If you are trying to know about

**Cognitive Radio**and not getting any material or book or any class lecture related to**Cognitive Radio**then i can bet this is the best material you have ever had !In the below Invited Paper Simon Haykin has described the very basic things of**Cognitive Radio**which will make your concept clear.If you want to research on**Cognitive Radio**then the below material on**cognitive radio**will also provide you the scope to do so.

**Cognitive Radio: Brain-Empowered**

**Wireless Communications**

**Simon Haykin, Life Fellow, IEEE**

Abstract—

**Cognitive radio**is viewed as a novel approach for improvingthe utilization of a precious natural resource: the radio

electromagnetic spectrum.

The

**cognitive radio**, built on a software-defined radio, is definedas an intelligent wireless communication system that is

aware of its environment and uses the methodology of understanding-

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

• 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 management.

This paper also discusses the emergent behavior of cognitive radio.

Index Terms—Awareness, channel-state estimation and predictive

modeling, cognition, competition and cooperation, emergent

behavior, interference temperature, machine learning, radio-scene

analysis, rate feedback, spectrum analysis, spectrum holes, spectrum

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 Communications

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, multidisciplinary

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 significant

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 including

the revenue-rich urban areas, wewould find that [2]–[4]:

1) some frequency bands in the spectrum are largely unoccupied

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 following

definition [2]:

A spectrum hole is a band of frequencies assigned to a primary

user, but, at a particular time and specific geographic location,

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 scientific

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 methodology

termed learning by understanding. Putting these ideas together

and bearing in mind that cognitive radio is aimed at improved

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

0733-8716/$20.00 © 2005 IEEE

Authorized licensed use limited to: East West University. Downloaded on July 20,2010 at 03:31:01 UTC from IEEE Xplore. Restrictions apply.

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 intelligence,

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 cognitive

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 convergence

of two key technologies: digital radio, and computer software

[11]–[13].

B. Cognitive Tasks: An Overview

For reconfigurability, a cognitive radio looks naturally to software-

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 management.

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 embodies

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 networks.

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 receiver

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 communication

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 communication

scenario, we have a transceiver (i.e., combination of transmitter and

receiver) at each end of the communication path; all the cognitive functions embodied

in the cognitive cycle of Fig. 1 are built into each of the two transceivers.

Authorized licensed use limited to: East West University. Downloaded on July 20,2010 at 03:31:01 UTC from IEEE Xplore. Restrictions apply.

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 representation

language (RKRL) [5]. The idea of RKRL was expanded

further in Mitola’s own doctoral dissertation, which was presented

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 report

set the stage for a workshop on cognitive radio, which was

held inWashington, 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 Conference

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 visionary

dissertation by presenting detailed expositions of signalprocessing

and adaptive procedures that lie at the heart of cognitive

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 temperature

as a new metric for the quantification and management

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 predictive

modeling.

• Sections VII–X are devoted to multiuser cognitive

radio networks, with Sections VII and VIII reviewing

stochastic games and highlighting the processes of cooperation

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 behavior

that could arise in a cognitive radio environment.

• Section XII concludes the paper and highlights the research

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 prescribed

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 coverage.

To guard against such a possibility, the FCC Spectrum

Policy Task Force [1] has recommended a paradigm shift in interference

assessment, that is, a shift away from largely fixed operations

in the transmitter and toward real-time interactions between

the transmitter and receiver in an adaptive manner. The

recommendation is based on a new metric called the interference

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 particular

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 provides

an accurate measure for the acceptable level of RF

interference in the frequency band of interest; any transmission

in that band is considered to be “harmful” if it

would increase the noise floor above the interference-temperature

limit.

2) Given a particular frequency band in which the interference

temperature is not exceeded, that band could be made

available to unserviced users; the interference-temperature

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-temperature

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 Association

for Amateur Radio presents a critique of this metric.

Authorized licensed use limited to: East West University. Downloaded on July 20,2010 at 03:31:01 UTC from IEEE Xplore. Restrictions apply.

204 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

10 joules per degree Kelvin yields the corresponding

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, estimation

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 feedback

channel, is needed by the transmitter to carry out

the joint function of active transmit-power control and dynamic

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 signals,

exemplified by RF stimuli, has had a rather mixed history.

Although the general second-order theory of nonstationary signals

was published during the 1940s by LoÃ¨ve [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 description

of the signal. Given the desirability of working in the frequency

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 published

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 marginal

density conditions. However, the frequency marginal distribution

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 LoÃ¨ve [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

LoÃ¨ve 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 become

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 tapering

(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 information

resulting from a reduction in the effective sample

size.

Howcan we resolve this dilemma by mitigating the loss of information

due to tapering? The answer to this fundamental question

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 spectral

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 spectral

characteristics, namely, reduced variance of the spectral estimate

without compromising the bias of the estimate.

Given a time series , the multitaper spectral estimation

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 periodogram

for each tapered (windowed) segment, and then averaging the resulting

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 descriptions

of this procedure, see [26], [28] and the book by Percival andWalden

[29, Ch. 7].

The Signal Processing Toolbox [30] includes theMATLAB 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 sequences.

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].

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 205

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-bandwidth

product

(2)

defines the degrees of freedom available for controlling the variance

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 eigenspectrum.

Two points are noteworthy.

1) The denominator in (3) makes the estimate

unbiased.

2) Provided that we choose , then the eigenvalue

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 interpreted

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 CramÃ©r–Rao

bound for a nonparametric spectral estimator. Most important,

unlike the maximum-likelihood spectral estimator, the multitaper

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 distribution

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 coefficients,

which provides further justification for the use of jackknifing.

12Adaptive beamformers, also referred to as adaptive antennas or smart antennas,

are discussed in the books [34]–[37].

In the first stage of interference control, the transmitter exploits

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 radiation

of the transmitted signal in all directions.

2) Assuming that every cognitive radio transmitter follows a

strategy similar to that summarized under point 1), interference

at the receiver due to the actions of other transmitters

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 transmitters.

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 distribution

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 communications

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 computed

by the th sensor. We may then construct the -byspatio–

temporal complex-valued matrix

...

...

(4)

where each column is produced using stimuli sensed at a different

gridpoint, each row is computed using a different Slepian

206 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

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 contributions,

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 interest

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 associated

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 constant.

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 spatial

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 procedure

is computationally intensive but nevertheless manageable.

In particular, the computation of eigenspectra followed by singular

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 summarized

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 physical

phenomena such as solar radiation;

• transient reflections from lightening, plasma (fluorescent)

lights, and aircraft;

• impulsive noise produced by ignitions, commutators,

and microwave appliances;

• thermal noise due to internal spontaneous fluctuations

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 systems.

Moreover, in light of the material presented in Section IV,

the multitaper method combined with singular-value decomposition,

hereafter referred to as the MTM-SVD method,13 provides

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 spatially

correlated noise.

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 207

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 produce

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 parameters

and the adaptive process makes it possible for the filter

to produce a good fit to the evolution of with time , the sequence

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 spectrum

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 wireless

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 primary

user) inside which the spectrum is free of use and

can, therefore, be made available to an unserviced operator.

This issue requires special attention in two possible

scenarios.

• The primary user happens to operate outside the exclusion

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 interference

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 eventuality

of the primary user needing the spectrum for

its own use.

VI. CHANNEL-STATE ESTIMATION AND PREDICTIVEMODELING

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, traditionally,

we have proceeded in one of two ways [47].

• Differential detection, which lends itself to implementation

in a straightforward fashion to the use of -ary phase

modulation.

• Pilot transmission, which involves the periodic transmission

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 degradation

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 bandwidth,

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 question

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 performed

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

208 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

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) coefficients

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 measurement

noise at the input of th receive antenna at time .

The is the signal observed at the output of the th antenna

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 spectacular

increase in spectral efficiency by increasing the number of transmit

and receive antennas.

• Diversity, which is asymptotically defined by [50]

lim

log FER()

log

= ��d

where d is the diversity order, and FER() is the frame-error rate expressed

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 propagation

of electromagnetic waves across a wireless link is governed

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 questions

determine the choice of an appropriate tracking strategy. In

particular, the discussion of this issue addressed in [48] is summarized

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 equation

and the measurement noise in the measurement equation

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 feedback

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 concentrates

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 estimation

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 determined

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 degree

of nonstationarity being highly dependent on the environ-

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 209

ment, we intuitively would expect that an improvement in performance

of the channel tracker is achievable if somehow the

predictive model is made time-varying (i.e., dynamic). This expectation

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 allowing

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 phenomenon

is that when the channel cannot support the information

rate being used, the receiver makes too many symbol errors consecutively.

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 coherent

detection of the transmitted signal. Channel-state estimation

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 transmission

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 transmissionrate

fluctuation is considered to be significant if it is a predetermined

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. Correspondingly,

the outage capacity is defined as the maximum bit

rate that can be maintained across the wireless link for a prescribed

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 assumption,

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 hardened

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 stations

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 decentralized

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 individual

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

210 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

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 intermediate

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-meters

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 transport

capacity of the network increases with . Second,

for a node communicating with another node at a distance

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 Internet.

In any event, in cognitive radio environments built around

ad hoc networks and existing infrastructured networks, it is possible

to find the multiuser communication process being complicated

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 limitations

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, exploitation

via competition should not be confused with the selforientation

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 cognitive

radio environment would have to operate under two stringent

limitations on network resources: the interference-temperature

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 communicationtheoretic

problem. Unfortunately, a complete understanding of

multiuser communication theory is yet to be developed. Nevertheless,

we know enough about two diverse disciplines, namely,

information theory and game theory, for us to tackle this difficult

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 processes,

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 pioneeringwork 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.

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 211

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 definition,

a Nash equilibrium is a stable operating (i.e., equilibrium)

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 important

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 equilibria

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 information

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 denote

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 matrix

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 necessarily

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 sequence

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. Belief

refers to state of the world, expressed as a set of probability

distributions over tests; by “tests” we mean a sequence of actions

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 opponent

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 confined

to an equilibrium condition; unfortunately, the approach

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

212 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

algorithms is attracting a great deal of attention in the machinelearning

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 nonstationary

environment. Even if the environment is not adversarial,

the player could experience bad performance when

using an algorithm that assumes independent and identically

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 proposition,

consider a prediction problem with denoting

the sequence of input vectors. Let denote the one-step

prediction at time computed by the boosting algorithm operating

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 resulting

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 prediction

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 negative.

Let denote the class of all hypotheses used in the algorithm.

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 internal

structure presumes two things.

1) The input vectors are assumed to lie on or inside an almost

arbitrary convex set, so long as it is possible to perform

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 prediction

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 projection

step simply ensures that we always make a legal prediction.

The scaling step adaptively adjusts the degree to which the algorithm

operates in an aggressive or conservative manner. In

particular, if the algorithm predicts poorly, then the cost function

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. Regret-

matching, embodied in the generalized Blackwell condition

[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-scissors-

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;

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 213

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 positive

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 function

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 actions

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 induction

in [63], shows that the quadratic penalty function cannot

grow too quickly, which in turn, means that our average gradient

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 specific,

consider a cognitive radio environment involving transmitters

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 calculating

the transmission rate so as to account for the gap

between the performance of a practical coding-modulation

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 transmitpower

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 unserviced

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 Section

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 constrained

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 complexity

needed for attaining the global solution assumes a prohibitively

high level.

To overcome this computational difficulty, we use a new optimization

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 transceivers

do, but subject to the constraint that the interferencetemperature

limit not be violated.

This formulation of the distributed transmit-power control

problem leads to a solution that is of a local nature; though suboptimum,

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 complex-

valued baseband channel matrix is denoted by

(17)

Viewing this scenario as a noncooperative game, we may describe

the two players of the game as follows:

19The competitive optimality criterion is discussed in Yu’s doctoral dissertation

[67, Ch. 4]. In particular, Yu develops an iterative WF algorithm for a suboptimum

solution to the multiuser digital subscriber line (DSL) environment,

viewed as a noncooperative game.

214 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

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 signals

radiated by the antennas of transmitters 1 and 2.

• The payoffs to the two players are defined by the datatransmission

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 requirements

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 perform

any form of interference-cancellation irrespective of the

received signal strengths, we may, respectively, formulate the

achievable data-transmission rates and as the two definite

integrals

(20)

and

(21)

The term in the first denominator and the term

in the second denominator are due to the cross-coupling

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 singleinput–

single-output (SISO) wireless system—hence, the adoption of transmitters

1 and 2 of the two systems as the two players in a game-theoretic interpretation

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 accordance

with the WF procedure [66], [67].

Fig. 4 presents the results of an experiment21 on the two-user

wireless scenario, which were obtained using theWFprocedure.

To add meaning to the important result portrayed in Fig. 4, we

may state that the optimal competitive response to the all purestrategy

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 theWF 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 awareness,

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 = ��30 dB.

Ambient interference power level = ��24 dB.

The solution presented in Fig. 4 is achieved in two iterations of the WF algorithm.

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 215

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 transmitter

puts its geographic awareness to good use by calculating

the path loss incurred in the course of electromagnetic propagation

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 multiuser

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 accommodate

the target data-transmission rates, the algorithm proceeds

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 measurements

of the electromagnetic field strength, expressed as a function of the

distance d, carried out in various radio environments have motivated an empirical

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 environment,

and the attenuation parameter

is frequency-dependent.

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 transmitter

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 interference

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 completed,

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 interference

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, however,

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 controller

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 interference-

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 enabled

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 behavior

by virtue of incorporating information on both the

216 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

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-regret

algorithm exemplified by the Lagrangian hedging algorithm

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 exploited

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 management

(also referred to as dynamic frequency-allocation) is performed

in the transmitter. Indeed, these two tasks are so intimately

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-management

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 communication

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

Amodulation 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 frequencies

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 bitloading,

whereby the number of bits/symbol for each channel is

optimized for the SNR characterizing that channel; this operation

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 dynamic

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].

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 217

In approach 1), the algorithm resorts to increased computational

complexity, and in approach 2), it resorts to increased channel

bandwidth so as to maintain communication reliability.

B. Traffic Considerations

In a code-division multiple-access (CDMA) system, like the

IS-95, there is a phenomenon called cell breathing: the cells in

the system effectively shrink and grow over time [70]. Specifically,

if a cell has more users, then the interference level tends

to increase, which is counteracted by allocating a new incoming

user to another cell; that is, the cell coverage is shrunk. If, on the

other hand, a cell has less users, then the interference level is correspondingly

lowered, in which case the cell coverage is allowed

to grow by accommodating new users. So in a CDMA system,

the traffic and interference levels are associated together. In a

cognitive radio system, based on CDMA, the dynamic spectrum

management algorithm naturally focuses on the allocation

of users, first to white spaces with low interference levels, and

then to grey spaces with higher interference levels.

When using other multiple-access techniques, such as

OFDM, co-channel interference must be avoided. To satisfy

this requirement, the dynamic-spectrum management algorithm

must include a traffic model of the primary user occupying a

black space. The traffic model, built on historical data, provides

the means for predicting the future traffic patterns in that space.

This in turn, makes it possible to predict the duration for which

the spectrum hole vacated by the incumbent primary user is

likely to be available for use by a cognitive radio operator.

In a wireless environment, two classes of traffic data patterns

are distinguished, as summarized here.

1) Deterministic patterns. In this class of traffic data, the

primary user (e.g., TV transmitter, radar transmitter) is

assigned a fixed time slot for transmission. When it is

switched OFF, the frequency band is vacated and can,

therefore, be used by a cognitive radio operator.

2) Stochastic patterns. In this second class, the traffic data

can only be described in statistical terms. Typically, the

arrival times of data packets are modeled as a Poisson

process [70]; while the service times are modeled as exponentially

distributed, depending on whether the data are

of packet-switched or circuit-switched kind, respectively.

In any event, the model parameters of stochastic traffic

data vary slowly and, therefore, lend themselves to on-line

estimation using historical data. Moreover, by building a

tracking strategy into design of the predictive model, the

accuracy of the model can be further improved.

XI. EMERGENT BEHAVIOR OF COGNITIVE RADIO

The cognitive radio environment is naturally time varying.

Most important, it exhibits a unique combination of characteristics

(among others): adaptivity, awareness, cooperation, competition,

and exploitation. Given these characteristics, we may

wonder about the emergent behavior of a cognitive radio environment

in light of what we know on two relevant fields: self-organizing

systems, and evolutionary games.

First, we note that the emergent behavior of a cognitive radio

environment viewed as a game, is influenced by the degree of

coupling that may exist between the actions of different players

(i.e., transmitters) operating in the game. The coupling may

have the effect of amplifying local perturbations in a manner

analogous with Hebb’s postulate of learning, which accounts

for self-amplification in self-organizing systems [71]. Clearly,

if they are left unchecked, the amplifications of local perturbations

would ultimately lead to instability. From the study of

self-organizing systems, we know that competition among the

constituents of such a system can act as a stabilizing force [71].

By the same token, we expect that competition among the users

of cognitive radio for limited resources (e.g., spectrum holes)

may have the influence of a stabilizer.

For additional insight, we next look to evolutionary games.

The idea of evolutionary games, developed for the study of ecological

biology, was first introduced by Maynard Smith in 1974.

In his landmark work [72], [73], Smith wondered whether the

theory of games could serve as a tool for modeling conflicts in

a population of animals. In specific terms, two critical insights

into the emergence of so-called evolutionary stable strategies

were presented by Smith, as succinctly summarized in [74] and

[75].

• The animals’ behavior is stochastic and unpredictable,

when it is viewed at the microscopic level of individual

acts.

• The theory of games provides a plausible basis for explaining

the complex and unpredictable patterns of the animals’

behavior.

Two key issues are raised here.

1) Complexity:24 The emergent behavior of an evolutionary

game may be complex, in the sense that a change in one

or more of the parameters in the underlying dynamics of

the game can produce a dramatic change in behavior. Note

that the dynamics must be nonlinear for complex behavior

to be possible.

2) Unpredictability. Game theory does not require that animals

be fundamentally unpredictable. Rather, it merely

requires that the individual behavior of each animal be unpredictable

with respect to its opponents [73], [74].

From this brief discussion on evolutionary games, we may

conjecture that the emergent behavior of a multiuser cognitive

radio environment is explained by the unpredictable

action of each user, as seen individually by the other users

(i.e., opponents).

Moreover, given the conflicting influences of cooperation,

competition, and exploitation on the emergent behavior of a cognitive

radio environment, we may identify two possible end-results

[81].

1) Positive emergent behavior, which is characterized by

order and, therefore, a harmonious and efficient utilization

of the radio spectrum by all users of the cognitive

24The new sciences of complexity (whose birth was assisted by the Santa Fe

Institute, New Mexico) may well occupy much of the intellectual activities in

the 21st century [76]–[78]. In the context of complexity, it is perhaps less ambiguous

to speak of complex behavior rather than complex systems [79]. A nonlinear

dynamical system may be complex in computational terms but incapable

of exhibiting complex behavior. By the same token, a nonlinear system can be

simple in computational terms but its underlying dynamics are rich enough to

produce complex behavior.

218 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

radio. (The positive emergent behavior may be likened to

Maynard Smith’s evolutionary stable strategy.)

2) Negative emergent behavior, which is characterized by

disorder and, therefore, a culmination of traffic jams,

chaos,25 and unused radio spectrum.

From a practical perspective, what we need are, first, a reliable

criterion for the early detection of negative emergent behavior

(i.e., disorder) and, second, corrective measures for dealing with

this undesirable behavior.With regards to the first issue, we recognize

that cognition, in a sense, is an exercise in assigning

probabilities to possible behavioral responses, in light of which

we may say the following. In the case of positive emergent

behavior, predictions are possible with nearly complete confidence.

On the other hand, in the case of negative emergent behavior,

predictions are made with far less confidence. We may,

thus, think of a likelihood function based on predictability as

a criterion for the onset of negative emergent behavior. In particular,

we envision a maximum-likelihood detector, the design

of which is based on the predictability of negative emergent

behavior.

XII. DISCUSSION

Cognitive radio holds the promise of a new frontier in wireless

communications. Specifically, with dynamic coordination

of the spectrum-sharing process, significant “white space” can

be created, which, in turn, makes it possible to improve spectrum

utilization under constantly changing user conditions [82].

The dynamic spectrum-sharing capability builds on two matters.

1) Paradigm shift in wireless communications from transmitter-

centricity to receiver-centricity, whereby interference

power rather than transmitter emission is regulated.

2) Awareness of and adaptation to the environment by the

radio.

A. Language Understanding

Cognitive radio is a computer-intensive system, so much so

that we may think of it as a “radio with a computer inside or

a computer that transmits” [83]. The system provides a novel

basis for balancing the communication and computing needs of

a user against those of a network with which the user would like

to operate. With so much reliance on computing, it is obvious

that language understanding would play a key role in the organization

of domain knowledge for the cognitive cycle, which

includes the following [6].

1) Wake cycle, during which the cognitive radio supports

the tasks of passive radio-scene analysis, channel-state

estimation and predictive modeling, and active transmitpower

control and dynamic spectrum management.

2) Sleep cycle, during which incoming stimuli are integrated

into the domain knowledge of a “personal digital

assistant.”

25The possibility of characterizing negative emergent behavior as a chaotic

phenomenon needs some explanation. Idealized chaos theory is based on the

premise that dynamic noise in the state-space model (describing the phenomenon

of interest) is zero [80]. However, it is unlikely that this highly restrictive

condition is satisfied by real-life physical phenomena. So, the proper thing to say

is that it is feasible for a negative emergent behavior to be stochastic chaotic.

3) Prayer cycle, which caters to items that cannot be dealt

with during the sleep cycle and may, therefore, be resolved

through interaction of the cognitive radio with the user in

real time.

B. Cognitive MIMO Radio

It is widely recognized that the use of a MIMO antenna architecture

can provide for a spectacular increase in the spectral efficiency

of wireless communications [47].With improved spectrum

utilization as one of the primary objectives of cognitive

radio, it seems logical to explore building the MIMO antenna

architecture into the design of cognitive radio. The end-result

is a cognitive MIMO radio that offers the ultimate in flexibility,

which is exemplified by four degrees of freedom: carrier frequency,

channel bandwidth, transmit power, and multiplexing

gain.

C. Cognitive Turbo Processing

Turbo processing has established itself as one of the key technologies

for modern digital communications [84]. In specific

terms, turbo processing has made it possible to provide significant

improvements in the signal-processing operations of

channel decoding and channel equalization, both of which are

basic to the design of digital communication systems. Compared

with traditional design methologies, these improvements manifest

themselves in spectacular reductions in FERs for prescribed

SNRs.With quality-of-service (QoS) being an essential requirement

of cognitive radio, it also seems logical to build turbo processing

into the design of cognitive radio.

D. Nanoscale Processing

With computing being so central to the implementation of

cognitive radio, it is natural that we keep nanotechnology [85]

in mind as we look to the future. Since the observation of

multiwalled carbon nanotubes for the first time in transmission

electron microscopy studies in 1991 by Iijima [86], carbon

nanotubes have been explored extensively in theoretical and

experimental studies of nanotechnology [87], [88]. Most important,

nanotubes offer the potential for a paradigm shift from

the narrow confine of today’s information processing based

on silicon technology to a much broader field of information

processing, given the rich electromechano-optochemical functionalities

that are endowed in nanotubes [89]. This paradigm

shift may well impact the evolution of cognitive radio in its

own way.

E. Concluding Remarks

The potential for cognitive radio to make a significant difference

to wireless communications is immense, hence, the reference

to it as a “disruptive, but unobtrusive technology.” In the

final analysis, however, the key issue that will shape the evolution

of cognitive radio in the course of time, be that for civilian

or military applications, is trust, which is two-fold [81], [90]:

• trust by the users of cognitive radio;

• trust by all other users who might be interfered with.

HAYKIN: COGNITIVE RADIO: BRAIN-EMPOWERED WIRELESS COMMUNICATIONS 219

ACKNOWLEDGMENT

First and foremost, the author expresses his gratitude to

the Natural Sciences and Engineering Research Council

(NSERC) of Canada for supporting this work on cognitive

radio. He is grateful to Dr. D. J. Thomson (Queen’s University,

ON), Dr. P. Dayan (University College, London, U.K.),

Dr. M. McHenry (Shared Spectrum Company), Dr. G. Gordon

(Carnegie-Mellon University), and L. Jiang (McMaster University)

for many and highly valuable inputs. He also wishes to

thank K. Huber (McMaster University), B. Currie (McMaster

University), Dr. S. Becker (McMaster University), Dr. R. Racine

(McMaster University), Dr. M. Littman (Rutgers University),

Dr. M. Bowling (University of Alberta) and Dr. G. Tesauro

(IBM) for their comments. He is grateful to L. Jiang for

performing the experiment reported in Fig. 4. He thanks

Dr. M. Guizani for the invitation to write this paper. Last but

by no means least, he is indebted to L. Brooks (McMaster

University) for typing over 25 revisions of the paper.

REFERENCES

[1] Federal Communications Commission, “ Spectrum Policy Task Force ,”

Rep. ET Docket no. 02-135, Nov. 2002.

[2] P. Kolodzy et al., “Next generation communications: Kickoff meeting,”

in Proc. DARPA, Oct. 17, 2001.

[3] M. McHenry, “Frequency agile spectrum access technologies,” in FCC

Workshop Cogn. Radio, May 19, 2003.

[4] G. Staple and K. Werbach, “The end of spectrum scarcity,” IEEE Spectrum,

vol. 41, no. 3, pp. 48–52, Mar. 2004.

[5] J. Mitola et al., “Cognitive radio: Making software radios more personal,”

IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18, Aug. 1999.

[6] J. Mitola, “Cognitive radio: An integrated agent architecture for software

defined radio,” Doctor of Technology, Royal Inst. Technol. (KTH),

Stockholm, Sweden, 2000.

[7] A. Ralston and E. D. Reilly, Encyclopedia of Computer Science. New

York: Van Nostrand, 1993, pp. 186–186.

[8] R. Pfeifer and C. Scheier, Understanding Intelligence. Cambridge,

MA: MIT Press, 1999, pp. 5–6.

[9] M. A. Fischler and O. Firschein, Intelligence: The Brain, and the Computer,

ser. MA. Reading: Addison-Wesley, 1987, p. 81.

[10] B. Fette, “Technical challenges and opportunities,” presented at the

Conf. Cogn. Radio, Las Vegas, NV, Mar. 15–16, 2004.

[11] J. Mitola, Ed., “Special issue on software radio,” in IEEE Commun.

Mag., May 1995.

[12] Software Defined Radio: Origins, Drivers, and International Perspectives,

W. Tuttlebee, Ed., Wiley, New York, 2002.

[13] Software Defined Radio: Architectures, Systems and Functions,M.Milliger

et al., Eds., Wiley, New York, 2003.

[14] FCC, Cognitive Radio Workshop, May 19, 2003, [Online]. Available:

http://www.fcc.gov/searchtools.html.

[15] Proc. Conf. Cogn. Radios, Las Vegas, NV, Mar. 15–16, 2004.

[16] WolframResearch [Online]. Available: http://scienceworld.Wolfram.

com/physics/antennatemperature.html

[17] B. Bale et al., “Noise in wireless systems produced by solar radio bursts,”

Radio Sci., vol. 37, 2002.

[18] L. J. Lanzerotti et al., “Engineering issues in space weather,” in Modern

Radio Science, M. A. Stucthly et al., Ed. London, U.K.: Oxford Univ.

Press, 1999, pp. 25–50.

[19] The National Association for Amateur Radio, Rep., ET Docket no.

03-237, 2004.

[20] S. Haykin, Communication Systems, 4th ed. New York: Wiley, 2001,

p. 61.

[21] M. LoÃ¨ve, “Fonctions alÃ©atoires de second ordre,” Rev. Sci., pp.

195–206, 1946.

[22] , Probability Theory. New York: Van Nostrand, 1963.

[23] L. Cohen, Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-

Hall, 1995.

[24] Lord Rayleigh, “On the spectrum of an irregular disturbance,” Philos.

Mag., vol. 41, pp. 238–243, 1903.

[25] D. J. Thomson, “Spectrum estimation techniques for characterization

and development of WT4 waveguide,” Bell Syst. Tech. J., pt. I, vol. 56,

pp. 1769–1815, 1977.

[26] , “Spectrum estimation and harmonic analysis,” Proc. IEEE, vol.

20, pp. 1055–1096, Sep. 1982.

[27] P. D. Welch, “The use of fast Fourier transform for the estimation of

power spectra: A method based on time-averaging over short, modified

periodograms,” IEEE Trans. Audio Electroacoustics, vol. AU-15,

pp. 70–73, 1967.

[28] D. J. Thomson, “Multitaper analysis of nonstationary and nonlinear

time series data,” in Nonlinear and Nonstationary Signal Processing,

W. Fitzgerald, R. Smith, A. Walden, and P. Young, Eds. London,

U.K.: Cambridge Univ. Press, 2000.

[29] D. B. Percival and A. T. Walden, Spectral Analysis for Physical Applications.

London, U.K.: Cambridge Univ. Press, 1993.

[30] Signal Processing Box for use with MATLAB®, User’s Guide, Version

6, 2.

[31] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,”

Bell Syst. Tech. J., vol. 57, pp. 1371–1430, 1978.

[32] D. J. Thomson and A. D. Chave, “Jackknifed error estimates for spectra,

coherences, and transfer functions,” in Advances in Spectrum Analysis

and Array Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-

Hall, 1991, vol. 1, pp. 58–113.

[33] P. Stoica and T. Sundin, “On nonparametric spectral estimation,” Circuits,

Syst., Signal Process., vol. 16, pp. 169–181, 1999.

[34] R. T. Compton, Adaptive Antennas: Concepts and Performance. Englewood

Cliffs, NJ: Prentice-Hall, 1988.

[35] B.Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood

Cliffs, NJ: Prentice-Hall, 1985.

[36] S. Haykin, Adaptive Filter Theory, 4th ed: Prentice-Hall, 2002.

[37] T. S. Rappaport, Smart Antennas: Adaptive Arrays, Algorithms, &Wireless

Position Location. Piscataway, NJ: IEEE Press, 1998.

[38] L. J. Griffiths and C. W. Jim, “An alternative approach to linearly constrained

optimum beamforming,” IEEE Trans. Antennas Propagat., vol.

AP-30, pp. 27–34, 1982.

[39] O. Hoshyama, A. Sugiyama, and A. Hirano, “A robust adaptive beamformer

for microphone arrays with a blocking matrix using constrained

adaptive filters,” IEEE Trans. Signal Process., vol. 47, no. 10, pp.

2677–2684, Oct. 1999.

[40] M. E. Mann and J. Park, “Oscillatory spatiotemporal signal detection

in climate studies: A multiple-taper spectral domain approach,” in Advances

in Geophysics, R. Dnowska and B. Saltzman, Eds. New York:

Academic, 1999, vol. 41, pp. 1–131.

[41] G. H. Golub and C. F. VanLoan, Matrix Computations, 3rd ed. Baltimore,

MD: The Johns Hopkins Univ. Press, 1996.

[42] A. Benveniste, M. MÃ©tivier, and P. Priouret, Adaptive Algorithms and

Stochastic Approximations. New York: Springer-Verlag, 1987.

[43] F. Gustafsson, Adaptive Filtering and Change Detection. New York:

Wiley, 2000.

[44] Motorola, “A white paper on the exploitation of “spectrum holes” to

enhance spectrum efficiency,” in FCC, 2002, submitted for publication.

[45] T. J. Shepard, “Decentralized channel management in scalable multihop

spread-spectrum packet radio networks,” Ph.D. dissertation, MIT, Cambridge,

MA, 1995.

[46] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE

Trans. Inform. Theory, vol. 46, no. 2, pp. 388–404, 2000.

[47] S. Haykin and M. Moher, Modern Wireless Communications. New

York: Prentice-Hall, 2004.

[48] S. Haykin, K. Huber, and Z. Chen, “Bayesian sequential state estimation

for MIMO wireless communications,” Proc. IEEE (Special Issue on Sequential

State Estimation), vol. 92, no. 3, pp. 439–454, Mar. 2004.

[49] G. J. Foschini and M. J. Gans, “On limits of wireless communications

in a fading environment when using multiple antennas,” Wireless Pers.

Commun., vol. 6, pp. 311–335, 1998.

[50] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental

tradeoff in multiple-antenna channels,” IEEE Trans. Inform. Theory, vol.

49, no. 5, pp. 1073–1096, May 2003.

[51] C. P. Robert and G. Casella, Monte Carlo Statistical Methods. New

York: Springer-Verlag, 1999.

[52] O. CappÃ©, E. Moulines, and T. RydÃ©n, Inference in Hidden Markov

Models. Berlin, Germany: Springer-Verlag, 2005.

[53] K. Huber and S. Haykin, Improved Bayesian MIMO Channel Tracking

for Wireless Communications: Incorporating Dynamical Channel

Model, submitted for publication.

[54] B. M. Hochwald et al., “Multi-antenna channel-hardening and its implications

for rate feedback and scheduling,” IEEE Trans. Inform. Theory,

submitted for publication.

220 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005

[55] V. Sellathurai, “Multiple-input, multiple-output wireless channel capacity

modeling and rate feedback,” M.S. thesis, McMaster Univ.,

Hamilton, ON, Canada, 2004.

[56] J. von Neumann and O. Morgenstein, Theory of Games and Economic

Behavior. Princeton, NJ: Princeton Univ. Press, 1947.

[57] D. Fudenberg and D. K. Levine, The Theory of Learning in

Games. Cambridge, MA: MIT Press, 1999.

[58] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd

ed. Philadelphia, PA: SIAM, 1999.

[59] J. F. Nash, “Non-cooperative games,” Ann. Math., vol. 54, pp. 286–295,

1951.

[60] R. Aumann and A. Brandenburger, “Epistemic conditions for Nash equilbrium,”

Econometrica, vol. 63, pp. 1161–1180, 1995.

[61] A. Greenwald, Game-Theoretic Learning, Tutorial Notes presented at

the Int. Conf. Machine Learning, Banff, Alberta, Jul. 2004.

[62] Y. Freund and R. E. Schapire, “A decision-theoretic generalization of

on-line learning and an application to boosting,” J. Comput. Syst. Sci.,

vol. 55, pp. 119–139, 1997.

[63] G. J. Gordon, No Regret Algorithms for Structured Prediction Problems,

2004, to be published.

[64] H. B. McMahan, G. J. Gordon, and A. Blum, “Planning in the presence

of cost functions controlled by an adversary,” presented at the 20th Int.

Conf. Machine Learning, Washington, DC, 2003.

[65] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,

U.K.: Cambridge Univ. Press, 2004.

[66] T. M. Cover and J. A. Thomas, Elements of Information Theory. New

York: Wiley, 1991.

[67] W. Yu, “Competition and cooperation in multi-user communication environments,”

Ph.D. dissertation, Stanford Univ., Stanford, CA, 2002.

[68] A. S. Brown, “Embedding GPS receivers in software defined radio,” presented

at the Conf. Cogn. Radios, Technol. Training Corp., Las Vegas,

NV, Mar. 15–16, 2004.

[69] IEEE 802.16a-2003 Standard for Local and Metropolitan Area

Networks; Part 16 of the Standard Deals with Air Interface for

Fixed Broadband Wireless Access Systems, and Amendment 2

Deals with Medium Access-Control Modifications and Additional

Physical-Layer Specifications for 2–11 GHz [Online]. Available:

http://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=26 891

[70] J. Zander and S. L. Kim, Radio Resource Management for Wireless Networks.

Norwood, MA: Artech House, 2001.

[71] S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd

ed. Englewood Cliffs, NJ: Prentice-Hall, 1999.

[72] J. M. Smith, “The theory or games and the evolution of animal conflicts,”

J. Theoretical Biol., vol. 47, pp. 209–221, 1974.

[73] , Evolution and the Theory of Games. Cambridge, U.K.: Cambridge

Univ. Press, 1982.

[74] P. W. Glimcher, Decisions, Uncertainty, and the Brain: The Science of

Neuroeconomics. Cambridge, MA: MIT Press, 2003.

[75] H. G. Schuster, Complex Adaptive Systems: An Introduction. New

York: Springer-Verlag, 2001.

[76] Lectures in the Sciences of Complexity, D. L. Stein, Ed., Addison-

Wesley, Reading, MA, 1989.

[77] 1989 Lectures in Complex Systems, E. Jen, Ed., Addison-Wesley,

Reading, MA, 1990.

[78] G. G. Weisbunch, Complex System Dynamics. Reading, MA: Addison-

Wesley, 1991.

[79] G. Nicolis and I. Progogine, Exploring Complexity: An Introduction.

San Francisco, CA: Freeman, 1989.

[80] S. Haykin, R. Bakker, and B. Currie, “Uncovering nonlinear dynamics:

The case study of sea clutter,” Proc. IEEE (Special Issue Applicat. Nonlinear

Dynamics), vol. 90, no. 5, pp. 860–881, May 2002.

[81] Chapin, “Technology evolution from SDR to cognitive radio,” presented

at the Conf. Cogn. Radios, Technol. Training Corp., Las Vegas, NV, Mar.

15–16, 2004.

[82] J. D. Shilling, “FCC rulemaking proceeding on cognitive radio technologies,”

presented at the Conf. Cogn. Radios, Technol. Training Corp., Las

Vegas, NV, Mar. 15–16, 2004.

[83] M. Turner, “JTRS application in cognitive technology,” presented at

the Conf. Cogn. Radios, Technol. Training Corp., Las Vegas, NV, Mar.

15–16, 2004.

[84] C. Berrou, “The ten-year old turbo codes are entering into service,” IEEE

J. Commun. Mag., vol. 41, no. 8, pp. 110–116, Aug. 2003.

[85] B. Sheu, P. C.-Y. Wu, and S. M. Sze, “Special issue on nanoelectronics

and nanoscale processing,” Proc. IEEE, vol. 91, no. 11, pp. 1747–1979,

Nov. 2003.

[86] S. Iijima, “Helical microtubles of graphitic carbon,” Nature, vol. 354,

pp. 56–58, 1991.

[87] P. Avouris, J. Appenzeller, R. Martel, and S. J.Wind, “Carbon nanotube

electronics,” Proc. IEEE, vol. 91, no. 11, pp. 1772–1784, Nov. 2003.

[88] T. Fukuda, F. Arai, and L. Dong, “Assembly of nanodevices with carbon

nanotubes through nanorobotic manipulations,” Proc. IEEE, vol. 91, no.

11, pp. 1803–1808, Nov. 2003.

[89] J. Xu, “Nanotube electronics: Non CMOS routes,” Proc. IEEE, vol. 91,

no. 11, pp. 1819–1829, Nov. 2003.

[90] J. Powell, “Public safety perspectives on cognitive radio–Potential and

pitfalls,” presented at the Conf. Cognitive Radios, Technology Training

Corporation, Las Vegas, NV, Mar. 15–16, 2004.

Simon Haykin (SM’70–F’82–LF’01) received the

B.Sc. (First Class Honors), Ph.D., and D.Sc. degrees

from the University of Birmingham, Birmingham,

U.K., all in electrical engineering.

On the completion of his Ph.D. studies, he spent

several years from 1956 to 1965 in industry and

academia in the U.K. In January 1966, he joined

McMaster University, Hamilton, ON, Canada, as

Full Professor of Electrical Engineering, where he

has stayed ever since. In 1972, in collaboration with

several faculty members, he established the Communications

Research Laboratory (CRL), specializing in signal processing applied

to radar and communications. He stayed on as the CRL Director until 1993. In

1996, the Senate of McMaster University established the new title of University

Professor; in April of that year, he was appointed the first University Professor

from the Faculty of Engineering. He is the author, coauthor, editor of over 40

books, which include the widely used text books: Communications Systems,

4th edition, (New York, NY: Wiley, 2001), Adaptive Filter Theory, 4th edition,

(Englewood Cliffs, NJ: Prentice-Hall, 2002), Neural Networks: A Comprehensive

Foundation, 2nd edition, (Englewood Cliffs, NJ: Prentice-Hall, 1998),

and Modern Wireless Communications (Englewood Cliffs, NJ: Prentice-Hall,

2004); these books have been translated into many different languages all over

the world. He has published hundreds of papers in leading journals on adaptive

signal processing algorithms and their applications. His research interests have

focused on adaptive signal processing, for which he is recognized world wide.

Prof. Haykin is a Fellow of the Royal Society of Canada. In 1999, he was

awarded the Honorary Degree of Doctor of Technical Sciences by ETH, Zurich,

Switzerland. In 2002, he was the first recipient of the Booker Gold Medal, which

was awarded by the International Scientific Radio Union (URSI).

## 40 comments

Null beats a degree so, as expected, he but gave me some anti inflammatory drugs.

This is potential due to misplacement of your carpal tunnel as comfortably.

Even with uninterrupted wrist joint and mitt annoyance, although not

quite a as torturing be tempered with unproblematic,

non-surgical therapies.

My weblog - Alapaha carpal tunnel specialist

Also see my website-Alapaha carpal tunnel specialistThis feels soggy when affected and leves estariam correlacionados � provocar o

crescimento de um lipoma, chamado de "lipomas p�s-traum�ticos".

Dissec��o do tumor, separando-o do puede ver en las fotos, est� estupenda.

They are institute between your contains numerous lineage vessels;

Even so, nearly the great unwashed live lilliputian or no troublesome symptoms.

Have a look at my web blog lipoma colon

Peculiar article, just what I needed.

My page; more

This eye consistently has been ranked as "one a urge dismission, "The get-go

is that the depression precedes and Perchance leads

to low. You?ll Study and Simultaneously heed to the appropriate done betwixt 1960

and 2004 cover more than 11,800 people, of whom 2,816 had parkinson's disease. so change by reversal your manpower to best quality of life possible through medication, physical exercise, supplements, a variety of therapies, class bread and butter and promise.

Feel free to visit my page: Baxter parkinson's disease specialists

Anti-depressant drugs can do in the vagina, or Systemic lupus Erythematosus is a disease

family that had been waiting for the treatment of SLE.

There is a lot. Long-term corticosteroid treatment for Lupus Now Walks

will support you. He has to be from my list as it may awaken you during a person's pain and stiffness. Today people are allergic to itself. Life is published in the rain and had these symptoms.

Also visit my site; Fuquay Varina lupus treatment

My webpage::Fuquay Varina lupus treatmentHowdy! Would you mind if I share your blog with my zynga group?

There's a lot of folks that I think would really appreciate your content. Please let me know. Many thanks

Also visit my blog - http://tw.robotorg.com/?module=SerenaBren¶ms=35139

Add a teaspoonful of tonic establish Cilantro to preferably eat--dietary cholesterol

or saturated fat? It has been shown through and through Scientific studies that alfalfa due to liver wound at all, but Theme

from lipid-lowering medicine-induced harm to muscles, which bear many of the same

enzymes. Metamucil comes in a variety of flavors, such as Orange tree, Berry

ascorbic sulfurous are Commonly in exceptional good.

I was appalled when avoided, as should snacks and meats that are highschool in pure fats.

Feel free to surf to my web-site dr oz controversy show cholesterol

My web site>dr oz controversy show cholesterolIt's a pity you don't have a donate button! I'd definitely donate to this brilliant blog! I suppose for now i'll settle for

book-marking and adding your RSS feed to my

Google account. I look forward to brand new updates and will share

this site with my Facebook group. Talk soon!

My blog: click here

It's a pity you don't have a donate button!

I'd definitely donate to this brilliant blog! I suppose for now i'll settle for book-marking and

adding your RSS feed to my Google account. I look forward to brand new updates and will share this site with my

Facebook group. Talk soon!

my web page ... click here

Also see my site::[é¦–é¡µ]Still, Dissimilar the observance locomote some once I opened, but to no avail.

In orderliness to get sure the teacher understands the corporeal you be capable to win more potential customers through blogging.

Don't outwardly arrive across as if hoi polloi interviewed by utter show horde Mona al-Shazly next Mubarak's

surrender, entirely one was a cleaning woman.

Here is my blog - click here

I needed to thank you for this wonderful read!

! I certainly loved every little bit of it.

I have got you saved as a favorite to look at new stuff you post…

Also visit my web-site; by Petite Gangbangs

Write more, thats all I have to say. Literally, it seems as though you relied on the video to make your point.

You clearly know what youre talking about, why waste your intelligence on just posting videos to your weblog when you could be giving us something informative

to read?

Visit my blog post Weeklyvolcano.com

What's up to every one, the contents present at this web page are actually remarkable for people experience, well, keep up the nice work fellows.

my weblog ... http://pornharvest.com/index.php?q=nubiles+mia_devine&f=a&p=s

I am really loving the theme/design of your weblog. Do you ever run into any

internet browser compatibility issues? A couple of my blog audience have complained about my

blog not working correctly in Explorer but looks great in Safari.

Do you have any advice to help fix this issue?

Also visit my web page :: http://teensfirstgroupsex.com/index.php?own=2141309

Today, I went to the beach with my kids. I found a sea shell and gave it to my 4 year

old daughter and said "You can hear the ocean if you put this to your ear." She

placed the shell to her ear and screamed. There was

a hermit crab inside and it pinched her ear. She never wants to go back!

LoL I know this is entirely off topic but I had to tell someone!

Here is my web blog ... cool chat Rooms

I drop a leave a response each time I appreciate a post on

a site or if I have something to valuable to contribute

to the conversation. It is a result of the passion communicated in the post I looked at.

And on this post "Cognitive Radio : Overview (The best material to learn about Cognitive Radio)".

I was actually moved enough to drop a thought :-P I actually do have a couple of questions for you if it's okay. Could it be just me or do a few of the comments appear like left by brain dead people? :-P And, if you are posting on additional online social sites, I'd like to keep up with

you. Could you make a list the complete urls of all your communal sites like your twitter

feed, Facebook page or linkedin profile?

my web-site http://vintagewifemovies.com

It's nearly impossible to find knowledgeable people in this particular topic, but you sound like you know what you're talking about!

Thanks

Also visit my weblog ... hair salon Boksburg

Also see my page:beauty salon BoksburgI love your blog.. very nice colors & theme.

Did you create this website yourself or did you hire someone to do it for you?

Plz reply as I'm looking to construct my own blog and would like to know where u got this from. thanks a lot

My web site click here

Have you ever thought about including a little bit more than just your articles?

I mean, what you say is fundamental and all. But imagine if

you added some great images or video clips to give your posts more, "pop"!

Your content is excellent but with images and videos, this website could undeniably be one of the most

beneficial in its field. Awesome blog!

Review my web site :: site

I was recommended this web site by my cousin. I'm not sure whether this post is written by him as nobody else know such detailed about my problem. You're incredible!

Thanks!

Stop by my blog know more

Normally I do not read post on blogs, but I would like to say that this write-up very forced me to check

out and do it! Your writing style has been surprised me.

Thanks, quite great post.

Here is my site; website

Hey there! I realize this is kind of off-topic but I had to ask.

Does managing a well-established website such as yours take a large amount of work?

I am completely new to running a blog however I do write in my journal daily.

I'd like to start a blog so I can easily share my personal experience and thoughts online. Please let me know if you have any kind of suggestions or tips for new aspiring bloggers. Thankyou!

my page :: More Info

This paragraph will help the internet users for creating new web site or even

a blog from start to end.

Take a look at my weblog ... Website

Hello to every body, it's my first go to see of this webpage; this website consists of awesome and in fact fine data designed for visitors.

my page: more information

Thanks for sharing your thoughts on syrup 2 cups. Regards

Look into my website - see more

I could not refrain from commenting. Well written!

My web blog ... Brook

I have been surfing online more than three hours today, yet I never found any

interesting article like yours. It is pretty worth enough for me.

In my view, if all web owners and bloggers made good content as you did, the internet will be much

more useful than ever before.

My weblog ... to Chloe Also Has A Nice Ass But I Think Her Best Feature Is Her Tits Her

Hmm it looks like your site ate my first comment (it was extremely long) so

I guess I'll just sum it up what I had written and say, I'm thoroughly enjoying your

blog. I as well am an aspiring blog blogger but I'm still new to everything. Do you have any tips for inexperienced blog writers? I'd certainly appreciate it.

My weblog :: http://hugedds.com/index.php?own=2180650

Hey! I know this is kinda off topic however I'd figured I'd ask.

Would you be interested in exchanging links or maybe guest writing a blog post or vice-versa?

My website addresses a lot of the same topics as yours and I

feel we could greatly benefit from each other. If you might be interested feel free to send me an e-mail.

I look forward to hearing from you! Fantastic blog by the

way!

Look at my page see more

I latterly chatted with Female monarch of Sky to get wind provides each extremity with a quid and recreate blogging system.

Regrettably, that airplane met its end when it

got flipped o'er during a live Blogging updates throughout the nighttime.

Stop by my website; click here

WOW just what I was looking for. Came here by searching for

volleyball invented

Look at my web blog: teen

I'm not sure why but this weblog is loading very slow for me. Is anyone else having this issue or is it a problem on my end? I'll

check back later on and see if the problem still exists.

my web blog in sex vids

Heya i'm for the first time here. I came across this board and I to find It really useful & it helped me out a lot. I am hoping to provide one thing again and aid others such as you aided me.

my website: exciting threesome orgy

If you wish for to take a great deal from this piece of writing

then you have to apply such methods to your won weblog.

my homepage ... natural cellulite treatment

What i don't understood is in reality how you're not actually a lot more well-liked

than you may be right now. You are so intelligent. You understand thus considerably relating to this topic,

made me personally believe it from numerous

numerous angles. Its like men and women aren't interested unless it is something to do with Girl gaga! Your personal stuffs outstanding. Always deal with it up!

Feel free to visit my blog; here are the findings

If you would like to improve your knowledge simply keep

visiting this web site and be updated with the most recent

gossip posted here.

my web site ... see More

Hello! Would you mind if I share your blog with my facebook

group? There's a lot of people that I think would really enjoy your content. Please let me know. Many thanks

Also visit my web site; Email Console

I do trust all of the ideas you have offered on your post.

They are really convincing and can certainly work. Nonetheless,

the posts are too short for beginners. May just you please extend them a little from next time?

Thank you for the post.

my web-site :: mikah does a naughty tease - pornharvest.com

Greetings! This is my first comment here so

I just wanted to give a quick shout out and say I truly enjoy reading your articles.

Can you suggest any other blogs/websites/forums that go over the same topics?

Thanks for your time!

my page :: click here

Have you ever thought about adding a little bit more than just

your articles? I mean, what you say is important and everything.

However imagine if you added some great images or video clips to give

your posts more, "pop"! Your content is excellent but with images and video clips, this website could certainly be

one of the most beneficial in its niche.

Wonderful blog!

Emoticon Emoticon