Capacity and Performance of Adaptive MIMO System Based on Beam-Nulling
aa r X i v : . [ c s . I T ] J u l Capacity and Performance of Adaptive MIMOSystem Based on Beam-Nulling
Mabruk Gheryani, Zhiyuan Wu, and Yousef R. ShayanConcordia University, Department of Electrical EngineeringMontreal, Quebec, H4G 2W1, Canadaemail: (m gherya, zy wu, yshayan)@ece.concordia.ca
Abstract
In this paper, we propose a scheme called “beam-nulling” for MIMO adaptation. In the beam-nulling scheme,the eigenvector of the weakest subchannel is fed back and then signals are sent over a generated subspace orthogonalto the weakest subchannel. Theoretical analysis and numerical results show that the capacity of beam-nulling isclosed to the optimal water-filling at medium SNR. Additionally, signal-to-interference-plus-noise ratio (SINR) ofMMSE receiver is derived for beam-nulling. Then the paper presents the associated average bit-error rate (BER) ofbeam-nulling numerically which is verified by simulation. Simulation results are also provided to compare beam-nulling with beamforming. To improve performance further, beam-nulling is concatenated with linear dispersioncode. Simulation results are also provided to compare the concatenated beam-nulling scheme with the beamformingscheme at the same data rate. Additionally, the existing beamforming and new proposed beam-nulling can beextended if more than one eigenvector is available at the transmitter. The new extended schemes are called multi-dimensional (MD) beamforming and MD beam-nulling. Theoretical analysis and numerical results in terms ofcapacity are also provided to evaluate the new extended schemes. Simulation results show that the MD schemewith LDC can outperform the MD scheme with STBC significantly when the data rate is high.
I. I
NTRODUCTION
Since the discovery of multiple-input-multiple-output (MIMO) capacity [1] [2], a lot of research effortshave been put into this field. It has been recognized that adaptive techniques proposed for single-input-single-output (SISO) channels [3] [4] can also be applied to improve MIMO channel capacity.The ideal scenario is that the transmitter has full knowledge of channel state information (CSI). Giventhis perfect CSI feedback, the original MIMO channel can be converted to multiple uncoupled SISOchannels via singular value decomposition (SVD) at the transmitter and the receiver [1]. In other words,the original MIMO channel can be decomposed into several orthogonal “spatial subchannels” with variouspropagation gains.To achieve better performance, various schemes can be implemented depending on the availability ofCSI at the transmitter [5]- [17]. If the transmitter has full knowledge of the channel matrix, i.e., fullCSI, the so-called “water-filling” (WF) principle is performed on each spatial subchannel to maximize thechannel capacity [1]. This scheme is optimal in this case. Various WF-based schemes have been proposed,such as [9] [11]. For the WF-based scheme, the feedback bandwidth for the full CSI grows with respectto the number of transmit and receive antennas and the performance is often very sensitive to channelestimation errors.To mitigate these disadvantages, various beamforming (BF) techniques for MIMO channels have alsobeen investigated intensively. In an adaptive beamforming scheme, the complex weights of the transmitantennas are fed back from the receiver. If only one eigenvector can be fed back, eigen-beamforming[12] is optimal. The eigen-beamforming scheme only applies to the strongest spatial subchannel but canachieve full diversity and high signal-to-noise ratio (SNR) [12]. Also, in practice, the eigen-beamformingscheme has to cooperate with the other adaptive parameters to improve performance and/or data rates suchas constellation and coding rate. There are also other beamforming schemes based on various criteria.Examples of such schemes are [12] - [22]. Note that the conventional beamforming is optimal in terms of maximizing the SNR at the receiver. However, it is sub-optimal from the MIMO capacity perspective,since only a single data stream, as opposed to parallel streams, is transmitted through the MIMO channel[13].In this paper, we propose a new technique called “beam-nulling” (BN). This scheme uses the samefeedback bandwidth as beamforming, that is, only one eigenvector is fed back to the transmitter. Thebeam-nulling transmitter is informed by the weakest spatial subchannel and, where both transmitter andreceiver know how to generate the same spatial subspace, sends signals over a generated spatial subspaceorthogonal to the weakest subchannel. Although the transmitted symbols are “precoded” according tothe feedback, beam-nulling is different from the other existing precoding schemes with limited feedbackchannel, which are independent of the instantaneous channel but the optimal precoding depends on theinstantaneous channel [14] [15].Using this new techniques instead of the optimal water-filling scheme, the loss of channel capacity canbe reduced. This paper also addresses the performance of beam-nulling. To achieve better performance,beam-nulling can be concatenated with the other space-time (ST) coding schemes, such as space-timetrellis codes (STTCs) [23], space-time block codes (STBCs) [24] [25] and linear dispersion codes (LDCs)[26]- [29], etc. For simplicity and flexibility, LDCs are preferable. We provide numerical and simulationresults are provided to demonstrate the merits of the new proposed scheme. Additionally, if more thanone eigenvector, e.g. k eigenvectors, can be available at the transmitter, the existing beamforming schemeand the proposed beam-nulling scheme can be further extended, respectively. The extended schemeswill exploit or discard k spatial subchannels and they will be referred to as “multi-dimensional (MD)”beamforming and “multi-dimensional” beam-nulling, respectively.This paper will be organized as follows. Our channel model is presented in Section II. In Section III, fourpower allocation strategies, i.e., equal power, water-filling, eigen-beamforming, and a new power allocationstrategy called “beam-nulling” are studied and compared in terms of channel capacity. In Section IV, biterror rate (BER) of the proposed beam-nulling scheme using MMSE detector is studied and verified.The proposed scheme is compared with the eigen-beamforming scheme at various data rates in terms ofBER. Beam-nulling concatenated with LDC is proposed and evaluated. In Section V, extended adaptiveframeworks, i.e., MD beamforming and MD beam-nulling, are proposed. Capacity and performance ofthese two schemes are discussed and compared. To improve performance further and maintain reasonablecomplexity, MD schemes concatenated with linear space-time codes, such as STBC and LDC, are proposedand evaluated. Finally, in Section VI, conclusions are drawn.II. C HANNEL M ODEL
In this study, the channel is assumed to be a Rayleigh flat fading channel with N t transmit and N r ( N r ≥ N t ) receive antennas. We denote the complex gain from the transmit antenna n to the receiverantenna m by h mn and collect them to form an N r × N t channel matrix H = [ h mn ] . The channel is knownperfectly at the receiver. The entries in H are assumed to be independent and identically distributed ( i.i.d. )symmetrical complex Gaussian random variables with zero mean and unit variance.The symbol vector at the N t transmit antennas is denoted by x = [ x , x , . . . , x N t ] T . According toinformation 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 variables with zero mean and unit variance, i.e., E ( x i ) = 0 and E | x i | = 1 .The singular-value decomposition of H can be written as H = UΛV H (1)where U is an N r × N r unitary matrix, Λ is an N r × N t matrix with singular values { λ i } on thediagonal and zeros off the diagonal, and V is an N t × N t unitary matrix. For convenience, we assume λ ≥ λ . . . ≥ λ N t , U = [ u u . . . u N r ] and V = [ v v . . . v N t ] . { u i } and v i are column vectors. Fromequation (1), the original channel can be considered as consisting of uncoupled parallel subchannels. Each subchannel corresponds to a singular value of H . In the following context, the subchannel is also referredto as “spatial subchannel”. For instance, one spatial subchannel corresponds to λ i , u i and { v i } .III. P OWER A LLOCATION A MONG S PATIAL S UBCHANNELS
We assume that the total transmitted power is constrained to P . Given the power constraint, differentpower allocation among spatial subchannels can affect the channel capacity tremendously. Depending onpower allocation strategy among spatial subchannels, four schemes are presented which are equal power,water-filling, eigen-beamforming, and the new power allocation which is beam-nulling.If the transmitter has no knowledge about the channel, the most judicious strategy is to allocate thepower to each transmit antenna equally, i.e., equal power. In this case, the received signals can be writtenas y = s PN t Hx + z (2) z is the additive white Gaussian noise (AWGN) vector with i.i.d. symmetrical complex Gaussian elementsof zero mean and variance σ z . The associated ergodic channel capacity can be written as [1] ¯ C eq = E " N t X i =1 log (cid:18) ρN t λ i (cid:19) (3)where E [ · ] denotes expectation with respect to H and ρ = Pσ z denotes SNR. If the transmitter has fullknowledge about the channel, the most judicious strategy is to allocate the power to each spatial subchannelby water-filling principle [1]. In water-filling 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 AWGN random variable with zero mean and σ z variance.Following the method of Lagrange multipliers, P i can be found [1] and the total ergodic channel capacityis ¯ C wf = E " N t X i =1 log P i σ z λ i ! (5)To save feedback bandwidth, beamforming can be considered. For the MIMO model, the optimalbeamforming is called “eigen-beamforming” [12], or simply beamforming. We assume one symbol, saying x , is transmitted. At the receiver, the received vector can be written as y = √ P Hv x + z (6)where z is the additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elementsof zero mean and variance σ z . The associated ergodic channel capacity can be written as ¯ C bf = E h log (cid:16) ρλ (cid:17)i (7)The eigen-beamforming scheme can save feedback bandwidth and is optimized in terms of SNR [22].However, since only one spatial subchannel is considered, this scheme suffers from loss of channel capacity[13], especially when the number of antennas grows. Ant-N t Ant-1 Ant-1Ant-N r Channel Estimation g g Nt,1 ~ x Nt-1 y x g g Nt-1 g g Nt,Nt-1 U H Generate Φ v Nt-1
Fig. 1. beam-nulling scheme.
A. Beam-Nulling
The eigen-beamforming scheme can save feedback bandwidth and is optimized in terms of SNR [22].However, since only a single spatial subchannel is considered, this scheme suffers from loss of channelcapacity [13], especially when the number of antennas grows. Inspired by the eigen-beamforming scheme,we will propose a new beamforming-like scheme called “beam-nulling” (BN). This scheme uses the samefeedback bandwidth as beamforming, that is, only one eigenvector is fed back to the transmitter. Unlikethe eigen-beamforming scheme in which only the best spatial subchannel is considered, the beam-nullingscheme discards only the worst spatial subchannel. Hence, in comparison with the optimal water-fillingscheme, the loss of channel capacity can be reduced.In this scheme as shown in Fig. 1, the eigenvector associated with the minimum singular value fromthe transmitter side, i.e., v N t , is fed back to the transmitter. A subspace orthogonal to the weakest spatialchannel is constructed so that the following condition is satisfied. Φ H v N t = (8)The N t × ( N t − matrix Φ = [ g g . . . g N t − ] spans the subspace. Note that the method to construct thesubspace Φ should also be known to the receiver.Here is an example of construction of the orthogonal subspace. We construct an N t × N t matrix A = [ v N t I ′ ] (9)where I ′ = [ I ( N t − × ( N t − ( N t − × ] T . Applying QR decomposition to A , we have A = [ v N t Φ ] · Γ (10)where Γ is an upper triangular matrix with the (1,1)-th entry equal to 1. Φ is the subspace orthogonal to v N t .At the transmitter, N t − symbols denoted as x ′ are transmitted over the orthogonal subspace Φ . Thereceived signals at the receiver can be written as y ′ = s PN t − HΦx ′ + z ′ = c Hx ′ + z ′ (11)where z ′ is additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elements ofzero mean and variance σ z and c H = q PN t − HΦ .Substituting (1) into (11) and multiplying y ′ by U H , results in e y = s PN t − Λ B0 T ! x ′ + e z (12) where e z is additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elements ofzero mean and variance σ z . With the condition in (8), V H Φ = B0 T ! (13)where B = v H g v H g . . . v H g N t − v H g . . . . . . ...... ... . . . ... v HN t − g . . . . . . v HN t − g N t − (14) B is an ( N t − × ( N t − unitary matrix. From (12), the available spatial channels are N t − . Sincethe weakest spatial subchannel is “nulled” in this scheme, power can be allocated equally among the other N t − subchannels. Equation (12) can be rewritten as e y ′ = s PN t − Λ ′ Bx ′ + e z ′ (15)where e y ′ and e z ′ are column vectors with the first ( N r − elements of e y and e z , respectively, and Λ ′ = diag[ λ , λ , . . . , λ ( N t − ] . From (15), the associated ergodic channel capacity can be found as ¯ C bn = E " N t − X i =1 log (cid:18) ρN t − λ i (cid:19) (16)As can be seen, the beam-nulling scheme only needs one eigenvector to be fed back. However, sinceonly the worst spatial subchannel is discarded, this scheme can increase channel capacity significantly ascompared to the conventional beamforming scheme. B. Comparisons Among the Four Schemes
In this section, we compare the new proposed beam-nulling scheme with the other schemes, i.e.,equal power, beamforming and water-filling schemes. Water-filling is the optimal solution among thefour schemes for any SNR.Differentiating the above ergodic capacities with respect to ρ respectively, we have ∂ ¯ C eq ∂ρ = E N t X i =1 ρ + N t λ i (17) ∂ ¯ C bf ∂ρ = E ρ + λ (18) ∂ ¯ C bn ∂ρ = E N t − X i =1 ρ + N t − λ i (19)The differential will also be referred to as “slope”. Since the second order differentials are negative, theabove ergodic capacities are concave and monotonically increasing with respect to ρ .With the fact that λ ≥ λ . . . ≥ λ N t , it can be readily checked that the slopes of ergodic capacitiesassociate with equal power and beam-nulling are bounded as follows. E N t ρ + N t λ ≥ ∂ ¯ C eq ∂ρ ≥ E N t ρ + N t λ Nt (20) E N t − ρ + N t − λ ≥ ∂ ¯ C bn ∂ρ ≥ E N t − ρ + N t − λ ( Nt − (21) C apa c i t y ( b i t/ s / H z ) Fig. 2. × Rayleigh fading channel.
For the case of N t = 2 , beamforming and beam-nulling have the same capacity for any ρ as can beseen from equations of capacity and slope. If ρ → , equivalently at low SNR, it can be easily found that ∂ ¯ C bf ∂ρ ≥ ∂ ¯ C bn ∂ρ ≥ ∂ ¯ C eq ∂ρ , ρ → (22)If ρ → ∞ , equivalently 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 dB. Hence, at medium SNR, ∂ ¯ C bn ∂ρ hasthe largest value compared to ∂ ¯ C bf ∂ρ and ∂ ¯ C eq ∂ρ . Therefore, for low, medium and high SNRs, beamforming,beam-nulling and equal power have the largest capacities, respectively.In Fig. 2, capacities of water-filling, beamforming, beam-nulling and equal power are compared over × Rayleigh fading channels, respectively. Note that since SNR is measured in dB, the curves becomeconvex. In these figures, “EQ” stands for equal power, “WF” stands for water-filling, “BF” stands forbeamforming and “BN” stands for beam-nulling. As can be seen, the water-filling has the best capacityat any SNR region. The other schemes perform differently at different SNR regions. At low SNR, thebeamforming is the closest to the optimal water-filling, e.g., the SNR region below . dB for × fadingchannel. Note that at low SNR, the water-filling scheme may only allocate power to one or two spatialsubchannels. At medium SNR, the proposed beam-nulling is the closest to the optimal water-filling, e.g.,the SNR region from . dB to dB for × fading channel. The beam-nulling scheme only discardsthe weakest spatial subchannel and allocates power to the other spatial subchannels. As can be seen fromthe numerical results, the beam-nulling scheme performs better than the other schemes in this case. Notethat at high SNR, the equal power scheme will converge with the water-filling scheme.IV. P ERFORMANCE OF B EAM - NULLING
A. MMSE Detector
The close-form error probability for the optimal ML receiver is difficult to establish. Other suboptimalreceivers can also be implemented. The MMSE detector is especially popular due to its low complexityand good performance [31] [32]. In the following context, BER of the MMSE detector is analyzed forthe beam-nulling scheme.
Let us define c H = q PN t − HΦ and ˆ h i is the i -th column of c H . Equation (11) can also be written as y ′ = ˆ h i x i + X j = i ˆ h j x j + z ′ (24)where x i is the i-th element of x ′ .Without loss of generality, we consider the detection of one symbol, say x i . We collect the rest of thesymbols into a column vector x I and denote c H I = [ˆ h , .., ˆ h i − , ˆ h i +1 , ..., ˆ h N t − ] as the matrix obtained byremoving the i -th column from c H .A linear MMSE detector [32] [33] is applied and the corresponding output is given by ˆ x i = w Hi y = x i + ˆ z i . (25)where ˆ z i is the noise term of zero mean. ˆ z i can be approximated to be Gaussian [32]. The corresponding w i can be found as w i = (cid:16) ˆ h i ˆ h Hi + R I (cid:17) − ˆ h i ˆ h Hi (cid:16) ˆ h i ˆ h Hi + R I (cid:17) − ˜ h i (26)where R I = c H I c H HI + σ z I . Note that the scaling factor h Hi ( ˆ h i ˆ h Hi + R I ) − ˆ h i in the coefficient vector of theMMSE detector w i is added to ensure an unbiased detection as indicated by (25). The variance of thenoise term ˆ z i can be found from (25) and (26) as ˆ σ i = w Hi R I w i (27)Substituting the coefficient vector for the MMSE detector in (26) into (27), the variance can be writtenas ˆ σ i = 1ˆ h Hi R − I ˆ h i (28)Then, the SINR of MMSE associated with x i is / ˆ σ i . γ i = 1ˆ σ i = ˆ h Hi R − I ˆ h i (29)The closed-form BER for a channel model such as (25) can be found in [34]. The average BER overMIMO fading channel for a given constellation can be found for beam-nulling as follows. BER av = E γ i " N t − X i BER ( γ i ) (30)The closed-form formula for the average BER in (30) depends on the distribution of γ i , which is difficultto determine. Here, the above average BER is calculated numerically. For example, the average BER for η -PSK is BER av = E γ i " N t − X i η Q (cid:18)q η γ i sin( π η ) (cid:19) (31)and the average BER for rectangular η -QAM is BER av = E γ i N t − X i η Q s η γ i η − (32)where Q ( · ) denotes the Gaussian Q -function.In Fig. 3, numerical and simulation results are compared for 8PSK over × Rayleigh fading channeland QPSK over × Rayleigh fading channel, respectively. As can be seen, the numerical and simulationresults match well. −6 −5 −4 −3 −2 −1 SNR BE R (a) × , 8PSK −7 −6 −5 −4 −3 −2 −1 SNR (dB) BE R QPSK−SimulationQPSK−Numerical (b) × , 4PSKFig. 3. Numerical and simulation results for beam-nulling scheme. −7 −6 −5 −4 −3 −2 −1 SNR (dB) BE R BF−8PSKBN−BPSK−MMSEBN−BPSK−ML (a) R=3 −7 −6 −5 −4 −3 −2 −1 SNR (dB) BE R BF−64QAMBN−QPSK−MMSE (b) R=6Fig. 4. Comparison over × Rayleigh fading channel.
B. Performance Comparison Between Beamforming and Beam-nulling
In Fig. 4, simulation results are compared for various data rates R over × Rayleigh fading channels.In the following simulations, a data rate R is measured in bits per channel use. The beamforming schemeis equivalent to a SISO channel using a maximum ratio combining (MRC) receiver [14]. For the beam-nulling scheme, the optimal ML receiver and the suboptimal MMSE receiver are used.From Fig. 4, if the data rate is low, i.e., constellation size is low, beamforming outperforms beam-nulling. If the data rate is high, i.e., constellation size is high, beam-nulling outperforms beamforming atlow and medium SNR, however at high SNR beamforming outperforms beam-nulling. Also, as can beseen, at the high data rate, even the beam-nulling scheme with suboptimal MMSE receiver outperformsthe beamforming scheme. C. Concatenation of Beam-nulling and LDC
To further improve the performance of beam-nulling with tractable complexity, we propose to concate-nate beam-nulling with a linear dispersion code. Note that to meet error-rate requirements, multiple levelsof error protection can be implemented. In this study, we focus on space-time coding domain.In this system, the information bits are first mapped into symbols. The symbol stream is parsed intoblocks of length L = ( N t − T . The symbol vector associated with one modulation block is denoted by −8 −7 −6 −5 −4 −3 −2 −1 SNR (dB) BE R BF−8PSKBN−BPSK−MMSEBN−BPSK−MLBL−BPSK−MMSEBL−BPSK−ML (a) R=3 −4 −3 −2 −1 SNR (dB) BE R BF−64QAMBN−QPSK−MMSEBL−QPSK−MMSE (b) R=6Fig. 5. Comparison over × Rayleigh fading channel. x = [ x , x , . . . , x L ] T with x i ∈ Ω ≡ { Ω m | m = 0 , , . . . , η − , η ≥ } , i.e., a complex constellation ofsize η , such as η -QAM). The average symbol energy is assumed to be , i.e., η η − P m =0 | Ω m | = 1 . Eachsymbol in a block will be mapped to a dispersion matrix of size N t × T (i.e., M i ) and then combinedlinearly to form ( N t − data streams over T channel uses. The output ( N t − data streams are transmittedonly over the subspace Φ orthogonal to the weakest spatial channel. The generation of the orthogonalsubspace Φ is described in Section III-A. The received signals can be written as y = s PN t − HΦ L X i =1 M i x i + z (33)where z is additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elements ofzero mean and variance σ z . It is worthy to note that the traditional beamforming scheme cannot workwith space-time coding since it can be viewed as a SISO channel. We compare the concatenated schemewith the original schemes at the same data rate.In Fig. 5, simulation results are compared for various data rates R over × Rayleigh flat fadingchannels. In the figure, “BL” denotes beam-nulling with LDC. As can be seen, beam-nulling with LDCoutperforms beam-nulling without LDC using the same receiver. The performance of beam-nulling withLDC using MMSE receiver is close to that of beam-nulling without LDC using the optimal ML receiver.Also it can be seen, if data rate is low, i.e., constellation size is low, the performance of beam-nullingwith LDC can approach that of beamforming at high SNR. If data rate is high, i.e., constellation sizeis high, beam-nulling with LDC outperforms beamforming even when the suboptimal MMSE receiver isused. V. E
XTENDED A DAPTIVE F RAMEWORKS
For the beamforming and beam-nulling schemes, only one eigenvector has been fed back to thetransmitter. If more backward bandwidth is available for feedback, e.g. k eigenvectors, can be sent tothe transmitter for adaptation. With the feedback of k eigenvectors, we can extend our frameworks, whichwill be called multi-dimensional (MD) beamforming and MD beam-nulling. The original schemes can bereferred to as 1D-beamforming and 1D-beam-nulling. To save bandwidth, k ≤ ⌊ N t ⌋ should be satisfied,where ⌊·⌋ denotes rounding towards minus infinity. That is, whether the strongest or the weakest k spatialsubchannels will be fed back according to the channel conditions. For example, at low SNR, k strongestspatial subchannels will be fed back. At medium SNR, k weakest spatial subchannels will be fed back. A. MD Beamforming
For MD beamforming, v , . . . , v k are fed back to the transmitter. k symbols, saying x k = [ x , x , . . . , x k ] T ,are transmitted. At the receiver, the received vector can be written as y k = s Pk H [ v . . . v k ] x k + z k (34)where z k is the additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elementsof zero mean and variance σ z .Consequently, the associated ergodic channel capacity can be found as ¯ C k,bf = E " k X i =1 log Pk σ z λ i ! (35)Let ρ = P/σ z denote SNR. It is readily checked that the capacity of MD beamforming is also concave andmonotonically increasing with respect to SNR ρ . Differentiating the above ergodic capacity with respectto ρ , we have ∂ ¯ C k,bf ∂ρ = E k X i =1 ρ + kλ i (36)If ρ → , equivalently at low SNR, it can be easily found that ∂ ¯ C ( k − ,bf ∂ρ > ∂ ¯ C k,bf ∂ρ , ρ → (37)If ρ → ∞ , equivalently at high SNR, it can be easily found that ∂ ¯ C k,bf ∂ρ > ∂ ¯ C ( k − ,bf ∂ρ , ρ → ∞ (38)Note that ¯ C k,bf = 0 for any k when ρ = 0 or minus infinity in dB. Hence, at low SNR, the capacity ofthe k -D beamforming scheme is worse than the ( k − -D beamforming scheme and while at high SNR,the capacity of the k -D beamforming scheme is better than the ( k − D beamforming scheme at the costof feedback bandwidth.
B. MD Beam-nulling
For MD beam-nulling, similar to D beam-nulling, by a certain rule, a subspace orthogonal to the k weakest spatial channel is constructed. That is, the following condition should be satisfied. v Hn Φ ( k ) = T , ∀ n = N t − k + 1 , . . . , N t . (39)The N t × ( N t − k ) matrix Φ ( k ) = [ g g . . . g N t − k ] spans the ( N t − k ) -dimensional subspace.At the transmitter, N t − k symbols denoted as x ( k ) are transmitted only over the orthogonal subspace Φ ( k ) . The received signals at the receiver can be written as y ( k ) = s PN t − k HΦ ( k ) x ( k ) + z ( k ) (40)where z ( k ) is additive white Gaussian noise vector with i.i.d. symmetrical complex Gaussian elements ofzero mean and variance σ z . From (40), the associated instantaneous channel capacity with respect to H can be found as ¯ C ( k ) bn = E N t − k X i =1 log P ( N t − k ) σ z λ i ! (41) SNR (dB) C apa c i t y ( b i t/ s / H z ) EQWF1D−BN2D−BN
Fig. 6. MD beam-nulling over × Rayleigh fading channel.
It is readily checked that the capacity of MD beam-nulling is also concave and monotonically increasingwith respect to SNR ρ . Let ρ = P/σ z denote SNR. Differentiating the above ergodic capacity with respectto ρ , we have ∂ ¯ C ( k ) bn ∂ρ = E N t − k X i =1 ρ + N t − kλ i (42)If ρ → , equivalently at low SNR, it can be easily found that ∂ ¯ C ( k ) bn ∂ρ > ∂ ¯ C ( k − bn ∂ρ , ρ → (43)If ρ → ∞ , equivalently at high SNR, it can be easily found that ∂ ¯ C ( k − 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, the capacity ofthe k -D beam-nulling scheme is better than the ( k − -D beam-nulling scheme at the cost of feedbackbandwidth and while at high SNR, the capacity of the k -D beam-nulling scheme is worse than the ( k − -Dbeam-nulling scheme.For example, in Fig. 6, capacities of 1D beam-nulling and 2D beam-nulling schemes are compared withWF and equal power scheme over × Rayleigh fading channel at different SNR regions. At relativelylow SNR, i.e., less than 13dB, the 2D beam-nulling scheme outperforms the 1D beam-nulling scheme interms of capacity at the price of feedback bandwidth. While at relatively high SNR, i.e., more than 13dB,the 1D-beam-nulling scheme outperforms the 2D beam-nulling scheme as predicted.
C. Capacity Comparison of MD Schemes
Here, over × Rayleigh fading channel, the MD schemes are compared with WF and equal powerschemes as shown in Fig. 7. It can be readily check that, at relatively low SNR, MD beamformingschemes are better than MD beam-nulling schemes; while at relatively high SNR, the results are opposite.Specifically, at very low SNR, i.e. less than 0dB, the 1D beamforming scheme outperforms the other MDschemes. At the SNR region between 0dB and 5.5dB, the 2D beamforming scheme outperforms the otherMD schemes. At the SNR region between 5.5dB and 12.7dB, the 2D beam-nulling scheme outperformsthe other MD schemes. At the SNR region between 12.7dB and 23dB, the 1D beam-nulling schemeoutperforms the other MD schemes. Again, when SNR is more than 23dB, the equal power schemeoutperforms the other suboptimal schemes. SNR (dB) C apa c i t y ( b i t/ s / H z ) EQWF1D−BF1D−BN2D−BF2D−BN
Fig. 7. Comparison over × Rayleigh fading channel.
Ant- t MD BF orMD BNLDCorOD
Fig. 8. Concatenated MD scheme.
D. MD Schemes Concatenated with Linear Space-Time Code
MD beamforming scheme and MD beam-nulling scheme make k and N t − k spatial subchannelsavailable, respectively. As a result, they can concatenate with space-time schemes to improve performance.For simplicity, space-time codes with linear structure, such as high-rate LDCs [26] and STBCs [25] (i.e.,orthogonal design), are preferable. It is worthy of noting that the 2D beamforming scheme in [12] is justa special case of MD beamforming. As shown in Fig. 8, we propose to concatenate an MD scheme withan LDC or an STBC. In these figures “OD” stands for orthogonal design.Over × Rayleigh fading channel, concatenated MD schemes are compared at various data rate. Inthe simulations, two eigenvectors can be fed back to the transmitter. For an MD scheme with LDC, asuboptimal linear MMSE receiver is applied. Since a MD scheme with STBC are orthogonal, a matchedfilter is applied, which is also optimal.In Fig. 9, MD beamforming scheme with STBC are compared with MD beamforming scheme withLDC in terms of BER when data rate is R = 2 . Also when R = 6 , Their BERs are shown in Fig. 11. Fromthese figures, it is shown that at high data rate, MD beamforming with LDC outperform MD beamformingwith STBC significantly even though a suboptimal MMSE receiver is applied. Specifically, when BER is − , the coding gain is about dB. At low data rate, MD beamforming with LDC performs slightly worsethan MD beamforming with STBC since the suboptimal receiver is applied. Specifically, when BER is − , the coding gain is about dB.In Fig. 10, MD beamforming scheme with STBC are compared with MD beamforming scheme withLDC in terms of BER when data rate is R = 3 . Also when R = 6 , Their BERs are shown in Fig. 11. Fromthese figures, it is shown that at high data rate, MD beam-nulling with LDC outperform MD beam-nullingwith STBC significantly even though a suboptimal MMSE receiver is applied. Specifically, when BER is − , the coding gain is about . dB. At low data rate, MD beam-nulling with LDC performs slightlyworse than MD beam-nulling with STBC since the suboptimal receiver is applied. Specifically, when BERis − , the coding gain is about . dB.In Fig. 11, four schemes are compared when data rate is R = 6 . As shown in the figure, MD beam-nulling with LDC has the best BER performance even suboptimal MMSE receiver is used. In summary, −8 −7 −6 −5 −4 −3 −2 −1 SNR BE R BF−LDC−2PSKBF−OD−4PSK
Fig. 9. BER of concatenated MD beamforming when R = 2 . −8 −7 −6 −5 −4 −3 −2 −1 SNR BE R BN−LDC−2PSKBN−OD−16QAM
Fig. 10. BER of concatenated MD beam-nulling when R = 3 . MD scheme with LDC outperforms MD scheme with STBC especially when the data rate is high. At lowdata rate, the performance will depend on the receiver. At high data rate, MD beam-nulling with LDCperform the best among the four schemes.VI. C
ONCLUSIONS
Based on the concept of spatial subchannels and inspired by the beamforming scheme, we proposed ascheme called “beam-nulling”. The new scheme exploits all spatial subchannels except the weakest oneand thus achieves significantly high capacity that approaches the optimal water-filling scheme at mediumsignal-to-noise ratio. The performance of beam-nulling with an MMSE receiver has been analyzed andverified by numerical and simulation results. It has been shown that if the data rate is low, beamformingoutperforms beam-nulling. If the data rate is high, beam-nulling outperforms beamforming at low and −8 −7 −6 −5 −4 −3 −2 −1 SNR BE R BF−LDC−8PSKBN−LDC−4PSKBF−OD−64QAMBN−OD−256QAM
Fig. 11. BER Comparison of concatenated MD schemes when R = 6 . medium SNR but beamforming outperforms at high SNR. To achieve better performance and maintaintractable complexity, beam-nulling was concatenated with a linear dispersion code and it was demonstratedthat if the data rate is low, beam-nulling with a linear dispersion code can approach beamforming at highSNR. If the data rate is high, beam-nulling outperforms beamforming even with a suboptimal MMSEreceiver. If more than one eigenvector can be fed back to the transmitter, new extended schemes based onthe existing beamforming and the proposed beam-nulling are proposed. The new schemes are called multi-dimensional beamforming and multi-dimensional beam-nulling, respectively. The theoretical analysis andnumeric results in terms of capacity are also provided to evaluate the new proposed schemes. Both ofMD schemes can be concatenated with an LDC or an STBC. It is shown that the MD scheme with LDCcan outperform the MD scheme with STBC significantly when the data rate is high. Additionally, at highdata rate, MD beam-nulling with LDC outperforms MD beamforming with LDC, MD beamforming withSTBC and MD beam-nulling with STBC. R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,”
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