Inattentive cognitive radios: Cognition with
Communication Networks (ComNets) Research Group
Abstract—A “true” cognitive radio is a sophisticated
A cognitive radio was originally envisioned as a
devise that is aware of, can adapt to, and even learn
sophisticated devise capable of becoming aware of,
from its environment. To do so, such device would need
adapting to, and even learning from its environment,
to perform many complex (parallel) tasks in minuscule
to achieve the most efficient possible operation
amounts of time, and within tight size, weight, heat, and
[2][3]. In order to accomplish that, such device
energy limitations. In any sophisticated computing system,communication and computation limit each other. Here
would need to perform many complex (parallel)
we discuss a formal model that abstract an information-
tasks in minuscule amounts of time, and within
processing limited rational entity (human or not) seeking
extremely tight size, weight, heat, and energy lim-
to maximise a performance function that depends on both
itations. In a “true” cognitive radio, as in any so-
an action under its control, and and an exogenous input.
phisticated computing system, communication and
The entity finds and optimal probabilistic response such
computation limit each other [1, P.4]. Thus, the
that the Shannon-theoretic mutual information betweeninput and output does not exceed an information capacity
list of factors constraining the device’s operation
constraint. The possible application to secondary dynamic
must also include limited information processing
spectrum access receives special attention, but the frame-
work is of interest whenever the practical limits on acognitive radio information processing capacity cannot be
The problem of information processing con-
neglected. Rather than proposing a particular algorithm
straints affects other rational entities, including hu-
and showing numerical results, our aims are to (i) in-troduce an important practical problem affecting “true”
man beings. For instance, in an economic environ-
cognitive radios, (ii) introduce a body of existing scientific
ment, a rational consumer or investor may have at its
literature not widely known in engineering, and (iii) discuss
disposal much more information that it can “digest”
potential applications of this literature to cognitive radio
on time to make a critical decision. This includes not
only information available by subscription from “ex-pert” sources, but also freely available information
obtainable from news broadcasts, popular media,
or from government sources. New telecommuni-
cation technologies such as the world-wide web
and social networking websites have increased the
availability of both paid and free information. Thus,
a rational investor or consumer need to find “sub-
limited. These become intertwined. . . ”[1,
optimal” strategies that yield choices under limited
information-processing capability in finite time.
Below we focus on an approach to the prob-
V. Rodriguez is now with the Signal & System Theory Group
at the Universität Paderborn, in Germany.
lem of rational decision-making under information-
processing constraints proposed in [4]∗. It consists
and I(X, Y ) denotes mutual information, defined as
of a formal model that abstract a rational entity
(human or machine) that seeks to maximise a per-
formance function that depends on both an action
under its control, and and an exogenous input. The entity finds and optimal probabilistic response
such that the mutual information between inputand output does not exceed a Shannon-theoretic
Above, f can be interpreted as an algorithm –
or a physical device — that takes input x and
We focus on the “tracking” performance function.
probabilistically produces and output y. One wants
Here “tracking” simply means that the performance
to design f to maximise the expected value of
function is monotonic in the absolute value of
U , while obeying(1d), which can be interpreted as
the difference between the value of the exogenous
an “information processing” constraint. To produce
signal and the entity’s choice, thus, the entity want
y from x in given time, an information carrying
its choice to “track” (that is, to stay near) the
signal (or “data”) must travel at a maximum rate
value of the external signal. A simple example is
C. The “optimal” f must not require or imply a
(x−y)2 with x the exogenous and y the endogenous
mutual information between X and Y that exceeds
variable. A function such as this can — as discussed
the information that can actually be delivered in the
below — be of interest in two communication
scenarios: jamming, and secondary radio-spectrum
A probabilistic response is common in another
mathematical model: Game Theory [6], [7]. Cer-
We continue by introducing and discussing the
tain games are known to have a “mixed strategy”
formal Shannon-theoretic model from [4]. Then we
equilibrium in which it is optimal for one or more
discuss briefly common dynamic-spectrum access
players to randomly choose among available actions
scenarios, and describe how ideas from the model
according to a probability distribution over those
can be useful in these scenarios. Finally, we provide
actions that satisfy some optimality condition(s);
that is, the player derives an optimal probabilitydistribution that produces the player’s response.
The solution to problem (1) also answers the
following information-theoretic question:
Among all communication channels ofcertain capacity, C, which one is optimalgiven a performance function, U (x, y) of
the channel’s input x and its output y?
To immediately develop some intuition on prob-
lem (1), let x be associated with the energy sensed
by a receiver coming from its desired transmitter
over a period of interest. Then, U (x, y) = −(x −
X and gY are the probability density func-
tions (pdf) associated with an information source,
ky)2 could be of interest to a nearby “jammer”,
with y denoting its energy output over the same
period, and k an appropriate constant. Solving (1)
with this U will yield an f such that, on average,
the transmitter and the jammer’s signals reach thevictim receiver with similar strength, which would
be most disruptive to the victim. Below we shall
The author of [4], Prof. Christopher A. Sims, is co-recipient of
the 2011 Nobel Prize in Economic Science [5].
see that a nearby secondary user, in a dynamic
spectrum access scenario, can behave constructively
search for radio spectrum bands that are temporarily
by following a strategy that is similar but “opposite”
unused ( “spectrum holes”), an utilise such bands as
1) First-order optimising conditions: It can be
1) The “spectrum interweave” scenario: In a
shown that at any (x, y) such that both f (x, y) > 0
common dynamic spectrum access (DSA) scenario
and gX(x) > 0, the first-order optimising conditions
(“spectrum interweave”), a secondary user with a
stream of data to transmit periodically senses cer-tain communication channels and only uses those
expected to be unused by primary users in the
with µ(x) and λ ≥ 0 Lagrange-multipliers asso-
2) Major challenges for instantaneous sensing
ciated with, respectively, (1b) (the source marginal
based approaches: However, even with an ex-
tremely accurate sensing system, the “hidden termi-
2) General solution form: For λ > 0, and with
nal problem” remains a major challenge. Further-
more, the possibility that a primary user starts ac-cessing a channel — or, conversely, stops using it —
immediately after the sensing operation has ended
could never be ruled out. Thus, recent work has
The desired conditional pdf is f (y\x) — not
proposed secondary channel-usage policies based on
f (x\y) — because gX is given, and f (x, y) =
f (y\x)gX(x). Nevertheless, (2b) provides valuable
3) Statistical approach to secondary DSA: Be-
cause of the challenges of relying solely on instan-
3) Particular solution forms: (2b) reveals that a
taneous sensing information for secondary spectrum
quadratic U (e. g., (x − y)2) will yield a conditional
access, several recent proposals seek secondary ac-
f (x\y) with the “Gaussian” form, irrespective of
cess strategies based on statistics obtained from his-
torical information about the primary user’s channel
Furthermore, as λ approaches zero; that is, as
access. For instance, [10] focuses on a wireless LAN
capacity becomes very large, the conditional proba-
scenario — in which the hidden terminal node is not
bility density ordinate becomes very large. Since the
an issue — and utilises empirical information to
total area under the density must equal one, a very
predict idle periods between bursty transmissions.
large “peak” implies that all the probability mass
On a non-contiguous OFDM scenario, [11] pro-
must be allocated about a single point. Thus, when
poses portfolio theory [12], for a secondary user
capacity is very large, the optimal relation between
to determine on the basis of historical information
inputs and outputs (that is, the underlying algorithm)
on which channels to “invest” communication re-
sources. Then, [13] amends [11]’s scheme to also
For given input x1, the optimal deterministic y is,
consider the interest of the primary user, while[14]
of course, the one that maximises φ(y) := U (x1, y).
C. Information-capacity limited secondary access
As conceptualised by [2], and discussed by [3],
As mentioned before in the discussion of the
a cognitive radio terminal is aware of, can adapt to,
jammer scenario, one can view the secondary user
and even learn from its environment, to achieve the
as the jammer’s antithesis. Thus, we can also apply
most efficient possible operation. However, most of
the framework introduced in section II to obtain a
the literature has focused in a much narrower con-
statistical secondary user access strategy. Specifi-
cept of “spectrum agile” radios[8]. These radios can
cally, we can find a solution to problem (1) which
the secondary user can utilise to randomly (but opti-
considering its information processing limitations.
mally) choose a transmission power. Of course, we
We have presented, motivated, and interpreted a
must specify an appropriate performance function,
formal information-theoretic framework from [4],
U , along with the exogenous parameters gX and C.
which is relevant to this scenario, and shown gen-
1) Single-channel scenario: If the radio is mon-
eral solution forms. Hundreds of publications have
itoring a single channel. The input x can be asso-
followed [4] nearly all devoted to macroeconomic
ciated with the energy sensed by a common energy
issues, reporting analysis, numerical results and
detector, which at least in the simplest case follows
empirical data. It seems sensible to examine this
a χ2 distribution[15]. The quadratic performance
vast literature, to identify those results that are,
function U (x, y) = (kx − y)2, with k a suitable
after suitable adaptation, potentially most useful to
If the secondary radio determines its energy usage
The ideal cognitive radios seem far from practical
by following the output of the optimal channel
realisation, and most of the scientific literature has
implied by f , it will tend to emit a high power
focused on “spectrum agile” radios that can find
when the input energy reading is low, and a low
“spectrum holes” for secondary dynamic spectrum
power when sensed energy is high. Notice that if the
access. Thus, we have also considered the possible
primary receiver uses successive interference can-
application of ideas from this framework to the
cellation, high interference (relative to the desired
signal strength) is also favourable.
However, just as the “true” cognitive radio is
Above model can be improved, by defining x as
much more than a mere “spectrum agile” radio,
the (predicted) actual energy that would be sensed
the issues discussed herein reach much farther than
if the sensing was done during channel usage (to
its application to the mundane “spectrum holes”
account for the fact that the primary user can en-
scenario. The central issue is that a “true” cognitive
ter/exit the channel after the sensing has just ended.
radio will “encounter both a computation limit and a
In this case, the pdf of x would be a function of that
communication limit”[1, P.4]. Thus, as this radio is
of the output of the energy sensor, which would be
asked to perform ever more sophisticated (parallel)
tasks within minuscule amounts of time, and within
2) Multiple secondary channels: If the secondary
the ever present size, weight, heat, and perhaps most
user has a “long queue” of data to transmit, then
importantly, energy limitations, it must consider its
it can use as many secondary channels as it can
information processing limits when devising opti-
find. Thus, it can apply the previous analysis on a
channel by channel basis, to transfer as much data
The previously discussed model does provide an
as possible. However, some consideration must be
insight that may serve as a guiding principle for
given to the power constraint, which must be shared
designers of “true” cognitive radios: This radio will
often find itself overwhelmed by the sheer amount
The secondary user sorts the target channels by
of information it needs to process in order to make
channel state, and determines the energy to be used
an “optimal” decision in minuscule time. Rather
in each channel sequentially, starting with the best
than finding the “true” (deterministic) optimum, the
channel. Thus, after choosing the power to be used
radio must content itself with making a decision at
on the best channel (as discussed above), the radio
random. At random, yes, but not haphazardly (as by
updates its power budget, and then choose the power
choosing uniformly over the set of actions). Rather,
for the second-best channel (as discussed above),
a decision judiciously random, through an optimised
and again update its power budget, and so on, until
probabilistic rule. This rule yields decisions that on
average optimise the desired performance function,while utilising not more information than what the
radio can actually process in the available decision
We have focused on a scenario in which a cog-
nitive radio must make an optimal choice while
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