This article has been accepted for inclusion in a future issue of this journal.Content is final as presented,with the exception of pagination. IEEE/ACM TRANSACTIONS ON NETWORKING YRx()should be larger than AspRmin/2+ApsTi/2;the distance 十 between any two primary transmitters is larger than ApRmin. ·YRxy Now consider the case that△sp<△(Note that if△sp> Ap:Ips(i)will be smaller,and the upper bound still holds.) Xgx() Then,all XjjT(p)is at least ApRmin/2 away from Y() B时 except X.First,consider the interference contributed byX P Fig.3.Analyzing the interference.Left plot shows an example for≤ii,and right plot for 2 ij. Next,consider the interference from some other primary trans- mitter Xi.Let Bii=B(Xi:ApRmin/2)n B(YRx(),Xi- denote the interference at receiver XRx(i)from the secondary YRx()c,as shown in Fig.3,then network,and dA be the area element dA I(=∑ CiP Bii IBiil Xj-YRx( 4CP dA (where Bjl is the area of disk Bij) JBy,△/2)|Y-XRx)F ≤ dA B IBminl Xj-Yix(Y (because dA=πA2号/4 JBYi,△sr/2 where |Bminl =min|Bijl ≤∑2CP dA AgD,居-XR下 P dA [Bminl (-YRx)Y s∑ 12CP dA B:1 T) )TA2 JD,X-XRx下 (because Xj-YRx(>ApRmin/2) (where X is the position vector of dA, 2P dA andlX -XRx(<Yj-XRx(l for X E Dij) Bmin K-Yr间T 12C1P dA 2+3P π△2 儿eDX-XRx间f1 △m dAx -YRx(l. Since (UjT(Dij)B(XRx()AspRi/2)=0.we have 12C1P dA Tosum up,lets()=∑jerp叭{xw(P/X-Yx(lP) Ip() T△2 X-XRx(下 ()≤ 2+3P ∑ dA |X-XRx(l2△pR:/2 π△华F jET(D)\X')Bis X-YRx() 12C1P 2xrdr 2+3P dA π△2 △pR:/2 24C1P nU,erok(B X-Yx() 2 Y-2 C2PR2-7. (since [Bii}are disjoint) △2(Y-2) 2y+3P dA ≤△F隔 C.Interference at Secondary Nodes K-小p画 X-YRx() Now we focus on the interference at secondary nodes.The 2+3P 2mrdr main challenge is to bound the uncontrollable interference from the primary network.Let Rmax(n)=max Ri(n).Rmin(n)= 不A早a… T 2 min Ri(n),and rmax(m)=maxri(m). Theorem 2:Under power assignment A(C)and the hybrid Combining the contribution from X protocol model,for any active link (Yi,YRx(),the interfer- 22+2 ence at YRx()from the primary network is upper-bounded by Ips()≤ PRin(n). C3PRmin(n)-7,for some constant c3. (y-2)△g Proaf:Denote byIp()=∑jET(P/IXi-YRx())） the interference at YRx()from the primary network.Pick We should also take into account the interference between as the interfering primary transmitter closest to YRx().From secondary links.Power assignment scheme A is well designed Corollary 1,the distance between any primary transmitter and so that it not only restricts the interference from the secondaryThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 IEEE/ACM TRANSACTIONS ON NETWORKING Fig. 3. Analyzing the interference. Left plot shows an example for ￾ , and right plot for . denote the interference at receiver from the secondary network, and be the area element because where is the position vector of and for Since , we have C. Interference at Secondary Nodes Now we focus on the interference at secondary nodes. The main challenge is to bound the uncontrollable interference from the primary network. Let , , and . Theorem 2: Under power assignment and the hybrid protocol model, for any active link , the interfer￾ence at from the primary network is upper-bounded by , for some constant . Proof: Denote by the interference at from the primary network. Pick as the interfering primary transmitter closest to . From Corollary 1, the distance between any primary transmitter and should be larger than ; the distance between any two primary transmitters is larger than . Now consider the case that . (Note that if will be smaller, and the upper bound still holds.) Then, all is at least away from except . First, consider the interference contributed by Next, consider the interference from some other primary trans￾mitter . Let , as shown in Fig. 3, then where is the area of disk where because To sum up, let since are disjoint Combining the contribution from We should also take into account the interference between secondary links. Power assignment scheme is well designed so that it not only restricts the interference from the secondary