A Geometric Analysis of the AWGN Channel with
a (σ, ρ)-Power Constraint
Varun Jog
Venkat Anantharam
EECS, UC Berkeley
Berkeley, CA-94720
Email: varunjog@eecs.berkeley.edu
EECS, UC Berkeley
Berkeley, CA-94720
Email: ananth@eecs.berkeley.edu
Abstract—We consider the additive white Gaussian noise
(AWGN) channel with a (σ, ρ)-power constraint, which is
motivated by energy harvesting communication systems. This
constraint imposes a limit of σ + kρ on the total power of
any k ≥ 1 consecutive transmitted symbols in a codeword. We
analyze the capacity of this channel geometrically, by considering
the set Sn (σ, ρ) ⊆ Rn which is the set of all n-length sequences
satisfying the (σ, ρ)-power constraints. For a noise power of
ν, we obtain an upper bound on capacity by considering the
volume of the Minkowski sum
√ of Sn (σ, ρ) and the n-dimensional
Euclidean ball of radius nν. We analyze this bound using
a result from convex geometry known as Steiner’s formula,
which gives the volume of this Minkowski sum in terms of the
intrinsic volumes of Sn (σ, ρ). We show that as n increases, the
logarithms of the intrinsic volumes of {Sn (σ, ρ)} converge to a
limit function under an appropriate scaling. An upper bound on
capacity is obtained in terms of the limit function, thus pinning
down the asymptotic capacity of the (σ, ρ)-power constrained
AWGN channel in the low-noise regime. We derive stronger
results when σ = 0, corresponding to the amplitude-constrained
AWGN channel.
Keywords: Additive white Gaussian noise, Shannon capacity,
energy harvesting, Steiner’s formula, intrinsic volumes.
I. I NTRODUCTION
The additive white Gaussian noise (AWGN) channel is one
of the most basic channel models in information theory. This
channel is represented by a sequence of channel inputs denoted
by Xi , and an input-independent additive noise Zi . The noise
variables Zi are assumed to be independent and identically
distributed as N (0, ν). The channel output Yi is given by
Yi = Xi + Zi for i ≥ 1.
(1)
The Shannon capacity of this channel is infinite when there
are no constraints on the channel inputs Xi ; however, practical
considerations always constrain the input in some manner. The
input constraints are often defined in terms of the power of the
input. For a channel input (x1 , x2 , . . . , xn ), the most common
power constraints encountered are:
(AP): An average power constraint of P > 0, which says
that
n
X
x2i ≤ nP.
i=1
(PP): A peak power constraint of A > 0, which says that
|xi | ≤ A, for all 1 ≤ i ≤ n.
The AWGN channel with the (AP) constraint was first
analyzed by Shannon [1]. Shannon showed that the capacity
C for this constraint 1 is given by
P
1
,
(2)
C = sup I(X; Y ) = log 1 +
2
ν
E[X 2 ]≤P
and the supremum is attained when X ∼ N (0, P ).
Compared to the (AP) constraint, fewer results exist about
the (PP) constrained AWGN. Smith [2] showed that the
channel capacity C is given by
C = sup I(X; Y ).
(3)
|X|≤A
Unlike the (AP) case, the supremum in equation (3) does not
have a closed-form expression. Smith used tools from complex
analysis to show that the supremum is achieved by a discrete,
finitely-supported distribution, and proposed an algorithm to
numerically evaluate the optimal distribution, and thereby the
capacity. Similar results were obtained for the quadrature
Gaussian channel by Shamai & Bar-David [3].
In this paper, we consider the following power constraint
called the (σ, ρ)-power constraint:
Definition 1. Let σ, ρ ≥ 0. A codeword (x1 , x2 , . . . , xn )
satisfies a (σ, ρ)-power constraint if
l
X
x2j ≤ σ + (l − k)ρ , ∀ 0 ≤ k < l ≤ n.
(4)
j=k+1
This constraint was first introduced in an earlier work [4],
and is motivated by an energy harvesting channel. For a
transmitter that harvests ρ units of energy per time slot and is
equipped with a battery of capacity σ, the power constraint on
transmitted sequences is precisely the (σ, ρ)-power constraint.
Let Sn (σ, ρ) be the set of all n-length sequences satisfying
the (σ, ρ)-power constraint. We look at the growth rate of the
volume of the sequence {Sn (σ, ρ)}, defined by
log Vol(Sn (σ, ρ))
,
(5)
n
where the limit exists due to subadditivity. Theorem 2 in [4]
established that the Shannon capacity C of a (σ, ρ)-power
v(σ, ρ) := lim
n→∞
1 All
logarithms in this paper are to base e
constrained AWGN channel with noise power ν satisfies
1
e2v(σ,ρ)
1
ρ
.
(6)
log 1 +
≤ C ≤ log 1 +
2
2πeν
2
ν
The upper bound is simply the channel capacity when σ = ∞
and the battery is initially empty [5], and the lower bound
is obtained via a simple application of the entropy power
inequality. The upper bound is not satisfactory as it does
not depend on σ. Furthermore, the lower and upper bounds
do not converge asymptotically as ν → 0: the lower bound
is v(σ, ρ) − 21 log 2πeν + O(ν) and the upper bound is
ρ
1
2 log ν + O(ν), and they differ by O(1). This implies that at
least one of these bounds is loose in the low-noise regime.
One might expect the upper bound to be loose, since it
disregards the effects of a finite value of σ on the capacity.
To rigorize this intuition, it is useful to think of coding with
the (σ, ρ)-constraints as trying to fit the largest number of
centers of noise balls into Sn (σ, ρ) such that the noise balls
are asymptotically approximately disjoint. As the noise power
ν decreases, so does the size of the noise balls, so one can
imagine a very efficient packing of the small balls such that
they occupy almost all the space. The total number of balls is
then roughly
# of balls ≈
Vol(Sn (σ, ρ))
,
Vol(Noise ball)
(7)
so that the capacity is roughly
1
Vol(Sn (σ, ρ)
1
log (# of balls ) = log
(8)
n
n
Vol(Noise ball)
1
≈ v(σ, ρ) − log 2πeν.
(9)
2
Expression (9) approximately equals the lower bound in (6)
for small values of ν. Turning this intuition into a proof is
nontrivial. Firstly, the intuitive explanation above ignores the
evolution of Sn (σ, ρ) with the dimension n. For instance,
consider the equations (7)-(9), where Sn (σ, ρ) is replaced by
the set Kn , given by
n−1 A A
1
1
Kn = − ,
× − n−1 , n−1 .
2 2
2
2
In this case, equations (7)-(9) might suggest that the asymptotic capacity is approximately log(A/2) − 12 log 2πeν, rather
than log A − 12 log 2πeν. Secondly, since we are only packing
the centers of noise balls in Sn (σ, ρ) and not balls themselves,
equation (8) is not entirely accurate.
In Section II of this paper, we make equations (7)-(9)
precise by establishing a new upper bound on capacity that is
asymptotically equal to expression (9). As in [4], our approach
involves a volume calculation; however, the improved upper
bound is in terms of the volume of the Minkowski sum of
Sn (σ, ρ) and a “noise ball.” We analyze this bound for the
special case of σ = 0 in Section III where it can be evaluated
in a closed form. We then consider the general case of σ > 0 in
Section IV. Although the upper bound does not admit a closed
form expression in this case, it is still possible to establish the
correctess of the asymptotic capacity expression in equation
(9) in the low-noise regime.
II. VOLUME BASED UPPER BOUND ON CAPACITY
√
Let B
√ of
√n ( nν) denote the n-dimensional Euclidean ball
sum of Sn (σ, ρ) and Bn ( nν)
radius nν. The Minkowski √
is denoted by Sn (σ, ρ) ⊕ Bn ( nν) and is the set
√
{xn + z n | xn ∈ Sn (σ, ρ), z n ∈ Bn ( nν)}.
Theorem 1. The capacity C of an AWGN channel with a
(σ, ρ)-power constraint and noise power ν satisfies
p
1
C ≤ lim lim sup log Vol Sn (σ, ρ) ⊕ Bn
n(ν + )
→0+ n→∞ n
1
− log 2πeν.
(10)
2
Proof: For n ∈ N, let Fn be the set of all probability
distributions supported on Sn (σ, ρ). The channel capacity
(Theorem 1, [4]) is
1
I(X n ; Y n ).
(11)
sup
C = lim
n→∞ n p n (xn )∈F
n
X
p
Let pX n (xn ) ∈ Fn . Denote Sn (σ, ρ) ⊕ Bn ( n(ν + ) ) by
Cn . Let > 0, and let δn := P (Y n ∈
/ Cn ) . By the law of large
numbers, we have δn → 0. Let χ be the indicator variable for
the event {Y n ∈ Cn }. Then
h(Y n ) = H(δn ) + δ̄n h(Y n |χ = 1) + δn h(Y n |χ = 0)
≤ H(δn ) + δ̄n log Vol(Cn ) + δn h(Y n |χ = 0). (12)
where ā = 1 − a. Since kX n k2 ≤ σ + nρ with probability 1,
we have the following bound on power of Y n :
E[kY n k2 ] = E[kX n k2 ] + E[kZ n k2 ] ≤ σ + nρ + nν.
This translates to the bound
E[kY n k2 | χ = 0] ≤
n(ρ + ν + σ/n)
,
δn
so
n
2πe(ρ + ν + σ/n)
log
.
2
δn
Substituting into inequality (12) and dividing by n gives
h(Y n | χ = 0) ≤
h(Y n )
H(δn )
log Vol(Cn ) δn
2πe(ρ + ν + σ/n)
≤
+ δ̄n
+
log
.
n
n
n
2
δn
Since this holds for any choice of pX n ∈ Fn , we obtain
H(δn )
log Vol(Cn )
1
h(Y n ) ≤
+ δ̄n
n
n
n
pX n ∈Fn
δn
2πe(ρ + ν + σ/n)
+
log
.
2
δn
Taking the limsup in n, we arrive at
sup
1
log Vol(Cn )
h(Y n ) ≤ lim sup
n
n
n→∞ pX n ∈Fn
n→∞
p
log Vol(Sn (σ, ρ) ⊕ Bn ( n(ν + ) ))
= lim sup
.
n
n→∞
lim sup sup
Taking the limit as → 0+ and noting that capacity is
limn→∞ suppX n ∈Fn n1 h(Y n ) − 12 log 2πeν, we arrive at the
bound in expression (10).
We define a function ` : [0, ∞) → R, as the growth rate of
volume of the Minkowski sum, as follows:
√
1
`(ν) := lim sup log Vol(Sn (σ, ρ) ⊕ Bn ( nν )). (13)
n→∞ n
The upper bound may be restated as
1
C ≤ lim `(ν + ) − log 2πeν .
→0+
2
(14)
Theorem 2 (Steiner’s formula). Let Kn ⊆ Rn be a compact
convex set and let Bn ⊆ Rn be the unit ball. Let µj (Kn )
denote j-th intrinsic volume Kn and j = Vol(Bj ). Then
n
X
µn−j (Kn )j tj ,
∀t ≥ 0.
(15)
j=0
Intrinsic volumes are fundamental to convex and integral
geometry. They describe the global characteristics of a set,
including the volume, surface area, mean width, and Euler
characteristic. For more details, we refer the reader to Schneider [7] and section 14.2 of Schneider
√ & Weil [8]. Steiner’s
formula states that Vol(Sn (σ, ρ)⊕Bn ( nν)) depends not only
on the volumes of these sets, but also on the intrinsic volumes.
Intrinsic volumes are notoriously hard to compute even for
simple sets such as polytopes [6], so it is optimistic to expect
a closed form expression for the intrinsic volumes of Sn (σ, ρ).
Furthermore, the sets {Sn (σ, ρ)} evolve with the dimension n,
so it is also important to keep track of the evolution of the
intrinsic volumes in order to compute the overall volume via
Steiner’s formula.
The case σ = 0 is special, since Sn (σ, ρ) is equal to the
√ √
cube [− ρ, ρ]n . Then the intrinsic volumes have a simple
closed form. We focus on this case in the next section.
III. T HE CASE OF σ = 0
n→∞
√
1
log Vol([−A, A]n ⊕ Bn ( nν)),
n
(16)
and the upper bound on channel capacity is as in inequality
(14). The main result of this section is as follows:
Theorem 3. The function `(ν) is continuous on [0, ∞). For
ν > 0, we have
`(ν) = H(θ∗ ) + (1 − θ∗ ) log 2A +
θ∗
2πeν
log ∗ ,
2
θ
since {An } and {Bn } themselves satisfy such an inclusion
property. Hence, the sequence {log Vol(Cn )} is super-additive,
n)
and `(ν) given by the limit limn log Vol(C
is well-defined.
n
Using Steiner’s formula, we express the volume of Cn as
n X
√
n
Vol(Cn ) =
(18)
(2A)n−j j ( nν)j .
j
j=0
Since the volume is a sum of n + 1 terms, the exponential
growth rate of the volume is determined by the growth rate of
the largest term amongst the n + 1 terms. To formalize this
notion, we define the function fnν (θ), for 0 ≤ θ ≤ 1 as
√
1
Γ(n + 1)
π nθ/2
nθ̄
nθ
log
(2A) ( nν)
,
n
Γ(nθ̄ + 1)Γ(nθ + 1) Γ(nθ/2 + 1)
where θ̄ denotes 1 − θ, and write
Vol(Cn ) =
n
X
ν
enfn (j/n) .
j=0
ν
nfn
(θ̂n )
The largest term is e
, where
θ̂n = arg max fnν (j/n).
j/n
(19)
We thus have
lim fnν (θ̂n ) = `(ν).
n
The key step in our proof is to show that the sequence {fnν }
converges uniformly on [0, 1] to a limit function f ν , given by
θ
2πeν
log
.
2
θ
This uniform convergence enables us to prove that
f ν (θ) = H(θ) + θ̄ log 2A +
(20)
`(ν) = lim fnν (θ̂n ) = max f ν (θ).
√
To simplify notation, we denote A := ρ. We consider
the scalar AWGN channel with noise power ν and an input
amplitude constraint of A. Let the capacity of this channel be
C. Recall that the function `(ν) is defined as
`(ν) = lim sup
(1 − θ∗ )2
2A2
=
.
πν
θ∗ 3
Proof of Theorem 3: We describe the proof strategy;
for
√
details, see [9]. Let An := [−A, A]n , Bn := Bn ( nν), and
Cn := An ⊕ Bn . Then
Cm × Cn ⊆ Cm+n , for all m, n ≥ 1,
Although ` is clearly monotonically increasing, it is not clear a
priori whether ` is continuous. Note that since `(0) = v(σ, ρ),
the continuity of ` at 0 leads to an asymptotic upper bound as
in expression (9). To study the properties of `, we use a result
from convex geometry known as Steiner’s formula [6]:
Vol(Kn ⊕ tBn ) =
where H is the binary entropy function and θ∗ ∈ (0, 1) is the
unique solution to
(17)
n
θ
To prove the continuity of ` at points ν > 0, note that
toggling ν slightly does not change f ν significantly. Thus, the
value of the supremum, which is `(ν), also does not change
significantly. To prove continuity at 0, we differentiate f ν and
obtain θ∗ (ν) = arg maxθ f ν (θ) as the solution to
2A2
(1 − θ∗ )2
=
.
(21)
3
πν
θ∗
From equation (21), we note that ν → 0 implies θ∗ → 0 at
the rate cν 1/3 , where c is a constant. Substituting this into the
expression for f ν , we obtain
`(ν) = log 2A + O(ν 1/3 ),
(22)
For n ≥ 1, denote the intrinsic volumes of Sn (σ, ρ) by
{µn (i)}ni=0 . Define Gn : R → R and gn : R → R as
5
Lower bound
4.5
New upper bound
Bits/Channel Use
4
Gn (t) = log
Old upper bound
3.5
n
X
µn (j)ejt , gn (t) =
j=0
3
2.5
2
1.5
1
0.5
0
−4
−3
−2
−1
0
1
2
3
4
log(1/⌫)
Fig. 1. Plot showing the capacity bounds from inequality (6) along with
the new upper bound from Theorem 1, for amplitude A = 1. The new upper
bound and the lower bound converge asymptotically as ν → 0.
Gn (t)
.
n
(26)
Let Λ be the pointwise limit of the sequence of functions {gn },
and let Λ∗ be the convex conjugate of Λ. Then the following
hold:
1) `(ν) is continuous on [0, ∞).
2) For ν > 0,
2πeν
θ
∗
.
(27)
`(ν) = sup −Λ (1 − θ) + log
2
θ
θ∈[0,1]
Proof of Theorem 4: We again provide a proof sketch
and refer to [9] for the details. We first check that Sn (σ, ρ) is
convex and has well-defined intrinsic volumes. Using Steiner’s
formula for Sn (σ, ρ), we obtain
n
X
√
√
µn (n − j)j ( nν)j . (28)
Vol(Sn (σ, ρ) ⊕ Bn ( nν )) =
j=0
implying the continuity of ` at 0.
Corollary 3.1. As ν → 0, the capacity C of an AWGN channel
with amplitude constraint A satisfies
C = log 2A −
n
X
√
ν
Vol(Sn (σ, ρ) ⊕ Bn ( nν)) =
enfn (j/n) ,
1
log 2πeν + O(ν 1/3 ).
2
1
log 2πeν + O(ν 1/3 ).
2
(23)
The lower bound from inequality (6) leads to
C ≥ log 2A −
1
log 2πeν + O(ν).
2
(24)
Inequalities (23) and (24) establish the corollary.
We may use Theorem 3 to numerically evaluate θ∗ and
plot the corresponding upper bound from Theorem 1. Figure 1
shows the resulting plot, along with the bounds from inequality
(6). Note that the upper bound from expression (6) (“old
upper bound”) is not asymptotically tight in the low-noise
regime, but the bound from Theorem 1 (“new upper bound”)
is asymptotically tight.
IV. T HE CASE OF σ > 0
In this section, we parallel the upper-bounding technique
used in Section III for σ > 0. The set Sn (σ, ρ) is no longer an
easily identifiable set and the intrinsic volumes of Sn (σ, ρ) do
not have a closed-form expression; however, it is still possible
to obtain similar results.
Our main theorem is the following:
Theorem 4. Define `(ν) as
`(ν) = lim sup
n→∞
√
1
log Vol(Sn (σ, ρ) ⊕ Bn ( nν )).
n
(29)
j=0
Proof: Using equation (22) along with Theorem 1 gives
C ≤ log 2A −
We argue that the growth rate of the volume depends only on
the growth rate of the largest amongst the n + 1 terms. As in
Section III, we would like to express the volume as
(25)
for a sequence of functions {fnν } defined over [0, 1]. Define a
function an (θ) by linearly interpolating the values of an (j/n),
where the value of an (j/n) is given by:
j
1
an
(30)
= log µn (n − j) for 0 ≤ j ≤ n.
n
n
The function bνn (θ) is given by
bνn (θ) =
1
π nθ/2
log
(nν)nθ/2 for θ ∈ [0, 1]. (31)
n
Γ(nθ/2 + 1)
Define fnν : [0, 1] → R as
fnν (θ) = an (θ) + bνn (θ).
(32)
This definition ensures that equation (29) is satisfied. The
may be shown in a
uniform convergence of {bνn } to θ2 log 2πeν
θ
straightforward manner. The challenge is to show that {an (·)}
uniformly converge to the limit function −Λ∗ (1 − θ), which
yields uniform convergence of {fnν } to the desired expression
in (27). This convergence is established as follows:
1. We use the following crucial property:
Sm+n (σ, ρ) ⊆ Sm (σ, ρ) × Sn (σ, ρ).
(33)
The sequence of intrinsic volumes of the product set
Sm (σ, ρ) × Sn (σ, ρ) is given by the convolution of the
sequences µn (·) and µm (·) [6]. The monotonicity of
intrinsic volumes applied to the inclusion in (33) yields
µm+n ≤ µm ? µn ,
(34)
By continuity of ` at 0, we have that as ν → 0,
which then implies
Gm+n ≤ Gm + Gn .
(35)
The pointwise convergence of {gn } to a limit function Λ
follows from the subadditivity in expression (35). We then
use the Gärtner-Ellis theorem [10] from large deviations
to show a “large deviations upper bound” on the measures
{µn/n }∞
n=1 , which are scaled versions of {µn } defined
by µn/n nj := µn (j). We show that
1
log µn/n (I) ≤ − inf Λ∗ (x),
(36)
x∈I
n→∞ n
for any closed set I ⊆ R.
2. Let γ = d σρ e. It was shown in [4] that the sequence
{Sn (σ, ρ)} satisfies another crucial structural property:
lim sup
Sm+n+2γ ⊇ [Sm × 0] × [Sn × 0] ,
(37)
where 0 is the zero vector of length γ. This property is
used to establish a “large deviations lower bound” on the
measures {µn/n }:
1
log µn/n (F ) ≥ − inf Λ∗ (x),
(38)
x∈F
n
for any open set F ⊆ R.
3. The steps above show that {µn/n } converge in the “large
deviation sense,” with rate function Λ∗ . Such convergence
does not generally imply uniform or even pointwise convergence of the linearly interpolated functions {an (·)}.
We use the fact that for each n, the sequence µn (·)
is log-concave [11]. This fact, combined with the large
deviations convergence, may be used to prove uniform
convergence of {an } to −Λ∗ (1 − θ).
Letting f ν (θ) = −Λ∗ (1 − θ) + θ2 log 2πeν
θ , we follow a
sequence of steps similar to those in Section III to arrive at
lim inf
n→∞
`(ν) = sup f ν (θ).
(39)
θ
The continuity of `(ν) for ν > 0 follows by the same
argument in the proof of Theorem 3. Proving continuity at
ν = 0 is more involved, since we do not have an explicit
expression for θ∗ = arg maxθ f ν (θ). We use the following
facts: (a) For all ν > 0, we have f ν (θ∗ ) ≥ f ν (0) = v(σ, ρ)
(b) f ν (θ∗ ) → −∞ as ν → 0 if θ∗ is bounded away from
0, and conclude that θ∗ → 0 as ν → 0. We then use the
continuity of Λ∗ and the fact that θ2 log 2πeν
≤ πν for all
θ
θ ∈ [0, 1] to obtain the limit f ν (θ∗ ) → v(σ, ρ) as ν → 0.
Corollary 4.1. The capacity C of a (σ, ρ)-power constrained
AWGN channel with noise power ν → 0 satisfies
1
C = v(σ, ρ) − log 2πeν + (ν),
(40)
2
where (·) is a function such that limν→0 (ν) = 0.
Proof: Using the lower bound in inequality (6), we have
1
e2v(σ,ρ)
C ≥ log 1 +
2
2πeν
1
= v(σ, ρ) − log 2πeν + O(ν).
(41)
2
`(ν) = v(σ, ρ) + (ν)
for some (·) satisfying limν→0 (ν) = 0. Then
1
C ≤ v(σ, ρ) − log 2πeν + (ν).
(42)
2
Our claim follows from inequalities (41) and (42). Unlike the
case of σ = 0, we cannot give a precise rate at which (ν) →
0.
V. C ONCLUSION
We have analyzed an AWGN channel with a power constraint motivated by energy harvesting communication systems. We approached the problem from a geometric viewpoint
and established an upper bound√on channel capacity in terms of
the volume of Sn (σ, ρ) ⊕ Bn ( nν), according to the intrinsic
volumes of Sn (σ, ρ). For the special case σ = 0, corresponding to the peak power constrained AWGN channel, we
explicitly evaluated the upper bound and derived asymptotic
capacity results in the low-noise regime. For the general case
of σ > 0, we exploited the geometric properties of {Sn (σ, ρ)}:
A : Sm+n ⊆ Sn × Sm ,
B : [Sm × 0] × [Sn × 0] ⊆ Sm+n+2γ (σ, ρ), when γ = d σρ e
and 0 is the zero vector of length γ,
to show that the appropriately normalized intrinsic volumes of
{Sn (σ, ρ)} converge to a continuous limit function. The upper
bound on channel capacity is expressed in terms of the limit
function, and the continuity of the same enabled us to prove
asymptotic capacity results in the low-noise regime.
ACKNOWLEDGEMENTS
The research of the authors was supported by NSF grant
ECCS-1343398 and the NSF Science & Technology Center
grant CCF-0939370, Science of Information.
R EFERENCES
[1] C. Shannon, “A mathematical theory of communications, I and II,” Bell
Syst. Tech. J, vol. 27, pp. 379–423, 1948.
[2] J. G. Smith, “The information capacity of amplitude-and varianceconstrained scalar gaussian channels,” Information and Control, vol. 18,
no. 3, pp. 203–219, 1971.
[3] S. Shamai and I. Bar-David, “The capacity of average and peakpower-limited quadrature Gaussian channels,” IEEE Transactions on
Information Theory, vol. 41, no. 4, pp. 1060–1071, 1995.
[4] V. Jog and V. Anantharam, “An energy harvesting AWGN channel with a
finite battery,” in International Symposium on Information Theory (ISIT)
2014, pp. 806–810, June 2014.
[5] O. Ozel and S. Ulukus, “Achieving AWGN capacity under stochastic
energy harvesting,” IEEE Transactions on Information Theory, vol. 58,
no. 10, pp. 6471–6483, 2012.
[6] D. A. Klain and G.-C. Rota, Introduction to geometric probability.
Cambridge University Press, 1997.
[7] R. Schneider, Convex bodies: the Brunn Minkowski theory, vol. 151.
Cambridge University Press, 2013.
[8] R. Schneider and W. Weil, Stochastic and integral geometry. Springer,
2008.
[9] V. Jog and V. Anantharam, “A geometric analysis of the AWGN channel
with a (σ, ρ)-power constraint,” arXiv e-prints, April 2015. Available
at http://arxiv.org/abs/1504.05182.
[10] A. Dembo and O. Zeitouni, Large deviations techniques and applications, vol. 2. Springer, 1998.
[11] P. McMullen, “Inequalities between intrinsic volumes,” Monatshefte für
Mathematik, vol. 111, no. 1, pp. 47–53, 1991.