1194 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009

Sensing in Cognitive Radio Channels: A Theoretical PerspectiveLionel Gueguen and Berna Sayrac

Abstract—The cognitive radio paradigm is based on the abilityof sensing the radio environment in order to make informeddecisions. This paper describes the effects of sensing on thecognitive radio channels capacity region. Sensing is modeled asa compression channel, which results in partial knowledge ofthe primary messages at the cognitive transmitter. This modelenables to impose constraints on the sensing strategy. First,the dirty paper channel capacity is derived when the channelencoder knows partially the side information. Then, the capacityarea of the Gaussian cognitive channel with partial informationis derived. Finally, numerical results illustrate the capacityreduction associated with constrained sensing, in comparison tothe capacity of the cognitive radio channel.

Index Terms—Cognitive radio, sensing, dirty paper coding,channel capacity.

I. INTRODUCTION

IN the current telecommunications area, concepts like dy-namic spectrum access, cognitive and flexible radio have

emerged and they are being intensively discussed in thewireless communication and information theory communities[1], [2]. A cognitive radio is capable to sense its radio environ-ment in order to adapt its transmission parameters such thatprimary users are not interfered. Recent advances in cognitivecommunications [1] refer to various solutions that seek tooverlay or interweave the cognitive user’s transmissions withprimary user. As part of interweave based solutions, a lot ofwork has focused on detecting and predicting whites spaces[3] to let a cognitive user transmit its own information. Recentwork in information theory [4], part of overlay solutions,has shown that the knowledge of the primary message couldsignificantly increase the overall channel capacity. The overlayapproach makes use of secondary transmitter to relay primarytransmissions and to transmit the secondary message. There-fore availability of the primary messages is a key issue in sucha framework.

Sensing, either to detect primary users or to get primarymessages, is crucial in a cognitive radio network. So far, con-siderable work has focused on sensing strategies for detectingwhite spaces. For example, compressed sensing techniques [5]have been used in [6] for the detection of free frequencies;partially observable Markov models have been used in [3] forthe detection and prediction of white spaces. From these earlystudies, it is obvious that radios with limited capabilities tryto reduce the sensing complexity, thus restraining the quantityof sensed information.

Manuscript received May 16, 2008; revised November 3, 2008; acceptedNovember 15, 2008. The associate editor coordinating the review of this letterand approving it for publication was A. Molisch.

The authors are with the Orange Labs, 38-40 rue du Général Leclerc, 92130,Issy-les-Moulineaux, France (e-mail: {lionel.gueguen, berna.sayrac}@orange-ftgroup.com).

Digital Object Identifier 10.1109/TWC.2008.080663

Actually, sensing is more than a detection process, andcan be viewed as an environment perception process. Forexample, from a restricted number of measurements, an orig-inal sparse signal can be recovered with some distortion byusing the compressive sensing paradigm [5]. Indeed, sensingor perception are strongly related to the rate-distortion theory[7]. In this study, we assume that the secondary user haspartial knowledge about the primary information, which canbe modeled by a compression channel between the primaryand the secondary user. As the primary signal does not containredundancies, we restrict the sensing process to a quantizationof the primary signal [8] which corresponds to a partialknowledge of the primary information according to the rate-distortion theory.

Since sensing is a corner stone in cognitive radio frame-work, we focus on the effects of sensing on the cognitivechannel described in [9]. In this paper, we aim at providing thecapacity area of the Gaussian cognitive channel, when a cog-nitive user senses partial information from primary messages.Recent works studied the partial decode and forward [10] orcompress and forward [11] strategies in a relay scheme whichis part of the cognitive channel. Although, these work seemsimilar to our goal, we stress that the secondary user is morethan a relay because it transmits its own information and triesto mitigate the induced interferences.

The paper is organized as follows. First, we derive insection II the capacity of the dirty paper channel with partialside information knowledge, where the partial knowledgeis modeled by a compression channel. In section III, afterreminding the rate area of the Gaussian cognitive channel,we derive the rate area of this channel when the cognitiveuser senses the primary transmission. In section IV-A, wegive numerical examples and application scenarios. Finally,we conclude in section V.

II. DIRTY PAPER CODING WITH PARTIAL SIDE

INFORMATION KNOWLEDGE

In this section, we first present the Dirty Paper channelmodel with partial side information obtained with a sensingstrategy which is modeled as a source encoder-decoder gen-erating distortion on the side information. Then, we derivethe capacity of this channel which depends on the sensingdistortion. In the following, random variables are denotedwith capital letters, and their realizations are denoted withthe corresponding lower case letters.

A. Channel model

We propose a channel model, where partial side informa-tion is known at the encoder. The partial side informationknowledge models the sensing process which consists in

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009 1195

a)

b)

Fig. 1. a) Channel encoding with partial observation of the side informationat the encoder. The partial observation is modeled as a compression channelgenerating some distortion on the side information S. b) Alternative represen-tation of the Dirty Paper coding with partial side information. E is the errorrandom variable modeling the distortion and is defined in (4). E is unknownat the encoder and is considered as additional noise on a DPC channel withside information S̃.

approximating the original information with a small amountof information.

Let X , S be two independent random variables whichrepresent respectively the source and the side information.Let Y be a random variable which represents the output ofthe ergodic channel modeled by:

Y = X + S + Z (1)

where Z is a zero mean white Gaussian noise, Z ∼ N (0, N).We suppose that X and S are power limited variables whichis expressed by:

E[X2

] ≤ PX ; E[S2

] ≤ PS (2)

We are interested in the case where the channel encoderknows partially the side information S and the channel en-coders, decoders have a perfect knowledge of the channelcharacteristics. Let S̃ be the random variable representing thepartial information obtained at the channel encoder. Let usconsider that S̃ is the best compressed representation of Swhich satisfies the following Euclidean distortion:

E

[(S − S̃)2

]= D (3)

Following the rate-distortion theory described in [12], [13]and considering that S is Gaussian, the sensing channel canbe expressed with the distortion random variable E as follows:

S = S̃ + E, E ∼ N (0, D) (4)

Quantization is well modeled by the previous equation,where the quantization step is linked to D [8]. It shouldbe underlined that the compression channel (4) differs froman AWGN channel: one is the dual of the other [13]. Thecompression channel and encoding scheme are depicted inFig.1.a, where the distortion generated on the side informationis represented as a compression channel.

B. Capacity of the dirty paper channel with partial sideinformation

The previous channel is very similar to the Dirty PaperCoding (DPC) channel [14]. In this section, we derive thechannel capacity of the link (X → Y ) with partial informationat the encoder defined in (3). From the well known Gel’fandand Pinsker formula [15], the capacity C is given by themaximization problem:

C(W ) = maxp(u|s̃),p(x|u,s̃)

I(U ; Y ) − I(U ; S̃) (5)

where U is a finite alphabet auxiliary random variable. Thechannel p(y | x, s), the side information p(s), and thecompression channel p(s̃ | s) are given. W is an averagedconstraint on X and S, which is compound of the powerconstraints described by (2).

Theorem 2.1: For a channel described by (1), some sideinformation S ∼ N (0, PS), a compressed version S̃ of S sat-isfying (3)-(4), and a cost constraint (2), the channel capacityC between the source X and the output Y is expressed by:

C =12

logPX + N + D

N + D(6)

Proof: As in the DPC theorem [14], we introduce therandom variable U defined by U = X + αS̃ and define therate R(α) = I(U ; Y ) − I(U ; S̃).

As S̃ is a compressed version of S which is Gaussian, weknow that E

[SS̃

]= E

[S̃2

]= PS̃ which is useful when

evaluating the rate R(α):

R(α) =1

2log

(PX + PS + N)PX

(PX + PS + N)(PX + α2PS̃) − (PX + αPS̃)2(7)

The maximum rate is obtained for α∗ expressed by:

α∗ =PX

PX + N + PS − PS̃

(8)

As S̃ represent partial and compressed information from S,we have PS = PS̃ + D and the maximum rate becomes:

R(α∗) =12

logPX + N + D

N + D(9)

When D equals zero, we obtain back the DPC rate. As a sim-ple explanation, the error generated on S which is unknown atthe channel encoder is simply viewed as an additive noise onthe channel and S̃ becomes the side information in the DPCtheorem (cf. Fig.1.b). Taking this problem reformulation andapplying the DPC theorem, it follows immediately that thechannel capacity C equals the maximum rate R(α∗).From the rate-distortion theory [12], S can be encoded at aminimum rate of R(D) = max { 1

2 log PS

D , 0}. This equationinduces a one-to-one link between the compression rate R andthe channel capacity C, which depends on the distortion D.Such a relation can be represented by a parametric function(R(D), C(D)), an example of which is displayed in Fig.2. AsD goes to zero, R goes to infinity while C tends to the fullchannel capacity CDPC = 1

2 log (1 + PX

N ). In addition, Fig.2shows that C grows linearly with small values of R.

1196 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Compression rate R in bits

Cap

acity

Cin

bits

Fig. 2. The parametric function defined between the compression rate Rand the channel capacity C. In this case PX = PS = 10 and N = 1. As Dgoes to zero, R goes to infinity while C tends to the DPC channel capacityCDPC represented by the straight line.

III. THE COGNITIVE CHANNEL WITH PARTIAL PRIMARY

INFORMATION KNOWLEDGE

In the following, we describe the Gaussian cognitive chan-nel and its corresponding rate area. Then, we derive the ratearea of this channel when the secondary transmitter sensesthe primary message in a non-causal manner. This yields toa cognitive channel with partial information knowledge at thesecondary transmitter.

A. The Gaussian cognitive channel

The cognitive channel was first introduced in [2], [9]. Aprimary transmitter and receiver pair (Tx1 → Rx1) uses acommunication medium which is also used by a cognitivetransmitter and receiver pair (Tx2 → Rx2). The cognitivechannel is differentiated from the interference channel byan asymmetric cooperation between transmitters.The secondtransmitter, making use of its cognitive capabilities, knowsperfectly the message sent by the primary transmitter. Thistransmission model is idealized by a non-causal knowledgeof the primary message at the cognitive transmitter, and isreferred as the genie-aided cognitive channel [2].

In the framework of information theory, channel capacitieshave been defined for such a transmission model. Thesechannel capacities are the upper bounds of the rates for whichthe messages can be reliably transmitted. Let (R1, R2) be therates between (Tx1 → Rx1) and (Tx2 → Rx2), respectively.These rate couples define an achievable rate region which isupper bounded by the capacity region.

As the cognitive transmitter knows the primary message,it can mitigate interference from Tx1 and relay the primarymessage at the same time, such that the overall channelcapacity is increased. In [2], the cognitive rate region is givenfor a Gaussian channel and is described by the relations:

Y1 = X1 + h12X2 + Z1 (10)

Y2 = h21X1 + X2 + Z2 (11)

where X1, X2 are the channel inputs, Y1, Y2 are the channelsoutputs, Z1 ∼ N (0, N1), Z2 ∼ N (0, N2) are the AWGN

Fig. 3. The cognitive channel with partial information obtained at thesecondary transmitter through a sensing operation. In this case, X̃1 denotesthe partially sensed information from the primary message X1.

components and h12, h21 are the channel crossover coef-ficients. In addition, X1, X2 comply with average powerconstraints expressed by E

[X2

i

] ≤ Pi, i ∈ {1, 2}. LetH1 = [1, h12], H2 = [h21, 1]. Using the framework of [16],we can define the transmit covariance matrix Bi that describesthe correlation of the message i between both transmitters. Thecovariance matrix set which admits the power and cognitiveconstraints is defined by:

B = {(B1, B2) | B1 � 0, B2 � 0,

B1 + B2 �[

P1 zz P2

], z2 ≤ P1P2, B2 =

[0 00 x

]}(12)

where B � 0 means B is positive definite and x > 0. Then,the cognitive region is defined as the convex hull of the ratepairs (R1, R2) which satisfy the following inequalities:

R1 ≤ 12

logH1(B1 + B2)HT

1 + N1

H1B2HT1 + N1

(13)

R2 ≤ 12

logH2B2H

T2 + N2

N2(14)

with (B1, B2) ∈ B. As a remark the second inequalitycorresponds to the dirty paper coding theorem [14] expressedwith the channel crossover coefficients. The cognitive userattributes a part of its power to transmit its own messageavoiding interference and the other part to relay the primarymessage. Obviously, the asymmetric cooperation betweentransmitters increases the overall channel capacity in compar-ison to the time sharing capacity region [17], where cognitiveusers transmit only when the primaries do not transmit.

B. The rate area of the cognitive channel with partial infor-mation

In this section, we are interested in the cognitive channelwhere the secondary transmitter is given partial informationfrom X1, denoted by X̃1, such that the compression rate Ris constant. This cognitive channel with partial primary infor-mation at the secondary transmitter is depicted in Fig.3. Thepartial information is obtained through a compression channeland models a sensing realized at the secondary transmitter.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009 1197

In the following, we derive the achievable rate area of acognitive Gaussian channel with partial primary informationat the cognitive transmitter. We restrict the distortion betweenX1 and X̃1 to be expressed by D = βE

[X2

1

]where β is

a constant. When β = 1, the distortion is maximum and nosensing is realized.

Theorem 3.1: For a cognitive channel described by (10)-(11), a compressed version X̃1 of X1 at the secondary trans-mitter with a distortion D = βE

[X2

1

], and power constraints

E[X2

i

] ≤ Pi, i ∈ {1, 2}, the achievable rate area is the convexhull of the rate pairs (R1,R2) which satisfy the followingequations:

R1 ≤ 12

logH1(L1 + L2)HT

1 + N1

H1L2HT1 + N1

(15)

R2 ≤ 12

logH2(M1 + M2)HT

2 + N2

H2M2HT2 + N2

(16)

where the covariance matrices L1, L2, M1, M2 are positivedefinite and satisfy the following conditions:

L1 =[

l111 l112l112 l122

]; L2 =

[0 00 l222

](17)

M1 =[

0 00 m1

22

]; M2 =

[βl111 00 0

](18)

P ′2 + m1

22 ≤ P2 ; l222 ≥ m122 + βP ′

2 (19)

L1 + L2 ≺[

P1 zz P2

]; z2 ≤ (1 − β)P1P2 (20)

Proof: We propose a two part proof, making use of thetheorem 2.1 and the cognitive rate area defined in (13)-(14).

First, we consider the pair (Tx2 → Rx2), where thecognitive encoder knows a compressed version X̃1 of theprimary source X1. As the compression rate is constant andequals − 1

2 log β, we know that D = βE[X2

1

]= βl111. As

in the cognitive channel case, the useful power of X2 fortransmitting its own message is restricted to m1

22. Integratingthe channel gains and rewriting Theorem 2.1 with matricesM1, M2 in (18), we obtain the inequality (16).

Secondly, we consider the pair (Tx1 → Rx1). As theprimary encoder does not know X2, the capacity definitiondoes not change and depends only on (B1, B2) which satisfysome modified constraints. Let (L1, L2) denote these newcovariance matrices expressed in (17). In the following, wederive the new constraints induced by the partial informationat the secondary encoder. In section III-A, B2 represents thepower attributed by Tx2 to transmit its own message, whichbecomes interference for the first transmitter. The remainingpower is used to relay the transmission of X1. In the casewhere Tx2 knows partially the first message, it acts as ifit knows it perfectly and produces interference because ofits partial knowledge. Therefore, the power P ′

2, attributed byTx2 to relay the transmission of X1, contains some noiseof variance βP ′

2. Thus, the interference power follows theinequality l222 ≥ m1

22 + βP ′2 where the relay power P ′

2

is restricted by the secondary message power m122 in (19).

Finally, the maximum correlation between X2 and X1, isrestricted by the compression loss:

l112 = E [X1X2] ≤ E

[X̃1X2

]≤ (1 − β)P1P2 (21)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

R1

R2

β = 0β = 0.25β = 0.50β = 0.75β = 1

Fig. 4. The channel is characterized by h12 = h21 = 0.55 and N1 =N2 = 1. The power constraints are P1 = P2 = 10. The rate area (R1, R2)depending on β represented in the plan.

Therefore, if the constraints on (L1, L2) are satisfied providingsome m1

22, the capacity bound of the primary pair is givenby equation (15). To summarize, the previous constraints on(L1, L2, M1, M2) and the inequalities (15)-(16) provides usthe rate region for fixed β.This theorem gives back the cognitive area defined by (13)-(14) if β and the distortion are null. Conversely, for a maxi-mum distortion (β = 1), the compression rate is null and therate area tends to the interference channel area [17].

IV. APPLICATIONS

A. Numerical results

In this numerical example, we propose to evaluate the ratearea for different compression rates parametrized with β. Theadmissible tuples (R1, R2, β) form a surface in R

3 and isrepresented in the plane of Fig.4.

We observe that the loss of information degrades the fullcognitive rate area. We can note that the rate area movescontinuously toward the interference channel area as β goesto 1. These results show that sensing still enables a capacityincrease in comparison to a time-sharing scheme. However,this analysis leads to a trade-off between the information lossor the compression rate and the capacity gain.

In order to have insight in this trade-off, we define the sumrate C̃γ(β) for the cognitive channel with partial informationas:

C̃γ(β) = max (R1 + R2) − γR(β) (22)

where γ is the trade-off parameter, R(β) = − 12 log β is

the compression rate and (R1, R2) belongs to the rate areaobtained for β. This sum rate represents the total quantity oftransmitted information minus the compression rate in propor-tion to γ, where γ relates to the sensing cost. For example, γcan represent the energy consumption of a quantization on acompression process. The sum rate C̃γ(β) function is plottedin Fig.5 for different values of γ.

1198 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

Compression rate R(β) in bits

C̃0(β)C̃0.25(β)C̃0.5(β)C̃0.75(β)C̃1(β)Cs

Fig. 5. The sum rate function C̃γ(β) defined in (22) versus the compressionrate R(β). It represents the maximum amount of transmitted informationminus the quantity of information sensed at the secondary transmitter. Cs isthe SISO capacity.

B. Cognitive Radio Scenarios

Let us consider a scenario of a dense network where twotransmitters Tx1, Tx2 are sufficiently close such that thechannel noise at the secondary transmitter is negligible. Weassume that the signal at the secondary transmitter is obtainedin a non-causal manner. In [9], protocols are proposed toovercome the problem of non-causality, while benefiting fromthe capacity gains. The second transmitter is required to limitits sensing process, whose cost is related to the trade-offparameter γ and to the quantity of sensed information. Tx2

quantizes the received signal of Tx1 such that the quantityof sensed information respect the sensing cost. Then, fromthis partial sensed information, Tx2 is able to recover thesignal of Tx1 with a certain distortion, and uses it to mitigateinterferences, resulting in an increase in the overall capacity.

In a second scenario, a primary base station Tx1 transmitsin advance through a high capacity wire link, a compressedversion of its information to Tx2. Then, the secondary basestation Tx2 makes use of the compressed signal in its codingscheme. The overall gain of such a scheme is given by (22),where γ is related to the cost of transmitting through the wirelink.

V. CONCLUSION

In this paper, we have investigated the capacity region ofthe cognitive channel with sensing realized at the secondarytransmitter. In a first step, we determined the capacity of the

dirty paper channel with partial side information available atthe encoder. Then, we used this result to derive the capac-ity region of the Gaussian cognitive channel where sensingwas modeled as a compression channel. Modeling sensingas a lossy compression provides an information theoreticalframework for analyzing the sensing or perception strategiesin a cognitive network. Finally, numerical results have beenobtained, showing that sensing is valuable for gaining incapacity through asymmetric transmitter cooperation.

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