, for example, for
optical fiber communication with low to intermediate power
and laser phase noise, see [20]–[22]. Since the sampling
of a continuous-time Wiener process yields a discrete-time
Wiener process (Gaussian random walk), it is tempting to use
the model (1) with {Θ} as a discrete-time Wiener process.
However, this ignores the effect of filtering prior to sampling.
It was pointed out in [20] that “even coherent systems relying
on amplitude modulation (phase noise is obviously a problem
in systems employing phase modulation) will suffer some
degradation due to the presence of phase noise”. This is
because the filtering converts phase fluctuations to amplitude
variations. It is worth mentioning that filtering is necessary
before sampling to limit the variance of the noise samples.
The model (1) thus does not fit1 the channel (2) and it is
not obvious whether a pre-log 1/2 is achievable. The model
that takes the effect of filtering into account is
Yk = Xk Hk + Nk

(3)

where {Hk } is a fading process. The model (3) falls in the
class of non-coherent fading channels, i.e., the transmitter
and receiver have knowledge of the distribution of the fading
process {Hk }, but have no knowledge of its realization. For
such channels, Lapidoth and Moser showed in [23] that, at
high SNR, the capacity grows double-logarithmically with the
SNR, when the process {Hk } is stationary, ergodic and has
finite differential entropy rate.
Rather than using a filter and sampling its output at the
symbol rate, we use a multi-sample receiver, i.e., a filter whose
output is sampled many times per symbol. We show that this
receiver achieves a rate that grows logarithmically with the
SNR if the number of samples per symbol grows with the
cubic root of the SNR. Furthermore, we show that a prelog of 1/2 is achievable through amplitude modulation. We
also show for an approximate multi-sample model, where the
approximation is to ignore the filtering effect over a sample,
that phase modulation contributes 1/4 to the pre-log factor at
high SNR if the oversampling factor grows with the square
root of SNR. We corroborate the results of the high-SNR
analysis with numerical simulations at finite SNR. This paper
collects and extends results presented in [24]–[26].
Benefits of oversampling were reported for other problems.
For example, in some digital storage systems, it was shown
by numerical simulations that oversampling can increase the
1

The model would fit if a very narrow pulse was used as the pulse-shaping
filter. Such pulses may be interesting for very large bandwidth applications.

information rate [27]. In a low-pass-filter-and-limiter (nonlinear) channel, it was found that oversampling (with a factor of
2) offers a higher information rate [28]. It was demonstrated
in [29] that doubling the sampling rate recovers some of the
loss in capacity incurred on the bandlimited Gaussian channel
with a one-bit output quantizer. For non-coherent block fading
channels, it was shown in [30] that by doubling the sampling
rate the information rate grows logarithmically at high SNR
with a pre-log 1 − 1/N rather than 1 − Q/N which is the prelog achieved by sampling at the symbol rate (N is the length
of the fading block and Q is the rank of the covariance matrix
of the discrete-time channel gains within each block).
This paper is organized as follows. The continuous-time
model is described in Sec. II and the discrete-time models
of the matched filter receiver and the multi-sample receiver
are described in Sec. III. We give a lower bound on the
capacity in Sec. IV based on amplitude modulation and show
that the pre-log factor is at least 1/2 if the oversampling
factor grows with the cubic root of the SNR. We also give
a lower bound on the rate achieved by phase modulation for
the approximate multi-sample discrete-time model. For such
a model, we show that phase modulation contributes at least
1/4 to the pre-log factor, resulting in an overall pre-log factor
of 3/4 when both amplitude and phase modulation are used.
This is achieved by using an oversampling factor that grows
with the square root of the SNR. We develop algorithms
to compute tight lower bounds on the information rates of
a multi-sample receiver in Sec. V. In Sec. VI, we report
the results of numerical simulations which demonstrate the
importance of increasing the oversampling factor with the SNR
to achieve higher rates. We discuss in Sec. VII our results,
related work, open problems and intuitions for our proofs.
Finally, we conclude with Sec. VIII.
II. WAVEFORM P HASE N OISE C HANNEL
√
We use the following notation: j = −1 , ∗ denotes
the complex conjugate, δD is the Dirac delta function, ⌈·⌉
is the ceiling operator. We use X k as a shorthand for
(X1 , X2 , . . . , Xk ). Suppose the transmit-waveform is x(t) and
the receiver observes
r(t) = x(t) ejθ(t) + n(t)

(4)

where n(t) is a realization of a white circularly-symmetric
complex Gaussian process N (t) with
E [N (t)] = 0
2
E [N (t1 )N ∗ (t2 )] = σN
δD (t2 − t1 ).

The phase θ(t) is a realization of a Wiener process
Z t
W (τ )dτ
Θ(t) = Θ(0) +

(5)

(6)

0

where Θ(0) is uniform on [−π, π) and W (t) is a real Gaussian
process with
E [W (t)] = 0
E [W (t1 )W (t2 )] = 2πβ δD (t2 − t1 )

(7)

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where 0 < β < ∞. The processes N (t) and Θ(t) are
independent of each other and independent of the input.
2
N0 = 2σN
is the single-sided power spectral density of the
additive noise. We define U (t) ≡ exp(jΘ(t)). The autocorrelation function of U (t) is
RU (t1 , t2 ) = E [U (t1 )U ∗ (t2 )] = exp (−πβ|t2 − t1 |)

(8)

and the power spectral density of U (t) is
Z ∞
1
β/2
SU (f ) =
RU (t, t + τ ) e−j2πf τ dτ =
.
π (β/2)2 + f 2
−∞
(9)
The spectrum is said to have a Lorentzian shape. We have
β = fFWHM = 2fHWHM where fFWHM is the full-width at halfmaximum and fHWHM is the half-width at half-maximum. One
also refers to β as the linewidth of the phase noise.
Let T be the transmission interval, then the transmitted
waveforms must satisfy the power constraint
" Z
#
1 T
2
E
|X(t)| dt ≤ P
(10)
T 0
where X(t) is a random process whose realization is x(t).
III. D ISCRETE - TIME M ODELS
A. Transmitter

r(t)

Fig. 2.

g ∗ (Ts − t)

y(t)

yk
kTs

decoder

x̂m

Matched filter receiver with symbol rate sampling.

where k = 1, . . . , b, ⋆ denotes convolution and nk is the part
of the output due to the additive noise. Therefore, we have the
discrete-time model

where
Hk ≡

Z

∞
−∞

Yk = Xk Hk + Nk

(13)

ejΘ(τ ) |g(τ − (k − 1)Ts )|2 dτ.

(14)

The transmitter and receiver know the distribution of the fading
process {Hk } but have no knowledge of its realization. We
strongly suspect that {Hk } in (14) is stationary, ergodic and
has a finite differential entropy rate, i.e., h({Hk }) > −∞.
Lapidoth and Moser showed in [23] that, at high SNR, the
capacity grows double-logarithmically with the SNR under the
aforementioned conditions on {Hk }.
A commonly-used approximation for Hk is
Z ∞
ejθ(τ ) |g(τ − (k − 1)Ts )|2 dτ
−∞
Z ∞
jθ(kTs )
≈e
|g(τ − (k − 1)Ts )|2 dτ
(15)
−∞

Let (x1 , x2 , . . . , xb ) be the codeword sent by the transmitter.
Suppose the transmitter uses a unit-energy pulse g(t) whose
time support is [0, Ts ] where Ts is the symbol interval, which
is finite and fixed. The waveform sent by the transmitter is
x(t) =

b
X

m=1

xm g(t − (m − 1)Ts ).

(11)

We remark that using pulses that are limited in time to one
symbol interval simplifies the analysis because it eliminates intersymbol interference at the receiver. We focus on rectangular
pulses for the high SNR analysis in Sec. IV. A disadvantage
of a rectangular pulses is the slow decay of the sidelobes in
frequency domain, which motivates us to also consider cosinesquared pulses for numerical rate computation in Sec. VI. The
sidelobes of a cosine-squared pulses, which is equivalent to a
(time-domain) raised-cosine pulse with roll-off factor 1, decay
much faster than those of a rectangular pulse. We also remark
that Martalò et al [31] extended some of our numerical results
for bandlimited pulses. However, their model does not include
fully the filtering effects of a continuous-time model.
B. Matched Filter Receiver
The received signal r(t) is fed to a matched filter and the
output y(t) of the filter is sampled at the symbol rate as
illustrated in Fig. 2. The k-th output sample is
yk = r(t) ⋆ g ∗ (Ts − t)|t=kTs
Z ∞
= xk
ejθ(τ ) |g(τ − (k − 1)Ts )|2 dτ + nk
−∞

(12)

which yields the approximate discrete-time model
Yk = Xk ejΘ(kTs ) + Nk .

(16)

We refer to (13) and (16) as the symbol-rate model with
filtering and without filtering, respectively. We also refer to
(16) as the approximate symbol-rate model. We do not claim
that this approximation is always accurate. We introduce the
approximate model to examine its validity by comparing
it with the discrete-time model based on the multi-sample
receiver. We remark that the differential entropy of {Θ(kTs )}
is finite because the linewidth β of Θ(t) and the symbol
interval Ts are finite and therefore, the pre-log of the model
(16) is 1/2.
C. Multi-sample Receiver
Consider next a multi-sample receiver (see Fig. 3). Let L
be the number of samples per symbol (L ≥ 1) and define the
sample interval as
Ts
.
(17)
L
The received waveform r(t) is filtered using an integrator over
a sample interval to give the output signal
Z t
y(t) =
r(τ ) dτ.
(18)
∆=

t−∆

The signal y(t) is a realization of a random process Y (t) that
is sampled at t = k∆, k = 1, . . . , n = bL. Hence, we have
the discrete-time model in which the k-th output sample is
Yk = X⌈k/L⌉ ∆ ejΘk Fk + Nk

(19)

JOURNAL, VOL. , NO. , MONTH YEAR

r(t)

Fig. 3.

R

y(t)

4

yk

decoder

kTs /L

x̂m

where the supremum is over all joint distributions of the input
symbols satisfying the power constraint. For a given input
distribution, the achievable rate R is

Multi-sample receiver.

R(SNR) = I(X; Y ) ≡ lim

b→∞

where Yk ≡ Y (k∆), Θk ≡ Θ((k − 1)∆),

 
 
Z
k
1 k∆
g τ−
− 1 Ts ej(Θ(τ )−Θk ) dτ
Fk ≡
∆ (k−1)∆
L
(20)
and
Nk ≡

Z

k∆

N (τ ) dτ.

(21)

(k−1)∆

The process {Nk } is an i.i.d. circularly-symmetric complex
Gaussian process with
E[Nk ] = 0

(22)

2
E[Nk Nℓ∗ ] = σN
∆ δK [ℓ − k]

E[Nk Nℓ ] = 0

(23)
(24)

where δK [·] is the Kronecker delta. The process {Θk } is the
discrete-time Wiener process
Θk = Θk−1 + Wk

(25)

for k = 2, . . . , n, where Θ1 is uniform on [−π, π) and {Wk }
is an i.i.d. real Gaussian process with mean 0 and E[|Wk |2 ] =
2πβ∆, i.e., the probability density function (pdf) of Wk is
2
pWk (w) = G(w; 0, σW
) where


(w − µ)2
1
2
exp −
(26)
G(w; µ, σ ) = √
2σ 2
2πσ 2
2
σW

and
= 2πβ∆. The random variable (Wk mod 2π) is a
wrapped Gaussian and its pdf is
pW (w; σ 2 ) =

∞
X

i=−∞

G(w − 2iπ; 0, σ 2 ).

(27)

Moreover, {Fk } and {Wk } are independent of {Nk } but not
independent of each other. Equations (10) and (11) imply the
power constraint
b
1 X
E[|Xm |2 ] ≤ P ≡ PTs .
b m=1

(28)

The signal-to-noise ratio SNR is defined as
SNR ≡

P
2 T
σN
s

=

P
2 .
σN

(29)

D. Information Rates
We use Yk as a shorthand for (Y1 , Y2 , . . . , Yk ) where
Yk ≡ (Y(k−1)L+1 , Y(k−1)L+2 , . . . , Y(k−1)L+L ).

(30)

The capacity of a point-to-point channel is given by
C(SNR) = lim

b→∞

1
sup I(X b ; Yb )
b

(31)

1
I(X b ; Yb ).
b

(32)

IV. H IGH -SNR I NFORMATION R ATES
In this section, we study rectangular pulses, i.e., we consider
 p
1/Ts , 0 ≤ t < Ts ,
g(t) ≡
(33)
0,
otherwise.
It follows that Fk in (20) becomes
Z
1 k∆
ej(Θ(τ )−Θk ) dτ.
Fk ≡
∆ (k−1)∆

(34)

We define XA ≡ |X| and ΦX ≡ arg(X) where arg(·)
denotes the phase angle (also called the argument) of a
complex number. We decompose the mutual information using
the chain rule into two parts:
b
b
I(X b ; Yb ) = I(XA
; Yb ) + I(ΦbX ; Yb |XA
)
b
where XA
=
(|X1 |, . . . , |Xb |)
(arg X1 , . . . , arg Xb ). Define

and

ΦbX

(35)
=

1
b
I(XA
; Yb )
b

(36)

1
b
I(ΦbX ; Yb |XA
).
b

(37)

I(X; Y ) = I(XA ; Y ) + I(ΦX ; Y |XA ).

(38)

I(XA ; Y ) ≡ lim

b→∞

and
I(ΦX ; Y |XA ) ≡ lim

b→∞

It follows from (32)–(37) that

The first term represents the contribution of the amplitude
modulation while the second term represents the contribution
of the phase modulation. The results of this section are
summarized in Theorem 1 and Theorem 2 which address the
contribution of amplitude modulation and phase modulation
to the information rate, respectively.
Before we proceed to the theorems, we introduce an important result that we use multiple times in the paper. Let X and
Y be the input and output of a channel, respectively and let
q(·|·) be a valid conditional pdf, which represents an “auxiliary
channel”. The following auxiliary-channel lower bound holds:
I(X; Y ) ≥ I(X; Y ) = E[log qY |X (Y |X) − log qY (Y )] (39)
where qY (·) is the output pdf obtained by connecting the true
input source to the auxiliary channel. Note that E[·] is the
expectation according to the true distribution of X and Y . The
lower bound is the rate achieved by a mismatched decoder. See
[32, pp. 6-7] for a recent comprehensive review on mismatched
decoding.
Theorem 1 (Amplitude Modulation): Consider the discretetime
p model in (19) with Fk as defined in (34). If L =
⌈ 3 (βTs )2 SNR⌉ and the process {|Xk |} is i.i.d. such that

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(|Xk |2 − P/2) is exponentially distributed with mean P/2,
then
1
1
π2
lim I(XA ; Y ) − log SNR ≥ −2 − log(8π) −
.
SNR→∞
2
2
45
(40)
Proof: See Appendix A. The lower bound in (40) is
achieved by the double-filtering receiver structure shown in
Fig. 4. We borrow the term “double-filtering” from [20].
Theorem 1 implies that the capacity pre-log satisfies
C(SNR)
1
≥
SNR→∞ log SNR
2
lim

(41)

which follows from (40) and C(SNR) ≥ I(X; Y ) ≥
I(XA ; Y ). The capacity thus grows logarithmically at high
SNR with a pre-log factor of at least 1/2 if the oversampling
factor L grows with the cubic root of SNR.
Consider next a simplified model that does not include the
effect of filtering modeled by {Fk }. More specifically, the kth
output of the simplified approximate model is
Ψk = X⌈k/L⌉ ∆ ejΘk + Nk

(42)

where Xk is kth input symbol and {Θk } and {Nk } are the
same processes defined in Sec. III-C.
Theorem 2 (Phase Modulation):
Consider the discrete-time
p
model in (42). If L = ⌈ πβTs SNR/2⌉ and the input X b
is i.i.d. with arg(Xk ) independent of |Xk | for k = 1, . . . , b
such that arg(Xk ) is uniformly distributed over [−π, π) and
(|Xk |2 − P/2) is exponentially distributed with mean P/2,
then
lim

SNR→∞

I(XA ; Ψ) −

1
1
log SNR ≥ −2 − log(8π)
2
2

lim

I(ΦX ; Ψ|XA ) −

1
1
1
log SNR ≥ − log(64πβTs ) − .
4
4
2
(44)

Proof: See Appendix B. The lower bounds (43) and (44)
are achieved by the receiver structure shown in Fig. 5.
Theorem 2 implies that the capacity pre-log for the approximate model (42) satisfies
lim

SNR→∞

C(SNR)
3
≥
log SNR
4

F INITE SNR

Xk = X⌈k/L⌉ g ((k mod L)∆) .

(47)

The information rate I(X; Y ) in (32) can be expressed as
1
I(Xn ; Y n )
(48)
b
which follows from (47) and (30). One difficulty in evaluating
(48) is that the joint distribution of {Fk } and {Wk } is
not available in closed form. Even the distribution of Fk is
not available in closed form (there is an approximation for
small linewidth, see (16) in [21]). However, we can numerically compute tight lower bounds on I(X; Y ) by using the
auxiliary-channel technique (cf. (39)). We use the following
algorithm [11].
1) Sample a long sequence (xn , y n ) according to the true
joint distribution of Xn and Y n ;
2) Compute qY n |Xn (y n |xn ) and
X
pXn (x̃n )qY n |Xn (y n |x̃n )
(49)
qY n (y n ) =
I(X; Y ) = lim

b→∞

x̃n

where pXn is the true distribution of Xn ;
3) Estimate I(X; Y ) using


qY n |Xn (y n |xn )
1
.
I(X; Y ) ≈ log
b
qY n (y n )

(50)

Auxiliary Channel I: Consider the auxiliary channel
Ψk = Xk ∆ ejΘk + Nk

(51)

where {Θk } and {Nk } are defined in Sec. III-C and Xk is
defined by (47). The channel output Ψ is the same as Y in
(19) except that Fk is replaced by 1 or more generally with
g ((k mod L)∆). The channel is described by the conditional
distribution
Z
pΘn ,Ψn |Xn (θn , y n |xn ) dθn
(52)
pΨn |Xn (y n |xn ) =
θn

where

(45)

which follows from (43)-(44) by using C(SNR) ≥ I(X; Ψ) =
I(XA ; Ψ)+I(ΦX ; Ψ|XA ). This shows that the capacity grows
logarithmically at high SNR with a pre-log factor of at least
3/4. We remark that Barletta and Kramer [33] extended the
result of Theorem 2 to the full model (19) that includes the
filtering effect,
√ i.e., they showed that C(SNR) ≥ 3/4 for (19)
if L = ⌈ SNR⌉. More generally, for L = ⌈SNRα ⌉, they
showed for the full model that

0 < α ≤ 1/3
 2α,
C(SNR)
(1 + α)/2, 1/3 ≤ α ≤ 1/2
≥
lim

SNR→∞ log(SNR)
3/4,
1/2 ≤ α < 1.
(46)

AT

We develop algorithms to compute the information rate
numerically. We use Xk to denote the kth input sample which
is defined by

(43)

and
SNR→∞

V. R ATE C OMPUTATION

pΘn ,Ψn |Xn (θn , y n |xn )
n
Y
=
pΘk |Θk−1 (θk |θk−1 ) pΨ|X,Θ (yk |xk , θk )

(53)

k=1

with

pΘk |Θk−1 (θ|θ̃) =



2
pW (θ − θ̃; σW
), k ≥ 2
1/(2π),
k=1

and
y − x∆ ejθ
1
pΨ|X,Θ (y|x, θ) =
exp −
2
2 ∆
πσN ∆
σN

2

(54)
!

.

(55)

The channel pΨn |Xn has continuous states θn , which implies
that step 2 of the algorithm can not be computed directly.
To compute the integrals, we quantize the states (the phase
noise). Finer quantization results in more accurate results

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6

R

r(t)

y(t)

yk
kTs /L

(·)2

integrate
and dump

Fig. 4.

vm

Σ

L↓

digital filter

downsample

Amplitude
decoder

x̂A,m

Double filtering receiver.

yk

(·)2

Σ

L↓

digital LTI filter

downsample

vm

Amplitude
decoder

x̂A,m

φ̂X,m

Phase
decoder

x̂m

x̂A ej φ̂X

ej θ̂k−1
Sample
delay

yk−1

÷

Symbol
delay

x̂m−1

Fig. 5. Two-stage decoding in a multi-sample receiver. The upper branch is part of the double filtering receiver for amplitude decoding. The lower branch
is a phase-noise tracker which outputs an estimate θ̂k−1 of the phase noise where θ̂k−1 = arg(yk−1 /x̂m−1 ) where m = ⌈k/L⌉ (cf. (157)). The phase
decoder uses the estimate of the phase noise θ̂k−1 and the estimated amplitude x̂A,m to produce an estimate φ̂X,m of the phase of xm .

but is also more computationally intensive. We approach this
issue by first computing an estimate of the rate at a small
number of quantization levels, then we increase the number of
quantization levels as long as there is a significant difference
between the estimated rates. When there is no signification
change in the rate estimates, we take that to be a (lower bound
on the) rate with the auxiliary channel Ψk . This approach is
demonstrated in Fig. 8 and Fig. 9, both in Sec. VI. However,
in Fig. 10 and Fig. 11, we show only the final result after
finding the appropriate number of quantization levels. Next,
we describe formally the quantization of the states and the
rate computation in that case.
Auxiliary Channel II: We use the following auxiliary
channel for the numerical simulations:
Υk = Xk ∆ ejSk + Nk .
The conditional probability is
X
pΥn |Xn (y n |xn ) =
pS n ,Υn |Xn (sn , y n |xn )

(56)

(57)

Next, we describe our choice of S and Q(·|·). We partition
[−π, π) into S intervals with equal lengths and pick the mid
points of these intervals to be the elements of S, i.e., we have
S = {ŝi : i = 1, . . . , S} where ŝi = i

π
2π
− − π.
S
S

(60)

The state transition probability Q(·|·) is chosen similar to [10]
and [12]:
Q(s|s̃) =

2π
S

Z

(φ,φ̃)∈R(s)×R(s̃)

2
pW (φ − φ̃; σW
) dφdφ̃ (61)

where R(s) = [s − π/S, s + π/S), i.e., R(s) is the interval
whose midpoint is s. The larger S and L are, the better the
auxiliary channel (56) approximates the actual channel (19).
We remark that the auxiliary channel gives a valid lower bound
on I(X; Y ) even for small S and L.

sn ∈S n

where S is a finite set and

pS n ,Υn |Xn (sn , y n |xn )
n
Y
=
pSk |Sk−1 (sk |sk−1 ) pΨ|X,Θ (yk |xk , sk )

A. Computing The Conditional Probability
(58)

k=1

where

pSk |Sk−1 (s|s̃) =



Q(s|s̃),
1/|S|,

k≥2
k = 1.

(59)

Suppose the input Xn has the distribution pXn . A Bayesian
network for Xn , S n , Υn is shown in Fig. 6. The probability
pΥn |Xn (y n |xn ) can be computed using
pΥn |Xn (y n |xn ) =

X
s∈S

ρn (s)

(62)

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with the initial value ψ0 (s) = 1/|S|. The set XL is
X1

X2

X3

X4

X5

X6

X7

X8

X9

Υ1

Υ2

Υ3

Υ4

Υ5

Υ6

Υ7

Υ8

Υ9

S1

Fig. 6.

S2

S3

S4

S5

S6

S7

S8

S9

Bayesian network for Xn , S n , Υn for n = 9.
X1

X2

X3

XL = {x · (g(∆), g(2∆), . . . , g(L∆)) : x ∈ X }.

(68)

χm,L (s, s̃, x̃) = pSmL ,Ym |S(m−1)L ,Xm (s, ym |s̃, x̃)

(69)

L

We remark that |XL | = |X | and not |X | . Next, we define
for s, s̃ ∈ S and x̃ ∈ XL . Computing χm,L (s, s̃, x̃) is similar
to computing ρn (see (64)). Intuitively, this is because a block
of L samples in Fig. 7 has a structure similar to Fig. 6. More
precisely, χm,L (s, s̃, x̃) can be computed recursively by using
χm,ℓ (s, s̃, x̃)
(70)
X

=
χm,ℓ−1 (ς, s̃, x̃) Q (s|ς) pΨ|X,Θ y(m−1)L+ℓ |x̃ℓ , s
ς∈S

X1

X2

X3

X4

X5

X6

X7

X8

X9

Υ1

Υ2

Υ3

Υ4

Υ5

Υ6

Υ7

Υ8

Υ9

S1

Fig. 7.

S2

S3

S4

S5

S6

S7

S8

S9

Bayesian network for Xn , S n , Υn for n = 9 and L = 3.

where we recursively compute
ρk (s) ≡ pSk ,Υk |Xn (s, y k |xn )
(a) X
=
pSk−1 ,Sk ,Υk |Xn (s̃, s, y k |xn )

(63)

s̃∈S

(b)

=

X
s̃∈S

=

X

s̃∈S

ρk−1 (s̃) pSk ,Υk |Sk−1 ,Υk−1 ,Xn (s, yk |s̃, y k−1 , xn )

ρk−1 (s̃) Q(s|s̃) pΨ|X,Θ (yk |xk , s)

(64)

with the initial value ρ0 (s) = 1/|S|. Step (a) is a marginalization, (b) follows from Bayes’ rule and the definition of ρk
in (63), while (64) follows from the structure of Fig. 6. We
remark that (64) is the same as with independent X1 , . . . , Xn ,
e.g., see equation (9) in [34, Sec. IV].
B. Computing The Marginal Probability
Define Xm ≡ (X(m−1)L+1 , X(m−1)L+2 , . . . , X(m−1)L+L ).
Suppose the input symbols are i.i.d. and Xm ∈ X where X is
a finite set. Therefore, pXn has the form
n

pX (x ) =
n

b
Y

pX (xm ).

(65)

m=1

A Bayesian network for Xn , S n , Υn is shown in Fig. 7. The
probability pΥn (y n ) can be computed using
X
pΥn (y n ) =
ψb (s)
(66)
s∈S

where ψm (s) ≡ pSmL ,Ym (s, ym ) which can be computed
using the recursion:
ψm (s)
(67)
X
X
pX (x̃)
ψm−1 (s̃) pSmL ,Ym |S(m−1)L ,Xm (s, ym |s̃, x̃)
=
x̃∈XL

s̃∈S

with the initial value

χm,0 (s, s̃, x̃) =



1, s = s̃
0, otherwise.

(71)

Therefore, computing pΥn (y n ) involves two levels of recursion: 1) a recursion over the symbols as described by (67) and
2) a recursion over the samples within a symbol as described
by (70).
VI. N UMERICAL S IMULATIONS
We use two pulses with a symbol-interval time support:
• A unit-energy square pulse
 
t
1
(72)
g1 (t) = √ rect
Ts
Ts
where

1, |t| ≤ 1/2,
(73)
rect(t) ≡
0, otherwise.
•

A unit-energy cosine-squared pulse
 
 
πt
1
t
cos2
g2 (t) = p
rect
.
Ts
Ts
Ts /2

(74)

The first step of the algorithm is to sample a long sequence
according to the true joint distribution of Xn and Y n . To
generate samples according to the original channel (19),
we must accurately represent digitally the continuous-time
waveform (4). We use a simulation oversampling rate Lsim =
1024 samples/symbol. After the filter (18), the receiver has L
samples/symbol distributed according to (19). Next, to choose
a proper sequence length, we follow the approach suggested
in [11]: for a candidate length, run the algorithm about 10
times (each with a new random seed) and check whether
all estimates of the information rate agree up to the desired
accuracy. We used b = 104 unless otherwise stated.
For efficient implementation of (64), pΨ|X,Θ (·|·, ·) can be
factored out of the summation to yield:

ρk (s) = pΨ|X,Θ (yk |xk , s)

zX

s̃∈S

ρ′k (s)

}|
{
ρk−1 (s̃) Q(s|s̃) .

(75)

Moreover, since Q(·|·) can be represented by a circulant matrix
due to symmetry, ρ′k (·) can be computed efficiently using the
Fast Fourier Transform (FFT). Similarly, the computation of
(70) can be done efficiently by factoring out pΨ|X,Θ (·|·, ·) and
by using the FFT.

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4

I(X; Y ) (bits/symbol)

3

2.5

2

3.5

3

I(X; Y ) (bits/symbol)

3.5

4

AWGN
L=16, S=64
L=16, S=32
L=16, S=16
L= 8, S=64
L= 8, S=32
L= 8, S=16
L= 4, S=64
L= 4, S=32
L= 4, S=16

1.5

2.5

2

1.5

1

1

0.5

0.5

0
−5

0

5

10

15

20

AWGN
L=16, S=64
L=16, S=32
L=16, S=16
L= 8, S=64
L= 8, S=32
L= 8, S=16
L= 4, S=64
L= 4, S=32
L= 4, S=16

0
−5

25

0

5

SNR (dB)
Fig. 8. Lower bounds on rates for 16-QAM, square transmit-pulse and multisample receiver at fHWHM Ts = 0.125.

10

15

20

25

SNR (dB)
Fig. 9. Lower bounds on rates for 16-QAM, cosine-squared transmit-pulse
and multi-sample receiver at fHWHM Ts = 0.125.
4

A. Very Large Linewidth
3.5

3

I(X; Y ) (bits/symbol)

Suppose fHWHM Ts = 0.125 and the input symbols are
independently and uniformly distributed (i.u.d.) 16-QAM. Fig.
8 shows an estimate of I(X; Y ) for a square transmit-pulse,
i.e., g(t) = g1 (t − Ts /2) and an L-sample receiver with
L = 4, 8, 16 and S = 16, 32, 64. The curves with S = 64
are indistinguishable from the curves with S = 32 over the
entire SNR range for all values of L, and hence it seems that
S = 32 is adequate up to 25 dB. Even S = 16 seems to be
adequate up to 20 dB. We remark that the non-monotonicity
of some lower bounds is an artifact of the auxiliary channel
being “far” from the true channel, specifically because the
number S of quantization levels of Auxiliary Channel II is
too low for some SNR values. The important trend in Fig.
8 is that higher oversampling rate L is needed at high SNR
to extract all the information from the received signal. For
example, L = 4 suffices up to SNR ∼ 10 dB, L = 8 suffices
up to SNR ∼ 15 dB but L ≥ 16 is needed beyond that. It was
pointed out in [11] that the lower bounds can be interpreted
as the information rates achieved by mismatched decoding.
For example, I(X; Y ) for L = 8 and S ≥ 32 in Fig. 8 is
essentially the information rate achieved by a multi-sample (8sample) receiver that uses maximum-likelihood decoding for
the simplified channel (51) when it is operated in the original
channel (19).
Fig. 9 shows an estimate of I(X; Y ) for a cosine-squared
transmit-pulse, i.e., g(t) = g2 (t − Ts /2) and an L-sample
receiver at L = 4, 8, 16 and S = 16, 32, 64. We find that
S = 32 suffices up to ∼ 25 dB. We see in Fig. 9 the same trend
as in Fig. 8: higher L is needed at higher SNR. Furthermore,
the square pulse is better than the cosine-squared pulse for
the same oversampling rate L. It might seem strange the the
cosine-squared pulse curves are substantially lower than the
square pulse curves despite taking into account the pulse shape
in the auxiliary channel model, see (56) and (47). One obtains

2.5

2

1.5

AWGN
L=16
L= 8
L= 4
L= 2
L= 1

1

0.5

0
10

12

14

16

18

20

22

24

26

28

30

SNR (dB)
Fig. 10. Lower bounds on rates for 16-PSK, square transmit-pulse and multisample receiver at fHWHM Ts = 0.0125.

insight on this effect with the following Gedankenexperiment.
Suppose we use a pulse that is narrow and peaky in time.
Then an integrate-and-dump receiver with oversampling puts
out essentially only one useful sample per symbol. One may
as well, therefore, use only one sample per symbol.
B. Large Linewidth
As the linewidth decreases, the benefit of oversampling
at the receiver becomes apparent only at higher SNR. For
example, for fHWHM Ts = 0.0125 and i.u.d. 16-PSK input, Fig.
10 shows an estimate of I(X; Y ) for a square transmit-pulse
and an L-sample receiver at L = 1, 2, 4, 8, 16 and S = 64. We
see that L = 4 suffices up to SNR ∼ 19 dB, L = 8 suffices up
to SNR ∼ 24 dB and only beyond that L ≥ 16 is necessary.

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4

Information Rate (bits/symbol)

3.5
3
2.5

16−QAM, AWGN
16−QAM, γ = 0.4, GK
16−QAM, γ = 0.4, MTR
16−QAM, γ = 0.4, ASM
QPSK, AWGN
QPSK, γ = 0.5, GK
QPSK, γ = 0.5, MTR
QPSK, γ = 0.5, ASM

2
1.5
1
0.5
0
−5

0

5

10

15

20

25

SNR (dB)
Fig. 11.

Comparison of information rates for different models.

We conclude from Fig. 8–10 that the required L depends
on 1) the linewidth fFWHM of the phase noise; 2) the pulse
g(t); and 3) the SNR.
C. Comparison With Other Models
We compare the discrete-time model of the multi-sample
receiver with other discrete-time models. The simulation parameters for our model (labeled GK) are b = 104 , L = 16
(with Lsim = 1024) and S = 64 for 16-QAM (S = 128
was too computationally intensive) and S = 128 for QPSK.
We set γ 2 = 2πβTs . In Fig. 11, we show curves for the
approximate symbol-rate model (16), which are labeled ASM.
The simulation parameters for this model are b = 105 and
S = 128.
We also show curves for the Martalò-Tripodi-Raheli (MTR)
model [34] in Fig. 11. For the sake of comparison, we adapt
the model in [34] from a square-root raised-cosine pulse to a
square pulse and write the “matched” filter output {Vm } as
Vm =

Lsim
X

Ψ(m−1)L+ℓ

(76)

ℓ=1

where m = 1, . . . , b and Ψk is defined in (51). The auxiliary
channel is
Ym = Xm ejΘm + Zm ,

m≥1

(77)

where the process {Zm } is an i.i.d. circularly-symmetric
2
complex Gaussian process with mean 0 and E[|Zm |2 ] = σN
Ts
while the process {Θm } is a first-order Markov process
(not a Wiener process) with a time-invariant transition probability, i.e., for k ≥ 2 and all θk , θk−1 ∈ [−π, π), we
have pΘk |Θk−1 (θk |θk−1 ) = pΘ2 |Θ1 (θk |θk−1 ). Furthermore, the
phase space is quantized to a finite number S of states and the
transition probabilities are estimated by means of simulation.
The probabilities are then used to compute a lower bound on
the information rate. The simulation parameters for the MTR
model are b = 105 , Lsim = 16 and S = 128.

We point out that MTR rates are essentially achievable rates
for the symbol-rate model with filtering (13) that are valid
up to some SNR that grows with the oversampling rate. We
observe from Fig. 11 the following:
1) The rates of ASM and MTR saturate well below the rate
achieved by the multi-sample receiver, which implies that
symbol-rate sampling results in information loss at high
SNR.
2) The multi-sample receiver rates are indistinguishable
from the MTR rates at low SNR which implies that
matched filtering with symbol-rate sampling incurs practically no information loss, i.e., using (16) to compute
rates at low/moderate SNR gives accurate results.
3) MTR is indistinguishable from ASM at low SNR which
implies that approximating (13) as (16) can be justified
at low SNR, i.e., the filtering effect may be ignored.
We remark that the saturation at high SNR of the rates
of the symbol-rate models at a value lower than the entropy
of the source should not be surprising. The impact of the
additive noise is smaller at high SNR, but the effect of phase
noise and/or filtering are not necessarily reduced. For example,
consider the memoryless channel Y = HX + Z and a fixedsize constellation (e.g. M -QAM). We have the following upper
bound (by using a genie-aided argument)
I(X; Y ) ≤ I(X; Y Z)
= H(X) − H(X|HX).

(78)

The term H(X|HX) represents impact of the ambiguity due to
the randomness of H. For the special case of Y = ejΘ X + Z,
we have
I(X; Y ) ≤ H(X) − H(ΦX |ΦX + Θ)

(79)

i.e., one may fully recover the amplitude of X but not the
phase of X because of the ambiguity due to Θ. Moreover, for
Y = ejΘ X + Z, there is a more general upper bound on the
contribution of phase modulation given by
I(ΦX ; Y |XA ) ≤ I(ΦX ; Y |XA , Z)
= I(ΦX ; ΦX + Θ|XA )
= h(ΦX + Θ|XA ) − h(Θ)
≤ log(2π) − h(Θ)

(80)

This upper bound is more general than the earlier one in
the sense that it is valid for any input constellation (of any
size). Unfortunately, we do not have a simple upper bound
expression similar to log(2π) − h(Θ) to predict the cap on the
information rate for the multi-sample receiver. Obtaining such
an upper bound is beyond the scope of this paper.
VII. D ISCUSSION
1) The authors of [21] treated on-off keying in the presence of
Wiener phase noise by using a double-filtering receiver, which
is composed of an intermediate frequency (IF) filter, followed
by an envelope detector (square-law device) and then a postdetection filter. They showed that by optimizing the IF receiver
bandwidth the double-filtering receiver outperforms the singlefiltering (matched filter) receiver. Furthermore, they showed

JOURNAL, VOL. , NO. , MONTH YEAR

10

via simulation that the optimum IF bandwidth increases with
the SNR. This is similar to our result in the sense that we
require the number of samples per symbol to increase with
the SNR in order to achieve a rate that grows logarithmically
with the SNR. We remark that Dallal and Shamai [35] derived
an upper bound on the error probability of (a variant of) the
double filtering receiver for systems employing noncoherent
demodulation, such as frequency shift keying (FSK), on-off
keying (OOK) and pulse-position modulation (PPM). Relevant
analysis can be found in [36], [37] and references therein. The
analysis typically involves moments of filtered Wiener phase
noise.

infinite number of projections per symbol are needed to obtain
sufficient statistics for all SNRs. For example, consider the
Karhunen-Loève expansion of U (t) = ejΘ(t) on t ∈ [0, Ts ],
for which we have2

2) The phase modulation pre-log of 1/4 in (44) requires
only 2 samples per symbol for which the time resolution
1/∆ grows as the square root of the SNR. In other words,
one does not need an analog-to-digital converter (ADC) with
sampling every Ts /L seconds. Instead, one may use two ADCs
that sample every Ts seconds and the sampling instants are
offset by Ts /L seconds. For large L, two ADCs with low
sampling rate would be less complex than one ADC with
a very high sampling rate. It is interesting to contemplate
whether another receiver, e.g., a non-coherent receiver, can
achieve the maximum amplitude modulation pre-log of 1/2
but requires only 1 sample per symbol. If so, one would need
only 3 samples per symbol to achieve a pre-log of 3/4.

where αi is the solution to

3) The paper [24] shows that a √
pre-log of 1/2 is achievable
for the model (19) when L ∝ SNR. Theorem√1 implies
3
that a pre-log of 1/2 can be achieved using L ∝ SNR
√ and
this perhaps can be reduced further. But using L ∝ SNR
may simplify the receiver design. Specifically, the amplitude
decoder designed for the auxiliary channel in [24] can be used
for any phase noise linewidth β because the auxiliary channel
is (99) with p
G̃ = 1 which is independent of β. In contrast,
using L = 3 β 2 SNR requires different amplitude decoders
for different linewidths because the auxiliary channel in this
case is (99) with G̃ = E[G] which depends on β.
4) For the multi-sample approximate model (42), Barletta and
Kramer [38] derived an upper bound on the rate and showed
that if
L = ⌈SNRα ⌉

(81)

where 0 < α ≤ 1, then the capacity
pre-log is at most (1 +
√
α)/2 which implies that L ∝ SNR is optimal for achieving
a pre-log of 3/4. We remark that the best known oversampling
factor to achieve a pre-log of 3/4 for the model (19) is L ∝
√
SNR. We also remark that the upper bound derived in [38]
was subsequently specialized to the additive-noise-free case in
[39] (cf. (141)-(143)).
5) We briefly discuss sufficient statistics for the continuoustime phase noise channel. One expects intuitively at any SNR
that
lim I(X b ; Y bL ) = I(x(t); r(t)).

L→∞

(82)

This is because at infinite sampling rate, one captures essentially the entire received waveform. In general, it seems that an

m.s.

U (t) =

∞
X

Ui fi (t)

(83)

i=0

where {fi (t)} are the orthonormal functions [40, Sec. 6.1]:
r

2
cos(αi (t − Ts /2)), i odd
(84)
fi (t) =
sin(αi (t − Ts /2)), i even
Ts + λi

2 arctan(αi /(βπ)) + αi Ts = iπ,

i = 0, 1, . . .

(85)

and the {Ui } are uncorrelated random variables with variance
λi where
λi =

2πβ
.
π 2 β 2 + α2i

(86)

Suppose that one symbol X is transmitted and consider a
rectangular pulse for simplicity. We have
r(t) = X ejΘ(t) + n(t) =

∞
X

XUi fi (t) + n(t).

(87)

i=0

We conclude, in this simplified case, that
I(X; {Γi }∞
i=0 ) = I(x(t); r(t))

(88)

where
Γi =

Z

0

Ts

r(t)fi∗ (t)dt,

i = 0, 1, . . .

(89)

i.e, {Γi }∞
i=0 are sufficient statistics for X. Therefore, an
infinite number of projections per symbol seems to be needed
to have sufficient statistics in general. Moreover, we believe
that a finite subset of {Γi }∞
i=0 captures essentially all the
information in the received waveform at a given SNR, and that
larger subsets would be needed at larger SNRs. The discretetime model based on the multi-sample receiver in Fig. 3 seems
to be easier to analyze than the discrete-time model (89).
6) Minimum energy per information bit is an important metric
at low SNR. For example, Fig. 12 plots the information rates
of Fig. 8 against Eb /N0 , where the energy per information bit
is Eb = P/R and R is the information rate in bits/symbol. We
notice a significant impact on the minimum Eb due to phase
noise. Furthermore, oversampling does not affect the rate at
low Eb /N0 , which suggests that using symbol-rate sampling
suffices and that several results available in the literature on
noncoherent fading channels and discrete-time phase noise
channels (with the commonly-used approximation) should
remain valid, e.g., see Fig. 2–Fig. 5 in [14].
2

m.s.

We use = to indicate convergence in mean-square sense.

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4
3.5

A PPENDIX A
A MPLITUDE M ODULATION

AWGN
L=16
L= 8
L= 4

This Appendix provides a proof of Theorem 1. We set Ts =
b
1 for simplicity. Since XA
is i.i.d., we have

I(X ; Y ) (bits/symbol)

3
2.5

(a)

b
; Yb ) =
I(XA

k=1

2

k−1
I(XA,k ; Yb |XA
)

b
(b) X
k−1
=
H(XA,k ) − H(XA,k |Yb XA
)

1.5
1

k=1
b
(c) X

I(XA,k ; Yk )

≥

I(XA,k ; Vk )

(90)

L
X

(91)

≥

k=1
b
(d) X

0.5
0
−5

b
X

0

5
Eb/N0 (dB)

10

15

k=1

where

Fig. 12. Rates vs Eb /N0 for 16-QAM, square transmit-pulse and multisample receiver at fHWHM Ts = 0.125.

Vk =

ℓ=1

VIII. C ONCLUSION
We studied a waveform channel impaired by Wiener phase
noise and AWGN. A discrete-time channel model based on
oversampling was derived taking into account filtering effects
on phase noise. At high SNR, the multi-sample receiver
achieves rates that grow logarithmically with SNR with at
least a 1/2 pre-log factor, if the number of samples per
symbol grows with the cubic root of the SNR. In addition,
we computed via numerical simulations lower bounds on
the information rates achieved by the multi-sample receiver.
We observed that the required oversampling rate depends on
the linewidth of the phase noise, the shape of the transmitpulse and the SNR. The results demonstrate that multi-sample
receivers increase the information rate for both strong and
weak phase noise at high SNR. We compared our results
with the results obtained by using other discrete-time models.
Finally, we showed for an approximate discrete-time model
of the multi-sample receiver that phase modulation achieves
a pre-log of at least 1/4 while amplitude modulation achieves
a pre-log of 1/2, if the number of samples per symbol grows
with the square root of the SNR. Thus, we demonstrated that
the overall capacity pre-log is 3/4 which is greater than the
capacity pre-log of the (approximate) discrete-time Wiener
phase noise channel with only one sample per symbol. The
impact of oversampling on the capacity of non-Wiener phase
noise channels, or multi-input multi-output (MIMO) phase
noise channels, are open problems.

ACKNOWLEDGEMENT
We thank the reviewers and the Associate Editor for useful
comments.

|Y(k−1)L+ℓ |2 .

Step (a) follows from the chain rule of mutual information,
(b) follows from the independence of XA,1 , XA,2 , . . . , XA,b ,
(c) holds because conditioning does not increase entropy, and
b
(d) follows from the data processing inequality. Since XA
b
is identically distributed, we find that V is also identically
distributed and we have, for k ≥ 2,
I(XA,k ; Vk ) = I(XA,1 ; V1 ).

(92)

In the rest of this section, we consider only one symbol
(k = 1) and drop the time index. By combining (91) and (19),
we have
V =

L
X
ℓ=1

2 2
XA
∆ |Fℓ |2 + 2XA ∆ℜ[ejΦX ejΘℓ Fℓ Nℓ∗ ] + |Nℓ |2

2
= XA
∆G + 2XA ∆Z1 + Z0



(93)

where G, Z1 and Z0 are defined as
L

G≡
Z1 ≡
Z0 ≡

1X
|Fℓ |2
L

(94)

ℓ=1

L
X

ℓ=1
L
X
ℓ=1

ℜ[ejΦX ejΘℓ Fℓ Nℓ∗ ]

(95)

|Nℓ |2 .

(96)

The second-order statistics of Z1 and Z0 are
2
E[Z1 ] = 0
Var[Z1 ] = E[G]σN
/2
2
4
E[Z0 ] = σN
Var [Z0 ] = σN ∆
E [Z1 (Z0 − E[Z0 ])] = 0.

(97)

By using the auxiliary-channel lower bound (cf. (39)), we
have
I(XA ; V ) ≥ E[− log qV (V )] − E[− log qV |XA (V |XA )] (98)

JOURNAL, VOL. , NO. , MONTH YEAR

12

where qV |XA (v|xA ) is the conditional distribution of the The inequality (106) follows from the non-negativity of the
auxiliary channel, which we choose to be
integrand. By combining (99), (107), (108) and making the
! change of variables x = xP ∆, we have
2 2
)
1
(v − x2A ∆G̃ − σN
qV |XA (v|xA ) = p
exp −
2
2
2
2
2
4xA ∆ σN
4πxA ∆2 σN
fV (v)


Z ∞ −x/(λ∆)
2 2
(99)
1
(v − x − σN
)
e
p
dx
exp −
=
2
2
λ∆
4x∆σN
4πx∆σN
0
with 0 < G̃ ≤ 1 and
Z ∞
1
= p
×
(100)
qV (v) =
pXA (xA )qV |XA (v|xA )dxA
2 )
λ∆(λ∆ + 4∆σN
0
"
#!
r
2
2
4∆σ
where pXA (·) is the (true) pdf of XA . The intuition behind
2
2
N
exp
v − σN
− |v − σN
| 1+
2
4∆σN
λ∆
the choice of (99) is given in Remark 3 below. It follows that
"
#
(109)
2
2 2
(V − XA
∆G̃ − σN
)
E[− log(qV |XA (V |XA ))] = E
2 ∆2 σ 2
4XA
where we used equation (140) in Appendix A of [41]:
N


1
1
Z ∞
2
2
 x 1
) + E[log(XA
)] (101)
+ log ∆ + log(4πσN
(u − x)2
1
2
2
√
dx
exp −
exp −
a
a
bx
πbx
0
"
#!
and by using (93) and (97), we have
r
#
"
1
2
b
2
2 2
u − |u| 1 +
.
(110)
=p
exp
(V − XA
∆G̃ − σN
)
b
a
a(a
+
b)
E
2
2
4XA ∆2 σN
i
h
Therefore, we have
1
1
2
2
=
+ 2 E[Z12 ]
E[X
]E
(G
−
G̃)
A
2
4σN
σN


i E[− log(fV (V ))]
h


1
1
1
2 2
+ 2 E[XA ]E (G − G̃)Z1
+
1
2
2 E X 2 E (Z0 − σN )
= log(∆2 (λ2 + 4λσN
4∆2 σN
σN
))
A


2
"
#
r


1
1
1
2
2
2
+
E Z1 (Z0 − σN
)
E[G − 1]E[Z0 − σN
]+
E
4σN
1
2
2
2
2
2∆σN
∆σN
XA
E[V − σN ] − E[|V − σN |] 1 +
−
2


2∆σN
λ
i
h
2
1
1
σN
1
"r
#
2
=
(102)
+
P
E
(G
−
G̃)
E[G]
+
E
2
(a)
2
2
4σN
1
4σN
2
4∆
XA
2
−1
≥ log(∆λ) + 2 E[V − σN ]
1+
2σN ∆
λ
where we also used
i
h
(b)
(103)
E (G − G̃)Z1 = 0.
≥ log(∆λ)
(111)
Substituting (102) into (101) and using E[G] ≤ 1 yield
E[− log(qV |XA (V |XA ))]
1
1
2
2
≤ log ∆ + log(4πσN
) + E[log(XA
)]
2
2


i 1 σ2
h
P
1
+ 2 E (G − G̃)2 + + N E
2 .
4σN
2 4∆
XA

where (a) holds because the logarithmic function is monotonic
and E[| · |] ≥ E[·], and (b) holds because
2
E[V − σN
]

(104)

2
XA
.

It is convenient to define XP ≡
The input distribution
of XP is
 1

xP −λ
, xP ≥ λ
λ exp − λ
(105)
pXP (xP ) =
0,
otherwise
where λ = P/2. It follows from (100) and (105) that


Z ∞
xP − λ
1
qV |XP (v|xP ) dxP
exp −
qV (v) =
λ
λ
λ
≤ e fV (v)
(106)
where
√
qV |XP (v|xP ) = qV |XA (v| xP )

(107)

2
2
= E[XA
]∆E[G] + 2E[XA ]∆E[Z1 ] + E[Z0 ] − σN

= P ∆E[G] ≥ 0.

(112)

The monotonicity of the logarithmic function and (106) yield
E[− log(qV (V ))] ≥ E [− log (e fV (V ))]
≥ log ∆ + log λ − 1

(113)

where the last inequality follows from (111). It follows from
(98), (104) and (113) that
1
1
2
2
I(XA ; V ) ≥ log λ − 1 − log(4πσN
) − E[log(XA
)]
2
2 

h
i 1 σ2
1
P
− 2 E (G − G̃)2 − − N E
2 . (114)
4σN
2 4∆
XA
We have

and
fV (v) ≡

Z

0

∞

 x 
1
P
exp −
qV |XP (v|xP )dxP .
λ
λ

(108)



1
E
XP



≤

2
P

(115)

JOURNAL, VOL. , NO. , MONTH YEAR

and

Z

13

∞

1 −(x−λ)/λ
e
log(x)dx
λ
λ
Z ∞
(a)
= log λ +
e−(u−1) log(u)du

E [log (XP )] =

1

(b)

≤ log λ + 1

(116)

where (a) follows by the change of variables u = x/λ, and
(b) holds because log(u) ≤ u − 1 for all u > 0. Substituting
into (114), we obtain
I(XA ; V ) −

1
1
1
log SNR ≥ −2 − log(8π) −
2
2
2SNR∆
i
h
1
− SNR E (G − G̃)2 . (117)
4

By combining (36), (90), (92) and (117), we have
I(XA ; Y ) −

1
1
1
log SNR ≥ −2 − log(8π) −
2
2
2SNR∆
i
h
1
− SNR E (G − G̃)2 . (118)
4

Since (118) is valid for any G̃ ∈ (0, 1], we choose G̃ to
maximize the right hand side of (118). The optimal G̃ is
h
i
arg min E (G − G̃)2 = E[G].
(119)
G̃

p
Since L = ⌈ 3 β 2 SNR⌉ and ∆ = 1/L, it follows that
1
lim SNR∆ = ∞ and lim SNR∆3 = 2 .
β
SNR→∞
SNR→∞

A PPENDIX B
P HASE M ODULATION
This Appendix provides a proof of Theorem 2. We assume
2
that Ts = 1 and σN
= 1 for simplicity. By using the chain
rule, we have
n
X
k−1
n
n
I(ΦnX ; Ψn |XA
)=
I(ΦX,k ; Ψn |XA
, ΦX
)
(a)

≥

(b)

≥

Ñk ≡

Remark 3: We motivate the auxiliary channel in (99). We
first express the auxiliary channel as Ṽ :
(123)

where G̃, Z̃1 and Z̃0 are defined as

Z̃1 ≡

L
X
ℓ=1

2
Z̃0 ≡ σN
.

ℜ[ejΦX ejΘℓ Nℓ∗ ]

(130)

N(k−1)L+1 e−jΘ(k−1)L+1
√
.
∆

(131)

The processes Ñk and Z̃k−1 are statistically independent and
Ñk ∼ NC (0, 1)
Z̃k−1

1
1
π2
log SNR ≥ −2 − log(8π) −
.
2
2
45
(122)

G̃ ≡ E[G]

∗
NkL
e−jΘkL
Xk∗ ∆

Z̃k ≡
and

Hence, for G̃ = E[G], we have

2
Ṽ = XA
∆G̃ + 2XA ∆Z̃1 + Z̃0

(127)

(129)

(120)

Var(G)
lim SNR Var(G) = lim SNR∆ · lim
∆→0
SNR→∞
SNR→∞
∆3
2
2
4π
1 4(πβ)
=
.
(121)
= 2·
β
45
45

SNR→∞

k=2

k−1
n
I(ΦX,k ; Ψ̃k |XA
, ΦX
)

where

3

I(XA ; Y ) −

k=2

n
X

k−1
n
I(ΦX,k ; Ψn |XA
, ΦX
)

k−1
n
where Ψ̃k is a deterministic function of (Ψn , XA
, ΦX
).
Inequality (a) follows from the non-negativity of mutual information and (b) follows from the data processing inequality.
We choose


Ψ(k−1)L+1 Ψ(k−1)L ∗
√
.
(128)
Ψ̃k =
Xk−1 ∆
∆
The intuition behind the choice of (128) is outlined in Remark
4 below. We express Ψ̃k as



√
Ψ̃k = |Xk | ∆ejΦX,k + Ñk
1 + Z̃k−1 ejW(k−1)L+1

Moreover, we have (see Appendix C)

lim

k=1
n
X

(124)
(125)
(126)

We make this choice for Ṽ because Z1 and Z0 converge
(in mean square) to Z̃1 and Z̃0 , respectively, as L → ∞.
Moreover, Z̃1 is Gaussian and Z̃0 is a constant, which are
easier quantities to handle than Z1 and Z0 .


{|Xk−1 | = |xk−1 |} ∼ NC 0,

1
|xk−1 |2 ∆



(132)
.

(133)

The notation (133) means that, conditioned on {|Xk−1 | =
|xk−1 |}, Z̃k−1 is a Gaussian random variable with mean 0 and
variance 1/(|xk−1 |2 ∆). Moreover, W(k−1)L+1 is statistically
independent of Ñk and Z̃k−1 . The choice of Ψ̃k in (128)
implies that
n
I(ΦX,k ; Ψ̃k |XA
, X k−1 ) = I(ΦX,k ; Ψ̃k |XA,k , Xk−1 ). (134)

Define Φ̃Ψ,k ≡ arg(Ψ̃k ) and
 exp(α cos(φy − φx ))
qΦ̃Ψ |ΦX φy φx ≡
2πI0 (α)

(135)

where I0 (·) is the zeroth-order modified Bessel function of the
first kind and α > 0. This distribution is known as Tikhonov
(or von Mises) distribution [42]. Furthermore, define

qΦ̃Ψ,k |XA,k ,Xk−1 φy |xk |, xk−1
Z π

≡
pΦX,k |XA,k ,Xk−1 φx |xk |, xk−1 qΦ̃Ψ |ΦX (φy |φx )dφx
−π

1
.
=
2π

(136)

JOURNAL, VOL. , NO. , MONTH YEAR

14

The last equality holds because X1 , . . . , Xn are statistically
independent and ΦX,k is independent of XA,k with a uniform
distribution on [−π, π). We have
I(ΦX,k ; Ψ̃k |XA,k , Xk−1 )
(a)

≥ I(ΦX,k ; Φ̃Ψ,k |XA,k , Xk−1 )
i
h
≥ E log qΦ̃Ψ |ΦX (Φ̃Ψ,k |ΦX,k )
h
i

− E log qΦ̃Ψ,k |XA,k ,Xk−1 Φ̃Ψ,k |Xk |, Xk−1
h
i
(c)
= log(2π) − log(2πI0 (α)) + αE cos(Φ̃Ψ,k − ΦX,k )
h
i
= − log(I0 (α)) + αE cos(Φ̃Ψ,k − ΦX,k )
h
i
(d) 1
≥ log α − α + αE cos(Φ̃Ψ,k − ΦX,k )
2
σ2
4α
1
(137)
≥ log α − α W −
2
2
SNR∆
(b)

where (a) follows from the data processing inequality, (b)
follows by extending the result of the auxiliary-channel lower
bound (cf. (39)), (c) follows from (135) and (136), (d) follows
from [43, Lemma 2]
√
ez
π ez
√ ≤ √
(138)
I0 (z) ≤
2
z
z
and the last inequality holds because
h
i
4
σ2
E cos(Φ̃Ψ,k − ΦX,k ) ≥ 1 − W −
2
SNR∆

(139)

for SNR∆ > 2, as we will show. Define

Sk ≡ 1 + Z̃k
√
Ψ̂k ≡ |Xk | ∆ejΦX,k + Ñk .

(140)
(141)

Therefore, we have
h
i
E cos(Φ̃Ψ,k − ΦX,k )
h
i
(a)
= E cos(W(k−1)L+1 + ΦS,k−1 + Φ̂Ψ,k − ΦX,k )
i

 h
(b)
= E cos(W(k−1)L+1 + ΦS,k−1 ) E cos(Φ̂Ψ,k − ΦX,k )
i

 h
− E sin(W(k−1)L+1 + ΦS,k−1 ) E sin(Φ̂Ψ,k − ΦX,k )
(142)
where ΦS,k ≡ arg(Sk ) and Φ̂Ψ,k ≡ arg(Ψ̂k ). Step (a) follows
from (129) while step (b) follows from the trigonometric
relation cos(A + B) = cos(A) cos(B) − sin(A) sin(B) and
because W(k−1)L+1 + ΦS,k−1 is independent of Φ̂Ψ,k and
ΦX,k .
By using Lemma 6 in Appendix D, we have


1
(143)
E cos(Φ̂Ψ,k − ΦX,k ) XA,k = |xk | ≥ 1 −
|xk |2 ∆


E sin(Φ̂Ψ,k − ΦX,k ) XA,k = |xk | = 0.
(144)

Equations (142)
and (144) imply that we do not need to

compute E sin(W(k−1)L+1 + ΦS,k−1 ) . We have


E cos(W(k−1)L+1 + ΦS,k−1 )


= E cos(W(k−1)L+1 ) E [cos(ΦS,k−1 )]


− E sin(W(k−1)L+1 ) E [sin(ΦS,k−1 )]
(145)

because W(k−1)L+1 and ΦS,k−1 are independent. Since
2
=
W(k−1)L+1 is Gaussian with mean 0 and variance σW
2πβ∆, the characteristic function of W(k−1)L+1 is


2
E ejW(k−1)L+1 = e−σW /2

(146)

and by using the linearity of expectation, we have


 

2
E cos(W(k−1)L+1 ) = ℜ E ejW(k−1)L+1 = e−σW /2
(147)



  jW
(k−1)L+1
=0
(148)
E sin(W(k−1)L+1 ) = ℑ E e

where ℜ[·] and ℑ[·] denote the real and imaginary parts,
respectively. By using Lemma 6 in Appendix D again, we
have


1
E cos(ΦS,k−1 ) Xk−1 = xk−1 ≥ 1 −
. (149)
|xk−1 |2 ∆
It follows from (145), (147), (148) and (149) that


E cos(W(k−1)L+1 + ΦS,k−1 )
 


2
1
1
.
≥ e−σW /2 1 − E
|Xk−1 |2 ∆

(150)

By combining (142), (143), (144) (150), we have (for P ∆ >
2)
h
i
E cos(Φ̃Ψ,k − ΦX,k )



 
 
2
1
1
1
1
≥ e−σW /2 1 − E
1
−
E
2
2
|Xk−1 | ∆
|Xk | ∆

2
(a)
2
2
≥ e−σW /2 1 −
P∆
(b)
2
4
≥ e−σW /2 −
P∆
(c)
σ2
4
≥ 1− W −
(151)
2
P∆
where (a) holds because |Xk |2 ≥ P/2 and |Xk−1 |2 ≥ P/2,
2
while (b) follows from e−σW /2 ≤ 1 and the non-negativity of
4/(P ∆)2 and (c) follows from e−x ≥ 1 − x.
It follows from (127), (134) and (137) that


b−1 1
4α
1
b
.
I(ΦbX ; Ψb |XA
)≥
log α − απβ∆ −
b
b
2
SNR∆
(152)
Hence, we have
I(ΦX ; Ψ|XA ) = lim

b→∞

≥

1
b
I(ΦbX ; Ψb |XA
)
b

1
4α
log α − απβ∆ −
.
2
SNR∆

(153)

JOURNAL, VOL. , NO. , MONTH YEAR

15

Since the lower bound (153) is valid for any α > 0, we have


1
4
I(ΦX ; Ψ|XA ) ≥ max log α − α πβ∆ +
α>0 2
SNR∆


1
8
1
−
= − log 2πβ∆ +
2
SNR∆
2


1
1
8L
2πβ
= − log
−
+
(154)
2
L
SNR
2
where we used ∆ = 1/L in the last equation. We maximize
the lower bound (154) over L to obtain 3
p
L = πβSNR/2.
(155)
p
By choosing an oversampling factor L = ⌈ πβSNR/2⌉, it
follows that
1
1
1
lim I(ΦX ; Ψ|XA ) − log SNR ≥ − log(64πβ) − .
SNR→∞
4
4
2
(156)
The proof of (43) is similar to the proof in Appendix A for
the full model and is omitted.4
Remark 4: We outline the intuition for choosing Ψ̃k in
k−1
n
(128). The basic idea is to process (Ψn , XA
, ΦX
) in three
steps: first estimate the phase noise, then cancel it, then
decode. We outline the motivation for each step:
1) Since the past inputs X k−1 and outputs Yk−1 are available,
one may use the simple estimator
b

e j Θk ′ ≡

Ψk ′
X⌈k′ /L⌉ ∆

(157)

to estimate Θk′ for k ′ = 1, . . . , (k − 1)L. This is a reasonable
estimate as long as the energy per sample is large relative to
the noise variance per sample, i.e., when SNR∆ ≫ 1.
2) Next, we use the previous estimates to cancel from Ynk the
phase noise that accumulated up to sample (k − 1)L. More
b (k−1)L . We
precisely, we use only the most recent estimate Θ
make this choice because the discrete-time Wiener process
{Θk } is a first-order Markov process so, conditioned on
ΘkL , the random variables ΘkL+1 ,...,ΘnL are independent of
Θ0 ,...,ΘkL−1 . This means that if the SNR is high enough so
b k′ is an accurate estimate of Θk′ for k ′ = 1,. . ., (k−1)L,
that Θ
b (k−1)L .
then it would suffice to use only Θ
3) Since the input is i.i.d. and there is no inter-symbol interference, we discard Ψnk+1 and use only Ψk (after canceling
the phase noise) for decoding ΦX,k . Moreover, we use only
the first sample Ψ(k−1)L+1 in Ψk . This is because, in absence
of additive noise, we have

A PPENDIX C
S UPPORTING L EMMA FOR A PPENDIX A
The following lemma is used in Appendix A.
Lemma 5: For G in (94) with Ts = 1, we have
4
Var(G)
=
(πβ)2 .
3
∆→0
∆
45
lim

(158)

Proof: Using the definition of G in (94), we have
!
L
1
1X
2
= Var(|F1 |2 )
Var(G) = Var
(159)
|Fℓ |
L
L
ℓ=1

because {Fk } is an i.i.d. process. The independence follows
because increments of a Wiener process over disjoint time
intervals are independent and because functions of independent
random variables are also independent. The identical distribution property follows from the rectangular pulse shape5 and
that the integrations are over increments of a Wiener process
with equal length. By using the formula in [44, Theorem
1] for computing moments of filtered Wiener phase noise
(specifically E[|F1 |2k ] where k is a positive integer), we have
E[|F1 |2 ] = 2

a − 1 − log a
(log a)2

(160)

and
E[|F1 |4 ]
783 − 784a + a4 + (540 + 240a) log a + 144(log a)2
.
=
18(log a)4
(161)
where
a ≡ e−πβ∆ .

(162)

It follows from (161) and (160) that
Var(|F1 |2 )

= E[|F1 |4 ] − (E[|F1 |2 ])2
711 − 640a − 72a2 + a4 + 12(33 + 32a) log a + 72(log a)2
=
18(log a)4
(163)

which implies that
4
Var(|F1 |2 )
=
.
a→1 (log a)2
45
lim

(164)

Finally, (158) follows from (164), (162) and L = 1/∆
(because Ts = 1).

I(ΦX,k ; Ψk )
= I(ΦX,k ; {ΦΨ,(k−1)L+1 : 1 ≤ ℓ ≤ L})
= I(ΦX,k ; ΦΨ,(k−1)L+1 , {W(k−1)L+ℓ : 2 ≤ ℓ ≤ L})

= I(ΦX,k ; ΦΨ,(k−1)L+1 )
= I(ΦX,k ; Ψ(k−1)L+1 ).

A PPENDIX D
S UPPORTING L EMMAS FOR A PPENDIX B
This appendix contains two lemmas. The main lemma is
Lemma 6 which is used in Appendix B. Lemma 7 is a
supporting lemma for Lemma 6.

3 Ignoring

L ∈ Z.
The inequality (118) is valid for the approximate model except that G is
set to 1 because Fk = 1 in the approximate model. Hence, by setting G̃ = 1,
the term −π 2 /45 in (40) does not appear in (43).
4

5 If the pulse is not rectangular, the process {F } may not be identically
k
distributed because the phase noise ejΘ(t) is weighted in a given sample
interval according to the pulse shape.

JOURNAL, VOL. , NO. , MONTH YEAR

16

Lemma 6: Let R be a fixed positive real number and let
Φ ≡ arg(Y ) where
Y =R+Z

(165)

and Z is a circularly-symmetric complex Gaussian random
variable with mean 0 and variance 1. Then, we have
E [cos(Φ)] ≥ 1 −

1
R2

E [sin(Φ)] = 0.

(166)
(167)

Proof: The probability density function of Φ is [45]
pΦ (φ) =

2
2
1
1 −R2
e
+ √ R cos(φ) e−R sin φ erfc (−R cos φ)
2π
4π
(168)

where erfc(·) is the complementary error function
Z ∞
2
2
erfc(z) ≡ √
e−t dt.
π z

(169)

Therefore, we have
E [cos(Φ)]
Z π
≡
cos(φ) pΦ (φ) dφ
−π
Z π
(a)
= 2
cos(φ) pΦ (φ) dφ


Z π0
2
2
1
(b)
=
cos(φ) √ R cos(φ) e−R sin φ erfc (−R cos φ) dφ
π
0
Z π/2
(c)
2
2
R
√ cos2 (φ) e−R sin φ erfc (−R cos φ) dφ
≥
π
0
Z π/2


(d)
2
2
2
2
R
√ cos2 (φ) e−R sin φ 2 − e−R cos φ dφ
≥
π
0
Z π/2
Z
2R
R −R2 π/2
2
−R2 sin2 φ
√ cos (φ) e
=
dφ − √ e
cos2 φdφ
π
π
0
0
√
Z π/2
2
2
2
π
2R
(e)
√ cos2 (φ) e−R sin φ dφ −
Re−R
(170)
=
π
4
0
where (a) follows because both pΦ (·) and
R π cos(φ) are even
functions, (b) follows from (168) and 0 cos φ dφ = 0,
(c) follows because the integrand is non-negative over the
interval φ ∈ (π/2, π], (d) holds because erfc (−R cos φ) ≥
2
2
2
2
2 − e−R cos φ (see Lemma 7) and cos2 (φ) e−R sin φ ≥ 0
and finally (e) follows by direct integration.
We bound the integral in (170) as follows
Z π/2
2
2
2R
√ cos2 φ e−R sin φ dφ
π
0
Z π/2
(a) 2R
2 2
(1 − φ2 ) e−R φ dφ
≥ √
π 0
Z
Z
2R π/2 2 −R2 φ2
2R π/2 −R2 φ2
e
dφ − √
φ e
dφ
= √
π 0
π 0
π 
π 
2 2
1
1
(b)
erf
R −
R + √ e−π R /4
= erf
2
2R2
2
2 πR
π 
(c)
1
(171)
R −
≥ erf
2
2R2

where erf(·) is the error function defined as
Z z
2
2
erf(z) ≡ √
e−t dt.
π 0
2

(172)

2

2

2

Step (a) follows from cos2 φ e−R sin φ ≥ (1 − φ2 )e−R φ
(see Lemma 7), (b) follows by using the definition of erf(·)
and [46, 3.321(5)]:
√

Z b
2 2
2 2
π
1
(173)
erf (ab) − ab e−a b
t2 e−a t dt = 3
2a
2
0
and (c) holds because erf(·) ≤ 1 and the last term is nonnegative. Substituting back in (170) yields
√
π 
2
1
π
−
R −
R e−R
(174)
E [cos(Φ)] ≥ erf
2
2R2
4 √
2 2
2
1
π
≥ 1 − e−π R /4 −
−
R e−R
(175)
2R2
4
where the last inequality follows from the relations
2 2
erf (πR/2) = 1 − erfc (πR/2) and erfc (πR/2) ≤ e−π R /4
(see [47, eq. (5)]). Therefore, we have
√  3/2
1
π 3
4
(176)
R2 E [cos(Φ)] ≥ R2 − 2 − −
π e 2
4
2e
because
 m m/2
(177)
x
2ae
for any a > 0 and positive integer m, which implies that
2
2 2
3/2
R3 e−R ≤ (3/2e)
and R2 e−π R /4 ≤ 4/π 2 e. We conclude
that
1
0.83
(178)
E [cos(Φ)] ≥ 1 − 2 ≥ 1 − 2 .
R
R
Finally, we have
Z π
E [sin(Φ)] ≡
sin(φ) pΦ (φ) dφ = 0
(179)
2

max xm e−ax =

−π

because the integrand is odd.
Lemma 7: The following bounds hold.
1) For z ≥ 0:

2

erfc(−z) ≥ 2 − e−z .
2) For a ≥ 0 and t ≥ 0:

2

cos2 t e−a sin

t

(180)

2

≥ (1 − t2 )e−at .

(181)

Proof:
1) Since for any real number z
erf(−z) = −erf(z)

erfc(z) = 1 − erf(z)

(182)
(183)

we have
erfc(−z) = 1 − erf(−z) = 1 + erf(z) = 2 − erfc(z).
(184)
To complete the proof, we use [47, eq. (5)]
2

erfc(z) ≤ e−z for z ≥ 0.

(185)

JOURNAL, VOL. , NO. , MONTH YEAR

17

2) The trigonometric identity cos2 t + sin2 t = 1 and the
bound sin t ≤ t (for t ≥ 0) imply that
cos2 t ≥ 1 − t2 .

(186)

Since the exponential function is monotone, we have
2

e−a sin

t

2

≥ e−at .

(187)

By combining the two inequalities, we obtain
cos2 t e−a sin

2

t

2

≥ (1 − t2 )e−at .

(188)

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