Channel Capacity Limits of Cognitive Radio in Asymmetric Fading Environments

Channel Capacity Limits of Cogniti v e Radio in Asymmetric F ading En vir onments Himal A. Suraweera ∗ , Jason Gao ∗ , Peter J. Smith † , Mansoor Shafi ‡ and Michael Faulkner ∗ ∗ School of Electrical Engineering, V ictoria University , Melbourne, Australia † Departmen t of Electrical an d Com puter Eng ineering , Uni versity of Can terbury , Christchurch, New Z ealand ‡ T elecom New Z ealand, PO Box 293 , W ellington, New Zealan d Email: him al.suraweera@vu.ed u.au, p.smith@elec.canterbury .ac.n z, mansoor .shafi@telecom.co.n z Abstract —Cognitiv e radio technology is an innovati ve radio design concept which aims to increase sp ectrum uti lization by ex- ploiting unused spectrum i n dynamically changing en vironments. By extending previous results, we in vestigate the capacity gains achiev able with this dynamic spectrum approach in asymmetric fading channels. M ore specifically , we allow the secondary-to- primary and secondary-to-secondary u ser channels to und ergo Rayleigh or Ri cian fading, with arbitrary link power . In order to compute th e capacity , we d eriv e the distributions of ratios of Rayleigh and Rician variables. Compared to the symmetric fading scenario, our results in dicate severa l i nteresting features of the capacity behav iour u nder both ave rage and peak receiv ed power constraints. Finally , t he impact of multiple primary u sers on the capacity under asymmetric fading h as also been studied. I . I N T R O D U C T I O N Conservati ve spectrum policies employed by regulato ry authorities have resulted in spectrum und erutilization of the overall available spe ctrum for wir eless comm unication s. Mea- surements perfo rmed by agen cies such as the Fed eral Commu - nications Commission [1] in the United States an d Of com [2] in the United Kingdo m have revealed that at any given time , large p ortions of spectrum are sparsely occupie d. Findings of such c ampaign s o n spectru m usage have challenged the traditional spectr um m anagemen t approaches. The co ncept of cognitive radio (CR) [3] r efers to a sm art ra - dio which can sense the e xternal electro magnetic en vironm ent and ada pt its transmission paramete rs ac cording to the curre nt state o f the en vironm ent. CRs can access pa rts of the sp ectrum for their informatio n tran smission, provided that they cause minimal interferen ce to th e pr imary u sers in th at ban d [ 5], [6]. T herefo re, spectr um shar ing am ong th e prim ary licen see and th e second ary CR m ust b e carried out in a contro lled fashion. In the technical literatur e, the interfer ence temperatur e introdu ced by K olodzy [ 7], [4] indicates the inter ference lev el at the primary licensee’ s rec eiv er . From the licensees’ point of view , the secondary access can b e contro lled in two ways. The to tal interf erence power can be require d to remain below a cer tain threshold (an interference temperature constraint) or the sign al-to-no ise-and-inter ference ( SINR) can be co nstrained. The capacity of wireless systems has b een exten si vely stud - ied under fixed spectrum access. For CR, this work is le ss ma- ture and many infor mation/co mmunicatio n theoretic p roblems and im plementatio n issues [8] r emain to be solved. Howe ver , se veral intere sting results on the cap acity , outage pr obabil- ity and through put o f CR systems h av e recen tly emerged. See for example, [10], [11], [9]. In [9], Gastpar derived the capacity of different non-fadin g additive-white-Gaussian- noise (A WGN) ch annels with the average r eceived-po wer at a primary receiver being constrained . In [10], Ghasem i an d Sousa showed that with the same limit o n the received- power lev el, channel capacity for a ran ge of fading mo dels (e.g., Rayleig h, Nakaga mi- m and log -norm al fading) exceeds that of the non-fading A WGN channel. In some scenarios, primary user spectral activity in the v icinity of the cog nitive transmitter may differ from that in th e vicin ity o f the cognitive receiver . Considering this, in [11], the cap acity of opp ortunistic spectrum acq uisition in the presenc e of distributed spe ctral activity has b een inv estigated. W e extend the work in [10] which assumed that fading condition s for the interfere nce path (CR transmitter-primary receiver) and th e desired path (CR tran smitter-CR r eceiver) are th e same. In pr actice, these two link s could experience different fading co ndition s ( types) and different link powers (due to path length or shadowing). Th is is refer red as a sym- metric fadin g in this paper . In this paper we con sider Rayleigh and Rician fading. Hence, we are ab le to q uantify th e effects of propaga tion paths consisting of b oth line-o f-sight (LoS) and scatter ed compon ents. Building on the work in [1 0], th is pa per makes the fo llowing main con tributions: 1) Th e secondar y capacity under average-received power and peak- power constraints is studied for asymmetr ic condition s inclu ding different fading types (Ray leigh and Rician) an d different link powers. Here we show that und er low in terference to the pr imary receiver , the secondary cap acity is sensiti ve to the fading type on the desired a nd in terferenc e paths. 2) Th e im pact of multiple primar y licensee rece iv ers in Rayleigh an d Rician fading is studied fo r pea k power constraints. 3) Closed- form expressions for the cumulative d istribution function (CDF) and th e pro bability d ensity fun ction (PDF) o f a ran dom variable (R V), g 1 /g 0 is deriv ed, for the c ases wher e ( √ g 1 , √ g 0 ) experie nce ( Rayleigh, Rician) and (Rician, Rayleigh) fading. This is needed to derive th e above mention ed capacities. secondary transmitter 0 g 1 g primary recei ver secondary receiver Fig. 1. Shared spectrum usage between primary and secondary users. This p aper is o rganized as follows. Th e system and chann el model is described in Section II. In Section I II we deri ve the exact PDFs for the r atio of Rayleig h an d Rician R Vs . In Section IV , th ese r esults ar e used to study the capacity gains un der average/peak rec eiv ed-power constra ints respec- ti vely . Ex tensions o f these results to multiple primary users are pr esented in Section V . Finally , some con clusions are drawn in Section VI. Through out the paper, the refer ence to average/peak rec eiv ed-power ref ers to as th e average/peak interferen ce power at th e pr imary re ceiv er . Th e CR link is also referred to as a secon dary link. I I . S Y S T E M A N D C H A N N E L M O D E L In this section , the system and channel mod el co nsidered in the paper are briefly o utlined (cf. Fig. 1) . The system model is borrowed f rom [1 0], however we ha ve considered asym metric fading scenarios. A point-to -point flat fading channel with perfect ch annel side informatio n available to b oth the receiv er and th e tran smitter is assumed. Let g 0 and g 1 denote the instantaneou s channel gains fr om the second ary tran smitter to the primary and seconda ry recei vers respecti vely . Fur thermor e, we denote the respective PDFs b y p g 0 ( g 0 ) and p g 1 ( g 1 ) . For a unit p ower channel ga in, the Rayleigh PDF is given by p √ g ( x ) = 2 xe − x 2 (1) for x ≥ 0 . For a Rician distribution the PDF is g i ven by p √ g ( x ) = 2 x (1 + K ) e − K − (1+ K ) x 2 I 0  2 x p K + K 2  (2) for x ≥ 0 , whe re K is the Rician K -factor defined as the ratio of s ignal power in dominan t compo nent over t he scattered power and I 0 ( · ) is the zeroth-order mod ified Bessel fun ction of the first kind. For K = 0 , Rayleigh fading is experienced and K = ∞ giv es the A WGN ( no fading) situation. V alu es of the K -f actor in indo or/outd oor land m obile applications nor mally range f rom 0 − 12 dB [ 12]. In a p ractical environment the CR transmitter to CR receiver link m ay n ot b e of the same length as th e CR in terference path to the primary recei ver . When the link powers, E { g 0 } an d E { g 1 } , d iffer , it can b e shown that capacity o nly dep ends on the p ower ra tio. Hence, we define the relativ e power para meter, c , b y c = E { g 1 } /E { g 0 } . Note that E {·} denotes the expectation operator . T w o imp ortant items of n otation shou ld be stressed at this point. Since the main results of the paper depend on the ratio g 1 /g 0 , we use the shorthan d notation Rayleigh/Rician to indicate that √ g 1 is Rayleigh and √ g 0 is Rician. Similarly , Rician/Rayleigh indica tes that √ g 1 is Rician and √ g 0 is Rayleigh. Th e second issue is that the secondary tran smitter must constrain its power so tha t the interferen ce at the p rimary is acceptab le. Hence the power con straints in this scenario are really interferenc e con straints. This is different to many other problem s where th e constra ints are for transmit power . I I I . C D F A N D P D F O F g 1 /g 0 Here, we an ticipate the results of Section IV , wh ere it is shown that capacity depen ds on the ratio, g 1 /g 0 . Hen ce, in this section the CDF and the PDF for a Rayleigh /Rician R V and a Rician/Rayle igh R V ar e derived. Consider the distribution of a Rayleigh /Rician R V , X = g 1 /g 0 . Mathematically , P ( X < x ) , i.e., the CDF of X , is giv en by P ( X < x ) = Z ∞ 0 P  g 1 g 0 < x | g 0  p g 0 ( g 0 ) dg 0 (3) Equation ( 3) can be simplified as P ( X < x ) = ( K + 1 ) e − K Z ∞ 0 (1 − e − xg 0 ) (4) · e − ( K +1) g 0 I 0 (2 p K ( K + 1) g 0 ) dg 0 The integral in (4) can be solved u sing [13, e q. 2.15 .5.4] and we o btain the CDF of X as F X ( x ) = 1 − K + 1 x + K + 1 e − K + K 2 + K x + K +1 (5) for x ≥ 0 . The PDF of X can be found by taking the derivati ve of (5) with respe ct to x , yielding p X ( x ) = ( K + 1) x + ( K + 1) 2 ( x + K + 1) 3 e − K + K 2 + K x + K +1 (6) for x ≥ 0 . As expected , for K = 0 th e PDF p X ( x ) is given by p X ( x ) = 1 / ( x + 1) 2 [10, eq. 11] . Now consider the distribution of Y = g 1 /g 0 when √ g 1 is Rician and √ g 0 is Rayleigh. Using the same approac h, P ( Y < y ) is g iv en by P ( Y < y ) = 1 − Z ∞ 0 Q 1  √ 2 K , p 2(1 + K ) yg 0  e − g 0 dg 0 (7) where Q 1 ( a, b ) = R ∞ b xe − a 2 + b 2 2 I 0 ( ax ) dx is the first-order Marcum Q -func tion which satisfies th e f ollowing iden tity [14, eq. 5 ] Q 1 ( a, b ) + Q 1 ( b, a ) = 1 + e − a 2 + b 2 2 I 0 ( ab ) (8) Using (8 ), we can express (7) a s shown in (9). The first integral in (9) can be e valuated in closed-fo rm using the result of [1 3, eq. 2. 15.5.4 ] an d is I 1 = e − K + K y + K 2 y y + K y +1 y + K y + 1 (10) P ( Y < y ) = − Z ∞ 0 e − ( y + K y +1) g 0 I 0 (2 p ( K y + K 2 y ) g 0 ) dg 0 + Z ∞ 0 Q 1 ( p 2( y + K y ) g 0 , √ 2 K ) e − g 0 dg 0 (9) The second integral in (9) can be ev aluated using th e result of [15, eq . 2 5] as I 2 = e − K y + K y +1 . Hen ce, P ( Y < y ) is given by F Y ( y ) = e − K y + K y +1 − e − K + K y + K 2 y y + K y +1 y + K y + 1 (11) for y ≥ 0 . After takin g the deriv ativ e of (11) with respect to y , we o btain the PDF of Y as p Y ( y ) = K ( 1 + K ) ( y + K y + 1) 2 e − K y + K y +1 (12) + (1 + K ) 2 (1 − K + y ) ( y + K y + 1) 3 e − K + K y + K 2 y y + K y +1 for y ≥ 0 . For Rayleigh/Rayleig h fading, the PDF p Y ( y ) simplifies to p Y ( y ) = 1 / ( y + 1) 2 [10, eq . 1 1]. T o confirm the deriv ations, the PDFs of (6) and ( 12) were validated by Monte Car lo simulatio ns, and a perfe ct match was ob tained. I V . C A PAC I T Y G A I N S O F S P E C T RU M S H A R I N G A. Capacity Und er an A verage Received-P ower Constraint In th is section, we in vestigate the capacity gain s achievable by the secon dary user un der an average received-power con- straint. In [ 10], the chann el capacity was expressed as C = Z Z B log 2  1 + g 1 P ( g 0 , g 1 ) N 0 B  p g 0 ( g 0 ) p g 1 ( g 1 ) dg 0 dg 1 (13) such tha t Z ∞ 0 Z ∞ 0 g 0 P ( g 0 , g 1 ) p g 0 ( g 0 ) p g 1 ( g 1 ) dg 0 dg 1 ≤ Q (14) where Q is the m aximum average interfer ence power tolerated by the primar y r eceiver 2 , B is the av ailable bandwid th, N 0 is the n oise power at th e secondary receiver and P ( g 0 , g 1 ) denotes th e optim al power allocation. Using the Lagrang ian technique , [10] h as fo und P ( g 0 , g 1 ) to be P ( g 0 , g 1 ) =  1 λ 0 g 0 − N 0 B g 1  + (15) where ( · ) + denotes ma x {· , 0 } . No te th at λ 0 is determined such th at the average receive p ower is equ al to Q . Th at is mathematically Z g 0 Z g 1  1 λ 0 − N 0 B g 0 g 1  + p g 1 ( g 1 ) p g 0 ( g 0 ) dg 1 dg 0 = Q ( 16) Hence, th e chan nel cap acity can be calculated from C = B Z ∞ 1 γ 0 log 2 ( γ 0 g 10 ) p g 1 g 0 ( g 10 ) dg 10 (17) 2 The qualit y of transmission at the primary recei ver can also be measured using the SINR. This require s a knowledg e of the primary transmitter to primary recei ver channel. where γ 0 = 1 / ( λ 0 N 0 B ) and p g 1 g 0 ( · ) de notes the PDF of g 1 /g 0 . T o the best of the authors’ kno wledge, there are no closed -form solutions f or the integral in ( 16) for the two fading scena rios considered in this paper . Rewriting (16) gives Z γ 0 0 ( γ 0 − x ) p ( x ) dx = Q N 0 B = α (18) where p ( x ) in (18) denotes the PDF of g 0 /g 1 . There fore, α is the a llow able inter ference-to -noise power ratio at the primary receiver . Using integration by p arts, (18) can be further simp lified as Z γ 0 0 F ( x ) dx = α (19) where F ( x ) den otes th e CDF o f g 0 /g 1 . Hence, usin g (19) we have calculated γ 0 numerically . Note th at the calculatio ns in (13)-(19) are g eneral and apply to both the equal po wer ( c = 1 ) and the un equal power ( c 6 = 1 ) case. I n Section I II the requ ired PDFs and CDFs were der iv ed f or the equal power ca se. When c 6 = 1 , it is a simp le process to repeat the steps in (13)-(19) and to show that using cα instead of α in (19) with the equal power results from Section III gives the correc t results. Henc e, we only require the PDF an d CDF of g 1 /g 0 , g 0 /g 1 for the equa l power case. T o obtain r esults f or the u nequa l power case, we simply use cα rather than α in (19). This is equivalent to replacing N 0 by N 0 /c , whic h makes intuitive sense sinc e a power r atio o f c implies that th e secon dary receiver r eceives a signal c times stronge r tha n th e primary . Hence, relativ e to the equal p ower case th e SNR is c times bigger and th e eq uiv alent noise level is N 0 /c . Figs. 2 and 3 show the seconda ry cap acity versus α and under an av erage r eceiv ed in terferen ce power con straint. I n all plots, A WGN r efers to the scena rio where g 0 and g 1 are equa l to unity all the time [10]. W e make the fo llowing no tew orthy observations: 1) Th e secon dary cap acity increases if the p rimary r eceiver can tolerate more interf erence. Th is is because th e secondary transmitter is able to tran smit with h igher power (probab ilistically). 2) Th e case of intere st in engineering practice is for a low value of α , i.e., when the ac ceptable CR in ter- ference is co rrespon dingly low . Here we see that the capacity can be sen siti ve to the ty pe o f fading an d indeed the symmetric fading , i.e, the Rayleigh/Rayleigh case significantly overestimates the capac ity comp ared to th e Rayleigh /Rician case in the low α regime. This observation is centr al to our contribution in this paper . The difference in capacity reduc es to almost zero when the accep table in terference at the primar y is large. 3) Th e capacity of Rician/Rayleigh fading (cf. Fig. 3) is not so sensitive to the K -factor (0 − 15 ) dB. For a given −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) Rayleigh/Rayleigh K = 0 dB K = 5 dB K = 15 dB AWGN Fig. 2. Capa city under an av erage recei ve d-po wer constrai nt against α in Raylei gh/Rici an fading. c = 0 dB. −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) Rayleigh/Rayleigh K = 0 dB K = 5 dB K = 15 dB AWGN Fig. 3. Capa city under an av erage recei ve d-po wer constrai nt against α in Ricia n/Rayle igh fading. c = 0 dB. α , the Rayleigh fading on the primar y link deter mines the tr ansmit power of th e secon dary user . Once th is is determined , th e resulting second ary user capacity is less sensiti ve to the K -factor with in the considered range of 0 − 15 dB. T his is in co ntrast with Rayleigh/Rician fading, (cf. Fig. 4), where we see that the K -factor induces an ap preciable capacity difference especially in the lo w α regime. As K -factor d ecreases and in th e low α r egime, m ore opp ortunities for th e secon dary user to transmit with relatively high power are cr eated. Howe ver , f or large α , the im pact of chang ing K -factor on th e seco ndary user tran smit power is reduc ed. 4) Und er fading, the second ary capacity is h igher than the A WGN ca se. Th is observation is consistent with the findings o f [10]. In a fadin g environment, the secondary user can transmit with hig h power , when its sign al received b y th e p rimary user is subject to d eep fades. −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) K = −10 dB K = 0 dB K = 5 dB K = 10 dB Rayleigh/Rayleigh AWGN Fig. 4. Capaci ty under a peak recei ved-po wer constra int again st α in Raylei gh/Rici an fading. c = 0 dB. −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) K = −10 dB K = 0 dB K = 10 dB Rayleigh/Rayleigh AWGN Fig. 5. Capaci ty under a peak recei ved-po wer constra int again st α in Ricia n/Rayle igh fading. c = 0 dB. B. Capacity Und er a P eak Received -P ower Constraint As discu ssed in [10], although average received-power is reasonable f or delay insensitive ap plications, in other cases it is desirab le to impose a p eak received-power constraint. Under the pea k rece i ved-power co nstraint [ 10] g 0 P ( g 0 , g 1 ) ≤ Q (20) and th e chan nel capacity was given in [ 10] as C = B Z ∞ 0 log 2 (1 + αx ) p g 1 g 0 ( x ) dx (21) Therefo re, u nder Rayleigh/Rician fading the c hannel capacity is obtained b y sub stituting the PDF in (6) into (21). This giv es C = ( K + 1) B Z ∞ 0 log 2 (1 + αx ) x + ( K + 1) 2 ( x + K + 1) 3 (22) × e − K + K 2 + K x + K +1 dx C = K (1 + K ) B Z ∞ 0 log 2 (1 + αx ) e − K x + K x +1 ( x + K x + 1) 2 dx + (1 + K ) 2 B Z ∞ 0 log 2 (1 + αx ) (1 − K + x ) ( x + K x + 1) 3 e − K + K x + K 2 x x + K x +1 dx (23) −20 −15 −10 −5 0 5 10 15 20 10 −3 10 −2 10 −1 10 0 10 1 α in dB channel apacity (bps/Hz) Rayleigh/Rician (average) Rician/Rayleigh (average) Rayleigh/Rician (peak) Rician/Rayleigh (peak) = 10 dB c = −10 dB c Fig. 6. Capaci ty under avera ge and peak recei ved -powe r constrai nts against α in Rician/ Raylei gh and Rayleigh/ Rician fad ing for tw o differ ent v alue s of c . K = 6 dB. Similarly , under Rician/Rayleigh fading the chann el capacity is given by ( 23) on the next page. The case wher e th e shadowing on the two link s is different can be de riv ed u sing the same argumen ts as above. Hen ce, nume rical results ar e obtained assum ing g 1 and g 0 have equal power but α is replaced b y cα . Figs. 4 and 5 show the secon dary cap acity versus α and under a peak received in terferen ce power c onstraint. W e make the following notew orthy observations: 1) Like the a verage interferen ce power case, the capacities increase if th e prim ary can tolerate more in terferenc e. 2) Th e secondary capacity is sensitiv e to the type of fading on the two links and d ependin g on the fading type on either link on e cou ld overestimate the capacity especially for low values of α and Rayleigh/Rician fading. 3) Fro m [ 10, Fig. 4] in symmetr ic fading co nditions, the capacity under a peak received power constraint is always highe r than the A WGN case. Howe ver in Rayleigh/Rician fading, we see tha t the capacity is higher/lower th an the A WGN case d epend ing on the α . Fig. 6 shows the impact of signal power differences o n CR capacity . Such differences u sually arise from shadowing and path len gth d ifferences. W e assume two values for th e power ratio between the links, c = 10 dB and c = − 10 d B. The effect on CR cap acity is a simple scaling by th e c p arameter . Hence, we have a simple an d efficient ap proach to in vestigating such asymmetric link s. V . E FF E C T O F M U LT I P L E P R I M A RY U S E R S When n > 1 primary users are p resent, the transmit/rec eiv e powers of th e seconda ry user would be sub ject to add itional −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) n = 1 n = 2 n = 3 AWGN Fig. 7. Capacity under a peak receiv ed-po wer constrai nt and Rician/Rayl eigh fadi ng for diffe rent numbers of primary receiv ers. K = 6 dB. −20 −15 −10 −5 0 5 10 15 20 10 −2 10 −1 10 0 10 1 α in dB channel capacity (bits/Hz) n = 1 n = 2 n = 3 AWGN Fig. 8. Capacity under a peak receiv ed-po wer constrai nt and Rayleigh/Ric ian fadi ng for diffe rent numbers of primary receiv ers. K = 6 dB. constraints. This leads to a ca pacity reductio n [10]. Let g 0 i denote the channe l g ain of th e seconda ry tran smitter to th e i -th pr imary receiver . In this case, the peak received-power constraint is ref ormulate d by the following constraint P ( g 01 , g 02 , . . . , g 0 n , g 1 ) ≤ min i Q g 0 i , i = 1 , . . . , n (24) The ch annel capa city is given by C = B Z ∞ 0 log 2 (1 + αz ) p Z ( z ) dz (25) where Z = g 1 / max i g 0 i . In Appen dix A we have derived the PDF f or Z when √ g 0 i , i, = 1 , . . . , n are ind ependen t and C = nB n − 1 X k =0 ( − 1) k  n − 1 k  Z ∞ 0 log 2 (1 + αx ) 1 (1 + k + (1 + K ) x ) 2 e − (1+ k ) K 1+ k +(1+ K ) x  1 + K + K ( K + 1) 2 x 1 + k + ( K + 1) x  dx (26) identically distributed (i.i.d) Rayleigh R Vs an d √ g 1 is a Rician R V . Substituting this PDF into (25) results in the capacity given by (2 6). Such a result can be extended to th e uneq ual power case by considering the maximum of in depend ent Rayleigh variables with differing m eans. Th is is p ossible using standard order statistic resu lts, but is beyo nd the scope of the paper . Unfortu nately , the PDF for the case wh en √ g 0 i , i = 1 , . . . , n are i.i.d Rician R Vs and √ g 1 is a Rayleigh R V could n ot be found in closed-for m. In stead we h av e resorted to time consu ming Mo nte-Carlo simulations to obtain the capacity . The average receiv ed-power case ap pears to be r ather complex and is not co nsidered here. Figs. 7 and 8 illustrate the CR capac ity for n = 1 , 2 , 3 and Rayleigh /Rician and Rician/Rayleigh fading respectively . In all cases, the capac ity reduces compare d to the A WGN case as n g ets larger . V I . C O N C L U S I O N S In th is p aper we have in vestigated the impact of asy mmetric fading on the seco ndary user capacity und er a verage and peak interferen ce power constrain ts. Compared to symme tric fading condition s assumed in the p revious literature, our analysis have added several new insigh ts, especially for a low value of α , i.e., the regime th at m ost CRs would expect to operate in practice. The results show th at under Rayleigh/Rician fading and low α , the capacity is significan tly lower than that in a sy mmetric Ray leigh/Rayleigh fadin g scenario, and as α increases, the imp act of K -factor on the cap acity is red uced. Under Rician/Rayleigh fading , the capacity resu lts ch ange only slightly with d ifferent K -factors within considered ra nge o f 0 − 15 dB. Th e cap acity results were also e xtende d to include the effects o f different p ower g ain and multiple pr imary users. A P P E N D I X A D E R I V AT I O N S W I T H M U LT I P L E P R I M A RY U S E R S Let √ g 0 i , f or i = 1 , . . . , n , be i.i.d Rayleigh R Vs a nd let √ g 1 , which is in depend ent of all g 0 i , have a Rician distribution. De fine g 0 = ma x i g 0 i for i = 1 , . . . , n and U = g 1 /g 0 . Then the CDF of U is given by P ( U < u ) = Z ∞ 0 P ( g 1 < g 0 u | g 0 ) p g 0 ( g 0 ) dg 0 (27) The PDF of g 0 , p 0 ( g 0 ) is g iv en by [16, eq. 9 .326 ] as p g 0 ( g 0 ) = n n − 1 X k =0 ( − 1) k  n − 1 k  e − (1+ k ) g 0 (28) Substituting (2 8) in to (2 7) we obtain P ( U < u ) = 1 − n n − 1 X k =0 ( − 1) k  n − 1 k  (29) × Z ∞ 0 Q 1  √ 2 K , p 2(1 + K ) ut  e − (1+ k ) t dt After solving the integral in (29), we express P ( U < u ) in closed-for m as F U ( u ) = 1 − n n − 1 X k =0 ( − 1) k 1 + k  n − 1 k  (30) ×  1 − (1 + K ) u 1 + k + (1 + K ) u e − (1+ k ) K 1+ k +(1+ K ) u  Finally , differentiation of P ( U < u ) with respect to u , yield s the PDF o f U . Therefo re, the PDF of U is giv en by p U ( u ) = n n − 1 X k =0 ( − 1) k  n − 1 k  1 (1 + k + (1 + K ) u ) 2 (31) × e − (1+ k ) K 1+ k +(1+ K ) u  1 + K + K ( K + 1) 2 u 1 + k + ( K + 1) u  A C K N O W L E D G E M E N T This research is suppo rted u nder the Australian Research Council’ s Discovery funding schem e (DP07 74689 ). R E F E R E N C E S [1] Feder al Comm unicat ions Comm ission (FCC), “Faci litat ing opportunitie s for flexible , ef ficient, and reliabl e spectrum use emplo ying cognit i ve radio techn ologies, ” ET Docket No. 03-108, Mar . 2005. [2] Cogni ti ve Radi o T echnology - A Study for Ofcom, [online] A v ailabl e: http:/ /www .ofcom.org.uk/researc h/technology/research/emer tech/ cograd/c ograd main.p d f [3] J. Mitola III, Cognit ive radio: An inte grated agent ar chit ectur e for softwar e defined radio , Ph.D Thesis, KTH Ro yal Institute of T echnolog y , Sweden, May 2000. [4] S. 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