2012 IEEE International Symposium on Information Theory Proceedings
On the Capacity of Additive White Alpha-Stable Noise Channels
Jihad Fahs, Ibrahim Abou-Faycal
Dept. of Elec. and Comp. Engineering, American University of Beirut
Beirut 1107 2020, Lebanon
jjf03@aub.edu.lb, iaf@alum.mit.edu
Abstract— Many communication channels are reasonably modeled
to be impaired by additive noise. Recent studies suggest that many of
these channels are affected by additive noise that is best explained
by alpha-stable statistics.
We study in this work such channel models and we characterize
the capacity-achieving input distribution for those channels under
fractional order moment constraints. We prove that the optimal input
is necessarily discrete with a compact support for all such channels.
Interestingly, if the second moment is viewed as a measure of
power, even when the channel input is allowed to have infinite second
moment, the optimal one is found to have finite power.
I. I NTRODUCTION
In modeling the noise effect in a communication channel, it
is common to assume that the noise is additive and Gaussian
distributed due to the Central Limit Theorem which in layman
terms states that, if the noise is due to multiple independent
sources, then their cumulative effect is asymptotically Gaussian distributed.
Recent studies however suggest that in some circumstances,
the additive noise is better explained by non-Gaussian statistics
as is the case when modeling the multiuser interference in
a network [1], [2]. This was also observed in the context
of modeling the Radio-Frequency interference in embedded
wireless laptop transceivers [3]. In both setups, the noise was
found to be better modeled as a Symmetric Alpha-Stable
(SαS ) random variable.
This noise model poses multiple challenges to system
designers and we intend to address one of them in this work:
• Despite the fact that the Probability Density Function
(PDF) of a SαS random variable was proven to exist and
exhibit rather “nice” properties, no closed-form expression is known except in two special cases: the Cauchy
and the Gaussian ones.
• Such noise distributions have infinite variance, which
implies that the received signal has potentially infinite
power and any analysis that is based on a Hilbert space
approach is not valid anymore.
The “classical” signal processing techniques used in
receiver design are hence not necessarily applicable, let
alone optimal.
• The channel capacity of a basic linear channel where the
output is simply a noisy version of the input is not known
and optimal signaling schemes are not known either. A
few attempts were made along this direction and as far
This work was supported by AUB’s University Research Board.
978-1-4673-2579-0/12/$31.00 ©2012 IEEE
299
as the authors know, only numerical evaluation of some
achievable rates have been conducted [4], [5]. In this
work, we attempt to partially answer these questions.
Earlier information theoretic studies of linear channels under general additive non-Gaussian noise have been conducted
as in [6], [7]. However, typically these studies either require
a finite noise variance which is not the case in this model, or
impose a peak input power condition, a constraint that we will
not impose in this work. It is worth noting as well that, it is
not clear whether the SαS noise model fits within the general
categories described in [7].
In this paper, we study the additive linear channel model,
where a SαS noise variable is added to the input that is
generally constrained to satisfy a fractional r-th order moment
constraint: E [|X|r ] ≤ a, for any real value of r > 1. Clearly,
the average power constrained channel is a special case of our
setup. More importantly, if r is chosen to be less than 2, then
the input is technically allowed to have infinite power similarly
to the noise and the received signal: a primary motivation
behind this particular choice of the input constraint. Since
the received signal is of infinite power, and if a transceiver’s
hardware is capable of coping with such a received signal,
then it should be able to cope with a similar transmitted one.
The rest of the paper is organized as follows: We first
describe the channel model in Section II and derive a lower
bound on the output PDF in Section III. Our main results are
presented in Section IV, and Section V concludes the paper.
Two Appendices I and II establish useful technical lemmas.
II. C HANNEL MODEL
We consider the real linear discrete-time memoryless channel:
Y = X + N,
(1)
where Y is the channel output and X is its input which
is subject to the fractional r-th order moment constraint:
E [|X|r ] ≤ a for some real-valued r > 1 and a > 0. The
additive noise variable N , which is independent of the input,
is assumed to behave statistically as an alpha-stable Random
Variable (RV) whose characteristic function is defined by:
α
Φ(ω) = eiγω−|cω| ,
where α ∈ [1, 2), c > 0 and γ ∈ R. We note first that, from an
information theoretic perspective, assuming the values γ = 0
and c = 1, does not affect the generality of the problem since
it relates to the equivalent channel where the output is scaled
(Y − γ)/c. Henceforth we assume that the noise, the PDF of
which is pN (·), is a standard SαS RV whose characteristic
α
function is e−|ω| .
For additive channels with absolutely continuous noise, the
output PDF exists [8]. We denote by pY (y) = p(y; F )† the
output density induced by an input probability distribution
function F . Note that, for α ∈ [1, 2),
Z
1
1
1
1
−|ω|α
< √ < 1,
Γ
e
dω =
pN (n) ≤
2π
πα
α
π
the bound in the two ranges follow similar steps. We present
them hereafter for y ≤ −y0 :
Z
pN y − x dF (x)
p(y; F ) ≥
1
r
Zx:|x|≤(2a)
≥
(1 − ǫ) g(y − x) dF (x)
(4)
1
for all n ∈ R and therefore, since pN (·) is continuous [9, p.11
Th.1.9], less than 1 and has R as support, so is pY (·).
where we used the lower bound in (2) to obtain (4), and (5) is
justified by the fact that g(y − x) is a decreasing function of
x on the given interval. The last inequality is an application
of Markov’s inequality and the r-th moment constraint.
III. P RELIMINARIES
In order to study the capacity of channel (1) and characterize
its achieving input, we derive in this section a lower bound
on the output PDF.
It is known that alpha-stable distributions possess heavy tails
that decay asymptotically as power laws. More specifically [9,
Th.1.12, p.14], let 0 < ǫ < 12 , there exists n1 > 0 such that
∀ |n| ≥ n1 ,
(1 − ǫ) g(n) < pN (n) < (1 + ǫ) g(n),
(2)
πα
−(α+1)
α
= |n|4kα+1
.
where g(n) = α
π sin( 2 )Γ(α)|n|
Additionally, alpha-stable distribution functions are known
to be analytic on R whenever α ∈ [1, 2) [10, p.183 Sec.36].
Whenever α > 1 the radius of convergence is ∞ and is no
less than one if α = 1. Hence, if Sδ = {z ∈ C : |ℑ(z)| < δ},
pN (·) is analytically extendable over Sδ for δ small enough.
Moreover, we prove in Appendix I the following novel bound:
there exists κ > 0 and n2 > 0 such that
κ
,
|pN (z)| ≤
|ℜ(z)|α+1
∀ z ∈ Sδ : |ℜ(z)| ≥ n2 ,
(3)
which we use to establish Lemma 2 in Appendix II. In the
remainder of this paper, we fix ǫ and a small enough δ, and
we define n0 = max{n1 , n2 } where n1 and n2 are as above.
Lemma 1. For any input probability distribution function, The
output PDF pY (y) of channel (1) is lowerbounded by

kα

 h(2a) r1 −yi(α+1) y ≤ −y0
h(y) =
kα
y ≥ y0 ,

i(α+1)
1
 h
(2a) r +y
1
where y0 = n0 + (2a) r .
Proof: First note that the statement of this lemma does
not hold for all inputs in the Gaussian case. The derivations of
†p
Y
(y) = p(y; F ) =
R
p(y|x) dF (x) =
R
x:|x|≤(2a) r
1 1
1
g y − (2a) r
≥ Pr |X| ≤ (2a) r
2
h
i−(α+1)
1
≥ kα (2a) r − y
,
IV. M AIN R ESULT
In the following we state and prove the main theorem of
this paper
Theorem. The capacity achieving input of channel (1), whenever r > 1 and α ∈ [1, 2) is compactly supported, discrete
with no accumulation point except possibly at zero.
Proof: With the sufficient topological and functional
requirements satisfied, we proceed as in [8] and we write the
Karush-Kuhn-Tucker (KKT) conditions as being necessary and
sufficient conditions for the unique optimal input to satisfy.
These conditions state that an input random variable X ∗ with
Cumulative Distribution Function F ∗ achieves the capacity C
of an r-th moment constrained channel if and only if there
exists ν ≥ 0 such that,
Z
r
ν(|x| − a) + C + H + pN (y − x) ln p(y; F ∗ ) dy ≥ 0, (6)
for all x ∈ R, with equality if x is a point of increase of F ∗ . H
is the entropy of the noise and is finite by virtue of equation (2)
along with the continuity of pN (·). Assume that the points of
increase of F ∗ have a positive accumulation point and let
Z
s(z) = ν(z r − a) + C + H + pN (y − z) ln p(y; F ∗ ) dy.
The function s(z) is analytic on Sδ \R− by virtue of Lemma 2
proven in Appendix II and by using for z r the principal
branch of the logarithm [11, p.85 Prop.1.6.4]. Since s(z) has
accumulating zeros on the positive real axis, by the identity
theorem [12], s(·) is identically null on Sδ \ R− . Therefore,
Z
ν(xr − a) + C + H = − pN (y) ln p(y − x; F ∗ ) dy, (7)
for all x ∈ R+ . Examining the integral, let
Z ∞
pN (y) ln p(y − x; F ∗ ) dy
t(x) = −
−∞
=
pN (y − x) dF (x).
300
(5)
I1 + I2 + I3
where the interval of integration is divided into three subintervals: (−∞, x − y0 ), [x − y0 , x + y0 ] and (x + y0 , ∞).
Using Lemma 1,
Z x−y0
pN (y) ln p(y − x; F ∗ ) dy
I1 = −
−∞
≤
− (ln kα ) [1 − Pr(N > x − y0 )] +
Z x−y0
h
i
1
pN (y) ln (2a) 2r + x − y dy.
(α + 1)
−∞
1
Using the fact that ln x ≤ x α+1 ,
Z x−y0
i
h
1
pN (y) ln (2a) 2r + x − y dy
−∞
Z x−y0
1
h
i α+1
1
pN (y) (2a) 2r + x − y
≤
dy
−∞
1 α+1
1
≤ EN (2a) 2r + x − N h
i
1
1
1
≤ (2a) 2r(α+1) + |x| α+1 + EN |N | α+1 .
(8)
Pn
Pn
β
We justify (8) by the hfact thati| i=1 Xi |β ≤
i=1 |Xi |
1
for β ≤ 1. Since EN |N | α+1 is finite [10, p.179 Sec.35
Th.3], An inspection of equation (8) along with the fact that
Pr(N > x − y0 ) ∼ 4kαα (x − y0 )−α as x → ∞ [9, Th.1.12,
p.14] shows that I1 is o(|x|r ). Similarly, it can be shown that
I3 = o(|x|r ). As for I2 ,
Z x+y0
pN (y) ln p(y − x; F ∗ ) dy
I2 = −
x−y0
≤
FN (x + y0 ) − FN (x − y0 ) max ln
|y|≤y0
1
,
pY (y)
where the maximum exists and is positive since pY (y) is
continuous and 0 < pY (y) < 1. I2 is also o(|x|r ) and hence
t(x) = o(xr ) for which equation (7) is impossible unless
ν = 0 which is non-sensible since the constraint is biding, and
can be formally ruled out [13]. This leads to a contradiction
and rules out the assumption of having a positive accumulation
point. Similarly, we rule out a negative accumulation point
and in conclusion, the set of points of increase of F ∗ has no
accumulation point except possibly at 0.
Compact support of the optimal input
Having ruled out the possibility that an optimal distribution
has a set of points of increase with a non-zero accumulation
point, we can immediately conclude that this set is countable
on R by virtue of the fact that R is Lindelöf, and X ∗ is
discrete. Denoting by {xi }i≥0 the points of increase of F ∗ (x)
and pi = Pr(X = xi ) > 0, the output probability density can
be lowerbounded by pY (y) > pi p(y|xi ), ∀y ∈ R; ∀i ≥ 0.
Examining the last term in the Left-Hand Side (LHS) of the
KKT condition (6), it can be proven to be o(|x|r ) as in the
preceding section. Hence, the LHS of (6), LHS(x) = ν|x|r +
o(|x|r ) diverges to infinity as |x| goes to infinity for any ν > 0.
301
However, LHS(xi ) = 0 for all points of increase xi . If these
were arbitrarily large, then ν will be zero which is not possible.
V. C ONCLUSION
We studied the channel capacity and the capacity-achieving
input distributions for channels affected by SαS additive noise.
We proved that the optimal input is discrete and compactly
supported for any fractional order input moment constraint
larger than one. We note that, even when imposing a fractional
lower order moment constraint on the input and allowing it to
be potentially of infinite variance, since the optimal input is
compactly supported, its variance is nevertheless finite. This
indicates that over these types of channels, surprisingly, it
does not increase the rates to transmit with infinite power.
We acknowledge that measuring the power of a signal via the
second moment might not be adequate in this context.
We also note that the methodology developed in this paper
may be readily used for Pareto distributions.
Finally, the results may be readily generalized for non-linear
channels of the form Y = sgn(X)|X|s + N , where N is a
SαS noise random variable, and where the input is subject to
E [|X|t ] ≤ a, t > s. An appropriate change of variable as
in [14] yields readily a similar result.
A PPENDIX I
We study in this appendix the rate of decay of standard
SαS distributions with α ∈ [1, 2) on the horizontal strip
S = {z ∈ C : |ℑ(z)| < 1}. We prove that |pN (z)| =
1
, for z ∈ S as |ℜ(z)| → ∞‡ .
O |ℜ(z)|
α+1
1
For the case α = 1, pN (z) = π1 1+z
2 which clearly satisfies
our assertion. When α > 1, pN (·) can be formally extended
on C as:
Z ∞
Z ∞
α
α
1
1
e−izt−|t| dt =
e−ixt+yt−|t| dt
pN (z) =
2π −∞
2π −∞
Z
∞
1 X y n ∞ n −ixt−|t|α
=
t e
dt,
(9)
2π n=0 n! −∞
where we wrote z = x + iy, and used Lebesgue’s Dominated
Convergence Theorem (DCT) for the interchange in (9). This
is justified since
N
∞
X yn
X
α
α
|y|n n −|t|α
n −ixt−|t|
t e
|t| e
= e|y||t|−|t| ,
≤
n!
n!
n=0
n=0
which is integrable for α > R1.
α
∞
We study next Jn (x) =
ˆ −∞ tn eixt−|t| dt†† . It is known
that limx→∞ xα+1 J0 (x) = 2 Γ(α + 1) sin πα
2 [15, p.134].
‡ By definition, given three positive real-valued functions f (.), g(.) and
h(.) we write f (z) = O (g(z)) as h(z) → ∞ if and only if there exist two
positive scalars, c > 0 and ho such that f (z) ≤ cg(z), ∀ h(z) ≥ ho .
†† Note that J (−x) = (−1)n J (x) and the n-th derivative of p (x) is,
n
n
N
(n)
pN (x) =
in
(−i)n
Jn (−x) =
Jn (x),
2π
2π
n ∈ N.
For n ≥ 1 and x > 0, we have
Z ∞
Z ∞
α
α
xtn eixt−t dt + (−1)n xtn e−ixt−t dt
xJn (x) =
0
0
Z
Z ∞
∞
α
n−1 ixt−|t|α
tn+α−1 eixt−t dt
t
e
dt − iα
= in
0
−∞
Z ∞
n+α−1 −ixt−tα
n−1
t
e
dt
(10)
+(−1)
0
iα (11)
In + (−1)n−1 In
xn+α
where equation (10) is obtained by integration by parts and
regrouping, and where In is the complex conjugate of In :
Z ∞
Z ∞
γ
αγ
α
eiv −ζv dv,
tn+α−1 eixt−t dt = γ
In = xn+α
= i n Jn−1 (x) −
0
0
1
where γ = n+α
, ζ = x−α and the change of variable is
v = (xt)n+α . Since as x → ∞, ζ → 0+ ,
Z ∞
γ
αγ
lim In = γ lim
eiv −ζv dv
x→∞
ζ→0 0
Z ∞
γ iγθ
αγ iαγθ
+iθ
= γ lim lim
eiv e −ζv e
dv (12)
ζ→0 θ→0 0
Z ∞
γ iγθ
eiv e +iθ dv
(13)
= γ lim
θ→0 0
Z
γ
= γ lim
lim
eiz dz,
θ→0 R→∞,ρ→0
L1
iθ
where z = ve for some small positive θ and L1 = {z ∈
C : z = veiθ , 0 < ρ ≤ v ≤ R}. Equation (12) is justified by
ζ αγ
DCT since the integrand’s norm is less than e− 2 v which is
integrable. Similarly, (13) is justified since the order between
the two limits is interchangeable and the integrand in (12) is
γ
upperboundedR by e−v sin γθ which is integrable. To evaluate
γ
the limit of L1 eiz dz, we use contour integration over C
shown in Figure 1. where C1 and C2 are arcs of radius R,
6
C2
Fig. 1.
L2
)
L1
1
C1
-
The contour C.
and ρ respectively between angles θ and ϕ =
ˆ
mod 2π.
Note that since we are interested in the limit as θ goes to
zero, we can always choose it small enough in order to have
the contour counter-clockwise. Finally, L2 is a line connecting
the extremities of the arcs.
γ
Now since f (z) = eiz is analytic on and inside C (by
choosing an appropriate branch cut in the plane), by Cauchy’s
theorem [11, p.111 Sec.2.2],
I
Z
Z
Z
Z
0=
f (z) dz =
f (z) +
f (z) +
f (z) +
f (z).
L1
C1
R→∞
θ R→∞
θ
=0
γ
where the interchange is valid because Re−R Rsin(γφ) is def (z) = 0.
creasing as 0 < γθ ≤ γφ ≤ π2 . Hence, limR→∞
C1
R
Similarly, it can be shown that limρ→0 C2 f (z) = 0. It
π
remains to evaluate the integral on L2 where z = tei 2γ ,
Z ∞
Z
π
γ iπ
ei 2γ eit e 2 dt
f (z)dz = −
lim
R→∞,ρ→0 L
0
2
Z ∞
γ
π
π 1
1
i 2γ
i 2γ
−t
= −e
.
Γ
dt = −e
e
γ
γ
0
π
In conclusion limx→∞ In = ei 2 (n+α) Γ(n + α), and by (11),
we can write for n ≥ 1
lim xn+α+1 Jn (x) − i nxn+α Jn−1 (x) = Wn
x→∞
π
π
=
ˆ − iαΓ(n + α) ei 2 (n+α) + (−1)n−1 e−i 2 (n+α) (14)
Since lim xα+1 J0 (x) exists and is equal to U0 = 2 Γ(α +
ˆ lim xn+α+1 Jn (x) exists for all
1) sin( πα
2 ), then Un =
x→∞
non-negative
n and is equal to,
h
i Un = inUn−1 + Wn =
Pn−1 ik
n
n! i U0 + k=0 (n−k)! Wn−k . Furthermore, for n ≥ 0,
#
"
n−1
X |Wn−k |
|Un | ≤ n! |U0 | +
(n − k)!
k=0
#
"
n−1
X
(n + 1 − k) = 2 n! n2 + 3n + 2 .
≤ 4 n! 1 +
k=0
The last inequality is justified using the fact that for α ∈ (1, 2),
Γ(α + l) is increasing for l ∈ N∗ .
Now using (9),
lim xα+1 |pN (z)|
Z
∞
X
1
y n ∞ n −ixt−|t|α α+1 =
lim x
t e
dt
2π x→∞
n! −∞
n=0
∞
1 X yn
lim xα+1 Jn (−x)
=
2π n=0 n! x→∞
x→∞
π
2γ
C
On C1 , we have:
Z
Z ϕ
iφ iRγ eiγφ
lim f (z)dz = lim iRe e
dφ
R→∞
R→∞
C1
θ
Z ϕ
Z ϕ
γ
γ
−R sin(γφ)
lim Re−R sin(γφ) dφ
Re
dφ =
≤ lim
L2
C2
302
≤
≤
(15)
∞
∞
1 X |y|n
1 X |y|n
lim xn+α+1 |Jn (x)| =
|Un |
2π n=0 n! x→∞
2π n=0 n!
∞
1X n 2
|y| n + 3n + 2 ,
π n=0
which is finite because |y| < 1, and where we used the
fact that f (x) = |x| is continuous. The interchange in (15)
is valid because the end result is finite. In conclusion,
lim xα+1 |pN (z)| < ∞ which concludes our proof.
x→∞
A PPENDIX II
In this appendix we prove the analyticity of the integral
function on a strip around the real axis.
Lemma 2. The function i(·; F ) : Sδ → C defined by:
Z ∞
pN (y − z) ln p(y; F ) dy,
z → i(z; F ) = −
(16)
−∞
is analytic over Sδ , for all F such that
R
|x|r dF (x) ≤ a.
Proof: To prove this lemma, we will make use of
Morera’s theorem:
a) We start first by proving the continuity of i(·; F ) on Sδ .
In fact, let z0 ∈ Sδ , ρ > 0. and z ∈ Sδ such that |z − z0 | ≤ ρ,
Z
pN (y − z) ln p(y; F ) dy
lim i(z; F ) = − lim
z→z
z→z0
Z 0
(17)
=−
lim pN (y − z) ln p(y; F ) dy
z→z0
Z
= − pN (y − z0 ) ln p(y; F )dy = i(z0 ; F ).
(18)
Equation (18) is justified by pN (y − z) being a continuous
function of z on Sδ , and (17) by DCT. Indeed, in what follows
we find an integrable function r(y) such that,
pN (y − z) ln p(y; F ) = −pN (y − z) ln p(y; F ) ≤ r(y),
for all z ∈ Sδ such that |z − z0 | ≤ ρ and for all y ∈ R.
Using (3), we upperbound first |pN (y − z)|. For z ∈ Sδ
such that |z − z0 | ≤ ρ and y ≤ −(n0 + |ℜ(z0 )| + ρ), then
y ≤ −n0 + ℜ(z0 ) − ρ and
κ
κ
|pN (y − z)| ≤
≤
max
|y − ℜ(z)|α+1
ζ∈Sδ :|ζ−z0 |≤ρ |y − ℜ(ζ)|α+1
κ
=
,
|y − ℜ(z0 ) + ρ|α+1
Similarly, for y ≥ (n0 + |ℜ(z0 )| + ρ) ≥ (n0 + ℜ(z0 ) + ρ),
κ
pN (y − z) ≤
.
|y − ℜ(z0 ) − ρ|α+1
Since p(y; F ) < 1, we upperbound − ln p(y; F ) using
Lemma 1 to obtain
#
"

1
r − y

(2a)
(α
+
1)κ


y ≤ −ψ
ln

1
α+1


 |y − ℜ(z0 ) + ρ|
kα(α+1)
−M ln M1
r(y) =
# |y| < ψ
"

1


r
(α
+
1)κ
(2a)
+
y


y ≥ ψ.

1
 |y − ℜ(z0 ) − ρ|α+1 ln
kα(α+1)
where
o
n
1
ψ = max n0 + |ℜ(z0 )| + ρ, n0 + (2a) r
M = max
max
|y|≤ψ ζ∈Sδ :|ζ−z0 |≤ρ
|pN (y − ζ)|
M1 = min py (y).
|y|≤ψ
303
Note that M is finite since the maximization of pN (·) is taken
over a compact set that is included in Sδ where it is analytic,
and 0 < M1 < 1 since py (·) is positive and continuous. The
fact that r(y) is integrable concludes the proof of continuity
of i(z; F ).
b) To continue the proof of analyticity, we need to integrate
i(·; F ) on the boundary ∂∆ of a compact triangle ∆ ⊂ Sδ .
We denote by |∆| its perimeter, η0 = minz∈∂∆ ℜ(z), η1 =
maxz∈∂∆ ℜ(z) and φ = max{n0 + max{|η0 |, |η1 |}, n0 +
1
(2a) r }. By similar arguments as above and using
the fact that
R
pN (·) is analytic on Sδ one can prove that ∂∆ i(z; F )dz =
0, which in addition to the continuity of i(·; F ) insure its
analyticity on Sδ by applying Morera’s theorem.
ACKNOWLEDGMENTS
The authors would like to thank AUB Professor Farouk
Abi-Khuzam for helpful discussions on upper-bounding the
analytic extension of the alpha-stable density function.
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