Transporting Information and Energy
Simultaneously
LavR.Varshney
Laboratory for Information and Decision Systems and Research Laboratory of Electronics
Massachusetts Institute of Technology
Abstract—The fundamental tradeoff between the rates at
which energy and reliable information can be transmitted over
a single noisy line is studied. Engineering inspiration for this
problem is provided by powerline communication, RFID systems,
and covert packet timing systems as well as communication
systems that scavenge received energy. A capacity-energy function
is defined and a coding theorem is given. The capacity-energy
function is a non-increasing concave ∩ function. Capacity-energy
functions for several channels are computed.
I. INTRODUCTION
The problem of communication is usually cast as one of
transmitting a message generated at one point to another
point. During the pre-history of information theory, a primary
accomplishment was the abstraction of the message to be
communicated from the communication medium. As noted,
“electricity in the wires became merely a carrier of messages,
not a source of power, and hence opened the door to new ways
of thinking about communications” [1]. This understanding
of signals independently from their physical embodiments led
to modern communication theory, but it also blocked other
possible directions. As Norbert Wiener said, “Information is
information, not matter or energy. No materialism which does
not admit this can survive at the present day” [2, p. 132]. This
separation of messages and media arguably led to the division
of electrical engineering i nto two distinct subfields, electric
power engineering and communication engineering.
Some have argued that the greatest inventions of civilization
either transform, store, and transmit energy or they transform,
store, and transmit information [3]. Although quite reasonable,
many engineering systems actually deal with both energy and
information. Representation of signals requires the modulation
of energy, matter, or some such thing. The separation of
messages and media is not always warranted.
Are there scenarios where one would want to transmit
energy and information simultaneously over a single line? If
there is a power-limited receiver that can harvest received
energy, then one should want both things. The earliest tele-
graphs, telephones, and crystal radios had no external power
sources [1], providing historical examples of such systems.
Modern communication systems that operate under severe
energy constraints may also benefit from harvesting received
energy [4]. A powerful base station or other special node
[5], [6], may effectively be used to recharge mobile devices.
In RFID systems, the energy provided through the forward
Work supported in part by an NSF Graduate Research Fellowship.
channel is used to transmit over the backward channel [7].
There are also extant mudpulse telemetry systems in the oil
industry where energy and information are provisioned to
remote instruments over a single line [J. Kusuma, personal
communication]. For a truly space-age application, one might
consider furnishing photons to spacecraft with space sails [8]
and optical receivers for both information and propulsion.
Back on earth, power line communication has received
significant attention [9], [10], but the literature has focused on
the informational aspect under the constraint that modulation
schemes not severely degrade power delivery. This need not
be the case in future engineering systems.
Except for papers on reversible computing [11], the fact
that matter/energy must go along with information does not
seem to have been considered in information theory. Sim-
ilarly, the information carried in power transmission seems
not to have been considered in power engineering. Though
not implemented in current systems, a receiver constructed
from reversible gates would allow received energy to perform
additional work and would need not be dissipated as heat [11].
Electricity, of course, is not the only commodity in which
signals can be modulated. Information can be physically
manifested in almost any substance. Examples include water,
railroad cars, and packets in communication networks (whose
timing is modulated [12]); the results presented apply equally
to these scenarios.
This work deals with the fundamental tradeoff between
transmitting energy and transmitting information over a single
noisy line. Although this tradeoff must be known to other
researchers, it does not seem to appear in the literature. A char-
acterization of communication systems that simultaneously
meet two goals:
1) large received energy per unit time, and
2) large information per unit time
is found. Notice that unlike traditional transmitter power
constraints, where small transmitted power is desired, here
large received power is desired. One previous study has looked
at maximum received power constraints [13].
II. C
APACITY-ENERGY FUNCTION
In order to achieve the first goal, one would want the most
energetic symbol received all the time, whereas to achieve
the second goal, one would want to use the unconstrained
capacity-achieving input distribution. This intuition is formal-
ized for discrete memoryless channels, following [14].
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
1612978-1-4244-2571-6/08/$25.00 ©2008 IEEE
A discrete memoryless channel (DMC) is characterized by
the input alphabet X, the output alphabet Y, and the transition
probability assignment Q
Y |X
(y|x). Furthermore, each output
letter y ∈Yhas an energy b(y), a nonnegative real number.
Channel inputs are described by random variables X
n
1
=
(X
1
,X
2
,...,X
n
) with distribution p
X
n
1
(x
n
1
); the correspond-
ing outputs are random variables Y
n
1
=(Y
1
,Y
2
,...,Y
n
) with
distribution p
Y
n
1
(y
n
1
). The average received energy is
E [b(Y
n
1
)] =
y
n
1
∈Y
n
b(y
n
1
)p(y
n
1
).
An optimization problem that precisely captures the tradeoff
between the two goals is as follows. Maximize information
rate under a minimum received power constraint. For each
n,thenth capacity-energy function C
n
(B) of the channel is
defined as
C
n
(B)= max
X
n
1
:E
[
b(Y
n
1
)
]
≥nB
I(X
n
1
; Y
n
1
).
An input vector X
n
1
is a test source; one that satisfies
E [b(Y
n
1
)] ≥ nB is B-admissible. The maximization is over
all n-dimensional B-admissible test sources. The set of B-
admissible p(x
n
1
) is a closed subset of R
|X|
n
and is bounded
since
p(x
n
1
)=1. Since the set is closed and bounded,
it is compact. Mutual information is a continuous function
of the input distribution and since continuous, real-valued
functions defined on compact subsets of metric spaces achieve
their supremums (see Theorem 5), defining the optimization
as a maximum is not problematic. The nth capacity-energy
functions are only defined for 0 ≤ B ≤ B
max
, where B
max
is the maximum element of b
T
Q; b is a column vector of the
b(y) and Q is Q
Y |X
.
The capacity-energy function of the channel is defined as
C(B)=sup
n
1
n
C
n
(B). (1)
A coding theorem can be proven that endows this informa-
tional definition with operational significance.
A code is a pair of mappings (f, g) where f maps a message
alphabet M to X and g maps Y to M. The rate of an n-
length block code is
1
n
log |M|.Ann-length block code with
maximum probability of error bounded by is an (n, )-code.
Definition 1: Given 0 ≤ <1, a non-negative number R
is an -achievable rate for the channel Q
Y |X
with constraint
(b, B) if for every δ>0 and every sufficiently large n there
exist (n, )-codes of rate exceeding R − δ for which b(y
n
1
) <
B implies g(y
n
1
) /∈M. R is an achievable rate if it is -
achievable for all 0 <<1. The supremum of achievable
rates is called the capacity of the channel under constraint
(b, B) and is denoted C
O
(B).
Theorem 1: C
O
(B)=C(B).
Proof: Follows by reversing the output constraint inequal-
ity in the solution to [15, P20 on p. 117]. See also [13].
III. PROPERTIESOFTHECAPACITY-ENERGY FUNCTION
The coding theorem provides operational significance to the
capacity-energy function. Some properties of this function may
also be developed.
It is immediate that C
n
(B) is non-increasing, since the fea-
sible set in the optimization becomes smaller as B increases.
The function is also concave ∩.
Theorem 2: C
n
(B) is a concave ∩ function of B for 0 ≤
B ≤ B
max
.
Proof: Let α
1
,α
2
≥ 0 with α
1
+ α
2
=1. The inequality
to be proven is that that for B
1
,B
2
≤ B
max
,
C
n
(α
1
B
1
+ α
2
B
2
) ≥ α
1
C
n
(B
1
)+α
2
C
n
(B
2
).
Let X
1
and X
2
be n-dimensional test sources distributed
according to p
1
(x
n
1
) and p
2
(x
n
1
) that achieve C
n
(B
1
) and
C
n
(B
2
) respectively. Denote the corresponding channel out-
puts as Y
1
and Y
2
. It follows that E[b(Y
i
)] ≥ nB
i
and
I(X
i
; Y
i
)=C
n
(B
i
) for i =1, 2.Define another source X
distributed according to p(x
n
1
)=α
1
p
1
(x
n
1
)+α
2
p
2
(x
n
1
) with
corresponding output Y . Then
E[b(Y )] = b
T
Qp = b
T
Q[α
1
p
1
+ α
2
p
2
] (2)
= α
1
b
T
Qp
1
+ α
2
b
T
Qp
2
= α
1
E[b(Y
1
)] + α
2
E[b(Y
2
)]
≥ n(α
1
B
1
+ α
2
B
2
),
where b and Q have been suitably extended. Thus, X is
(α
1
B
1
+ α
2
B
2
)-admissible. Now, by definition of C
n
(·),
I(X; Y ) ≤ C
n
(α
1
B
1
+ α
2
B
2
). However, since I(X; Y ) is
a concave ∩ function of the input probability,
I(X; Y ) ≥ α
1
I(X
1
; Y
1
)+α
2
I(X
2
; Y
2
)
= α
1
C
n
(B
1
)+α
2
C
n
(B
2
).
Linking the two inequalities yields the desired result:
C
n
(α
1
B
1
+ α
2
B
2
) ≥ I(X; Y ) ≥ α
1
C
n
(B
1
)+α
2
C
n
(B
2
).
It can also be shown that C
1
(B)=C(B).
Theorem 3: For any DMC, C
n
(B)=nC
1
(B) for all n =
1, 2,... and 0 ≤ nB ≤ nB
max
.
Proof: Let X =(X
1
,...,X
n
) be a B-admissible test
source with corresponding output Y that achieves C
n
(B),so
E[b(Y )] ≥ nB and I(X; Y )=C
n
(B). Since the channel is
memoryless, I(X; Y ) ≤
n
i=1
I(X
i
; Y
i
). Let B
i
= E[b(Y
i
)],
then
n
i=1
B
i
=
n
i=1
E[b(Y
i
)] = E[b(Y )] ≥ nB.Bythe
definition of C
1
(B
i
), I(X
i
; Y
i
) ≤ C
1
(B
i
). Now since C
1
(B)
is a concave ∩ function of B, by Jensen’s inequality,
1
n
n
i=1
C
1
(B
i
) ≤ C
1
1
n
n
i=1
B
i
= C
1
1
n
E[b(Y )]
.
But since
1
n
E[b(Y )] ≥ B and C
1
(B) is a non-increasing
function of B,
1
n
n
i=1
C
1
(B
i
) ≤ C
1
1
n
E[b(Y )]
≤ C
1
(B),
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
1613
that is,
n
i=1
C
1
(B
i
) ≤ nC
1
(B).
Combining yields C
n
(B) ≤ nC
1
(B).
For the reverse, let X be a random variable with correspond-
ing output Y that achieves C
1
(B).Thatis,E[b(Y )] ≥ B
and I(X; Y )=C
1
(B).NowletX
1
,X
2
,...,X
n
be i.i.d.
according to p(X) with outputs Y
1
,...,Y
n
. Then
E[b(Y
n
1
)] =
n
i=1
E[b(Y
i
)] ≥ nB.
Moreover by memorylessness,
I(X
n
1
; Y
n
1
)=
n
i=1
I(X
i
; Y
i
)=nC
1
(B).
Thus, C
n
(B) ≥ nC
1
(B).SinceC
n
(B) ≥ nC
1
(B) and
C
n
(B) ≤ nC
1
(B), C
n
(B)=nC
1
(B).
The theorem implies that single-letterization is valid: C(B)=
C
1
(B).
IV. T
HREE BINARY CHANNELS
Closed form expressions of the capacity-energy function
for some particular channels may provide insight. Here, three
binary channels with output alphabet energy function b(0) = 0
and b(1) = 1 are considered. Such an energy function
corresponds to discrete particles and packets, among other
commodities.
Consider a noiseless binary channel. The optimization prob-
lem is solved by the maximum entropy method, hence the
capacity-achieving input distribution is in Gibbsian form. It is
easy to show that the capacity-energy function is
C(B)=
log(2), 0 ≤ B ≤
1
2
h
2
(B),
1
2
≤ B ≤ 1,
where h
2
(·) is the binary entropy function. The capacity-
energy functions for other discrete noiseless channels are
similarly easy to work out using maximum entropy methods.
Consider a binary symmetric channel with crossover prob-
ability ω. It can be shown that the capacity-energy function
is
C(B)=
log(2) − h
2
(ω), 0 ≤ B ≤
1
2
h
2
(B) − h
2
(ω),
1
2
≤ B ≤ 1 − ω.
Recall that for the unconstrained problem, equiprobable inputs
are capacity-achieving, which yield output power
1
2
.ForB>
1
2
, the distribution must be perturbed so that the symbol 1 is
transmitted more frequently. The maximum power receivable
through this channel is 1 − ω, when 1 is always transmitted.
A third worked example is the Z-channel; the unconstrained
capacity expression and associated capacity-achieving input
distribution [16] are used. Consider a Z-channel with 1 to 0
crossover probability ω. The capacity-energy function is
C(B)=
⎧
⎨
⎩
log
1 − ω
1
1−ω
+ ω
ω
1−ω
, 0 ≤ B ≤ (1 − ω)π
∗
h
2
(B) −
B
1−ω
h
2
(ω), (1 − ω)π
∗
≤ B ≤ 1 − ω,
where
π
∗
=
ω
ω
1−ω
1+(1− ω)ω
ω
1−ω
.
A Z-channel models quantal synaptic failure [17] and other
“stochastic leaky pipes” where the commodity may be lost en
route.
V. A G
AUSS IAN CHANNEL
Attention now turns to discrete-time, continuous-alphabet,
memoryless channels. The coding theorem (Theorem 1) can
be extended in the usual way [18]. Continuous additive noise
systems have the interesting property that for the goal of
received power, noise power is actually helpful, whereas for
the goal of information, noise power is hurtful. In discrete
channels, such an interpretation is not obvious. When working
with real-valued alphabets, some sort of transmitter constraint
must be imposed so as to disallow arbitrarily powerful signals.
Hard amplitude constraints that model rail limitations in power
circuits are suitable. Assume that the channel transition pdf
Q(y|x) exists.
Rather than working with the output energy constraint
directly, it is convenient to think of the output energy function
b(y) as inducing costs on the input alphabet X:
ρ(x)=
Q(y|x)b(y)dy.
By construction, this cost function preserves the constraint:
E[ρ(X)] =
ρ(x)dF (x)=
dF (x)
Q(y|x)b(y)dy
=
Q(y|x)b(y)dF (x)dy = E[b(Y )],
where F (x) is the input distribution function. Basically, ρ(x)
is the expected output energy provided by input letter x.
Consider the AWGN channel N(0,σ
2
N
) and b(y)=y
2
. Then
ρ(x)=
∞
−∞
y
2
σ
N
√
2π
exp
−
(y−x)
2
2σ
2
N
dy = x
2
+ σ
2
N
,
that is, the output power is just the sum of the input power
and the noise power.
Since Theorem 3 extends directly to continuous alphabet
channels,
C(B)= sup
X:E[ρ(X)]≥B
I(X; Y ). (3)
Consider the AWGN channel, N(0,σ
2
N
), with input alpha-
bet X =[−A, A] ⊂ R, and energy function b(y)=y
2
.
Denote the capacity-energy function as C(B; A). Following
lockstep with Smith [19], [20], it is shown that the capacity-
energy achieving input distribution consists of a finite number
of mass points. Before proceeding, two optimization theorems
are quoted [20]:
Theorem 4: Let Ω be a convex metric space, and f and
g concave ∩ functionals on Ω to R; assume there exists an
x
1
∈ Ω such that g(x
1
) < 0 and let
D
sup
x∈Ω:g(x)≤0
f(x).
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
1614
If D
is finite, then there exists a constant λ ≥ 0 such that
D
=sup
x∈Ω
[f(x) − λg(x)].
Moreover if the supremum in the first equation is achieved by
x
0
and g(x
0
) ≤ 0, then the supremum is achieved by x
0
in
the second equation and λg(x
0
)=0.
Theorem 5: Let f be a continuous, weakly-differentiable,
strictly concave ∩ map from a compact, convex, topological
space Ω to R.Define
D sup
x∈Ω
f(x).
Then the following two properties hold:
1) D =maxf(x)=f(x
0
) for some unique x
0
∈ Ω,and
2) A necessary and sufficient condition for f(x
0
)=D is
for f
x
0
(x) ≤ 0 for all x ∈ Ω, where f
x
0
is the weak
derivative.
Let F
A
be the space of input probability distribution
functions having all points of increase on the finite interval
[−A, A].
Lemma1([19]): F
A
is convex and compact in the Levy
metric.
Since the channel is fixed, mutual information can be written
as a function of the input distribution, I(F ).
Lemma2([19]): Mutual information I : F
A
→ R is a
concave ∩, continuous, weakly differentiable functional.
Let us denote the input squared value under input distribu-
tion F as
σ
2
F
A
−A
x
2
dF (x).
Recall the energy constraint
B ≤ E[ρ(X)] = E[x
2
+ σ
2
N
]=σ
2
N
+ σ
2
F
,
which is equivalent to B − σ
2
N
− σ
2
F
≤ 0.Nowdefine the
functional J : F
A
→ R as
J(F ) B −σ
2
N
−
A
−A
x
2
dF (x).
Lemma 3: J is a concave ∩, continuous, weakly differen-
tiable functional.
Proof: Clearly J is linear in F (see (2) for basic argu-
ment). Moreover, J is bounded as B−σ
2
N
−A
2
≤ J ≤ B−σ
2
N
.
Since J is linear and bounded, it is concave ∩, continuous,
and weakly differentiable.
Returning to the optimization problem to be solved,
Theorem 6: There exists a constant λ ≥ 0 such that
C(B; A)= sup
F ∈F
A
[I(F ) − λJ(F )].
Proof: The result follows from Theorem 4 since I is a
concave ∩ functional (Lemma 2), J is a concave ∩ functional
(Lemma 3), since capacity is finite whenever A<∞ and
σ
2
N
> 0, and since there is obviously an F
1
∈F
A
such that
J(F
1
) < 0.
Theorem 7: There exists a unique capacity-energy achiev-
ing input X
0
with distribution function F
0
such that
C(B; A)= max
F ∈F
A
[I(F ) − λJ(F )] = I(F
0
) − λJ(F
0
).
Moreover, a necessary and sufficient condition for F
0
to
achieve capacity-energy is
I
F
0
(F ) − λJ
F
0
(F ) ≤ 0 for all F ∈F
A
. (4)
Proof: Since I and J are both concave ∩, continuous,
and weakly differentiable (Lemmas 2, 3), so is I −λJ.Since
F
A
is a convex, compact space (Lemma 1), Theorem 5 applies
and yields the result.
For our function I −λJ, the optimality condition (4) is,
A
−A
[i(x; F
0
)+λx
2
]dF (x) ≤ I(F
0
)+λ
x
2
dF
0
(x),
for all F ∈F
A
, where i is
i(x; F )=
Q(y|x)log
Q(y|x)
p(y; F )
dy
and is variously known as the marginal information density
[20], the Bayesian surprise [21], or without name [22, Eq. 1].
This follows since the mutual information weak derivative is
I
F
1
(F
2
)=
A
−A
i(x; F
1
)dF
2
(x) − I(F
1
)
and the energy weak derivative is J
F
1
(F
2
)=J(F
2
) −J(F
1
).
If
x
2
dF
0
(x) >B−σ
2
N
, then the moment constraint is trivial
and the constant λ is zero, thus the optimality condition can
be written as
A
−A
[i(x; F
0
)+λx
2
]dF (x) ≤ I(F
0
)+λ[B − σ
2
N
]. (5)
The optimality conditions (5) may be rejiggered to a con-
dition on the input alphabet.
Theorem 8: Let F
0
be an arbitrary distribution function in
F
A
satisfying the energy constraint. Let E
0
denote the points
of increase of F
0
on [−A, A]. Then F
0
is optimal if and only
if, for some λ ≥ 0,
i(x; F
0
) ≤ I(F
0
)+λ[B − σ
2
N
− x
2
] for all x ∈ [−A, A],
i(x; F
0
)=I(F
0
)+λ[B − σ
2
N
− x
2
] for all x ∈ E
0
.
Proof: If both conditions hold for some λ ≥ 0, F
0
must
be optimal and λ is the one from Theorem 7. This is because
integrating both sides of the conditions by an arbitrary F yield
satisfaction of condition (4).
For the converse, assume that F
0
is optimal but that the
inequality condition is not satisfied. Then there is some x
1
∈
[−A, A] and some λ ≥ 0 such that i(x
1
; F
0
) >I(F
0
)+λ[B −
σ
2
N
− x
2
1
].LetF
1
(x) be the unit step 1(x − x
1
) ∈F
A
;but
then
A
−A
[i(x; F
0
)+λx
2
]dF
1
(x)=i(x
1
; F
0
)+λx
2
1
>I(F
0
)+λ[B − σ
2
N
].
ISIT 2008, Toronto, Canada, July 6 - 11, 2008
1615
This violates (5), thus the inequality condition must be valid
with λ from Theorem 7.
Now assume that F
0
is optimal but that the equality con-
dition is not satisfied, i.e. there is set E
⊂ E
0
such that the
following is true.
E
dF
0
(x)=δ>0 and
E
0
−E
dF
0
(x)=1−δ,
i(x; F
0
)+λx
2
<I(F
0
)+λ[B − σ
2
N
] for all x ∈ E
,
i(x; F
0
)+λx
2
= I(F
0
)+λ[B − σ
2
N
] for all x ∈ E
0
− E
.
Then,
0=
[i(x; F
0
)+λx
2
]dF
0
(x) − I(F
0
) − λ[B −σ
2
N
]
=
E
0
[i(x; F
0
)+λx
2
]dF
0
(x) − I(F
0
) − λ[B −σ
2
N
]
<δ[I(F
0
)+λ(B − σ
2
N
)]+(1− δ)[I(F
0
)+λ(B − σ
2
N
)]
− I(F
0
) − λ[B −σ
2
N
]=0,
a contradiction. Thus the equality condition must be valid.
At a point like this, one might try to develop measure-
matching conditions like Gastpar et al. [22] for undetermined
b(·), but this path is not pursued here. To show that the input
distribution is supported on a finite number of mass points
requires Smith’s reductio ab absurdum argument (see [23] for
a slight correction).
Theorem 9: E
0
is a finite set of points.
The proof uses optimality conditions from Theorem 8 to derive
a contradiction using the analytic extension property of the
marginal entropy density h(x; F ),
h(x; F )=−
Q(y|x)logp(y; F )dy.
Since the capacity-energy achieving input distribution is a
pmf, a finite numerical optimization algorithm may be used
[24]. Consider the AWGN channel N(0, 1) and find the point
C(B =0;A =1.5). The capacity achieving input density is
p(x)=
1
2
δ(x +1.5) +
1
2
δ(x −1.5). The achieved rate is
C(0; 1.5) =
+1
−1
2/3
√
2π(1−y
2
)
e
−
(1−(4/9) tanh
−1
(y))
2
8/9
log(1+y)dy.
The achieved output power is E[Y
2
]=3.25.Infact,thisis
the maximum output power possible over this channel, since
E[Y
2
]=E[X
2
]+σ
2
N
,andE[X
2
] cannot be improved over
operating at the edges {−A, A}. Thus,
C(B;1.5) = C(0; 1.5), 0 ≤ B ≤ B
max
=3.25.
For this particular channel, there actually is no tradeoff be-
tween information and power: antipodal signaling should be
used all the time. This is not a general phenomenon, however.
This is not true for the same noise, but for say A ≥ 1.7 rather
than A =1.5 [19].
VI. C
ONCLUSION
Information is patterned matter-energy. In this work, the
fundamental tradeoff between the rate of transporting a com-
modity between one point and another and the rate of simulta-
neously transmitting information by modulating in that com-
modity has been studied. As extensions, one might consider a
“wideband regime” formulation [25], a multiterminal problem
where different users have different energy and information
requirements, or a deeper look into reversible decoding [11].
A
CKNOWLEDGEMENT
Comments from V. K. Goyal & S. K. Mitter are appreciated.
R
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