Adaptive Sum Power Iterative Waterfilling for MIMO Cognitive Radio Channels

Adapti v e Sum Po wer Iterati v e W aterfill ing for MIMO Cogniti v e Radio Channels Raji v Soundarara jan and Sriram V ishwanath Departmen t of Electrical and Co mputer E ngineer ing, Uni versity of T exas at Austin 1 Un i versity Station C0803 , Austin, TX 78712 , USA Email: sou ndara,sr iram@ece.utexas.edu Abstract —In th is paper , the sum capacity of th e Gaussian M ul- tiple Input Mul tiple Outp ut (M IMO) Cognitive Radio Channel (MCC) is expressed as a conv ex problem wi th finite numb er of linear constraints, allowing for polynomial time interior point techniques to find the solution. In addition, a sp ecialized class of sum power iterative waterfilling algorithms is determined that ex ploits the inherent structu re of the sum capacity prob- lem. These algor ithms not only determine the maximizing sum capacity value, but also the transmit poli cies that achiev e this optimum. The paper conclu des by p ro viding nu merical results which demonstrate that the algorithm takes very few iterations to con ver ge to the optimum. I . I N T RO D U C T I O N In recent years, the study of co gnitive ra dios from an informa tion theoretic perspective has ga ined pro minence [1]. As the Federal Communica tions C ommission (FCC) determines the ways and band s in which cog nitive radios can be used, it is imperative that we u nderstan d the fundamental limits of the se radios to ben chmark gains from their desig n and deployment. Cognitive ra dio c hannels ref er to those media o f commu nication in which c ognitive rad ios op erate, thereby efficiently utilizing av ailable resources. Moreover , since mo st wireless systems th ese day s use mu ltiple antenn as at the transmitter and receiver , it is imp ortant that we stud y the limits in a Multip le Inp ut Multiple Output ( MIMO) setting. T he situation considered in this pa per is different from the tra ditional class of MIMO pr oblems on accou nt of the in telligent and ad aptive capabilities of the cog nitive radios. Based on the model proposed in [2], the co gnitive radio channel is an in terferen ce chann el [3][4][ 5][6] with degraded message sets in which the transmitter with a single m essage is called the “primary ” or “licensed” user while the transmitter with both message sets is called the “secondary ” or “cog nitiv e” user . In this pap er, we study the sum capacity of cog nitive radios in a MIM O settin g where both the pr imary and seco ndary transmitter and r eceivers have multiple an tennas and the no ise is Gau ssian. The sum capacity en ables the design o f a MIMO cognitiv e radio system by specify ing the sum r ate req uired fo r the primary and the secondar y users. Recen tly , an achiev able region was found an d shown to be op timal for the sum rate o f the primary and secondar y users [7] under certain cond itions on channel parameters. T houg h an a chiev able codin g strategy ba sed on Costa’ s dirty p aper co ding [8] was sh own to b e op timal, an optimization over transmit covariances is required . The optimization to dete rmine the sum capacity is in general a nonco n vex pro blem an d is hence computatio nally difficult [9]. In this paper , we find th at the no nconve x problem formu lation is similar to a MIMO Bro adcast Chan nel (BC) sum capacity problem form ulation [10]. W e therefore transform the nonco n vex problem into a co n vex pro blem by using “d uality”’ techniques as detailed in [10]. As a result we obtain a co n vex-concave g ame wh ich can be solved in polyno mial time. W e prop ose efficient algorith ms to find th e saddle poin t of the p roblem and hence compu te the sum capacity and op timal tran smit po licies. The conv ex-concave gam e for mulation of the sum capa city of the MIMO Cognitive Channel (MCC) is a minimax problem in which the inner maximization corr esponds to computin g th e sum capa city of a MIMO Multiple Access Channel (MAC ) subject to a sum power con straint. There are many efficient algorithm s in literature that solve saddle point problem s and they are analogo us to c onv ex optimization technique s like interior poin t and bundle metho ds [11]. Howe ver th ese algorithms are both much more inv o lved than our algorithm and offer limited intuition about the structure of the optimal value. Our algorithm is based o n the sum power iterativ e waterfilling algo rithm fo r BC chann els [ 12], but is significantly different as the waterle vel is no longer given, but h as to be discovered th rough adap tation . W e thus call this strategy , the ad aptive sum power waterfilling algo rithm, to ach iev e ou r objective of solving the min imax pr oblem. The rest o f the p aper is o rganized as follows. In Section II, the system m odel is descr ibed and in Section III, we state the sum capacity prob lem for MCC. In Section IV, the convex problem form ulation is described . In Section V, we p ropo se the algorithm and numerical results are given in Section VI. W e co nclude th e paper in Section VII. I I . S Y S T E M M O D E L W e use boldface letters to denote vectors and matr ices. | H | denotes th e determ inant of the matrix H and T r( H ) denotes the trace. For any general matrix S , S † denotes the c onjugate transpose. I n is the n × n identity matrix and H  0 denotes that the squ are matrix H is positive sem idefinite. If E is a set, then Cl( E ) an d Co( E ) refer to the closure and conv ex hull o f E resp ectiv ely . W e conside r the MCC illustrated in Fig. 1. The primar y transmitter and r eceiv er hav e n p,t and n p,r antennas while the cognitive tran smitter and receiver h av e n c,t and n c,r antennas respectively . Cognitive Receiver Licensed Source Power Constraint Cognitive Source Power Constraint Licensed Receiver P p X n p ( m p ) X n c ( m p , m c ) Y n p Y n c Z n p Z n c H c,c H c,p H p,c H p,p P c Fig. 1. MIMO Cogniti ve Radio System Model Let X p ( i ) ∈ C n p,t × 1 be the vector sign al tran smitted b y the pr imary user and X c ( i ) ∈ C n c,t × 1 be the vector signal transmitted by the cognitive user in time slot i . Let H p , p , H p , c , H c , p and H c , c be constant ch annel gain matrice s as shown in Fig. 1. It is assumed th at th e licensed receiver kno ws H p , p and H c , p , the licensed transmitter kn ows H p , p , the cognitive tra nsmitter knows H p , c , H c , p and H c , c and the cognitive receiver k nows H p , c and H c , c . T hese a ssumptions are made so as to make th e pro blem tractab le and thereby provide a bench mark on per forman ce of the system. Let Y p ( i ) ∈ C n p,r × 1 and Y c ( i ) ∈ C n c,r × 1 be the signal receiv ed by the pr imary receiver and cognitive receiver respectively . The additive noise at the primar y and seco ndary receivers are Gaussian, indep endent acr oss time symbols and r epresented by Z p ( i ) ∈ C n p,r × 1 and Z c ∈ C n c,r × 1 respectively , where Z p ( i ) ∼ N (0 , I n p,r ) and Z c ( i ) ∼ N (0 , I n c,r ) . Z p ( i ) and Z c ( i ) can b e arbitrarily cor related between them selves. The received sign al is math ematically rep resented as Y p ( i ) = H p , p X p ( i ) + H c , p X c ( i ) + Z p ( i ) Y c ( i ) = H p , c X p ( i ) + H c , c X c ( i ) + Z c ( i ) . The covariance m atrices of the primar y and cogn iti ve input signals ar e Σ p ( i ) and Σ c ( i ) . The pr imary and second ary transmitters are subject to average power constraints P p and P c respectively . Thu s, n X i =1 T r( Σ p ( i )) ≤ nP p n X i =1 T r( Σ c ( i )) ≤ nP c . I I I . P RO B L E M S TA T E M E N T In this sectio n, we restate the sum c apacity of the MCC as in [7]. Before doin g so, we de velop the requ ired no tation for the same. Let G = [ H p , p H c , p ] . The set R ach is defin ed as R ach =                         ( R p , R c ) , Σ p , Σ c , p , Σ c , c , Q  : R p , R c ≥ 0 , Σ p , Σ c , p , Σ c , c  0 R p ≤ lo g( | I + GΣ p , net G † + H c , p Σ c , c H † c , p | | I + H c , p Σ c , c H † c , p | ) R c ≤ log ( | I + H c , c Σ c , c H † c , c | ) Σ p , net =  Σ p Q Q † Σ c , p  , T r( Σ p ) ≤ P p , T r( Σ c , p + Σ c , c ) ≤ P c                        where Σ p , net is a ( n p,t + n c,t ) × ( n p,t + n c,t ) covariance matrix while Σ c , c is a n c,t × n c,t covariance matrix. Σ p and Σ c , p are principal subm atrices of Σ p , net of dimensions n p,t × n p,t and n c,t × n c,t respectively . The set o f all rate pairs R in is giv en by R in = Cl Co ( ( R p , R c ) : ∃ Σ p , Σ c , p , Σ c , c  0 and Q :  ( R p , R c ) , Σ p , Σ c , p , Σ c , c , Q  ∈ R ach )! . It is shown in [7] that R in is an inner bound on the capacity region of the M CC. Let G α = [ H p , p H c , p √ α ] and K α = [ 0 H c , c √ α ] . The set R α part is defined as R α part =                 ( R p , R c ) , Q p , Σ c , c , Q  : R p ≥ 0 , R c ≥ 0 , Q p , Σ c , c  0 R p ≤ log ( | I + G α Q p G † α + 1 α H c , p Σ c , c H † c , p | | I + 1 α H c , p Σ c , c H † c , p | ) R c ≤ lo g( | I + 1 α H c , c Σ c , c H † c , c | ) T r( Q p ) + T r ( Σ c , c ) ≤ P p + αP c                and the set R α part, out is d efined as R α part, out = Cl Co ( ( R p , R c ) : ∃ Q p , Σ c , c  0 such that (( R p , R c ) , Q p , Σ c , c ) ∈ R α part )! . It is also shown in [7] that R α part, out is an outer bou nd on the cap acity region that includ es the sum rate wh en certain co nditions that depend on the channel pa rameters ar e satisfied. W e comp ute th e sum cap acity of the MCC u nder those co nditions. W e now restate Theo rem 3.3 from [ 7] which paves the way for th e MCC sum ca pacity p roblem f ormulatio n. For any µ ≥ 1 , max ( R p ,R c ) ∈R in µR p + R c = inf α> 0 max ( R p ,R c ) ∈R α part,out µR p + R c . Therefo re, the sum capacity of th e MCC (denoted by C M C C ( G α , H c , c ) ) is expressed as C M C C ( G α , H c , c ) = inf α> 0 max ( R p ,R c ) ∈R α part,out R p + R c . (1) I V . F O R M U L AT I O N A S A C O N V E X P RO B L E M The inner maximization in the sum capacity of the MCC, stated above, co rrespon ds to comp uting th e sum capac ity of a degrad ed broa dcast chann el in which th e transmitters cooper ate with a sum power co nstraint. This is illustrated in Fig. 2. W e ob serve fro m the expression for the sum capacity in (1) that th e inner max imization problem is not a co ncave function of the cov ariance matrices Q p and Σ c , c . Th us it is d ifficult to solve the entire prob lem using numerica l technique s. Howe ver , as in [12], we can use “dua lity” to transform the inner max imization p roblem into a sum capacity problem for the M A C with th e same sum power constraint. This can be done because it is sho wn in [10] that sum capacity of th e BC is exactly eq ual to the sum capacity of the dual MA C. These results enable us to convert the orig inal problem to a conve x-con cave gam e. Cognitiv e Rec eiv er Y n p Z n p K α Joint Source P p + αP c Power Constraint X n ( m p , m c ) Z n c Y n c G α Primary Receiver Fig. 2. MIMO Broadcast Channel Let Q c =  0 0 0 Σ c , c  , where the zero matrices have ap- propr iate dim ensions such that Σ c , c has dimen sion n c,t × n c,t and Q c has dim ension ( n p,t + n c,t ) × ( n p,t + n c,t ) . The sum capacity of th e MCC can theref ore be expressed a s C M C C = inf α> 0 max S 1 , S 2 log | I + G † α S 1 G α + K † α S 2 K α | subject to S 1 , S 2  0 , T r( S 1 ) + T r( S 2 ) ≤ P p + αP c (2) where the maximizatio n is perform ed over co variance matrices S 1 and S 2 , which are ob tained usin g BC-to-MA C transforma - tions of Q p and Q c such th at the sum o f the traces of S 1 and S 2 satisfies the sum power constraint P p + αP c . This new prob lem is conca ve in the cov ariance matrices S 1 and S 2 and conve x in the scalar α with linea r power co nstraints and is thus a con vex-concave game. The proof of the fact that the pro blem is con vex in α is given in th e append ix. This m in-max p roblem can be solved by using interior point methods o r other eq uiv alent con vex optimization meth ods for saddle point problems which have polynomial time complexity [11]. Once the optimal S 1 and S 2 are obtain ed, we can ap ply the MAC-to-BC transform ation [10] to o btain the op timal transmit policies of ou r original pro blem. T he MA C-to- BC transform ation, takes a set of MAC cov ariance matrices and outputs a set of BC cov ariance m atrices which achie ve the same sum ra te as the MAC cov ariance matrices. V . A DA P T I V E S U M P O W E R I T E R A T I V E W AT E R - FI L L I N G W e pro pose a class of algo rithms to co mpute the sum capacity of the MCC. T hese algo rithms a re motivated by the sum power water-filling algo rithms de veloped for co mputing the sum c apacity an d obtaining the o ptimal tra nsmit policies for the MIMO BC. As in [12], we o btain a d ual conve x problem correspo nding to a MA C. This dual MAC pr oblem is a conv ex pro blem and thus can b e solved using the multitude of conve x solvers in polyno mial time. Our intentio n, as that in [9][1 3] is to exploit the p roblem’ s stru cture to yield a m ore intuitive and easy-to-im plement algorithm for this prob lem. Specifically , it is to derive an algorith m th at reflects and conv erges to the KKT cond itions c orrespo nding to cognitive radio sum cap acity . Individual power iterative waterfilling w as fo und to achieve the cap acity of a MAC ch annel with separ ate power constraints per user in [9]. This was then extended to BCs by the sum p ower iterative waterfilling alg orithm in [13], which is based o n the d ual MAC ’ s KKT conditio ns. While n either of these two alg orithms works d irectly for the MIMO cog nitiv e radio chann el, we use an alogou s principles to derive an ad aptive sum power iterative algorithm. Before pro ceeding further, we revie w the waterfilling alg o- rithm for the single user po int to point MIMO case. Consider the p roblem, max T r( S ) ≤ P , S  0 1 2 log | I + H † SH | . (3) Let the eigen v alues of the cov ariance matrix S of size n × n be λ 1 , λ 2 , . . . , λ n and the singular values of the chann el ma trix H be σ 1 , σ 2 , . . . , σ n . Then th e waterfilling solution is g iv en by λ i + 1 /σ 2 i = K , if 1 /σ 2 i < K (4) λ i = 0 , if 1 /σ 2 i ≥ K , (5) where K is a con stant such th at P i λ i = P . KKT co nditions, along with comp lementary slackness yield c ondition s (4 ) an d (5) [14][15]. W e now reca ll th e sum power iter ativ e water- filling solu tion fo r the MIMO MA C as studied in [1 3]. Th e problem is stated be low: max T r( P S i ) ≤ P, S i  0 1 2 log | I + X i H † i S i H i | (6) The KKT cond itions for this problem are similar to the point to po int case except that in the MAC case, th e channel H in (3) for User i is replaced by the e ffecti ve channel matrix H i,ef f = H i ( I + P j 6 = i H † j S j H j ) . In the sum p ower waterfilling alg orithm, we waterfill for all the users simu ltaneously , while in ind ividual power iterati ve waterfilling, user waterfilling is seque nced [1 2]. The pro blem consider ed in this pap er has a min-m ax for- mulation as gi ven in (2) unlike the pure max formulation s for the MA C and BC capac ity prob lems [12][13]. The idea behind our algorithm is to start with a feasible choice for α . For a given α , the joint water-filling on th e two users is the optimal strategy for the inner maxim ization pr oblem. In (2), we perfo rm a joint water-fill exactly once for the in ner maximization , use this solution to solve the outer minimization with respect to α and iterate betwee n the tw o. T hus, we end up with one max imization problem (for a choice o f α ) given by S 1 , S 2 = arg max T 1 , T 2 log | I + G † α T 1 G α + K † α T 2 K α | subject to T 1 , T 2  0 , T r( T 1 ) + T r( T 1 ) ≤ P p + αP c and a minim ization pro blem (for a gi ven ch oice of covari- ances) a s: α = argmin β > 0 log | I + G † β γ S 1 G β + K † β γ S 2 K β | where γ = P p + β P c P p + α ( n − 1) P c is a scalar and α ( n − 1) is the p revious iterate of α. The solution of o ne is fed to the other, and th e p rocess is r epeated u ntil co n vergence. T his fo rms th e co re o f th e adaptive sum power iterative waterfilling algo rithm. W e refer to the p rocedu re detailed above as Algorithm 1. As the m inimization with respect to α can be so mewhat in volved (even thoug h α is a scalar ), we constru ct ano ther algorithm we call Alg orithm 2. In Algor ithm 2, we only obtain a d escent at each iteration by a simp le line search like N ewton search. W e do not solve the outer minimization problem completely at every iteration as in Algorithm 1 , which furth er simp lifies the overall alg orithm. In Section VI, we illustrate thro ugh examples that th is highly simplified algorithm , with one waterfill and one descent in each iter ation has nearly as good a co n vergence rate as exact solutions at each step . I n th e f ollowing, n refers to the iter ation n umber . Main Alg orithm ( Algorithm 1): 1) Initialize α (0) to any numbe r greater than 0 . 2) Gener ate the effecti ve channels as G ( n ) α,ef f = G α ( n − 1)  I + K † α ( n − 1) S ( n − 1) 2 K α ( n − 1)  − 1 / 2 K ( n ) α,ef f = K α ( n − 1)  I + G † α ( n − 1) S ( n − 1) 1 G α ( n − 1)  − 1 / 2 . 3) Obtain covariance matrices S ( n ) 1 and S ( n ) 2 by per formin g a joint waterfill with power P p + α ( n − 1) P c . { S ( n ) 1 , S ( n ) 2 } = argmax T 1 , T 2 log | I + G † α ( n − 1) T 1 G α ( n − 1) + K † α ( n − 1) T 2 K α ( n − 1) | subject to T 1 , T 2  0 and T r( T 1 ) + T r( T 1 ) ≤ P p + α ( n − 1) P c . 4) Use S ( n ) 1 and S ( n ) 2 from Step 3 to obtain α ( n ) by solving the fo llowing univ ariate optimizatio n prob lem. α ( n ) = arg min β > 0 log | I + G † β γ S ( n ) 1 G β + K † β γ S ( n ) 2 K β | where γ = P p + β P c P p + α ( n − 1) P c is a scalar . 5) Return to Step 2 until param eters conver ge. As mentio ned befo re, the ab ove algorithm is a very intuitiv e extension to the sum power waterfilling algorithm fo r the MIMO BC chann el. Th e intuitio n arises from the fact that at the saddle po int, the KKT con ditions must be satisfied f or both the max an d the min prob lems. Although we may no t be able to always guar antee convergence of the algorithm to th e optimal solution, when it do es con verge, the algor ithm takes very few iteration s to do so. V I . N U M E R I C A L R E S U LT S In this sectio n we present numerical r esults to compare the be havior of Algor ithms 1 an d 2 . In Fig. 3, we plot the sum rate versu s the num ber o f iterations for a M CC with two antennas at bo th the primar y and cognitive receiver and on e antenna each at the p rimary and cognitiv e tran smitter with power constraints P p = P c = 5 . The chann el matrices are G α =  − 0 . 432 6 − 1 . 665 6 0 . 1253 0 . 2877  and K α =  0 − 1 . 1465 0 1 . 190 9  when α = 1 . W e find that both A lgorithm 1 an d Algorith m 2 conv erge to the same sum rate. Howe ver th is may not always happen depen ding on the initial cond itions ch osen f or the Newton search. W e also o bserve that in some cases Alg orithm 1 converges in fewer iterations when compared to Algorithm 2. V I I . C O N C L U S I O N In this paper, we prop osed a class of algor ithms to compute the sum capacity and the optimal transmit policies of the MCC. This was made p ossible by tran sformin g the MCC sum capac- ity problem as a conv ex p roblem using M A C-BC (o therwise called as u plink-d ownlink) du ality . T he algorith m pe rforms a sum po wer waterfill at each iteration, while simultaneously adapting the waterlevel at each iteration . 0 5 10 15 20 25 30 35 40 45 50 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Number of iterations Sum Rate (in bps/Hz) Algorithm1 Algorithm2 Fig. 3. Conv erg ence of Algorit hm 1 and Algorit hm 2 to the sum rate A P P E N D I X Pr oo f of conve xity of th e objective in (2) in α : Let F ( α ) = log | I + G † α S 1 G α + K † α S 2 K α | . Hence, F = lo g      I + H † p , p S 1 H p , p H † p , p S 1 H c , p √ α H † c , p S 1 H p , p √ α I + H † c , p S 1 H c , p + H † c , c S 2 H c , c α      = log | I + H † p , p S 1 H p , p | + log      I + H † c , p S 1 H c , p + H † c , c S 2 H c , c α − H † c , p S 1 H p , p ( I + H † p , p S 1 H p , p ) − 1 H † p , p S 1 H c , p α      . Thus F is of the for m F = c + log    I + A α    where c = log | I + H † p , p S 1 H p , p | and A = H † c , p S 1 H c , p + H † c , c S 2 H c , c − H † c , p S 1 H p , p ( I + H † p , p S 1 H p , p ) − 1 H † p , p S 1 H c , p . From m atrix theory [16, Cha p. 7] , we know that I + A α is po siti ve semidef- inite for all α > 0 sin ce I + G † α S 1 G α + K † α S 2 K α and I + H † p , p S 1 H p , p are positive semidefinite for all α > 0 . Let λ i , i = 1 , 2 , 3 , . . . , n c,t be the e igenv alu es of A . T herefor e, for every i , 1 + λ i α ≥ 0 for all α > 0 which im plies λ i ≥ − α for all α > 0 . Hence λ i ≥ 0 for i = 1 , 2 , 3 , . . . , n c,t and A is positive semid efinite. ∂ 2 F ∂ α 2 is g iv en by ∂ 2 F ∂ α 2 = T r " 2 A α 3  I + A α  − 1 # − T r " A α 2  I + A α  − 1 A α 2  I + A α  − 1 # = T r " A α 3  I + A α  − 1  2 I + A α  I + A α  − 1 # . Using m atrix theory results [16, Chap. 7] we can fur ther show that ∂ 2 F ∂ α 2 ≥ 0 for all α > 0 . Thu s F ( α ) is conve x in α . A C K N O W L E D G M E N T This work was sup ported in part by grants from THECB- ARP an d ARO YIP . R E F E R E N C E S [1] A. Jovi cic and P . 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