Brain empowered

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005 201
Cognitive Radio: Brain-Empowered
Wireless Communications
Simon Haykin, Life Fellow, IEEE
Invited Paper
Abstract—Cognitive radio is viewed as a novel approach for im-
proving the utilization of a precious natural resource: the radio
electromagnetic spectrum.
The cognitive radio, built on a software-defined radio, is de-
fined as an intelligent wireless communication system that is
aware of its environment and uses the methodology of under-
standing-by-building to learn from the environment and adapt
to statistical variations in the input stimuli, with two primary
objectives in mind:
• highly reliable communication whenever and wherever
needed;
• efficient utilization of the radio spectrum.
Following the discussion of interference temperature as a new
metric for the quantification and management of interference, the
paper addresses three fundamental cognitive tasks.
1) Radio-scene analysis.
2) Channel-state estimation and predictive modeling.
3) Transmit-power control and dynamic spectrum manage-
ment.
This paper also discusses the emergent behavior of cognitive radio.
Index Terms—Awareness, channel-state estimation and predic-
tive modeling, cognition, competition and cooperation, emergent
behavior, interference temperature, machine learning, radio-scene
analysis, rate feedback, spectrum analysis, spectrum holes, spec-
trum management, stochastic games, transmit-power control,
water filling.
I. INTRODUCTION
A. Background
THE electromagnetic radio spectrum is a natural resource,
the use of which by transmitters and receivers is licensed
by governments. In November 2002, the Federal Communica-
tions Commission (FCC) published a report prepared by the
Spectrum-Policy Task Force, aimed at improving the way in
which this precious resource is managed in the United States [1].
The task force was made up of a team of high-level, multidis-
ciplinary professional FCC staff—economists, engineers, and
attorneys—from across the commission’s bureaus and offices.
Among the task force major findings and recommendations, the
second finding on page 3 of the report is rather revealing in the
context of spectrum utilization:
Manuscript received February 1, 2004; revised June 4, 2004.
The author is with Adaptive Systems Laboratory, McMaster University,
Hamilton, ON L8S 4K1, Canada (e-mail: haykin@mcmaster.ca).
Digital Object Identifier 10.1109/JSAC.2004.839380
“In many bands, spectrum access is a more signifi-
cant problem than physical scarcity of spectrum, in large
part due to legacy command-and-control regulation that
limits the ability of potential spectrum users to obtain such
access.”
Indeed, if we were to scan portions of the radio spectrum in-
cluding the revenue-rich urban areas, we would find that [2]–[4]:
1) some frequency bands in the spectrum are largely unoc-
cupied most of the time;
2) some other frequency bands are only partially occupied;
3) the remaining frequency bands are heavily used.
The underutilization of the electromagnetic spectrum leads us
to think in terms of spectrum holes, for which we offer the fol-
lowing definition [2]:
A spectrum hole is a band of frequencies assigned to a pri-
mary user, but, at a particular time and specific geographic lo-
cation, the band is not being utilized by that user.
Spectrum utilization can be improved significantly by making
it possible for a secondary user (who is not being serviced) to
access a spectrum hole unoccupied by the primary user at the
right location and the time in question. Cognitive radio [5], [6],
inclusive of software-defined radio, has been proposed as the
means to promote the efficient use of the spectrum by exploiting
the existence of spectrum holes.
But, first and foremost, what do we mean by cognitive radio?
Before responding to this question, it is in order that we address
the meaning of the related term “cognition.” According to the
Encyclopedia of Computer Science [7], we have a three-point
computational view of cognition.
1) Mental states and processes intervene between input
stimuli and output responses.
2) The mental states and processes are described by
algorithms.
3) The mental states and processes lend themselves to scien-
tific investigations.
Moreover, we may infer from Pfeifer and Scheier [8] that the
interdisciplinary study of cognition is concerned with exploring
general principles of intelligence through a synthetic method-
ology termed learning by understanding. Putting these ideas to-
gether and bearing in mind that cognitive radio is aimed at im-
proved utilization of the radio spectrum, we offer the following
definition for cognitive radio.
Cognitive radio is an intelligent wireless communication
system that is aware of its surrounding environment (i.e., outside
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202 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
world), and uses the methodology of understanding-by-building
to learn from the environment and adapt its internal states to
statistical variations in the incoming RF stimuli by making
corresponding changes in certain operating parameters (e.g.,
transmit-power, carrier-frequency, and modulation strategy) in
real-time, with two primary objectives in mind:
• highly reliable communications whenever and wherever
needed;
• efficient utilization of the radio spectrum.
Six key words stand out in this definition: awareness,1 in-
telligence, learning, adaptivity, reliability, and efficiency.
Implementation of this far-reaching combination of capabilities
is indeed feasible today, thanks to the spectacular advances
in digital signal processing, networking, machine learning,
computer software, and computer hardware.
In addition to the cognitive capabilities just mentioned, a cog-
nitive radio is also endowed with reconfigurability.2 This latter
capability is provided by a platform known as software-defined
radio, upon which a cognitive radio is built. Software-defined
radio (SDR) is a practical reality today, thanks to the conver-
gence of two key technologies: digital radio, and computer soft-
ware [11]–[13].
B. Cognitive Tasks: An Overview
For reconfigurability, a cognitive radio looks naturally to soft-
ware-defined radio to perform this task. For other tasks of a
cognitive kind, the cognitive radio looks to signal-processing
and machine-learning procedures for their implementation. The
cognitive process starts with the passive sensing of RF stimuli
and culminates with action.
In this paper, we focus on three on-line cognitive tasks3:
1) Radio-scene analysis, which encompasses the following:
• estimation of interference temperature of the radio
environment;
• detection of spectrum holes.
2) Channel identification, which encompasses the following:
• estimation of channel-state information (CSI);
• prediction of channel capacity for use by the
transmitter
3) Transmit-power control and dynamic spectrum manage-
ment.
Tasks 1) and 2) are carried out in the receiver, and task 3) is
carried out in the transmitter. Through interaction with the RF
1According to Fette [10], the awareness capability of cognitive radio em-
bodies awareness with respect to the transmitted waveform, RF spectrum,
communication network, geography, locally available services, user needs,
language, situation, and security policy.
2Reconfigurability provides the basis for the following features [13].
• Adaptation of the radio interface so as to accommodate variations in the
development of new interface standards.
• Incorporation of new applications and services as they emerge.
• Incorporation of updates in software technology.
• Exploitation of flexible heterogeneous services provided by radio net-
works.
3Cognition also includes language and communication [9]. The cognitive
radio’s language is a set of signs and symbols that permits different internal
constituents of the radio to communicate with each other. The cognitive task of
language understanding is discussed in Mitola’s Ph.D. dissertation [6]; for some
further notes, see Section XII-A.
Fig. 1. Basic cognitive cycle. (The figure focuses on three fundamental
cognitive tasks.)
environment, these three tasks form a cognitive cycle,4 which is
pictured in its most basic form in Fig. 1.
From this brief discussion, it is apparent that the cognitive
module in the transmitter must work in a harmonious manner
with the cognitive modules in the receiver. In order to maintain
this harmony between the cognitive radio’s transmitter and re-
ceiver at all times, we need a feedback channel connecting the
receiver to the transmitter. Through the feedback channel, the
receiver is enabled to convey information on the performance
of the forward link to the transmitter. The cognitive radio is,
therefore, by necessity, an example of a feedback communica-
tion system.
One other comment is in order. A broadly defined cognitive
radio technology accommodates a scale of differing degrees of
cognition. At one end of the scale, the user may simply pick a
spectrum hole and build its cognitive cycle around that hole.
At the other end of the scale, the user may employ multiple
implementation technologies to build its cognitive cycle around
a wideband spectrum hole or set of narrowband spectrum holes
to provide the best expected performance in terms of spectrum
management and transmit-power control, and do so in the most
highly secure manner possible.
C. Historical Notes
Unlike conventional radio, the history of which goes back to
the pioneering work of Guglielmo Marconi in December 1901,
the development of cognitive radio is still at a conceptual stage.
Nevertheless, as we look to the future, we see that cognitive
radio has the potential for making a significant difference to the
way in which the radio spectrum can be accessed with improved
utilization of the spectrum as a primary objective. Indeed, given
4The idea of a cognitive cycle for cognitive radio was first described by Mitola
in [5]; the picture depicted in that reference is more detailed than that of Fig. 1.
The cognitive cycle of Fig. 1 pertains to a one-way communication path, with
the transmitter and receiver located in two different places. In a two-way com-
munication scenario, we have a transceiver (i.e., combination of transmitter and
receiver) at each end of the communication path; all the cognitive functions em-
bodied in the cognitive cycle of Fig. 1 are built into each of the two transceivers.

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 203
its potential, cognitive radio can be justifiably described as a
“disruptive, but unobtrusive technology.”
The term “cognitive radio” was coined by Joseph Mitola.5 In
an article published in 1999, Mitola described how a cognitive
radio could enhance the flexibility of personal wireless services
through a new language called the radio knowledge represen-
tation language (RKRL) [5]. The idea of RKRL was expanded
further in Mitola’s own doctoral dissertation, which was pre-
sented at the Royal Institute of Technology, Sweden, in May
2000 [6]. This dissertation presents a conceptual overview of
cognitive radio as an exciting multidisciplinary subject.
As noted earlier, the FCC published a report in 2002, which
was aimed at the changes in technology and the profound impact
that those changes would have on spectrum policy [1]. That re-
port set the stage for a workshop on cognitive radio, which was
held in Washington, DC, May 2003. The papers and reports that
were presented at that Workshop are available at the Web site
listed under [14]. This Workshop was followed by a Confer-
ence on Cognitive Radios, which was held in Las Vegas, NV, in
March 2004 [15].
D. Purpose of this Paper
In a short section entitled “Research Issues” at the end of his
Doctoral Dissertation, Mitola goes on to say the following [6]:
“‘How do cognitive radios learn best? merits attention.’
The exploration of learning in cognitive radio includes the
internal tuning of parameters and the external structuring
of the environment to enhance machine learning. Since
many aspects of wireless networks are artificial, they may
be adjusted to enhance machine learning. This dissertation
did not attempt to answer these questions, but it frames
them for future research.”
The primary purpose of this paper is to build on Mitola’s vi-
sionary dissertation by presenting detailed expositions of signal-
processing and adaptive procedures that lie at the heart of cog-
nitive radio.
E. Organization of this Paper
The remaining sections of the paper are organized as follows.
• Sections II–V address the task of radio-scene analysis,
with Section II introducing the notion of interference tem-
perature as a new metric for the quantification and man-
agement of interference in a radio environment. Section III
reviews nonparametric spectrum analysis with emphasis
on the multitaper method for spectral estimation, followed
by Section IV on application of the multitaper method
to noise-floor estimation. Section V discusses the related
issue of spectrum-hole detection.
• Section VI discusses channel-state estimation and predic-
tive modeling.
• Sections VII–X are devoted to multiuser cognitive
radio networks, with Sections VII and VIII reviewing
stochastic games and highlighting the processes of co-
operation and competition that characterize multiuser
networks. Section IX discusses an iterative water-filling
(WF) procedure for distributed transmit-power control.
5It is noteworthy that the term “software-defined radio” was also coined by
Mitola.
Section X discusses the issues that arise in dynamic
spectrum management, which is performed hand-in-hand
with transmit-power control.
• Section XI discusses the related issue of emergent be-
havior that could arise in a cognitive radio environment.
• Section XII concludes the paper and highlights the re-
search issues that merit attention in the future development
of cognitive radio.
II. INTERFERENCE TEMPERATURE
Currently, the radio environment is transmitter-centric, in the
sense that the transmitted power is designed to approach a pre-
scribed noise floor at a certain distance from the transmitter.
However, it is possible for the RF noise floor to rise due to
the unpredictable appearance of new sources of interference,
thereby causing a progressive degradation of the signal cov-
erage. To guard against such a possibility, the FCC Spectrum
Policy Task Force [1] has recommended a paradigm shift in in-
terference assessment, that is, a shift away from largely fixed op-
erations in the transmitter and toward real-time interactions be-
tween the transmitter and receiver in an adaptive manner. The
recommendation is based on a new metric called the interfer-
ence temperature,6 which is intended to quantify and manage
the sources of interference in a radio environment. Moreover,
the specification of an interference-temperature limit provides
a “worst case” characterization of the RF environment in a par-
ticular frequency band and at a particular geographic location,
where the receiver could be expected to operate satisfactorily.
The recommendation is made with two key benefits in mind.7
1) The interference temperature at a receiving antenna pro-
vides an accurate measure for the acceptable level of RF
interference in the frequency band of interest; any trans-
mission in that band is considered to be “harmful” if it
would increase the noise floor above the interference-tem-
perature limit.
2) Given a particular frequency band in which the interfer-
ence temperature is not exceeded, that band could be made
available to unserviced users; the interference-tempera-
ture limit would then serve as a “cap” placed on potential
RF energy that could be introduced into that band.
For obvious reasons, regulatory agencies would be responsible
for setting the interference-temperature limit, bearing in mind
the condition of the RF environment that exists in the frequency
band under consideration.
What about the unit for interference temperature? Following
the well-known definition of equivalent noise temperature of a
receiver [20], we may state that the interference temperature is
measured in degrees Kelvin. Moreover, the interference-tem-
perature limit multiplied by Boltzmann’s constant
6We may also introduce the concept of interference temperature density,
which is defined as the interference temperature per capture area of the
receiving antenna [16]. The interference temperature density could be made
independent of the receiving antenna characteristics through the use of a
reference antenna.
In a historical context, the notion of radio noise temperature is discussed in the
literature in the context of microwave background, and also used in the study of
solar radio bursts [17], [18].
7Inference temperature has aroused controversy. In [19], the National Asso-
ciation for Amateur Radio presents a critique of this metric.

10 joules per degree Kelvin yields the cor-
responding upper limit on permissible power spectral density
in a frequency band of interest, and that density is measured in
joules per second or, equivalently, watts per hertz.
III. RADIO-SCENE ANALYSIS: SPACE–TIME PROCESSING
CONSIDERATIONS
The stimuli generated by radio emitters are nonstationary
spatio–temporal signals in that their statistics depend on both
time and space. Correspondingly, the passive task of radio-scene
analysis involves space–time processing, which encompasses
the following operations.
1) Two adaptive, spectrally related functions, namely, es-
timation of the interference temperature, and detection
of spectrum holes, both of which are performed at the
receiving end of the system. (Information obtained on
these two functions, sent to the transmitter via a feed-
back channel, is needed by the transmitter to carry out
the joint function of active transmit-power control and dy-
namic spectrum management.)
2) Adaptive beamforming for interference control, which is
performed at both the transmitting and receiving ends of
the system in a complementary fashion.
A. Time-Frequency Distribution
Unfortunately, the statistical analysis of nonstationary sig-
nals, exemplified by RF stimuli, has had a rather mixed history.
Although the general second-order theory of nonstationary sig-
nals was published during the 1940s by Loève [21], [22], it has
not been applied nearly as extensively as the theory of stationary
processes published only slightly previously and independently
by Wiener and Kolmogorov.
To account for the nonstationary behavior of a signal, we have
to include time (implicitly or explicitly) in a statistical descrip-
tion of the signal. Given the desirability of working in the fre-
quency domain for well-established reasons, we may include
the effect of time by adopting a time-frequency distribution of
the signal. During the last 25 years, many papers have been pub-
lished on various estimates of time-frequency distributions; see,
for example, [23] and the references cited therein. In most of
this work, however, the signal is assumed to be deterministic.
In addition, many of the proposed estimators of time-frequency
distributions are constrained to match time and frequency mar-
ginal density conditions. However, the frequency marginal dis-
tribution is, except for a scaling factor, just the periodogram
of the signal. At least since the early work of Rayleigh [24],
it has been known that the periodogram is a badly biased and
inconsistent estimator of the power spectrum. We, therefore, do
not consider matching marginal distributions to be important.
Rather, we advocate a stochastic approach to time-frequency
distributions which is rooted in the works of Loève [21], [22]
and Thomson [25], [26].
For the stochastic approach, we may proceed in one of two
ways.
1) The incoming RF stimuli are sectioned into a continuous
sequence of successive bursts, with each burst being short
enough to justify pseudostationarity and yet long enough
to produce an accurate spectral estimate.
2) Time and frequency are considered jointly under the
Loève transform.
Approach 1) is well suited for wireless communications. In any
event, we need a nonparametric method for spectral estimation
that is both accurate and principled. For reasons that will be-
come apparent in what follows, multitaper spectral estimation
is considered to be the method of choice.
B. Multitaper Spectral Estimation
In the spectral estimation literature, it is well known that
the estimation problem is made difficult by the bias-variance
dilemma, which encompasses the interplay between two points.
• Bias of the power-spectrum estimate of a time series, due
to the sidelobe leakage phenomenon, is reduced by ta-
pering (i.e., windowing) the time series.
• The cost incurred by this improvement is an increase in
variance of the estimate, which is due to the loss of infor-
mation resulting from a reduction in the effective sample
size.
How can we resolve this dilemma by mitigating the loss of infor-
mation due to tapering? The answer to this fundamental ques-
tion lies in the principled use of multiple orthonormal tapers
(windows),8 an idea that was first applied to spectral estimation
by Thomson [26]. The idea is embodied in the multitaper spec-
tral estimation procedure.9 Specifically, the procedure linearly
expands the part of the time series in a fixed bandwidth
to (centered on some frequency ) in a special family of
sequences known as the Slepian sequences.10 The remarkable
property of Slepian sequences is that their Fourier transforms
have the maximal energy concentration in the bandwidth
to under a finite sample-size constraint. This property,
in turn, allows us to trade spectral resolution for improved spec-
tral characteristics, namely, reduced variance of the spectral es-
timate without compromising the bias of the estimate.
Given a time series , the multitaper spectral estima-
tion procedure determines two things.
1) An orthonormal sequence of Slepian tapers denoted by
.
8Another method for addressing the bias-variance dilemma involves dividing
the time series into a set of possible overlapping segments, computing a pe-
riodogram for each tapered (windowed) segment, and then averaging the re-
sulting set of power spectral estimates, which is what is done in Welch’s method
[27]. However, unlike the principled use of multiple orthogonal tapers, Welch’s
method is rather ad hoc in its formulation.
9In the original paper by Thomson [36], the multitaper spectral estimation
procedure is referred to as the method of multiple windows. For detailed de-
scriptions of this procedure, see [26], [28] and the book by Percival and Walden
[29, Ch. 7].
The Signal Processing Toolbox [30] includes the MATLAB code for Thomson’s
multitaper method and other nonparametric, as well as parametric methods of
spectral estimation.
10The Slepian sequences are also known as discrete prolate spheroidal se-
quences. For detailed treatment of these sequences, see the original paper by
Slepian [31], the appendix to Thomson’s paper [26], and the book by Percival
and Walden [29, Ch. 8].

2) The associated eigenspectra defined by the Fourier
transforms
(1)
The energy distributions of the eigenspectra are concentrated
inside a resolution bandwidth, denoted by . The time-band-
width product
(2)
defines the degrees of freedom available for controlling the vari-
ance of the spectral estimator. The choice of parameters and
provides a tradeoff between spectral resolution and variance.11
A natural spectral estimate, based on the first few eigenspectra
that exhibit the least sidelobe leakage, is given by
(3)
where is the eigenvalue associated with the th eigenspec-
trum. Two points are noteworthy.
1) The denominator in (3) makes the estimate
unbiased.
2) Provided that we choose , then the eigen-
value is close to unity, in which case
Moreover, the spectral estimate can be improved by the
use of “adaptive weighting,” which is designed to minimize the
presence of broadband leakage in the spectrum [26], [28].
It is important to note that in [33], Stoica and Sundin show
that the multitaper spectral estimation procedure can be inter-
preted as an “approximation” of the maximum-likelihood power
spectrum estimator. Moreover, they show that for wideband
signals, the multitaper spectral estimation procedure is “nearly
optimal” in the sense that it almost achieves the Cramér–Rao
bound for a nonparametric spectral estimator. Most important,
unlike the maximum-likelihood spectral estimator, the multi-
taper spectral estimator is computationally feasible.
C. Adaptive Beamforming for Interference Control
Spectral estimation accounts for the temporal characteristic
of RF stimuli. To account for the spatial characteristic of RF
stimuli, we resort to the use of adaptive beamforming.12 The
motivation for so doing is interference control at the cognitive
radio receiver, which is achieved in two stages.
11For an estimate of the variance of a multitaper spectral estimator, we may
use a resampling technique called Jackknifing [32]. The technique bypasses
the need for finding an exact analytic expression for the probability distribu-
tion of the spectral estimator, which is impractical because time-series data
(e.g., stimuli produced by the radio environment) are typically nonstationary,
non-Gaussian, and frequently contain outliers. Moreover, it may be argued that
the multitaper spectral estimation procedure results in nearly uncorrelated coef-
ficients, which provides further justification for the use of jackknifing.
12Adaptive beamformers, also referred to as adaptive antennas or smart an-
tennas, are discussed in the books [34]–[37].
In the first stage of interference control, the transmitter ex-
ploits geographic awareness to focus its radiation pattern along
the direction of the receiver. Two beneficial effects result from
beamforming in the transmitter.
1) At the transmitter, power is preserved by avoiding radia-
tion of the transmitted signal in all directions.
2) Assuming that every cognitive radio transmitter follows a
strategy similar to that summarized under point 1), inter-
ference at the receiver due to the actions of other trans-
mitters is minimized.
At the receiver, beamforming is performed for the adaptive
cancellation of residual interference from known transmitters,
as well as interference produced by other unknown transmit-
ters. For this purpose, we may use a robustified version of the
generalized sidelobe canceller [38], [39], which is designed to
protect the target RF signal and place nulls along the directions
of interferers.
IV. INTERFERENCE-TEMPERATURE ESTIMATION
With cognitive radio being receiver-centric, it is necessary
that the receiver be provided with a reliable spectral estimate of
the interference temperature. We may satisfy this requirement
by doing two things.
1) Use the multitaper method to estimate the power spectrum
of the interference temperature due to the cumulative dis-
tribution of both internal sources of noise and external
sources of RF energy. In light of the findings reported in
[33], this estimate is near-optimal.
2) Use a large number of sensors to properly “sniff” the RF
environment, wherever it is feasible. The large number of
sensors is needed to account for the spatial variation of the
RF stimuli from one location to another.
The issue of multiple-sensor permissibility is raised under
point 2) because of the diverse ways in which wireless commu-
nications could be deployed. For example, in an indoor building
environment and communication between one building and
another, it is feasible to use multiple sensors (i.e., antennas)
placed at strategic locations in order to improve the reliability
of interference-temperature estimation. On the other hand, in
the case of an ordinary mobile unit with limited real estate, the
interference-temperature estimation may have to be confined to
a single sensor. In what follows, we describe the multiple-sensor
scenario, recognizing that it includes the single-sensor scenario
as a special case.
Let denote the total number of sensors deployed in the RF
environment. Let denote the th eigenspectrum com-
puted by the th sensor. We may then construct the -by-
spatio–temporal complex-valued matrix
...
...
(4)
where each column is produced using stimuli sensed at a dif-
ferent gridpoint, each row is computed using a different Slepian

taper, and the represent variable weights accounting
for relative areas of gridpoints, as described in [40].
Each entry in the matrix is produced by two contribu-
tions, one due to additive internal noise in the sensor and the
other due to the incoming RF stimuli. Insofar as radio-scene
analysis is concerned, however, the primary contribution of in-
terest is that due to RF stimuli. An effective tool for denoising
is the singular value decomposition (SVD), the application of
which to the matrix yields the decomposition [41]
(5)
where is the th singular value of matrix ,
is the associated left singular vector, and is the associ-
ated right singular vector; the superscript denotes Hermitian
transposition. In analogy with principal components analysis,
the decomposition of (5) may be viewed as one of principal
modulations produced by the external RF stimuli. According to
(5), the singular value scales the th principal modulation
of matrix .
Forming the -by- matrix product , we find
that the entries on the main diagonal of this product, except for
a scaling factor, represent the eigenspectrum due to each of the
Slepian tapers, spatially averaged over the sensors. Let the
singular values of matrix be ordered
. The th eigenvalue of is
. We may then make the following statements.
1) The largest eigenvalue, namely, , provides an
estimate of the interference temperature, except for a con-
stant. This estimate may be improved by using a linear
combination of the largest two or three eigenvalues:
, ,1,2.
2) The left singular vectors, namely, the , give the spa-
tial distribution of the interferers.
3) The right singular vectors, namely, the , give the
multitaper coefficients for the interferers’ waveform.
To summarize, multitaper spectral estimation combined with
singular value decomposition provides an effective procedure
for estimating the power spectrum of the noise floor in an RF
environment. A cautionary note, however, is in order: the proce-
dure is computationally intensive but nevertheless manageable.
In particular, the computation of eigenspectra followed by sin-
gular value decomposition would have to be repeated at each
frequency of interest.
V. DETECTION OF SPECTRUM HOLES
In passively sensing the radio scene and thereby estimating
the power spectra of incoming RF stimuli, we have a basis for
classifying the spectra into three broadly defined types, as sum-
marized here.
1) Black spaces, which are occupied by high-power “local”
interferers some of the time.
2) Grey spaces, which are partially occupied by low-power
interferers.
3) White spaces, which are free of RF interferers except for
ambient noise, made up of natural and artificial forms of
noise, namely:
• broadband thermal noise produced by external phys-
ical phenomena such as solar radiation;
• transient reflections from lightening, plasma (fluo-
rescent) lights, and aircraft;
• impulsive noise produced by ignitions, commuta-
tors, and microwave appliances;
• thermal noise due to internal spontaneous fluctu-
ations of electrons at the front end of individual
receivers.
White spaces (for sure) and grey spaces (to a lesser extent) are
obvious candidates for use by unserviced operators. Of course,
black spaces are to be avoided whenever and wherever the RF
emitters residing in them are switched ON. However, when at a
particular geographic location those emitters are switched OFF
and the black spaces assume the new role of “spectrum holes,”
cognitive radio provides the opportunity for creating significant
“white spaces” by invoking its dynamic-coordination capability
for spectrum sharing, on which more is said in Section X.
A. Detection Statistics
From these notes, it is apparent that a reliable strategy for
the detection of spectrum holes is of paramount importance to
the design and practical implementation of cognitive radio sys-
tems. Moreover, in light of the material presented in Section IV,
the multitaper method combined with singular-value decom-
position, hereafter referred to as the MTM-SVD method,13 pro-
vides the method of choice for solving this detection problem
by virtue of its accuracy and near-optimality.
By repeated application of the MTM-SVD method to the RF
stimuli at a particular geographic location and from one burst
of operation to the next, a time-frequency distribution of that
location is computed. The dimension of time is quantized into
discrete intervals separated by the burst duration. The dimension
of frequency is also quantized into discrete intervals separated
by resolution bandwidth of the multitaper spectral estimation
procedure.
Let denote the number of largest eigenvalues considered to
play important roles in estimating the interference temperature,
with denoting the th largest eigenvalue produced by
the burst of RF stimuli received at time . Let denote the
number of frequency resolutions of width , which occupy
the black space or gray space under scrutiny. Then, setting the
discrete frequency
where denotes the lowest end of a black/grey space, we
may define the decision statistic for detecting the transition from
such a space into a white space (i.e., spectrum hole) as
(6)
13Mann and Park [40] discuss the application of the MTM-SVD method to the
detection of oscillatory spatial-temporal signals in climate studies. They show
that this new methodology avoids the weaknesses of traditional signal-detection
techniques. In particular, the methodology permits a faithful reconstruction of
spatio–temporal patterns of narrowband signals in the presence of additive spa-
tially correlated noise.

Spectrum-hole detection is declared if two conditions are
satisfied.
1) The reduction in from one burst to the next exceeds
a prescribed threshold on several successive bursts.
2) Once the transition is completed, assumes minor
fluctuations typical of ambient noise.
For a more refined approach, we may use an adaptive filter
for change detection [42], [43]. Except for a scaling factor, the
decision statistic provides an estimate of the interference
temperature as it evolves with time discretized in accordance
with the burst duration. The adaptive filter is designed to pro-
duce a model for the time evolution of when the RF emitter
responsible for the black space is switched ON. Assuming that
the filter is provided with a sufficient number of adjustable pa-
rameters and the adaptive process makes it possible for the filter
to produce a good fit to the evolution of with time , the se-
quence of residuals produced by the model would ideally be the
sample function of a white noise process. Of course, this state of
affairs would hold only when the emitter in question is switched
ON. Once the emitter is switched OFF, thereby setting the stage
for the creation of a spectrum hole, the whiteness property of the
model output disappears, which, in turn, provides the basis for
detecting the transition from a black space into a spectrum hole.
Whichever approach is used, the change-detection procedure
would clearly have to be location-specific. For example, if the
detection is performed in the basement of a building, the change
in from a black space to a white space is expected to be
significantly smaller than in an open environment. In any event,
the detection procedure would have to be sensitive enough to
work satisfactorily, regardless of location.
B. Practical Issues Affecting the Detection of Spectrum Holes
The effort involved in the detection of spectrum holes and
their subsequent exploitation in the management of radio spec-
trum should not be underestimated. In practical terms, the task
of spectrum management (discussed in Section X) must not only
be impervious to the modulation formats of primary users, but
also several other issues.14
1) Environmental factors: Radio propagation across a wire-
less channel is known to be affected by the following
factors.
• Path loss, which refers to the diminution of received
signal power with distance between the transmitter
and the receiver.
• Shadowing, which causes the received signal power
to fluctuate about the path loss by a multiplication
factor, thereby resulting in “coverage” holes.
2) Exclusive zones: An exclusion zone refers to the area (i.e.,
circle with some radius centered on the location of a pri-
mary user) inside which the spectrum is free of use and
can, therefore, be made available to an unserviced oper-
ator. This issue requires special attention in two possible
scenarios.
• The primary user happens to operate outside the ex-
clusion zone, in which case the identification of a
14The issues summarized herein follow a white paper submitted by Motorola
to the FCC [44].
spectrum hole must not be sensitive to radio inter-
ference produced by the primary user.
• Wireless scenarios built around cooperative relay
(ad hoc) networks [45], [46], which are designed to
operate at very low transmit powers. The dynamic
spectrum management algorithm must be able to
cope with such weak scenarios.
3) Predictive capability for future use: The identification of
a spectrum hole at a particular geographic location and a
particular time will only hold for that particular time and
not necessarily for future time. Accordingly, the dynamic
spectrum management algorithm in the transmitter must
include two provisions.
• Continuous monitoring of the spectrum hole in
question.
• Alternative spectral route for dealing with the even-
tuality of the primary user needing the spectrum for
its own use.
VI. CHANNEL-STATE ESTIMATION AND PREDICTIVE MODELING
As with every communication link, computation of the
channel capacity of a cognitive radio link requires knowledge
of channel-state information (CSI). This computation, in turn,
requires the use of a procedure for estimating the state of the
channel.
To deal with the channel-state estimation problem, tradition-
ally, we have proceeded in one of two ways [47].
• Differential detection, which lends itself to implementa-
tion in a straightforward fashion to the use of -ary phase
modulation.
• Pilot transmission, which involves the periodic transmis-
sion of a pilot (training sequence) known to the receiver.
The use of differential detection offers robustness and simplicity
of implementation, but at the expense of a significant degrada-
tion in the frame-error rate (FER) versus signal-to-noise ratio
(SNR) performance of the receiver. On the other hand, pilot
transmission offers improved receiver performance, but the use
of a pilot is wasteful in both transmit power and channel band-
width, the very thing we should strive to avoid. What then do
we do, if the receiver requires knowledge of CSI for efficient
receiver performance? The answer to this fundamental ques-
tion lies in the use of semi-blind training of the receiver [48],
which distinguishes itself from the differential detection and
pilot transmission procedures in that the receiver has two modes
of operations.
1) Supervised training mode: During this mode, the receiver
acquires an estimate of the channel estimate, which is per-
formed under the supervision of a short training sequence
(consisting of two to four symbols) known to the receiver;
the short training sequence is sent over the channel for a
limited duration by the transmitter prior to the actual data
transmission session.
2) Tracking mode: Once a reliable estimate of the channel
state has been achieved, the training sequence is switched
off, actual data transmission is initiated, and the receiver
is switched to the tracking mode; this mode of operation

is performed in an unsupervised manner on a continuous
basis during the course of data transmission.
A. Channel Tracking
The evolution of CSI with time is governed by a state-space
model comprised of two equations [48].
1) Process equation:
The state of a wireless link is defined as the minimal
set of data on the past behavior of the link that is needed
to predict the future behavior of the link. For the sake of
generality, we consider a multiple-input–multiple-output
(MIMO) wireless link15 of a narrowband category. Let
denote the channel coefficient from the th transmit
antenna to the th receive antenna at time , with
and . We may then describe
the scalar form of the state equation as
(7)
where the are time-varying autoregressive (AR) coef-
ficients and is the corresponding dynamic noise, both
at time . The AR coefficients account for the memory of
the channel due to the multipath phenomenon. The upper
limit of summation in (7) namely, , is the model order.
(The symbol used here should not be confused with the
symbol used to denote the time-bandwidth product in
Section III.)
2) Measurement equation:
The measurement equation for the MIMO wireless
link, also in scalar form, is described by
(8)
where is the encoded symbol transmitted by the th
antenna at time , and is the corresponding measure-
ment noise at the input of th receive antenna at time .
The is the signal observed at the output of the th an-
tenna at time .
15The use of a MIMO link offers several important advantages [47].
• Spatial degree of freedom, defined by N = minfN ; N g, where N
and N denote the numbers of transmit and receive antennas, respectively
[49].
• Increased spectral efficiency, which is asymptotically defined by [49]
lim
C(N)
N
= constant
where C(N) is the ergodic capacity of the link, expressed as a function of
N = N = N. This asymptotic property provides the basis for a spec-
tacular increase in spectral efficiency by increasing the number of transmit
and receive antennas.
• Diversity, which is asymptotically defined by [50]
lim
log FER()
log
= 0d
where d is the diversity order, and FER() is the frame-error rate ex-
pressed as a function of the SNR .
These benefits (gained at the expense of increased complexity) commend the
use of MIMO links for cognitive radio, all the more so considering the fact that
the primary motivation for cognitive radio is the attainment of improved spectral
efficiency. Simply put, a MIMO wireless link is not a necessary ingredient for
cognitive radio but a highly desirable one.
The state-space model comprised of (7) and (8) is linear. The
property of linearity is justified in light of the fact that the prop-
agation of electromagnetic waves across a wireless link is gov-
erned by Maxwell’s equations that are inherently linear.
What can we say about the AR coefficients, the dynamic
noise, and measurement noise, which collectively characterize
the state-space model of (7) and (8)? The answers to these ques-
tions determine the choice of an appropriate tracking strategy. In
particular, the discussion of this issue addressed in [48] is sum-
marized here.
1) AR model: A Markovian model of order on offers
sufficient accuracy to model a Rayleigh-distributed
time-varying channel.
2) Noise processes: The dynamic noise in the process equa-
tion and the measurement noise in the measurement equa-
tion can both assume non-Gaussian forms.
The finding reported under point 1) directly affects the design
of the predictive model, which is an essential component of the
channel tracker. The findings reported under point 2) prompt
the search for a tracker outside of the classical Kalman filters,
whose theory is rooted in Gaussian statistics.
A tracker that can operate in a non-Gaussian environment is
the particle filter, whose theory is rooted in Bayesian estimation
and Monte Carlo simulation [51], [52]. Each particle in the filter
may be viewed as a Kalman filter merely in the sense that its
operation encompasses two updates:
• state update;
• measurement update;
which bootstrap on each other, thereby forming a closed feed-
back loop. The particles are associated with weights, evolving
from one iteration to the next. In particular, whenever the few
particles whose weights assume negligible values, they are
dropped from the computation. Thereafter, the filter concen-
trates on particles with large weights. In particular, on the next
iteration of the filter, each of those particles is split into new
particles whose multiplicity is determined in accordance with
the weights of the parent particles. From this brief description,
it is apparent that the computational complexity of a particle
filter is in excess of that of a Kalman filter, but the particle
filter makes up for it by being readily amenable to parallel
computation.
In [48], the superior performance of the particle filter over
the classical Kalman filter and other trackers (in the context of
wireless channels) is demonstrated for real-life data. In light of
the detailed studies reported in [48], we may conclude that the
semi-blind estimation procedure, embodying the combined use
of supervised training and channel tracking, offers an effective
and efficient method for the extraction of channel-state estima-
tion for use in a cognitive radio system.
The predictive AR model used in [48] is considered to be
time-invariant (i.e., static) in that the model parameters are de-
termined off-line (i.e., prior to transmission) and remain fixed
throughout the tracking process. However, recognizing that a
wireless channel is in actual fact nonstationary, with the de-
gree of nonstationarity being highly dependent on the environ-

ment, we intuitively would expect that an improvement in per-
formance of the channel tracker is achievable if somehow the
predictive model is made time-varying (i.e., dynamic). This ex-
pectation has been demonstrated experimentally in [53] using
MIMO wireless data. Specifically, the dynamic channel tracker
accommodates a time-varying wireless channel by modeling the
channel parameters themselves as random walks, thereby al-
lowing them to assume a time-varying form.
Naturally, the maintenance of tracking a wireless channel in
a reliable manner is affected by conditions of the channel. To
be specific, we have found experimentally that when in the case
of a MIMO wireless communication system the determinant of
the channel matrix goes near zero, the particle filter experiences
difficulty in tracking the channel. The reason for this phenom-
enon is that when the channel cannot support the information
rate being used, the receiver makes too many symbol errors con-
secutively. This undesirable situation, in turn, causes the particle
filter and, therefore, the receiver to loose track. Monitoring of
the determinant of the channel matrix may, therefore, provide
the means to prevent the loss of channel tracking.
B. Rate Feedback
Channel-state estimation is needed by the receiver for co-
herent detection of the transmitted signal. Channel-state esti-
mation is also needed for calculation of the channel capacity
required by the transmitter for transmit-power control, which is
to be discussed in Section IX. To satisfy this latter requirement,
the receiver first uses Shannon’s information capacity theorem
to calculate the instantaneous channel capacity , but rather
then send directly, the practical approach is to quantize
and feed the quantized transmission rate back to the transmitter,
hence, the term rate feedback. A selection of quantized trans-
mission rates is kept in a predetermined list, in which case the
receiver picks the closest entry in the list that is less than the
calculated value of [54]; it is that particular entry in the list
that forms the rate feedback.
In wireless communications, we typically find that there are
significant fluctuations in the transmission rate. A transmission-
rate fluctuation is considered to be significant if it is a predeter-
mined fixed percentage of the mean rate for the channel. In any
event, the transmitter would like to know the transmission-rate
fluctuations. In particular, if the transmission rate is greater than
the channel capacity, then there would be an outage. Corre-
spondingly, the outage capacity is defined as the maximum bit
rate that can be maintained across the wireless link for a pre-
scribed probability of outage.
There are two other points to keep in mind.
1) Rate-feedback delay: There is always some finite
time-delay in transmitting the quantized rate across
the feedback channel. During the rate-feedback delay,
the channel capacity would inevitably vary, raising the
potential possibility for an outage by picking too high a
transmission rate. To mitigate this problem, prediction
of the outage capacity becomes a necessary requirement,
hence, the need for building a predictive model into the
design of rate-feedback system in the receiver [55].
2) Higher order Markov model: Typically, a first-order
Markov model is used to calculate the outage capacity
of a MIMO wireless system. By definition, a first-order
Markov model relies on information gained from the
state immediately proceeding the current state; in other
words, information pertaining to other previous states
is considered to be of negligible importance. This as-
sumption, usually made for mathematical tractability,
is justified for a slow-fading wireless link. However,
in the more difficult case of a fast-fading wireless link,
the channel fluctuates more rapidly, which means that a
higher order (e.g., second-order) Markov model is likely
to provide more useful information about the current
state than a first-order Markov model. Moreover, as the
diversity order is increased, the channel becomes hard-
ened quickly, in that variance of the channel capacity,
relative to its mean, decreases rapidly [54]. For this same
reason, we expect the fractional information gain about
the current state due to the use of a higher order model to
increase with decreasing diversity order [55].
VII. COOPERATION AND COMPETITION IN MULTIUSER
COGNITIVE RADIO ENVIRONMENTS
In this section, we set the stage for the next important task:
transmit-power control.
In conventional wireless communications built around base
stations, transmit-power levels are controlled by the base sta-
tions so as to provide the required coverage area and thereby
provide the desired receiver performance. On the other hand, it
may be necessary for a cognitive radio to operate in a decentral-
ized manner, thereby broadening the scope of its applications. In
such a case, some alternative means must be found to exercise
control over the transmit power. The key question is: how can
transmit-power control be achieved at the transmitter?
A partial answer to this fundamental question lies in building
cooperative mechanisms into the way in which multiple access
by users to the cognitive radio channel is accomplished. The
cooperative mechanisms may include the following.
1) Etiquette and protocol. Such provisions may be likened
to the use of traffic lights, stop signs, and speed limits,
which are intended for motorists (using a highly dense
transportation system of roads and highways) for their in-
dividual safety and benefits.
2) Cooperative ad hoc networks. In such networks, the
users communicate with each other without any fixed
infrastructure. In [45], Shepard studies a large packet
radio network using spread-spectrum modulation. The
only required form of coordination in the network is
that of pairwise between neighboring nodes (users) that
are in direct communication. To mitigate interference,
it is proposed that each node create a transmit-receive
schedule. The schedule is communicated to a nearest
neighbor only when a source node’s schedule and that of
the neighboring node permit the source node to transmit
it and the neighboring node to receive it. Under some
reasonable assumptions, simulations are presented to
show that with this completely decentralized control, the
network can scale to almost arbitrary numbers of nodes.
In an independent and like-minded study [46], Gupta
and Kumar considered a radio network consisting of
identical nodes that communicate with each other. The

nodes are arbitrarily located inside a disk of unit area. A
data packet produced by a source node is transmitted to a
sink node (i.e., destination) via a series of hops across in-
termediate nodes in the network. Let one bit-meter denote
one bit of information transmitted across a distance of one
meter toward its destination. Then, the transport capacity
of the network is defined as the total number of bit-me-
ters that the network can transport in one second for all
nodes. Under a protocol model of noninterference, Gupta
and Kumar derive two significant results. First, the trans-
port capacity of the network increases with . Second,
for a node communicating with another node at a dis-
tance nonvanishingly far away, the throughput (in bits per
second) decreases with increasing . These results are
consistent with those of Shephard. However, Gupta and
Kumar do not consider the congestion problem identified
in Shepard’s work.
Through the cooperative mechanisms described under 1) and 2)
and other cooperative means, the users of cognitive radio may
be able to benefit from cooperation with each other in that the
system could end up being able to support more users because
of the potential for an improved spectrum-management strategy.
The cooperative ad hoc networks studied by Shepard [45]
and Gupta and Kumar [46] are examples of a new generation
of wireless networks, which, in a loose sense, resemble the In-
ternet. In any event, in cognitive radio environments built around
ad hoc networks and existing infrastructured networks, it is pos-
sible to find the multiuser communication process being com-
plicated by another phenomenon, namely, competition, which
works in opposition to cooperation.
Basically, the driving force behind competition in a multiuser
environment lies in having to operate under the umbrella of lim-
itations imposed on available network resources. Given such an
environment, a particular user may try to exploit the cognitive
radio channel for self-enrichment in one form or another, which,
in turn, may prompt other users to do likewise. However, ex-
ploitation via competition should not be confused with the self-
orientation of cognitive radio which involves the assignment of
priority to certain stimuli (e.g., urgent requirements or needs).
In any event, the control of transmit power in a multiuser cog-
nitive radio environment would have to operate under two strin-
gent limitations on network resources: the interference-temper-
ature limit imposed by regulatory agencies, and the availability
of a limited number of spectrum holes depending on usage.
What we are describing here is a multiuser communication-
theoretic problem. Unfortunately, a complete understanding of
multiuser communication theory is yet to be developed. Never-
theless, we know enough about two diverse disciplines, namely,
information theory and game theory, for us to tackle this diffi-
cult problem in a meaningful way. However, before proceeding
further, we digress briefly to introduce some basic concepts in
game theory.
VIII. STOCHASTIC GAMES
The transmit-power control problems in a cognitive-radio
environment (involving multiple users) may be viewed as a
game-theoretic problem.16 In the absence of competition, we
would then have an entirely cooperative game, in which case
the problem simplifies to an optimal control-theoretic problem.
This simplification is achieved by finding a single cost function
that is optimized by all the players, thereby eliminating the
game-theoretic aspects of the problem [58]. So, the issue of
interest is how to deal with a noncooperative game involving
multiple players. To formulate a mathematical framework
for such an environment, we have to account for three basic
realities:
• a state space that is the product of the individual players’
states;
• state transitions that are functions of joint actions taken by
the players;
• payoffs to individual players that depend on joint actions
as well.
That framework is found in stochastic games [57], which, also
occasionally appear under the name “Markov games” in the
computer science literature.
A stochastic game is described by the five-tuple
, where
• is a set of players, indexed ;
• is a set of possible states;
• is the joint-action space defined by the product set
, where is the set of actions available to
the th player;
• is a probabilistic transition function, an element of
which for joint action satisfies the condition
• , where is the payoff for the th
player and which is a function of the joint actions of all
players.
One other notational issue: the action of player is denoted
by , while the joint actions of the other players in the
set are denoted by . We use a similar notation for some
other variables.
Stochastic games are supersets of two kinds of decision pro-
cesses, namely, Markov decision process and matrix games, as
illustrated in Fig. 2. A Markov decision process is a special case
of a stochastic game with a single player, that is, . On the
other hand, a matrix game is a special case of a stochastic game
with a single state, that is, .
A. Nash Equilibria and Mixed Strategies
With two or more players17 being an integral part of a game,
it is natural for the study of cognitive radio to be motivated by
certain ideas in game theory. Prominent among those ideas for
finite games (i.e., stochastic games for which each player has
only a finite number of alternative courses of action) is that of a
Nash equilibrium, so named for the Nobel Laureate John Nash.
16In a historical context, the formulation of game theory may be traced back to
the pioneering work of John von Neumann in the 1930s, which culminated in the
publication of the coauthored book entitled “Theory of Games and Economic
Behavior” [56]. For modern treatments of game theory, see the books under [57]
and [58].
17Players are referred to as agents in the machine learning literature.

Fig. 2. Highlighting the differences between Markov decision processes,
matrix games, and stochastic games.
A Nash equilibrium is defined as an action profile (i.e., vector
of players’ actions) in which each action is a best response to
the actions of all the other players [59]. According to this def-
inition, a Nash equilibrium is a stable operating (i.e., equilib-
rium) point in the sense that there is no incentive for any player
involved in a finite game to change strategy given that all the
other players continue to follow the equilibrium policy. The im-
portant point to note here is that the Nash-equilibrium approach
provides a powerful tool for modeling nonstationary processes.
Simply put, it has had an enormous influence on the evolution of
game theory by shifting its emphasis toward the study of equi-
libria as a predictive concept.
With the learning process modeled as a repeated stochastic
game (i.e., repeated version of a one-shot game), each player
gets to know the past behavior of the other players, which may
influence the current decision to be made. In such a game, the
task of a player is to select the best mixed strategy, given infor-
mation on the mixed strategies of all other players in the game;
hereafter, other players are referred to as “opponents.” A mixed
strategy is defined as a continuous randomization by a player
of its own actions, in which the actions (i.e., pure strategies) are
selected in a deterministic manner. Stated in another way, the
mixed strategy of a player is a random variable whose values
are the pure strategies of that player.
To explain what we mean by a mixed strategy, let de-
note the th action of player with . The
mixed strategy of player , denoted by the set of probabilities
, is an integral part of the linear combination
(9)
Equivalently, we may express as the inner product
(10)
where
is the mixed strategy vector, and
is the deterministic action vector. The superscript denotes ma-
trix transposition. For all , the elements of the mixed strategy
vector satisfy the following two conditions:
1)
(11)
2)
(12)
Note also that the mixed strategies for the different players
are statistically independent.
The motivation for permitting the use of mixed strategies is
the well-known fact that every stochastic game has at least one
Nash equilibrium in the space of mixed strategies but not neces-
sarily in the space of pure strategies, hence, the preferred use of
mixed strategies over pure strategies. The purpose of a learning
algorithm is that of computing a mixed strategy, namely a se-
quence over time .
It is also noteworthy that the implication of (9) through (12) is
that the entire set of mixed strategies lies inside a convex simplex
or convex hull, whose dimension is and whose vertices
are the . Such a geometric configuration makes the selection
of the best mixed strategy in a multiple-player environment a
more difficult proposition to tackle than the selection of the best
base action in a single-player environment.
B. Limitations of Nash Equilibrium
The formulation of Nash equilibrium assumes that the players
are rational, which means that each player has a “view of the
world.” According to Aumann and Brandenburger [60], mutual
knowledge of rationality and common knowledge of beliefs is
sufficient for deductive justification of the Nash equilibrium. Be-
lief refers to state of the world, expressed as a set of probability
distributions over tests; by “tests” we mean a sequence of ac-
tions and observations that are executed at a specific time.
Despite the insightful value of the Aumann–Brandenburger
exposition, the notion of the Nash equilibrium has two practical
limitations.
1) The approach advocates the use of a best-response
strategy (i.e., a strategy whose outcome against an op-
ponent with a similar goal is the best possible one), but
in a two-player game for example, if one player adopts
a nonequilibrium strategy, then the optimal response of
the other player is of a nonequilibrium kind too. In such
situations, the Nash-equilibrium approach is no longer
applicable.
2) Description of a noncooperative game is essentially con-
fined to an equilibrium condition; unfortunately, the ap-
proach does not teach us about the underlying dynamics
involved in establishing that equilibrium.
To refine the Nash equilibrium theory, we may embed learning
models in the formulation of game-theoretic algorithms. This
new approach provides a foundation for equilibrium theory, in
which less than fully rational players strive for some form of
optimality over time [57], [61].
C. Game-Theoretic Learning: No-Regret Algorithms
Statistical learning theory is a well-developed discipline for
dealing with uncertainty, which makes it well-suited for solving
game-theoretic problems. In this context, a class of no-regret

algorithms is attracting a great deal of attention in the machine-
learning literature.
The provision of “no-regret” is motivated by the desire to
ensure two practical end-results.
1) A player does not get unlucky in an arbitrary nonsta-
tionary environment. Even if the environment is not adver-
sarial, the player could experience bad performance when
using an algorithm that assumes independent and identi-
cally distributed (i.i.d.) examples; the no-regret provision
guarantees that such a situation does not arise.
2) Clever opponents of that player do not exploit dynamic
changes or limited resources for their own selfish benefits.
The notion of regret can be defined in different ways.18 One
particular definition of no regret is basically a rephrasing of
boosting, the original formualation of which is due to Freund
and Schapire [62]. Basically, boosting refers to the training of
a committee machine in which several experts are trained on
data sets with entirely different distributions [62], [71]. It is a
general method that can be used to improve the performance of
any learning model. Stated in another way, boosting provides a
method for modifying the underlying distribution of examples
in such a way that a strong learning model is built around a set
of weak learning modules.
To see how boosting can also be viewed as a no-regret propo-
sition, consider a prediction problem with de-
noting the sequence of input vectors. Let denote the one-step
prediction at time computed by the boosting algorithm oper-
ating on this sequence. The prediction error is defined by the
difference . Let denote a convex cost function
of the prediction error ; the mean-square error is an example
of such a cost function. After processing examples, the re-
sulting cost function of the boosting algorithm is given by
(13)
If, however, the prediction was to be performed by one of
the experts using some fixed hypothesis to yield the predic-
tion error , the corresponding cost function would have the
value
(14)
The regret for not having used hypothesis is the difference
(15)
We say that the regret is negative if the difference is neg-
ative. Let denote the class of all hypotheses used in the al-
gorithm. Then the overall regret for not having used the best
hypothesis is given by the supremum
(16)
18In a unified treatment of game-theoretic learning algorithms, Greenwald
[61] identifies three regret variations:
• External regret
• Internal regret
• Swap regret
External regret coincides with the notion of boosting as defined by Freund and
Schapire [62].
A boosting algorithm is synonymous with no-regret algorithms
because the overall regret is small no matter which particular
sequence of input vectors is presented to the algorithm.
Unfortunately, most no-regret algorithms are designed on the
premise that the hypotheses are chosen from a small, discrete
set, which, in turn, limits applicability of the algorithms. To
overcome this limitation, Gordon [63] expands on the Freund-
Schapire boosting (Hedge) algorithm by considering a class of
prediction problems with internal structure. Specifically, the in-
ternal structure presumes two things.
1) The input vectors are assumed to lie on or inside an al-
most arbitrary convex set, so long as it is possible to per-
form convex optimization; for example, we could have a
-dimensional polyhedron or -dimensional sphere, were
is dimensionality of the input space.
2) The prediction rules (i.e., experts) are purposely designed
to be linear.
An example scenario that has the internal structure embodied
under points 1) and 2) is that of planning in a stochastic game
described by a Markov decision process, in which state-action
costs are controlled by an adversarial or clever opponent after
the player in question fixes its own policy. The reader is referred
to [64] for such an example involving a robot path-planning
problem, which may be likened to a cognitive radio problem
made difficult by the actions of a clever opponent.
Given such a framework, we can always make a legal pre-
diction in an efficient manner via convex duality, which is an
inherent property of convex optimization [65]. In particular, it
is always possible to choose a legal hypothesis that prevents the
total regret from growing too quickly (and, therefore, causes the
average regret to approach zero).
By exploiting this internal structure, Gordon derives a new
learning rule referred to as the Lagrangian hedging algorithm
[63]. This new algorithm is of a gradient descent kind, which
includes two steps, namely, projection and scaling. The projec-
tion step simply ensures that we always make a legal prediction.
The scaling step adaptively adjusts the degree to which the al-
gorithm operates in an aggressive or conservative manner. In
particular, if the algorithm predicts poorly, then the cost func-
tion assumes a large value on the average, which, in turn, tends
to make the predictions change slowly.
The algorithms derives its name from a combination of two
points.
1) The algorithm depends on one free parameter, namely, a
convex hedging function.
2) The hypothesis of interest can be viewed as a Lagrange
multiplier that keeps the regret from growing too fast.
To expand on the Lagrangian issue under point 2), consider the
case of a matrix game using a regret-matching algorithm. Re-
gret-matching, embodied in the generalized Blackwell condi-
tion [61], means that the probability distribution over actions
by a player is proportional to the positive elements in the regret
vector of that player. For example, in the so-called “rock-scis-
sors-paper” game in which rock smashes scissors, scissors cut
paper, and paper wraps the rock, if we currently have a vector
made up as follows:
• regret 2 versus rock;
• regret versus scissors;
• regret 1 versus paper;

then we would play rock 2/3 of the time, never play scissors,
and play paper 1/3 of the time. The prediction at each step of
the regret-matching algorithm is a probability distribution over
actions. Ideally, we desire the no-regret property, which means
that the average regret vector approaches the region where all
of its elements are less than or equal to zero. However, at any
finite time, in practice, the regret vector may still have posi-
tive elements. (The magnitudes of these elements are bounded
by theorems presented in [63].) In such a situation, we cannot
achieve the no-regret condition exactly in finite time. Rather, we
apply a soft constraint by imposing a quadratic penalty function
on each positive element of the regret vector. The penalty func-
tion involves the sum of two components, one being the hedging
function and the other being an indicator function for the set of
unnormalized hypotheses using a gradient vector. The gradient
vector is itself defined as the derivative of the penalty function
with respect to the regret vector, the evaluation being made at the
current regret vector. It turns out that the gradient vector is just
the regret vector with all negative elements set equal to zero. The
desired hypothesis is gotten by normalizing this vector to form
a probability distribution of actions, which yields exactly the
regret-matching algorithm. In choosing the distribution of ac-
tions in the manner described herein, we enforce the constraint
that the regret vector is not allowed to move upwards along the
gradient. Gordon’s gradient descent theorem, proved by induc-
tion in [63], shows that the quadratic penalty function cannot
grow too quickly, which in turn, means that our average gra-
dient vector will get closer to the negative orthant, as desired.
In short, the Lagrangian hedging algorithm is a no-regret
algorithm designed to handle internal structure in the set of
allowable predictions. By exploiting this internal structure,
tight bounds on performance and fast rates of convergence
are achieved when the provision of no regret is of utmost
importance.
IX. DISTRIBUTED TRANSMIT-POWER CONTROL: ITERATIVE
WATER-FILLING
As an alternative to game-theoretic learning exemplified by a
no-regret algorithm, we may look to another approach, namely,
water-filling (WF) rooted in information theory [66]. To be spe-
cific, consider a cognitive radio environment involving trans-
mitters and receivers. The environmental model is based on
two assumptions.
1) Communication across a channel is asynchronous, in
which case the communication process can be viewed as
a noncooperative game. For example, in a mesh network
consisting of a mixture of ad hoc networks and existing
infrastructured networks, the communication process
from a base station to users is controlled in a synchronous
manner, but the multihop communication process across
the ad hoc network could be asynchronous and, therefore,
noncooperative.
2) A signal-to-noise ratio (SNR) gap is included in calcu-
lating the transmission rate so as to account for the gap
between the performance of a practical coding-modula-
tion scheme and the theoretical value of channel capacity.
(In effect, the SNR gap is large enough to assure reliable
communication under operating conditions all the time.)
In mathematical terms, the essence of transmit-power control
for such a noncooperative multiuser radio environment is stated
as follows.
Given a limited number of spectrum holes, select the transmit-
power levels of unserviced users so as to jointly maximize
their data-transmission rates, subject to the constraint that the
interference-temperature limit is not violated.
It may be tempting to suggest that the solution of this problem
lies in simply increasing the transmit-power level of each un-
serviced transmitter. However, increasing the transmit-power
level of any one transmitter has the undesirable effect of also
increasing the level of interference to which the receivers of all
the other transmitters are subjected. The conclusion to be drawn
from this reality is that it is not possible to represent the overall
system performance with a single index of performance. (This
conclusion further confirms what we said previously in Sec-
tion VIII.) Rather, we have to adopt a tradeoff among the data
rates of all unserviced users in some computationally tractable
fashion.
Ideally, we would like to find a global solution to the con-
strained maximization of the joint set of data-transmission rates
under study. Unfortunately, finding this global solution requires
an exhaustive search through the space of all possible power
allocations, in which case we find that the computational com-
plexity needed for attaining the global solution assumes a pro-
hibitively high level.
To overcome this computational difficulty, we use a new opti-
mization criterion called competitive optimality19 for solving the
transmit-power control problem, which may now be restated as
follows.
Considering a multiuser cognitive radio environment viewed
as a noncooperative game, maximize the performance of each
unserviced transceiver, regardless of what all the other trans-
ceivers do, but subject to the constraint that the interference-
temperature limit not be violated.
This formulation of the distributed transmit-power control
problem leads to a solution that is of a local nature; though sub-
optimum, the solution is insightful, as described next.
A. Two-User Scenario: Simultaneous WF is Equivalent to
Nash Equilibrium
Consider the simple scenario of Fig. 3 involving two
users communicating across a flat-fading channel. The com-
plex-valued baseband channel matrix is denoted by
(17)
Viewing this scenario as a noncooperative game, we may de-
scribe the two players of the game as follows:
19The competitive optimality criterion is discussed in Yu’s doctoral disserta-
tion [67, Ch. 4]. In particular, Yu develops an iterative WF algorithm for a sub-
optimum solution to the multiuser digital subscriber line (DSL) environment,
viewed as a noncooperative game.

Fig. 3. Signal-flow graph of a two-user communication scenario.
• The two players20 are represented by transmitters 1 and 2.
• The pure strategies (i.e., deterministic actions) of the two
players are defined by the power spectral densities
and that, respectively, pertain to the transmitted sig-
nals radiated by the antennas of transmitters 1 and 2.
• The payoffs to the two players are defined by the data-
transmission rates and , which are, respectively,
produced by transmitters 1 and 2.
From the discussions presented in Section IV, we recognize
that the noise floor of the RF radio environment is characterized
by a frequency-dependent parameter: the power spectral density
. In effect, defines the “noise floor” above which
the transmit-power controller must fit the transmission-data re-
quirements of users 1 and 2.
Define the cross-coupling between the two users in terms of
two new real-valued parameters and by writing
(18)
and
(19)
where is the SNR gap. Assuming that the receivers do not per-
form any form of interference-cancellation irrespective of the
received signal strengths, we may, respectively, formulate the
achievable data-transmission rates and as the two defi-
nite integrals
(20)
and
(21)
The term in the first denominator and the term
in the second denominator are due to the cross-cou-
pling between the transmitters and receivers. The remaining
two terms and are noise terms defined by
(22)
20In the two-user example of Fig. 3, each user is represented by a single-
input–single-output (SISO) wireless system—hence, the adoption of transmit-
ters 1 and 2 of the two systems as the two players in a game-theoretic inter-
pretation of the example. In a MIMO generalization of this example, each user
has multiple transmitters. Nevertheless, there are still two players, with the two
players being represented by the two sets of multiple transmitters.
and
(23)
where and are, respectively, the particular
parts of the noise-floor’s spectral density that define the
spectral contents of spectrum holes 1 and 2.We are now ready to
formally state the competitive optimization problem as follows.
Given that the power spectra density of transmitter 2
is fixed, maximize the transmission-data of (20), subject to
the constraint
where is the prescribed interference-temperature limit and
is Boltzmann’s constant. A similar statement applies to the
competitive optimization of transmitter 2.
Of course, it is understood that both and remain
nonnegative for all . The solution to the optimization problem
described herein follows the allocation of transmit power in ac-
cordance with the WF procedure [66], [67].
Fig. 4 presents the results of an experiment21 on the two-user
wireless scenario, which were obtained using the WF procedure.
To add meaning to the important result portrayed in Fig. 4, we
may state that the optimal competitive response to the all pure-
strategy corresponds to a Nash equilibrium. Stated in another
way, a Nash equilibrium is reached if, and only if, both users
simultaneously satisfy the WF condition [67].
An assumption implicit in the WF solution presented in Fig. 4
is that each transmitter of cognitive radio has knowledge of its
position with respect to the receivers in its operating range at all
times. In other words, cognitive radio has geographic aware-
ness, which is implemented by embedding a global positioning
21Specifications of the experiment presented in Fig. 4 are as follows.
Narrowband channels (uniformly spaced in frequency) available to the two
users:
• user 1: channels 1, 2, and 3;
• user 2: channels 4, 5, and 6.
Modulation Strategy: orthogonal frequency-division multiplexing (OFDM)
Multiuser path-loss matrix
0:5207 0 0 0:0035 0:0020 0:0024
0 0:5223 0 0:0030 0:0034 0:0031
0 0 0:5364 0:0040 0:0015 0:0035
0:0036 0:0002 0:0023 0:7136 0 0
0:0028 0:0029 0:0011 0 0:6945 0
0:0022 0:0010 0:0034 0 0 0:7312
:
Target data transmission rates:
• user 1: 9 bits/symbol;
• user 2: 12 bits/symbol;
Power constraint (imposed by interference-temperature limit) = 0 dB:
Receiver noise-power level = 030 dB.
Ambient interference power level = 024 dB.
The solution presented in Fig. 4 is achieved in two iterations of the WF algo-
rithm.

Fig. 4. Two-user profile, illustrating two things. 1) The spectrum-sharing
process performed using the iterative WF algorithm. 2) The bit-loading curve
shown “bold-faced” at the top of the figure.
satellite (GPS) receiver in the system design [68]. The trans-
mitter puts its geographic awareness to good use by calculating
the path loss incurred in the course of electromagnetic propaga-
tion of the transmitted signal to each receiver in the transmitter’s
operating range, which, in turn, makes it possible to calculate
the multiuser path-loss matrix of the environment.22
B. Multiuser Scenario: Iterative WF Algorithm
Emboldened by the WF solution illustrated in Fig. 4 for a
two-user scenario, we may formulate an iterative two-loop WF
algorithm for the distributed transmit-power control of a mul-
tiuser radio environment. The environment involves a set of
transmitters indexed by and a corresponding set
of receivers indexed by . Viewing the multiuser
radio environment as a non cooperative game and assuming the
availability of an adequate number of spectrum holes to accom-
modate the target data-transmission rates, the algorithm pro-
ceeds as follows [67].
1) Initialization: Unless some prior knowledge is available,
the power distribution across the users is set equal to
zero.
22Let d denote the distance from a transmitter to a receiver. Extensive mea-
surements of the electromagnetic field strength, expressed as a function of the
distance d, carried out in various radio environments have motivated an empir-
ical propagation formula for the path loss, which expresses the received signal
power P in terms of the transmitted signal power P as follows [47]:
P =

d
P
where the path-loss exponent m varies from 2 to 5, depending on the environ-
ment, and the attenuation parameter

is frequency-dependent.
Considering the general case of n transmitters indexed by i, and n receivers
indexed by j, let h denote the complex-valued channel coefﬁcient from trans-
mitter i to receiver j. Then, in light of the empirical propagation formula, we
may write
jh j =
P
P
=

d
; i = 1;2;. . . ;n j = 1;2;. . . ;n
where d is the distance from transmitter i to receiver j. Hence, knowing

,
m, and d for all i and j, we may calculate the multiuser path-loss matrix.
2) Inner loop (iteration): Given a set of allowed channels
(i.e., spectrum-holes):
• User 1 performs WF, subject to its power constraint.
At first, the user employs one channel; but if its target
rate is not satisfied, it will try to employ two channels,
and so on. The WF by user 1 is performed with only
the noise floor to account for.
• Then, user 2 performs the WF process, subject to its
own power constraint. At this point, in addition to the
noise floor, the WF computation accounts for interfer-
ence produced by user 1.
• The power-constrained WF process is continued until
all users are dealt with.
3) Outer loop (iteration): After the inner iteration is com-
pleted, the power allocation among the users is adjusted:
• If the actual data-transmission rate of any user is found
to be greater than its target value, the transmit power
of that user is reduced.
• If, on the other hand, the actual data-transmission rate
of any user is less than the target value, the transmit
power is increased, keeping in mind that the interfer-
ence temperature limit is not violated.
4) Confirmation step: After the power adjustments, up or
down, are completed, the transmission-data rates of all the
users are checked:
• If the target rates of all the users are satisfied, the
computation is terminated.
• Otherwise, the algorithm goes back to the inner loop,
and the computations are repeated. This time, how-
ever, the WF performed by every user, including user
1, must account for the interference produced by all the
other users.
In effect, the outer loop of the distributed transmit-power con-
troller tries to find the minimum level of transmit power needed
to satisfy the target data-transmission rates of all users.
For the distributed transmit-power controller to function
properly, two requirements must be satisfied.
• Each user knows, a priori, its own target rate.
• All the target rates lie within a permissible rate region;
otherwise, some or all of the users will violate the inter-
ference-temperature limit.
To distributively live within the permissible rate region, the
transmitter needs to be equipped with a centralized agent that
has knowledge of the channel capacity (through rate-feedback
from the receiver) and multiuser path-loss matrix (by virtue of
geographic awareness). The centralized agent is thereby en-
abled to decide which particular sets of target rates are indeed
attainable.
C. Iterative WF Algorithm Versus No-Regret Algorithm
The iterative WF approach, rooted in communication theory,
has a “top-down, dictatorially controlled” flavor. In contrast,
a no-regret algorithm, rooted in machine learning, has a
“bottom-up” flavor. In more specific terms, we may make the
following observations.
1) The iterative WF algorithm exhibits fast-convergence be-
havior by virtue of incorporating information on both the

Fig. 5. Illustrating the notion of dynamic spectrum-sharing for OFDM based on four channels, and the way in which the spectrum manager allocates the requisite
channel bandwidths for three time instants t t t , depending on the availability of spectrum holes.
channel and RF environment. On the other hand, a no-re-
gret algorithm exemplified by the Lagrangian hedging al-
gorithm relies on first-order gradient information, causing
it to converge comparatively slowly.
2) The Lagrangian hedging learner has the attractive feature
of incorporating a regret agenda, the purpose of which
is to guarantee that the learner cannot be deceptively ex-
ploited by a clever player. On the other hand, the iterative
WF algorithm lacks a learning strategy that could enable
it to guard against exploitation.
In short, the iterative WF approach has much to offer for
dealing with multiuser scenarios, but its performance could
be improved through interfacing with a more competitive,
regret-conscious learning machine that enables it to mitigate
the exploitation phenomenon.
X. DYNAMIC SPECTRUM MANAGEMENT
As with transmit-power control, dynamic spectrum manage-
ment (also referred to as dynamic frequency-allocation) is per-
formed in the transmitter. Indeed, these two tasks are so inti-
mately related to each other that we have included them both
inside a single functional module, which performs the role of
multiple-access control in the basic cognitive cycle of Fig. 1.
Simply put, the primary purpose of spectrum management is
to develop an adaptive strategy for the efficient and effective
utilization of the RF spectrum. Specifically, the spectrum-man-
agement algorithm is designed to do the following.
Building on the spectrum holes detected by the radio-scene
analyzer and the output of transmit-power controller, select a
modulation strategy that adapts to the time-varying conditions
of the radio environment, all the time assuring reliable commu-
nication across the channel.
Communication reliability is assured by choosing the SNR
gap large enough as a design parameter, as discussed in
Section IX.
A. Modulation Considerations
A modulation strategy that commends itself to cognitive radio
is the OFDM23 by virtue of its flexibility and computational
efficiency. For its operation, OFDM uses a set of carrier fre-
quencies centered on a corresponding set of narrow channel
bandwidths. Most important, the availability of rate feedback
(through the use of a feedback channel) permits the use of bit-
loading, whereby the number of bits/symbol for each channel is
optimized for the SNR characterizing that channel; this opera-
tion is illustrated by the bold-faced curve in Fig. 4.
As time evolves and spectrum holes come and go, the
bandwidth-carrier frequency implementation of OFDM is
dynamically modified, as illustrated in the time-frequency
picture in Fig. 5 for the case of four carrier frequencies. The
picture illustrated in Fig. 5 describes a distinctive feature of
cognitive radio: a dynamic spectrum-sharing process, which
evolves in time. In effect, the spectrum-sharing process satisfies
the constraint imposed on cognitive radio by the availability
of spectrum holes at a particular geographic location and their
possible variability with time. Throughout the spectrum-sharing
process, the transmit-power controller keeps an account of the
bit-loading across the spectrum holes in use. In effect, the
dynamic spectrum manager and the transmit-power controller
work in concert together, thereby fulfilling the multiple-access
control requirement.
Starting with a set of spectrum holes, it is possible for the dy-
namic spectrum management algorithm to confront a situation
where the prescribed FER cannot be satisfied. In situations of
this kind, the algorithm can do one of two things:
1) work with a more spectrally efficient modulation strategy,
or else;
2) incorporate the use of another spectrum hole, assuming
availability.
23OFDM has been standardized; see the IEEE 802.16 Standard, described in
[69].

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