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New Beamforming Schemes with Optimum Receive

Combining for Multiuser MIMO systems

Sang-Rim Lee, Seokhwan Park, Sung-Hyun Moon and Inkyu Lee

School of Electrical Eng., Korea University, Seoul, Korea

Email: {srlee, shpark, shmoon}@wireless.korea.ac.kr and inkyu@korea.ac.kr

Abstract—In this paper, we present a new beamforming

scheme for a downlink of multiuser multiple-input multiple-

output (MIMO) communication systems. Recently, a block-

diagonalization (BD) algorithm has been proposed for the mul-

tiuser MIMO downlink where both a base station and each

user have multiple antennas. However, the BD algorithm is not

efficient when the number of supported streams per user is

smaller than that of receive antennas. Since the BD method

utilizes the nullspace based on the channel matrix without

considering the receive combining, the degree of freedom for

beamforming cannot be fully exploited at the transmitter. In

this paper, we optimize the receive beamforming vector under

a zero forcing (ZF) constraint, where all inter-user interference

is driven to zero. We propose an efficient algorithm to find the

optimum receive vector by an iterative procedure. The proposed

algorithm requires two phase values feedforward information for

the receive combining vector. Also, we present another algorithm

which needs only one phase value by using a decomposition of

the complex general unitary matrix. Simulation results show that

the proposed beamforming scheme outperforms the conventional

BD algorithm in terms of error probability and obtains the

diversity enhancement by utilizing the degree of freedom at the

base station.

I. INTRODUCTION

Multiple-input multiple-output (MIMO) channels for future

wireless communication systems have attracted considerable

attention since the use of multiple antennas at both transmitter

and receiver was shown to provide extraordinary high data

rates compared to single-input single-output (SISO) systems

[1]. More recently, as an interest is being shifted to MIMO

Gaussian broadcast channels (BC), the sum capacity of these

MIMO Gaussian BC has been extensively studied by several

approaches [2]. It is well known that dirty paper coding (DPC)

introduced in [3] achieves this capacity. However, the encod-

ing process is interference-dependent and largely information-

theoretic. Thus, DPC is quite complex and requires techniques

which may be incompatible with practical systems.

In this paper, we consider a multiuser MIMO downlink

where a base station (BS) uses multiple antennas to communi-

cate with several co-channel users. One of the main challenges

is to develop a transmission scheme in such multiuser MIMO

This work was supported in part by the Ministry of Information and

Communication (MIC), Korea, under the Information Technology Research

Center (ITRC) support program, supervised by the Institute of Information

Technology Assessment (IITA) and in part by grant No. R01-2006-000-

11112-0 from the Basic Research Program of the Korea Science and En-

gineering Foundation.

systems which considers the co-channel interference of other

users. In the case where the mobile users are equipped with

a single antenna, channel inversion is one of simple beam-

forming techniques for eliminating all inter-user interference

[4]. A generalization of channel inversion to the system where

each user has multiple antennas, called block diagonalization

(BD), was proposed in [5]. The BD algorithm provides a

block-diagonal solution, which is less strict than the channel

inversion where a complete diagonalization is enforced. The

key idea of the BD is to eliminate multiuser interference by

placing all the unintended users at nullspace.

In this paper, we investigate a practical beamforming tech-

nique which fully utilizes the degree of freedom to improve

the performance under a zero forcing (ZF) constraint. We

focus on a case where the number of supported streams per

user is smaller than the number of receive antennas. This is the

case where the performance of the BD degrades. Since in the

BD method the nullspace is obtained based on the channel

matrix without considering the receive combining, the BD

cannot fully utilize the degree of freedom for diversity at the

BS. Our beamforming method transmits the data along the

nullspace of the other users’ effective channel matrix while

the receive combining process is taken into account. As a

result, diversity can be preserved by ensuring that there is no

interference on the activated spatial modes.

The problem to optimize the receive combining vector

is non-convex since both the intended user channel gain

and the interference for other users should be considered

simultaneously. In related work [6], an iterative descent al-

gorithm using QR-update is proposed, where the combining

gain obtained from cooperation among receive antennas at

each user is neglected since a diagonal matrix is employed

as a receive beamforming matrix. In contrast, we assume a

complex combining receive vector in order to improve the

performance by organizing the cooperation among receive

antennas. We propose an efficient iterative algorithm using

the complex rotation matrix introduced in [7] which achieves

the orthogonality between two complex vectors. The optimum

solution of the proposed algorithm can be represented by two

phase values feedforward information sent from the BS to

each user. Also, we present a second algorithm which needs

only one phase value by using a decomposition of the general

complex unitary matrix. The simulation results show that

the proposed algorithm achieves diversity improvement and

provides a significant performance gain compared to the BD

with comparable complexity.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

978-1-4244-2075-9/08/$25.00 ©2008 IEEE

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Throughout this paper, normal letters represent scalar quan-

tities, boldface letters indicate vectors and boldface uppercase

letters designate matrices. For any complex notation c, we

denote the real and imaginary part of c by ?[c] and ?[c],

respectively. For any matrix A, [A]m,ndenotes the (m,n)th

entry of A. A∗, AHand ATrepresent the complex conjugate,

the conjugate transpose and the transpose of A, respectively.

Tr(A) indicate the trace of A. I and 0 denote an identity

matrix and a zero matrix, respectively, and i indicates√−1.

II. SYSTEM MODEL

In this section, we consider a multiuser MIMO downlink

system where the BS is transmitting to K independent users

simultaneously and generates co-channel interference at all

users. In this system, the BS is equipped with M transmit

antennas and user j has nj≥ 2 receive antennas, referred to

in the following as {n1,··· ,nK} × M. Lj spatial streams

are supported for the jth user. In the discrete-time complex

baseband MIMO case, the channel from the BS to the jth

user is modeled as the nj× M channel matrix Hj, where

[Hj]p,q represents the channel gain from antenna q at the

BS to antenna p at user j. We assume that Hj is modeled

as a block-fading channel, which indicates that the channel

is constant during one block. Let xj represent the Lj× 1

transmit data symbol vector for user j. Each user receives a

combination of all L =?K

given as Nr=?K

Denoting Bj∈ CM×Ljas the transmit beamforming matrix

for user j, the received signal at the jth user can be written

as

i=1Lisymbol streams through its

own channel. The total number of antennas at all receivers is

j=1nj. We assume that Lj is less than or

equal to njand M is greater than L.

yj=

K

?

i=1

HjBixi+wj= HjBjxj+

?

∈

i?=j

HjBixi+wj (1)

where yj

[wj,1,··· ,wj,nj]T∈ Cnjare the received signal and noise

vectors, respectively. The components wj,iof the noise vector

wj are independently and identically distributed (i.i.d.) with

zero mean and variance σ2

1,··· ,nj. Note that the term

interference from the other users i ?= j to user j.

Applying the receive matrix Aj∈ CLj×njin equation (1),

the filter output zjfor the jth user can be expressed as

?

=[yj,1,··· ,yj,nj]T

Cnj

and wj

=

wfor j = 1,··· ,K and i =

?

i?=jHjBixi indicates the

zj= Ajyj= Aj

Hj

K

?

i=1

Bixi+ wj

?

.

(2)

For ease of exposition, we define the network channel

Hs, the receive beamforming matrix As and the transmit

beamforming matrix Bsas

Hs=?

Bs=?

HT

1

HT

2

...

HT

K

?T∈ CNr×M,

?∈ CM×L,

As= diag{A1,··· ,AK} ∈ CL×Nr,

B1

B2

···

BK

respectively. Then, the corresponding filter output at all the

users can be arranged as

where xs

= [xT

[wT

In order to satisfy the power constraint, we construct the

unnormalized signal xsand Bssuch that

zs=

z1

z2

...

zK

=AsHsBsxs+Asws=HeBsxs+Asws

1,xT

2,··· ,wT

(3)

2,··· ,xT

K]T

∈ CL×1, and ws

=

1,wT

K]T∈ CNr×1, and He= AsHs.

E?Bsxs?2≤ ρ

restrictive which

E(?Bsxs?2|Hs) ≤ ρ where ρ represents the total transmitted

power.

isa less powerconstraintthan

III. REVIEW OF BLOCK DIAGONALIZATION

In this section, we briefly describe the BD algorithm for

multiuser MIMO systems presented in [5]. To eliminate all

the multiuser interference, we impose a constraint as

HiBj= 0

for all

i ?= j

and

1 ≤ i,j ≤ K.

(4)

In order to satisfy the ZF constraint (4), Bj should lie in

the nullspace of?Hjwhere

Denoting?Ljas?Lj= rank(?Hj), we define the singular value

?Hj=?Uj?Λj[?V

matrix?Λjconsist of ordered singular values of?Hj, the matrix

the matrix˜V

j

holds the last (M−?Lj) right singular vectors.

we can construct the beamforming matrix Bjfor the jth user

using a linear combination of columns of?V

for transmission to the jth user to take place under the ZF

constraint,¯Ljshould be greater than or equal to 1. A sufficient

condition for¯Lj≥ 1 is that at least one row of Hjis linearly

independent of the other rows of?Hj. Assuming that all rows

be satisfied.

After applying the nullspace of other users’ channel matrix

˜V

j

to the jth user’s channel Hj, the received signal can be

written as

?Hj= [HT

decomposition (SVD) of?Hjas

where the unitary matrix?Ujcontains left singular vectors, the

?V

Since?V

Let¯Ljrepresent the rank of the product Hj˜V

1

··· HT

j−1HT

j+1

···

HT

k]T∈ C(Nr−nj)×M.

(1)

j

?V

(0)

j]H

(1)

j

is composed of the first˜Lj right singular vectors, and

(0)

(0)

j

forms an orthogonal basis for the nullspace of?Hj,

(0)

j.

(0)

j. In order

of Hj are linearly independent of?Hj, this condition would

(0)

yj= He,jxj+ wj

where He,j= Hj˜V

stream without interference from other users, any scheme for

single user MIMO systems can be applied.

(0)

j. Since the jth user receives its own data

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Page 3

In what follows, we analyze the dimension of the nullspace

utilized for beamforming considering a receive combining

matrix. We defineˆHjas

Aj−1Hj−1

Aj+1Hj+1

...

AKHK

ˆHj=

A1H1

...

∈ C(L−Lj)×M.

(5)

Then, the dimension of the nullspace ofˆHjis M−(L−Lj). In

other words, we can employ the M−L+Ljorthonormal basis

vectors for transmit beamforming corresponding to the jth

user. In contrast, the BD algorithm does not consider the num-

ber of actual supported streams for each user in the process of

obtaining the nullspace of?Hj. Since the receive combining

fully utilize the degree of freedom for beamforming. In the

BD, the number of orthonormal basis vectors for the jth user

is M−(Nr−nj). Since L−Ljshould be less than or equal to

Nr−nj, the BD has a smaller dimension for determining the

beamforming matrix. For instance, for the case of {2,2} × 4

where one stream is assumed to be supported per user, each

user with a receive combining matrix is assigned 3(=4-2+1)

dimensions in determining the beamforming matrix, while the

BD algorithm only utilizes 2(=4-4+2) dimensions. This added

degree of freedom in computing the receive combining matrix

allows us to design a system with a better performance, and it

will be confirmed in the simulation section that the increased

dimension for beamforming improves the diversity order.

process is not included in the BD method, the BD cannot

IV. NEW BEAMFORMING SCHEME

In this section, we present a procedure for finding the

optimal receive matrix Aswhich minimizes the power used

for eliminating all multiuser interference. We propose an

efficient iterative algorithm based on the result in [6] using

the rotation matrices in [7]. For simplicity, we consider the

case where each user is equipped with two receive antennas

(nj = 2) and assume that the channel matrix Hs is known

perfectly at the BS. Note that the proposed algorithm can be

generalized to the case with nj > 2 receive antennas. Also,

we assume that one stream is supported for each user. Since

the receive matrix Aj reduces to a row vector, we will use

ajto denote the receive combining vector from now on.

In order to satisfy the ZF constraint, we can select a

pseudo-inverse of the effective channel matrix considering

the receive combining vector at all users. Thus, the overall

transmit beamforming matrix Bsfor (3) can be obtained as

?

Applying the transmit beamforming matrix Bsin equation

(3), the filter output vector zscan be rewritten as

1

√γxs+ Asws

Bs= HH

e

HeHH

e

?−1

.

zs=

where γ is referred to as the power loss factor. From the power

constraint, γ is computed as

γ =1

ρTr

Our objective is to obtain the optimum receive vector for

minimization of the power loss factor γ. This problem is

formulated as

??

HeHH

e

?−1?

.

(a1,··· ,aK)opt= arg

As observed in (6), this joint optimization problem is a

complicated non-convex problem.

In what follows, we present an iterative algorithm, which

will be referred to as Algorithm 1, to solve the optimization

problem. First, we will show that the permutation of the

effective channel matrix does not change the metric. From

(3) and (5), we define Pj as the permutation matrix which

satisfies the following condition

min

a1,···,aKTr

??

HeHH

e

?−1?

.

(6)

PjHe,perm= He

where the permuted effective channel matrix He,permis given

by

He,perm?

?

ˆHj

ajHj

?

.

Then, the optimization metric from the above relation can

be represented as

??

= Tr

P−1

j

He,permHH

Tr

HeHH

e

?−1?

?

=Tr

??

(PjHe,perm)(PjHe,perm)H?−1?

?−1

?

e,perm

P−H

j

?

.

Using the properties Tr(AB) = Tr(BA) and PjPH

above metric is solved as

??

From (7), we can see that the metric is not affected by the

permutation of the effective channel matrix.

Next, we obtain the optimum solution of (6) by utilizing the

result in [6]. Due to non-convexity of the optimum problem

(6), it is difficult to solve directly the overall receive beam-

forming matrix As. To overcome this difficulty, we calculate

iteratively the optimum combining vector ajfor the jth user

by fixing the receive combining vectors of the other users.

Instead of performing joint optimization of (6), a solution for

aj with other ai’s for i ?= j fixed can be computed as (see

Theorem 1 in [6] for details)

??

= e1

I + HjQR−1R−HQHHH

?

where the QR factorization ofˆH

QR, and e1(Γ) refers to the eigenvector corresponding to

j= I, the

Tr

HeHH

e

?−1?

= Tr

??

He,permHH

e,perm

?−1?

.

(7)

argmin

aj

Tr

He,permHH

e,perm

?−1?

??

×

j

?−1

HjHH

j− HjQQHHH

j

??

(8)

H

j

is defined asˆH

H

j

=

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Page 4

the maximum eigenvalue of Γ. After obtaining aj for j =

1,··· ,K from (8) and repeating this procedure iteratively,

we can determine the optimum vector set which minimizes

the power loss factor.

To summarize, the proposed algorithm 1 can be described

as follows:

1) Initialize ajas aj= [1 0] for j = 1,··· ,K.

2) Set iter count = 0 for outer loop.

3) Set j = 0 for inner loop.

4) ObtainˆHjand Hjfrom He.

5) Compute Q and R by performing the QR factorization

ofˆH

6) Determine ajas the unit-norm eigenvector corresponding

to the maximum eigenvalue of the matrix

?

7) Go back to step 4 with j ← j + 1 unless j > K.

8) Go back to step 3 with iter count ← iter count + 1 if

iter count < max iteration.

As stated in [6], this iterative algorithm may converge to

a local optimum due to nature of a non-convex problem.

Complexity can be reduced by employing simpler QR-updates

instead of the QR-factorization at step 5 in the above algo-

rithm. Note that unlike [6], our algorithm works directly on

complex-valued matrices.

Once the receive beamforming vector aj is computed, the

given ajcan be represented using a general complex unitary

matrix expressed by [7]

?

where we define cos(α,β) and sin(α,β) as

H

j.

I+ HjQR−1R−HQHHH

j

?−1?

HjHH

j−HjQQHHH

j

?

.

C(θ1,θ2) ?

cos(θ1,θ2)

−sin∗(θ1,θ2)

sin(θ1,θ2)

cos∗(θ1,θ2)

?

(9)

cos(α,β)

sin(α,β)

=

=

cos(α)cos(β) + isin(α)sin(β),

sin(α)cos(β) + icos(α)sin(β).

In [7], the above complex rotation matrix is used for the com-

plex vector orthogonalization which provides a constructive

basis for beamforming in single user MIMO channels. In our

formulation, we employ the row vectors of C(θ1,θ2) as a

general expression for complex-valued unit-norm vectors. In

this paper, we use the first row of this matrix for receive

beamforming, although the second row of the matrix can be

employed as well. Then, we define the receive beamforming

vector ajas

aj=?

To compute two phase values θj,1and θj,2which identify

aj, we utilize the relation between the components of the

above equation. Denoting a1and a2as the first and the second

element of aj, respectively, we can easily compute θj,1and

θj,2as

θj,1= arctan(?[a2]/?[a1]),

θj,2= arctan(?[a2]/?[a1])).

Thus, after the maximum number of iterations is reached in

Algorithm 1, θj,1and θj,2can be obtained from (11). Then,

cos(θ1,θ2) sin(θ1,θ2)

?.

(10)

(11)

these values are transmitted to the jth user to identify ajusing

(10).

Now we will explain why a performance gain is expected

with the proposed aj in (10) in comparison to the solution

in [6]. Unlike our scheme, the receive combining filter in [6]

is extended to a matrix form in the case with two receive

antennas, which is given as

?

By using this combining matrix, two data streams are allocated

for each user. However, since the modulation is restricted to

one-dimensional PAM for each symbol in [6], the spectral

efficiency is the same as our scheme using two-dimensional

QAM.

Applying the above Ajto equation (2) and taking the real

part of the result, the filter output zjfor the scheme in [6] is

obtained as

Aj=

ejθj,1

0

0

ejθj,2

?

.

(12)

zj= ?[Ajyj] =

?

?[ejθj,1yj,1]

?[ejθj,2yj,2]

?

where zj = [zj,1zj,2]Tand yj= [yj,1yj,2]T. By utilizing

only the real part of complex-valued symbols for transmission,

the scheme in [6] eliminates the inter-user interference which

is confined to the imaginary part of the filter output. However,

as off-diagonal elements in (12) are zero, this receive combin-

ing matrix cannot obtain a performance gain from cooperation

among receive antennas. In the simulation section, we will

show that the proposed scheme outperforms the scheme in

[6] by allowing non-zero off-diagonal elements in Aj. By

adopting a more general form of the receive combining matrix

compared to [6], the proposed scheme exhibits a 2-3dB

gain over the method in [6] with the same computational

complexity, as will be demonstrated in the simulation section.

In Algorithm 1, two phase values should be transmitted

for each user. To reduce this overhead, we present a simpler

algorithm which requires only one phase value at the receiver,

which will be refered to as Algorithm 2. The simplification is

carried out based on the fact that the complex rotation matrix

in (9) can be decomposed into two matrices, i.e.,

?

By choosing one of two matrices above for the receive

combining, we can reduce the feedforward information to only

one phase value. Since any choice between two matrices does

not affect the overall performance, we use the first row of the

first matrix as the receive combining vector. Then, the receive

combining vector for the jth user for Algorithm 2 can be

expressed as

aj=?

Compared to the complex solution (10), aj in (13) has only

real-valued elements.

Now we will discuss how to determine the real-valued

optimum combining vector for Algorithm 2. In [6], the result

C(θ1,θ2) =

cos(θ1)

−sin(θ1)

sin(θ1)

cos(θ1)

??

cos(θ2)

isin(θ2)

isin(θ2)

cos(θ2)

?

.

cos(θj)sin(θj)

?.

(13)

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Page 5

(8) was derived using the following well-known result

?

ajM2aH

j

In the real-valued representation, the above equation is equiv-

alent to

?¯ aj¯ M1¯ ajH

where

¯ aj=?

for i = 1,2. Since ajis a real-valued vector (?[aj] = 0) for

Algorithm 2, the solution of (14) reduces to

??[M1]−1?[M2]?.

Employing the above result, we can obtain Algorithm 2 by

making a slight change in step 6 of Algorithm 1 as

6) Determine ajas the unit-norm eigenvector corresponding

to the maximum eigenvalue of the matrix

? ?

Similarly, after ajis determined from the iterative algorithm,

θj for the receive combining vector aj is easily computed

from (13). In the following simulation section, we will see

that the performance gap between Algorithm 1 and 2 is small.

In our proposed algorithms, for decoding the signal at the

receiver, the transmitter needs to send information about the

power loss factor γ and phase values to each user. Note that

this amount of information is inevitable in any beamforming

scheme for multiuser systems. Since the optimization and the

training can be done once per transmission block, this may

incur no additional penalty in this regard.

argmin

aj

ajM1aH

j

??? ?aj? = 1

?

= e1

?M−1

1M2

?.

argmin

¯ aj

¯ aj¯ M2¯ ajH

??? ? ¯ aj? = 1

?[aj]

?

?

= e1

?

¯ M−1

1

¯ M2

?

(14)

?[aj]

?,¯ Mi=

?[Mi]

?[Mi]

−?[Mi]

?[Mi]

?

e1

?

I+ HjQR−1R−HQHHH

j

??−1

?

? ?

HjHH

j−HjQQHHH

j

??

.

V. SIMULATION RESULTS

In this section, we present the simulation results for the

proposed beamforming schemes to compare the performance

with conventional algorithms. Throughout the simulation, the

number of iterations is set to 2 for Algorithm 1 and 2.

Figure 1 shows the simulation results for {2,2}×4 systems

in terms of bit error rate (BER) with respect to signal-to-noise

ratio (SNR) in dB at 2bps/Hz/user and 4bps/Hz/user. The

proposed beamforming scheme using Algorithm 1 provides

about a 5 dB power gain at a BER of 10−4over the BD

algorithm. Note that the diversity improvement is achieved

by transmitting the data along the nullspace of the other

users’ effective channel matrix while optimizing the receive

combining vector. That is, the proposed scheme can fully

utilize the remaining nullspace dimension at the BS for the

diversity improvement. As can be shown in this plot, the

performance loss of Algorithm 2 is less than 0.5dB compared

to Algorithm 1. In comparison to the scheme proposed in

[6], we can see that the utilization of cooperation among

receive antennas provides a performance gain up to 3dB. It

should be emphasized that with reduced feedforward overhead

compared, the proposed algorithm 2 achieves a 2dB gain over

the scheme in [6].

02468 10

SNR (dB)

12 141618 2022

10

-4

10

-3

10

-2

10

-1

{2,2} x 4

BER

BD (2bps/Hz/User)

Precoding [10] (2bps/Hz/User)

Algorithm 2 (2bps/Hz/User)

Algorithm 1 (2bps/Hz/User)

BD (4bps/Hz/User)

Precoding [10] (4bps/Hz/User)

Algorithm 2 (4bps/Hz/User)

Algorithm 1 (4bps/Hz/User)

Fig. 1.

4bps/Hz/user

Bit error probability of different methods at 2bps/Hz/user and

VI. CONCLUSION

In this paper, we have proposed a new beamforming scheme

with the optimum receive combining vector for multiuser

MIMO downlink systems where each user has more than

one antenna. Considering the receive combining process, the

proposed scheme can fully utilize the remaining degrees of

freedom for diversity. Also, unlike conventional schemes, we

can achieve a performance improvement from the cooperation

among receive antennas. An efficient iterative algorithm for

computation of the receive combining vector has been pro-

posed. Also, we present a simplified algorithm which requires

only one phase value per user at the expense of a small

performance loss. The simulation results confirm that the

proposed schemes outperform the conventional BD method.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.