Power Allocation Strategy for MIMO System Basedon Beam-Nulling

Mabruk Gheryani, Zhiyuan Wu, and Yousef R. ShayanConcordia University, Department of Electrical Engineering

Montreal, Quebec, H4G 2W1, Canadaemail: (m gherya, zy wu, yshayan)@ece.concordia.ca

Abstract— In this paper, we propose a scheme called “beam-nulling” using the same feedback bandwidth as beamforming butwith higher capacity. In the beam-nulling scheme, the eigenvectorof the weakest subchannel is fed back and then signals are sentover a generated subspace orthogonal to the weakest subchannel.Hence, the scheme can achieve high capacity. The capacities ofwater-filling, equal power, beamforming and beam-nulling arecompared through theoretical analysis and numerical results.It is shown that at medium signal-to-noise ratio, beam-nullingapproaches the optimal water-filling scheme.

I. INTRODUCTION

It has been recognized that adaptive techniques, proposedfor single-input-single-output (SISO) channels [1] [2], can alsobe applied to improve MIMO channel capacity [3]- [5]. Theideal scenario in adaptive schemes is that the transmitter hasfull knowledge of channel state information (CSI) which isfed back from the receiver. With such a perfect CSI feedback,the original MIMO channel can be converted to multipleuncoupled SISO channels via singular value decomposition(SVD) at the transmitter and the receiver [3]. In other words,the original MIMO channel can be decomposed into several or-thogonal “spatial subchannels” with various propagation gains.To achieve better performance, various strategies to allocateconstrained power to these subchannels can be implementeddepending on the availability of CSI at the transmitter [6]- [8].

If the transmitter has full knowledge about channel matrix,i.e., full CSI, the so-called “water-filling” (WF) principle isperformed on each spatial subchannel to maximize the channelcapacity [3]. This scheme is optimal in this case. Various WF-based schemes have been proposed, such as [9]- [12]. Forthe WF-based scheme, the feedback bandwidth for the fullCSI grows with respect to the number of transmit and receiveantennas and the performance is often very sensitive to channelestimation errors.

Various beamforming techniques for MIMO channels havealso been investigated extensively, which can mitigate theabove disadvantages. In an adaptive beamforming scheme,complex weights of the transmit antennas are fed back fromthe receiver. If only one eigenvector can be fed back, eigen-beamforming [7] is optimal. The eigen-beamforming schemeonly applies to the strongest spatial subchannel but can achievefull diversity and high signal-to-noise ratio (SNR) [7]. Also, inpractice, the eigen-beamforming scheme has to cooperate withthe other adaptive parameters to improve performance or/and

data rate, such as constellation and coding rate. There are alsoother beamforming schemes based on various criteria, such as[7]- [14]. Note that the conventional beamforming is optimalin terms of maximizing the SNR at the receiver. However,it is sub-optimal from a MIMO capacity point of view, sinceonly one data stream, instead of parallel streams, is transmittedthrough the MIMO channel [15].

In this paper, we propose a new technique called “beam-nulling” (BN). This scheme uses the same feedback bandwidthas beamforming. That is, only one eigenvector is fed backto the transmitter. The beam-nulling transmitter is informedwith the weakest spatial subchannel and then sends signalsover a generated spatial subspace orthogonal to the weakestsubchannel. Note that both transmitter and receiver shouldknow how to generate the same spatial subspace. Hence,the loss of channel capacity as compared to the optimalwater-filling scheme can be reduced. Although the transmittedsymbols are “precoded” according to the feedback, the beam-nulling scheme is different from the other existing precodingschemes with limited feedback channel, which are independentof the instantaneous channel but the optimal precode dependson the instantaneous channel [16] [17].

II. CHANNEL MODEL

In this study, the channel is assumed to be a Rayleigh flatfading channel with Nt transmit and Nr (Nr ≥ Nt) receiveantennas. We denote the complex gain from transmit antennan to receiver antenna m by hmn and collect them to forman Nr × Nt channel matrix H = [hmn]. The channel isknown perfectly at the receiver. The entries in H are assumedto be independently identically distributed (i.i.d.) symmetricalcomplex Gaussian random variables with zero mean and unitvariance.

The symbol vector at the Nt transmit antennas is denoted byx = [x1, x2, . . . , xNt ]

T . According to information theory [5],the optimal distribution of the transmitted symbols is Gaussian.Thus, the elements {xi} of x are assumed to be i.i.d. Gaussianvariables with zero mean and unit variance, i.e., E(xi) = 0and E|xi|2 = 1.

The singular-value decomposition of H can be written as

H = UΛVH (1)

where U is an Nr×Nr unitary matrix, Λ is an Nr×Nt matrixwith singular values {λi} on the diagonal and zeros off the

diagonal, and V is an Nt×Nt unitary matrix. For convenience,we assume λ1 ≥ λ2 . . . ≥ λNt , U = [u1u2 . . .uNr ] andV = [v1v2 . . .vNt

]. {ui} and vi are column vectors.From equation (1), the original channel can be considered as

consisting of uncoupled parallel subchannels. Each subchannelcorresponds to a singular value of H. In the following context,the subchannel is also referred to as “spatial subchannel”. Forinstance, one spatial subchannel corresponds to λi, ui and{vi}.

III. POWER ALLOCATION AMONG SPATIAL SUBCHANNELS

We assume that the total transmitted power is constrainedto P . Given the power constraint, different power allocationamong spatial subchannels can affect the channel capacitytremendously. Depending on power allocation strategy amongspatial subchannels, four schemes are presented which haveequal power, water-filling, eigen-beamforming, and the newpower allocation which is beam-nulling.

If the transmitter has no knowledge about the channel,the most judicious strategy is to allocate the power to eachtransmit antenna equally, i.e., equal power. In this case, thereceived signals can be written as

y =√

P

NtHx + z (2)

z is the additive white Gaussian noise (AWGN) vector withi.i.d. symmetrical complex Gaussian elements of zero meanand variance σ2

z . The associated ergodic channel capacity canbe written as [3]

Ceq = E

[Nt∑

i=1

log(

1 +ρ

Ntλ2

i

)](3)

where E[·] denotes expectation with respect to H and ρ = Pσ2

z

denotes SNR. If the transmitter has full knowledge about thechannel, the most judicious strategy is to allocate the powerto each spatial subchannel by water-filling principle [3]. Inwater-filling scheme, the received signals can be written as

yi =√

Piλixi + zi (4)

whereNt∑i=1

Pi = P as a constraint and zi is the AWGN

random variable with zero mean and σ2z variance. Following

the method of Lagrange multipliers, Pi can be found [3] andthe total ergodic channel capacity is

Cwf = E

[Nt∑

i=1

log(

1 +Pi

σ2z

λ2i

)](5)

To save feedback bandwidth, beamforming can be consid-ered. For the MIMO model, the optimal beamforming is called“eigen-beamforming” [7] [18], or simply beamforming. Weassume one symbol x1, is transmitted. At the receiver, thereceived vector can be written as

y1 =√

PHv1x1 + z1 (6)

Ant-Nt

Ant-1

Ant-1

Ant-Nr

Channel

Estimation

g1,1

gNt,1

~

xNt-1

y1

x1 g1

gNt-1

g1,Nt-1

gNt,Nt-1

UH

Generate

Φ vNt-1

Fig. 1. beam-nulling scheme.

where z1 is the additive white Gaussian noise vector withi.i.d. symmetrical complex Gaussian elements of zero meanand variance σ2

z . The associated ergodic channel capacity canbe written as

Cbf = E[log

(1 + ρλ2

1

)](7)

The eigen-beamforming scheme can save feedback band-width and is optimized in terms of SNR [18]. However,since only one spatial subchannel is considered, this schemesuffers from loss of channel capacity [15], especially when thenumber of antennas grows.

A. Beam-Nulling

Inspired by the eigen-beamforming scheme, we propose anew beamforming-like scheme called “beam-nulling” (BN).This scheme uses the same feedback bandwidth as beamform-ing. That is, only one eigenvector is fed back to the transmitter.Unlike the eigen-beamforming scheme in which only the bestspatial subchannel is considered, in the beam-nulling scheme,only the worst spatial subchannel is discarded. Hence, the lossof channel capacity can be reduced as compared to the optimalwater-filling scheme.

In this scheme as shown in Fig. 1, the eigenvector associatedwith the minimum singular value from the transmitter side, i.e.,vNt , is fed back to the transmitter. A subspace orthogonal tothe weakest spatial channel is constructed so that the followingcondition is satisfied.

ΦHvNt = 0 (8)

The Nt × (Nt − 1) matrix Φ = [g1g2 . . .gNt−1] spans thesubspace. Note that the method to construct the subspace Φshould also be known to the receiver.

An example to construct the orthogonal subspace is pre-sented as follows. We construct an Nt ×Nt matrix

A = [vNtI′] (9)

where I′ = [I(Nt−1)×(Nt−1)0(Nt−1)×1]T . Applying QR de-composition to A, we have

A = [vNtΦ] ·R (10)

where R is an upper triangular matrix with the (1,1)-th entryequal to 1. Φ is the subspace orthogonal to vNt .

At the transmitter, Nt − 1 symbols denoted as x′ aretransmitted over the orthogonal subspace Φ. The receivedsignals at the receiver can be written as

y′ =√

P

Nt − 1HΦx′ + z′ (11)

where z′ is additive white Gaussian noise vector with i.i.d.symmetrical complex Gaussian elements of zero mean andvariance σ2

z .Substituting (1) into (11) and multiplying y′ by UH , we

have

y =√

P

Nt − 1Λ

(B0T

)x′ + z (12)

where z is additive white Gaussian noise vector with i.i.d.symmetrical complex Gaussian elements of zero mean andvariance σ2

z . With the condition in (8),

VHΦ =(

B0T

)(13)

where

B =

vH1 g1 vH

1 g2 . . . vH1 gNt−1

vH2 g1

. . . . . ....

......

. . ....

vHNt−1g1 . . . . . . vH

Nt−1gNt−1

(14)

B is an (Nt − 1)× (Nt − 1) unitary matrix. As can be seenfrom (12), the available spatial channels are Nt−1. Since theweakest spatial subchannel is “nulled” in this scheme, powercan be allocated equally among the other Nt−1 subchannels.Equation (12) can be rewritten as

y′ =√

P

Nt − 1Λ′Bx′ + z′ (15)

where y′ and z′ are column vectors with the first(Nr − 1) elements of y and z, respectively, and Λ′ =diag[λ1, λ2, . . . , λ(Nt−1)]. From (15), the associated ergodicchannel capacity can be found as

Cbn = E

[Nt−1∑

i=1

log(

1 +ρ

Nt − 1λ2

i

)](16)

As can be seen, the beam-nulling scheme only needs oneeigenvector to be fed back. However, since only the worstspatial subchannel is discarded, this scheme increases channelcapacity significantly as compared to the conventional beam-forming scheme.

IV. COMPARISONS AMONG THE FOUR SCHEMES

In this section, we compare the new proposed beam-nullingscheme with the other schemes. Water-filling is the optimalsolution among the four schemes for any SNR.

Differentiating the above ergodic capacities with respect toρ respectively, we have

∂Ceq

∂ρ= E

[Nt∑

i=1

1ρ + Nt

λ2i

](17)

∂Cbf

∂ρ= E

[1

ρ + 1λ2

1

](18)

∂Cbn

∂ρ= E

[Nt−1∑

i=1

1ρ + Nt−1

λ2i

](19)

The differential will also be referred to as “slope”. Sincethe second order differentials are negative, the above ergodiccapacities are concave and monotonically increasing withrespect to ρ.

For the case of Nt = 2, beamforming and beam-nullinghave the same capacity for any ρ as can be seen from equationsof capacity and slope. If ρ → 0, i.e., at low SNR, it can beeasily found that

∂Cbf

∂ρ≥ ∂Cbn

∂ρ≥ ∂Ceq

∂ρ, ρ → 0 (20)

If ρ →∞, i.e., at high SNR, it can be easily found that

∂Ceq

∂ρ≥ ∂Cbn

∂ρ≥ ∂Cbf

∂ρ, ρ →∞ (21)

Note that Cbf = Cbn = Ceq = 0 when ρ = 0 or minusinfinity in dB. Hence, at medium SNR, ∂Cbn

∂ρ has the largest

value compared to ∂Cbf

∂ρ and ∂Ceq

∂ρ . Therefore, for low, mediumand high SNRs, beamforming, beam-nulling and equal powerhas largest capacity, respectively.

In Fig. 2, capacities of water-filling, beamforming, beam-nulling and equal power are compared over 5 × 5 Rayleighfading channels, respectively. Note that since SNR is measuredin dB, the curves become convex. In these figures, “EQ” standsfor equal power, “WF” stands for water-filling, “BF” standsfor beamforming and “BN” stands for beam-nulling. As canbe seen, the water-filling has the best capacity at any SNRregion. The other schemes perform differently at differentSNR regions. At low SNR, the beamforming is the closestto the optimal water-filling, e.g., the SNR region below 3.5dB for 5×5 fading channel. 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 isthe closest to the optimal water-filling, e.g., the SNR regionfrom 3.5 dB to 16 dB for 5 × 5 fading channel. The beam-nulling scheme only discards the weakest spatial subchanneland allocates power to the other spatial subchannels. As canbe seen from the numerical results, the beam-nulling schemeperforms better than the other schemes in this case. Note thatat high SNR, the equal power scheme will converge with thewater-filling scheme.

0 2 4 6 8 10 12 14 160

2

4

6

8

10

12

SNR (dB)

Cap

acity

(bi

t/s/H

z)

5x5

EQWFBFBN

Fig. 2. 5× 5 Rayleigh fading channel.

V. CONCLUSIONS

Based on the concept of spatial subchannels and inspiredby the beamforming scheme, we proposed a novel schemecalled “beam-nulling”. Using the same feedback bandwidth asbeamforming, the new scheme exploits all spatial subchannelsexcept the weakest one and thus achieves significant highcapacity near the optimal water-filling scheme at mediumsignal-to-noise ratio. The comparison showed that at lowsignal-to-noise ratio, beamforming is the closest to the optimalwater-filling, at medium signal-to-noise ratio, beam-nulling isthe closest to the optimal solution, and at high signal-to-noiseratio, equal power is the closest to the optimal solution.

REFERENCES

[1] J. K. Cavers, “Variable-rate transmission for Rayleigh fading channels,”IEEE Transactions on Communications, COM-20, pp.15 - 22, 1972.

[2] A. J. Goldsmith and S.-G. Chua, “Variable rate variable power MQAMfor fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 12181230, Oct. 1997.

[3] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans.Telecom., vol 10, pp. 585 - 595, Nov. 1999.

[4] G. J. Foschini, M. J. Gans, “On limits of wireless communications ina fading environment when using multiple antennas,” Wireless PersonalCommunications, vol. 6, no. 3, pp. 311 - 335, 1998.

[5] C. E. Shannon, “A mathematical theory of communication, Bell Syst.Tech. J., vol. 27, pp. 379 423 (Part one), pp. 623 - 656 (Part two), Oct.1948, reprinted in book form, University of Illinois Press, Urbana, 1949.

[6] S. Zhou,and G. B. Giannakis, “Adaptive modulation for multiantennatransmissions with channel mean feedback,” IEEE Trans. WirelessComm., vol.3, no.5, pp. 1626 - 1636, Sep. 2004.

[7] S. Zhou and G. B. Giannakis, “Optimal transmitter eigen-beamformingand space-time block coding based on channel mean feedback” IEEETransactions on Signal Processing, vol. 50, no. 10, October 2002.

[8] P. Xia and G. B. Giannakis, “Multiantenna adaptive modulation withbeamforming based on bandwidth-constrained feedback,” IEEE Transac-tions on Communications, vol. 53, no. 3, March 2005.

[9] Z. Luo, H. Gao,and Y. Liu,J. Gao “Capacity Limits of Time-VaryingMIMO Channels,” IEEE International Conference On Communications,vol.2, Mar. 2003.

[10] Z. Shen, R. W. Heath, Jr., J. G. Andrews, and B. L. Evans, “Comparisonof Space-Time Water-filling and Spatial Water-filling for MIMO FadingChannels,” in Proc. IEEE Int Global Communications Conf., vol. 1, pp.431 435, Nov. 29-Dec. 3, 2004, Dallas, TX, USA.

[11] Z. Zhou and B. Vucetic “Design of adaptive modulation using imperfectCSI in MIMO systems,” 2004 Eelectronics Letters, vol. 40, no. 17, Aug.2004.

[12] X. Zhang and B. Ottersten, “Power allocation and bit loading for spatialmultiplexing in MIMO systems,” IEEE Int. Conf.on Acoustics, Speech,and Signal Processing, 2003. Proceedings (ICASSP ’03), vol. 5, pp. 54- 56, Apr. 2003.

[13] B. Mondal and R. W. Heath, Jr., “Performance analysis of quantizedbeamforming MIMO systems,” IEEE Transactions on Signal Processing,vol. 54, no. 12, Dec. 2006.

[14] B. Chalise, S. Shahbazpanahi, A. Czylwik, and A. Gershman“Robustdownlink beamforming based on outage probability specifications,” IEEETrans. Wireless Comm., vol. 6, no. 10, Oct. 2007

[15] S. Zhou and G. B. Giannakis, “How accurate channel prediction needsto be for transmit-beamforming with adaptive modulation over RayleighMIMO channels,” IEEE Trans. Wireless Comm., vol.3, no.4, pp. 12851294, July 2004.

[16] D. J. Love, R. W. Heath, Jr. and T. Strohmer, “Grassmannian beam-forming for multiple-input multiple-output wireless systems,” IEEE Trans.Inform. Theory, vol. 49, no. 10, pp. 2735 2747, Oct. 2003.

[17] J. Zheng and B. D. Rao, “Capacity analysis of MIMO systems usinglimited feedback transmit precoding schemes,” IEEE Trans. on SignalProcessing, vol. 56, no. 7, pp. 2886 - 2901, July 2008.

[18] J. K. Cavers, “Single-user and multiuser adaptive maximal ratio trans-mission for Rayleigh channels,” IEEE Trans. Veh. Technol., vol. 49, no.6, pp. 2043 - 2050, Nov. 2000.