Performance Analysis of SDMA with Inter-tier Interference Nulling in HetNets

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1 Performance Anaysis of SDMA with Inter-tier Interference Nuing in HetNets Yamuna Dhungana and Chintha Teambura Department of Eectrica and Computer Engineering University of Aberta, Edmonton, Aberta T6G H9, Canada Abstract The downink performance of two-tier macro/pico muti-antenna ceuar heterogeneous networks HetNets empoying space division mutipe access SDMA technique is anayzed in this paper. The number of users simutaneousy served with SDMA by each BS in a resource bock depends on user distribution, unike previous studies which assume the number to be any arbitrary vaue. By expoiting the feasibiity of depoying arger number of antennas at macro BS, we propose to utiize the excess spatia degrees of freedom for interference nuing to pico users from their corresponding nearest dominant macro BSs. Biased-nearest-distance based user association scheme is proposed as those introduced in previous studies are unsuitabe for anayzing the proposed muti-antenna scheme. Coverage probabiity and average data rate of a typica user are then evauated. Our resuts demonstrate that the proposed interference nuing scheme has strong potentia to improve performance. However, the system parameters such as association bias, and number of dedicated antennas at each macro BS for serving its own users must be carefuy tuned. I. INTRODUCTION Network densification and muti-antenna techniques are we-known for their tremendous potentia to increase spectra efficiency of wireess systems. The cost effective approach to network densification is co-channe depoyment of a diverse set of ow power base-stations BSs within the areas covered by macro ceuar infrastructure, thus forming a heterogeneous network HetNet []. BSs equipped with mutipe antennas can utiize the additiona spatia degrees of freedom for mutipexing, signa power boosting through beamforming, interference suppression or a combination of these. Whie muti-antenna techniques in genera have been very we studied, attempts to study their coexistence with HetNets has started ony recenty [2] [5], and choosing the right technique is not quite understood yet. In this paper, we deveop an anaytica framework to evauate the performance of space division mutipe access SDMA in ceuar HetNets. In [3], the performance of zeroforcing ZF precoding based cosed-oop SDMA technique is compared against singe-user-beamforming SU-BF and singe-antenna communication for muti-tier HetNets through stochastic ordering approach. However, each ce of a tier is assumed to be serving the same number of users with SDMA, say, and it can be any arbitrary integer in the interva[,k i ], where K i is the number of antennas in a BS of ith tier. This assumption is not suitabe for ceuar networks since the number of users in a ce, which depends on user distribution, is different from another ce, in genera. Simiar anaysis with emphasis on user association rues is done in [4]. An open-oop SDMA with each antenna serving an independent data stream to its user is anayzed in [5] for two-tier ceuar HetNets under the imiting requirement that the number of users in each ce to be at east equa to the number of antennas. In this paper, rather than fixing the number of users served with SDMA to an arbitrary vaue, we ony set the imit on maximum number of users served, say max. If the tota number of users in a ce is beow the imit, a the users are served; otherwise ony max randomy chosen users are served. One of the key chaenges in ceuar HetNets is inter-ce interference coordination ICIC. Due to the huge disparities in transmit power between macro and sma-ce nodes such as picos and femtos, ICIC between macro and pico/femto tier is very important as the performance of the sma-ce users on the ce edge coud be severey degraded, otherwise. Whie there can be a number of approaches to ICIC such as resource aocation and user scheduing [6], [7], we anayze precoding based interference nuing method in this paper. Due to the physica size of macro BSs, it is more feasibe to have arger number of antennas at macro BS than at ow-power BSs. Thus, the idea is to utiize the extra degrees of freedom at macro BSs to suppress the interference from macro tier to sma-ce users through nuing. Whie SU-BF with interference nuing in singe-tier ceuar networks is studied in [8], [9], to the best of our knowedge, this is the first work to study user-distribution dependent SDMA scheme with interference nuing in ceuar HetNets. The user association rue proposed in [4] is based on the assumption that a deterministic fixed number of users are served with SDMA in each ce. Hence, it does not appy to our proposed SDMA scheme, and a different association rue is introduced in this paper. Notations: We now present integer-partition notations used in this paper. The set of a possibe partitions of a positive integer n is represented by Ω n. For exampe: Ω 3 = {{3},{2, },{,, }}, where each partition represents a way to express n = 3 as a sum of positive integers. The number of a possibe partitions of n is represented by Pn, thus, P3 = 3. For the ith partition p n i, ωn i represents the number of eements, µ n ij is the number of positive integer j {,2,...,n} and a n ik is the kth eement k {,2,...,ωn i }.

2 We have, n j= jµn ij = n and n j= µn ij = ωn i. Exampe: for p 3 2 = {2,} in Ω 3, ω2 3 = 2, µ 3 2 =, µ 3 22 =, µ 3 23 = 0, a 3 2 = 2, a3 22 =. II. SYSTEM MODE We consider the downink of a two-tier muti-antenna Het- Net comprising macro and pico BSs spatiay distributed in R 2 pane as independent homogeneous PPPs Φ m with density λ m and Φ p with density λ p, respectivey. The macro BSs are equipped with K m transmit antennas and pico BSs with K p antennas. Simiary, users are assumed to be distributed according to an independent PPP Φ u with density λ u, and each has a singe receive antenna. The two network tiers share the same spectrum with universa frequency reuse. The transmission scheme is SDMA with ZF precoding appied at each BS to serve mutipe users simutaneousy in each resource bock RB. We assume ony one RB per time sot. As BSs and users are independenty distributed in R 2 pane, the number of users in different ces are different. Thus, in our proposed SDMA scheme, a typica active macro ce with N m users serves M m = minn m, M max users simutaneousy in a given time sot, where M max is the maximum number of users it can serve. When N m > M max, the BS choses M max users for service randomy, ese, a N m users are served. Simiary, M p = minn p, P max users are simutaneousy served by a typica active pico ce in a given time sot, which has N p users and P max is the maximum number it can serve. The macro and pico BSs transmit to each of their users with power P m and P p, respectivey. A. User Association According to the user association rue introduced in [4], a typica user at the origin is associated with the nearest pico BS if P p p τ p Xp α P m m τ m Xm α, otherwise associated with the nearest macro BS, where X m = min x m and x m Φ m X p = min x p are the distances from the origin to the x m Φ p nearest macro and pico BSs, respectivey. If associated with the macro tier, m is the average desired channe gain from the nearest macro BS and τ m is the average interference channe gain from the nearest pico BS. Simiary, p and τ p are the corresponding parameters, if associated with the pico tier. The average gains of the desired and interference channes depend on the number of users served with SDMA. This association rue is thus not suitabe for our proposed SDMA scheme, where the number of users served with SDMA in each ce is a function of number of users in that ce. The number of users, on the other hand, is determined by the association rue. The above rue however can be equivaenty expressed as: a user is associated with the pico tier ony if Pm α α X m Xp X m ρx p, η P p whereρ = Pm P p η α, η = pτ p mτ m. This rue can be perceived as biased nearest distance association, where the biasing is due to the difference in transmit power and average channe gains between macro and pico tier, and for oad baancing as we. We wi investigate the optima vaue of η for average data rate in Section V, which in turn determine the optima ρ. Using the fact thatx m andx p are Rayeigh RVs with mean /2 λ m and /2 λ p, respectivey [0], the probabiity that a typica user at the origin is associated with pico tier is A p = PX m ρx p = λ p λ p +λ m ρ2, 2 and the probabiity that it is associated with macro tier is A m = A p. These tier association probabiities are aso vaid for any randomy seected user. As per the given association scheme, the tota set of users in the network, Φ u can be divided into two disjoint subsets: Φ m u and Φ p u, the set of macro and pico users, respectivey. A m and A p can be interpreted as average number of users beonging to Φ m u and Φp u, respectivey. As we are interested in the number of users in a typica ce, rather than the actua ocations of users, Φ m u and Φp u can be equivaenty modeed as independent PPPs with density A m λ m and A p λ p respectivey. Since each macro user is aways associated with the nearest macro BS and each picouser with the nearest pico BS, the network can be viewed as a superposition of two independent Voronoi tesseations of macro and pico tier. The distribution of the number of users associated with a typica macro and pico BS is derived next. emma. et the number of users in a randomy chosen macro and pico ce be denoted by U m and U p, respectivey. Their probabiity mass functions PMFs are given by PU = n = Γ3.5+nA λ u /λ n Γ3.5n!A λ u /λ +3.5 n+3.5,n 0, {m,p}, 3 Proof: The proof is simiar to that of emma 2 in [7]. A BS without any user associated does not transmit at a and is inactive. Given in the foowing coroary are the PMFs of the number of users in a typica ce of macro and pico tier, under the condition that it is active. Coroary. The PMFs of the number of users in a randomy chosen active ce of macro and pico tier are given by PN = n = PU = nn p, {m,p}, 4 where p m and p p are the probabiities that a typica BS of macro and pico tier, respectivey, is active, and are given by p = PU = 0 = +3.5 A 3.5 λ u, λ {m,p}. 5 et the sets of active macro and active pico BSs be denoted by Ψ m and Ψ p respectivey.ψ m and Ψ p are thinned versions of the origina PPPs Φ m and Φ p, respectivey, and hence are independent PPPs with densitiesp m λ m andp p λ p, respectivey.

3 By using coroary, the PMFs of the number of users simutaneousy served by a typica active macro/pico BS in a given time sot for max > can be obtained as PN = n, n < max PM = n = max PN = k, n = max, k= For max =, PM = =, {m,p}. B. Interference Nuing {m,p}. 6 We assume that K m is typicay much arger than K p. The additiona spatia degrees of freedom that macro BSs have can be utiized to suppress the strong interference they impose on pico users through nuing strategy. To accompish this, we propose that each served pico user requests its nearest active macro BS to perform interference nuing. As interference nuing costs macro BSs their avaiabe degrees of freedom for their own users, we assume that each macro BS can hande at most K m requests ony, which ensures that a macro BS has at east M max antennas dedicated for serving its own users. Hence, if Q m requests are received by a typica active macro BS, it wi perform interference nuing to B = minq m,k m pico users. ForQ m > K m, the BS wi randomy choose K m pico users. Hence, not a interference nuing requests are satisfied. The number of interference nuing requests Q m received by a typica active macro BS is equa to the number of served pico users within a typica Voronoi ce Υ of the tesseation formed by Ψ m. Athough, the number of pico users served by a typica active pico BS cannot exceed p max,q m is unbounded because the number of active pico BSs within Υ is Poisson distributed with mean p p λ p /p m λ m. In order to derive the PMF of Q m, we first derive E[M p ] = A p ϑ p λ u /p p λ p, where ϑ p = p maxp p λ p A p λ u Γ3.5 p max k= [ Γ3.5+n n! A pλ u /λ p n p max k A p λ u /λ p +3.5 n+3.5 ], 7 then approximate the set of pico users requesting interference coordination Ψ p u as a PPP with density A pϑ p λ u so that E[Q m ] = A p ϑ p λ u /p m λ m. By using this approximation, the PMF of Q m is derived as foows. emma 2. The PMF of the number of interference nuing requests received by a typica active macro BS is given by n Apϑ pλ u Γ3.5+n PQ m = n = n+3.5,n 0. 8 Apϑ Γ3.5n! pλ u +3.5 C. Channe Mode and Precoding Matrices For independent unit-mean Rayeigh mutipath fading between any transmit-receive antenna pair and standard power aw path-oss with exponent α, the received signa z m at a typica user u ocated at the origin if u Φ m u is given by z m = P m D α 2 m h b m,w bm s bm + Pq x q α 2 g W xq, x q s xq +n m, 9 q {m,p} x q Ψ q\b m where b m is the serving BS at a distance D m, which is serving M m other users simutaneousy, h bm, CN0 Km, I Km and g xq CN0 Kq, I Kq are the desired and interference channe vectors from the tagged BS b m and the interfering BS at x q, respectivey, n m CN0,σ 2 is the additive white Gaussian noise AWGN, s bm = [s bm,i] i M m + C M m + is the signa vector transmitted from b m to its M m + served users with the symbo s b m, intended for u, and W bm = [w bm,i] i M m + C Km M m + is the corresponding precoding matrix. et the channe vectors from the tagged BS b m to its M m users other than u be represented by [ h bm,i] 2 i M m +, and the interference channe vector from the tagged BS to B = minq m,k m pico users chosen for interference nuing by F = [ f i ] i B. Under perfect channe state information CSI assumption, ZF precoding vectors W bm = [w bm,i] i M m + are designed such that h b m,jw bm,j 2 is maximized for each j =,2,...,M m +, whie satisfying the orthogonaity conditions h b w m,j b m,i = 0 for i j and f i w bm,j = 0, i =,2,...,B, j =,2,...,M m +. This can be achieved by choosing w bm,i in the direction of the projection of h bm,i on Nu [ h bm,j] j M m +,j i,[f i ] i B. The nuspace is K m M m B dimensiona. Under Rayeigh fading, the desired channe power β bm = h b w m, b m, 2 Gamma m,, where m = K m M m B []. Given that an interfering macro BS at x m is servingm m users simutaneousy, the interference channe powerζ xm = g x W m, x m 2 GammaM m, [3]. One possibe choice of W bm = [w bm,i] i M m + is the normaized coumns of the pseudo inverse of H b m, i.e., W bm = H bm H b m H bm, where H bm = [ h bm,i] i M m + C Km M m +, h bm,i = I Km FF F F h bm,i being the projection of h bm,i on the nuspace of F = [ f i ] i B. Simiary, the received signa z p at u when u Φ p u is z p = P p D α 2 p h b p,w bp s bp +ξ + Pq x q α 2 g W xq, x q s xq +n p, 0 where q {m,p} ξ = x q Ψ q\{v m,b p} { 0, if u χ Pm V α 2 m g v m, W v m s vm, if u / χ, b p is the serving BS at a distance D p, which is serving M p other users simutaneousy, n p CN0,σ 2 is AWGN, v m is the nearest active macro BS to u at a distance V m, χ is the set of pico users to which interference from their corresponding nearest active macro BSs are nued. The ZF precoding vectors W bp = [w bp,i] i M p + are given by

4 the normaized coumns of H bp H b p H bm, where H bp = [h bp,i] i M p + C Kp M p + is the channe matrix from the tagged BS b p to its M p + served pico users. The desired channe power β bp = h b p, w b p, 2 Gamma p,, where p = K m M p and the interference channe power ζ xp = g x p, W x p 2 GammaM p,. The signa-to-interference-and-noise ratio SINR of a typica user u when it beongs to Φ u can now be expressed as SINR = P β b D α I b,m +I b,p +σ2, {m,p}, 2 where I bm,m = P m ζ xm x m α, I bm,p = P p ζ xp x p α, x m Ψ m\b m x p Ψ p P m ζ xm x m α if u χ x I bp,m = m Ψ m\v m P m ζ xm x m α if u / χ, x m Ψ m I bp,p = P p ζ xp x p α. 3 x p Ψ p\b p To compute the desired performance metrics in the foowing sections, we require Pu χ which is derived next. emma 3. The probabiity that a typica user u Φ p u requesting interference nuing to its nearest active macro BS gets its request fufied, i.e., ϕ = Pu χ is given by ϕ = K m p m λ m +3.5 A 3.5 pϑ p λ u A p ϑ p λ u p m λ m Apϑ pλ u i Km K m T min Γ3.5+i i Γ3.5 i+3.5. Apϑ i= i! pλ u Proof: et the BS v m receives Q m other requests aong with the one from u. Then, conditioned on Q m, ϕ = if Q m + K m, otherwise, ϕ = K m /Q m +. By using the fact that the conditiona PDF f Y y of the area of a Voronoi ce given that a typica user beong to it is equa to cyf Y y, where f Y y is the unconditiona PDF and c is a constant such that f o Y ydy = [6], the PMF of Q m can be derived as PQ m = n = n + PQ m = n+/e[q m ], n 0. Eqn. 4 then foows easiy. III. COVERAGE PROBABIITY A user is in coverage if its SINR is greater than a predefined threshod γ. The coverage probabiity of a typica user u at the origin is given by Pγ = P m γa m +P p γa p. 5 We first evauate P p γ, the coverage probabiity of a typica pico user as P p γ = p max PM p = kpsinr p > γ u Φ p u,m p = k. }{{} P p M γ k p Theorem. The coverage probabiity of a typica pico user u in interference imited scenario, i.e., σ 2 = 0 is given by P p γ = T γϕ+t 2 γ ϕ, 6 where T γ = PSINR p > γ u Φ p u,u χ and T 2 γ = PSINR p > γ u Φ p u,u / χ are given by λ p max K p ρ p k T γ = 2p m λ m PM p A = k p θ=0 =0 γ! θα+ P c o p m λ m δ q Ξ m q δ,θ,γ+ p pλ µ p θ αq 2Ξp q,,γ oq o= q= ω o Γω o +2 p m λ m Ξ m 0 δ,θ,γ+p pλ p θ 2 Ξ p 0,,γ + p m λ m ρ 2 θ 2 + p p λ p θ 2 ω o +2 dθ, 7 T 2 γ = λ p A p pm λ m δ q q= p max ρ αq 2 Ξm q ω o Γω o + K p k PM p = k =0 γ P! o= c o δ, ρ,γ +p p λ p Ξ pq,,γ µ oq p m λ m ρ 2 Ξ m 0 δ, ρ,γ +p p λ p Ξ p 0,,γ + p m λ m ρ 2 + p p λ p ω o +, 8 where δ = Pm P p, c o =! ω o k= a ok! q= µ oq!, and Ξ q ς,κ,ε = max i= [2F i+q, 2 α 2 +q, α ] i q 2 α q α 2 α PM = i q Proof. The proof is given in the appendix. α +q, ςκα ε, {m,p}, 9 Remark. The number of other users served by the BS which is serving the typica user u Φ u is given by M = minu, max, where U is the number of other users in the Voronoi ce to which the user u beongs, whose PMF is given by PU = n = n+pu = n+/e[u ]. The PMF of M for max > is thus given by PU = n, 0 n < max PM = n = max 2 k= PU = k, n = max, For max =, PM = 0 =, {m,p}. The coverage probabiity of a typica macro user {m,p}. 20 m max P m γ = PM m = kpsinr m > γ u Φ m u,m m }{{ = k } P m M γ k m

5 in interference imited scenario can be simiary derived, where P m M m γ k = λ m A m P c o o= q= K m k n= k n P m = n M m = k p m λ m Ξ m ρ αq 2 q,,γ+p p λ p δ q Ξ p q ω o Γω o + p m λ m Ξ m 0,,γ+ p pλ p =0 γ! δ,ρ,γ µ oq ρ 2 Ξp 0 δ,ρ,γ + p m λ m + p p λ ω o p + ρ 2, 2 P m = n M m = k K m T min PQ m = v, n = k = v=0 PQ m = K m k n, k + n K m k. 22 IV. AVERAGE DATE RATE With adaptive moduation so that the Shannon imit can be achieved and interference treated as noise, the data rate of a typica user is given by R = S W og 2 +SINR u Φ u, 23 {m,p} where S = min max/u +, is the share of resources received by u when u Φ u and W is the tota bandwidth. The average date rate R thus can be computed as foows. The proof is omitted for brevity and wi be provided in a journa version of the paper. R = A m Rm +A p Rp, 24 where R m and R p are average data rates of a typica macro and pico user, receptivey, and are given by R = [ O P n2 0 +y M y max max 2 + P M y kpm = k dy, {m,p}, 25 where O = max λ A λ u max Γ3.5 n= A λ u /λ Γ3.5+n A λ u λ n max n! A λ u λ n. 26 V. NUMERICA RESUTS In this section, we present some numerica anaysis of our anaytica resuts. Uness otherwise stated, we set δ = P m /P p = 00, λ m = BS/Km 2, and W = MHz. In Figure, the average data rate 24 is vaidated via Monte Caro simuations for different system configurations. We can Average Rate bits/sec Simuation Anaytica λ p = 4λ m, η = 0 db 6, 2, 6, 3, 2 2, 4, 8, 4, 2 8, 2, 4, 4, 2 Anaytica λ p = 3λ m, η = 8 db Anaytica λ p = 2λ m, η = 3 db λ u /λ m Fig.. Vaidation of the average user data rate 24 via Monte Caro simuations for different vaues of λ p, η and K m, m max,,k p, p max. Coverage Probabiity Pγ λ u = 6λ m η = 5 db λ u = 0λ m η = 20 db = 7 = 0 = K m = 0 = K m = Threshod γ db Fig. 2. Impact of on coverage probabiity: K m = 4, m max = 4,Kp = 6, p max = 4,λ p = 6λ m,α = 3.5 observe cose match between the anaytica and simuation resuts. The average data rate decrease with increase in user density λ u due to the increase in interference and decrease in user s share of resources. Next, we anayze the impact of interference nuing on coverage probabiity of a typica user in Figure 2, where = K m impies no interference nuing empoyed. We can observe that with propery chosen, the coverage probabiity can be improved with interference nuing. The performance gain is higher for smaer vaues of threshods, which indicates that interference nuing significanty improves the SINRs of poor ce edge pico users. If we compare the curves for two different sets of λ u and η, the performance gain is ower for higher vaues of λ u and η, and it is due to insufficient resources for interference nuing. In Figure 3, the average data rate with and without interference nuing are potted against η for different pico densities. The average rate with an arbitrary and optimum Tmin are potted to iustrate the significance of seection on performance. The optima pair,η is found to be 8,0 db and 6, db for pico density λ p = 4λ m and λ p = 6λ m, respectivey. VI. CONCUSIONS The downink performance of muti-antenna HetNets with SDMA is anayzed, in which the precoding matrix at macro

6 BS aso considers interference nuing to certain pico users. Further, the number of users served with SDMA in each ce is a function of user distribution. Our resuts show that the SINRs of victim pico users those suffering strong interference from macro BS can be significanty improved with the proposed interference nuing scheme. However, since this costs macro BSs their avaiabe degrees of freedom for serving their own users, must be carefuy chosen. The optima choice of is couped with association bias as it determines the number of users to be served and interference coordination requests to be fufied. The optima,η, that maximized the average data rate is investigated in this paper. APPENDIX A. Proof of Theorem PSINR p > γ u Φ p u,m p = k,u χ = r=0 r =ρr K p k =0 s d! ds I bp s s= γrα Pp f Vm D p r rf Dp rdr dr, 27 where Ibp s = Ibp,m s Ibp,p s is the apace transform T of I bp = I bp,m+i bp,p. The PDF of the distance D p from u to the nearest pico BS b p, given that u Φ p u is f Dp r = 2πλ p A p rexp πλ m ρ 2 +λ p r Simiary, the PDF of the distance V m to the nearest active macro BS v m, given that u Φ p u and its distance to the serving pico BS b p is D p = r, is f Vm D p r r = 2πp m λ m r exp πp m λ m r 2 ρ2 r 2, r > ρr. 29 The proofs for these PDFs are omitted for brevity. The T Ibp, s = E[exp si bp,], {m,p} can be derived as [ ] Ibp, s = E Ψ exp sp ζ x x α. 30 x Ψ \b p E ζx Given M, ζ x GammaM,. By performing the expectation over this conditiona distribution, foowed by the probabiity generating functiona of PPP with density p λ, and finay the expectation over the PMF of M, we have [ Ibp, s = exp πp λ 2 Ξ 0,, P ] s, 3 α where is the ower bound on distance to the cosest interferer from u in the tier {m, p}. Thus, m = r and p = r. et ys = e πs, and ts = p m λ m r 2 Ξ m 0,, Pm r α s + p p λ p r 2 Ξ p 0,, Pp r α s Then Ibp s = e πpmλmr2 +ppλpr2 yts. The th derivative of a composite function yts can be evauated as foows, which is a simpified form of Faà di Bruno s formua. P y s ts = c oy ω o o= ts ts q=. t q s s µ oq, 32 Average Rate bits/sec x 0 6 Optimum Arbitrary = K m = 0 λ p = 6λ m λ p = 4λ m * T = 8 min = 4 * T = 6 min η db Fig. 3. Effect of pico ce density λ p on the optima choices of and η: λ u = 6λ m, K m = 2, m max = 4, K p = 4, p max = 4, α = 4. where y k ts ts represents kth derivative of function yts with respect to ts, the summation is over a possibe partitions of integer. The notations for integer-partition are presented in Section I. The fina resut for T γ in 7 is obtained by appying 32 into 27 where d q ds qξ 0,, P s α = P α q Ξ q,, P s α, 33 foowed by changing the order of integration, substituting r r θ, r r, then integrating with respect to r, and finay deconditioning with respect to the PMF of M p. T 2 γ is derived in the same way as T γ. However, it does not require integration over f Vm r as m = ρr in this case, given that D p = r. REFERENCES [] Quacomm Incorporated, TE Advanced: Heterogeneous networks, white paper, Jan. 20. [2] R. W. Heath, J. M. Kountouris, and T. Bai, Modeing heterogeneous network interference with using poisson point processes, IEEE Trans. on Signa Processing., vo. 6, no. 6, pp , Aug [3] H. Dhion and J. Kountouris, M.and Andrews, Downink MIMO hetnets: Modeing, ordering resuts and performance anaysis, IEEE Trans. on Wireess Commun., vo. 2, no. 0, pp , Oct [4] A. Gupta, H. Dhion, S. Vishwanath, and J. Andrews, Downink mutiantenna heterogeneous ceuar network with oad baancing, IEEE Trans. on Commun., vo. 62, no., pp , Nov [5] S. Veeti, K. Kuchi, and R. Ganti, Spatia mutipexing in heterogeneous networks with MMSE receiver, in Proc. IEEE GOBECOM, Austin, TX, Dec. 204, pp [6] S. Singh and J. Andrews, Joint resource partitioning and offoading in heterogeneous ceuar networks, IEEE Trans. Wireess Commun., vo. PP, no. 99, pp. 4, Dec [7] Y. Dhungana and C. Teambura, Muti-channe anaysis of ce range expansion and resource partitioning in two-tier heterogeneous ceuar networks, IEEE Trans. Wireess Commun., 205. [8] J. Zhang and J. G. Andrews, Adaptive spatia interce interference canceation in mutice wireess networks, IEEE J. Se. Areas in Commun., vo. 28, no. 9, pp , Dec [9] C. i, J. Zhang, M. Haenggi, and K. B. etaief, User-centric interce interference nuing for downink sma ce networks, IEEE Trans. on Commun., vo. 63, no. 4, pp , Apr [0] S. N. Chiu, D. Stoyan, W. S. Kenda, and J. Mecke, Stochastic Geometry and its Appications, 3rd ed. John Wiey and Sons, 203. [] N. Jinda, J. G. Andrews, and S. Weber, Rethinking MIMO for wireess networks: inear throughput increases with mutipe receive antennas, in Proc. IEEE ICC, Dresden, Dec. 2009, pp. 6.