LINK ADAPTATION FOR INTERFERENCE ALIGNMENT WITH IMPERFECT CSIT

Transcription

1 2014 IEEE Internatonal Conference on Acoustc, Speech and Sgnal Processng (ICASSP) LINK ADAPTATION FOR INTERFERENCE ALIGNMENT WITH IMPERFECT CSIT Mohsen Rezaee, Mehrdad Tak, Maxme Gullaud Insttute of Telecommuncatons, Venna Unversty of Technology, Venna, Austra e-mal: Electrcal Engneerng Department, Unversty of Isfahan, Isfahan, Iran e-mal: ABSTRACT In ths paper lnk adaptaton for nterference algnment (IA) based on mperfect channel state nformaton (CSI) s nvestgated. We consder a MIMO nterference channel where the transmt and receve spaces are determned by IA. We look at mzng a weghted sum of the average rates provded that a certan set of bt-error-rate and power constrants are satsfed by dynamcally adjustng codng, modulaton and powers. The problem s qute general and ntractable. We resort to some approxmatons and provde smulatons to show the accuracy of the approxmatons n the regmes wth practcal nterest. 1. INTRODUCTION Interference algnment (IA) has receved a lot of attenton due to ts promsng performance mprovement over orthogonal schemes. It has been shown that as long as a set of feasblty condtons are satsfed, IA can acheve the mum achevable DoF n a K-user constant MIMO IC [1]. A few papers have nvestgated the effect of mperfect channel state nformaton at the transmtter (CSIT) on the performance of IA [2 6]. However, an mproved IA-based scheme that accounts for the mperfect CSIT s stll mssng. Even though methods have been proposed to reduce the CSIT requrement [7], we need to contan the effect of the resdual nterference caused by mperfect CSIT and look for methods that can deal wth ths resdual nterference properly. Lnk adaptaton methods can be very helpful to deal wth resdual nterference and explotng the drect channels. Lnk adaptaton mproves the performance of a wreless lnk by adjustment of the rate and/or power at the TX based on the estmated CSI fed back from the RX [8]. Extensve research has been done on lnk adaptaton for pont-to-pont lnks. Dscrete rate lnk adaptaton for practcal systems s ntroduced n [8] by usng adaptve modulaton (AM) and s extended to adaptve modulaton and codng (AMC) n [9]. In ths paper we consder a MIMO IC where the transmt and receve spaces are determned by IA. Consderng practcal modulaton schemes, we use adaptve modulaton, codng and power (AMCP) to mze a weghted sum of the average rates whle havng a sumpower constrant across the users and ensurng that the bt-error-rate (BER) for decodng every stream n the network remans below a certan threshold. Smulaton results are provded to compare the proposed scheme wth the orthogonal transmsson scheme. 1 Ths work was supported by the FP7 projects HIATUS (grant ) and Newcom# Network of Excellence n Wreless Communcatons and by the Austran Scence Fund (FWF) through grant NFN SISE (S106). 1 Notaton: Nonbold letters represent scalar quanttes (talcs denote ran- 2. SYSTEM MODEL Consder an nterference channel n whch K TXs communcate wth ther respectve RXs over a shared medum. For smplcty we assume that every TX s equpped wth N t antennas whle every RX has N r antennas. Data symbols are spatally precoded at the TXs. Let d denote the number of data streams sent by each TX. Our results can be extended to more general confguratons as long as the feasblty condtons for IA [1] are satsfed. The receved sgnal at RX reads y = H V x + H jv jx j + n (1) j n whch H j C Nr N t s the channel matrx between TX j and RX, V j C Nt d and x j C d are the precodng matrx and the data vector of TX j, respectvely. Furthermore, n CN (0, σ 2 I Nr ) s the addtve nose at RX. Assumng E{x x H } = dag( P (1),..., P (d)), = 1,..., K, the covarance matrx of the sgnal transmtted by user s gven by Q = V dag( P (1),..., P (d))v H. We assume that all V are truncated untary matrces, therefore the transmt power for user s tr (Q ) = d P l=1 (l) = P. 3. INTERFERENCE ALIGNMENT WITH IMPERFECT CSIT Assumng that perfect global CSI s avalable at every TX, the precoders V, = 1... K should be desgned to algn the nterference at each RX nto a N r d dmensonal space, n order to acheve d nterference-free dmensons per user. A soluton to the IA problem exsts (see [1] and more recently [10, 11] for feasblty crtera here we wll assume that the dmensons and the consdered channel realzatons are such that the problem s feasble) ff there exst full rank precodng matrces V j, j = 1,..., K and projecton matrces U C Nr d, = 1,..., K such that U H H jv j = 0, j {1,..., K}, j, ) rank (U H H V = d. dom varables), boldface lowercase and uppercase letters ndcate vectors and matrces, respectvely. The trace, conjugate, transpose, Hermtan transpose of a matrx or vector are denoted by tr( ), ( ), ( ) T, ( ) H respectvely. The expectaton operator s represented by E{ }. A dagonal matrx s represented by dag( ). Absolute value of a scalar s denoted by. The mth element of a vector a and the (m, n)th element of a matrx A are denoted by a(m) and A(m, n) respectvely. The complex crcularly symmetrc Gaussan dstrbuton wth mean r and varance σ 2 s denoted by CN (r, σ 2 ). Fnally var(.) shows the varance of ts argument random varable. (2) /14/$ IEEE 6208

2 We focus on the frst condton n (2) snce the second condton s satsfed almost surely when the channel entres are drawn ndependently from a contnuous dstrbuton. It s clear that any truncated untary matrx that has the same column space as V also fulflls (2) and the same argument holds for the RX flters. Ths means that we can assume wthout loss of generalty that the equvalent drect channels,.e., U H H V for = 1,..., K are dagonal. When the CSIT s not perfect at the TXs, the IA flters desgned based on mperfect CSIT are denoted by truncated untary matrces ˆV and Û where Û H Ĥ j ˆVj = 0, j {1,..., K}, j, (3) where the estmated channels are denoted by Ĥj. Smlar to the perfect CSI scenaro, we assume that the flters are such that the drect equvalent channels whch we denote by Ĝ ÛH Ĥ ˆV for = 1,..., K, are dagonal. Usng the desgned precoders, the receved sgnal after projecton by the receve flters can be wrtten as r = ÛH y = G x + G jx j + ÛH n (4) j where G = ÛH H ˆV and G j = ÛH H j ˆVj. Clearly the second term s due to the nterference leakage from other users whch s caused by desgnng the flters based on mperfect CSI. Also the equvalent channel G for each user s not dagonal anymore whch ntroduces nter-stream nterference. To smplfy the notaton, we denote the qth dagonal elements of G and Ĝ by Gq and Ĝq respectvely, for q = 1,..., d. 4. STATISTICAL MODEL OF THE CSIT We assume that the entres of the true channels H j and the channel estmates avalable at the TX sde Ĥj are modeled as complex Gaussan random varables wth zero mean and varance j. We assume the followng CSI mperfecton model, H j = ρ 0 Ĥ j + 1 ρ 2 0Ej (5) where we have assumed that the entres of the perturbaton matrx E j have the same dstrbuton as the true channel and they are ndependent from the entres of Ĥj. Wth ths model the parameter ρ 0 represents the correlaton between the true channel elements and the estmated ones. Denotng the real and magnary parts of H j by X j and Y j respectvely and that of Ĥj by ˆX j and Ŷj respectvely, due to the crcular symmetry, the entres of the real matrces are ndependent from the magnary matrces. Further, we have E{H j(m, n) Ĥj(m, n)} = ρ0ĥj(m, n) E{ H j(m, n) 2 Ĥj(m, n)} = (1 ρ0)2 j + ρ 2 0 Ĥj(m, n) IA WITH AMCP USING IMPERFECT CSIT In the context of lnk adaptaton, transmsson rate and power of the TX s adjusted dependng on the nstantaneous channel condton. There are two dfferent approaches for lnk adaptaton n the lterature namely, contnuous and dscrete lnk adaptaton. In the frst case, capacty achevng codes wth vanshng error probablty are employed and t s assumed that the rate can be adjusted contnuously accordng to the channel condton. In ths case f we denote the equvalent channel gan, the average power of nose (and nterference) and the transmsson power by h eq, δ eq and p respectvely, then the spectral effcency of the lnk s equal to log 2 (1 + ps eq) (bt/sec/hz) (6) where s eq = heq 2 δ eq s the equvalent SINR of the lnk. In the context of dscrete lnk adaptaton, adaptve codng s used along wth adaptve modulaton whch mproves the performance sgnfcantly. In ths method, M + 1 transmsson modes are consdered where every mode m corresponds to a par of modulaton and codng confguraton and fnally provdes a transmsson rate denoted by R m. The rates for dfferent modes are sorted as 0 = R 0 < R 1 <... < R M. Mode 0 represents the case where no nformaton s transmtted. In mode m the BER s approxmated by the followng expresson [12]: p e(ps eq, R m) = A mexp( A mps eq), m = 0,..., M, s eq 0 (7) where A m and A m are constants that are determned by the transmsson mode. Ths model s based on approxmatng the BER curve whch s realzed usng Monte Carlo smulaton over a large range of SNR. The process s repeated for every mode m whch yelds the parameters A m and A m va curve fttng. In transmsson mode m, the mnmum requred SINR to ensure a mum BER of B 0 s denoted by g B0 (m),.e., where g B0 (m) 1 A m ps eq g B0 (m) p e(ps eq, m) B 0 (8) ln( B 0 A m ) for B 0 mn (1/2, A m). If the CSI s not perfect, drect use of mperfect CSI nstead of the true CSI n IA and AMCP wll be hghly sub-optmal. Based on our assumptons, snce the avalable CSI at each TX s a degraded verson of the true CSI, therefore the IA and lnk adaptaton methods should be revsted takng the effect of the resdual nterference nto account. Here we present a new scheme where AMCP s performed usng mperfect CSIT to mprove IA performance Problem Formulaton Accordng to Secton 2, every TX has d data streams. For stream q of user, we denote the average power of nterference and nose whch affects ths stream by δ q. Furthermore, the nstantaneous true SINR and the estmate of the nstantaneous SINR (whch s avalable at the transmtters) are denoted respectvely by γ q = Gq 2 δ q and ˆγ q = Ĝq 2 δ q. The nstantaneous rate and power for stream q of user are denoted by k q = Φ(ˆγ q ) and pq = Ψ(ˆγ q ) respectvely, for 1 K and 1 q d. Based on the avalable CSIT at the TXs (Ĥj,, j), the TXs choose the precoders ˆV j accordng to (3), whle the rate and power are determned by the mappngs Φ( ) and Ψ( ) whch should be desgned such that the average sum-power s at most P 0 and the BER at each RX s less or equal to B 0, whle a weghted sum of the average rates s mzed. The weghts for dfferent users are denoted by ω, = 1,..., K. Wth these defntons, the 6209

3 overall optmzaton problem can be formulated as follows K d ω Φ( ),Ψ( ) =1 q=1 K =1 q=1 Eˆγ q {Φ(ˆγq )} d q Eˆγ {Ψ(ˆγq )} P0 q Eˆγ,γq{pe(Ψ(ˆγq )γq, Φ(ˆγq ))} B0,, q. Solvng the exact optmzaton problem n (9) s ntractable especally snce the optmzaton s performed usng mperfect nformaton about the channels. We dvde the optmzaton problem n (9) nto the followng three steps Effect of Imperfect CSI on IA Here we nvestgate the propertes of the resdual nterference affectng the decodng of every stream when the precoders are desgned usng IA. For smplcty we defne ˆv,q and û,q as the qth column of ˆV and Û respectvely. Therefore from (4) we have r (q) = G q x(q) + I,q + I,q + n q where G q = û H,qH ˆv,q s the qth dagonal element of G, I,q = 1 l d û H,qH ˆv,l x (l) l q s the nterference from other streams of the same user, I,q = j û H,qH j ˆVjx j (9) s the nterference from other users and n q = û H,qn s the equvalent nose for ths stream. If the channels are perfectly known (Ĥj = Hj,, j), then the terms I,q and I,q wll be zero due to the algnment equatons. In the followng, we derve the statstcal propertes of G q (gven the nformaton about Ĝ q ) whch can be used to derve the statstcs of the true SINR. Then we derve the power of each nterference term dependng on the level of mperfectness of the channels controlled by ρ Statstcs of the drect channel power gan It can be shown that G q 2 and Ĝq 2 are approxmately dstrbuted as exponental random varables (ths holds exactly n the snglestream case) [13]. Henceforth we work under the assumpton that G q 2 and Ĝq 2 are exponentally dstrbuted wth parameter 1. Note that the value of Ĝq s the estmaton of the drect channel gan at the transmtter and ths value s used to have an nstantaneous estmate of the SINR Calculatng the power of nterference plus nose In ths part we calculate the power of nterference plus nose for every stream at each RX. Clearly, assumng Û, ˆV, Ĥj, j are gven, I,q s a sum of d 1 random varables and I,q s a sum of K 1 random varables. It can be shown that every two dstnct elements of I,q, I,q and n q are ndependent. Ths gves δ q = E{ I,q 2 } + E{ I,q 2 } + E{ n q 2 }. (10) It can be shown that the power of the nterference terms s E{ I,q 2 } = (1 ρ 2 0) P (l), E{ I,q 2 } = (1 ρ 2 0) j 1 l d l q j 1 l d (11) P (l). (12) Also t s clear that the nose varance wll not change after projecton by the (truncated untary) receve flters Statstcs of γ q condtoned on ˆγ q In the consdered lnk adaptaton method, transmsson rate and power are desgned based on the SINR of the lnk. Snce only ˆγ q s avalable at the TX, n order to do lnk adaptaton, we need to fnd the relatonshp between γ q and ˆγ q or, more precsely, the statstcs of γ q gven ˆγ q should be determned. We know that G q 2 and Ĝq 2 and hence γ q and ˆγ q have exponental dstrbuton approxmately. The pdf of two correlated exponental random varables reads { } f q (x1 x2) = 1 ρ x 2 γ ˆγq Γ q exp (13) (1 ρ) 1 ρ ˆΓ q { } { } 2 ρ x2 x1 I 0 1 ρ Γ qˆγ x1 exp q Γ q (1 ρ) where Γ q E{γ q } and ˆΓ q E{ˆγ q }, therefore Γq = ˆΓ q =. Also t can be shown that the correlaton σ 2 +E{ I,q 2 }+E{ I,q 2 } coeffcent s equal to ρ = ρ 2 0 [13] Pont-to-Pont Lnk wth Imperfect CSIT Pont-to-Pont Lnk wth average power and BER constrants Denotng the true SINR and the estmated SINR by γ and ˆγ respectvely and Γ and ˆΓ ther respectve expectatons, the condtonal dstrbuton f γ ˆγ s accordng to (13). The goal s to desgn the transmsson scheme usng adaptve power and rate p = Ψ(ˆγ) and k = Φ(ˆγ) whch are functons of the estmated SINR. The desgn parameters are the functons Ψ( ) and Φ( ) such that the average rate s mzed whle the average power and average BER constrants are satsfed as formulated n the followng optmzaton problem E{Φ(ˆγ)} Φ( ),Ψ( ) E{Ψ(ˆγ)} P (C 1) E γ,ˆγ {p e(ψ(ˆγ)γ, Φ(ˆγ))} B 0. (C 2) (14) Condton (C 2) s complcated and nstead we enforce the same condton for any nstance of ˆγ,.e., E γ{p e(pγ, k) ˆγ} B 0. By satsfyng ths condton, (C 2) s always satsfed. Lemma 1 Defnng A 0 = (A 1,, A M), to satsfy the BER condton (C 2) t s suffcent to have g B 0 (m) Q 1 ˆγ+Q 2 Ψ(ˆγ), where Q 1 = ρ 2 0 ΓˆΓ and Q 2 = (1 ρ 2 0)Γln( B 0 A 0 ). Proof: See [13]. Consderng the lmtaton on the transmt power (condton C 1), Lemma 1 gves Ψ(ˆγ) = g B 0 (m) Q 1 ˆγ+Q 2. Clearly when ρ 0 1 (more accurate channels), we get (8) whch was derved for perfect channels. Smlar to [9], where lnk adaptaton s performed usng perfect SINR estmates, we dvde the range of ˆγ usng thresholds t m, 0 m M such that when ˆγ [t m, t m+1) then the transmsson rate s chosen to be R m. Therefore the optmzaton problem (14) can be reformulated as {t m} M m=0 m=0 M tm+1 R m fˆγ (ˆγ)dˆγ M g B0 () =0 t m t+1 t fˆγ(ˆγ) Q 1ˆγ + Q 2 dˆγ P. (15) 6210

4 Note that we have mplctly ncluded the BER constrant. Usng the Lagrangan method we can show that the optmal thresholds are gven by t m = λ (gb0 (m) g B0 (m 1)) R m R m 1 Q 2, 0 m M (16) Q 1 The Lagrangan multpler λ s computed such that the power constrant s satsfed wth equalty Approxmaton of the average rate n a pont-to-pont lnk Inspred by the Gaussan channel, we approxmate the average rate achevable n the lnk by E{k} = L 1 log(1 + L 2ΓP ) where L 1 and L 2 are functons of the transmsson modes m, mum error probablty B 0 and the CSI uncertanty parameter ρ 0. We have used a curve-fttng method to fnd the values for L 1 and L 2. We verfed the accuracy of ths approxmaton compared to the actual average rate va extensve smulatons (see [13]) Optmzaton of the approxmate average rates Usng the approxmated closed-form expresson for the average rate we equvalently look at the followng problem P (q), 1 K, 1 q d K =1 q=1 K =1 ω d K (q) q=1 (17) d P (q) P 0 where K (q) = L 1 log(1 + L 2Γ q P (q)) E{k q }. Note that Γq s also a functon of the average powers. The above problem s a non-convex constraned optmzaton problem. We resort to numercal optmzaton methods to fnd a (locally) optmum soluton. The optmzaton problem s solved usng the actve-set method [14]. We used the standard MATLAB mplementaton. Therefore, we fnd the average powers to be assgned to dfferent streams of dfferent users. After fndng the average powers, the nstantaneous rates and powers can be chosen ndependently for each stream smlar to the pont-topont scenaro dscussed n the prevous subsecton. 6. ORTHOGONAL TRANSMISSION IN THE K-USER MIMO IC USING AMCP Due to the mperfectness of the CSI and sub-optmalty of IA, t s natural to ask whether we are better off performng IA rather than smply usng an orthogonal resource-sharng transmsson scheme. In orthogonal transmsson we assume that the channel resources (tme/bandwdth) are dvded equally among dfferent users n the network. In order to have a far comparson wth the proposed scheme, we consder a smlar problem where we look for mzng a weghted sum of the average rates of the users whle havng a total transmsson power constrant P 0 and a mum average BER of B 0. Here we have K pont-to-pont MIMO lnks where for each lnk d = mn(n t, N r) parallel streams are transmtted. Therefore we wll have a smlar optmzaton problem except that the flters are not determned by IA equatons but they are chosen such that the drect channels are dagonalzed. It should be noted that n ths case every RX only has to consder nter-stream nterference when computng the SINR dstrbuton. Smlar to the case of nterferng transmsson we can fnd the statstcs of the SINR and proceed wth solvng the problem. Sum of spectral effcences (bt/sec/hz) IA, α=3 Orthogonal, α=3 IA, α=5 Orthogonal, α=5 IA, α=7 Orthogonal, α= Doppler frequency Fg. 1. Rate comparson for dfferent values of Doppler and dstance, K = 5, N t = N r = 3 7. SIMULATION RESULTS To evaluate the performance of the proposed scheme, we use the AMC modes defned n the IEEE a standard. In ths settng there are 8 dfferent modes each of whch s assocated to a partcular modulaton and codng par. These modes provde the set of rates {0,.5,.75, 1, 1.5, 2, 3, 4}. The parameters of the BER estmaton accordng to (7) are extracted from [12]. We assume that the TXs and the RXs are placed such that the dstance between TX j and RX equals α 2 + ( j) 2 β 2 and the varance of the channel entres equals E{ H j 2 Ψ } = j = 0 where Ψ (α 2 +( j) 2 β 2 ) 0, α 1.5 and β are constants. α s the dstance between a par of TX and RX. We assume that the TXs and the RXs are equpped wth the same number of antennas and that the weghts for dfferent users are equal (ω j = 1, j). Also we set β = 1 and a total power constrant of P 0 = 1 s consdered and the nose varance s 0.1. Even though the uncertanty model ntroduced prevously s qute general, we assume n ths secton that the mperfecton s ntroduced by feedback latency and Doppler effects. The mum Doppler frequency s denoted by f d and the delay between the current transmsson and channel estmaton at the RX s denoted by T d. Accordng to the Jakes model [15], the coeffcent ρ 0 s derved from ρ 0 = J 0(2πf d T d ) where J 0( ) s the Bessel functon of the frst knd of order 0. Further, we assume that α s related to T d as T d = 10 4 α n our settng. In Fgure 1, assumng K = 3, N t = N r = 4 and d = 2, the sum rate s plotted for dfferent values of Doppler f d and for dfferent values of dstance α. Clearly, by ncreasng the Doppler and dstance, the CSI becomes more outdated and t degrades the overall performance. It s evdent that the proposed scheme outperforms the orthogonal scheme. 8. CONCLUSION Interference algnment based on mperfect CSI was nvestgated. We consdered a MIMO nterference channel where the transmt and receve spaces are determne by IA. We looked at mzng a weghted sum of the average rates provded that a certan set of bterror-rate and power constrants are satsfed by choosng approprate modulaton codng and powers. 6211

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