Clean relaying aided cognitive radio under the coexistence constraint
Pin-Hsun Lin, Shih-Chun Lin, Hsuan-Jung Su, Y.-W. Peter Hong
aa r X i v : . [ c s . I T ] A p r Clean relaying aided cognitive radio under thecoexistence constraint
Pin-Hsun Lin, Shih-Chun Lin, Hsuan-Jung Su and Y.-W. Peter Hong
Abstract
We consider the interference-mitigation based cognitive radio where the primary and secondary users cancoexist at the same time and frequency bands, under the constraint that the rate of the primary user (PU) mustremain the same with a single-user decoder. To meet such a coexistence constraint, the relaying from the secondaryuser (SU) can help the PU’s transmission under the interference from the SU. However, the relayed signal in theknown dirty paper coding (DPC) based scheme is interfered by the SU’s signal, and is not “clean”. In this paper,under the half-duplex constraints, we propose two new transmission schemes aided by the clean relaying from theSU’s transmitter and receiver without interference from the SU. We name them as the clean transmitter relaying(CT) and clean transmitter-receiver relaying (CTR) aided cognitive radio, respectively. The rate and multiplexinggain performances of CT and CTR in fading channels with various availabilities of the channel state informationat the transmitters (CSIT) are studied. Our CT generalizes the celebrated DPC based scheme proposed previously.With full CSIT, the multiplexing gain of the CTR is proved to be better (or no less) than that of the previous DPCbased schemes. This is because the silent period for decoding the PU’s messages for the DPC may not be necessaryin the CTR. With only the statistics of CSIT, we further prove that the CTR outperforms the rate performance ofthe previous scheme in fast Rayleigh fading channels. The numerical examples also show that in a large class ofchannels, the proposed CT and CTR provide significant rate gains over the previous scheme with small complexitypenalties.
I. I
NTRODUCTION
Efficient spectrum usage becomes a critical issue to satisfy the increasing demands for high data rateservices. Recent measurements from the Federal Communications Commission (FCC) have indicated thatninety percent of the time, many licensed frequency bands remain unused and are wasted. Cognitive radio[1] is a promising technique to cope with such problems by accessing the unused spectrum dynamically.This new technology is capable of dynamically sensing and locating unused spectrum segments in atarget spectrum pool, and communicating via the unused spectrum segments without causing harmfulinterference to the primary users. The primary user (PU) is the user who communicates in the licensed
Pin-Hsun Lin and Hsuan-Jung Su are with Department of Electrical Engineering and Graduate Institute of Communication Engineering,National Taiwan University, Taipei, Taiwan 10617. Shih-Chun Lin and Yao-Win Peter Hong are with Institute of Communications Engineering,National Tsing Hua University, HsinChu, Taiwan, 30013. Emails: { [email protected], [email protected], [email protected],[email protected] } . The material in this paper was presented in part at the Annual Conference on Information Sciences and Systems,2010. This work was supported by the National Science Council, Taiwan, R.O.C., under grant NSC 98-2219-E-002-016. band using existing commercial standards, while the user who uses the cognitive radio technology is calledthe secondary user (SU). Originally, the cognitive radio adopts the interference avoidance methodology,that is, if a PU demands the licensed band, the SU should vacate and find an alternative one. Recently,the concept of interference mitigation was proposed for the cognitive radio [2], where the SU and PUcan coexist and simultaneously transmit at the same time and frequency bands to further improve thespectrum efficiency. The key is to allow cooperations between the transmitters of the SU and PU. Tomake the interference-mitigation based cognitive radio in [2] more practical, the coexistence constraint was further proposed in [3]. The cognitive radio is forced to maintain the same PU rate performance asif it is silent, under the constraint that the decoder of PU must be a single-user decoder, such as theconventional minimum distance decoder. Assuming that the PU’s message is known by the SU, in [3], theSU’s transmitter not only transmits its own signal but also relays the PU’s signal to meet the coexistenceconstraint. Moreover, by precoding with the celebrated dirty paper coding (DPC) [4], the SU’s receivercan decode as if the interference from the PU does not exist. Indeed, such a transmission scheme is provedto be capacity-achieving in some channel conditions [3].However, there are still some deficiencies and impractical assumptions in the cognitive radio proposedin [3] which motivate our work. First, in [3], the relayed PU’s signal and the SU’s own signal aresimultaneously transmitted. Since the SU’s signal is an interference to the PU’s receiver, it pollutes therelaying and may cause power inefficiency. Second, the DPC requires that the SU’s transmitter knowsthe PU’s message. It may be hard to satisfy this requirement, especially when the channel between thetransmitters of the PU and SU is not good enough. Finally, the perfect channel state information at thetransmitter (CSIT) may not always be available, epically when the channel is fast faded. Without full CSIT,the DPC used in [3] suffers [5]. To solve these problems, we propose two new transmission schemes forcognitive radio which are aided by the “clean” relaying to the PR’s receiver without the interferencefrom the SU. Under the half-duplex constraint, the clean relaying comes from the transmitter or/and thereceiver of SU, thus we name the proposed schemes as the clean transmitter relaying (CT) and the cleantransmitter-receiver relaying (CTR) aided cognitive radio, respectively.Our main contributions are proposing the new CT and CTR to improve the performance in [3]. OurCT generalizes the DPC-precoded cognitive radio in [3]. Moreover, our CTR can also avoid the last twoproblems mentioned in the previous paragraph since it does not require the DPC. The cooperation methodof the CTR makes it face a multiple-access channel (MAC) with common message, and we adopt theoptimal signaling for this channel from [6] in the CTR. We also invoke the channel coding theorem in [7] to ensure that the coexistence constraint is met under the relaying. With full CSIT and high signal-to-noiseratio (SNR), we find that the multiplexing gain performance of the CTR is better than (or at least no lessthan) that of [3]. This is due to the fact that the silent period spent on decoding the PU’s messages forthe DPC in [3] may not be necessary in the CTR. When there is only the statistics of CSIT, the CTR iseven more promising. We observe that the DPC used in [3] fails in fast Rayleigh fading channels, thatis, the rate performance of the SU is the same as that of treating the interference from the PU as purenoise at the SU’s receiver. Then the CTR always has better rate performance than that of [3] for all SNRregimes. We also identify the structure of the optimal common message relaying ratio for the CTR byexploring the corresponding stochastic rate optimization problem. Simulation results verify the superiorityof the proposed CT and CTR over methods in [3] in terms of rates and multiplexing gains under a largeclass of channels. Finally, the complexity of the CTR is lower than that in [3], while the complexity ofthe CT is approximately the same as that in [3]. The former is because the signaling from [6] adopted inthe CTR is much easier to implement in practice than the complicated DPC [8].The cognitive channel model studied in the paper is related to [9] [10], where cooperations in interferencechannels were studied. However, the coexistence constraints were not imposed in these papers, and thusthe relay strategies could be more flexible to obtain better rate performance compared with ours. As notedin [3], the capacity results for these less restricted channels can serve as the performance outer bounds forour setting. Moreover, full CSIT is usually assumed in the literatures [2] [3] [9] [10] (also in our previouswork [11]), while this work also considers the partial CSIT case. With only the statistics of CSIT, weshow that our CTR outperforms the DPC based schemes in [3] [5] in fast Rayleigh fading channels. Inaddition, the CT and multiplexing gain analysis also are new, and did not appear in our previous works[11].The paper is organized as following. The system model is discussed in Sec. II. In Sec.III and IV, wepresent the proposed CT and CTR and their rate and multiplexing gain performances with full CSIT,respectively. The performance analysis and the optimal common message relaying ratio with only thestatistics of CSIT in fast Rayleigh fading channels are given in Sec. V. We provide numerical examplesin Sec. VI. Finally, Sec. VII concludes this paper.II. S YSTEM M ODEL
A. Notations
In this paper, the superscript ( . ) H denotes the transpose complex conjugate. Identity matrix of di-mension n is denoted by I n . A block-diagonal matrix with diagonal entries A , . . . , A k is denoted by diag ( A , . . . , A k ) ; while | A | and | a | represent the determinant of a square matrix A and the absolute value of a scalar variable a , respectively. The mutual information between two random variables is denoted by I ( ; ) . We define C ( x ) , log ( + x ) (the base of log function is 2), and the function ( x ) + as ( x ) + = x if x ≥
0, otherwise, ( x ) + =
0. Also the indicating function A is one if the event A is valid, and is zerootherwise. B. Cognitive channel model
As shown in Fig. 1, in the considered four-node cognitive channel, Node 1 and 2 are the transmittersof PU and SU while Node 4 and 3 are the corresponding receivers, respectively. For the t -th symbol timewhere t is the discrete time index, the received signals Y ( t ) , Y ( t ) and Y ( t ) at Node 2, 3 and 4 can berespectively represented by Y ( t ) Y ( t ) Y ( t ) = h ( t ) h ( t ) h ( t ) h ( t ) h ( t ) h ( t ) X ( t ) X ( t ) X ( t ) + Z ( t ) Z ( t ) Z ( t ) , (1)where the channel gain between node i and j is denoted by h i j ( t ) , and Z i ( t ) is the additive white Gaussiannoise process at node i . Each time sample of Z i ( t ) is independent and identically distributed (i.i.d.)circularly-symmetric complex Gaussian, i.e., Z i ∼ C N ( , ) . Signals transmitted from Node 1, 2 and 3are denoted as X ( t ) , X ( t ) , and X ( t ) with long term average power constraints ¯ P , ¯ P , and ¯ P , respectivelyas 1 n n (cid:229) t = [ | X i ( t ) | ] ≤ ¯ P i , for i = , , , (2)where n is the number of coded symbols in a codeword. Note that all nodes are half-duplex .In this paper, we consider two cases with different channel knowledge of h i j ( t ) at the transmitter, whilethe channel gains h i j ( t ) are always assumed perfectly known at the corresponding receivers. In the firstcase, h i j ( t ) = h i j = | h i j | e j q i j , ∀ ≤ t ≤ n , where q i j is the channel phase. As for the CSIT assumptions,we assume that Node 1 knows h , Node 3 knows h , and Node 2 knows all channel gains based onthe method proposed in [3]. The second case is the fast Rayleigh fading channel, where each h i j ( t ) isvarying at each t . We assume that h i j ( t ) are i.i.d. generated according to a random variable H i j , and H i j is complex Gaussian distributed with zero mean and variance s i j . Moreover, due to the limited channelfeedback bandwidth, we assume that the channel realizations h i j ( t ) are unknown at the transmitters.However, Node 1 knows the statistics of H , Node 3 knows the statistics of H , and Node 2 knows thestatistics of all channels by applying the methods in [5] [3]. The SU also knows the target rate of the PUby using the methods in [5, Sec. II]. We restrict the decoder of PU at Node 4 as a single-user decoder. A single-user decoder D s is defined tobe any decoder which performs well on the point-to-point channel with perfect channel state knowledgeat the decoder [3]. Without loss of generality, we set the decoder to be the maximum-likelihood decoderfor fading channel with temporal independent Gaussian noise as in [12] (minimum-distance decoder). Wethen define the achievable rate under such decoder as the following. Definition 1:
A rate R is single-user achievable for the PU if there exists a sequence of ( nR , n ) encoders E n that encodes PU’s message w , such that the average probability of error vanishes to zero as n → ¥ when the receiver uses a single user decoder D s .Denote the set of all primary encoders that map primary messages to the transmitted signals as E n , wethen have the following definition. Definition 2:
A cognitive radio code with rate R and length n consists of an encoder to encode the SU’smessage w with output X n = { X ( ) , . . . , X ( n ) } as E n : E n × { , . . . , nR } × { , . . . , nR } → X n , where k X n k / n ≤ ¯ P , and a decoder to decode message w from the received signal Y n = { Y ( ) , . . . , Y ( n ) } .Based on Definition 2, we have the following definition for the achievable rate of the cognitive radiounder the coexistence constraint [3] . Definition 3:
The coexistence constraint means that for a given PU’s rate R T , the SU must take R T asa rate target and ensure that under its own transmissions, R T is still single-user achievable for the PU asdefined in Definition 1. A rate R is achievable for the SU if there exists a sequence of ( nR , n ) cognitiveradio codes defined in Definition 2 such that under the coexistence constraint, the average probability oferror vanishes to zero as n → ¥ .III. C LEAN T RANSMITTER R ELAYING IN C HANNELS WITH F ULL
CSITFor simplicity, we will introduce the CT aided cognitive radio and its performance in channels withfull CSIT first. Then we will discuss the CTR which further allows the relaying from the SU’s receiver inSection IV. As shown in Fig. 2 (a), the new three-phase CT is a generalization of the two-phase cognitiveradio in [3], by introducing an additional “clean” relay link (without interference from the SU) from Node2 in the third phase. As will be shown later, to know the PU’s message w for the DPC operation, Node 2needs Phase 1 to be long enough to correctly decode w from the received signal. However, this phase isneglected in [3] and most of the existing works. It is also clear from Fig. 2 (a) that due to the half-duplexconstraint, the transmission scheme must be multi-phase since Node 2 cannot receive and transmit at thesame time. As will be clarified later, the multi-phase transmission will cause SNR changes at Node 4 indifferent phases. To deal with this new problem, we need to invoke the upcoming Lemma 1 to meet the coexistence constraint.The detailed CT signaling method of each phase in Fig. 2 (a) comes as the following. To simplifythe notations, we omit the time index of the signals in (1) to represent the corresponding signals in theShannon random coding setting [13]. For example, X corresponds to X ( t ) . Assume that each three-phasetransmission occupies n symbol times, which forms a codeword. We have Phase 1 : Within the first ⌊ t n ⌋ symbols, Node 2 listens to and decodes the PU’s message w . Here t i isthe portion of time of Phase i , i = , , Phase 2 : If the decoding of w is successful, within the next ⌊ t n ⌋ symbols, Node 2 sends the DPCencoded signal X D using side-information h X plus the relaying of X as X = X D + r a P P e j ( q − q ) X , (3)where message w is conveyed in X D , a is the relay ratio from Node 2 to maintain the rate performance R T of the primary link, while P and P are the transmitted power of X and X in Phase 2, respectively. Phase 3 : For the remaining t n = n − ⌊ t n ⌋ − ⌊ t n ⌋ symbols, the clean relaying is transmitted from Node2 to assist decoding at Node 4 as X = p P / P e j ( q − q ) X . (4)To meet average power constraints (2), the power P and P are set as P = ¯ P and ( − t ) P = ¯ P . (5)After Phase 1, Node 2 knows the PU’s message w . Since Node 2 also knows the PU’s codebook fromDefinition 2, the PU’s transmitted codeword and then the interference at Node 3 h X is known atNode 2. The DPC results in [4] can be applied by using h X as the non-causally known transmitterside-information which is unknown at Node 3.Note that the received SNR of X at Node 4 changes in different phases (different block of symbols).To meet the coexistence constraint in Definition 3 under this phenomena, we introduce a Lemma as Lemma 1:
For a block of n transmissions over the channel Y n = H n X n + Z n , where n × X n and Y n are the transmit and received signals respectively, the diagonal channel matrix H n is known at thereceiver, and Z n is a Gaussian random sequence with diagonal covariance matrix K Z n (each element of Z n may not be identically distributed), the coding rate R is single-user achievable for Gaussian codebooks if R < n log | H n K X n ( H n ) H + K Z n || K Z n | , (6)where the covariance matrix K X n of the transmitted signal X n satisfies the power constraint. In Lemma 1, the n × n channel matrix H n is a collection of scalar time-domain channel coefficients overthe n transmissions. It is different to the spatial-domain channel matrix over single transmission in themultiple-antenna system [14], where the vector channels are assumed to be i.i.d in time. The proof of thislemma follows the steps in [7] where the asymptotic equipartition property for arbitrary Gaussian processis invoked to prove that the right-hand-side (RHS) of (6) is achievable by the suboptimal jointly typicaldecoder. Then it is also achievable by the optimal maximum-likelihood decoder defined in Definition 1.The detail is omitted. Then we have the following achievable rate result for the CT with the proof givenin Appendix A Theorem 1:
With full CSIT and the transmitted power setting in (5), the following rate of SU isachievable by the CT R ≤ max t , a t C ( | h | ( − a ) P ) , (7)which is subject to the constraint for coexistence with R T < t C ( | h | P ) + t C (cid:18) ( | h |√ P + | h |√ a P ) + | h | ( − a ) P (cid:19) + ( − t − t ) C (cid:16)(cid:0) | h |√ P + | h |√ P (cid:1) (cid:17) , (8)and the constraint for Node 2 to successfully decode PU’s message as t > R T / C ( | h | P ) , (9)where the intervals of Phase 1 and 2 are ⌊ t n ⌋ and ⌊ t n ⌋ , respectively, and the relaying ratio a ∈ [ , ] .Note that in [3], there is no Phase 3 ( t =
0) and the relaying from SU is always “noisy” (interfered bySU’s own signal X D ) from (3). When the channel gain | h | is large, much of the SU’s available powerare used to overcome the interference from SU’s own signal and the transmission may not be efficient( a is high). Also in [3], the assumption | h | ≫ | h | or I ( X ; Y ) ≫ R T is made to ensure that t can beessentially neglected ( t = t = t = The optimization problem in (7) is not convex in ( t , a ) and the analytical solution is hard to obtain.However, for fixed t , one can easily show that the optimal a (function of t ) is a ∗ ( t ) = −| h |√ P + p ( − | h | P + | h | P ) K ( t ) + ( + | h | P ) K ( t ) | h |√ P ( + K ( t )) ! , (10)where K ( t ) = t ( R T − t C ( | h | P ) − ( − t − t ) C ( | h | P + | h | P ) ) − . Note that if t = t = K ( ) = P , then a ∗ ( ) from (10) equals to the one derived in [3]. Since 0 < t ≤ − t , it is easy to find the optimal t maximizing (7) by line search.Now we study the multiplexing gain (or the pre-log factor) [14] of the SU, which is defined by m = lim ¯ P c → ¥ R / log ¯ P c , (11)where ¯ P c is the average transmission power utilized by the SU. For the CT, ¯ P c = ¯ P . The reason forintroducing ¯ P c is to fairly compare the performance of CT and CTR, of which the ¯ P c is defined in theupcoming (23). We focus only on the multiplexing gain of the SU since that of the PU is unchanged withand without the existence of SU due to the coexistence constraint. With (7) and (5), the upper bound ofthe multiplexing gain of the CT can be easily found as m ≤ − t = − C ( | h | ¯ P ) C ( | h | ¯ P ) ! + . (12)That is, the multiplexing gain is limited by the decoding time of Phase 1, which is small when | h | / | h | is small. This motivates us to develop the CTR discussed in the next section.IV. C LEAN T RANSMITTER -R ECEIVER R ELAYING IN C HANNELS WITH FULL
CSITAlthough the proposed CT is more practical and expected to outperform the cognitive radio in [3]when | h | is large, there are still some disadvantages. First, when | h | / | h | is small due to a deep fadefrom Node 1 to Node 2 or a blockage in this signal path, from (12), the CT may fail since t approaches1. In addition, the complexity of practical DPC implementation [8] may still be inhibitive in currentcommunication systems. These problems motivate us to include the clean relaying from Node 3, the SU’sreceiver, and develop the CTR aided cognitive radio. We will show that with full CSIT, the multiplexinggain of the CTR is no less than that of the CT (also the special case [3]) with lower implementationcomplexity. With only the statistics of the CSIT, the rate performance of CTR is even more promisingfor fast Rayleigh faded channels, as will be shown in Section V. Again, due to the half duplex constraint,the CTR transmission is multi-phase since Node 2 and 3 cannot transmit/receive at the same time.The equivalent channel of each phase in the proposed CTR is depicted in Fig. 2 (b). The basic designconcept comes as follows. After Phase 1, the PU’s message w is known by the SU, and can be treated as a common message for the PU and SU. Thus in Phase 2, Node 3 faces an asymmetric MAC with acommon message [6], since Node 3 also needs to decode w to enable clean relaying in Phase 3. Here theword “asymmetric” comes from the fact that the PU in this two-user MAC can only transmit the commonmessage w . The signaling method (upcoming (13)) in Phase 2 is then inspired from the optimal signalingproposed in [6]. Two independent codebooks are used to transmit the private and common messages w and w from Node 2, respectively. Note that we can not use the signaling designed for the conventionalinterference channels without the coexistence constraint such as [9], where the PU’s receiver needs todecode part of the SU’s messages to get good rate performance. The detailed signaling method for eachphase comes as the following. Phase 1 : In the first ⌊ t n ⌋ symbols, Node 2 and 3 listen to the PU’s message w . Node 2 decodes w . Phase 2 : Within the next ⌊ t n ⌋ symbols, Node 2 transmits X = U + r a P P e j q X , (13)where U is the signal bearing SU’s message w and is independent of X , while a and q are the relayingratio and phase for the common message w , respectively. Node 3 decodes both w and w . Phase 3 : For the remaining t n = n − ⌊ t n ⌋ − ⌊ t n ⌋ symbols, the clean relaying signals are transmittedfrom Node 2 and 3 as X = q P ( ) / P e j ( q − q ) X , X = p P / P e j ( q − q ) X , (14)respectively, where P ( ) and P are the transmitted power of Node 2 and 3 at Phase 3 respectively.To satisfy the power constraint (2), we have P = ¯ P , t P + t P ( ) = ¯ P , and t P = ¯ P . (15)It was shown in [6] that other than the complicated scheme in [17], the simple signaling (13) is alsooptimal for the Gaussian MAC with common message. The low complexity advantage of our CTR is theninherited from [6].To calculate the achievable rate of the CTR, first note that the received SNRs of X and U at Node3 both change at Phase 1 and 2. Then we need the following Lemma from [18]. Although Lemma 2 isan extension of the achievable rate in Lemma 1 to the MAC setting, in Lemma 1, we need to furtherprove that the rate is single-user achievable to meet the coexistence constraint in Node 4. However, suchrequirement is not needed for Node 3 where Lemma 2 is applied. Lemma 2:
For a block of n transmissions over the MAC Y n = H nx X n + H nu U n + Z n , where the channelmatrices H nx and H nu are diagonal and known perfectly at the receiver, and Z n is a Gaussian random sequence with covariance matrix K Z n , the rate pair ( R , R ) is achievable for Gaussian codebooks if R ≤ n log | H nx K x n ( H nx ) H + K z n || K z n | , (16) R ≤ n log | H nu K u n ( H nu ) H + K z n || K z n | , R + R ≤ n log | H nx K x n ( H nx ) H + H nu K u n ( H nu ) H + K z n || K z n | , (17)where the covariance matrices K x n and K u n of the transmitted signals X n and U n satisfy the powerconstraints, respectively.By combining the results in [6] and Lemma 2 as well as using Lemma 1, we can choose PU’s andSU’s codebooks which can simultaneously ensure successful decoding at Node 3, and meet the coexistenceconstraint at Node 4. We then have the following achievable rate of the CTR in Theorem 2. Here therate R ′ T can be treated as, after Phase 1, the residual information flow of w to be decoded at Node 3;while u Tx and u Rx in the end of the theorem statement indicate whether the relaying from transmitter andreceiver are possible, respectively. Theorem 2:
With full CSIT and the transmitted power setting as (15), the following rate of the SU isachievable by the CTR R ≤ max q , t , a n min h t · C (cid:0) | h | ( − a ) P (cid:1) , t · C (cid:16)(cid:12)(cid:12) h p a P / P e j q + h (cid:12)(cid:12) P + | h | ( − a ) P (cid:17) − R mT io , (18)where R mT = min n R ′ T , t · C (cid:16)(cid:12)(cid:12) h p a P / P e j q + h (cid:12)(cid:12) P (cid:17)o with R ′ T , R T − t C ( | h | P ) , and is subjectto the constraint for coexistence with R T ≤ t · C ( | h | P ) + t · C (cid:12)(cid:12) h + h p a P / P e j q (cid:12)(cid:12) P + | h | ( − a ) P ! + ( − t − t ) C (cid:18) | h |√ P + | h | q P ( ) + | h | p P (cid:19) ! , (19)where the common message relaying ratio and phase are a and q , and the time fractions of Phase 1 and 2are t and t , respectively. Moreover, let u Tx = t > R T / C ( | h | P ) , and u Rx = t C ( | h √ a P / P e j q + h | P ) ≥ R ′ T , t , a , and P ( ) are all zero when u Tx =
0, while P = u Rx = Proof:
We first consider the case where u Tx = u Rx =
1. In this case, both Node 2 and 3 arecapable of relaying with a ≥ P ( ) ≥ P ≥
0. As explained in the beginning of Section IV, Node 3faces an asymmetric MAC with common message w and private message of SU w . From [6], we knowthat one should choose X and U independent and Gaussian distributed with variance P and ( − a ) P ,respectively. The codebooks of PU and SU are generated according to X and U with rate R T and R ,respectively. As suggested in [6], the equivalent channel at Node 3 is similar to a common two-user MAC without common message as in [13]. However, as explained previously, the difference between this MACand that in [13] is that both the SNRs of X and U at Node 3 vary during Phase 1 and 2. Then we needLemma 2 which is more general than [13] to ensure correct decoding, with K u n = ( − a ) P I n , K x n = P I n , H nu = diag (cid:0) · I ⌊ t n ⌋ , h I ⌊ t n ⌋ , · I ⌊ t n ⌋ (cid:1) , H nx = diag (cid:16) h I ⌊ t n ⌋ , (cid:16) h p a P / P e j q + h (cid:17) I ⌊ t n ⌋ , · I ⌊ t n ⌋ (cid:17) , and K z n = I n , where (13) in Phase 2 and the channel model in (1) are used. Then from (17) in Lemma 2,the following rate constraints apply for the correctly decoding of ( w , w ) at Node 3 in Phase 2 R ≤ t · C (cid:0) | h | ( − a ) P (cid:1) , R + R T ≤ t · C ( | h | P ) + t · C (cid:16) | h p a P / P e j q + h | P + | h | ( − a ) P (cid:17) . (20)With the above two inequalities, we have (18) with R mT = R T − t C ( | h | P ) . Since u Rx = R mT = R ′ T = R T − t C ( | h | P ) by construction. Similarly, with u Rx = t C ( | h p a P / P e j q + h | P ) ≥ R ′ T ,inequality (16) in Lemma 2 is met by applying the above procedure. The decoding of ( w , w ) will thenbe successful. To ensure the coexistence, by invoking Lemma 1, (13), (14), and (1), and following thesteps in Appendix A, one can obtain (19).Now we consider the case u Tx = u Rx =
0. It happens when R T is too large for the MAC decoderin Node 3 to successfully decode w . Then there is only relaying from Node 2 and no relaying from Node3 at Phase 3 ( P = X and U as described previously, Node 3 treats the PU’s signal X as pureGaussian noise when decoding w . The achievable rate R is then t · C | h | ( − a ) P + (cid:12)(cid:12) h p a P / P e j q + h (cid:12)(cid:12) P ! . (21)Note that (21) can be rearranged as the second argument of the minimum in (18) with R mT = t C ( (cid:12)(cid:12) h p a P / P e j q + h (cid:12)(cid:12) P ) . (22)When u Rx =
0, our definition of R mT in the Theorem statement will make (22) valid. Also the minimumin (18) always equals to (21) since t · C (cid:0) | h | ( − a ) P (cid:1) is always larger than (21), and (21) equals tothe second argument in the minimum of (18) with (22).Finally, we consider u Tx = u Rx =
1, which results in a = t = P ( ) = X . However, as long as the clean relaying from Node 3 can satisfy the coexistence constraint with P > u Tx = u Rx = u Tx = u Rx = P must be zero to satisfy (19) sincethere is no relaying a = P ( ) = P =
0. The SU’s rate is zero from (18), and this concludes the proof. The optimization problem in Theorem 2 is non-convex even when t is given. However, since all variablesare bounded, the complexity of numerical line search is still acceptable.Note that our CTR uses different coding scheme compared with the CT, and does not always guaranteerate advantage over CT under full CSIT assumption. However, unlike the CT, even if (9) is violated and u Tx =
0, the CTR may still meet the coexistence constraint with only the relaying from Node 3 ( u Rx = u Tx =
1, if Node 2 needs too much time to decode w , setting t = P c in (11) equals to the sum of the average transmitted power from Node 2 and 3 (or total energyconsumption of the SU, equivalently). From (15), ¯ P c equals to¯ P c = t P + t P ( ) + t P = ¯ P + ¯ P . (23)Now we have the following Corollary with the proof given in Appendix B. Corollary 1:
With full CSIT, the following multiplexing gain of the SU is achievable by the CTR underthe power constraints (15) max m , − C ( | h | ¯ P ) C ( | h | ¯ P ) ! + , (24)where m = ( − t , for any t ∈ (cid:16) , − (cid:16) C ( | h | ¯ P ) / C ( | h | ¯ P ) (cid:17)i , when | h | < | h | , , otherwise. (25)Indeed, according to Appendix B, the multiplexing gains m and ( − C ( | h | ¯ P ) / C ( | h | ¯ P )) + cor-respond to the CTR using pure receiver ( t =
0) and pure transmitter ( t =
0) relaying, respectively.Comparing (24) and (12), we know that with full CSIT, the multiplexing gain of the CTR is larger (or atleast no less) than that of the CT (also its special case in [3]). In the next section, we will investigate theperformance of the CTR and CT in fast Rayleigh faded channels with only the statistics of CSIT. TheCTR is even more promising in this setting.V. P
ERFORMANCE IN F AST R AYLEIGH F ADING C HANNELS WITH S TATISTICS OF
CSITWe will first show that the performance of CT (and its special case [3]) has rate performance worse thanthat of the CTR. Then we focus on the CTR and its achievable rate. The optimal common message relayingratio a will also be investigated. First, for the precoding for the CT, it was shown that the linear-assignmentGel’fand-Pinsker coding (LA-GPC) [19] outperforms the DPC in Ricean-faded cognitive channels withthe statistics of CSIT [5]. This is because the LA-GPC, which includes the DPC as a special case, doesnot need the full CSIT as the DPC in designing the precoding paramteters. However, for Rayleigh fading channels with only the statistics of CSIT, we observe that even the more general LA-GPC results in arate performance the same as that of treating interference as noise. So the CTR will outperform the DPCbased CT in this channel setting. With a little abuse of notations, the above observation can be found asthe following proposition with the proof given in Appendix C. Proposition 1:
With only the statistics of CSIT, for the ergodic Rayleigh faded channel Y = H X + H X + Z with transmitter side-information X and power constraints E [ | X | ] ≤ ¯ P , E [ | X | ] ≤ ¯ P , themaximal achievable rate of the LA-GPC coded X is the same as the rate obtained by treating theinterference H X as noise, which is E " C | H | ¯ P + | H | ¯ P ! . (26)It is easy to use Proposition 1 to calculate the achievable rate of CT, which equals to the rate oftreating H X at Node 3 in Phase 2 as noise. Then the CTR always performs better than the CT in thefast Rayleigh fading channels according to the following intuitions. In the CTR, Node 3 will face a twouser MAC in Phase 2, and the rate pair from treating H X as noise while decoding the SU’s message isalways in the rate region of this MAC. Thus we only describe the CTR and its achievable rate in detailas follows Phase 1 : In the first ⌊ t n ⌋ symbols, Node 2 and 3 listen to the PU’s message w . Node 2 decodes w . Phase 2 : Within the next ⌊ t n ⌋ symbols, Node 2 transmits X = U + r a P P X , (27)where a is the relaying ratio for the common message w . Node 3 listens to and decodes w and w . Phase 3 : For the rest of ⌊ t n ⌋ symbol time, the clean relaying signals are transmitted from Node 2 and3 respectively as X = q P ( ) / P X and X = p P / P X . (28)Note that one of the differences compared with the full CSIT case in Section IV is that now the CTRcannot chose the phase in (27) and (28) since the channel phase realizations are unknown at Node 2.The achievable rate of the CTR in fading channels is presented in the following Theorem. Comparedwith the conventional fast fading channels, now the channel fading statistics will vary in different phases(block of symbols) at Node 3 and 4. This new problem corresponds to the SNR variation problem inSection IV, and can be solved by Lemma 1 and 2 as well as the channel ergodicity. The detailed proofis given in Appendix D. Theorem 3:
With the statistics of CSIT and the transmitted power meeting (15), the following rate ofthe SU is achievable by the CTR in the fast Rayleigh faded channel R ≤ max t , a min ( t E (cid:2) C (cid:0) | H | ( − a ) P (cid:1)(cid:3) , t E " C (cid:12)(cid:12)(cid:12)(cid:12) H + r a P P H (cid:12)(cid:12)(cid:12)(cid:12) P + | H | ( − a ) P ! − R mT ) , (29)where R mT = min n R ′ T , t · E h C (cid:16)(cid:12)(cid:12) H p a P / P + H (cid:12)(cid:12) P (cid:17)io with R ′ T , R T − t E [ C ( | H | P )] , and issubject to the constraint for coexistence with R T ≤ t E [ C ( | H | P )] + t E C (cid:12)(cid:12)(cid:12) H + q a P P H (cid:12)(cid:12)(cid:12) P + | H | ( − a ) P + ( − t − t ) E " C (cid:12)(cid:12)(cid:12)(cid:12) H + q P ( ) / P H + p P / P H (cid:12)(cid:12)(cid:12)(cid:12) P ! , (30)where the a , t and t are defined as those in Theorem 2, respectively. Moreover, let u Tx = t > R T / E [ C ( | H | P )] and u Rx = t E [ C ( | H √ a P / P + H | P )] ≥ R ′ T , t , a , and P ( ) are all zero if u Tx =
0, while P = u Rx = a as inthe following Corollary. The key observation is that the pointwise minimum of the two rate functions in(29) can be shown to be monotonically decreasing with a . Note that we can not get similar results forthe full CSIT case, the discussions are given right after the proof of this Corollary. Corollary 2:
Given t and t , the optimal common message relaying ratio a in Theorem 3 will validatethe equality in the constraint for coexistence (30). Proof:
To get the desire result, first we prove that both arguments of the pointwise minimum min { , } in (29) are monotonically decreasing with a given t . We focus on the second argument first and rearrangeit as t E (cid:20) log (cid:18) + | H | P + | H | P + { H H ∗ }√ P P √ a (cid:19)(cid:21) − R cT = t max { f ( a ) , f ( a ) } , (31)where the equality comes from the definition of R mT in Theorem 3, with f ( a ) and f ( a ) defined as f ( a ) , E (cid:20) log (cid:18) + | H | P + | H | P + { H H ∗ }√ P P √ a (cid:19)(cid:21) − R ′ T t , (32) f ( a ) , E (cid:20) log (cid:18) + | H | P + | H | P + { H H ∗ }√ P P √ a (cid:19)(cid:21) − E h C (cid:16)(cid:12)(cid:12) H p a P / P + H (cid:12)(cid:12) P (cid:17)i , (33)respectively. In the following, we will respectively show that f ( a ) and f ( a ) are both monotonicallydecreasing of a . Since the pointwise maximum of the two monotonically decreasing functions is still amonotonically decreasing function, from (31), the second argument of the min { , } in (29) is a monotoni-cally decreasing function of a . Now we show the monotonically decreasing properties of f ( a ) and f ( a ) . As for the f ( a ) in (32),note that from the definition of R ′ T in Theorem 3, only the first term in the RHS of (32) is related to a .This term can be further represented by E | H | , | H | (cid:20) E q , q (cid:20) log (cid:18) + | H | P + | H | P + √ P P √ a Re { H H ∗ } (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | H | , | H | (cid:21)(cid:21) , (34)where the property of the conditional mean is applied. We will show that given realizations | H | = | h | and | H | = | h | , the conditional mean E q , q (cid:2) ( . ) (cid:12)(cid:12) | H | = | h | , | H | = | h | (cid:3) in (34) is a monotonically decreasingfunction of a . Then so are (34) and f ( a ) . This conditional mean equals to E q , q (cid:20) log (cid:18) + | h | P + | h | P + √ P P √ a | h || h | cos ( q − q ) (cid:19)(cid:21) . (35)Since | H | , | H | and q , q are independent, given | H | = | h | and | H | = | h | , both q and q are still independent and uniformly distributed in ( , p ] , respectively. Then cos ( q − q ) is zero mean.Together with the fact that the log function is concave, we know that (35) is monotonically decreasingwith respect to a from [20, P.115]. As for f ( a ) , note that the term E [ C ( | h p a P / P + h | P )] in(33) is monotonically increasing in a . Since the first terms of the RHS of (33) and (32) are the same,from the previous results, we establish the monotonically decreasing property of f ( a ) .As for t E (cid:2) C (cid:0) | H | ( − a ) P (cid:1)(cid:3) , the first argument of the min { , } in (29), it is clear that this term ismonotonically decreasing with a given t . Then from the fact that the minimum of two monotonicallydecreasing functions results in a monotonically decreasing function, we prove the monotonically decreasingproperty of the pointwise minimum in (29). Finally, it is easy to see that the RHS of (30) monotonicallyincreases with a given t and t . Then the optimal a must validates the equality in (30).Note that the optimization problem with full CSIT in Theorem 2 is much more complicated than that inTheorem 3, and the simple result in Corollary 2 can not be obtained. Depending on the combinations of q , q and q , the second argument of the min { , } in (18) may increase with a . That is, more commonmessage relaying from Node 2 can increase the sum rate of the MAC at Node 3. The monotonicallydecreasing property does not always exist in the RHS of (18), and the SU’s rate may increases in acertain range of a . However, the unknown channel phase at Node 2 prohibits the SU to adjust q , andthe common message relaying is blind and always harmful at Node 3. One should just use the minimumpower which meets the constraint for coexistence for the common message relaying.Now we show the multiplexing gain. The proof is similar to that of Corollary 1 and is omitted. Corollary 3:
With the statistics of CSIT, the CTR can achieve the following multiplexing gain underthe power constraints (15), max m , − E [ C ( | H | ¯ P )] E [ C ( | H | ¯ P )] ! + , (36)where m = ( − t , for any t ∈ (cid:18) , − E [ C ( | H | ¯ P )] E [ C ( | H | ¯ P )] (cid:21) , when E [ C ( | H | ¯ P )] < E [ C ( | H | ¯ P )] , , otherwise.VI. S IMULATION R ESULTS
Here we provide simulation results to show the performances of our clean-relaying aided cognitiveradios. In the following discussions and the simulation figures, we will abbreviate the results from [3], orCT with t =
0, as JV. The noise variances at the receivers are set to unity, and the average transmittedSNR of PU ( ¯ P in (2)) is set to 20 dB. We assume that the SU in both CT (including JV) and CTRhave the same average transmission SNR ¯ P c , which can be computed according to (5) ( ¯ P c = ¯ P ) and (23),respectively. We set the PU’s rate R T as that when the interference from the SU is absent, that is, as C ( | h | P ) and E [ C ( | H | P )] in the full and statistics of CSIT cases, respectively.We first show the rate comparisons for channels with full CSIT. The channel gain of each figure islisted in Table I where the unit of the phase is radian. The t in both CT and JV are R T / C ( | h | P ) . InFig. 3, we can see that with large enough | h | as specified in Table I, the clean relaying from Node 3makes the CTR have the best rate performance. Next we consider the case where | h | is weaker in Fig. 4.When | h | is smaller than | h | , the CTR may prefer clean relaying from Node 2 rather than from Node3, that is, P ( ) > P =
0. It is easy to check that in this case, the optimal a for the CTR is also feasiblefor the CT. Then comparing (18) and (7), we know that the CT performs better than the CTR as in Fig. 4.Moreover, in Fig. 3 and 4, the clean relaying of the CTR and CT yields significant gains over the JV,respectively. Next, we show how the SU’s rate changes with | h | in Fig. 5. We can find out that there arethree regions. In Region 1, where | h | < | h | , we find that the CT and JV coincide. This is consistentwith [3], where JV is proved to be optimal in this region when relaying from Node 3 is prohibited. InRegion 2 and 3, | h | > | h | , the JV wastes lots of power on the relaying since the SU produces largeinterference at Node 4. The CT performs better than the JV due to the clean relaying. In Region 2, | h | < | h | , the CTR performs better than the CT since the CTR can use a better relaying path than thatof CT in Phase 3. In Region 3, | h | > | h | , the CT performs the best according to previously discussionsfor Fig. 4. However, the CT and CTR have the same performance due to the following reasons. In thechannel setting for Fig. 5 listed in Table I, we find that the first term of the min { . } in (18) is selected,which is the same as (7). Moreover, since this term is independent of q , the relaying phase q for theCTR is chosen as q − q from (19). Together with the power allocation as in the discussions for Fig. 4, the constraints for coexistence (19) and (8) are the same in this simulation. The optimal a of CTR andCT are also the same, and the CTR and CT have the same rate performance.Next we consider the rate performance in the fast Rayleigh faded channels with the statistics of CSIT.The channel variance of each link is listed in Table II. As shown in Fig. 6, the CTR outperforms theCT and JV, which is consistent with the discussions under Proposition 1 in Section V. The t in the JVis set to R T / E [ C ( | h | P )] . When the SU’s transmitted SNR is low, the CT (also JV) can only supportvery low rate as shown in Fig. 6. This is because that the PU’s transmitted SNR is set to 20 dB, then theinterference at Node 3 is relatively large for the SU when the SU’s transmitted SNR is small. Accordingto Proposition 1, the SU of CT (also JV) can only treat interference from the PU as noise, which degradesthe rate performance a lot. However, the MAC decoder of CTR at Node 3 can avoid this problem. InFig. 7 we show an example to verify the results in Corollary 2. We can find that the optimal a whichmaximizes the SU’s rate also make the equality in the constraint for coexistence (30) valid. That is, theoptimal a is the minimum a which makes the PU’s rate with the interference from SU the same as theinterference-free rate.Finally, we show the multiplexing gain comparisons in the following. Following the spirit of [21], weuse the generalized multiplexing gain (GMG) of the SU, which is defined as R / log ¯ P c , as the performancemetric for finite SNR. As ¯ P c approaches infinity, the GMG will approach the multiplexing gain definedin (11). We first show the full CSIT cases in Fig. 8 and 9 with channels specified in Table I respectively.In our simulation, we set a lower bound for t as 0.01 when t =
0, and m in Corollary 1 will be upper-bounded by 1-0.01=0.99. We then use the multiplexing gain in (24) with m = .
99 as the GMG upperbound in Fig. 8 and 9. With large | h | as in Table I, the GMG advantages of the CTR over the JV canbe seen from Fig. 8. When the transmitted SNR is larger than 40 dB, we can find that the curve of CTRdiverges from those of the CT and JV. This is because the CTR selects pure receiver relaying in this SNRregion. Since | h | < | h | in this simulation, according to discussions under Corollary 1, the CTR withpure receiver relaying has larger GMG than those of the CT and JV when t is small and the SNR is largeenough. Also when the SNR increases, the GMG of the CTR will approach the upper bound (24). Notethat we plot the figures according to the transmitted SNR not the common received SNR in most of theliteratures. The transmitted SNR is much larger than the received SNR since the | h | of Fig. 8 in Table Iis small. It then takes larger transmit SNR than the common received SNR for the GMG to approach theupper bound (multiplexing gain). In Fig. 9, we show the case with small | h | . The CT performs the bestwhile the CTR performances the worst. However, as predicted by Corollary 1, even though the CTR has the worst GMG, it will approach the GMGs of CT and JV as the SNR increases. The GMG results forthe fading channels with the statistics of CSIT are shown in Fig. 10. The GMG upper bound is computedfrom Corollary 3 with m = .
99 as in Fig. 8. According to the discussions for Fig. 6, the CT and JValways have worse GMG than that of the CTR according to Proposition 1.VII. C
ONCLUSION
In this paper, we considered the interference-mitigation based cognitive radio where the SU must meetthe coexistence constraint to maintain the rate performance of the PU. We proposed two new transmissionschemes aided by the clean relaying named as the clean transmitter relaying and the clean transmitter-receiver relaying aided cognitive radio, respectively. Compared with the previous DPC-based cognitiveradio without clean relaying, the proposed schemes provide significant rate gains in a variety of channelswith different levels of CSIT. Moreover, the implementation complexity of the CTR is much lower thanthat of the DPC-based cognitive radio. A
PPENDIX
A. Proof of Theorem 1
Let X be zero mean Gaussian with variance P , the PU then generates its random codebook accordingto the distribution of X with rate R T . From [22] we know that the fractional decoding interval mustsatisfy t > R T / I ( X ; Y ) to ensure the successful decoding of w using the received symbols from Node2 in Phase 1. It then results in the constraint (9) from (1).We now invoke Lemma 1 to derive the coexistence constraint. From (3) in Phase 2, (4) in Phase 3and the channel model (1), we know that within the n -symbol time, K X n = P I n , the equivalent channelat Node 4 H n = diag h I ⌊ t n ⌋ , (cid:18) | h | + | h | r a P P (cid:19) e j q I ⌊ t n ⌋ , (cid:16) | h | + | h | p P / P (cid:17) e j q I ⌊ t n ⌋ ! and the equivalent noise has covariance matrix K Z n = diag (cid:0) I ⌊ t n ⌋ , ( + | h | ( − a ) P ) I ⌊ t n ⌋ , I ⌊ t n ⌋ (cid:1) , sincethe DPC encoded X D is Gaussian with variance ( − a ) P and independent of X [4]. Then by invokingLemma 1, we have (8) to ensure that R T is single-user achievable. Finally, since Node 2 uses h X asthe noncausal side-information at the transmitter in Phase 2, by applying the well-known DPC result [4]we have (7). B. Proof of Corollary 1
We will consider two cases, that is, pure receiver and pure transmitter relaying. These two schemescan achieve multiplexing gains m and ( − C ( | h | ¯ P ) / C ( | h | ¯ P )) + , respectively. When the channelsconditions | h | > | h | and | h | > | h | are both valid, both schemes are feasible and the CTR can achievable the best multiplexing gain of these two schemes as (24). If only one of the channel conditionsis valid, the multiplexing gain of the corresponding feasible scheme will be chosen by (24).We first show that if | h | > | h | , as (25), the multiplexing gain 1 − t is achievable by the pure receiverrelaying. In this scheme, t = a = P ( ) =
0, and t = − t , then we may set P = P = ¯ P c from (23).Without loss of generality, we can set R T = C ( | h | ¯ P ) in the following analysis since R T ≤ C ( | h | ¯ P ) from the channel capacity theorem [13]. With the above parameter selections, the constraint for coexistence(19), and the constraint t C ( | h p a P / P e j q + h | P ) ≥ R ′ T to validate u Rx = C ( | h | ¯ P ) < t · C | h | ¯ P + | h | ¯ P c ! + ( − t ) C (cid:18) | h | q ¯ P + | h | q ¯ P c (cid:19) ! , and t ≥ C ( | h | ¯ P ) C ( | h | ¯ P ) . (37)When ¯ P c → ¥ , we can find that the range of t to validate (37) is C ( | h | ¯ P ) C ( | h | ¯ P ) ≤ t < . Therefore we need t ∈ ( , − C ( | h | ¯ P ) / C ( | h | ¯ P )] to meet the constraints. From (18), (11) and the fact that R mT = R ′ T since u Rx =
1, it is easy to see that the multiplexing gain t = − t is achievable, and (25) is valid. Notethat our selection of t and a is definitely a suboptimal choice with respect to (18). If | h | ≤ | h | and t =
0, there will be no relaying in this case since Node 3 can not decode w before the end of Phase 2.Then the multiplexing gain is zero for pure receiver relaying as in (25).Now we show that when | h | < | h | , the multiplexing gain 1 − C ( | h | ¯ P ) / C ( | h | ¯ P ) in (24) isachievable with only transmitter relaying ( t = t = C ( | h | ¯ P ) C ( | h | ¯ P ) , t = − t and q = q − q . Together with the setting R T = C ( | h | ¯ P ) as describe previously, thecoexistence constraint in (19) then becomes C ( | h | P ) < C (cid:0) | h | + | h | p a P / P (cid:1) P + | h | ( − a ) P ! . (38)With t P = ¯ P c from (23) and P = ¯ P from (15), as ¯ P c → ¥ , (38) becomes | h | ¯ P < a − a . Then we have a > | h | ¯ P / ( + | h | ¯ P ) to meet the constraint for coexistence. It can be easily seen that with theselected a , q , and t , when ¯ P c → ¥ , R mT = R ′ T in (18). Therefore, from (18) and (11) we can find that themultiplexing gain t = − C ( | h | ¯ P ) / C ( | h | ¯ P ) is achievable. Finally, when | h | ≥ | h | the function ( . ) + in (24) will force the multiplexing gain to be zero. In this case, the coexistence constraint is violatedsince Node 2 cannot relay without correct knowledge of w . C. Proof of Proposition 1
From [5], by treating X as non-causally known transmitter side-information, the following rate isachievable by the LA-GPCmax b { E (cid:2) log (cid:0) ( | H | ¯ P + | H | ¯ P + ) ¯ P (cid:1)(cid:3) − f ( b ) } , (39) where f ( b ) , E (cid:2) log (cid:0) ¯ P ¯ P | H − b H | + ¯ P + | b | ¯ P (cid:1)(cid:3) , and b ∈ C is the precoding coefficient of theLA-GPC. Note that solving (39) over b is the same as minimizing f ( b ) . In the following we will showthat f ( ) is the minimal. We know that for any b f ( ) = E (cid:2) log (cid:0) ¯ P ¯ P | H | + ¯ P (cid:1)(cid:3) ≤ E (cid:2) log (cid:0) ¯ P ¯ P ( + | b | s / s ) | H | + ¯ P (cid:1)(cid:3) = E (cid:2) log (cid:0) ¯ P ¯ P | H − b H | + ¯ P (cid:1)(cid:3) , (40)where the last equality comes from the fact that since H and H are independent zero-mean Gaussiandistributed with variance s and s , respectively, H − b H is also zero-mean Gaussian distributedwith variance s + | b | s . Thus ( + | b | s / s ) | H | and | H − b H | have the same distribution.Moreover, for any b , E (cid:2) log (cid:0) ¯ P ¯ P | H − b H | + ¯ P (cid:1)(cid:3) ≤ E (cid:2) log (cid:0) ¯ P ¯ P | H − b H | + ¯ P + | b | ¯ P (cid:1)(cid:3) = f ( b ) . Combining the above equation with (40), we know that b = f ( b ) and thus maximizes (39).Substituting b = D. Proof of Theorem 3
To meet the coexistence constraint, we invoke Lemma 1 again. Following the steps for proving (19) inTheorem 2, from (27), (28), (1), and Lemma 1, to ensure that the target PU’s rate is single-user achievable R T ≤ n ⌊ t n ⌋ (cid:229) t = log (cid:0) + | h ( t ) | P (cid:1) + n ⌊ t n ⌋ + ⌊ t n ⌋ (cid:229) t = ⌊ t n ⌋ + log + (cid:12)(cid:12)(cid:12) h ( t ) + q a P P h ( t ) (cid:12)(cid:12)(cid:12) P + | h ( t ) | ( − a ) P ) + n n (cid:229) t = n −⌊ t n ⌋ + log + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( t ) + s P ( ) P h ( t ) + r P P h ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P , (41)where h i j ( t ) is the realization of the random channel H i j at time t . When n is large enough, the first termof the RHS of (41) can be rewritten as1 n ⌊ t n ⌋ (cid:229) t = log (cid:0) + | h ( t ) | P (cid:1) = t ⌊ t n ⌋ ⌊ t n ⌋ (cid:229) t = log (cid:0) + | h ( t ) | P (cid:1) = t E [ log ( + | H | P )] , (42)where the last equality comes from the assumption that the channel coefficients are i.i.d. and applyingthe ergodicity property. After applying the same steps to the rest two terms of the RHS of (41), we havethe constraint for coexistence (30).The achievable rate of the SU in (29) can be obtained similarly. As for the steps to obtain (41), westill invoke Lemma 2 but modify the proof steps of Theorem 2 with the channel coefficients replaced by h i j ( t ) . Then we invoke the channel ergodicity as the proof steps in (42) to reach (29). The details areomitted. R EFERENCES [1] J. Mitola, “Cognitive radio: An integrated agent architecture for software defined radio,” Ph.D. dissertation, KTH Royal Inst. Technology,Stockholm, Sweden, 2000.[2] N. Devroye, P. Mitran, and V. Tarokh, “Achievable rates in cognitive radio channels,”
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HANNEL GAINS IN (1) U
SED IN THE S IMULATIONS (F ULL
CSIT)Figure h h h h h h . e . j . e . j . e − . j . e − . j . e − . j e − . j . e − . j . e . j . e . j . e − . j . e − . j e . j . e − . j varying | h | , q = p . e − . j . e − . j . e . j e . j . e − . j . e . j . e . j . e . j . e . j e − . j . e − . j . e . j . e . j . e − . j . e − . j e . j TABLE IIC
HANNEL V ARIANCES OF R AYLEIGH F ADING C HANNELS IN (1)U
SED IN THE S IMULATIONS (S TATISTICS OF
CSIT)Figure s s s s s s PU-TX
12 34
SU-TX PU-RXSU-RX h ( t ) h ( t ) h ( t ) h ( t ) h ( t ) h ( t ) Fig. 1. Cognitive channel model, where the TX and RX are the abbreviations of the transmitter and receiver, respectively. X X X X (cid:172) (cid:188) nt (cid:172) (cid:188) (cid:172) (cid:188) ntnt (cid:14) nPhase Phase Phase
PU-TX PU-RX PU-TX PU-RX PU-TX PU-RXSU-TX SU-RX SU-TX SU-RX SU-TX SU-RX X X X X X X nPhase Phase Phase
PU-TX PU-RX PU-TX PU-RX PU-TX PU-RXSU-TX SU-RX SU-TX SU-RX SU-TX SU-RX X (a)(b) (cid:172) (cid:188) (cid:172) (cid:188) ntnt (cid:14) (cid:172) (cid:188) nt Fig. 2. The signaling methods of the (a) CT and (b) CTR aided cognitive radio.
10 15 20 25 3000.20.40.60.811.21.41.61.822.2
Trasmitted SNR of SU (dB) R a t e o f S U ( bp c u ) CTRCTJV
Fig. 3. Comparison of the rate performance of the SU with full CSIT, under the coexistence constraint, and channels with large | h | asspecified in Table I. The rate is measured in bit per channel use (bpcu). Transmitted SNR of SU (dB) R a t e o f S U ( bp c u ) CTCTRJV
Fig. 4. Comparison of the rate performance of the SU with full CSIT, under the coexistence constraint, and channels with the | h | smallerthan the | h | as specified in Table I. | h | R a t e o f S U ( bp c u ) CTRCTJV
Region 1 Region 2 Region 3
Fig. 5. Comparison of the rate performance of the SU with full CSIT for different | h | , with 20 dB transmitted SNR and channel gainsspecified in Table I. Region 1 is the one where | h | < | h | , Region 2 is the one where | h | < | h | < | h | , and Region 3 is the one where | h | < | h | , respectively. Transmitted SNR of SU R a t e o f S U ( bp c u ) CTRCTJV
Fig. 6. Comparison of the rate performance of the SU under the coexistence constraint, and fast Rayleigh fading channels with the staticsof CSIT. The channel variances are listed in Table II. α R a t e ( bp c u ) SU’s ratePR’s rate with interference from SUPR’s target rate (interference free)
Fig. 7. Rate performance of the SU of CTR versus the common message relaying ratio a , under fast Rayleigh fading channels with thestatics of CSIT. The transmit SNR of SU is 20 dB and the channel variances are listed in Table II.
30 40 50 60 70 80 90 1000.10.20.30.40.50.60.70.80.91
Transmitted SNR of SU G ene r a li z ed M u l t i p l e x i ng G a i n o f S U Upper bound CTR CT JV
Fig. 8. Comparison of the generalized multiplexing gain performance of the SU with full CSIT, under channels with large | h | as specifiedin Table I. The upper bound is computed by (24) with m = .
30 40 50 60 70 80 90 1000.350.40.450.50.550.60.65
Transmitted SNR of SU G ene r a li z ed M u l t i p l e x i ng G a i n o f S U Upper boundCTJVCTR
Fig. 9. Comparison of the generalized multiplexing gain performance of the SU with full CSIT, under channels with small | h | as specifiedin Table I. The upper bound is computed by (24) with m = .
30 40 50 60 70 80 90 10000.10.20.30.40.50.60.70.80.91
Transmitted SNR of SU G ene r a li z ed m u l t i p l e x i ng ga i n o f S U Upper boundCTRCTJV
Fig. 10. Comparison of the generalized multiplexing gain performance of the SU, under fast Rayleigh fading channels with the statisticsof CSIT. The channel variances are listed in Table II. The upper bound is computed by (36) with m = ..