Limited Feedback for Multi-Antenna Multi-user Communications with Generalized Multi-Unitary Decomposition

Limited Feedback for Multi-Antenna Multi-user Communications with Generalized Multi-Unit ary Decomposition Wee Seng Chua 1 , Chau Yuen 2 , Yong Liang Guan 1 , and Francois Chin 2 1 Nanyang Technological University, 2 Institute for Infocomm Research, Singapore Email: chua0159@ntu. edu.sg, cyuen@i2r.a-star.edu.sg Abstract – In this paper, we propose a decomp osition method called Generalized Mu lti-Unitary Decomposition (GMUD) which is useful in multi-user MIMO precoding. This decomposition transforms a complex matrix mn × ∈ H ^ into H ii = H P RQ , where R is a special matrix whose first row contains on ly a non-zero user def ined value at the left-mos t position, i P and i Q are a pair of unitary matrices. The major at traction of our proposed GMUD is we can obtain multiple solutions of i P and i Q . With GMUD, w e propose a precoding method for a MIMO multi-user system th at does not require full channel state information (CSI ) at the transmitter. The proposed precodin g method uses the multiple unitary matrices property to compensate the inaccur ate feedback information as the transmitte r can steer the transmission beams o f individual users such that t he inter-user interference is kept minim um. I. INTRODUCTION Multiple input multiple ou tput (MIMO) schemes have assumed grea t importance i n current an d next generat ion broadban d wireless networks. T hey have been used for increasing spatial dive rsity, spatial multiplexing, beamform ing and precodi ng. In a multi-user broadcast communication syste m, where the base station with multiple an tennas simultaneously sends one data stream to each user with one rec eive antenna, the regularize-inve rse precoding pro posed in [1] can im prove the probability o f bit errors, and co mbine it with vector - perturbation as shown in [2] can furthe r reduce the energy of the transmitted signal so as to ach ieve excellent sum-rate with low bit errors. In addition, we show that by having additional receive anten nas at the user, which is feasible in many future c ommunicat ion systems e.g. 3G pp LTE, the performance in probability of bit errors can significantly improve if the transm itter can select the optimum receive antenna per user when sendi ng one data stream per user. However, this selection can only be performed if ful l CSI knowledge is available at the transmitter, and it is a tricky task to feedback full CSI from all users to the transmitter. Thus, it is importan t to consider both the probability of bi t error rate and t he amount of feedba ck informati on in the design of the precod ing matrix. Apparently, there is a tradeoff between t he performance and the am ount of feedback information, and it is a challenge to maximize the advantage of having add itional antenn a to improve the performance when the fee dback information of t he channel information is not complete. In this paper , we propose a preco ding technique that makes use of the additio nal antenna without in creasing the feedback information. This is analogous to what is mentioned in [3], when the increasing number of receive antennas is not used t o increase the number of data stream s received at each user, it can i ncrease the channel estimation quality. II. SIGNAL MODEL Given N T transmit antennas at the base station servicing one data stream to each of a pool of K users with N R receive antennas each, the received si gnal at k th user is gi ven as kk k = + yH x n (1) where x represents the transmit signal vector, k H is a N R × N T matrix containing the channel information between the base station and k th user, n k is the additive white Gaussian noise vector such that H2 E kk σ = nn I . The transmitter signal x from the transmitter can be represented as () γ = xG u (2) where G is a N T × K precoding matrix, [] T 1 K uu = u " whose element u k is the intended d ata for k th user, and 2 γ = Gu is used to normalize the transmitted signal. In a multi-user with one receive antenna commu nication, the regularize-inve rse precodi ng in [1] arranges the channel information into a K × N T matrix such that each individual row of H  represents i ts correspondin g channel i nformation between the base station and its resp ective user. The precoding matrix G is () 1 HH reg-inv α − =+ GH H H I   (3) where 2 K K αρ σ == as derived in [1], I is a K × K identity matrix and [.] H denotes com plex conjugate. Although thi s precoding matrix introduces inter-user interference due to ( ) 1 HH α − +≠ HH HH I I   , it reduces the normalization constant, γ , significantly, which results in a stronger signal of higher SINR at the receiver. The amount of interference is determined by α . When more than one receive antenna is used to receive one data stream , the communi cation channel can be arranged i nto K × N T matrix, whose rows represen t the corresponding channel in formation bet ween the N T transmit antennas and one of the N R receive antennas of its respective user, and there are a total of () K R N different c ombinations. The regularize-inverse precoding matrix is thus beco ming () 1 HH reg-in v ˆ ˆˆ ˆ α − =+ GH H H I (4) where ˆ H is one of the () K R N different com binations. The received signal vector [] T 1K = yy y " becomes () 1 HH eq ˆˆ ˆˆ ˆ α γ − + =+ = + HH HH I u yn H u n (5) where () 1 HH 1,1 1, K eq K,1 K,K ˆˆ ˆˆ ˆˆ ˆ ˆˆ hh hh α γ − ⎡⎤ + ⎢⎥ == ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ HH HH I H " % . The regularize-inverse precodi ng with receive antenna selection can be obtained by m aximizing mi n signal to interference plus noise ratio of the received si gnals. 2 , reg- inv,optimal 2 ˆ 1 , =1, ˆ ˆ max min ˆ mm K mK mn nn m h h γ ρ ≤≤ ≠ ⎛⎞ ⎜⎟ = ⎜⎟ ⎜⎟ + ⎝⎠ ∑ H G (6) III. GENERALIZE D MULTI-UNITARY DECOMPOSITION (GMUD) Given a com plex mat rix channel H , it can transform into various form s using differ ent decomposi tion technique s such as Singula r Value Decomposit ion (SVD) [4], Geometric Mean Decom position (GMD) [5,6] and Geometric Triangular Decomposition (GTD) [7]. In this paper, we propose a new decom position technique called Generalized Multi-Unitary Decomposition (GMUD) , that transforms H to H ii PR Q , where R can be a R×R lower triangular matrix or a special R×R matrix with a prescribed value at the first element an d zeros for the rest of the elements in the first row, i P and i Q are a group of di fferent unitary m atrices. It is a general decompositi on method that includes SVD, GMD, and GTD as part of the solut ions. Both GMUD and GTD allow the use r to prescribe the diagonal value s of the R matrices. However , GMUD can produce m ore than one pairs of different unitary mat rices, as opposed to one pair produced by GTD and the other decompositio ns. A. Derivation of R Consider a rank R complex H with singular values 1 R r λ λ ≤≤ . R can be defined a s a special R × R matrix in the form of 21 22 2 R 1 R1 R2 RR 00 where R r cc c r cc c λ λ ⎡⎤ ⎢⎥ =≤ ≤ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ R " " ## # # " (7) In the first row of R , there is only one non-zero positive element r at the (1,1) positio n. The remaining elements at the other rows are calculated based on r and the singular values. R can also be define d as a lower triangular m atrix with user pre -defined diag onal elements. This can be achieved by assigning the next remai ning row in t he same way after the previous row has been assi gned and let all the entries on the right of the diagonal entries to be zero. Thu s this leads to 1 21 2 R1 R2 R 00 00 r cr cc r ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ R " ## # # " (8) where the values of the diag onal values follo w the condition as described i n [7]. From this point onwards , for simplicity of illustratio n, we consider R = 2, howeve r GMUD can be ext ended to hig her rank. The channel H is first transforms into H = HU Λ V using SVD. The matrix R with a pre-assigned value 1 R r λ λ ≤ ≤ at the (1,1) position can be arranged in the form of H = 00 RU Σ V , where 0 U and 0 V are unitary matrices assigned using Givens rotations, Λ is the sam e diagonal matrix of H containing th e singular values. HH HH , cs ab s c ba ⎡⎤ ⎡⎤ == ⎢⎥ ⎢⎥ − − ⎣⎦ ⎣⎦ 00 UV (9) where 2 1 ba = − , and 2 1 s c = − . As a result, r , z 1 and z 2 can be defined in term s of a , b , c , s , 1 λ and 2 λ . H H HH 1 HH 12 2 12 1 2 12 1 2 1 2 00 0 0 r cs ab zz s c ba ra c b s a s b c z z bc as bs ac λ λ λλ λ λ λλ λ λ = ⎡⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ = ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ − − ⎣⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ +− ⎡⎤ ⎡ ⎤ = ⎢⎥ ⎢ ⎥ −+ ⎣⎦ ⎣ ⎦ 00 RU Λ V (10) The value of a and c can be derived from the first row of (10), after substituting b and s from (9), a nd subsequent ly 0 U and 0 V can be determined. 12 1 2 ,0 ac bs r as bc λ λλ λ + =− = (11) The remaining values o f R can be found by substituting the values of a and c 11 2 21 2 , z bc as z bs ac λ λλ λ = −= + (12) Next, replacing H = 00 Λ UR V into the singular value decomposition of H yields () ( ) H H == 00 0 0 HU U R V V P R Q (13) Since ,, a n d 00 UU V V are unitary matrices, 0 P and 0 Q will also be unitary matrices. B. Derivation of P i and Q i In order to get multiple different P and Q , we include i M , a diagonal matrix w ith some unity phase at th e diagonal elements, and its conjugate transpose H i M to (13), where H ii = M Λ M Λ . 11 22 H 00 , 00 ii ii jj ii jj ee ee θθ θθ − − ⎡⎤ ⎡ ⎤ == ⎢⎥ ⎢ ⎥ ⎣⎦ ⎣ ⎦ MM (1 4) where 1 i θ and 2 i θ can be any value from 0 to 2 π . After the inclusion of i M and H i M , H becom es () ( ) H H ii i i == 00 H U MU R V MV P R Q (15) From (15), R is independen t to the value of i M , hence it remains the same. Since i P and i Q include the inform ation of i M , it is apparent that i P and i Q are varying with 1 i θ and 2 i θ . In addition, ,, a n d 00 UU V V are unitary matrices, the combination of other unitary matrices i M and H i M , will result in i P and i Q being unitary matrices as well. Figure 1: The principle and second eigen-vector s of SVD and the first column vectors of the unitary matrix i Q of GMUD with different i M are plotted on a uniform 3-D sphere. It shows that with the sam e R , different 1 i θ or 2 i θ produces different vectors in i Q , which form a cone whose center is the principle eigen-vector of SVD. For the case of dime nsion of two, we only need to kn ow the principle eigenvector, i.e. the first row of V 0 . The exac t second eigenvector is not im portant, as eventually the rotation matrix i M will make a rotation and generate a series of vectors. He nce the decomposit ion of a 2-by-2 matrix only can be performe d with any vector that is orthogonal to the pri ncipal ei genvector, not necessary with the exact second eigenvect or. So we can generate a n arbitrary second eigenvector that is orthogonal to the principal eigenvector, as sh own in Figure 1. An exampl e is shown in Fig ure 1, differe nt 1 i θ with values ranging fr om 0 to 2 π , produce different i M , which result in different ve ctors. These vector s form a cone surrounding the principle eige n-vector of SVD which is located at the center of the cone. The value of r determines t he radius of the cone, i.e. the closer the value of r to the largest singular value 1 λ , the smaller the cone radius. If the value of r is equal to 1 λ , the first column vector of i Q becomes t he principle eigenvector of SVD. The lines with square m arkers represent GMUD vectors with a smaller r = 0.75 1 λ , while the lines with asterisk mark ers represent GMUD vectors with a larger r = 0.95 1 λ . IV. PREC ODING BASED ON GMUD In the scenario where there a re N T transmit antennas at the base station sendi ng one data stream to each of the K users with N R receive antennas each. The received signal per user is given in (1) and its corresponding ch annel matrix k H is decomposed u sing GMUD, mult iple different pairs of i P and i Q matrices can be generated with the same R matrix, and each user’s k H can be decom posed into H kk k k = HP R Q (16) where we replace r in R from (7) and 1 i θ in i M from (14) (for the case of 2x2 matrices, the optimization of single parameter 1 i θ is enough) with opti mizing parameters r k and θ k respectively, and we denote the optimize d i P , i Q and R as k P , k Q and k R respectively. The first colum n vector of , kk r θ Q is considered as an indivi dual transm ission beam for that user. I n other wo rds, it is im portant to ha ve many different k Q m atrices, which represent d ifferent transmission beams, because the tra nsmitter can steer the beam s of every users to make them as orthogonal as possible. If all the users’ transmission beams are orthogonal to each othe r, there will be zero multi-user s interferen ce. Hence by changing r k and θ k , multi ple beamforming ve ctors “listening” t o different di rection can be obta ined. The precoding matrix G can be formed by assigning each column of the matrix as th e first column vectors of k Q of each users. From (15) and (16), N 1, 2 , k k kk k θ ⎡⎤ ⎢⎥ == ⎢⎥ ⎣⎦ 0 g QV M V q q (17) where 1, k q and 2, k q are the first and second column vectors respectively. For simplicity, we shall focus o n the example of having K = N T = N R = 2 for the rest of the docum ent. The idea is extendable to any nu mber of transmit antennas or any users with any number of r eceive a ntennas. We let 1, 1, , kk l l = = gq g q (18) and the precoding matrix is [] kl = Gg g (19) When we combine (1), (16) a nd (19), the received si gnal for user k becomes [] 1, H 2, 1, 1, 2, 2, 1 k k kkk k kk k l k k kk k kl l kk k kk k k l l uu uu γγ γ ⎡⎤ =+ = + ⎢⎥ ⎢⎥ ⎣⎦ ⎡⎤ + =+ ⎢⎥ + ⎢⎥ ⎣⎦ H H HH HH q Gu u yP R Q nP R g g n q qg qg PR n qg qg (20) where 2 γ = Gu is u sed to normalized the transmitted signal. Given k R is a special matrix as sh own in (7) and ( 8), (20) can be reduced to () 1, 1, 1 kk k k k l l kk k ru u γ ε ⎡⎤ + =+ ⎢⎥ ⎢⎥ ⎣⎦ HH qg qg yP n (21) where () ( ) 21 , 1 , 1 , 22, 2, 2, kk k k k l l k k k k k l l cu u c u u ε =+ + + HH H H q g q g qg qg , and 21, k c and 22, k c refer to the elements of R k in (7). In order to re duce the BER, the interference terms must be minimized by choosing 1, k q and l g to be as orthogonal to each other as pos sible. However, it is more appropriate to use the cost functi on of maxim izing the signal t o interference- plus-noise ratio (SI NR) to find the precoding matrix G where the signal is kk ru γ and the interfe rence is 1, kk l l ru γ H qg . This is because reducing the dot product between two u sers’ 1 q vectors may produces a G that ha s a large norm alization constant γ which results in a weaker received signal and inc reases th e probability of bit error at the receiver. Thus, t he cost function of finding G becom es 22 22 22 , H2 H2 1, 1, 22 22 22 ,, , H2 H2 1, 1, 1, 1, max mi n , max min , kl kl k l kl kk l ll k kl rr kk l ll k rr rr rr rr θθ σγ σγ σγ σ γ ⎛⎞ ⎜⎟ = ⎜⎟ ++ ⎝⎠ ⎛⎞ ⎜⎟ = ⎜⎟ ++ ⎝⎠ gg G qg q g qq q q (22) The performance of the system improves if we choose different power loadin g factor to k g and l g and G bec omes [ ] kl αβ = Gg g (23) Thus the cost f unction becomes 22 22 22 22 2H 2 2 H 2 ,, , , , 1, 1, 1, 1, max min , kl k l kl rr a b kk l ll k rr rr θθ αβ β σγ α σγ ⎛⎞ ⎜⎟ = ⎜⎟ ++ ⎝⎠ G qq q q (24) Since we assume the receiver only feedback the principal eigenvector, in (22 ) and (24) the precodi ng matrix generate d by the transmit ter does not ma ke use of the second column vector of i Q . In other words, th e receiver decodes the received signal without the use of ε as shown i n (21). Hence the propose d GMUD precoding onl y requires partial CSI feedb ack, i.e. it only requires the receiver to feedba ck the eige nvalue s and eigenvectors. For the case of 2x2, the feedback information can be further reduced to t he singular values and pri ncipal eigenvector. V. SIMULATION RESULTS AND DIS CUSSION The performance of the MIMO multi-user system using GMUD precod ing and regularize-i nverse precodi ng [1] with or without antenna selection is compared under perfect and quantized limited feed back information in this section. Given N T transmit antennas at the base stati on sending one data stream to each of the K users with N R receive antennas each. For the regularize-invers e precoding with antenna selection at the transmitter, the full chann el matrix of the k th user, H k , in (1) is needed to be feed back from all users to the transmitter, this can be done by normalizing its row or column vector first, an d fee dback the no rmalizing scalars and unit norm vectors. For regula rize-inverse precoding without antenna selection, th e user needs to feedbac k one row of H to the transmitter, likewise this can be done by normalizing this row vector and feedback the normalizing scalar and the unit norm vect or. For the proposed pr ecoding based on GMUD, the user will send back the singu lar values and the first column vector of V . The number of equi valent scalars needed to be fee dback per user for the above three different schemes are summarized in Table 1 for the case of a MIMO system with two transmit antennas at the transmitter, and two receive a ntennas per user. In the simulation, we consider a system wit h two transmit antennas, and two users with two receive antennas each. We are using the same total number of feedback bits (besides the unquantized cas e) for each scheme to test their vulnerability to imperfect feedb ack channel information. In the case of regularize-inverse precoding with antenna selection, we are usi ng N bits and 2 N bits to quantize each of the real normalized elemen t and normalizing constant respectively. If there is no antenna selection , we are using double number of bits for all the information com pared to the antenna selection case. For GM UD precoding, we a re allocating 2N bits for all the information. We are using a total of 12 N bits to quan tize the feedback info rmation. For example, if we are using 48 feedback bits, N will become 4. Table 1: Allocation o f the bits used to quanti ze the feedback information Feedback Info Precoding Scheme No. of equivale nt scalars for unit norm vector (Bits used) No. of normali zing scalar or singular values (Bits used) Reg-Inv wi th antenna selection 8 ( N bits) 2 (2 N bits) Reg-Inv wi thout antenna selection 4 (2 N bits) 1 (4 N bits) GMUD 4 (2 N bits) 2 (2 N bits) 0 2 4 6 8 10 12 14 16 18 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 SN R BER Number of user: 2 wit h 2 ant ennas and c ons t el lat i on s i z e: 2. N umber of ant ennas at bas e s t at i on: 2. GM UD prec oding - per f ec t f eedback GM UD prec oding - 48 f eedbac k bi t s GM UD prec oding - 24 f eedbac k bi t s R eg- I nv w /o selection - per f ec t f eedbac k R eg- I nv w /o selection - 48 f eedbac k bi ts R eg- I nv w /o selection - 24 f eedbac k bi ts R eg- I nv w / selec tion - per f ec t f eedbac k R eg- I nv w / selec tion - 48 f eedbac k bi ts R eg- I nv w / selec tion - 24 f eedbac k bi ts Figure 2: Performance of probability of bit error for 2 users using QPSK symbols given perfect or quan tized feedback infor mation (48, 24 bits) available at the trans mitter. 0 5 10 15 20 25 30 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 SN R BER Number of user: 2 wit h 2 ant ennas and c ons t el lat ion s i z e: 4. Number of antennas at bas e s t at i on: 2. GM UD prec oding - per f ec t f eedback GM UD prec oding - 48 f eedbac k bi t s GM UD prec oding - 24 f eedbac k bi t s R eg- I nv w /o selec tion - per f ec t f eedbac k R eg- I nv w /o selec tion - 48 f eedbac k bits R eg- I nv w /o selec tion - 24 f eedbac k bi ts R eg- I nv w / selec tion - per f ec t f eedbac k R eg- I nv w / selec tion - 48 f eedbac k bi ts R eg- I nv w / selection - 24 f eedbac k bi ts Figure 3: Performance of probability of bit error for 2 users using 16QAM symbols given perfect or quantized feedback information ( 48, 24 bits) available at the transmitter. Figure 2 and F igure 3 show t hat under the assum ption of having perfect CSI at the transmi tter, regularize-in verse precoding with an tenna selection requires d ouble the feedback information than regularize-inverse prec oding without antenna syste m to obtain a gain of around 4.1dB and 10.75dB respectiv ely, while GMUD precod ing uses nearly half feedback info rmation to achieve a gain of arou nd 1.95dB an d 8.8dB respect ively. When t he three schem es have the sam e number of feedbac k bits as show n in Table 1, the BER performance of regularize-i nverse precoding wit h antenna selection deteriorates tre mendously if limited feedback information is available at the transmitter. As shown in Figure 3 , when there are only 48 bits available, the performance of re gularize-inve rse precoding suffers a 1.75dB loss as compare to perfect CSI, the lost widen as the feedback information becomes m ore scarce, and eventually an error floor c an be seen at BER of 0.05. Howeve r, GMUD precoding does not de pend too much on the accuracy of the feedback, it suffers a negligible loss of when limited feedback is avail able, thus making it a robust scheme for limited feed back comm unication. We would like to mentio n that the number of bits u sed in this paper is just for illustration purposes. In practice, one can use other ways t o feedback the channel matrix, e.g. codebook based or c ompression based on Givens rotatio n. Our intention is to show that with limited feedb ack constraint, it is important to feedback less in formation but with better quality, rather than feedback a lot of information but with poor quality. VI. CON CLUSION GMUD precoding provides an alternative to feedb ack less information through mathe matical decomposition method, w hile maintai ning a high level o f BER performance. It can decom pose a mn × channel matrix into multiple mm × and nn × unitary matrices, and a mn × triangular matrix with pres cribed diagonal elements. 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