A MIMO system with finite-bit feedback based on fixed constellations

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1 . RESEARCH PAPER. SCIENCE CHINA Information Sciences June 013, Vol : :14 doi: /s A MIMO system with finite-bit feedback based on fixed constellations WANG HaiQuan & ZHAO ZhiJin School of Communications Engineering, Hangzhou Dianzi University, Hangzhou , China Received June ; 010; accepted July 0, 011; published online April 5, 01 Abstract In this paper, a MIMO system with finite-bit feedback based on fixed constellations is considered. Based on performance analysis of the system, an optimal operating system with maximum-likelihood decoding is demonstrated. Surprisingly, this operation reveals that the optimal way for the system to transmit signals needs to invoke the multimode scheme. We propose designing criteria for this scheme and methods for pre-codebooks, and a method to determine the number of modes for any specific channel. Furthermore, to reduce encoding complexity at the receiver side, we also develop a fast encoding algorithm. Theoretical analysis and simulations show that the proposed systems offer considerable gain over existing systems. Moreover, these systems also have much lower encoding complexity. Indeed, for the case of a MIMO system with two pairs of transmitting and receiving antennas, a properly designed system with a transmission rate of 8 bits per channel use and with 6-bit feedback can provide about a 1.5 db performance gain over a beamforming system. Keywords beamforming, fixed constellations, finite-bit feedback, multiple-input, multiple-output systems, space-time coding, union bound Citation Wang H Q, Zhao Z J. A MIMO system with finite-bit feedback based on fixed constellations. Sci China Inf Sci, 013, 56: 06303(14), doi: /s Introduction Consider a multiple-input, multiple-output (MIMO) system with a finite-bit feedback channel. Assume that the system has M transmitting antennas and N receiving antennas with an error free feedback channel having r-bit capacity. Suppose that the receiver knows the channel matrix H and can send r-bit information about H to the transmitter using the feedback channel. For this MIMO system, there are two basic problems. The first one is to understand what kind of information about the instant channel H is needed to be sent back. When the transmitter receives the channel information, it needs to adapt its transmission way by following the information. The second basic problem is finding a means to efficiently use the feedback information at the transmitter side. On the first problem, the existing literature gives essentially two different considerations. One is to send back statistical information about the channel; for example, see [1 5], the comprehensive review paper [6], and the references therein. The other is to feedback information about the instant channel H. This paper will focus on the latter in detail. Corresponding author ( wanghq33@gmail.com) c Science China Press and Springer-Verlag Berlin Heidelberg 01 info.scichina.com

2 Wang H Q, et al. Sci China Inf Sci June 013 Vol : An ideal way is to send the whole information of the matrix H back to the transmitter. However, this approach is impossible because the channel has infinite states and the feedback channel has only r-bit capacity. A feasible way, which has been adopted in recent works, is to divide the whole channel space, consisting of all matrices, into ordered r parts, and then, the index to which the instant channel H belongs is sent back. Following up this idea, the first basic problem reduces to how to divide the matrix space into r parts. Dealing with this problem is related to finding out the way the transmitter uses this r-bit information. In turn that depends on the setup of the design obectives for the system. Most existing work can be classified according to one of the following four obectives. The first is to use the information to increase capacity, or more precisely, to increase mutual information; see [7 13] and the references therein. The second is to decrease the outage probability. The work in this direction can be found in [14]. The third is to increase the receiving SNR; the original work can be found in [15] and further developed in [16,17]. In [16], it was proposed that the design of the pre-codes should be implemented on a Grassmannian manifold, and moreover, packing theory on this manifold is employed. However, these papers essentially work on a multiple-input, single-output system, although in [16], it is claimed that all results can be generalized to a MIMO system. The last obective, by following traditional and convenient method, is to decrease error probability of the whole system. Works guided by this obective can be found in the[18 1]. Nevertheless, most of these studies are based on a special constellation, for example, quadrature amplitude modulation (QAM) or phase-shift keying, or based on a linear receiver, for example, zero-forcing or minimum mean square error receivers. Moreover, only the multiple-input, single-output system is considered. Recently, in [], Wang and Yang considered a general system, which has any number of receiving antennas and the maximum-likelihood (ML)-decoder. Based on a performance analysis in [], criteria for designing pre-codes are given. Furthermore, criteria and methods of designing codes are induced. In this paper, we will adopt the latter as our design goal. For convenience, we will assume that the transmitted signal X has a form (s 1,s,...,s i )V, where s is taken from a fixed constellation S, and V is an M i unitary matrix, which is taken from a pre-designed codebook. We denote this codebook by V and call an element of V a pre-code. Based on the performance analysis of this system with ML-decoder, we will obtain the optimal transmission scheme. Following this optimal scheme, it turns out that the proper design of pre-codes is required and a multimode transmission scheme is needed. Indeed, in this scheme, those sub-channels with small channel gain generated by the singular values of the channel matrix H will be abandoned. This scheme is very different from that considered in [] where all N channels are used simultaneously and the codes are adapted to fit sub-channel situations. In this paper, we will also reconsider and obtain criteria for designing pre-codes with an aim to minimize error probability. Guided by these criteria, a designing method is given. In fact, by this method, all designs of pre-codes are turned into parking problems on CP M, which is the simplest Grassmannian manifold. Hence, it is reasonable to believe that our proposed method is much easier than doing parking on a general Grassmannian manifold. Furthermore, this method surprisingly, brings about a fast encoding procedure at the receiver side. To reduce the complexity of this encoding further, we will also provide a fast and easy method to determine the mode number i. For the multimode scheme, there are also several existing studies. In [3], the concept of multimode is proposed and the methods to determine mode number i are given based on different decoding methods, for example, ML-decoding and linear receivers. Notice that the given methods are not optimal (in the sense of minimizing error probability) and the complexity to determine the number i is high. In [10] and [4], a multimode scheme based on maximizing mutual information has been suggested. If the receiver is linear, Refs. [5] and [6] provided schemes to obtain the number i based on bit error ratio (BER) analysis. The organization of this paper is as follows. In the next section, we will describe the system model and formalize problems. In Section 3, designing criteria and method are given. Also, in this section, a fast encoding scheme and a fast method for determining i are stated. In Section 4, results of simulations are shown. The last section concludes the paper. Notation. Matrices are represented by Roman capital letters. I t denotes the unit matrix of size t and

3 Wang H Q, et al. Sci China Inf Sci June 013 Vol :3 A denotes the conugate transpose of A. R and C denote the real and complex field, respectively. Problem formulation In this section, we will set up the model of the system and state the problems to be considered more precisely and completely. Consider a MIMO system with M transmitting and N receiving antennas operating on a quasi-static and flat Rayleigh fading channel. To state our results smoothly, we assume that M N. All results still applied if M < N. The transmission rate of the system is R bits per channel use. We assume that there is an error-free feedback channel with a capacity of r bits from the receiver to the transmitter. The delay time T is always assumed to be 1. The basic transmit-receive equation is assumed to be as follows: Y = ρ M SV H +W, (1) where Y, an N-dimensional complex vector, is the received signal; S is an i (1 i N)-dimensional complex vector that carries data information; V is an M i unitary matrix, which is taken from the pre-designed codebook V i (1 i N), called a pre-codebook. We also call an element in this precodebook as pre-codeword, or simply, pre-code; H represents an instant channel realization. Each H is taken from a random, M N matrix H, for which the components have a standard complex Gaussian distribution. W represents noise. It is also a N-dimensional complex vector, whose component is taken from the standard complex Gaussian random variable. Put S = (s 1,s,...,s i ). The energy restriction requires that S MT = M. (3) In this paper, we assume that each s t is taken from a constellation S (i) t, where S (i) t is a finite subset on a complex two-dimensional lattice. For brevity, we assume this set to be a subset of QAM. Furthermore, S (i) t and S s (i), for all 1 t,s N,t s, are independent. Thus, S i = S (i) 1 S (i) S (i) i. Obviously, S (i) 1 S(i) S(i) i = R = L, where S (i) denotes the size of the set S (i). We denote the minimal distance of the set S (i) by d (i) m, for = 1,,...,i. Notice that in the model above, we do not fix the index i, that is, i may take any value between 1 to N. It depends on the instant channel H. The result, given in the next section, shows that this change is in fact necessary to minimize the error probability of the whole system. Now let us review a general operating strategy for the MIMO system with finite-bit feedback above. We adopt the following steps. First, based on an analysis of the instant channel H, which is known to the receiver, values of two indices i and are sent back to the transmitter from the receiver using the r-bit feedback channel. Second, when the transmitter receives values i and, in the following coherent time period, it transmits data using the codebook S i and a pre-codeword V (i) V i. Finally, at the receiver side, the ML decoder decodes received signal Y into arg min ρ S S i Y M S(V (i) ) H. (4) In this operating strategy, index i represents the number of channels used to transmit data whereas index is used to decide which pre-codeword in the codebook V i is assigned to weaken the divergence caused by the instant channel H. Therefore, the first problem for this operating strategy is how to determine these indices for a given H; in other words, how to define a function g from the space H, consisting of all M N matrices, to index pairs. Let us look at the range of these indices. As there are N sub-channels, log N bits are needed to indicate different sub-channels. Let r = log N. Because

4 Wang H Q, et al. Sci China Inf Sci June 013 Vol :4 the capacity of the feedback channel is r bits, there are r 1 ( = r r ) bits to represent the size of the pre-codebook V i. Put J 1 = r1 and J = r. Thus, the problem involves defining the function Set = g 1 ((i,)), that is g : H {1,,...,N} {1,,...,J 1 }. (5) = {H H : g(h) = (i,)}. (6) Obviously, werequirethat the unionofall is equalto the whole spacehandno pairofsetsintersects. The second problem is how to design the pre-codebooks V i. From the setup of the system above, we know that each of these consists of J 1 M i unitary matrices, that is, V i = {V (i) 1 (i),v,...,v (i) J 1 } (7) and (V (i) ) V (i) = I i for = 1,,...,J 1. Notice that the size of the pre-codeword is related to index i. For a fixed index i, there are many existing contributions to the construction of V i. Different design criteria are proposed and different methods of design are implemented. As stated in the introduction, these criteria and methods are based on different obectives of the designing system. In this paper, our obective is to minimize the overall decoding error probability. We denote this probability as P e and need to calculate this probability. Obviously, where p (i) e, H P e = N J 1 i=1 =1 p (i) e, P(i) e,, (8) is the probability of H belonging to H(i), and P(i) e, is the conditional error probability given. Thus, calculationsofp(i) arein order. To this end, the density function ofthe random e, and P(i) e, matrixh isrequired. Becausethe entriesh i ofh areindependent, zero-meancomplexgaussianrandom variables of variance 1/ per dimension, the density function is Thus, for any a subset A in H, the measure of A can be defined as meas(a) = f(h)dh, f(h) = 1 π MN exp( tr(hh )). (9) where dh is defined as dh = M u=1 N v=1 dh uv,r dh uv,i, and h uv = h uv,r +h uv,i is the (u,v)-th entry of matrix H. Hence, A p (i) e, = meas(h(i) H ) = f(h)dh. (10) (i) For the error probability P (i) e,, it is difficult to get an exact formula, but we can use a union bound to obtain its estimation. Assume that H. Then, the codebook S i and the pre-codeword V (i) V i are adopted, and ( (i) ) d (H) Q P (i) e, (H) 1 L where Q is the standard Q-function. L ( (i) d,uv Q (H) ) ( (i) d (H) (L 1)Q ), (11) t=1 t s (d (i),uv (H)) = ρ M tr(( S(i),uv ) S (i),uv HH ), (1) where S (i),uv = (s(i) 1,u 1 s (i) 1,v 1,s (i),u s (i),v,...,s (i) i,u i s (i) i,v i )V (i) and d (i) (H) is the minimum of d (i),uv (H) on S (i),uv for any (u 1,...,u i ) (v 1,...,v i ) and for given H and V (i).

5 Wang H Q, et al. Sci China Inf Sci June 013 Vol :5 Performing average on H in the formula (11) and noticing the fact that p (i) e, P(i) e, = P (i) e, (H)f(H) dh, we have N J 1 i=1 =1 ( (i) d (H) N J 1 Q )f(h)dh P e (L 1) i=1 =1 ( (i) d (H) Q )f(h)dh. (13) Clearly, when SNR is large enough, both the lower and upper bounds in (13) are asymptotically tight. Therefore, in the region of high SNR, to minimize P e, one should find and V (i) so that is minimized subect to C1. N J1 i=1 C. each V (i) is unitary. =1 meas(h(i) N J 1 P w = i=1 =1 ( (i) d (H) Q )f(h)dh (14) ) = 1 and meas(h(i) k H(i) t ) = 0 for 1 k t J 1 ; In summary, we have the following optimization problem. min V i min P w = min V i min N J 1 i=1 =1 subect to the conditions C1, C. ( (i) d (H) Q )f(h)dh (15) Let us clarify the main differences between the optimization problem above and one stated in []. In [], the optimization is on the sets V i,, and codebooks, which is similar to constellations S i, while the constellations S i are given in this paper. As a result in [], these codebooks are designed to be packing points in an ellipsoid that is determined by the singular values of H. This design for the codebook results in that the optimal solution needs to utilize all channels generated by the singular values of the instant channel H. In contrast, it will be seen that the optimal solution should abandon some of the worse channels in the current case. In the following section, our obective is to figure out optimization (15). 3 Designing criteria and designing methods It is difficult to obtain an analytic solution to the problem (15). Notice that the minimization involves two variables V i and. Hence, we first try to find out the relationship between the optimal solutions V i and. More specifically, this step optimizes H(s) t (s = 1,,...,N,t = 1,,...,J 1 ) for given V i (i = 1,,...,N). By doing so, we can find that, in fact, the optimal solution is determined by the optimal solution V i. Thus, the optimization of (15) can be changed into the optimality on V i only. Based on this result, we next try to get criterions and methods of design for pre-codebooks V i. 3.1 Methods of determination for H (s) t for given pre-codebooks Assume that the pre-codebooks V i (i = 1,,...,N) are given. For the determination of the index i and designs of H (s) t (s = 1,,...,N,t = 1,,...,J 1 ), we have the following results. Theorem 1. (1) If the pre-codebooks V i (i = 1,,...,N) are given, the optimal solution of H (s) t is determined as follows H (s) t = {H d (s) t (H) > d (u) v (H), (s,t) (u,v), u = 1,,...,N,v = 1,,...,J 1 }, (16) where d (s) t (H) is defined on the below line of the formula (1). () For a specific channel realization H, if H H (s0) t 0, then, i = s 0 and = t 0, which means that if a channel realization H belongs to region H (s0) t 0, the optimal choices in the corresponding coherent time for the pre-codeword and the codebook are V (s0) t 0 and S s0, respectively.

6 Wang H Q, et al. Sci China Inf Sci June 013 Vol :6 The proof of this theorem is simple. Result () is a direct consequence of result (1). To prove result (1), it is sufficient to notice that in (15), P w is a decreasing function of d (s) t (H) since the Q-function is decreasing with d (s) t (H). The results of this theorem reveal the following facts. The optimal solutions for and V i in (15) are essentially related. Once V i is determined, are all obtainable for i = 1,,...,N and = 1,,...,J 1. That is, are functions of V i. Thus, the optimal problem (15) can be simplified as follows: min V i N J 1 i=1 =1 ( (i) d (H) Q )f(h)dh (17) subect to the conditions C1, C, and (16). The second result in this theorem tell us that index i is a function of the channel realization H. That is, Eq. (16) determines the number of sub-channels generated by singular values of H used. Not all subchannels are used, as considered in [], and not only the sub-channel generated by the largest singular value is used, as for beamforming. It depends on H. This difference might result in differences in precodebook designs. We have to reconsider the designing criteria and methods of the pre-codebooks. Let us begin with the former. 3. Criteria of designing pre-codebook V i for given H (s) t Recall that V i = {V (i) 1,V (i),...,v (i) J 1 }, where each V (i) is an M i unitary matrix. Let us assume temporarily that subsets H (s) t,s = 1,,...,N,t = 1,,...,J 1 are given. Under this assumption, the optimization (15) can be simplified as follows: to design an m i unitary matrix V (i), such that the following integral is minimized: ( (i) d (H) Q )f(h)dh, (18) where indices i and are fixed. Notice that d (i) (H) is a function of V (i) and H. Because the Q-function has no analytical expression, the upper bound exp( x /)/ of Q(x) is substituted for it in the integral. Recall the fact that f(h) = exp( tr(hh )). The integral (18) has upper-bound 1 π MN exp ( (d(i) (H)) ) tr(hh ) dh. (19) Moreover, by the definition of d (i) (H), the integral above is equal to ( 1 π MN exp ρ ) 4M tr(( S(i) ks ) S (i) ks V (i) HH V (i) ) tr(hh ) dh, (0) where S (i) ks = s (i) k s(i) s and this pair of codes achieves the minimum of d(h) for this specific H and V (i). Because S (i) t for t = 1,,...,i are QAM, the t-th component of the vector S uv (i) for any pair (u,v) belongs to the lattice d (i) t A, where d (i) t is the minimal distance of S (i) t, and A = {n 1 +n : n 1,n are integers}. (1) For brevity, we omit superscripts and subscripts where no confusion arises. Thus, the integral above can be simply represented as ( 1 π MN exp ρ ) 4M tr(s S V HH V +HH ) dh, () H where 0 S d (i) 1 A d(i) A d(i) i A. Let us look at the term tr(s S V HH V +HH ) more specifically.

7 Wang H Q, et al. Sci China Inf Sci June 013 Vol :7 Denote the s-th column of the matrix V as v s and write S as (s 1,s,...,s i ). Introduce the singular value decomposition of H as UDQ, where U is an M N unitary matrix, D = diag(λ 1,λ,...,λ N ) and λ 1 λ λ N 0 are the singular values of H and Q is an N N unitary matrix. Denote the t-th column of U as u t, t = 1,,...,N. Using this notation, we have and Therefore, tr(s S V HH V +HH ) = tr(s S V UD U V +UD U ) = tr((u V S S V U +I N )D ) = tr(((s V U) (S V U)+I N )D ) S V U = (s 1,s,...,s i )(v 1,v,...,v i ) (u 1,u,...,u N ) = ((s 1 v 1 + +s iv i )u 1,...,(s 1 v 1 + +s iv i )u N). tr(s S V HH V +HH ) = N λ t(1+ (s 1 v 1 + +s iv i )u t ). (3) t=1 Our obective is to minimize the integral (), which forces us to enlarge (3). More precisely, our obective is to enlarge the worst case of (3), where the worst case means that the value of (3) is minimized. Since 0 (s 1,...,s i ) d (i) 1 A d(i) i A, one worst case is that s k = 0 for k = 1,,...,i,k q for which (3) becomes N N N λ t(1+ s q v qu t ) = λ t + s q λ t v qu t. (4) t=1 t=1 Thelastterm in the aboveis, infact, the squareofthe weightedlengthofthe proectionofthe vectorv q on the N-dimensional subspace U generated by {u 1,...,u N } with weights (λ 1,...,λ N ). To enlarge this term is equivalent to pushing the vector v q towards subspace U, which turns out to require subspace V generated by vectors {v 1,...,v i } closing to U, because index q can be arbitrarily chosen from 1,,...,i. Recalling the fact that the dimension of these two spaces might be different, the above can be stated as follows: the maximal angle from the set of all angles between any vector in V and the subspace U, should be as small as possible. This consideration produces the following design criterion for pre-codebook V i : to design M i unitary matrices V (i), = 1,,...,J 1 such that the minimal angle between any pair of spaces generated by the column vectors of these matrices is as large as possible. Two remarks are necessary. The first one is that this criterion is independent to, although the assumption on a given is made at the beginning of the subsection. Thus, we can remove this assumption and regard this criterion as a general criterion of designing V i. The second is that following this criterion to design pre-codebooks is not easy because even it might be difficult to get the minimal angle. Although there is much work guided by similar criteria, for example, using Riemannian distance or choral distance, there is none that design pre-codebooks by following the criterion above exactly, to the best of our knowledge. Therefore, we will propose more practical criterion of the design. Todoso,letusgobacktotheformula(4)andnoticethatthevectorsu 1,...,u N areorthogonaltoeach other. Duetothisorthogonally,itisimpossibleforafixedindexq toenlarge v qu i forallisimultaneously. This fact forces us to enlarge one from the set { v q u 1, v q u,..., v q u N }. By considering that q can be arbitrary chosen from 1,,...,i, a reasonable way to proceed is to enlarge v q u q for q = 1,,...,i. This way guides the following design criterion. Before writing down this criterion clearly, some notation is needed. For given unit vectors {v 1,v,...,v k }, the Voronoi cell U t for each v t is defined as U t = {p C M ; p = 1, p v t p v l,l = 1,...,k;l t}. (5) t=1

8 Wang H Q, et al. Sci China Inf Sci June 013 Vol :8 Moreover, Voronoi cells Q t (t = 1,,...,J 1 ) for given M i unitary matrices V 1,V,...,V J1 is defined as the set consisting of all unitary matrices in the set U t,1 U t,i, where U t, is the Voronoi cell of v t, corresponding to the unit vectors {v 1,,v,,...,v J1,}, and v t, is the -th column of the matrix V t. Also we define the numbers α,s for = 1,,...,J 1 and s = 1,,...,i as and α,s = max{α : u t v,s α for any U Q and u t is the t-th column of U} (6) α = min{α,s : s = 1,,...,i}. (7) Thus, the design criterion can be stated as to design J 1 M i unitary matrices {V 1,V,...,V J1 } such that the numbers α are as large as possible, or, such that the number α is as large as possible, where α = min{α 1,α,...,α i }. (8) The basic idea behind this criterion is that for any M N unitary matrix U, there always exists a unitary matrix, sayv, in the designed M iunitary matricesv 1,V,...,V J1, suchthat V U approaches the i N unit, wherethe unit means that the elements takesvalue 1 in the main diagonaland 0 otherwise. In fact, we have v,t u t α t α, t = 1,,...,i, where v,t is the t-th column of V and u t is the t-column of U. Guided by this criterion, a feasible design method can be introduced that is given in the next subsection. Indeed, by comparing with the design criterion given in [], we find that we have reached the same design criterion, although we have different environments for the systems. Thus, in the next subsection, we adopted this design method. 3.3 Method of designing pre-codebook V i As the first step in the design, let us take i positive integersr 11,r 1,...,r 1i such that r 11 +r 1 + +r 1i = r 1 and put J 1s = r1s for s = 1,,...,i. Asthe secondstep, noticingthat onlythe valueu t v,t isrelatedtoparameterα,t, whereu i andv,t are M-dimensionalcomplexunitvectors,weselectJ 11 M-dimensionalcomplexunitvectorsv 1,1,v 1,,...,v 1,J11 such that the maximal value among v 1,t v 1,s (1 t s J 11 ) is as small as possible. As the third step, because v 1,s is a vector in C M, we can find M M unitary matrices Q 1s for s = 1,,3,...,J 11, such that v 1,s = Q 1s v 1,1, (30) where Q 11 = I M. Denoting Q 1 = {v C M : v v 1,1 = 0}, then Q 1 is a subspace which is orthogonal to v 1,1. As the fourth step, we select J 1 points v,1,v,,...,v,j in the subspace Q 1, such that the maximum among the values v,t v,s (1 t s J 1 ) is as small as possible. Set v,s,t = Q 1s v,t, 1 s J 11, 1 t J 1. (31) Notice that for any fixed integer s, we have v,s,t v,s,l = v,t v,l because Q 1s is unitary. As the fifth step, define Q = {v C M : v v 1,1 = 0,v v,1 = 0}. Thus, Q is also a subspaceorthogonalto the vectorsv 1,1 and v,1. Also we can find unitary matrices Q s (s = 1,...,J 1 ) with Q 1 = I M, such that Q s Q 1 = Q 1 for every s and Q s v,1 = v,s (s = 1,...,J 1 ). Repeating the above steps, we can obtain the vectors v t,1,,..., t for 1 t i. Using these vectors, we can obtain unitary matrices V as follows V φ(1,,..., i) = (v 1,1,v,1,,...,v i,1,,..., i ), (3)

9 Wang H Q, et al. Sci China Inf Sci June 013 Vol :9 where 1 t J 1t and φ is any biection from the set {( 1,,..., i ) 1 t J 1t } to the set {1,,...,J 1 }. Following this method, we know that designing pre-codebooks is equivalent to performing a packing on complex proective spaces CP k (k = 1,,...,M). Because CP k has a simpler structure than a general Grassmannian manifold, it is reasonable to expect that packing on this space should become easier than that for a Grassmannian manifold, which has been proposed in constructing pre-codebooks in some previous reports; for example, Refs. [7 9], on packing in CP M. Another merit of the above designing method is that, because of the designing structure, the encoding complexity on receiver side is reduced greatly. More precisely, the exponential encoding complexity will become linear. The details will be explained in the following subsections. A similarity exists between the method above and the sequential vector quantization (SVQ) given in [30]. Let us here clarify some differences between these two methods. First, the SVQ is a designing method of pre-codes aimed to increase mutual information, while our goal of the design is to decrease the error probability of the system. Second, the SVQ is dependent on the channel realization H. That is, different channel matrices need a different pre-code design. Thus, the design cannot be implemented on an off-line state, while our design can be done off-line. Therefore, the encoding complexity at the receiver side will be high in the SVQ. Third, when the transmitter receives the feedback information, it needs to construct matrices which depend on this information. Therefore, this encoding complexity is also high. In contrast, in the method above, encoding entails ust picking some vectors. 3.4 Optimal operating system Up to this point, we have on the one hand the criterion and design method for pre-codes V (i) (i = 1,,...,N). Thus, by following both, the pre-codebooks can be designed. On the other hand, according tothe firstconclusionin Theorem1, is also determined. Therefore, in principle, we have the following operating system. The optimal operating system. Step 1 According to the transmission rate R, for each i, 1 i N, decide the constellations S (i) = S (i) 1 S (i) S (i) i. Step Design the pre-codebooks V (i) with size J 1 for each i, 1 i N. Step 3 Construct the codebooks C (i), 1 i N = 1,,...,J 1 as follows C (i) = {(s 1,s,...,s i )(V (i) ) : (s 1,s,...,s i ) S i }. (33) Step 4 The transmitter sends training signals and the receiver estimates the instant channel H while it receives these training signals. Step 5 The receiver determines the indices i and by using the formula (16), and sends these indices back to the transmitter. Step 6 The transmitter transmits data using the codebook C (i) when it receives the indices i and. Step 7 The receiver decodes data using ML-decoding based on the codebook C (i). Although this operating system is theoretically optimal, the complexity in determining index pair (i,) is very high. Let us look at Step 5. To evaluate i and for a channel realization, the receiver needs to obtain the pair of codewords that achieves the minimal distance for each t (t = 1,,...,N) and s (s = 1,,...,J 1 ); then, among these pairs, select maximum values of i and. Thus, for a system with parameters M,N,R,J 1, 1 NJ 1 R ( R 1) calculations are needed. For example, in the system with M = N =,R = 8, and r = 6, = comparing calculations are needed. Obviously, this complexity increases exponentially with R and r. This situation forces us to figure out a way to reduce this encoding complexity. To do so, a reasonable method is to determine indexes i and separately, which is stated in the following subsections.

10 Wang H Q, et al. Sci China Inf Sci June 013 Vol : Direct method to determine index i To develop a direct method to determine i for a given instant channel H, let us go back to the analysis given in formulae (3) and (4). We adopt the idea given there and still consider the worst case. From (4), a lower-bound for one worst case for any index i (i = 1,,...,N) is as follows N t=1 λ t(1+ s (i) q v (i) q u t ) N t=1 λ t + s (i) q λ qα N t=1 λ t +α (d (i) m,q) λ q, (34) where the first inequality is from (9) and the second is from the fact that d (i) m,q is the minimal distance on S q (i). Notice that the last term is not related to the vector V (i). Thus, deciding on an index i can be made independently. To determine the index i, we need to choose the best among the worst cases. For a fixed index, say t, from the formula above, the minimum of the last term above is N k=1 λ k +α (min{d (t) m,1 λ 1,d (t) m, λ,...,d (t) m,tλ t }). (35) Therefore, i should be chosen as the largest one among the those above for t = 1,,...,N. That is, i = arg max t=1,,...,n min{d(t) m,1 λ 1,...,d (t) m,tλ t }. (36) Obviously, the complexity in determining the index i by the formula above is very low. Let us now turn to determine index. 3.6 Method with low complexity to determine index Deciding on a value for is equivalent to choosing a unitary matrix V (i) from the pre-codebook V i, once the index i is given for a specific channel realization H. Suppose that the pre-codebook is designed by following the prescription given in the Subsection 3.3. Notice that the construction of this pre-codebook is finished column by column. Thus, the matrix V (i) can also be selected column by column. i) Let UDQ be the SVD of H and put U = (u 1,u,...,u N ), where u t is the t-th column of U. ii) Calculate values (v (i) 1,t ) u 1 for t = 1,,...,J 11, and identify the index, say 1, such that (v (i) 1, 1 ) u 1 is maximal. iii) Calculate values (v (i), 1,t ) u for t = 1,,...,J 1, and identify the index, say, such that (v (i), 1, ) u is maximal. iv) Repeating the above, until values 1,,..., i are obtained. v) Put V (i) = (v 1,1,v,1,,...,v i,1,,..., i ). Clearly, to determine V, it is enough to do J 11 + J J 1i calculations of inner products of vectors, while it needs J 1 (= J 11 J 1 J 1i ) calculations in general. 3.7 Operating system with lower encoding complexity We can summarize the above and describe an operate system with lower encoding complexity: Operating system with lower encoding complexity: Step 1 According to transmission rate R, for each i, 1 i N, determine the constellations S i = S (i) 1 S (i) S (i) i. Step Design pre-codebooks V i with size J 1 for each i, 1 i N. Step 3 The transmitter sends training signals. The receiver estimates the instant channel H once on receiving training signals from the transmitter, and then performs a SVD for H, that is, H = UDQ. Step 4 The receiver determines the index value i according to formula (36). Step 5 For this index value i, the receiver chooses index using the procedure stated in Subsection 3.6.

11 Wang H Q, et al. Sci China Inf Sci June 013 Vol :11 Step 6 The receiver sends both i and values back to the transmitter. Step 7 The transmitter transmits data in the coherent time period by using codebook C (i) on receiving the i and values. Step 8 The receiver decodes data using ML-decoding based on the codebook C (i). Compared with the optimal operating system given in Subsection 3.3, the complexity of obtaining i and for a given channel H is dramatically reduced. In this system, we determine i based on the formula (36) independently of. After that, is determined according to the criterion and the special construction given in Subsection 3.6. Thus, the complexity of the encoding at the receiver side is linear with feedback rate. As a cost in reducing the complexity, performance will suffer. However this loss is not significant because our proposed encoding methods above are based on an analysis of the performance. In fact, this point will be confirmed again by results of simulations given in the next section. Finally, we make some comparison between our method and that given in [3]. In that paper, based on a maximization of the receive minimal distance, a method to determine indices i and is given as follows (The details can be found in formulae (14) and (15) on page 3677 of the paper) F t = argmax V V t λ t (V H), i = arg max s=1,,...,n λ s (F s H) (d (s) m,s s ), V = F i, (37) where λ t (A) denotes the t-th singular value. In comparison with our method, this method has obviously greater complexity. It needs to calculate the singularvaluesofthe matrixv H foreachpre-codebookv = V s (t) fort = 1,,...,N ands = 1,,...,J 1. In contrast, in our method, i is easily determined. After that, to determine V, it is sufficient to do J 11 +J 1 + +J 1i calculations of inner products of vectors. 4 Simulations In this section, we illustrate results of simulations to show that our proposed scheme exhibits better performance than the beamforming scheme or other schemes with a fixed number of channels used. Also, we will confirm that our scheme with fast encoding has comparable performance to the optimal scheme and almost the same performance as the scheme proposed in [3], even though our scheme has much lower encoding complexity. Simulation 1. This simulation will highlight the differences among the beamforming case (denoted as Case 1), the case that all channels (in fact, channels) are used (denoted as Case ) and the case that the channels used depend on the channel matrix H (denoted as Case 3). The simulation environment is as follows: assume that the system has two transmitting and two receiving antennas, that is, M = and N =, and R = 8, r = 6. Thus, r = 1 and r 1 = 5. In Case 1, the constellation S 1 consists of 56 complex numbers. A reasonable choice is 56-QAM with total energy 51; thus, d (1) m,1 = In this case, essentially, only the first channel is used corresponding to beamforming. For the design of r (= 64) 1 pre-codewords v, = 1,,...,64, we follow the method given in Subsection 3.3, that is, to find 64 1 vectors, such that the maximal value on the set { v t v s : 1 t < s 64} is as small as possible. To obtain these, because these vectors are in the space C, we can employ the method given in a thesis [8]. More specifically, it can be constructed as follows: Put ( ) ( ) S 1 =, S =, and take the 64 packing points a s = (a 0,s,a 1,s,a,s ),(a 0,s 1) (s = 1,,...,64) from the real - dimensional unit ball S R 3, such that the minimal distance among these 64 points is as large as

12 Wang H Q, et al. Sci China Inf Sci June 013 Vol :1 Figure 1 Simulation results on three different schemes. Figure Simulation results on the optimal and asymptotically optimal operating systems. possible. Let t s = arccos(a 0,s )/ and design the complex matrices A s as follows A s = cos(t s )I +sin(t s )T s (s = 1,,...,64), where Define precodes v as follows: T s = v=1 a v,s S v. 1 a 0,s v s = A s (1 0) t (s = 1,,...,64). In Case, the constellation S consists of 56 -dimensional complex vectors. Each component of each vector is taken from a 16-QAM; the total energy of the 16-QAM is 16. Thus, S = S () 1 S () = 16 QAM 16 QAM, and d () m,1 = d() m, = In this case, in fact, the two channels generated by the system are used simultaneously. The pre-codebook V consists of matrices A, = 1,,...,64 for the case 1 above. Distinct from the two cases above, we follow the operating system given in 3.7 for Case 3. When i = 1, the constellation is the same as S 1 above. The pre-codebook, which consists of r1 = 3 1 vectors, is constructed by the same method as the one above. When i =, the constellation is also the same as S above, and the pre-codebook, r1 = 3 matrices, is also obtained by the same method as that above. In this case, the determination of i is simple. Because d () m,1 = d() m, and λ 1 λ, we have Hence, by following formula (36), min{d () m,1 λ 1,d () m, λ } = d () m, λ. i = argmax{d (1) m,1 λ 1,d () m, λ }. Therefore, i = 1 when λ 1 /λ d () m, /d(1) m,1 =.9143, and otherwise, i =. The results of the simulations above are displayed in Figure 1. Obviously, from Figure 1, Case 3 exhibits the lowest error probability. It has almost 1.5 db gain over the beamforming case and about 4 db gain over Case. Therefore, the results show that the multimode scheme proposed in this paper has the best performance, which confirms the analysis given in the Section 3. Simulation. In this simulation, we tried to show the differences among the optimal operating systems, the method given in [3], and the operating system given in Subsection 3.7. The system environment is M = N =,T = 1 and R = 4,r = 6. If i = 1, a 16-QAM is the constellation whereas for i =, the

13 Wang H Q, et al. Sci China Inf Sci June 013 Vol :13 constellation is 4-QAM 4-QAM. The pre-codebooks V 1 and V of size 3 are generated by following the method in [8]. Figure shows that the proposed operating system given in Subsection 3.7 has about 0.8 db loss in the high SNR region compared with that for the optimal case in this special environment. Also, the figure shows that our proposed method has almost the same performance as the one shown in [3]. However, the encoding complexity of our proposed method is much lower than that given in [3], as explained in Subsection Conclusions In this paper, we investigated a MIMO system based on fixed constellations with finite-bit feedback. For this system, we propose an optimal operating system based on a performance analysis. Furthermore, to reduce the encoding complexity, an operating system with lower encoding complexity is established. Theoretical analysis and simulations show that our systems have considerable benefits over beamforming systems or systems with fixed channel use given identical situations of decoding complexity. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos , ), Natural Science Foundations of Zheiang Province (Grant No. Y ), and Scientific Research Foundation for Returned Scholars, Ministry of Education of China. References 1 Goldsmith A, Jafar S A, Jindal N, et al. Capacity limits of MIMO channels. IEEE J Sel Area Comm, 003, 1: Visotsky E, Madhow U. Space-time transmit precoding with imperfect feedback. IEEE Trans Inform Theory, 003, 47: Blum R S. MIMO with limited feedback of channel state information. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing. Hong Kong: IEEE, Zhou S, Giannakis G B. Adaptive modulation for multiantenna transmissions with channel mean feedback. IEEE Trans Wirel Commun, 004, 3: Zhang L, Wu G, Li S Q. Capacity bounds of transmit beamforming over MISO time-varying channels with imperfect feedback. Sci China Inf Sci, 010, 53: Love D J, Heath R W, Lau V K N, et al. An overview of limited feedback in wireless communication systems. IEEE J Sel Area Comm, 008, 6: Lau K N, Liu Y, Chen T A. On the design of MIMO blockfading channels with feedback-link capacity constraint. IEEE Trans Commun, 004, 5: Roh J C, Rao B D. Multiple antenna channels with partial channel state information at the transmitter. IEEE Trans Wirel Commun, 004, 3: Roh J C, Rao B D. Transmit beamforming in multiple-antenna systems with finite rate feedback: a VQ-based approach. IEEE Trans Inform Theory, 006, 5: Roh J C, Rao B D. Design and analysis of MIMO spatial multiplexing systems with quantized feedback. IEEE Trans Signal Proces, 006, 54: Zheng J, Duni E R, Rao B D. Analysis of multiple-antenna systems with finite-rate feedback using high-resolution quantization theory. IEEE Trans Signal Proces, 007, 55: Xia P, Giannakis G B. Design and analysis of transmit-beamforming based on limited-rate feedback. IEEE Trans Signal Proces, 006, 54: Jafar S A, Srinivasa S. On the optimality of beamforming with quantized feedback. IEEE Trans Commun, 007, 55: Mukkavilli K K, Sabharwal A, Erkip E, et al. On beamforming with finite rate feedback in multiple-antenna systems. IEEE Trans Inform Theory, 003, 49: Narula A, Lopez M J, Trott M D, et al. Efficient use of side information in multiple-antenna data transmission over fading channels. IEEE J Sel Area Comm, 1998, 16:

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