Capacity and Performance of Adaptive MIMO System Based on Beam-Nulling

1 Capacity and Performance of Adapti v e MIMO System Based on Beam-Nulling Mabruk Gheryani, Zhiyuan W u, and Y ousef R. Sha yan Concordia Uni versit y , De partment of Electrical Engineering Montreal, Quebec, H4G 2W1, Canada email: (m gherya, zy wu, yshayan)@ece .concordia.ca Abstract In this paper, we prop ose a schem e called “beam-nullin g” fo r MIMO adaptation. In the beam-nulling scheme, the eigenvector of the weakest subchann el is fed back and then signals are sent over a generated subspace orthogo nal to the weakest subchannel. Theoretical analysis and numerical results show that the capacity of b eam-nu lling is closed to the o ptimal w a ter-filling at me dium SNR. Additionally , signal-to-interf erence-p lus-noise ratio ( SINR) of MMSE recei ver is deri ved fo r bea m-nulling . Th en the paper p resents the associated average bit-err or rate (BER) o f beam-nu lling numerica lly which is v erified by simulation. Simulation results ar e also provided to comp are beam- nulling with beamfo rming. T o im prove p erform ance f urther, beam -nulling is con catenated with linear dispersio n code. Simulation results are also provid ed to compar e the c oncatenated beam-n ulling scheme with the beamfo rming scheme at the same data rate. Additiona lly , the existing beamf orming and new pro posed beam-n ulling can be extended if m ore than on e eigenv ector is available at the transmitter . The new extended schemes are called multi- dimensiona l (MD) beamform ing an d MD be am-nulling . Theo retical analysis and numerical results in term s of capacity are also p rovided to evaluate th e new extended schemes. Simulation results show that the MD scheme with LDC can outper form the MD scheme with STBC sign ificantly when the d ata rate is h igh. I . I N T R O D U C T I O N Since the d iscovery of multip le-input-multi ple-output (MIM O) capacity [1] [2], a lot of research efforts hav e been put into t his field. It has been recognized that adaptiv e techniques proposed for sing le-input- single-output (SISO) channels [3] [4] can also be applied to i mprove MIMO channel capacity . The i deal scenario is that the transmit ter has full kn owledge of channel state information (CSI). Given this perfect CSI feedback, the o riginal MIMO channel can b e con verted to multiple u ncoupled SISO channels via singular value d ecomposition (SVD) at the transmitter and the receiv er [1]. In other words, the origi nal MIMO channel can be decompo sed into severa l orthogon al “spatial s ubchannels” with various propagation gains. T o achie ve better performance, various schem es can be implemented depending on the a va ilabilit y of CSI at the transmitter [5]- [17]. If the transm itter has full k nowledge of the channel matrix, i.e., full CSI, the so-called “water -filling” (WF) pri nciple is performed on each spatial s ubchannel to maximize the channel capacity [1]. Thi s scheme is opti mal in thi s case. V arious WF-based schemes hav e been proposed, such as [9] [11]. For the WF- based scheme, t he feedback bandwid th for the full CSI grows wi th respect to the number of transm it and receiv e antennas and the performance is often very sensitive to channel estimation errors. T o mitigate these disadvantages, various beamformi ng (BF) techniques for MIMO channels have also been in vestigated int ensiv ely . In an adaptive beamforming scheme, the complex weights of th e transmit antennas are fed b ack from the receiver . If only on e eigenv ector can be fed back, eigen-beamforming [12] is optimal. The eigen-beamforming scheme only appl ies to the strongest s patial subchannel but can achie ve full diver sity and high signal -to-noise ratio (SNR) [12]. Also, in practice, the eigen-beamforming scheme has to cooperate with the ot her adaptive parameters to i mprove performance and/or d ata rates such as con stellation and coding rate. Th ere are also other b eamforming s chemes based on various criteria. Examples of such schemes are [12] - [22]. Not e that the con ventional beamforming is optimal in terms 2 of m aximizing the SNR at the receiv er . Howe ver , it is sub-optim al from the MIMO capacity perspective, since only a si ngle data stream, as opposed to parallel st reams, is transmitt ed through the MIMO channel [13]. In this paper , we propose a new technique call ed “beam-nul ling” (BN). This scheme us es t he same feedback bandwid th as beamformin g, that is, only one eigen vector is fed back to the transmit ter . The beam-nulling transmitter is i nformed by the weakest spatial subchannel and, wh ere both transmitter and recei ver know how to generate the same spati al s ubspace, sends si gnals over a generated spatial su bspace orthogonal to the weakest subchannel. Although t he transmitted sym bols are “precoded” acc ording to the feedback, b eam-nulling is dif ferent from the other existing p recoding schemes with limited feedback channel, which are i ndependent of the i nstantaneous channel but the optim al precoding depends on the instantaneous channel [14] [15]. Using t his new t echniques in stead of the optimal water-fi lling schem e, t he l oss of channel capacity can be reduced. This paper also addresses th e performance of beam-nulli ng. T o achie ve better performance, beam-nulling can be concatenated with the oth er space-tim e (ST) codi ng schemes, such as space-time trellis codes (STTCs) [23], s pace-time block codes (STBCs) [24] [25] and l inear d ispersion codes (LDCs) [26]- [29], etc. For si mplicity and flexibilit y , LDCs are preferable. W e provide numerical and simulation results are provided to demonstrate the merits of the ne w proposed scheme. Additionally , if more than one eigen vector , e.g. k eigen vectors, can be a vailable at the transmitter , the existing beamformin g scheme and the proposed beam-nulling scheme can be further extended, respectively . The e xtended schemes will exploit or discard k spati al s ubchannels and t hey will be referred to as “multi-dim ensional (MD)” beamforming and “multi-dimensional” beam-nulling, respectively . This paper will be or ganized as follo ws. Our channel model is presented in Section II. In Sec tion III, four power allocation strategies, i.e., equal powe r , water -filling, eigen-beamforming, and a new po wer allocation strategy called “beam-nulli ng” are s tudied and compared in t erms of channel capacity . In Sec tion IV, bit error rate (BER) of the proposed beam-nulling s cheme us ing MMSE detector is st udied and verified. The propo sed scheme is compared wi th the eigen-beamform ing scheme at various data rates in terms of BER. Beam-nulling concatenated wi th L DC is proposed and ev aluated. In Section V, extended adaptive frame works, i.e., MD beamforming and M D beam-nulling, are proposed. Capacity and performance of these two schemes are discussed and compared. T o improve performance further and maint ain reasonable complexity , MD schemes concatenated with linear space-time codes, such as STBC and LDC, are proposed and ev aluated. Finally , in Section VI, conclusi ons are dra wn. I I . C H A N N E L M O D E L In this s tudy , the channel is assu med to be a Rayleigh flat fading channel with N t transmit and N r ( N r ≥ N t ) receive antennas. W e d enote the compl ex gain from the transm it antenna n to the receiver antenna m by h mn and collect them to form an N r × N t channel matrix H = [ h mn ] . The channel is known perfectly at the receiv er . The entries in H are assum ed to be i ndependent and identicall y distributed ( i.i.d. ) symmetrical complex Gaussi an random variables with zer o mean and unit var iance. The s ymbol v ector at the N t transmit antennas is denoted by x = [ x 1 , x 2 , . . . , x N t ] T . According to information theory [30], the optimal distribution of the transmitted symbols is Gaussian. Thus, the elements { x i } of x are assumed to be i .i.d. Gaussian v ariables with zero mean and uni t v ariance, i.e., E ( x i ) = 0 and E | x i | 2 = 1 . The singul ar -va lue decompos ition of H can be written as H = UΛV H (1) where U is an N r × N r unitary matrix, Λ i s an N r × N t matrix with singular v alues { λ i } on the diagonal and zeros off the diagonal, and V is an N t × N t unitary matrix. For con venience, we assume λ 1 ≥ λ 2 . . . ≥ λ N t , U = [ u 1 u 2 . . . u N r ] and V = [ v 1 v 2 . . . v N t ] . { u i } and v i are column vectors. From equation (1), the origi nal channel can be consi dered as cons isting of uncoupled parallel subchannels. Each 3 subchannel corresponds to a singular value of H . In the following context, the subchannel is also referred to as “spatial subchannel”. F or instance, one spatial subchann el corresponds to λ i , u i and { v i } . I I I . P OW E R A L L O C A T I O N A M O N G S PA T I A L S U B C H A N N E L S W e ass ume that the to tal transmitted po wer is constrained to P . Given the power constraint, diffe rent power allocation am ong spatial subchannels can aff ect the channel capacity tremendous ly . Depending on power allocation s trategy among s patial subchannels, four schem es are presented whi ch are equal power , water -filli ng, eigen-beamforming , and the new power allocation whi ch is beam-nul ling. If the transmit ter has no knowledge abo ut the channel, t he most judicious strategy is to allocate the power to each t ransmit antenna equally , i.e., equal power . In this case, the receiv ed signals can be written as y = s P N t Hx + z (2) z is t he additive white Gaussian noise (A WGN) vector with i.i.d. symmetrical complex Gaussian element s of zero mean and v ariance σ 2 z . The associated ergodic channel capacity can be written as [1] ¯ C eq = E " N t X i =1 log  1 + ρ N t λ 2 i  # (3) where E [ · ] denotes expectation with respect to H and ρ = P σ 2 z denotes SNR. If the transmitter has full knowledge about the channel, the most judicious strategy is to a llocate the po wer to each spatial subchannel by water -filling principle [1]. In water -fillin g scheme, the received signals can be written as ˜ y i = q P i λ i x i + ˜ z i (4) where N t P i =1 P i = P as a constraint and ˜ z i is the A WGN random variable w ith zer o mean and σ 2 z var iance. Follo wing the meth od of Lagrange m ultipli ers, P i can be found [1] and the total er godic channel capacity is ¯ C w f = E " N t X i =1 log 1 + P i σ 2 z λ 2 i !# (5) T o s a ve feedback bandwi dth, beamforming can be considered. F or the MIMO model, the optimal beamforming is called “eigen-beamforming” [12], or simpl y beamforming. W e ass ume one symbo l, saying x 1 , is transmitted. At the receiver , the receiv ed vector can be written as y 1 = √ P Hv 1 x 1 + z 1 (6) where z 1 is the additiv e wh ite Gauss ian noise v ector wi th i.i.d. symmetrical complex Gaussian elements of zero mean and v ariance σ 2 z . The associated ergodic channel capacity can be written as ¯ C bf = E h log  1 + ρλ 2 1 i (7) The eigen-beamforming scheme can sa ve feedback bandwi dth and is opti mized in terms of SNR [22]. Howe ver , since only one spatial subchannel is considered, this scheme suf fers from loss of channel c apacity [13], especiall y when the n umber of antennas grows. 4 Ant-N t Ant-1 Ant-1 Ant-N r Channel Estimation g 1,1 g Nt,1 ~ x Nt-1 y 1 x 1 g 1 g Nt-1 g 1,Nt-1 g Nt,Nt-1 U H Generate Φ v Nt-1 Fig. 1. beam-nulling scheme. A. Beam-Nulling The eigen-beamforming scheme can sa ve feedback bandwi dth and is opti mized in terms of SNR [22]. Howe ver , since only a single spatial subchannel is considered, this scheme suffer s from loss of channel capacity [13], especially when the number of antennas grows. Inspired by the eigen-beamforming schem e, we will propose a new beamform ing-like scheme called “beam-null ing” (BN). This scheme uses th e sam e feedback b andwidth as beamforming, that i s, o nly one eigen vector is fed back t o the transmit ter . Unlike the eigen-beamformin g scheme in which only th e best spatial su bchannel is considered, the beam-nulling scheme discards onl y the worst spatial su bchannel. Hence, in compariso n with the optim al water -filling scheme, t he loss o f channel capacity can be reduced. In this scheme as s hown in Fig. 1, the eigen vector associated with the minim um singular v alue from the transmi tter side, i.e., v N t , i s fed back to the transm itter . A su bspace orthogonal t o the weakest spatial channel is constructed so that the fol lowing condition is satisfied. Φ H v N t = 0 (8) The N t × ( N t − 1) matrix Φ = [ g 1 g 2 . . . g N t − 1 ] spans the subspace. Note t hat the method to cons truct the subspace Φ should also b e known to th e receiv er . Here is an example of cons truction of t he orth ogonal subspace. W e con struct an N t × N t matrix A = [ v N t I ′ ] (9) where I ′ = [ I ( N t − 1) × ( N t − 1) 0 ( N t − 1) × 1 ] T . App lying QR decom position to A , we have A = [ v N t Φ ] · Γ (10) where Γ i s an upper triangular m atrix with t he (1,1)-th entry equal to 1. Φ is the subspace orthogonal to v N t . At the transmitter , N t − 1 symbols denoted as x ′ are transmit ted over the orthogon al subspace Φ . The recei ved signals at the recei ver can be written as y ′ = s P N t − 1 HΦx ′ + z ′ = c Hx ′ + z ′ (11) where z ′ is additiv e white Gauss ian noise vector with i.i.d. sym metrical com plex Gaussian elem ents of zero mean and variance σ 2 z and c H = q P N t − 1 HΦ . Substitutin g (1) into (11) and multiply ing y ′ by U H , result s in e y = s P N t − 1 Λ B 0 T ! x ′ + e z (12) 5 where e z is addi tiv e white Gauss ian noise vector with i.i.d. sy mmetrical complex Gauss ian elements of zero mean and variance σ 2 z . W ith th e condition in (8), V H Φ = B 0 T ! (13) where B =        v H 1 g 1 v H 1 g 2 . . . v H 1 g N t − 1 v H 2 g 1 . . . . . . . . . . . . . . . . . . . . . v H N t − 1 g 1 . . . . . . v H N t − 1 g N t − 1        (14) B is an ( N t − 1 ) × ( N t − 1) unitary matrix. From (12), the av ailable spatial channels are N t − 1 . Since the weakest spatial sub channel is “nulled” in t his scheme, power can be allocated equally amon g the other N t − 1 subchannels. Equation (12) can be rewritten as e y ′ = s P N t − 1 Λ ′ Bx ′ + e z ′ (15) where e y ′ and e z ′ are column vectors with the first ( N r − 1) elements of e y and e z , respectiv ely , and Λ ′ = diag [ λ 1 , λ 2 , . . . , λ ( N t − 1) ] . From (15), the associated ergodic channel capacity can be found as ¯ C bn = E " N t − 1 X i =1 log  1 + ρ N t − 1 λ 2 i  # (16) As can be seen, the beam-nullin g schem e only n eeds one eigen vector t o be fed back. Howe ver , sin ce only t he worst spatial subchannel is dis carded, thi s scheme can increase channel capacity significantly as compared to the con ventional b eamforming scheme. B. Compariso ns Among the F our Schemes In this section, we compare the new proposed beam-nulling scheme wi th the other schemes, i .e., equal power , b eamforming and water -fillin g schemes. W ater- filling is th e optimal solution among the four schemes for any SNR. Diffe rentiating the abov e ergodic capacities with respect to ρ respecti vely , we ha ve ∂ ¯ C eq ∂ ρ = E   N t X i =1 1 ρ + N t λ 2 i   (17) ∂ ¯ C bf ∂ ρ = E   1 ρ + 1 λ 2 1   (18) ∂ ¯ C bn ∂ ρ = E   N t − 1 X i =1 1 ρ + N t − 1 λ 2 i   (19) The dif ferential will also be referred to as “slope”. Since the second order differentials are negati ve, the above ergodic capacities are conca ve and mo notonically increasing with respect to ρ . W ith the fact th at λ 1 ≥ λ 2 . . . ≥ λ N t , it can be readily checked that th e sl opes of er go dic capacities associate wi th equal p owe r and beam-nulling are bounded as foll ows. E   N t ρ + N t λ 1   ≥ ∂ ¯ C eq ∂ ρ ≥ E   N t ρ + N t λ N t   (20) E   N t − 1 ρ + N t − 1 λ 1   ≥ ∂ ¯ C bn ∂ ρ ≥ E   N t − 1 ρ + N t − 1 λ ( N t − 1)   (21) 6 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 SNR (dB) Capacity (bit/s/Hz) 5x5 EQ WF BF BN Fig. 2. 5 × 5 Rayleigh fading channe l. For the case of N t = 2 , beamformin g and beam-nulling have the same capacity for any ρ as can be seen from equations of capacity and s lope. If ρ → 0 , equivalently at low SNR, i t can be easily found that ∂ ¯ C bf ∂ ρ ≥ ∂ ¯ C bn ∂ ρ ≥ ∂ ¯ C eq ∂ ρ , ρ → 0 (22) If ρ → ∞ , equi valently at high SNR, it can be easily found that ∂ ¯ C eq ∂ ρ ≥ ∂ ¯ C bn ∂ ρ ≥ ∂ ¯ C bf ∂ ρ , ρ → ∞ (23) Note that ¯ C bf = ¯ C bn = ¯ C eq = 0 when ρ = 0 or minus infinity in d B. Hence, at medium SNR, ∂ ¯ C bn ∂ ρ has the l ar gest v alue compared to ∂ ¯ C bf ∂ ρ and ∂ ¯ C eq ∂ ρ . Therefore, for low , medium and high SNRs, beamforming, beam-nulling and equal power ha ve the largest capacities, respecti vely . In Fig. 2, capacities of water -filling, beamforming, beam-nullin g and equal power are compared over 5 × 5 Rayleigh fading channels, respectiv ely . No te that since SNR is measured in dB, the curves become con vex. In these figures, “EQ” st ands for equal power , “WF” stands for water -filling, “BF” stands for beamforming and “BN” stands for beam-nulling. As can be seen, the water -fill ing has the best capacity at any SNR region. The other schemes perform differently at diffe rent SNR regions. At l ow SNR, the beamforming is the closest to t he o ptimal water-fi lling, e.g., t he SNR region below 3 . 5 dB for 5 × 5 fading channel. Note that at l ow SNR, t he water -filling scheme may only allocate power to one or two spatial subchannels. A t m edium SNR, the proposed beam-nulling i s t he clo sest to the optimal water -filling, e.g., the SNR region from 3 . 5 dB to 1 6 dB for 5 × 5 fading channel. The beam-nu lling scheme only discards the weakest spatial subchannel and all ocates p owe r to the other s patial sub channels. As can be seen from the numerical results , the beam-nulling scheme performs better than the other schemes in this case. Note that at high SNR, the equ al power scheme wi ll con ver ge with the water-filling scheme. I V . P E R F O R M A N C E O F B E A M - N U L L I N G A. MMSE Det ector The close-form error probability for the optimal ML receiver is diffi cult t o establis h. Other suboptimal recei vers can also be implemented. The MMSE detector is especiall y pop ular due to its low complexity and g ood performance [31] [32]. In the following context, BER of the MMSE detector is analyzed for the beam-nu lling scheme. 7 Let us define c H = q P N t − 1 HΦ and ˆ h i is the i -th column of c H . Equatio n (11) can also be written as y ′ = ˆ h i x i + X j 6 = i ˆ h j x j + z ′ (24) where x i is the i-th element of x ′ . W ithou t loss of generality , we con sider the detection of one symbol, say x i . W e coll ect th e rest of th e symbols into a column vector x I and denote c H I = [ ˆ h 1 , .., ˆ h i − 1 , ˆ h i +1 , ..., ˆ h N t − 1 ] as t he m atrix obtained by removing t he i -th column from c H . A linear MMSE detector [32] [33] is applied and th e correspondin g output is gi ven by ˆ x i = w H i y = x i + ˆ z i . (25) where ˆ z i is the noise t erm of zero mean. ˆ z i can be approxim ated to be Gaussian [32]. The corresponding w i can be found as w i =  ˆ h i ˆ h H i + R I  − 1 ˆ h i ˆ h H i  ˆ h i ˆ h H i + R I  − 1 ˜ h i (26) where R I = c H I c H H I + σ 2 z I . Note th at the scaling factor 1 ˆ h H i ( ˆ h i ˆ h H i + R I ) − 1 ˆ h i in the coeffi cient vector of th e MMSE detector w i is added to ensure an unbiased detection as ind icated by (25). The variance of the noise term ˆ z i can be found from (25) and (26) as ˆ σ 2 i = w H i R I w i (27) Substitutin g th e coeffi cient vector for the MM SE detector in (26) into (2 7), the variance can be written as ˆ σ 2 i = 1 ˆ h H i R − 1 I ˆ h i (28) Then, the SINR of MM SE associated with x i is 1 / ˆ σ 2 i . γ i = 1 ˆ σ 2 i = ˆ h H i R − 1 I ˆ h i (29) The clos ed-form BER for a channel model such as (25) can be found in [34]. The av erage BER over MIMO fading channel for a given const ellation can b e found for beam-nulling as follows. B E R av = E γ i " 1 N t − 1 X i B E R ( γ i ) # (30) The closed-form formula for the average B ER in (30) d epends on the distribution of γ i , which i s diffic ult to determine. Here, the above aver age BER is calculated numerically . For example, the av erage BER for 2 η -PSK is B E R av = E γ i " 1 N t − 1 X i 2 η Q  q 2 η γ i sin( π 2 η )  # (31) and th e av erage BER for rectangular 2 η -QAM is B E R av = E γ i   1 N t − 1 X i 4 η Q   s 3 η γ i 2 η − 1     (32) where Q ( · ) denotes the Gaussian Q -function. In Fig . 3, numerical and sim ulation results are compared for 8 PSK over 3 × 3 Rayleigh fading channel and QPSK over 4 × 4 Rayleigh fading channel, respectively . As can be seen, t he n umerical and simulat ion results match well. 8 0 5 10 15 20 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR BER 8PSK−Simulation 8PSK−Numerical (a) 3 × 3 , 8PSK 0 5 10 15 20 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) BER QPSK−Simulation QPSK−Numerical (b) 4 × 4 , 4PSK Fig. 3. Numerical and simulation results for beam-nulling scheme. 0 5 10 15 20 25 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) BER BF−8PSK BN−BPSK−MMSE BN−BPSK−ML (a) R =3 0 2 4 6 8 10 12 14 16 18 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) BER BF−64QAM BN−QPSK−MMSE (b) R=6 Fig. 4. Comparison o ver 4 × 4 Rayleigh fading channel. B. P erforman ce Comparison Between Beamforming and Beam-nul ling In Fig. 4, si mulation results are compared for va rious data rates R over 4 × 4 Rayleigh fading channels. In t he following simul ations, a data rate R is measured i n bits per channel u se. The beamformin g scheme is equiv alent t o a SISO channel us ing a m aximum ratio combining (MRC) receiv er [14]. For the beam- nulling scheme, the optimal ML recei ver and the suboptimal MMSE rece iv er are used. From Fig. 4, if the data ra te is lo w , i.e., con stellation size is low , beamformin g o utperforms beam- nulling. If t he data rate is hi gh, i.e., constell ation size is high, beam-nulling outperforms beamforming at low and medium SNR, howe ver at high SNR beamformi ng outperforms beam-nulling. A lso, as can be seen, at the high data rate, e ven th e beam-nulling scheme with suboptimal MMSE recei ver outperforms the beamform ing scheme. C. Concatenation of Beam-nulling and LDC T o further improve t he p erformance of beam-nulli ng wit h tractable com plexity , we propos e t o concate- nate beam-null ing with a linear dis persion code. Not e that to meet error -rate requirement s, multip le levels of error protection can be implemented. In this study , we focus on space-time coding domain. In this system, the information b its are first m apped into symbols. T he symbol stream is parsed into blocks of length L = ( N t − 1) T . T he symbol vector associated with one modul ation block is denoted by 9 0 5 10 15 20 25 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) BER BF−8PSK BN−BPSK−MMSE BN−BPSK−ML BL−BPSK−MMSE BL−BPSK−ML (a) R =3 0 2 4 6 8 10 12 14 16 18 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) BER BF−64QAM BN−QPSK−MMSE BL−QPSK−MMSE (b) R=6 Fig. 5. Comparison o ver 4 × 4 Rayleigh fading channel. x = [ x 1 , x 2 , . . . , x L ] T with x i ∈ Ω ≡ { Ω m | m = 0 , 1 , . . . , 2 η − 1 , η ≥ 1 } , i.e., a comp lex const ellation of size 2 η , such as 2 η -QAM). The average symbol ener gy is assumed to be 1 , i.e., 1 2 η 2 η − 1 P m =0 | Ω m | 2 = 1 . Each symbol in a bl ock will b e mapped t o a dispersion matrix of size N t × T (i .e., M i ) and then combi ned linearly to form ( N t − 1) data streams over T channel us es. The output ( N t − 1) data streams are transmi tted only over th e subspace Φ orthogonal to th e weakest spatial channel. Th e generation of the orthogonal subspace Φ is described i n Section III-A. The recei ved signals can be w ritten as y = s P N t − 1 HΦ L X i =1 M i x i + z (33) where z is additive white Gaussian noise vector with i.i.d. s ymmetrical complex G aussian elements of zero mean and variance σ 2 z . It is worthy to note that t he tradition al beamform ing scheme cannot work with space-time coding since i t can be viewe d as a SISO channel. W e compare the concatenated scheme with the original schemes at the same data rate. In Fig. 5, simulation results are compared for v arious data rates R o ver 4 × 4 Rayleigh flat fading channels. In t he figure, “BL” denotes beam-nulling with L DC. As can be seen, beam-nulling with LDC outperforms beam-nulling without LDC using the same receiv er . The performance of beam-null ing with LDC using M MSE receiv er is close to that of beam-nul ling wi thout LDC us ing the opt imal ML recei ver . Also it can b e seen, i f data rate is low , i.e., cons tellation size is low , th e performance of b eam-nulling with LDC can approach t hat of beamforming at high SNR. If data rate is high, i.e., constellatio n size is high, beam-nu lling wi th LDC out performs beamforming even when the suboptimal M MSE receiv er is used. V . E X T E N D E D A DA P T I V E F R A M E W O R K S For the beamforming and beam-nulling schemes, only one eigen vector has been fed back to the transmitter . If more backward bandwidth is a vailable for fee dback, e.g. k eigen vectors, can be sent t o the transmitter for adaptation. W it h the feedback o f k eigen vectors, we can extend our frameworks, which will b e called multi -dimensional (MD) beamforming and MD beam-nul ling. T he ori ginal schem es can be referred to as 1D-beamforming and 1 D-beam-nulling. T o sa ve bandwi dth, k ≤ ⌊ N t 2 ⌋ should be satisfied, where ⌊·⌋ denotes rounding towards min us infinit y . That is, whether the strongest or the weakest k spatial subchannels will be fed back according to the channel condit ions. For example, at lo w SNR, k s trongest spatial sub channels will be fed back. At medium SNR, k weakest sp atial subchannels will be fed b ack. 10 A. MD Beamf orming For MD beamforming, v 1 , . . . , v k are fed back to the transmitter . k symbols, saying x k = [ x 1 , x 2 , . . . , x k ] T , are t ransmitted. At t he receiver , the recei ved vector can b e writt en as y k = s P k H [ v 1 . . . v k ] x k + z k (34) where z k is the add itive whi te Gauss ian noise vector wit h i.i.d. symmetrical complex Gaussian element s of zero mean and v ariance σ 2 z . Consequently , the associated er god ic channel capacity can be found as ¯ C k ,bf = E " k X i =1 log 1 + P k σ 2 z λ 2 i !# (35) Let ρ = P /σ 2 z denote SNR. It is readily checked t hat the capacity o f M D beamforming is also concav e and monotonicall y i ncreasing wi th respect to SNR ρ . Differentiating the abov e ergodic capacity wi th respect to ρ , we hav e ∂ ¯ C k ,bf ∂ ρ = E   k X i =1 1 ρ + k λ 2 i   (36) If ρ → 0 , equi va lently at l ow SNR, it can be easily found that ∂ ¯ C ( k − 1) ,bf ∂ ρ > ∂ ¯ C k ,bf ∂ ρ , ρ → 0 (37) If ρ → ∞ , equi valently at high SNR, it can be easily found that ∂ ¯ C k ,bf ∂ ρ > ∂ ¯ C ( k − 1) ,bf ∂ ρ , ρ → ∞ (38) Note that ¯ C k ,bf = 0 for any k when ρ = 0 or min us i nfinity in dB. Hence, at l ow SNR, the capacity of the k -D beamformi ng scheme is worse than the ( k − 1) -D beamforming scheme and whi le at high SNR, the capacity of the k -D beamform ing scheme is better than the ( k − 1 ) D b eamforming s cheme at t he cost of feedback bandwidth. B. MD Beam-nu lling For MD beam-nulli ng, similar to 1 D beam-null ing, by a certain rule, a sub space orthogonal to the k weakest spatial channel is constructed. That is, the fol lowing conditi on should be satisfied. v H n Φ ( k ) = 0 T , ∀ n = N t − k + 1 , . . . , N t . (39) The N t × ( N t − k ) matrix Φ ( k ) = [ g 1 g 2 . . . g N t − k ] spans the ( N t − k ) -dimensional subspace. At t he transmitter , N t − k symbols denoted as x ( k ) are t ransmitted only over the orthogonal subspace Φ ( k ) . The recei ved si gnals at the recei ver can b e written as y ( k ) = s P N t − k HΦ ( k ) x ( k ) + z ( k ) (40) where z ( k ) is additive white Gaussi an no ise vector with i.i.d . s ymmetrical com plex Gaussian elements of zero mean and v ariance σ 2 z . From (40), th e associated i nstantaneous channel capacity wi th respect t o H can be found as ¯ C ( k ) bn = E   N t − k X i =1 log 1 + P ( N t − k ) σ 2 z λ 2 i !   (41) 11 0 5 10 15 20 25 0 5 10 15 20 25 SNR (dB) Capacity (bit/s/Hz) EQ WF 1D−BN 2D−BN Fig. 6. MD beam-nulling o ver 5 × 5 Rayleigh fading channel. It i s readily checked that the capacity of M D b eam-nulling is also concave and monotonically increasing with respect to SNR ρ . Let ρ = P /σ 2 z denote SNR. Differentiating t he above ergodic capacity with respect to ρ , we hav e ∂ ¯ C ( k ) bn ∂ ρ = E   N t − k X i =1 1 ρ + N t − k λ 2 i   (42) If ρ → 0 , equi va lently at l ow SNR, it can be easily found that ∂ ¯ C ( k ) bn ∂ ρ > ∂ ¯ C ( k − 1) bn ∂ ρ , ρ → 0 (43) If ρ → ∞ , equi valently at high SNR, it can be easily found that ∂ ¯ C ( k − 1) bn ∂ ρ > ∂ ¯ C ( k ) bn ∂ ρ , ρ → ∞ (44) Note that ¯ C k ,bn = 0 for any k when ρ = 0 or minus infinity in dB. Hence, at low SNR, t he capacity of the k -D beam-nulling scheme is better than the ( k − 1) -D beam-nu lling scheme at t he cost of feedback bandwidth and while at high SNR , the capacity of the k -D beam-nulling scheme is worse than the ( k − 1) -D beam-nulling scheme. For example, in Fig . 6, capacities of 1D beam-nulling and 2D beam-nu lling schemes are compared with WF and equal power scheme over 5 × 5 Rayleigh f ading channel at diff erent SNR regions. At relati vely low SNR, i.e., less th an 13dB, the 2D beam-nulling scheme outperforms the 1 D b eam-nulling scheme in terms of capacity at th e price of feedback b andwidth. While at relative ly high SNR, i.e., more than 1 3dB, the 1D -beam-nulling scheme ou tperforms the 2D beam-nulling scheme as predicted. C. Capacity Comparison o f MD Sc hemes Here, ov er 5 × 5 Rayleigh fading channel, the MD schemes are compared wi th WF and equal po wer schemes as shown in Fig. 7. It can be readily check that, at relatively lo w SNR, MD beamformin g schemes are better than M D b eam-nulling schemes; whil e at relatively high SNR, the results are opposit e. Specifically , at very low SNR, i.e. less than 0dB, the 1D beamforming schem e ou tperforms the ot her MD schemes. At t he SNR region between 0d B and 5.5 dB, the 2D beamformi ng scheme out performs the ot her MD schemes. At the SNR region between 5.5dB and 12.7dB, the 2 D beam-nul ling scheme outperforms the other MD schemes. At the SNR region between 12.7dB and 23dB, t he 1D beam-nulling scheme outperforms the other MD schemes. Again, wh en SNR is more than 23dB, the equal power scheme outperforms th e other s uboptimal schemes. 12 0 5 10 15 20 25 0 5 10 15 20 25 SNR (dB) Capacity (bit/s/Hz) EQ WF 1D−BF 1D−BN 2D−BF 2D−BN Fig. 7. Comparison o ver 5 × 5 Rayleigh fading channel. Ant- 1 Ant-N t MD BF or MD BN LDC or OD 1 Nt-k Fig. 8. Concatenated MD scheme. D. MD Schemes Concatenated with Linear Space-T ime Code MD beamforming scheme and MD beam-nulling scheme make k and N t − k spatial subchannels a vailable, respectiv ely . As a result, they can concatenate with space-time schem es t o imp rove per formance. For si mplicity , s pace-time codes with li near structure, su ch as high-rate LDCs [26] and STBCs [25] (i.e., orthogonal design ), are preferable. It is worthy of noting that the 2D beamformi ng scheme in [12] is just a sp ecial case of MD beamformi ng. As shown in Fig. 8, we propose to concatenate an MD scheme with an LDC or an STBC . In these figures “OD” stands for orthogonal desi gn. Over 5 × 5 Ra yleigh fading channel, concatenated M D schem es are compared at various data rate. In the sim ulations, two eigen vectors can be fed back to the transmitter . For an MD scheme with LDC, a suboptim al linear MMSE receiver i s applied. Since a M D scheme with STBC are orthogonal, a matched filter is applied, which is also optim al. In Fig. 9, MD beamform ing scheme wi th STBC are compared with MD beamforming schem e wit h LDC i n terms of BER when d ata rate is R = 2 . Al so when R = 6 , Their BERs are sh own in Fig. 11. From these figures, it is s hown that at high data rate, MD beamforming with LDC outperform MD beamforming with STBC sign ificantly ev en though a subop timal MMSE recei ver i s applied. Specifically , when BER is 10 − 5 , the codi ng gain i s about 4 dB. At low data rate, MD beamformin g with LDC performs sl ightly worse than MD beamfo rming wi th STBC s ince t he s uboptimal recei ver is applied. Specifically , when BER is 10 − 5 , th e coding gain is about 1 dB. In Fig. 10, MD beamforming scheme with STBC are compared with MD beamform ing scheme wit h LDC i n terms of BER when d ata rate is R = 3 . Al so when R = 6 , Their BERs are sh own in Fig. 11. From these figures, it is shown th at at high data rate, MD beam-nu lling with LDC outperform M D beam-null ing with STBC sign ificantly ev en though a subop timal MMSE recei ver i s applied. Specifically , when BER is 10 − 5 , the coding g ain is about 6 . 8 dB. At low data rate, MD beam-nulli ng with LDC performs slightly worse than MD beam-nul ling w ith STBC si nce t he subopt imal receiv er is appli ed. Specifically , when BER is 10 − 5 , th e coding gain is about 1 . 5 dB. In Fig. 11, four schemes are compared when data rate is R = 6 . As sho wn in the figure, M D beam- nulling with LDC has the best BER performance e ven subop timal MMSE recei ver i s u sed. In summary , 13 0 2 4 6 8 10 12 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR BER BF−LDC−2PSK BF−OD−4PSK Fig. 9. BER of con catenated MD beamforming when R = 2 . 0 2 4 6 8 10 12 14 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR BER BN−LDC−2PSK BN−OD−16QAM Fig. 10. BER of co ncatenated MD beam-nulling when R = 3 . MD scheme wi th LDC outperforms MD schem e with STBC especially when the data rate is h igh. At l ow data rate, th e performance will depend on the recei ver . At high data rate, MD beam-nulli ng with LDC perform th e best among the four schemes. V I . C O N C L U S I O N S Based on the concept of spatial su bchannels and inspired by the beamforming scheme, we p roposed a scheme called “beam-nulling”. The ne w scheme exploits all spati al subchannels e xcept the weakest one and thus achiev es significantly high capacity that approaches the optimal water-fi lling scheme at medium signal-to-noise ratio. The performance of beam-nulling wit h an MMSE receiv er has been analyzed and verified by num erical and sim ulation results. It has been sho wn t hat if the data rate is lo w , beamforming outperforms b eam-nulling. If the data rate is hi gh, beam-nul ling ou tperforms beamformi ng at lo w and 0 2 4 6 8 10 12 14 16 18 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR BER BF−LDC−8PSK BN−LDC−4PSK BF−OD−64QAM BN−OD−256QAM Fig. 11. BER Comparison of concatenated MD schemes when R = 6 . 14 medium SNR b ut beamforming outperforms at high SNR. T o achieve better performance and maintain tractable complexity , beam-nulling was concatenated with a l inear dispersion code and it was demon strated that if the data rate is low , beam-nullin g with a linear dispersion code can approach beamforming at hi gh SNR. If the data rate is high, beam-nul ling o utperforms beamform ing ev en with a suboptimal MM SE recei ver . If more than o ne eig en vector can be fed back to the transmit ter , new extended schemes based on the existing beamforming and th e proposed beam-nul ling are propo sed. The new schemes are called mult i- dimensional beamform ing and m ulti-dimensi onal beam-nul ling, respectiv ely . Th e theoretical analy sis and numeric results in terms of capacity are also provided to ev aluate the new p roposed schemes. Both of MD schemes can be concatenated with an LDC or an STBC. It is s hown that the MD scheme with LDC can outperform the MD s cheme wi th STBC sign ificantly when the data rate is high . 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