Optimal Power Control over Fading Cognitive Radio Channels by Exploiting Primary User CSI

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1 Otimal Power Control over Fading Cognitive Radio Channels by Exloiting Primary User CSI Rui Zhang Abstract arxiv: v2 [cs.it] 7 Feb 2009 This aer is concerned with sectrum sharing cognitive radio networks, where a secondary user (SU) or cognitive radio link communicates simultaneously over the same frequency band with an existing rimary user (PU) link. It is assumed that the SU transmitter has the erfect channel state information (CSI) on the fading channels from SU transmitter to both PU and SU receivers (as usually assumed in the literature), as well as the fading channel from PU transmitter to PU receiver (a new assumtion). With the additional PU CSI, we study the otimal ower control for the SU over different fading states to maximize the SU ergodic caacity subject to a new roosed constraint to rotect the PU transmission, which limits the maximum ergodic caacity loss of the PU resulted from the SU transmission. It is shown that the roosed SU ower-control olicy is suerior over the conventional olicy under the constraint on the maximum tolerable interference ower/intererferecne temerature at the PU receiver, in terms of the achievable ergodic caacities of both PU and SU. Index Terms Cognitive radio, ergodic caacity, fading channel, ower control, sectrum sharing. I. INTRODUCTION In this aer, we are concerned with the newly emerging cognitive radio (CR) tye of wireless communication networks, where the secondary users (SUs) or the so-called CRs communicate over the same frequency band that has been allocated to the existing rimary users (PUs). For such scenarios, the SUs usually need to deal with a fundamental tradeoff between maximizing the secondary network throughut and minimizing the erformance loss of the rimary network resulted from the SU transmissions. One commonly known technique used by the SUs to rotect the PUs is oortunistic sectrum access (OSA) [], whereby the SUs decide to transmit over the channel of interest only if the PU transmissions are detected to be off. Many algorithms have been reorted in the literature for detecting the PU transmission status, in general known as sectrum sensing (see, e.g., [2]-[4] and references therein). In contrast to OSA, another general oeration model of CRs is known as sectrum sharing (SS) [5], where the SUs are allowed to transmit simultaneously with the PUs rovided that the interferences from the SUs to PUs Rui Zhang is with the Institute for Infocomm Research, A*STAR, Singaore ( rzhang@i2r.a-star.edu.sg).

2 2 will cause the resultant PU erformance loss to be within an accetable level. Under the SS aradigm, the cognitive relay idea has been roosed in [6], [7], where the SU transmitter is assumed to know a riori the PU s messages and is thus able to comensate for the interferences to the PU receiver resulted from the SU transmission by oerating as an assisting relay to the PU transmission. As an alternative method for SS, the SU can rotect the PU transmission by regulating the SU to PU interference ower level to be below a redefined threshold, known as the interference temerature (IT), [8], [9]. This method is erhas more ractical than the cognitive relay method since only the SU to PU channel gain knowledge is required to be known to the SU transmitter. In this aer, we focus our study on the SS (as oosed to OSA) -based CR networks. It is known that in wireless networks, dynamic resource allocation (DRA) whereby the transmit owers, bit-rates, bandwidths and/or antenna beams of users are dynamically allocated based uon the channel state information (CSI) is essential to the achievable network throughut. For the case of single-antenna PU and SU channels, adative ower control for the SU is an effective means of DRA, and has been studied in, e.g., [0], [], to maximize the SU s transmission rate subject to the constraint on the maximum tolerable average IT level at the PU receiver. In [2], the authors roosed both otimal and subotimal satial adatation schemes for the SUs equied with multile antennas under the given IT constraints at different PU receivers. DRA in the multiuser CR networks based on the rincile of IT has also been studied in, e.g., [3], [4]. It is thus known that most existing works in the literature on DRA for the SUs are based on the IT idea. Although IT is a ractical method to rotect the PU transmission, its otimality on the achievable erformance tradeoff for the SU has not yet been carefully addressed in the literature, to the author s best knowledge. In this aer, we consider a simlified fading CR network where only one air of SU and PU is resent and all the terminals involved are equied with a single antenna. We assume that not only the CSI on the SU fading channel and that from the SU transmitter to PU receiver is known to the SU (as usually assumed in the literature), but is also the CSI on the PU fading channel (a new assumtion made in this aer). In ractice, CSI on the SU s own channel can be obtained via the classical channel training and feedback methods, while CSI from the SU transmitter to PU receiver can be obtained by the SU transmitter via estimating the reversed channel from PU receiver under the assumtion of channel

3 3 recirocity. The more challenging task is erhas on obtaining the CSI on the PU fading channel by the SU, for which more sohisticated techniques are required, e.g., via eavesdroing the feedback from the PU receiver to PU transmitter [5], or via a cooerative secondary node located in the vicinity of the PU receiver [6]. With the additional PU CSI, the SU is able to transmit with large owers when the PU fading channel is in inferior conditions such as dee fading, since under such circumstances the resultant PU erformance loss is negligible, nearly indeendent of the exact interference level at the PU receiver. In contrast, for the case where the IT constraint is alied, the SU s transmit ower is determined by the channel gain from the SU transmitter to PU receiver, while it is indeendent of the PU CSI. Motivated by these observations, this aershows that there in fact exists a better means to rotect the PU transmission as well as to maximize the SU transmission rate than the conventional IT constraint or more secifically, the average interference ower constraint (AIPC) over the fading states at the PU receiver. This new roosed method ensures that the maximum ergodic caacity loss of the PU resulted from the SU transmission is no greater than some rescribed threshold, thus named as rimary caacity loss constraint (PCLC). Clearly, the PCLC is more directly related to the PU transmission than the AIPC. In this aer, we will formally study the otimal ower-control olicy for the SU under the new roosed PCLC, and show its erformance gains over the AIPC in terms of imroved ergodic caacities of both PU and SU. The rest of this aeris organized as follows. Section II resents the system model. Section III resents the conventional ower-control olicy for the SU based on the AIPC. Section IV introduces the new PCLC, and derives the associated otimal ower-control olicy for the SU. Section V rovides numerical examles to evaluate the achievable rates of the roosed scheme. Finally, Section VI gives the concluding remarks. Notation: denotes the Euclidean norm of comlex number. E[ ] denotes the statistical exectation. The distribution of a circularly symmetric comlex Gaussian (CSCG) random variable (RV) with mean x and variance y is denoted by CN(x,y), and means distributed as. ( ) + = max(0, ). The notations, ( ) T, and ( ) H denote the matrix determinant, transose, and conjugate transose, resectively. I denotes an identity matrix. denotes the Euclidean norm of comlex vector.

4 4 II. SYSTEM MODEL As shown in Fig., we consider a SS-based CR network where a SU link consisting of a SU transmitter (SU-Tx) and a SU receiver (SU-Rx) transmits simultaneously over the same narrow band with a PU link consisting of a PU transmitter (PU-Tx) and a PU receiver (PU-Rx). The fading channel comlex gains from SU-Tx to SU-Rx and PU-Rx are reresented by ẽ and g, resectively, and from PU-Tx to PU-Rx and PU-Rx by f and õ, resectively. We assume a block-fading (BF) channel model and denote i as the joint fading state for all the channels involved. Furthermore, we assume coherent communications for both the PU and SU links and thus only the fading channel ower gains (amlitude squares) are of interest. Let e i denote the ower gain of ẽ at fading state i, i.e., e i = ẽ(i) 2 ; similarly, g i, f i, and o i are defined. It is assumed that e i, g i, f i, and o i are indeendent RVs each having a continuous robability density function (PDF). It is also assumed that the additive noises at both PU-Rx and SU- Rx are indeendent CSCG RVs each CN(0, ). Since we are interested in the information-theoretic limits, it is assumed that the otimal Gaussian codebook is used by both the PU and SU. Consider first the PU link. Since the PU may adot an adative ower control based on its own CSI, the PU s transmit ower at fading state i is denoted by q i. It is assumed that the PU s ower-control olicy, P (f i ), is a maing from the PU channel ower gain f i to q i, subject to an average transmit ower constraint reresented by E[q i ] Q. Examles of PU ower control are the constant-ower (CP) olicy q i = Q, i () and the water-filling (WF) olicy [7], [8] q i = ( d f i ) + (2) where d is a constant water-level with which E[q i ] = Q. In this aer, we assume that the PU is oblivious to the existence of the SU, and any interference from SU-Tx is treated as additional Gaussian noise at PU-Rx. Thus, the ergodic caacity of the PU channel is exressed as [ ( C = E log + f )] iq i +g i i (3)

5 5 where i denotes the SU s transmit ower at fading state i (to be more secified later). It is worth noting that the maximum PU ergodic caacity, denoted as C max = E[log(+f i q i )], is achievable only if f i q i +g i i = f i q i, i. (4) From (4), it follows that g i i = 0 if f i q i > 0 for any i. In other words, to achieve C max for the PU, the SU transmission must be off when the PU transmission is on, which is the same as the OSA with erfect sectrum sensing. Next, consider the SU link. The SU is also known as CR since it is aware of the PU transmission and is able to adat its transmit ower levels at different fading states based on all the available CSI between the PU and SU to maximize the SU s average transmit rate and yet rovide a sufficient rotection to the PU. In this aer, we assume that the CSI on both g i and f i is erfectly known at SU-Tx for each fading state i. For notational convenience, we combine the Gaussian-distributed interference from PU-Tx with the additive noise at SU-Rx, and define the equivalent SU channel ower gain h at fading state i as h i := e i +o i q i, which is also assumed to be known at SU-Tx for each i. Thus, the SU s ower-control olicy can be exressed as P s (h i,g i,f i ), subject to an average transmit ower constraint, E[ i ] P. By assuming that SU-Rx treats the interference from PU-Tx as additional Gaussian noise, the SU s achievable ergodic caacity is then exressed as C s = E[log(+h i i )]. (5) Note that the maximum SU ergodic caacity, denoted by Cs max, is achievable when P s maximizes C s with no attemt to rotect the PU transmission. In this case, the otimal P s is the WF olicy exressed ( ) +, as i = d s h i where ds is the water-level with which E[ i ] = P. Remark 2.: It is imortant to note that in the assumed system model, we have deliberately excluded the ossibility that the PU s allocated ower at fading state i is a function of the received interference ower from SU-Tx, g i i. If not so, the SU s ower control needs to take into account of any redictable reaction of the PU uon receiving the interference from SU-Tx, e.g., the PU may change transmit ower that will also result in change of the interference ower level at SU-Rx. Such feedback loo over the SU s and PU s ower adatations will make the design of the SU transmission more involved even for the deterministic channel case. This interesting henomenon will be studied in the future work.

6 6 III. SU POWER CONTROL UNDER AIPC Existing rior work in the literature, e.g., [0], has considered the eak/average interference ower constraint over fading states at PU-Rx as a ractical means to rotect the PU transmission. In this section, we first resent the SU ower-control olicy to maximize the SU ergodic caacity under the constraint that the average interference ower level over different fading states at PU-Rx must be regulated below some redefined threshold, thus named as average interference ower constraint (AIPC). The associated roblem formulation is similar to that in [0], but with an additional constraint on the SU s own transmit ower constraint, and is exressed as (P) : Maximize { i } E[log(+h i i )] Subject to E[g i i ] Γ (6) E[ i ] P (7) i 0, i (8) where Γ 0 is the redefined threshold for AIPC. It is easy to verify that (P) is a convex otimization roblem, and thus by alying the standard Karush-Kuhn-Tucker (KKT) otimality conditions [20], the otimal solution of (P), denoted as { () i }, is obtained as ( () i = ν () g i +µ ) + (9) () h i where ν () and µ () are non-negative constants. Note that the ower-control olicy given in (9) is a modified version of the standard WF olicy [7], [8]. Comared with the standard WF olicy, (9) differs in that the water-level is no longer a constant, but is instead a function of g i. Interestingly, similar variable water-level WF ower control has also been shown in [9] for multi-carrier systems in the resence of multiuser crosstalk. If g i = 0, (9) becomes the standard WF olicy with a constant water-level /µ (), since in this case the SU transmission does not interfere with PU-Rx. On the other hand, if g i, from (9) it follows that the water-level becomes zero and thus () i = 0 regardless of h i, suggesting that in this case no SU transmission is allowed since any finite SU transmit ower will result in an infinite interference ower at PU-Rx. Numerically, ν () and µ () can be obtained by, e.g., the ellisoid method [2]. This method utilizes the sub-gradients Γ E[g i i[n]] and P E[ i[n]] to iteratively udate ν[n+] and µ[n+] until they converge to ν () and µ (), resectively, where { i[n]} is obtained from (9) for some given ν[n] and µ[n] at the nth iteration.

7 7 Furthermore, it is observed from (9) that the ower control under AIPC does not require the PU CSI, f i, which is desirable from an imlementation viewoint. However, there are also drawbacks of this ower control exlained as follows. Suosing that f i q i = 0, i.e., the PU transmission is off, the SU can not take this oortunity to transmit if g i haens to be sufficiently large such that (9) results in that () i = 0. On the other hand, if f i q i haens to be a large value, suggesting that a substantial amount of information is transmitted over the PU channel, such transmission may be corruted by a strong interference from the SU if in (9) g i and h i result in a large interference ower g i () i (though it is uer-bounded by ν () ) at PU-Rx. Clearly, the above drawbacks of the AIPC-based ower control are due to the lack of joint exloitation of all the available CSI at SU-Tx, which will be overcome by the roosed ower-control olicy in the next section. It is worth mentioning that although the AIPC-based ower control is non-otimal, the AIPC still guarantees an uer bound on the maximum PU ergodic caacity loss, as given by the following theorem: Theorem 3.: Under the given AIPC threshold, Γ, the PU ergodic caacity loss due to the SU transmission, defined as C max C, is uer-bounded by log(+γ), regardless of P (f i ), P s (h i,g i,f i ), and the distributions of f i, h i, and g i. Proof: The roof is based on the following equality/inequalities: [ ( (a) C = E log + f )] iq i +g i i (b) E[log(+f i q i )] E[log(+g i i )] (c) E[log(+f i q i )] log(+e[g i i ]) (d) C max log(+γ) where (a) is due to (3); (b) is due to g i i 0, i; (c) is due to the concavity of the function log(+x) for x 0 and Jensen s inequality (see, e.g., [8]); and (d) is due to the definition of C max and the inequality (6). IV. SU POWER CONTROL UNDER PCLC In this section, we roose a new SU ower-control olicy by utilizing all the CSI on h i, g i, and f i, known at SU-Tx. This new olicy is based on an alternative constraint of AIPC to rotect the PU

8 8 transmission, named as rimary caacity loss constraint (PCLC). PCLC and AIPC are related to each other: From Theorem 3., it follows that the AIPC in (6) with a given Γ imlies that C max log(+γ), while the PCLC directly alies the constraint C max C C C δ, where C δ is a redefined value for the maximum tolerable ergodic caacity loss of the PU resulted from the SU transmission. 2 The ergodic caacity maximization roblem for the SU under the PCLC and the SU s own transmit ower constraint is exressed as (P2): Maximize { i } Subject to E[log(+h i i )] C max C C δ (0) (7),(8). Note that (P) and (P2) only differ in the constraints, (6) and (0). Since C max is a fixed value given Q, the distribution of f i, and the PU ower control P (f i ), using (3) we can rewrite (0) as [ ( E log + f )] iq i C 0 () +g i i wherec 0 = C max C δ. Unfortunately, the constraint () can be shown to be non-convex, rendering (P2) to be also non-convex. However, under the assumtion of continuous fading channel gain distributions, it can be easily verified that the so-called time-sharing condition given in [23] is satisfied by (P2). Thus, we can solve (P2) by considering its Lagrange dual roblem, and the resultant duality ga between the original and the dual roblems is zero. Due to the lack of sace, we ski here the detailed derivations and resent the solution of (P2) directly as follows: ( ) (2) i = λ i ( (2) i )ν (2) g i +µ + (2) (2) h i where similarly like (P), ν (2) and µ (2) are nonnegative constants that can be obtained by the ellisoid method. Comared to the AIPC-based ower-control olicy in (9), the new olicy in (2) based on PCLC has an additional multilication factor in front of the term ν (2) g i, which is further exressed as λ i ( (2) i ) = ( +g i (2) i f i q )( i ). (3) +g i (2) i +f i q i 2 Since most communication systems in ractice emloy some form of ower margin and/or rate margin (see, e.g., [22]) for the receiver to deal with unexected interferences, the PCLC is a valid assumtion if the PU belongs to such systems.

9 9 Note that λ i is itself a (decreasing) function of the otimal solution (2) i. Thus, the ower-control olicy (2) can be considered as a self-biased WF solution. From (2) and (3), we obtain Theorem 4.: The otimal solution of (P2) is (2) i = { 0 if λ i (0)ν (2) g i +µ (2) h i 0 z 0 otherwise, where z 0 is the unique ositive root of z in the following equation: (4) z = λ i (z)ν (2) g i +µ. (5) (2) h i An illustration of the unique ositive root z 0 for the equation (5) is given in Fig. 2. Note that F(z) λ i (z)ν (2) g i +µ (2) is an increasing function of z for z 0, and F(0) h i, F( ) = µ (2). As shown, z 0 is obtained as the intersection between a 45-degree line starting from the oint (0, h i ) and the curve showing the values of F(z). Numerically, z 0 can be obtained by a simle bisection search [20]. Some interesting observations are drawn on the PCLC-based ower control (4) as follows: First, from (2) and (3) it is observed that what is indeed required at SU-Tx for ower control at each fading state is the received signal ower at PU-Rx, f i q i, instead of the exact PU channel CSI, f i. Note that f i q i may be more easily obtainable by the SU than f i in some cases, e.g., when f i q i is estimated and then sent back by a collaborate secondary node in the vicinity of PU-Rx. Second, from (3) and (4) it is inferred that (2) i > 0 for any i if and only if f i q i +f i q i ν (2) g i +µ (2) < h i. For given ν (2), µ (2), and h i, it then follows that (2) i > 0 only when g i and/or f i q i are sufficiently small. This is intuitively correct because they are indeed the cases where the SU will cause only a negligible PU caacity loss. In the extreme case of g i = 0 and/or f i q i = 0, the condition for (2) i > 0 becomes > µ (2) h i, the same as the standard WF olicy. V. NUMERICAL EXAMPLES The achievable ergodic caacity airs of PU and SU, denoted by (C,C s ), over realistic fading channels are resented in this section via simulation. f, ẽ, g, and õ are assumed to indeendent CSCG RVs CN(0,), CN(0,), CN(0,0.5), and CN(0,0.0), resectively. It is also assumed that P = Q = 0, corresonding to an equivalent average signal-to-noise ratio (SNR) of 0 db for both PU and SU channels (without the interference between PU and SU). The following cases of (C,C s ) are then considered:

10 0 PCLC : The SU emloys the roosed ower-control olicy (4). C s s are obtained from (5) by substituting i s that are solutions of (P2) with different values of C δ, while the corresonding C s are obtained from (3). AIPC : The SU emloys the conventional ower-control olicy (9). C s s are obtained from (5) by substituting i s that are solutions of (P) with different values of Γ, while the corresonding C s are obtained from (3). AIPC, Lower Bound : The SU emloys the AIPC-based ower-control olicy (9). C s is obtained same as that in the second case, while for a given value of Γ, C is obtained as C max log(+γ). Note that log(+γ) is shown in Theorem 3. to be a caacity loss uer bound for the PU and, thus, C in this case corresonds to a PU caacity lower bound. MAC, Uer Bound : An auxiliary 2-user fading Gaussian multile-access channel (MAC) is considered here to rovide the caacity uer bounds for the PU and SU. In this auxiliary MAC, all the channels are the same as those given in Fig. excet that PU-Rx and SU-Rx are assumed to be collocated such that the received signals from PU-Tx and SU-Tx can be jointly rocessed. Let h (i) = [ f(i) õ(i)] T and h s (i) = [ g(i) ẽ(i)] T. Considering the auxiliary MAC, for a given PU ower-control olicy, P (f i ), it can be shown that the uer bounds on the PU and SU achievable rates belong to the following set [24] { (C,C s ) : C E [ log(+q i h (i) 2 ) ],C s E [ log(+ i h s (i) 2 ) ], P s(h i,g i,f i ):E[ i ] P C +C s E [ log I +qi h (i)h H (i)+ i h s (i)h H s (i) ]} (6) which can be efficiently comuted by alying the methods given in [25] and the details are thus omitted here for brevity. Fig. 3 and Fig. 4 show the achievable PU and SU ergodic caacities for the aforementioned cases when the PU ower-control olicy is the CP olicy in () and the WF olicy in (2), resectively. It is observed that in both figures, the caacity gains by the roosed SU ower-control olicy using the PCLC are fairly substantial over the conventional olicy using the AIPC. For examle, when the PU caacity loss resulted from the SU transmission is 5%, i.e., C = 0.95 C max, the SU caacity gain by the roosed olicy over the conventional olicy is around 28% in the CP case, and 50% in the

11 WF case. Another interesting observation is that for the roosed SU ower control, the SU caacity reaches its minimum value of zero in the CP case when the PU caacity attains its maximum value, C max, while in the WF case the SU caacity has a non-zero value at C max. In general, caacity gains of the roosed SU ower control are more significant in the case of WF over CP PU ower control. This is because the WF olicy results in variable PU transmit owers based on the PU CSI and thus makes the roosed SU ower control that is designed to exloit the PU CSI more beneficial. VI. CONCLUDING REMARKS In this aer, we studied the fundamental caacity limits for sectrum sharing based cognitive radio networks over fading channels. In contrast to the conventional ower-control olicy for the SU that alies the interference-ower or interference-temerature constraint at the PU receiver to rotect the PU transmission, this aer roosed a new olicy based on the constraint that limits the maximum ermissible PU ergodic caacity loss resulted from the SU transmission. This new olicy is more directly related to the PU transmission than the conventional one by exloiting the PU CSI. It was verified by simulation that the roosed olicy can lead to substantial caacity gains for both the PU and SU over the conventional olicy. Many extensions of this work are ossible. First, this aer considers the ergodic caacity as the erformance limits for both PU and SU, while similar results can be obtained for non-ergodic PU and SU channels where the outage caacity should be is a more aroriate measure. Second, results in this aer can also be extended to the cases with imerfect/quantized PU CSI. Last, the roosed scheme in this aer is alicable to the general arallel Gaussian channel with sufficiently large number of subchannels, e.g., the broadband channel that is decomosable into a large number of arallel narrow-band channels via multi-carrier modulation and demodulation. REFERENCES [] Joseh Mitola, Cognitive radio: an integrated agent architecture for software defined radio, PhD Dissertation, KTH, Stockholm, Sweden, Dec [2] D. Cabric, S. M. Mishra, and R. W. Brodersen, Imlementation issues in sectrum sensing for cognitive radios, IEEE Asilomar Conference on Sig., Sys., and Com., [3] A. Ghasemi and E. S. Sousa, Collaborative sectrum sensing for oortunistic access in fading environments, IEEE Int. Sym. New Front. Dyn. Sectrum Access Networks (DySPAN),. 3-36, Nov [4] S. M. Mishra, A. Sahai, and R. W. Brodersen, Cooerative sensing among cognitive radios, Proc. IEEE ICC, Jun

12 2 [5] Q. Zhao and B. M. Sadler, A survey of dynamic sectrum access, IEEE Sig. Proc. Mag., vol. 24, no. 3, , May [6] N. Devroye, P. Mitran, and V. Tarokh, Achievable rates in cognitive radio channels, IEEE Trans. Inf. Theory, vol. 52, no. 5, , May [7] A. Jovičić and P. Viswanath, Cognitive radio: An information-theoretic ersective, Proc. IEEE Int. Sym. Inf. Theory (ISIT), Jul [8] S. Haykin, Cognitive radio: brain-emowered wireless communications, IEEE J. Sel. Areas Commun., vol. 23, no. 2, , Feb [9] M. Gastar, On caacity under receive and satial sectrum-sharing constraints, IEEE Trans. Inf. Theory, vol. 53, no. 2, , Feb [0] A. Ghasemi and E. S. Sousa, Fundamental limits of sectrum-sharing in fading environments, IEEE Trans. Wireless Commun., vol. 6, no. 2, , Feb [] S. Srinivasa and S. A. Jafar, Soft sensing and otimal ower control for cognitive radio, Proc. IEEE GLOBECOM, Dec [2] R. Zhang and Y. C. Liang, Exloiting multi-antennas for oortunistic sectrum sharing in cognitive radio networks, IEEE J. S. Toics Sig. Proc., vol. 2, no., , Feb [3] J. Huang, R. Berry, and M. L. Honig, Sectrum sharing with distributed interference, in Proc. IEEE Sym. New Frontiers in Dynamic Sectrum Access Networks (DySPAN), Nov [4] R. Zhang, S. Cui, and Y. C. Liang, On ergodic sum caacity of fading cognitive multile-access and broadcast channels, submitted to IEEE Trans. Inf. Theory. Available [Online] at arxiv: [5] K. Eswaran, M. Gastar, and K. Ramchandran, Bits through ARQs: sectrum sharing with a rimary acket system, in Proc. IEEE International Symosium on Information Theory (ISIT), , Jun [6] G. Ganesan and Y. Li, Cooerative sectrum sensing in cognitive radio networks, IEEE Int. Sym. New Front. Dyn. Sectrum Access Networks (DySPAN), , Nov [7] A. Goldsmith and P. P. Varaiya, Caacity of fading channels with channel side information, IEEE Trans. Inf. Theory, vol. 43, no. 6, , Nov [8] T. Cover and J. Thomas, Elements of Information Theory, New York: Wiley, 99. [9] W. Yu, Multiuser water-filling in the resence of crosstalk, in Proc. IEEE Information Theory Alications Worksho, , Jan [20] S. Boyd and L. Vandenberghe, Convex otimization, Cambridge University Press, [2] R. G. Bland, D. Goldfarb, and M. J. Todd, The ellisoid method: A survey, Oerations Research, vol. 29, no. 6, , 98. [22] T. Starr, J. M. Cioffi, and P. J. Silverman, Understanding digital subscriber line technology, Englewood Cliffs, NJ: Prentice-Hall, 999. [23] W. Yu and R. Lui, Dual methods for nonconvex sectrum otimization of multicarrier systems, IEEE Trans. Commun., vol. 54. no , Jul [24] D. Tse and S. Hanly, Multi-access fading channels-part I: olymatroid structure, otimal resource allocation and throughut caacities, IEEE Trans. Inf. Theory, vol. 44, no. 7, , Nov [25] M. Mohseni, R. Zhang, and J. Cioffi, Otimized transmission of fading multile-access and broadcast channels with multile antennas, IEEE J. Sel. Areas Commun., vol. 24, no. 8, , Aug

13 3 PSfrag relacements f PU-Tx ẽ g õ PU-Rx SU-Tx SU-Rx Fig.. System model for the CR network. µ (2) F(z) PSfrag relacements z 0 h i 45 z 0 z Fig. 2. Illustration of the unique ositive root z 0 for the equation (5).

14 C s (bits/comlex dimension) PCLC AIPC AIPC, Lower Bound MAC, Uer Bound C (bits/comlex dimension) Fig. 3. Achievable rates of PU and SU when the PU ower-control olicy is the CP olicy given by ().

15 C s (bits/comlex dimension) PCLC AIPC AIPC, Lower Bound MAC, Uer Bound C (bits/comlex dimension) Fig. 4. Achievable rates of PU and SU when the PU ower-control olicy is the WF olicy given by (2).