FRACTIONAL SPATIAL REUSE PRECODING FOR MIMO DOWNLINK NETWORKS Ahmed Medra and Tmothy N. Davdson Department of Electrcal and Computer Engneerng McMaster Unversty, Hamlton, Ontaro, Canada ABSTRACT A lnear precodng scheme s developed for unbounded MIMO downlnk networks wth quas-statc channels that have a hexagonal cell archtecture. In the scheme developed heren, the equvalent channel model s structured to be decomposable, and the lnear precoders at the base statons are desgned to be decomposable as well. The proposed scheme s based the prncples of fractonal spatal reuse precodng. Spatal reuse precodng (SRP) s a precodng scheme that explots the fact that nterferng sources that employ the same structured precoder arrve n the same subspace, regardless of the partcular channel matrces between the nterferng sources and the recever. The proposed scheme s fractonal n the sense that each cell s parttoned and dfferent precoders, wth dfferent power levels, are assgned n each partton. The proposed fractonal SRP scheme enables the elmnaton of the domnant sources of nterference wthout requrng cooperaton between base statons. Index Terms nterference algnment, spatal reuse, ter, cellular network, Kronecker product, decomposable, MIMO IBC. 1. INTRODUCTION One of the fundamental aspects of wreless communcaton networks s nterference management. The smplest approach to nterference management s to avod nterference by transmttng sgnals orthogonally n tme (TDMA) or frequency (FDMA). More sophstcated approaches that actvely manage the nterference offer the potental for hgher data rates [1, 2. One such approach s nterference algnment (IA) [3 5, whch has been shown to acheve more degrees of freedom (DoF) than those that can be acheved usng nterference avodance. One typcal assumpton n the desgn of IA schemes s the presence of global and perfect channel state nformaton (CSI). In small networks, the gans from IA may outwegh the cost of provdng ths CSI, but as the network sze ncreases, the amount of CSI to be communcated ncreases rapdly, whch can result n dmnshng returns n terms of achevable rates. Another related ssue s the need for a central processng unt for the lnear precoder desgn, and the correspondng back-haul requrement. Thus, t would be desrable f each base staton could desgn ts own lnear precoder wthout cooperaton wth other base statons n the network and usng only local feedback. Another typcal assumpton n the desgn of IA schemes s that of a fully connected network. In networks wth a small number of cells, the recevers are often presumed to be close enough to all the transmtters for the nterference to be deemed to be sgnfcant. In that scenaro, examnng the DoF of the network generates consderable nsght. However, for larger networks, at moderate SNRs we can often neglect the power of nterference from dstant transmtters, and practcal precodng schemes (and CSI feedback schemes) Ths work was supported n part by NSERC. Fg. 1. Parttonng of a cell for such SNRs ought to take advantage of ths partal connectvty; see also [6, 7. Wth those perspectves n mnd, the goal of ths paper s to develop a lnear precodng scheme for large unbounded networks that provdes mproved performance over exstng schemes. We seek to desgn ths scheme wthout requrng base statons cooperaton and usng only local feedback. To acheve these goals, we develop a class of lnear precoders that we call fractonal spatal reuse precoders. Spatal Reuse Precodng (SRP) [8 descrbes a network precodng scheme that can be desgned so that the sgnals from the domnant nterferng sources at each recever algn n a reduced dmensonal subspace. Ths enables the recever to elmnate the nterference usng a smple proecton operaton. Ths IA s desgned to be acheved regardless of the exact values of the channel matrces between the nterferng sources and the user experencng that nterference. The proposed scheme s fractonal n the sense that each cell s parttoned as llustrated n Fgure 1, and dfferent precoders are used n each partton. Ths s analogous to the noton of fractonal frequency reuse [9, 10. The parttonng of the cells enables each base staton to assgn hgher power levels to the cell edge users, and also enables dense reuse patterns of the precoders used n the central parttons of each cell. The proposed fractonal SRP scheme s desgned to elmnate the domnant sources of nterference so that rates hgher than those of some exstng schemes can be acheved, and s desgned to do so usng only local feedback. 2. SYSTEM MODEL The G-cell MIMO nterference broadcast channel (IBC) conssts of G transmtters or base statons (BSs), each of whch has M transmt antennas and communcates to K users, where each user has N receve antennas. The kth user n the th cell, user (, k), receves one data stream, and the receved sgnal at that user can be modelled as ỹ,k = G =1,k,k x + ñ,k, where ỹ,k C N, C N M s the channel matrx between BS and user (, k), x s the transmtted sgnal from BS and s subect to the average power constrant E [ x 2 P, and ñ,k represents the addtve nose. The channel matrces,k are assumed to be full rank, but are otherwse unre-
strcted. The sgnalng schemes that we consder are based on blocks of T c channel uses, over whch the channels are assumed to be constant. Defnng y,k = [ ỹ,k [1 T, ỹ,k [2 T,..., ỹ,k [T c T T as the concatenaton of sgnals receved over a block, and defnng x and n,k analogously, the receved sgnal over a block can be wrtten as y,k = H,k x + n,k, (1) Fg. 2. Cell-arrangement n a lnear model where H,k s a block dagonal matrx, wth dagonal blocks whch can be expressed as: H,k = I Tc,k,,k, (2) where denotes the Kronecker product and I Tc s the dentty matrx of sze T c. In ths paper, we consder lnear precodng schemes n whch the sgnal transmtted from BS takes the form x = F s = k f k s k, (3) where f k s the transmt beamformer for user (, k), and s k s the data symbol for that user. Further, we assume that each cell s dvded nto a number of parttons γ as shown n Fgure 1. We wll take a factorzed approach to the desgn of the beamformers f k s k that depends on the cell ndex, the user ndex k, and the ndex of the partton n cell n whch user k les, whch we wll denote by l(k). The beamformers take the form f k = p l(k) v k,l(k), (4) where v k,l(k) s scaled so that v k,l(k) has unt norm. In ths factorzed form, p l(k) denotes the power allocated to the users n partton l(k), s a matrx desgned so that recevers n partton l(k) can elmnate the domnant sources of nterference due to transmssons to users n other cells and due to transmsson to users n the same cell, but n other parttons, and v k,l(k) s desgned to elmnate the ntra-partton nterference wthn partton l(k). To model the nput to the decoder, we let w,k C N s the unt norm receve beamformer used at user (, k), we let P,l denote the set of ndces, k, of users n partton l of cell. That sgnal can be wrtten as ŷ,k = w,k H,k + w,k H,k + w,k H,k + w,k H,k v k,l(k)s k µ P,l(k),µ k vµ,l(k) sµ m l(k) Φm µ P,m v µ,m sµ µ f µ sµ + ñ,k, (5) Here, the frst term represents the desred sgnal, the second term represents the nterference term due to transmssons to users n the same partton, the thrd term represents the nterference term due to transmssons to users n the same cell, but dfferent parttons, the fourth term represents the nterference term due to transmssons to users n other cells, and ñ,k = w,k n,k s the ectve nose. One of the goals of out approach to the desgn of f k,l(k) s to enable the recevers to elmnate the nter-cell (and nter-partton) nterference, wthout the need for nter-cell CSI at the BSs. To enable that, the desgn s performed sequentally. Frst, each BS constructs the matrces Φ l so that the domnant sources of nter-partton nterference and nter-cell can be elmnated by the recevers. The constructon of these matrces wthout CSI s the key contrbuton of ths paper. Next, each user desgns ts lnear receve beamformer w,k to elmnate the nter-partton nterference and nter-cell nterference. (Varatons on the choce of w,k are dscussed n Secton 3.4.) Wth Φ l and w,k desgned n ths way, the operaton of each partton resembles that of an solated sngle-cell downlnk wth ectve channels H,k = w,k H,k. (6) Therefore, each recever feeds back H,k to ts servng BS and that BS desgns the transmt beamformng vectors v k,l(k) as f t were servng a sngle-cell downlnk; e.g., zero-forcng beamformng [11, a qualty of servce desgn [12, 13 or one of many other choces. To develop the proposed SRP scheme, we wll frst consder an solated 3-cell network wth lnear arrangement of cells. We wll consder a lnear IA scheme for ths network that does not requre BS cooperaton, and nvolves only a modest number of channel extensons. Then we present the applcaton of the underlyng concept of fractonal spatal reuse precodng n an unbounded network wth hexagonal arrangement of cells. 3. SPATIAL REUSE PRECODING In the proposed sgnallng schemes, the matrces Φ l are chosen to be the Kronecker product of two matrces. That s, Φ l = T l,1 T l,2, (7) where the matrces T l,1 C Tc β and T l,2 C M M are randomly and ndependently generated from a contnuous dstrbuton and T c > β. By constructon, the generc rank of Φ l s βm. In the conceptual development n ths secton we wll use abstract partallyconnected models for the networks. In those models weak connectons are modelled as beng absent. 3.1. A Motvatng Example We begn wth a smple example based on the 3-cell lnear downlnk network llustrated n Fgure 2. For the ease of exposton, let us consder a case n whch each cell conssts only of one partton,.e., γ = 1. For that case, we can consder beamformers of the form f k = pφ v k. If the power transmtted by each BS s controlled so that users that are close to ther BSs do not suffer from sgnfcant nterference, then the abstract partally-connected network model s rather sparse, wth only cell-edge users sufferng sgnfcant nterference from one nterferng source (users n the brown area n Fgure 2). One way to mprove the performance of these users s to structure the transmssons n such a way that each user can cancel one source of nterference at ts sde. As one example, the precoders desgned for the solated 2-cell case n [14, 15 provde ths structure. Now, let us consder the case n whch each BS ncreases ts power (n order to better servce ts assgned users) to the pont that
each user receves non-neglgble nterference from all ts neghborng cells. In ths case, users n cell 1 and cell 3 wll suffer sgnfcant nterference from BS 2, and those n cell 2 wll suffer sgnfcant nterference from both BS 1 and BS 3. As such, t appears that although applyng sgnalng technques from the solated 2-cell model may be ectve strateges for users n cell 1 and 3, they mght not be ectve for those n cell 2. Indeed, for users n cell 2 we observe that f each BS uses beamformers of the form f k = pφ v k wth Φ constructed accordng to (7) wth T c = β + 1, and the vectors {v k } k beng lnearly ndependent, then the subspace spanned by the nterference at user (2, k) s the column span of Z 2,k = [ H 2,k 1 Φ 1 H 2,k 3 Φ 3 = [ T 1 1 2,k 1 T 2 1 T 1 3 (8a) 2,k 3 T 2 3. (8b) Unfortunately, the nterference matrx Z 2,k s genercally full rank and the recever cannot desgn a receve beamformer w 2,k that can elmnate the nterferng sgnals. However, f BSs 1 and 3 use the same precoder, Φ odd, and f Φ odd = T 1 o T 2 o s desgned accordng to (7), then Z 2,k takes the form Z 2,k = [ T 1 o Ĥ2,k 1 T 1 o Ĥ2,k 3, (9) where Ĥ2,k 2,k 1 = 1 T2 o and Ĥ2,k 2,k 3 = 3 T2 o. Usng the same precoder Φ odd at cell 1 and 3 algns the subspaces spanned by the sgnals arrvng from BSs 1 and 3, regardless of the partcular channel matrces 1 and 3. To show that ths results n Z,k beng rank de- 2,k 2,k fcent, we begn wth the fact that T 1 o C Tc β. If T c = β + 1, then T 1 o s rank defcent wth null space of dmenson 1, and the recever can desgn a vector q to le n ts null space;.e., q T 1 o = 0. Now, the recever desgns ts receve beamformer w 2,k as w 2,k = q u, where u C M s a degree of desgn freedom at the recever, and w 2,k Z 2,k = w 2,k [ T 1 o Ĥ2,k 1 T 1 o Ĥ2,k 3 = [ q T 1 o u Ĥ 2,k 1 q T 1 o u Ĥ 2,k 3 = 0. (10) Thus, Z 2,k s rank defcent, wth null space of dmenson at least one. Accordngly any user n cell 2 can elmnate the nter-cell nterference at ts sde, wthout any cooperaton between the BSs. Users n cell 1 and 3 can do the same. The last step n the lnear precoder desgn s the desgn of the transmt beamformers v k,l(k). Each recever feeds back H,k = w,k H,k Φ to ts servng BS and that BS desgns the matrces v k,l(k) to elmnate the ntra-cell nterference. Ths s the only component of the desgn that uses feedback, and that feedback s only local. 3.2. Defntons The above example conures a noton of Spatal Reuse Precodng (SRP), a precodng scheme desgned to explot the fact that the sgnals transmtted by nterferng sources that use the same precoder algn together at the unntended recevers [8. The system s desgned so that that algnment s n a reduced dmensonal subspace, and hence nter-cell nterference can be removed wthout requrng the knowledge of the nter-cell channels at the nterferng BSs. In a complementary way, we can defne the noton of Spatal Reuse Factor (SRF) as the rate at whch the same precoder s used n the network. We wll show below that the structured precodng scheme presented n (7) can be modfed n such a way to allow fractonal SRP n unbounded downlnk networks that are hexagonal n structure. Fg. 3. A spatal reuse precodng scheme for a hexagonal cell arrangement 3.3. Unbounded hexagonal network Ths secton proposes the man contrbuton of the paper: fractonal SRP for the hexagonal arrangement of cells llustrated n Fgure 3, n whch, each cell s dvded nto 3 parttons. As s mplct n our earler dscusson, the key aspect of the proposed approach s the constructon of the proecton matrces. For the hexagonal network n Fgure 3 we construct nne such matrces, one for each partton n three cells denoted A, B, and C. These precoders are then re-used n the network accordng to the pattern n Fgure 3, whch has an SRF of 3. We begn wth the constructon of the precoders for the outer two parttons, Φ 3 = T 3,1 T 3,2, (11) Φ 2 A = T md T 2 A, Φ 2 B = T md T 2 B, Φ 2 C = T md T 2 C, (12) where T 3,m C Tc β, T md C Tc β and T 2 A,B,C C M M are randomly and ndependently generated matrces from a contnuous dstrbuton and T c = 3β + 1. Fnally, we construct the precoders for the frst partton as follows Φ 1 A = T 1 A T 2 center, Φ 1 B = T 1 B T 2 center, Φ 1 C = T 1 C T 2 center, (13) where T 2 center C M M s a randomly generated matrx, whle T 1 A,B,C C Tc β are desgned accordng to T 1 A = T 3,1 B + T3,1 C, T1 B = T 3,1 A + T3,1 C, T1 C = T 3,1 A + T3,1 B. (14) Here we clam that the above 3-partton Kronecker-structured SRP scheme enables each user to elmnate the domnant sources of nterference. Due to space lmtatons, we wll focus on the nterference experenced by a user at the cell edge n cell A,.e., at the edge of the thrd partton n cell A. The proposed precodng scheme helps that user elmnate the nterferng sgnals due to transmsson to the other parttons n cell A and due to transmsson to the second and thrd parttons n any cell ndexed as B and C. To verfy that clam, we examne the nterference matrx for that user, Z = [ H A Φ1 A H A Φ2 A H B Φ2 B H B Φ3 B H = [ T 1 A T 3,1 B A T2 center T md B T3,2 B T md A T2 A T md C T2 C T 3,1 C C Φ2 C H C B T2 B C T3,2 C Φ3 C. (15) Carefully examnng the structure of Z, we fnd that we can desgn a vector q n the null space of Q = [ T md T 3,1 B T 3,1 C C (3β+1) 3β,.e., q Q = 0. By constructon, q T 1 A = 0, as well. Therefore, the recever can desgn a receve beamformer w = q u that les n the null space of Z, where the recever s free to
Table 1. Break ponts n pece-wse lnear path loss model Dstance α loss 200m 2 500m 3 2km 4 10km 5 choose u C M. Although the nterferng sgnals due to transmssons to users n the frst parttons n cells B and C, and the nterference due to transmssons to thrd parttons n any other cell ndexed A, cannot be elmnated, the power assgned to the frst partton s lower than the power assgned for the other parttons, because users n the frst partton are close to the base staton and do not suffer sgnfcant nterference except at hgh SNRs. Furthermore, the archtecture of the SRP scheme assgns the same precoder to cells that are one cell apart from each other. Ths reduces the nterferng sgnal power experenced by a cell edge user n cell A, due to transmssons to the users n the thrd partton of other cells ndexed by A. 3.4. Varatons on the theme The nsght that drove the development of the proposed scheme was based on constructng the proecton matrces so that the recevers can elmnate nter-cell/partton nterference. However, the recever s not compelled to completely elmnate that nterference, and may choose to employ an alternate nterference mtgaton technque, such as the maxmum SINR receve beamformer, e.g., [16, w,k (Q,k ) 1 H,k v,l(k). k (16) In the proposed system, the ntra-partton beamformng vectors v l are desgned after the ectve channels are fed back to the assgned BSs and hence are not avalable when w,k s desgned. Although an teratve desgn scheme along the lnes of [16 can be envsoned, our smulatons suggest that a substantal performance gans can be obtaned by determnng w,k n (16) as f all the ntra-partton beamformng matrces n the network are dentty matrces. 4. SIMULATION RESULTS We evaluate the performance of the proposed scheme n the case of a network wth the hexagonal arrangement of cells shown n Fgure 3 and a cell radus of 500m. The ect of the dstance between any transmtter and any recever s captured by a pece-wse lnear path loss model [17, 18, where the path loss exponent α loss vares wth dstance accordng to lnear nterpolaton between ponts n Table 1. The BSs and termnals each have four antennas. We wll compare the performance of the proposed scheme (prop), aganst schemes based on desgns for solated sngle cell [2, 2-cell [14 and 3-cell [19 networks. In all of the consdered networks, there s no cooperaton between BSs and only local feedback s employed. In each cell, K = 12 users are served, and the nterpartton nterference s elmnated by havng the recevers feed back ther ectve channels, and then choosng v k,l(k) to be the approprate column of the zero-forcng beamformng matrx [11. Furthermore, and for all schemes, we assgn dfferent power levels for each partton n the cell. In partcular, the powers for the frst and second parttons are 20dB and 13dB below that assgned for the thrd partton, respectvely. In the proposed fractonal SRP scheme, the matrces Φ l are desgned wth β = 1 and the SRP pattern n Fgure 3. Ths results Fg. 4. Achevable rates of varous users n a hexagonal cellular network under four dfferent sgnallng schemes. n a block length of T c = 4. Each recever employs the Max-SINR receve beamformer n (16). The scheme based on nsght from the solated sngle-cell case (1-cell) gnores (nter-cell) nterference. The matrces Φ l are random matrces of sze MT c,1-cell K, and hence a block length T c,1-cell = 3 s chosen to enable K = 12 users to be served n each cell. Here, there s no spatal reuse; each BS (randomly) chooses ts Φ matrx ndvdually. Snce the 1-cell desgn gnores the ntercell nterference, the receve beamformer w,k s chosen to be the matched flter,.e., w,k s algned wth the left sngular vector of that corresponds to the largest sngular value. In the schemes based on the solated 2-cell case (2-cell) and 3- cell case (3-cell), the matrces Φ l at each BS are chosen usng the subspace IA technque n [14 and [19 respectvely. For K = 12 users per cell, ths results n block lengths T c,2-cell = 4 and T c,3-cell = 7, respectvely. In an solated 2-cell network, ths choce enables each recever to proect out the nterference that t receves from the other BS. To extend that noton to the case of an unbounded network, each recever chooses ts receve beamformer w,k to proect out the H,k domnant nterference source,.e., w,k N (H,k Φl ), where and l ndex of the domnant nterferng sgnal, and N ( ) denotes the null-space. Smlar concepts apply to the 3-cell case, where each recever proects out the two domnant sources of nterference. In Fgure 4, we compare the achevable rates of three users n the network under the four schemes descrbed above. The frst user (Ucenter) s located close to ts servng BS, the second user (Umd) s located half way to the cell edge, and the thrd user (Uedge) s located at the cell edge (at the center of a face of a hexagon). Fgure 4 shows that at hgh SNRs the proposed fractonal SRP scheme has a sgnfcant mpact on the rates that can be acheved by users, especally, the cell edge users. In partcular, for a cell edge user, t provdes 122% and 220% ncreases over the achevable rates of the schemes based on the 2-cell and 3-cell schemes, respectvely, and more than a 10 fold ncrease over the 1-cell scheme (whch does not manage nterference). Moreover, the performance of the cell-edge user n the proposed scheme s better than the performance of a user n the mddle of the cell usng the other schemes.
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