information-theoretic aspects of cognitive radio

Intro CT model Det channel IT for interference IT Cog Channel

Information-Theoretic Aspectsof Cognitive Radio

Petar PopovskiAalborg University, Denmark

Cognitive Radios and Networks: Theory and PracticePh. D. course at CTVR, Trinity College Dublin

May 12-16 2013

Petar Popovski, AAU, DK Information-Theoretic Aspects of Cognitive Radio


Table of contents

1 Introductory Notes

2 Elementary communication-theoretic model

3 A Simple Deterministic Binary Channel

4 Informaton-Theoretic Model of Interference

5 Information-theoretic Cognitive Channel



The problem of spectrum shortage

The concept of cognitive radio has emerged as a potentialsolution to the problem of spectrum shortage

But to understand this shortage,we first need to understand what a spectrum is.

(Broadly) Spectrum is a physical, electromagnetic resourcethat can be used for transferring information.Frequency is one characteristic of a certain communicationsignal, but there are other characteristics.Why the spectrum allocation is done by frequency division?What could be the alternatives?What does the term ”allocation of a frequency band” means?



The problem of spectrum shortage

Spectrum usage is a process that has evolved through decades

The spectrum is populated by different ”device civilizations”.Backward compatibility, respect for legacy systems.Ethics in the air: good behavior when using the spectrum.

Observation

Spectrum shortage largely occurs due to the fact thatat the time of specifying spectrum usage policywe are uncertain about the future devices and servicesthat will use that spectrum,as well as their technical capabilities.



Technology axioms behind the spectrum policy

Spectrum policy and regulation must makebasic assumptions about the available technologyand prescribe rules that can be implemented by thattechnology.

Due to cost/size/complexity/battery constrain, not everydevice is supposed to use all known wireless technologies.

We therefore need to carefully set the minimal technologyrequirements and expectations from the other devices.

Those can be observed as technology axioms.Examples include: Listen Before Talk (LBT),duty cycle regulation,frequency hopping, etc.



A simple utopian algorithm for ideal spectrum usage

1 Monitor the new radio technologies;

2 If a new wireless technology X emerges,evaluate the spectrum usage that would occuras if all devices have access to technology X .

Not all the devices need to use the technology X directly,but they need to be designed to coexist with the devices thatuse technology X .

3 Rebuild/Reprogram all the devices in the world to complywith the technology X .

4 Goto step 1.

Since Step 4 is infeasible, we will always have to live with someprimary or legacy systems that will set our design constraint.



Example of idealized spectrum usage

A group of devices operates in unlicensed spectrum usinga limit on the duty cycle to control the spectrum access.

The default operation is to use a duty cycle of at most 5 % inorder to coexist with proximate devices and systems.

SIGNAL

INTERFERENCE




But, assume that each device can receive a certain commandthat can shut off the duty cycle and operate as instructedby a hypothetic superordinate system.

SIGNAL

INTERFERENCE

CONFIG. SIGNAL




Several years later, a reconfigurable system with multipleantennas, strong processing capability starts to use the samespectrum.

The new system uses the opportunity to take control over thelegacy devices and impose a different operation.

SIGNAL

INTERFERENCE

CONFIG. SIGNAL




The new system can allocate resources for the legacy ones,perform relaying, help to cancel interference, do a processingon behalf of it, etc.It can never be worse than the original legacy system, as dutycycle operation is a special case of all th operation modes thatcan be enforced by the new system.

SIGNAL

INTERFERENCE

CONFIG. SIGNAL

NEW SIGNAL



What is a Cognitive Radio (CR)

Cognitive radio is an idea how to repopulate the spectrum byrespecting the ”civil rights” of the existing devices andsystems.

Conceptually, the cognitive radio needs to model all theconstraints coming from the legacy devices and ensure thebest possible operation for itself, and perhaps the legacysystems.

. . . but the research question we need to also start to ask is

How to specify the systems and spectrum policies such thatafter several years they can act as smart legacy devicesand enable sustainable introduction of new wireless technologyin the same spectrum?



A more technical definition of CR

A definition that covers the widespread understanding ofcognitive radio:

A cognitive radio system is aware about its environment andadapts its operation in order to maximize the efficiency ofsecondary spectrum usage, while satisfying the constraints ofthe legacy systems that appear as primary spectrum users.

Awareness is a rather broad term,while communication/information theory requires precisedefinition of what it means.



A simple communication-theoretic model of a CR

We consider a very simple model with a singleprimary and a single secondary link.

PTX is a primary transmitter.

PRX is a primary receiver.

STX is a secondary transmitter.

SRX is a secondary receiver.

This model is far from general

Intra-system complexity is minimal(brought to a single link)

Yet, it is sufficient to illustrate severalimportant communication paradigms andtechniques that can arise in a CR setup.

PTX PRX

STX SRX



The primary link

Knowledge that is a side information usedby the cognitive (secondary) system.

traffic patternssignaling procedureschannel conditionscodebooks(part of the) messages sent in the legacysystem.

Need for suitable assumptions about whatthe primary (legacy) system knows anddoes.

PTX PRX

STX SRX



Accommodating secondary transmissions in a legacy band

Two general ways to allow secondary communication in agiven spectrum are:

Usage of certain unused margin in the operation of the primarysystem.The CR disrupts the normal operation of the legacy system,but uses part of its resources to help the operation of thelegacy system.

Margin-type CR paradigms are interweave CR andunderlay CR

Help-type CR paradigm is the overlay CR



Interweave CR

Primary margin in terms of unusedtime/frequency resources.

NB: sometimes this type of CR iscalled “overlay” in thenon-information-theoreticliterature.

CR needs to detect the spectrumhole and use it (sensing, prediction,database, explicit signaling, etc. )

Secondary-to-primary interferenceis avoided

Opportunistic, orthogonaltransmission

PTX PRX

STX SRX

time fre

q



Underlay CR

Interference towards the legacy systemis not avoided, but is controlled.

Primary margin in terms ofallowed interference

The interference constraint met by

Using multiple antennas to beamform and guide the interferenceaway from the primary receiver;

Spreading the signal in order to gobelow the noise floor (e. g.Ultra-WIdeband (UWB)).

The problem is that the exactinterference situation at theprimary receiver is not known.

PTX PRX

STX SRX

time

freq

primary

primary

primary



Overlay CR

In this model the CR knowsmore about the legacy system:codebooks used, messages, etc.

Q: Is the primary codebookknowledge a securitythreat?

The knowledge of the primarymessages is harder to justify.

Becomes viable if weconsider retransmittedmessages.

The interference is contained,mitigated or even turned intoan advantage for the primary. time

freq

primary

primary

PTX PRX

STX SRX

sec

sec



A simple communication-theoretic model of a CR (contd.)

In the most general case there are six relevantlinks in this model.

PTX PRX

STX SRX



What we will do with the communication-theoretic model

Under the actual communication andconsidering the intended transfer ofinformation, some of the links are directed.

We are going to:

investigatethe impact of the individual links;

relate itto the three CR paradigms;

introducesuitable communication strategies.

PTX PRX

STX SRX



A communication channel

In practice, a link or a communication channel can be designed ina large variety of ways considering types of modulation, codes,frequency bandwidth, etc.

Shannon’s general model of a communication system:

INFORMATIONSOURCE

MESSAGE

TRANSMITTER

SIGNAL RECEIVEDSIGNAL

RECEIVER

MESSAGE

DESTINATION

NOISESOURCE

Fig. 1—Schematic diagram of a general communication system.

a decimal digit is about 3 13 bits. A digit wheel on a desk computing machine has ten stable positions andtherefore has a storage capacity of one decimal digit. In analytical work where integration and differentiationare involved the base e is sometimes useful. The resulting units of information will be called natural units.Change from the base a to base b merely requires multiplication by logb a.

By a communication system we will mean a system of the type indicated schematically in Fig. 1. Itconsists of essentially five parts:

1. An information sourcewhich produces a message or sequence of messages to be communicated to thereceiving terminal. The message may be of various types: (a) A sequence of letters as in a telegraphof teletype system; (b) A single function of time f t as in radio or telephony; (c) A function oftime and other variables as in black and white television — here the message may be thought of as afunction f x y t of two space coordinates and time, the light intensity at point x y and time t on apickup tube plate; (d) Two or more functions of time, say f t , g t , h t — this is the case in “three-dimensional” sound transmission or if the system is intended to service several individual channels inmultiplex; (e) Several functions of several variables— in color television the message consists of threefunctions f x y t , g x y t , h x y t defined in a three-dimensional continuum— we may also thinkof these three functions as components of a vector field defined in the region — similarly, severalblack and white television sources would produce “messages” consisting of a number of functionsof three variables; (f) Various combinations also occur, for example in television with an associatedaudio channel.

2. A transmitter which operates on the message in some way to produce a signal suitable for trans-mission over the channel. In telephony this operation consists merely of changing sound pressureinto a proportional electrical current. In telegraphy we have an encoding operation which producesa sequence of dots, dashes and spaces on the channel corresponding to the message. In a multiplexPCM system the different speech functions must be sampled, compressed, quantized and encoded,and finally interleaved properly to construct the signal. Vocoder systems, television and frequencymodulation are other examples of complex operations applied to the message to obtain the signal.

3. The channel is merely the medium used to transmit the signal from transmitter to receiver. It may bea pair of wires, a coaxial cable, a band of radio frequencies, a beam of light, etc.

4. The receiver ordinarily performs the inverse operation of that done by the transmitter, reconstructingthe message from the signal.

5. The destination is the person (or thing) for whom the message is intended.

We wish to consider certain general problems involving communication systems. To do this it is firstnecessary to represent the various elements involved as mathematical entities, suitably idealized from their

2




A good way to think about the communication channel is that it isa part of the system that we are unwilling or unable to change.

+ + + + +

+ +x

xx

y

yy

h

hh

z

zz

v

w

(a) (b)

(c)

MAP DMAP




A communication channel defines certain stochastic relationbetween channel input x and channel output y .

Two common channel models:

Discrete Memoryless Channel (DMC)

The channel input/output belong to discrete sets x ∈ X , y ∈ YDescribed through the probability distribution p(y |x).

Additive White Gaussian Noise (AWGN) channel

The channel input/output are continuous variables, real orcomplex.

Y = X + Z

where Z is a Gaussian noise.



Primary vs. secondary communication channel

The primary communication channel isdesigned as if the secondary channel does notexist.

PTX PRX

The restrictions coming from the operation ofthe primary are part of the secondarycommunication channel.

STX SRX



A Simple Deterministic Binary Channel

1 is sent by transmitting power

0 is sent by staying silent

No noise and random errors.

If two or more transmitters are sending to the same receiverthen 1 is received.

The above is sufficient to specify the primary channel. But, tospecify the secondary channel, we need to fix the way the primarychannel is used.

Objective

Find as high as possible error-free secondary rate provided that theprimary operates error-free.



Some notation

All the inputs and outputs arebinary:

XP ,XS ,YP ,YS ∈ {0, 1}

PTX PRX

STX SRX

XP

XS

YP

YS



The Trivial Case: Noninterfering links

Both links can operate at 1 bit/c. u.

This means that XP and XP get values 0 or 1with equal probability 0.5.

PTX PRX

STX SRX



STX interferes at PRX

Channel specification:

XP XS YP YS

0 0 0 00 1 1 11 0 1 01 1 1 1

We need to specify how the primary operates.

If the primary communicates with 1 bit/c.u.then the secondary rate is zero.

PTX PRX

STX SRX




Let now the primary be less efficient:

Each data bit is 0 or 1 with equalprobability

Each data bit is repeated 3 times and thereceiver decides based on majority voting.

The primary data rate is 13 bits/c.u.

What does this mean for the secondary?

PTX PRX

STX SRX




Primary data:XP : 000,111,000,000,111,111,000, . . .

Let the secondary send at 1 bit/c.u:XS : 010,001,110,010,001,010,111, . . .

The primary receiver receives two erroneous bits: YP :010,111,110,010,111,111,111, . . .

The secondary receiver gets all the secondary bits correctly.

Conclusion:The secondary can send at most a single 1 every three symbols.




New secondary coding strategy:

Instead of transmitting one channel use at the time, it groupsthree channel uses into a single symbol of a new channel(!).

The possible inputs to the new channel are: 000,001,010,100;the outputs are identical.

This channel can transport two bits per single channel use ofthe new channel, but in terms of the channel use of theprevious channel is 2

3 bits/c.u.

Example data-to-transmission mapping:

00→ 000 01→ 001 10→ 010 11→ 100



A more capable STX interferes at PRX

Let us now assume that STX non-causallyknows the message that the PRX wants tosend.

Questionable how realistic it is, but agood theoretical model of cognition :)

Key observation

When the primary sends 111,in those three channel usesSTX can send any bit combination.

PTX PRX

STX SRX




When the primary signal is 111, the allowed secondary symbols are:

000, 001, 010, 100, 011, 101, 110, 111

such that STX-SRX can communicate at 1 bit/c.u.

When the primary signal is 000, the allowed secondary symbols are:

000, 001, 010, 100

such that STX-SRX can communicate at 23 bit/c.u.

The problem is that PTX decides which of these channels will takeplace in a given triplet of channel uses.




To treat the easier case, we make even braverassumption:Also SRX knows the message of PTXnon-causally.

PTX PRX

STX SRX

A possible communication strategy:

When PTX sends 000, STX takes two new data bits andmodulates them as follows:

00→ 000 01→ 001 10→ 010 11→ 100

When PTX sends 111, STX takes three new data bits andsends them as they are.




Primary data:XP : 000,111,000,000,111,111,000, . . .

Let SRX receive: YS : 010,010,100,000,101,111,001, . . .

Since SRX knows which channel is activated when, it decodes:10, 010, 11, 00, 101, 111, 01

But, if SRX does not know the primary message,then it cannot know whether to interpret010 as 10 or as 010.




If SRX does not know the primary message, then the STX mustuse part of its resources to tell which channel is being used in agiven triplet. Here is one way to approach it:

We can think that STX-SRX is a channel with errors, with 8inputs and 8 outputs.

If one of 000, 001, 010, 100 is sent, it arrives without errors.

If 011, 101, 110, 111 is sent, then if PTX sends 111, it arriveswithout errors.

But, if 011 is sent and PTX is 000, then we can think as ifSTX is forced to make an error and send 000 instead.



New secondary channel model

000

001

010

100

011

101

110

111

000

001

010

100

011

101

110

111

0.5

0.5

1



Achievable rates when STX interferes to PRX

When STX does not know the primary data:

2

3= 0.667 bit/c.u.

When STX and SRX know the primary data:

1

2· 2

3+

1

2· 1 = 0.833 bit/c.u.

When STX knows the primary data, but SRX does not:

2.1831

3= 0.7277 bit/c.u.

NB: This rate is achieved by sending one of 000, 001, 010, 011 withprobability 0.19, the probability of sending one of the others011, 101, 110, 111 with probability 0.6.



PTX interferes to SRX

PTX PRX

STX SRX

000

001

010

100

011

101

110

111

000

001

010

100

011

101

110

111

0.5

0.5

1



Complete interference channel

STX can only use000,001,010,100. But now thePTX can erase these symbolsinto 111.Secondary channel:

000

001

010

100

000

001

010

100

111

0.5

0.5

PTX PRX

STX SRX

The highest data rate is13 = 0.33 bit/c.u..



Complete interference channel

PTX PRX

STX SRX

Does more cognitivity help?

The problem is that when we canuse more transmission symbols,they are all erased!Even if both STX and SRX knowthe primary message, it does nothelp. The best we can do is theone shown on the previous slide.



AWGN channel

If not stated otherwise, we will assume Gaussian (AWGN) channels.

We use a slightly more general form:

Y = hX + z

where h is a coefficient that contains information about thepropagation/receiving conditions.

To put this model to work, we need to assume that X has alimited power P. It is common to limit the average power:

1

n

n∑i=1

|Xi |2 < P

Each i is a different channel use, an atomic unit of time for thecommunication model in question.



AWGN channel

The main conceptual result of information theory is that efficientcommunication is achieved by jointly designing the transmissionstrategy over many channel uses.



AWGN channel

A communication strategy is defined as follows.

Fix n and one codeword takes n channel uses.

Define a set of possible source messages

W ∈ [1 : 2nR ] = {1, 2, . . . 2nR}

Each W is mapped to a codeword with n channel uses

xn(W ) = (x1(W ), x2(W ), . . . xn(W ))

where the codeword satisfies the power constraint.

For each possible received version of the codeword yn, thereceiver makes a decision which message has been sent

W = g(yn)

Error occurs when W 6= W .



Channel capacity

If the probability of error is 0, then the data rate at which thetransmitter communicates to the receiver is:

log2(# of messages)

n=

nR

n= R [bits/c.u.]

The capacity of the channel is the maximal data rate that can beachieved if we are allowed to use infinitely many channel uses percodeword. Equivalently,

C = max I (X ;Y )

where I (; ) is the mutual information between the random variablesX and Y .

The remarkable result of Shannon

What can be achieved by the best strategy over n channel uses isgiven by the maximal mutual information for a single channel use.



Capacity of AWGN channel

If we define Signal-to-Noise Ratio (SNR) as

γ =|h|2PN

where N is the noise variance.

Capacity of a real AWGN channel with SNR of γ

C (γ) =1

2log2(1 + γ) [bits/c .u.]

This capacity is achieved by infinitely long codewords, where thedistribution of the constituting symbols {xi} of each codeword xn

is Gaussian.

For a complex Gaussian channel Y = X + Z , C (γ) doubles.



Time, frequency, degrees of freedom

In reality a channel use takes a certain time. The minimaltime between two channel uses is related to the frequencybandwidth of the signal (recall the sampling theorem).

If the bandwidth is B, then the number of channel uses persecond is 2B samples per second.

Let the power density of the noise be N02 , while now the power

P is the total power used in the whole band B.

Then the capacity is:

C = B log2

(1 +

h2P

BN0

)[bits/second]

The factor that multiples the logarithm is a measure for thedegrees of freedom in the system.In the sequel, unless stated otherwise, we assumea normalized bandwidth B = 1.



Fading and time-varying channels

The capacity C (γ) can be achieved if:

The SNR γ is constant throughout all the channel uses of asingle codeword;The transmitter knows the SNR.

These assumptions may be violated due to the change of thechannel coefficient h.

We will always use a block fading model, where h stays constantduring the whole codeword.

Under a block fading, if h is not known at the transmitter, then:

The transmitter selects the rate R based on e. g. thestatistics of γ (which depends on the statistics of h);The event of outage occurs if the rate R cannot be supportedby the current channel realization:

R > log2(1 + γ)



Back to the primary link

In the interweave and underlay CR paradigms,the primary system is not supposed to changethe behavior, i. e. it should operate as if thesecondary communication is absent.

As mentioned before, the usual operation ofthe primary system must involve some marginin order to enable entrance for the secondarysystem.

PTX PRX

STX SRX



Non-interfering primary and secondary communication

This may correspond to several cases

The primary/secondary operate in thesame band,but PRX is far away from STX andSRX is far away from PTX;

Interweave CR:

STX-SRX operates either in a differentfrequency band compared to PTX-PRX;STX-SRX operates in the samefrequency band with PTX-PRX, but atthe times when the link PTX-PRX is notused (time-sharing).

PTX PRX

STX SRX

This implies that both the secondary and the primary can use anordinary codebook for a Gaussian channel and does not need tocare about the codebooks of the legacy system.



Non-interfering primary and secondary communication

If time-sharing is used and a fraction of λchannel uses are available for secondarytransmission, while the SNR of the secondarylink is γs , then the capacity of the secondarylink is

λ log2(1 + γs) = λC (γs)

It must be noted that another information flowis required to make the time- orfrequency-interweaved communicationpossible.

STX-SRX can adapt to operate in aninterweaved way only if it has informationor senses the operation of the primarysystem.

PTX PRX

STX SRX



Inclusion of the interfering links

Let us now consider a model that involvesinterference between the primary and thesecondary system.

This can be used both for the underlayand the overlay CR paradigm.

The primary and the secondary may have avarious level of synchronization.

We will again simplify as much as possibleand assume complete synchrony;

The rationale is that the secondary(upcoming) system can put an effort tosynchronize and align the signals with theprimary (legacy) system.

PTX PRX

STX SRX



The Gaussian interference channel

In a single channel use, the system can bedescribed with the following equations:

Yp = hppXp + hspXs + Zp

Ys = hpsXp + hssXs + Zs

Two source signals, Xp and Xs , imply thatthere are two relevant rates in the system andthat is why we characterize it by the rate pair

(Rp,Rs)

PTX PRX

STX SRX

hpp

hss

hps

hsp

The setup for a classical interference channel

PTX has a message Wp intended for PRX only

STX has a message Ws intended for SRX only

There is neither cooperation PTX-STX nor PRX-SRX.Petar Popovski, AAU, DK Information-Theoretic Aspects of Cognitive Radio


The Gaussian interference channel (contd.)

Yet, there is additional knowledge:

PTX and STX know all four channel coefficientsPTX knows that there is a link STX-SRX and that theinterference from PTX is through a channel hpsSimilar observation for STX.

Noise variance N is assumed equal at both receivers. Two SNRs:

γp =|hpp|2P

Nγs =

|hss |2PN

and two INRs (Interference-to-Noise Ratio):

γps =|hps |2P

Nγsp =

|hsp|2PN

The capacity of this channel is unknown in general. But there arespecial cases where it is known, as well as interesting strategies forgetting higher achievable rates.



Interference channel as two coupled Multiple AccessChannels (MACs)

If we observe each receiver separately then each of them acts as areceiver over a multiple access channel.

PTX PRX

STX

hpp

hsp

PTX

STX SRX hss

hps

Yp = hppXp + hspXs + Zp Ys = hpsXp + hssXs + Zs

An important difference with the interference channel is that in aMAC both transmitters want to send their messages to the samereceiver.

Here, for the MAC defined at Yp, STX has, in principle, nointerest to send any information to PRX.



[Digression] Capacity of a MAC

Take the one with the receiver Yp


C (γp) - the capacity of the AWGN channel Yp = hppXp + Zp

C (γsp) - the capacity of the AWGN channel Yp = hspXs + Zp

Rp ≤ log2(1 + γp)

Rs ≤ log2(1 + γsp)

Rp + Rs ≤ log2(1 + γp + γsp)

Rp

Rs

C(γp)

C(γsp)



Interference cancellation

In the setup of the interference channel, PRX (SRX) is notinterested in Xs(Xp). So why to decode it?

If each receiver treats the unwanted signal as a Gaussian noisethen the maximal rates are:

Rp ≤ log2

(1 +

γp1 + γsp

)Rs ≤ log2

(1 +

γs1 + γps

)However, in a regime of strong interference, when γsp > γp andγps > γs the capacity is known and it is achieved by letting firsteach decoder decode the unwanted signal!

This implies that each of the two messages should becommon, i. e. decodable by both receivers.

It turns out that, in general channel conditions, it is useful ifpart of the message of each transmitter is common, for bothreceivers.



Interference channel as two coupled Gaussian BroadcastChannels (BCs)

This perspective is motivated by the common messages.

PTX PRX

SRX

hpp

hps

PRX

STX SRX hss

hsp

Yp = hppXp + Zp Yp = hspXs + Zp

Ys = hpsXp + Zs Ys = hssXs + Zs

The total rate of the transmitter is split into a rate for each ofthe receivers, respectively.In a Gaussian channel this implies that power must be split. Away to do it is via superposition coding.



Rate splitting: The scheme of Han & Kobayashi

PTX PRX

STX SRX

hpp

hss

hps

hsp

WP-‐C WP-‐P

WS-‐C WS-‐P

WP-‐C WP-‐P

WS-‐C

WP-‐C WS-‐C WS-‐P

Note that, although SRX decodes WP-C, the commonsub-message of PTX, the data bits of this reception are not part ofthe useful data rate in the system.

On the other hand, these bits become part of the usefulcontributions by being received at PRX.



Relation of the interference channel to the CR models

The interference channel is not directly suitable to for the CRmodels.

An important assumption is violated: In a CR setup, PTXdoes not know (or care) if an interfering transmission fromSTX is taking place.

We can not expect PTX to adapt its signaling in order toalign with the requirements of the interference channel.

But we can expect that STX and SRX can and should adaptto the conditions set by the legacy system.



Interference margin

Let us at first neglect the interference from PTX to SRX (hps = 0)and focus on the interference from STX to PRX.


If the primary rate is selected to be Rp = log2(1 + γp),then there is no room for secondary (cognitive) transmission, i. e.it must be Xs = 0.

In order to allow secondary transmission, the primary system mustselect its data rate with a certain margin.

Rp < log2(1 + γp)

Then STX can choose its power in order to satisfy:

Rp = log2

(1 +

γp1 + γsp

)⇒ γsp =

γp2Rp − 1

− 1



Power selection at STX

In order to select the right power level, STX must knowγp,Rp and hsp.

In absence of collaboration from PTX, we need additionalassumptions on the possible knowledge of these parameters.

A solution can be overhearing of receiver’s transmittedpackets and using reciprocity

Guessing the interference created by measuring thethroughput achieved by at the primary receiver

Overhear ACK/NACK sent by the primary receiver andmeasure the success ratio.Overhear the retransmitted packets by the primary receiver.



Some plausible scenarios for knowing the interference atPRX

Primary system that uses a fixed data rate Rp.

In that case a relevant measure for the primary performance isthe probability of outage.

STX/SRX overhear the NACK/ACK from the PRX and thusmeasure the probability of outage at PRX .

If this probability is lower than the nominal (e. g.QoS-dependent) probability of outage, then secondarytransmission takes place.

Note that each secondary transmission also has the role toprobe the level of interference caused to the PRX.



Multiple secondary transmittersSRX 1

PTX PRX

STX 1

STX 2

STX 3

SRX 2

SRX 3

Controlling the interference at PRX becomes more complex.

Required knowledge at STXsNon-competitive allocation of power to the secondaries

Fairness, opportunistic allocation, individual power constraints,max sum rate, . . .

Competitive allocation of powerGame-theoretic models.



Use of multiple antennas

Single cognitive link, but the cognitive devices STX, SRX havemultiple antennas.

Cognitive MIMO (Multiple-Input Multiple Output) link.

STX-PRX is a MISO (Multiple-Input Single-Output) channel.

A tradeoff between the following two requirements:

1 In absence of the interference constraint, spatial multiplexingmaximizes the secondary rate: the transmitter beamformsmultiple streams in the direction of its channel.

2 Transmission in a direction orthogonal to the receiver PRXremoves the worry of causing interference to the primary,regardless of the power used by the STX. This would be aninterweave CR with multi-antenna interference avoidance.Otherwise, in the underlay CR multiple antennas at STX areused to satisfy the interference constraint.



Gaussian vs. Non-Gaussian interference

We have so far treated the interference created towards PRX as aGaussian noise.

In practice, different interference waveforms can cause significantlydifferent impact at the signal received at the PRX, while using thesame power.

Example: constant-envelope modulations, such as PSK, usedby the STX and non-coherent receiver at the PRX.



Gaussian vs. Non-Gaussian interference (contd.)

However, regulation need to be rather general, not makingprovisions for any conceivable pair of interfering modulation vs.interfered modulation.

Minimal parameters to be specified:

The maximal interference power level η perceived by an activePRX

The probability ζ that this power level is violated.



Interference at the SRX

Assume now that STX has figured out what is the maximalallowed power P it can use in order to satisfy the interferenceconstraint at PRX.

The interference over the link hsp then becomes irrelevant.

Question

What is the best that STX+SRXcan do in order to improve theperformance of the cognitive link,knowing that the primary systemis not disturbed?

PTX PRX

STX SRX

hpp

hss

hps



Opportunistic Interference Cancellation (OIC)

The idea is that, under some conditions SRX can decode andcancel the interference arriving from PTX, and thus achieve ahigher data rate from STX.

There is an important difference with the interference channel:

PTX does not make any provision for the decoding capabilityof SRX and selects the rate Rp according to its usual criteria;

However, if it happens that the INR γps is strong enough,then SRX can decode the primary signal.



Opportunistic Interference Cancellation (OIC)

The opportunistic interference cancellation (OIC) works as follows:

If it happens that Rp and γps are such that

Rp ≤ log2(1 + γs)

then SRX applies OIC to remove Xp and improve the rate Rs .

Else, ifRp > log2(1 + γs)

then the interference from PTX is treated as noise at SRX.



Achievable rates with OIC

This graphs is interpreted differently from the MAC rate region:

For given channels and given choice of Rp, this is the functionthat gives the maximal possible secondary rate Rs

Rp

Rs

C(γp)

C(γs)

~ ~



Illustration of a CR in OIC with a primary cellular system

D - range of primary system

d-position of the SRX

PTX STX

SRX



The next frontier: STX knows the primary messageAsymmetrically cooperating cognitive radio channels

Why and how should STX knows the message that PTX wants tosend?

The channel PTX-STX is extremely good, PTX knows thatSTX can help and can send the message to it in a very shorttime. After that STX uses this knowledge to both help PTXand send its own message. Here PTX must be aware aboutthe existence of a helping CR system.

PTX sends the message to PRX, it gets erroneously receivedat PRX. Yet, STX has received it correctly and can use thisknowledge when PTX retransmits. Here PTX does not needto be aware about the help received from STX.

(other suggestions are welcome)



The next frontier: STX knows the primary messageAsymmetrically cooperating cognitive radio channels

If the knowledge of the primary message brings benefit to the linkPTX-PRX, then PTX should, in principle, be open to provide it.

Example: emergency message.

This has a potential to change the future radio regulation.



Non-causal knowledge of the primary message

In all the cases listed above the primary message Wp is knownnon-causally at the STX. This means the following:

When PTX starts to transmit the codeword that correspondsto Wp, STX knows exactly what PTX will send in eachchannel.

It is assumed that PRX has no prior information about Wp.

Thus, it appears to PRX that both PTX and STX know Wp

in exactly the same way.



Side information: competition, cognition, cooperation

In the interference channel PTX and STXare competing for the channel resources.If PTX is not aware about thecompetition, then SRX may applyopportunistic interference cancellation.

If STX knows the message, then this is acognitive channel, where informationflows also from STX to PRX.

If PTX and STX exchange theirmessages, then this is a cooperativeregime, where PTX and STX act as amulti-antenna transmitter.

PTX PRX

STX SRX

PTX PRX

STX SRX

PTX PRX

STX SRX



How should the primary message be used by STX?

STX splits its power in two parts and the n−channel-use codewordsent is

X ns = X n

s,1 + X ns,2

where

The total power of X ns,1 and X n

s,2 is Ps .

The message contained in X ns,1 is intended for SRX.

The message contained in X ns,2 is intended for PRX



Cognitive channel in a standard form

Let us write our original model of a real interference channel as:


Ys = hps Xp + hss Xs + Zs

where

The power limit for PTX (STX) is Pp(Ps);

The noise variance at PRX (SRX) is Np(Ns)

By specifying

a =hsp√

Ns

hss

√Np

b =hps

√Np

hpp√Ns

Pp =h2ppPp

Np

Ps =h2ss Ps

Ns

we can convert the cognitive channel into a standard form (nextslide).



Cognitive channel in a standard form (contd.)

Yp = Xp + aXs + Zp

Ys = bXp + Xs + Zp

where

The power limit for PTX (STX) is Pp(Ps);

The noise variance at PRX (SRX) is 1.

Note that, without losing generality, we can take that both a and bare positive.



The signal at PRX

The message from STX for PRX is selected as:

X ns,2 =

√αPs

PpX np

The power of this signal is αPs .The power of X n

s,2 is (1− α)P.

The received signal at PRX is (with n channel uses):

Y np = X n

p + a(X ns,1 +

√αPs

PpX np ) + Zn

p

or with some rearranging

Y np =

(1 + a

√αPs

Pp

)X np + aX n

s,1 + Znp



The signal at PRX (contd.)

Note that PRX is oblivious w. r. t. the secondary transmission andmust treat Xs,1 as a noise.

Then the maximal rate Rp is given by:

Rp ≤1

2log2

(1 +

(√

Pp + a√αPs)2

1 + a2(1− α)Ps

)

Exercise: Derive this equation.



The signal at SRX

Y ns =

(b +

√αPs

Pp

)X np + X n

s,1 + Znp

The main trick here is that STX(SRX) does not need to treat X pn

as an “ordinary” noise.

STX knows perfectly the interference(b +

√αPsPp

)X np

received at SRX.

One idea could be to use part of the power (1− α)Ps

contained in X ns,1 to cancel the interference.

A much better idea is the one of Dirty-Paper Coding, a. k. a.Costa coding and Gel’fand-Pinsker coding.

The main concept of Dirty Paper Coding is to code aroundthe interference, not against the interference.



Illustration of dirty paper coding

Define a 2−bit constellation with limited power.

00 01 10 11

real 0

Replicate the constellation.



Illustration of dirty paper coding (contd).

Assume that 01 needs to be sent.

Observe the interference.

Apply a signal that added with the interference brings you to theclosest representative of 01.



Rates achieved with dirty paper coding

Costa has proven that with dirty paper coding, the channel canoperate at rates as if the interference was not there at all!

In our case, the maximal rate of the secondary link is:

Rs ≤1

2log2(1 + (1− α)Ps)

In summary, the capacity region of the cognitive channel is:

0 ≤ Rp ≤ 1

2log2

(1 +

(√

Pp + a√αPs)2

1 + a2(1− α)Ps

)0 ≤ Rs ≤ 1

2log2(1 + (1− α)Ps)



Illustration of the capacity regionsInformation Theoretical Limits on Cognitive Radio Networks

! !"# $ $"# % %"#!

!"#

$

$"#

%

%"#

&$

&%

'()*+,-./+01-2+01+3*4560-2078&0$!90-%$:-

$%:!"##

0

0

;<;=0.14->(-620()-55+/

?435*2*,+0()-55+/

<52+1@+1+5(+0()-55+/

A*B+!6)-1*53

Figure 1.8: Capacity region of the Gaussian 2 × 1 MIMO two receiver broadcastchannel (outer), cognitive channel (middle), achievable region of the interferencechannel (second smallest) and time-sharing (innermost) region for Gaussian noisepowers N1 = N2 = 1, power constraints P1 = P2 = 10 at the two transmitters, andchannel parameters h12 = 0.55, h21 = 0.55.

that of partial message knowledge in [MYK07]. While the above channel assumesnon-causal message knowledge, a variety of two-phase half-duplex causal schemeshave been presented in [DMT06a, KG07], while a full-duplex rate region was stud-ied in [aXL07]. Many achievable rate regions are derived by having the primarytransmitter exploit knowledge of the exact interference seen at the receivers (e.g.dirty-paper coding in AWGN channels). The performance of dirty-paper codingwhen this assumption breaks down has been studied in the context of a compoundchannel in [MDT06] and in a channel in which the interference is partially known[GS07].

Cognitive channels have also been explored in the context of multiple nodesand/or antennas. Extensions to channels in which both the primary and secondarynetworks form classical multiple-access channels has been considered in [DMT05a,CYZ+]. Cognitive versions of the X channel [MAMK06] have been consideredin [DS07, JS08], while cognitive transmissions using multiple-antennas, withoutasymmetric transmitter cooperation has been considered in [ZL].

Finally, while we have outlined some results on the exact rate regions for cogni-tive radio channels, how these rate scale at high signal to noise ratio (SNR → ∞)is also a measure of interest. The multiplexing gain, m, of a cognitive network8 de-

8Multiplexing gain, degrees of freedom and pre-log are all terms which are used interchangeably

26

From N. Devroye, “Information Theoretical Limits on Cognitive Radio Networks,” inCognitive Radio Communications and Networks; Principles and Practice, A.M. Wyglinski, M.Nekovee and Y.T. Hou Ed., Elsevier, 2009.



Summary and outlook

We have reviewed some basic communication-theoretic andinformation-theoretic models for cognitive radio.

The models are deceptively simple with respect to the“reality” of cognitive radio;

Yet, they give rise to powerful new ideas and communicationschemes;

The main messages:

Cognition is a lot about the knowledge that the secondary hasabout the primary;The more cognition the merrier – the best results are obtainedif the secondary knows completely the behavior of the PTX,including the message it is sending;Interference is not a noise - find a way to use this fact.



Literature

N. Devroye, “Information Theoretical Limits on CognitiveRadio Networks,” in Cognitive Radio Communications andNetworks; Principles and Practice, A.M. Wyglinski, M.Nekovee and Y.T. Hou Ed., Elsevier, 2009.

A. Goldsmith, S. Jafar, I. Maric, and S. Srinivasa, Breakingspectrum gridlock with cognitive radios: An informationtheoretic perspective, Proceedings of the IEEE, vol. 97, pp.894 914, May 2009.

P. Popovski, H. Yomo, K. Nishimori, R. Di Taranto, ”RateAdaptation for Cognitive Radio under Interference fromPrimary Spectrum User”,http://arxiv.org/abs/0705.1227

... and the references therein


http://arxiv.org/abs/0705.1227

information-theoretic aspects of cognitive radio

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