On the Fundamental Limits of Interweaved Cognitive Radios
OOn the Fundamental Limits of InterweavedCognitive Radios
G. Chung, and S. Vishwanath
Wireless Networking and Communication GroupUniversity of Texas at AustinAustin, TX 78712, USAEmail: { gchung,sriram } @ece.utexas.edu C. S. Hwang
Communication Lab., SAITSamsung Electronics Co. Ltd.Yongin, KoreaEmail: [email protected]
Abstract — This paper considers the problem of channel sensingin cognitive radios. The system model considered is a set of N parallel (dis-similar) channels, where each channel at any giventime is either available or occupied by a legitimate user. Thecognitive radio is permitted to sense channels to determine eachof their states as available or occupied. The end goal of thispaper is to select the best L channels to sense at any giventime. Using a convex relaxation approach, this paper formulatesand approximately solves this optimal selection problem. Finally,the solution obtained to the relaxed optimization problem istranslated into a practical algorithm. I. I
NTRODUCTION
As the number and types of wireless (multimedia) applica-tions increase, so do the stringent requirements they impose onthe wireless medium. Thus, it is essential that we determineefficient means of utilizing limited spectral resources avail-able to us. Currently, bandwidth resources are divided intofrequency bands and allocated to different users exclusivelyin order to insure the quality of service (QoS) of multiplewireless systems, and the FCC’s frequency allocation chart[1] shows that almost all frequency bands are currently dividedand allocated to different groups for varying purposes. Also,according to recent surveys [2] and [3], most of this allocatedradio frequency spectrum is vastly under-utilized. Cognitiveradios are emerging as promising solutions to enable betterutilization of spectrum especially in bands that are currentlyunderutilized [4]. The classical example of a cognitive radio isone that employs “interweave” cognition [4]. These interweavecognitive radios are permitted to occupy a channel (frequencyband) only when it is not occupied by a user licensed touse that band. If the presence of other radios can be sensedaccurately and quickly, then such a policy can help ensure thatcognitive users cause little to no interference to the licensedradios in the system. A majority of existing literature oncognitive radios focuses on such interweaved radios. For ananalysis of other classes of cognitive radios, see [5], [6] and[7]. One of the main issues under study in the interweavedcognitive radio domain is the so-called ”sensing problem”,where we desire to determine, as accurately and efficiently This work is supported by a grant from Samsung Advanced Institute ofTechnology. as possible, if a given channel is occupied at any given time[8], [9], and [11]. For example, [8] describes a simple energydetection scheme for additive white Gaussian noise channel.In [9], the performance of energy detection schemes in amultipath environment is analyzed, and in [11], the impact ofadditional side information is considered in determining theperformance of cognitive sensing. Overall, channel sensing isone of the better established fields of research on cognitivecommunication. In this paper, our goal is significantly differentfrom that of channel-sensing literature.
Given a fairly accuratesensing algorithm, we desire to determine which channelsshould be sensed when . In addition, we desire to perform aresource-allocation problem across multiple channels whichmay or may not be available to the cognitive radio. Overall,we ask the question ”Given that there are multiple dissimilarchannels available for you to sense, which channels shouldyou sense and, if they are available, what rate/power shouldyou assign to them?”The dissimilarity between different channels arises fromvarious factors. The properties of the propagation environmentdepend on frequency and thus can be significantly differentfrom channel to channel. Some channels may suffer from“extraneous interference” from non-legitimate sources (suchas in the industrial, scientific and medical (ISM) bands) thatreduce the channel quality. Thus, just as any other multibandradio, the cognitive radio must allocate resources across dif-ferent bands it uses while simultaneously determining whichones it is permitted to exploit. Note that, in isolation, theproblem of channel selection for cognitive radios [12], [13] iswell studied. Also, by itself, the resource allocation problemfor muti-band radios is also well-understood [14]. However,bringing the two together is both important and challenging asthey are tightly coupled in the context of interweaved cognitiveradios. A simple explanation of this strong interdependencebetween sensing (channel selection) and resource allocation isas follows: Let us say that the system is such that “noisy”(poor) channels are less frequently used by licensed usersthan “clear” (good) channels. If the sensing mechanism wereto choose to sense the infrequently-used channels, it willpresent the cognitive radio with available channels that are all“poor” resulting in a low rate. On the flip side, if the resourceallocation mechanism were to assign high rates to the “good” a r X i v : . [ c s . I T ] O c t ime (slot) Leg i t i m a t e C hanne l : Unoccupied Time-Frequency Channel Block: Occupied Time-Frequency Channel Block . . . N T N Fig. 1. Channel Model channels, the sensing mechanism may find that they are notavailable for use and then again sustain a very poor rate. Thus,designing channel selection and allocation jointly is essentialfor cognitive radios. Note that this paper’s focus is on thefundamental limits of joint selection and resource allocationin cognitive networks to provide a benchmark on performance.Thus, aspects such as sensing error, delay, device and networknon-linearities etc. are not incorporated into the analysis.The rest of this is organized as follows. The next sectiondetails the system model and notations used in the paper.In Section III, we find the fundamental limit of the givensystem model. In Section IV, we propose an algorithm forjoint channel selection and power allocation, and we concludewith Section IV.II. S
YSTEM M ODEL AND P ROBLEM S TATEMENT
The channel model is shown in Fig. 1. We consider N parallel legitimate channels with equal bandwidth. In each timeslot, a channel n, ≤ n ≤ N , is occupied by a legitimate userwith probability q n . There are one cognitive transmitter andone cognitive receiver. The cognitive transmitter is allowed totransmit over channel n , if it is not occupied by any licenseduser. In legitimate channel n , cognitive radio’s channel is: Y n = X n + Z n where Z n is additive Gaussian noise of variance σ n . Note thatthis noise variance can be different from channel to channel, asit represents the fading state of that particular channel. Beforethe start of cognitive radio’s transmission using the legitimatechannels, the cognitive transmitter should know whether theyare occupied by the licensed users or not. Thus, at the start ofevery time slot, the cognitive transmitter is allowed to sensea subset of channels, and is allowed to exploit those channelsthat are unoccupied; in this paper, we assume that the sensingis performed perfectly. Also, the cognitive transmitter is notallowed to transmit using the channel which is not sensed inorder to guarantee the transmission of the licensed users. As N is assumed to be large, it is impractical to allow the cognitiveradio the ability to sense all of them at the start of every slot.Instead, we require it to cleverly choose a subset of bands onwhich to focus its efforts. The capacity of the cognitive radiodepends on which channels to sense from N parallel channels, and power allocation among the available parallel channels.Average total transmission power of cognitive transmitter isconstrained to P .First, define the I n ( t ) and I E,n ( t ) to be the indicatorfunction for selected channel to be sensed and an indicatorfunction for the unoccupied channel respectively, i.e., I n ( t ) = (cid:26) if channel n is not to be sensed if channel n is to be sensed (1)and I E,n ( t ) = (cid:26) if channel n is occupied if channel n is unoccupied . (2)Denote the time average capacity of the cognitive radio withthe selection of the sensing channel I n ( t ) and power allocation P n ( t ) in one time block as C ( I n ( t ) , P n ( t )) . Then, C ( I n ( t ) , P n ( t )) = 1 T N (cid:88) n =1 T (cid:88) t =1 I n ( t ) I E,n ( t )2 log (cid:18) P n ( t ) σ n (cid:19) , (3)where T is the number of time slot in each time block.In our model, we assume two constraints on the cognitiveradio:a. An average power constraint on the cognitive transmitter of P ,b. The number of channels that can be sensed by the cognitiveradio at any given time is L ≤ N .Note that if the cognitive radio could sense all channels, L = N , this problem has a fairly trivial solution. At the start of eachtime slot, the cognitive radio would determine all availablechannels and waterfill its power over them [10].If the number of channels that cognitive radio can sense isless than N , i.e. L < N , the resulting optimization problemcan be stated as follows: max P n ( t ) ,I n ( t ) C ( I n ( t ) , P n ( t )) (4a)such that T N (cid:88) n =1 T (cid:88) t =1 I n ( t ) I E,n ( t ) P n ( t ) ≤ P, (4b) N (cid:88) n =1 I n ( t ) ≤ L, (4c)and P n ( t ) ≥ ,I n ( t ) ∈ { , } ,I E,n ( t ) ∈ { , } . (4d)The optimization problem given by (4) determines themaximum empirical average rate achieved by the cognitiveradio given constraints on the system. Note that it is an integerprogramming (IP) due to the constraints in (4d), and multi-dimensional due to its dependence on time t .The next section studies the optimization problem given by(4) in an ergodic policy setting.II. O PTIMAL P OWER A LLOCATION AND S ELECTION OF S ENSING C HANNEL
As a first step, we assume that our policy is ergodic and“static” , i.e., that our sensing and power allocation policies areonly functions of the channel statistics and do not evolve withtime. This results in the following (simplified) optimizationproblem: max P n ,I n N (cid:88) n =1 I n q n (cid:18) P n σ n (cid:19) (5a)such that N (cid:88) n =1 I n q n P n ≤ P, (5b) N (cid:88) n =1 I n ≤ L, (5c)and P n ≥ ,I n ∈ { , } . (5d)Since P n = 0 where I n = 0 , constraints (5b) can further berelaxed to N (cid:88) n =1 q n P n ≤ P. (5e)Denoting the optimal selection of channels to be sensed andpower allocation for channel n as I ∗ n and P ∗ n respectively, theoptimum solution for (5) is given by the following theorem: Theorem 1:
The joint channel selection & rate allocationproblem (characterized by the optimization problem in (5) ismaximized when: I ∗ n = arg max I n N (cid:88) n =1 q n I n (cid:24) log λσ n (cid:25) + P ∗ n = (cid:6) λ − σ n (cid:7) + I ∗ n , where N (cid:88) n =1 (cid:6) λ − σ n (cid:7) + I ∗ n q n = P N (cid:88) n =1 I ∗ n = L, and (cid:100) w (cid:101) + is maximum value of 0 and w . Proof:
Note that (5a) is a concave function over P n for aparticular choice of I n . The following Lagrangian describesthe optimization of (5a) with respect to P n for a given I n : L = (cid:80) Nn =1 I n q n log (cid:16) P n σ n (cid:17) − λ (1) (cid:16)(cid:80) Nn =1 q n P n − P (cid:17) + (cid:80) Nn =1 λ (1) n P n (6)Taking the derivative of (6) and setting it to zero, we get: N N q I q I q I N N q I
N N q I P P N P Fig. 2. power allocation with given I n ( I = 1 , I = 1 , I = 0 , ..., I N = 1 ) ∂ L ∂P n = I n q n log e P ∗ n + σ n ) − λ (1) q n + λ (1) n = 0 . (7) P ∗ n = (cid:24) I n log e λ (1) − σ n (cid:25) + = (cid:6) λ − σ n (cid:7) + I n , (8)where λ = log e λ (1) .From (5e) we obtain, N (cid:88) n =1 (cid:6) λ − σ n (cid:7) + I n q n = P. (9)Fig. 2. provides a graphical representation for the powerallocation strategy in (8). Note that it is similar to the water-filling solution, with the main difference that the each channelhas different width, q n I n . We refer to the policy in (9) as modified water-filling throughout this paper. Given that weunderstand the structure of the power allocation policy thatoptimizes (5a), we now desire to determine I ∗ n . Note again thatthe optimization problem in (5) with respect to I n is an IP.It can be found by an exhaustive search, but computationallyvery hard to solve. Moreover, the power allocation strategy in8, specifically, the water-level λ is tightly coupled with thechoice of I n . In the next section, we present an algorithmicframework that approximates I ∗ n (and thus the water-level λ )using low-complexity iterative techniques.IV. J OINT S ELECTION AND P OWER C ONTROL
A typical integer program is non-polynomial in complexity.Although multiple techniques exist for obtaining approximatesolutions to such a program (such as branch and bound [15],relaxation), such techniques apply to any integer program anddo not take the structure of the problem into consideration.Our focus is on developing an algorithm customized to thisproblem setting. We perform this in two steps, which we call“coarse” and “fine” optimization. The coarse optimization stepdetermines a set of L channels to be utilized by the cognitiveradio. It gives us the lowest possible waterlevel, λ min . The fineptimization step uses λ min which we obtained from coarseoptimization to further optimize the choice of the L channels.First, we describe the coarse optimization step: Coarse Optimization : We iteratively find the channels tosense along with modified water-filling which incur the lowestwater level. Let λ min denote the lowest water level, and I cn and P cn indicate the selection of the channel and power allocationwhich result in λ min . Detailed procedures to find λ min , I cn ,and P cn is described in the following four steps.Step I: Start with L initial channels. We can choose L channels with the largest q n as initial channels, for example. I n, = (cid:26) if q n is among L largests otherwise (10) S = { n ∈ [1 , N ] | I n, = 1 } (11) j = 1 (12)Step II: Perform the modified water-filling with I n,j − , j ≥ ,such that N (cid:88) n =1 (cid:6) λ j − σ n (cid:7) + I n,j − q n = P. (13)Step III: Calculate q n ( λ j − σ n ) , and select the largest L channels. I n,j = if q n ( λ j − σ n ) > q n ( λ j − σ n ) is among L largests otherwise (14) S j = { n ∈ [1 , N ] | I n,j = 1 } (15)Step IV: If S j = S j − , terminate the iteration, and set thepower allocation and channel selection values. λ min = λ n (16) I cn = I n,j (17) P cn = (cid:0) λ j − σ n (cid:1) I n,j . (18)Otherwise, j = j + 1 and repeat from step II.The coarse optimization is performed for two reasons. Oneis that the performance of coarse optimization is very closeto the optimum. This will be shown from the simulationresult in the next section. Here, the optimality of the coarseoptimization in one special case will be stated and proven. Lemma 1:
Define S c to be the set of the channels whichare selected from coarse optimization; S c = { n ∈ [1 , N ] | I cn = 1 } . If the noise variances of all the channels which are not selectedin the coarse optimization are greater than the lowest waterlevel λ min , i.e., σ n ≥ λ min , ∀ n ∈ [1 , N ] , n / ∈ S c then the coarse optimization is optimal. Proof:
Define S ∗ to be the set of channels from optimalselection. S ∗ = { n ∈ [1 , N ] | I ∗ n = 1 } . From definition, max P n (cid:88) n ∈ S ∗ q n (cid:18) P n σ n (cid:19) ≥ max P n (cid:88) n ∈ S c q n (cid:18) P n σ n (cid:19) . (19)Let’s assume that there exist at least one legitimate channelwith noise variance higher than the lowest water level whichis included in the optimal channel selection. ∃ n ∈ S ∗ , σ n ≥ λ min . Define S (cid:48) = S ∗ ∪ S c , and allow the number of channel tosense to be M = | S ∗ ∪ S c | , which are strictly larger than L .Then, max P n (cid:88) n ∈ S (cid:48) q n (cid:18) P n σ n (cid:19) ≥ max P n (cid:88) n ∈ S ∗ q n (cid:18) P n σ n (cid:19) . (20)Note that S c ⊆ S (cid:48) , and σ n ≥ λ min for all n / ∈ S c . Modifiedwater-filling of M channels in S (cid:48) will lead to P n =0 for all n / ∈ S c . Thus, max P n (cid:88) n ∈ S (cid:48) q n (cid:18) P n σ n (cid:19) = max P n (cid:88) n ∈ S c q n (cid:18) P n σ n (cid:19) . Combine the above result with (20), we obtain max P n (cid:88) n ∈ S c q n (cid:18) P n σ n (cid:19) ≥ max P n (cid:88) n ∈ S ∗ q n (cid:18) P n σ n (cid:19) . (21)Above result contradict (19), unless S c is the optimal. Thisconcludes the proof.The other reason for performing the coarse optimizationis that it provides the essential information, λ m in , which isnecessary for further fine optimization. Upon the followingassumption, fine optimization is optimal. Conjecture 1:
If the noise variance σ n is greater than thewater level in the coarse optimization ( σ n > λ min ), then thechannel n is not likely to be sensed in the optimal strategy, oreven if it is included it will not increase the capacity much; Intuition:
The following gives the intuition for the aboveconjecture. Define S + to be the set of channels with noisevariance greater then or equal to the λ min and S − to be theset of channels with noise variance less then λ min but notincluded in S c ; S + = { n ∈ [1 , N ] | σ n > λ min } ,S − = { n ∈ [1 , N ] | σ n ≤ λ min , n / ∈ S c } . We have the set of channel S c which incur the lowestwaterlevel. Lemma(1) shows that average capacity cannotincrease by exchanging elements in S + with elements in c . Thus, for elements in S + to be included in S ∗ , optimalchannel selection, elements in S − should be included also.By exchanging elements in S − with elements in S c thewaterlevel rises up. Elements in S + can only be in optimalselection if exchanging S + with elements in S c lower thewaterlevel which increased due to the inclusion of channels in S − effectively. Intuition is that channels in S c are the channelswhich can lower the waterlevel effectively already. Thus, effectof lowering the waterlevel with channels in S + will not affectmuch in increasing the average capacity. The validity of thisconjecture is shown from the numerical analysis. Fine Optimization : From lemma(1), if the number of chan-nels that is selected to sense from the coarse optimizationis less than L , it is optimal, and no further optimization isneeded. Otherwise, further optimization will be required. FromConjecture(1), we reconstruct the problem, so that we canoptimize the selection of the channel over the channels withnoise variance smaller than or equal to lambda m in only. werearrange the useful channels by indexing from 1 to M , where M is the number of channels that has noise variance smallerthan λ min ; M = | S c ∪ S − | (22) σ n − λ min /le ∀ n ∈ [1 , M ] . (23)Then, the optimization problem can be rewritten as follows; max λ,I n C ( λ, I n ) = max λ,I n M (cid:88) n =1 q n (cid:32) (cid:6) λ − σ n (cid:7) + I n σ n (cid:33) (24) = max λ,I n M (cid:88) n =1 q n I n (cid:32) (cid:6) λ − σ n (cid:7) + σ n (cid:33) (25) = max λ,I n M (cid:88) n =1 q n I n (cid:24) log λσ n (cid:25) + , (26) ( a ) = max λ,I n M (cid:88) n =1 q n I n λσ n , (27)such that M (cid:88) n =1 (cid:6) λ − σ n (cid:7) + I n q n ( b ) = (cid:0) λ − σ n (cid:1) I n q n = P, (28) M (cid:88) n =1 I n ≤ L, (29) λ ≥ λ min (30) I n ∈ { , } , (31)where ( a ) and ( b ) result from constraining λ ≥ λ min . Thenthe optimal channel selection and power allocation can bedetermined by using the following theorem. Theorem 2: λ = (cid:80) Mn =1 q n I n σ n + P (cid:80) Mn =1 q n I n I n = 0 if λ > σ n e − σ nλ Proof:
Relax the constraint on I n , such that the I n can takethe value in the region [0 , . Construct the objective function C ( λ, f ( I n )) such that it is concave over the region of I n and λ , and f (0) = 0 and f (1) = 1 . Consider the function f ( I n ) = I kn , then f (0) = 0 , f (1) = 1 , and C ( λ, f ( I n )) = M (cid:88) n =1 q n I kn λσ n . (32)Concavity of C ( λ, f ( I n )) can be found as follows; (cid:34) ∂ C ( λ,f ( I n )) ∂I n ∂ C ( λ,f ( I n )) ∂I n ∂λ∂ C ( λ,f ( I n )) ∂λ∂I n ∂ C ( λ,f ( I n )) ∂ λ (cid:35) (33) = (cid:34) q n k ( k − I k − n log λσ n q n kI k − n λ log eq n kI k − n λ log e − (cid:80) Mj =1 q j I kj λ log e (cid:35) . (34)Since the matrix is symmetric, if the determinant, (1 , and (2 , components of the matrix take the negative values, thematrix is negative semi-definite. Thus, k ( k − ≤ (35) k ( k −
1) log λσ n + k log e ≤ (36)are the condition for C ( λ, f ( I n )) to be a concave function.We can find k such that the condition can be satisfied; k = min σ n (cid:32) log λ min σ n log λ min σ n + log e (cid:33) . (37)Now that we verified the concavity of the objective function,we can construct the according Lagrangian multiplier: L = (cid:80) Mn =1 q n I kn log (cid:16) λσ n (cid:17) − µ (cid:16)(cid:80) Mn =1 q n I ki ( λ − σ n ) − P (cid:17) − µ (cid:16)(cid:80) Mn =1 I ki − L (cid:17) + (cid:80) Nn =1 µ ,i I i − (cid:80) Nn =1 µ ,i ( I i −
1) + µ ( λ − λ min ) . (38)Solving the optimization, ∂ L ∂I n = kq n I k − n log λσ n − µ q n kI k − n ( λ − σ n ) − µ kI k − n + µ ,i − µ ,i = 0 (39) ∂ L ∂λ = M (cid:88) n =1 q n I kn log eλ − µ M (cid:88) n =1 q n I kn + µ = 0 (40) µ (cid:32) M (cid:88) n =1 q n I kn ( λ − σ n ) − P (cid:33) = 0 , (41) µ (cid:32) M (cid:88) n =1 I kn − L (cid:33) = 0 , (42) µ ,i I n = 0 , (43) ,i ( I n −
1) = 0 , (44) µ ( λ − λ min ) = 0 , (45)where µ , µ , µ ,i , µ ,i , and µ are non-negative values. Fromthe condition (40), λ = (cid:80) Mn =1 q n I kn log eµ (cid:80) Mn =1 q n I kn − µ . (46)Note that (cid:80) Mn =1 q n I kn log e , (cid:80) Mn =1 q n I kn , and µ are the non-negative value. Thus, µ should be positive number in orderfor λ to be positive. Then, from the condition (41), M (cid:88) n =1 q n I kn ( λ − σ n ) = P (47)Thus, λ = (cid:80) Mn =1 q n I kn σ n + P (cid:80) Mn =1 q n I kn , (48)and from rearranging the equation (46), µ = log eλ + µ (cid:80) Mn =1 q n I kn . (49)From the condition (39), we find that I n = (cid:16) q n log λσ n − µ q n ( λ − σ n ) − µ (cid:17) kµ ,i − µ ,i − k (50)If I n is not either 0 or 1, from conditions (43) and (44), µ ,i and µ ,i becomes 0, which will result in making I n infinite number. Thus, I n takes either 0,1 value, which givesthe desirable solution, such that the optimization of the re-laxed condition coincides with the condition for the originalproblem, and λ = (cid:80) Mn =1 q n I n σ n + P (cid:80) Mn =1 q n I n , (51)We set µ to be zero, then from (49) and (50), we obtain I n = (cid:32) q n log λσ n e − σ nλ − µ (cid:33) kµ ,i − µ ,i − k . (52)As a result, I n can be 1 only if λ > σ n e − σ nλ . (53)This conclude the proof. With the Theorem(2), we can designiterative algorithm to find the optimal selection of channels tosensed iteratively.Step I: Set the channels from coarse optimization to be theinitial channels. I n, = (cid:26) if n ∈ S c otherwise (54) S = { n ∈ [1 , N ] | I n, = 1 } (55) j = 1 (56)Step II: Calculate the waterlevel λ j from (51) λ j = (cid:80) Mn =1 q n I n,j − σ n + P (cid:80) Mn =1 q n I n,j − . (57)Step III: Calculate λ j − σ n e − σ nλj , and select the largest L channels. I n,j = (cid:40) if λ j − σ n e − σ nλj is among L largests otherwise (58) S j = { n ∈ [1 , N ] | I n,j = 1 } (59)Step IV: If S j = S j − , terminate the iteration, and set thepower allocation and channel selection values. λ f , I fn , and P fn are the waterlevel, channel selection, and power allocationfrom fine optimization, then λ f = λ n (60) I fn = I n,j (61) P fn = (cid:0) λ j − σ n (cid:1) I n,j . (62)Otherwise, j = j + 1 and repeat from step II. Followingfrom theorem( ?? ) this algorithm gives the optimal selectionof the channels to be sensed and power allocation, with theassumption that the channels with noise variance greater than λ min do not affect the optimization.V. N UMERICAL A NALYSIS
In this section, we present numerical example of capacitiesfor coarse and fine optimization along with optimal solution. Inthis example, dissimilarity between channels is implementedby adapting the multi-path fading, which will incur frequencyselective channel. Also, occupation of the legitimate channelis modeled by having q n to be uniform in [0 , and identi-cally distributed. Sixteen legitimate channels are considered, N = 16 , and cognitive radio is allowed to select and senseeight channels from all the legitimate channels, L = 8 . Fig.3. compares the capacities of sub-optimal algorithms withthe optimal one, where the performance of optimal channelselection is obtained from the exhaustive search. The graphshows that performance of the fine optimization meet with thatof optimal one. Thus, it can be stated that the Conjecture (1) infine optimization is valid. Coarse optimization also performsoptimally in the low SNR region. In the low SNR region, itis likely that σ n ≥ λ min , ∀ n ∈ [1 , N ] , n / ∈ S c , because thereare not much power to waterfill. From the Lemma (1), coarseoptimization is optimal in such case. It is also worthwhile tonote that coarse optimization perform as well as the optimalone. SNR(dB) C apa c i t y Exhaustive SerachCourse OptimizationFine Optimization
Fig. 3. Perforamance Analysis
VI. C
ONCLUSION
In this paper, fundamental limits of interweaved cognitiveradio has been verified. In the case that there are large numberof legitimate channels and only limited number of them can besensed, the capacity has been analyzed. However, it requiresexhaustive search over the combination of all the legitimatechannels, which is not practical in terms of complexity. Thus,two steps of sub-optimal solutions have been developed.Coarse optimization is developed, and verified to be optimalin the low SNR cases, and further optimization is performedto ensure the performance in the high SNR region also.A
CKNOWLEDGMENT
The authors would like to thank Illsoo Sohn for usefuldiscussions and comments.R
Asilomar Conference on Signals, Systems, andComputers , Asilomar, CA, Oct. 2006.[5] N. Devroye, P. Mitran, and V. Tarokh, “Achievable Rates in CognitiveRado Channels,”
IEEE Trans. Inform. Theory , vol. 52, pp. 1813-1827,May 2006.[6] W. Wu, S. Vishwanath, and A. Arapostathis, “Capacity of a Classof Cognitive Radio Channels: Interference Channels With DegradedMessage Sets,”
IEEE Trans. Inform. Theory , vol. 53, pp. 4391-4399, Nov.2007.[7] W. Wang, T. Peng and W. Wang, “Optimal Power Control under In-terference Temperature Constraints in Cognitive Radio Network,”
IEEEWireless Communications & Networking Conference , Hong Kong, Mar.2007.[8] H. Urkowitz, “Energy detection of unknown deterministic signals,”
Proc.IEEE, vol. 55, pp. 523.531, Apr. 1967 [9] F. Digham, M. Alouini, and M. Simon, “On the energy detection ofunknown signals over fading channels,”
IEEE Trans. Communications, vol. 55, pp. 21.24, Jan. 2007.[10] A. Goldsmith and P. Varaiya, “Capacity of fading channels with channelside information,”
IEEE Trans. Inform. Theory, vol. 43, pp. 1986-1992,Nov. 1997.[11] S. Hong, M. Vu, and V. Tarokh, “Cognitive Sensing Based on SideInformation,”
IEEE Sarnoff Symposium , Princeton, NJ, Apr. 2008.[12] Y. Song, Y. Fang, and Y. Zhang, “Stochastic Channel Selection inCognitive Radio Networks,”
IEEE Global Communications Conference ,Washington, DC, Nov. 2007.[13] X. Yang, Z. Yang, and D. Liao, “Adaptive Spectrum Selection for Cog-nitive Radio Networks,”
International Conference on Computer Scienceand Software Engineering , Wuhan, China, Dec. 2008.[14] D. Huang, C. Miao, C. Leung and Z. Shen, “Resource Allocation ofMU-OFDM Based Cognitive Radio Systems Under Partial Channel StateInformation,” http://arxiv.org/abs/0808.0549[15] P. Somol, P. Pudil, and J. Kittler, “Fast branch & bound algorithms foroptimal feature selection,”