Interference Reduction by Beamforming in Cognitive Networks

Transcription

1 Interferene Reduton by Beamformng n Cogntve Networks Smon Yu, Ma Vu, and Vahd Tarokh Shool of Engneerng and Appled Senes, Harvard Unversty, Cambrdge, MA, USA Emal: {smony, mavu, vahd}@seas.harvard.edu Abstrat We onsder beamformng n a ogntve network wth multple prmary users and seondary users sharng the same spetrum. In partular, we assume that eah seondary transmtter has antennas and transmts data to ts sngleantenna reever usng beamformng. The beamformer s desgned to maxmze the ogntve user s sgnal-to-nterferene rato (SIR), defned as the rato of the reeved sgnal power at the desred ogntve reever to the total nterferene reated at all the prmary reevers. Usng mathematal tools from random matrx theory, we derve both lower and upper bounds on the average nterferene at the prmary reevers and the average SIR of the ogntve user. We further analyze and prove the onvergene of these two performane measures asymptotally as the number of antennas or prmary users N p nreases. Spefally, the average nterferene per prmary reever onverges to the expeted value of the path loss n the network whereas the average SIR of the seondary user deays as 1/ when = N p/. In the speal ase of = N p,the average total nterferene approahes 0 and the average SIR approahes. I. INTRODUCTION The Federal Communaton Commssons (FCC) frequeny alloaton hart [1] ndates multple alloatons over all the frequeny bands under 3 GHz. The ntense ompetton for the use of spetrum at frequenes below 3 GHz reates the onepton of spetrum shortage. However, studes by FCC show that the usage of the lensed spetrum s vastly underutlzed [2]. Ths motvates researh n ogntve networks for the opportunst use of the spetrum. A ogntve network usually onssts of the prmary users who have the legay prorty aess to the spetrum and the seondary users who use the spetrum only f ommunaton does not reate sgnfant nterferene to the lensed prmary users. Therefore, the unlensed seondary users often employ ogntve rados for transmsson to ensure non-nterferng oexstene wth the prmary users [3]. Ths an be aheved n several ways as dsussed n [4] and referenes theren. For example, the ogntve user an transmt onurrently wth the prmary users under an enfored spetral mask. Another strategy s to have the ogntve users montor the spetrum and aess t when an unused slot s deteted. Beamformng s a well-known spatal flterng tehnque whh an be used for ether dreted transmsson or reepton of energy n the presene of nose and nterferene [5]. In multple-antenna systems, beamformng explots hannel knowledge at the transmtter to maxmze the sgnal-to-nose rato (SNR) at the reever by transmttng n the dreton of the egenvetor orrespondng to the largest egenvalue of the hannel [6]. Beamformng an also be used n the uplnk or downlnk of multuser systems to maxmze the sgnal-tonterferene-plus-nose rato (SINR) of a partular user [7]. In ths paper, we study the effet of beamformng n ogntve networks, n whh the prmary and seondary users are unformly dstrbuted n a rular ds. The seondary transmtters are allowed to transmt onurrently wth the prmary transmtters. To mnmze the nterferene aused to the prmary reevers, the seondary transmtters are equpped wth multple antennas and employ beamformng for transmsson. The beamformng vetor of eah ogntve transmtter s desgned suh that t maxmzes the desred sgnal power at ts orrespondng reever whle mnmzng the total nterferene aused to all prmary reevers. The rato of the reeved sgnal power to the nterferene s referred to as the sgnal-tonterferene rato (SIR) of the ogntve user. Sne nreasng the number of antennas mproves the spatal dretvty of sgnal energy, one would then expet a hgher average SIR and a lower average nterferene. On the other hand, wth onstant number of antennas, nreasng the number of prmary users n the network has the opposte effet. Therefore, there s an nterestng trade-off between these parameters. We nvestgate ths trade-off by studyng the average SIR of the ogntve users and the average nterferene reated at all prmary reevers. In partular, by employng some known results n random matrx theory, we provde analytal bounds for these two performane measures. We prove that the average nterferene per prmary reever onverges to the average path loss of the network. The average SIR of eah ogntve user par, on the other hand, deays as 1/ as = N p /, where N p and denote the number of prmary reevers and the number of beamformng antennas at eah ogntve transmtter, respetvely. In the extreme when = N p,the lower bound of the average total nterferene approahes 0 and the upper bound of the average SIR approahes. Ths mples that we an potentally reate lttle to no nterferene to the prmary users by employng as many antennas n the ogntve transmtter as the number of prmary reevers. Organzaton: Ths paper s organzed as follows. In Seton II, we ntrodue the system model of the proposed transmsson sheme. We formulate the beamformng optmzaton problem n Seton III. In Seton IV, we study the average total nterferene and the average SIR and derve bounds for these two terms. We present smulaton and numeral results n Seton V, and draw some onlusons n Seton VI /08/$ IEEE. 1

2 A. Network Model II. NETWORK AND CHANNEL MODELS Consder a ogntve network n whh N p prmary users and N ogntve (seondary) users are unformly dstrbuted n a rular ds wth radus R. A prmary user has a transmtter P T and a reever P R, 1 N p. Smlarly, a ogntve user has a transmtter C k T and a reever Ck R, 1 k N. Furthermore, we assume that eah reever (ether prmary or seondary) has a proteted radus of ɛ>0 wthout any other nterferng transmtter nsde. Ths assumpton nhbts nfnte nterferene at any reever. Fg. 1 shows an example of a network n whh a C T s loated at the enter of the ds. R ɛ C T C R P T Fg. 1. Network model wth N =1and N p =20. C T s loated at the enter of the ds. In suh a mult-user network, there wll be nterferene at both the prmary and ogntve reevers. In ths paper, we are manly onerned wth the nterferene at the prmary reevers reated by the ogntve transmtters. Defne the average total nterferene reated by the k th ogntve transmtter C k T as1 N p E[I k ]=E Interferene at P R 2 C k T transmts, (1) =1 and the average SIR of the ogntve user par C k T -Ck R as2 [ ] Desred sgnal at CR k E[SIR k ]=E I k 2 P R C k Ttransmts, (2) where the expetaton s taken over the spatal dstrbuton of C k T, Ck R, and P R,. We observe that E[I k] and E[SIR k ] are ndependent of k beause C k T and Ck R are unformly dstrbuted n the ds and have the same statstal propertes. Therefore, we an drop the dependent of k n (1) and (2) and onsder only the smplfed model of N =1and an arbtrary N p.as a onsequene, nreasng the densty of the ogntve users would only nrease E lnearly by a fator of N. 1 In ths paper, bold upper ase and lower ase letters denote matres and vetors, respetvely. [ ] H, E[ ], δ( ), j 1,, ln( ), Im( ),, and dag{x} denote Hermtan transposton, statstal expetaton, the Dra delta funton, the magnary unt, the absolute value of a saler, the natural logarthm, the magnary part of a omplex number, mathematal equvalene, and a dagonal matrx wth the elements of x n ts man dagonal, respetvely. In addton, I N, [X],j, λ mn (X),andλ max(x) refer to the N N dentty matrx, the element n row and olumn j of matrx X, and the mnmum and maxmum egenvalue of matrx X, respetvely. 2 I k and SIR k wll be formally defned n (6) and (8). B. Channel and Sgnal Models We assume that the ogntve transmtter C T s equpped wth unorrelated antennas whereas the ogntve reever C R and the prmary reevers P R, 1 N p are equpped wth only a sngle antenna 3. Denote the 1 hannel vetor from C T to P R as h and from C T to C R as g. The elements of h and g are modeled as and h n = 1 d α/2 h n, 1 N p, 1 n, (3) g n = 1 d α/2 gn, 1 n, (4) respetvely, where α s the path loss exponent and h n and g n are ndependent and dentally dstrbuted (..d.) zeromean omplex Gaussan random varables wth unt varane (Raylegh fadng). We assume that α = 2 n ths paper. The dstanes d and d are..d. random varables whh represent the dstane from C T to P R and to C R, respetvely. By the reeverproteted radus assumpton, d, d >ɛ. Fnally, n order to provde theoretal bounds for the onsdered network, t s assumed that C T has global hannel state nformaton (CSI) of the network,.e., omplete knowledge of h and g. The ogntve transmtter C T employs a beamformng vetor w wth dmenson 1 for transmsson of ts data symbol x. The orrespondng reeved sgnal at C R and P R are gven by respetvely. r C = w H gx and r P = w H h x, (5) III. BEAMFORMING FORMULATION As prevously mentoned, n ths paper, we fous on the nterferene reated by the ogntve transmtter to the prmary reevers and nose s not onsdered. However, t s straghtforward to norporate nose nto our hannel model and study the SINR nstead. We assume that the data symbols x n (5) are..d. taken from an M-ary symbol alphabet wth unt energy. Therefore, the nstantaneous total nterferene s gven by N p rp =1 I = N p 2 = w H h h H w = w H HTH H w = w H Rw, =1 where [H] n, = h n, (6) T = dag{d 2 1,..., d 2 N p }, (7) and R HTH H. The nstantaneous SIR of C T -C R s gven by r C 2 SIR = Np 2 = wh Gw w H Rw, (8) =1 rp 3 Note that for the problem under onsderaton, the number of antennas at P T s not mportant, f. (1) and (2) /08/$ IEEE. 2

3 where the defnton G gg H s used. Therefore, the maxmum SIR beamformer an be obtaned formally from the followng optmzaton problem w opt = argmax w H w=1 {SIR}. (9) The optmal soluton to the above optmzaton problem s the egenvetor orrespondng to the maxmum egenvalue of the followng generalzed egenvalue problem [7] Gw = λrw R 1 Gw = λw. (10) It s assumed that N p and therefore, R s nvertble. If ths ondton s not mposed, there wll be N p zero egenvalues and t s theoretally possble for C T to form a null n all the dreton where P R are loated resultng n I =0. We note that the above optmzaton problem (9) s losely related to the uplnk and downlnk beamformng problem onsdered n [7]. Fnally, wth the beamformng vetor n (9), the nstantaneous SIR n (8) beomes SIR = λ max {R 1 G}, (11) and the orrespondng nstantaneous total nterferene n (6) beomes I = w H optrw opt. (12) Sne G s a rank 1 matrx, R 1 G has only one nonzero egenvalue. Next, we turn our attenton to E[SIR] and E and provde bounds for these two average performane measures. IV. INTERFERENCE AND SIR ANALYSIS We analyze the average nterferene and average SIR n ths seton and derve the upper and lower bounds. These two measures depend on the path loss matrx T n (7). We frst study E for a speal ase of T, then for a general T, by usng some known random matrx results n the lterature. At the end of ths seton, we provde some dsusson on the mplaton of the results. A. Interferene Analyss: Speal Case: T = I Np Reall that the entres of H are..d. Gaussan random varables wth zero mean and unt varane. Therefore, f T s an dentty matrx 4,the matrx R (1/ )HH H s a omplex Wshart matrx wth N p degrees of freedom and ovarane matrx Np I. We note that t s a ustomary prate to onsder the matrx R nstead of R = HH H n the lterature. We shall apply the results obtaned for R to R at the end of ths subseton. The Wshart matrx has been studed extensvely n the lterature and n partular, t s known that the empral dstrbuton funton (e.d.f.) of ts egenvalues defned as F Nt R (x) =Number of egenvalues of R x (13) 4 Ths orresponds to the ase where all prmary reevers have the same dstane from the seondary transmtter. onverges almost surely, as N p / >0 as,to a nonrandom dstrbuton funton 5 F R(x) = lm Nt E[F Nt (x)] (14) R whose probablty densty funton (p.d.f.) s the famous Marčenko-Pastur law [8] df R(x) dx (x = f (x) =(1 ) + a)+ (b x) δ(x)+ + 2πx, (15) where (z) + = max(0,z), a =(1 ) 2, and b =(1+ ) 2. Clearly, the regon of support assoated wth (15) s smply the regon where f (x) 0. By nspeton, we an see that the support s ( 1) 2 x ( +1) 2. Invokng the Raylegh s prnple [9], we have λ mn ( R) wh Rw w H w λ max( R). (16) The bulk lmt n (15) suggests λ mn ( R) ( 1) 2 and λ max ( R) ( +1) 2. Indeed, f the entres n H has fnte fourth moment, t has been proven n [10] that lm λ max( R) =( +1) 2, (17) whereas [11] has results on the smallest egenvalue lm λ mn( R) =( 1) 2. (18) Therefore, the average total nterferene E =E[w H Rw] an be bounded by [12] ( 1) 2 E ( +1) 2, (19) where the fator omes from the fat that R = R. B. Interferene Analyss: T wth Known Dstrbuton Reall that T s a dagonal matrx whose dagonal entres are gven by [T ], = d 2 and d (d >ɛ) s the dstane from C T to P R. Sne the loaton of C T and P R are both unformly dstrbuted n a rular ds, the dstrbuton of the dstane d annot be obtaned trvally n losed-form expresson. Here, we make a smplfed assumpton that C T s always loated at the enter of the ds whereas P R are unformly dstrbuted n the rular ds. We lam that ths always results n a larger average total nterferene E and therefore, serves as an upper bound for E where C T s random. Due to spae lmtaton, the proof s omtted and an be found n [13]. Smlar to the last subseton, we bound the egenvalues of R = (1/ )HTH H (and therefore, also the nterferene: w H Rw, R = R) by the support of ts lmtng e.d.f. (f. (14)). An effent tool to determne the lmtng dstrbuton s the so-alled Steltjes transform. Spefally, the Steltjes transform of a dstrbuton funton F R(x) s gven by 1 m R(z) = x z df R(x), z D {z C, Im z>0}. (20) 5 Note that >0 an be trvally satsfed beause 1 s vald for our problem as we assume N p /08/$ IEEE. 3

4 The above ntegral s over the support of F R(x) whh wll be on x 0 n our ase beause R s a postve semdefnte matrx wth all of ts egenvalues beng non-negatve. The p.d.f. an be unquely determned by the Steltjes-Perron nverson formula [14] df R(x) dx = 1 π lm η 0 Im m R(ξ + jη). (21) It has been shown n [15] (see also [8]) that f the matres H and T satsfy the followng four ondtons 6 : 1) H s a N p matrx whose entres are..d. omplex random varables wth zero mean and unt varane. 2) N p s a funton of wth N p / >0 as. 3) T s a dagonal matrx wth real random entres and the e.d.f. of the entres {τ 1,...,τ Np } onverges almost surely n dstrbuton to a probablty dstrbuton funton F T (τ) as. 4) H and T are ndependent. Then, almost surely, the e.d.f. of R = (1/Nt )HTH H, namely F Nt (x), onverges n dstrbuton to a nonrandom R dstrbuton funton F R(x) whose Steltjes transform m = m R(z) s the unque soluton to the followng equaton ( ) 1 τdft (τ) m = z. (22) 1+τm The above equaton has a unque nverse, gven by z R(m) = 1 m + τdft (τ) 1+τm, m m R(D). (23) For the problem at hand, the e.d.f. of the entres n T onverges to a nonrandom dstrbuton funton, namely, the dstrbuton of the random varable τ = d 2 where d s the dstane between the enter of a ds wth radus R and a random loaton n the ds. It s straghtforward to show that the p.d.f. of τ s df T (τ) dτ = Substtutng (24) nto (23) yelds z R(m) = 1 m + 1 (R 2 ɛ 2 )τ 2, 1 R 2 τ 1 ɛ 2. (24) R 2 ɛ 2 ln ( m + R 2 m + ɛ 2 ). (25) To determne the spetral densty of R usng (21), m = m R(z) n (25) has to be solved expltly. It s generally dffult, f not mpossble, to obtan an analytal or even an easy numeral soluton for the densty of an arbtrary dstrbuton. However, as shown n [16], muh of the analyt behavor of F R(x) an be nferred from (22)-(23) and n partular, the methodology presented n [16] an be used to fnd the support of F R(x) and an be summarzed n the followng four steps: 1) Defne B {m R : m 0, m 1 S T } where S T denotes the omplement of the support of F T (τ). 6 We note that the orgnal proof n [15], [8] onsders matrx n the general form B = A + HTH H and there are 5 ondtons wth an addtonal ondton onernng the requrement of the matrx A. 2) Plot (25) on B,.e., z R(m), m B. 3) Delete the range of values where the dervatve z R(m) 0. 4) The remanng range of values s the support of F R(x). Sne S T = {1/R 2 m 1/ɛ 2 }, B = {m R : m 0,m < R 2,m > ɛ 2 }. As mentoned before, R s a postve semdefnte matrx wth non-negatve egenvalues and therefore, we only need to onsder the postve range of z R(m),.e., z R(m) 0. It an be shown that z R(m) < 0 holds for m R 2 and the proof an be found n [13]. As a onsequene, for our problem, we only have to plot (25) on B {m> ɛ 2 }. A typal z R(m) plot s shown n Fg. 2. For ths fgure, we assume R =10, ɛ =3, and =50. We hoose a relatvely large ɛ n ths example for llustratve purpose only. In our smulatons to be presented n Seton V, we pk ɛ =0.05. In Fg. 2, we an see that for m> ɛ 2, there s a vertal asymptote at m =0and there s a loal mnmum and a loal maxmum on the left and on the rght of the vertal asymptote, respetvely. These rtal ponts serve as the boundary ponts of the support of F R(x) hghlghted n bold lne n the vertal axs gven by z R(m 1 ) and z R(m 2 ) (z R(m 1 ) >z R(m 2 ), m 1 <m 2 ), where m 1 and m 2 are the two zeros of the dervatve of z R(m). Byremovng the rrelevant terms n z R(m), the zeros an be obtaned by solvng the followng seond-order polynomal, z R(m) =0 (1 )m 2 +(R 2 + ɛ 2 )m + ɛ 2 R 2 =0. (26) In general, onvergene n dstrbuton of F Nt (x) does not R mply that the extreme egenvalues of R,.e., λ mn ( R) and λ max ( R), onverge to the mnmum and maxmum of the support of F R(x). However, t has been shown that f the maxmum (mnmum) egenvalue of T onverges to the largest (smallest) number n the support of F T (τ), then the largest (smallest) egenvalue of R onverges almost surely to the largest (smallest) number n the support of F R(x) [17]. Clearly, the egenvalues of T are bounded by the support of F T (τ) n our ase, f. (24). Consequently, by reallng R = R, the average total nterferene E =E[w H Rw] may be bounded by z R(m 2 ) E z R(m 1 ).For performane omparson, t s more nsghtful to onsder the average nterferene per prmary reever defned as E a E/N p, z R(m 2 ) E a z R(m 1 ). (27) C. SIR Analyss The upper and lower bounds for E[SIR] are readly avalable by makng use of the results obtaned from the last subseton. In partular, the average SIR gven by E[SIR] = E[λ max (R 1 G)] an be bounded by E[λ mn (R 1 )]E[λ(G)] E[SIR] E[λ max (R 1 )]E[λ(G)]. (28) We use λ(g) to ndate the only nonzero egenvalue of G. In partular, E[λ(G)] s equal to E[d 2 ] where E[d 2 ] s gven by E[d 2 ]= ɛ 2 R 2 τ (R 2 ɛ 2 )τ 2 dτ = ln(r2 /ɛ 2 ) R 2 ɛ 2. (29) /08/$ IEEE. 4

5 Upper bound of E a Lower bound of E a 3 2 Ea 10 0 z R(m) E[SIR] Upper bound of E[SIR] Lower bound of E[SIR] m Fg. 2. z R(m) vs. m for R =10, ɛ =3,and =50. Support of F R(x) s hghlghted n bold lne on the vertal axs where z R(m 1 )=1.859 and z R(m 2 )= We note that E[λ mn (R 1 )] and E[λ max (R 1 )] are equvalent respetvely to E [ (λ max (R)) 1] and E [ (λ mn (R)) 1].As mentoned n the prevous subseton, λ max (R) and λ mn (R) onverge almost surely to the maxmum and the mnmum support of F R (x), respetvely. Therefore, we arrve at the followng bounds for E[SIR] ln(r 2 /ɛ 2 ) (R 2 ɛ 2 )z R(m 1 ) E[SIR] ln(r 2 /ɛ 2 ) (R 2 ɛ 2 )z R(m 2 ). (30) In the next subseton, we shall present results on E a and E[SIR] as 1 and. They orrespond respetvely to the two extreme ases where N p = and N p. D. Dsusson Clearly, E a and E[SIR] are both funtons of R, ɛ, and = N p /. In ths subseton, we provde some nsghts on how the two performane measures sale as 1 and for a gven R and ɛ. For 1, we note that (26) redues to a lnear funton and has only one root. Ths root orresponds to the upper bound of (27). Therefore, for large, t s theoretally possble to aheve E a =0and E[SIR] = f there are as many antennas at the ogntve transmtter as the prmary reevers,.e., = N p. In the other extreme where the number of antennas at the ogntve transmtter s muh less than the number of prmary reevers,.e.,, the roots of (26) are approxmately m 1,m 2 ±ɛr 1. Substtutng the resultng root nto (25), t an be shown that the maxmum and mnmum support of F R(x) gven by z R(m 1 ) and z R(m 2 ) onverges to z R(m 1 ) z R(m 2 ) ln(r2 /ɛ 2 ) R 2 ɛ 2 = E[d 2 ]. (31) Applyng the above result to (27) and (30), mmedately, we see that for E a E[d 2 ] and E[SIR] 1. (32) The above result an be also obtaned by onsderng E dretly. By applyng the law of large numbers for large N p, R = HTH H s approxmately a dagonal matrx wth Fg. 3. Upper and lower bounds of E a and E[SIR] for R =10and ɛ =0.05. [R] n,n = N p =1 h n 2 d 2 1 Np. Consequently, N p =1 h n 2 d 2 approahes ts expeted value gven by E[d 2 ] as N p. The above results suggest that for very dense network, the average nterferene per prmary reever depends only on the average path loss from C T to P R and the average SIR dereases exponentally wth nreasng N p.infg.(3),we plot the lower and upper bounds of (27) and (30) for , R =10and ɛ =0.05. As expeted, the lower and upper bounds for both (27) and (30) onverge aordng to (32) as. Note also that for =1, the lower bound of (27) and the upper bound of (30) are 0 and, respetvely. V. NUMERICAL AND SIMULATION RESULTS In ths seton, we present some numeral and smulaton results. For all results shown we assume R =10and ɛ =0.05. In Fg. 4, the smulated E a and E[SIR] are plotted as a funton of for N p = 100 and Wehaveshown the results for both random C T and fxed C T (where C T s plaed at the enter of the ds). For omparson, the lower bound of E a (27) and the upper bound of E[SIR] (30) are also depted. Clearly, the average nterferene s smaller for random C T whh s n aordane wth our dsusson n Seton IV-B. Also as expeted, nreasng N p nreases E a and results n a lower E[SIR]. On the other hand, nreasng the number of antennas at C T has the opposte effet. The smulaton results are qute lose to the theoretal lmts gven by the lower bound of E a and the upper bound of E[SIR]. The results for the upper bound of E a and the lower bound of E[SIR] are not shown beause the objetve of the beamformer s to mnmze the nterferene and maxmze the SIR, f. (9). In fat, the upper bound of E a and the lower bound of E[SIR] are qute loose for the relatvely small onsdered n ths fgure. Ths s a good ndaton that the beamformng vetors are performng well n the small regon of. As we have seen n Fg. 3, as nreases, the system beomes saturated and the upper and lower bounds of E a and E[SIR] onverge to the same value. In other words, hoosng a random beamformng vetor s as good as usng the optmal one obtaned from (9) for. In general, the bounds obtaned for E a and E[SIR] n (27) and (30) are asymptot bounds for N p / >0 as /08/$ IEEE. 5

6 Ea E[SIR] 10 1 N p = 1000 E a (Fxed C T loaton) E a (Random C T loaton) Lower bound of E a 10 0 N p = 100 E[SIR] (Fxed C loaton) T E[SIR] (Random C loaton) T 10 1 N p = 1000 N p = 100 Upper bound of E[SIR] 10 3 Fg. 4. Smulatons of E[SIR] and E a as a funton of and dfferent N p. Fxed C T (squares). Random C T (trangles). The lower bound of E a (27) and the upper bound of E[SIR] (30) are also plotted (rles). Ea E[SIR] 10 1 E a (Fxed C T loaton) = 100 E a (Random C T loaton) Lower bound of E a 10 0 E[SIR] (Fxed C T loaton) E[SIR] (Random C T loaton) 10 1 =10 = 100 =10 Upper bound of E[SIR] Fg. 5. Smulatons of E[SIR] and E a as a funton of for dfferent. Fxed C T (squares). Random C T (trangles). The lower bound of E a (27) and the upper bound of E[SIR] (30) are also plotted (rles).. Therefore, a natural queston to ask s how well these bounds perform n fnte regon of. Ths queston s answered n Fg. 5. In partular, the smulated E a and E[SIR] are plotted as a funton of for =10and 100 ( s kept fxed by varyng N p for dfferent s). Agan, we onsder both random and fxed C T. For referene, we have also plotted the lower bound of E a (27) and the upper bound of E[SIR] (30). Note that the bounds are onstant for fxed value of beause they depend only on the rato = N p / and not the atual values of N p and, f. (26), (27), and (30). The smulaton results for E[SIR] do not devate muh wth fxed and varyng and the bounds work well also for small values of. On the other hand, for =10and < 6, the smulaton results for E a depend on the atual values of and N p beause they devate even f the rato = N p / s fxed to a onstant. Ths s not surprsng, beause the asymptot bounds assume large. Nevertheless, when 6, E a beomes also dependent only on the rato = N p / and not the atual values of N p and. In general, we fnd that when s large enough, the smulaton results for E a depend also only on the rato even for small values of, f. the E a urves for = 100. VI. CONCLUSION In ths paper, we onsder a ogntve network whh onssts of multple prmary users and multple ogntve users. The seondary ogntve transmtters are allowed to transmt onurrently wth the prmary lensed transmtters. To mtgate nterferene, the seondary users transmt sgnals usng multple antennas wth a beamformng vetor. The beamformng vetor s desgned to maxmze the SIR of the seondary user. We derve bounds and provde asymptot analyses for the average SIR and the average nterferene aused to all prmary reevers. In partular, we have shown that f the number of antennas at the seondary transmtters an be of the same as the number of prmary reevers, the nterferene aused to the prmary reevers an be made zero, reatng an nfnte SIR at the ogntve user. On the other hand, f the number of prmary reevers outgrows the number of antennas at the seondary transmtter, then both the average nterferene and the average SIR approah fxed lmts. These analyses an be useful n dedng the number of antennas to deploy n the seondary transmtters. ACKNOWLEDGMENT Ths researh s supported n part by ARO MURI grant number W911NF and a NSERC PDF. The vews expressed n ths paper are those of the authors alone and not of the sponsors. REFERENCES [1] Unted States frequeny alloatons: The rado spetrum. U.S. Department of Commere, Natonal Teleommunatons and Informaton Admnstraton, Offe of Spetrum Management, Otober [2] Spetrum poly task fore report. Federal Communatons Commson Teh. Rep , November [3] J. Mtola. Cogntve Rado. PhD thess, Royal Insttute of Tehnology (KTH), [4] S. Srnvasa and S. A. Jafar. The throughput potental of ogntve rado: A theoretal perspetve. IEEE Communatons Magazne, 45(5):73 79, May [5] B. D. V. Veen and K. M. Bukley. 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Inequaltes: Theory of Majorzaton and Its Applatons. Aadem Press, [13] S. Yu, M. Vu, and V. Tarokh. Interferene and nose reduton by beamformng n ogntve networks. In preparaton. [14] N. I. Akhezer. The lassal moment problem and some related questons n analyss. Hafner Pub. Co., New York, [15] J. W. Slversten and Z. D. Ba. On the empral dstrbuton of egenvalues of a lass of large dmensonal random matres. Journal of Multvarate Analyss, 54(2): , August [16] J. W. Slversten and S.-I. Cho. Analyss of the lmtng spetral dstrbuton of large dmensonal random matres. Journal of Multvarate Analyss, 54(2): , August [17] Z. D. Ba and J. W. Slversten. No egenvalues outsde the support of the lmtng spetral dstrbuton of large dmensonal sample ovarane matres. The Annals of Probablty, 26(1): , January /08/$ IEEE. 6