Design of Spatial Decoupling Scheme

Design of Spatial Decoupling Scheme using Singular Value Decomposition for Multi-User Systems

Abstract– In this paper, we present the use of a polynomial singular value decomposition (PSVD) algorithm to examine a spatial decoupling based block transmission design for multiuser systems. This algorithm facilitates joint and optimal decomposition of matrices arising inherently in multiuser systems. Spatial decoupling allows complex multichannel problems of suitable dimensionality to be spectrally diagonalized by computing a reduced-order memoryless matrix through the use of the coordinated transmit precoding and receiver equalization matrices.

A primary application of spatial decoupling based system can be useful in discrete multitone (DMT) systems to combat the induced crosstalk interference, as well as in OFDM with intersymbol interference. We present here simulation-based performance analysis results to justify the use of PSVD for the proposed algorithm.

Index Terms-polynomial singular value decomposition, paraunitary systems, MIMO system.

1. INTRODUCTION

Block transmission based systems allows parallel, ideally noninterfering, virtual communication channels between multiuser channels. Minimally spatial decoupling channels are needed whenever more than two transmitting channels are communicate simultaneously. The channel of our interest here, is the multiple input multiple output channels, consisting of multiple MIMO capable source terminals and multiple capable destinations.

This scenario arises, obviously, in multi-user channels. Since certain phases of relaying involves broadcasting, it also appears in MIMO relaying contexts. The phrase ‘MIMO broadcast channel’ is frequently used in a loose sense in the literature, to include point-to-multipoint unicast (i.e. ‘private’) channels carrying different messages from a single source to each of the multiple destinations (e.g. in multi-user MIMO). Its use in this paper is more specific, and denotes the presence of at least one ‘common’ virtual broadcast channel from the source to the destinations.

The use of iterative and non-iterative spatial decoupling techniques in multiuser systems to achieve independent channels has been investigated, for instance in [1]-[9].

Their use for MIMO broadcasting, which requires common multipoint-to-multipoint MIMO channels is not much attractive, given the fact that the total number of private and common channels is limited by the number of antennas the source has.

Wherever each receiver of a broadcast channel conveys what it receives orthogonally to the same destination, as in the case of pre-and post-processing block transmission, the whole system can be envisaged as a single point-to-point MIMO channel.

Block transmission techniques have been demonstrated for point-to-point MIMO channels to benefit the system complexities. Other advantages includes: (i) channel interference is removed by creating $K$ independent subchannels; (ii) paraunitarity of precoder allows to control transmit power; (iii) paraunitarity of equalizer does not amplify the channel noise; (iv) spatial redundancy can be achieved by discarding the weakest subchannels.

Though the technique outperform the conventional signal coding but had its own demerits.  Amongst many, it shown in cite{Ta2005,Ta2007} that an appropriate additional amount of additive samples

still require individual processing, e.g. per- tone equalisation, to remove ISI, and  the receiver does not exploit the case of structured noise.

However, the choice of optimal relay gains, although known for certain cases (e.g. [10], [11]), is not straightforward with this approach. Since the individual equalization have no non-iterative means of decoding the signals, this approach cannot be used with decode-and-forward (DF), and code-and-forward (CF) relay processing schemes.

The use of zero-forcing at the destination has been examined [12], [13] as a mean of coordinated beamforming, since it does not require transmitter processing. The scheme scales to any number of destinations, but requires each destination to have no less antennas than the source.

Although not used as commonly as the singular value decomposition (SVD), generalized singular value decomposition (GSVD) [14, Thm. 8.7.4] is not unheard of in the wireless literature. It has been used in multi-user MIMO transmission [15], [16], MIMO secrecy communication [17], [18], and MIMO relaying [19]. Reference [19] uses GSVD in dual-hop AF relaying with arbitrary number of relays. Since it employs zero-forcing at the relay for the forward channel, its use of GSVD appears almost similar to the use of SVD in [1].

Despite GSVD being the natural generalization of SVD for two matrices, we are yet to see in the literature, a generalization of SVD-based beamforming to GSVD-based beamforming. Although the purpose and the use is somewhat different, the reference [17, p.1] appears to be the first to hint the possible use of GSVD for beamforming. In present work, we illustrate how GSVD can be used for coordinated beamforming in source-to-2 destination MIMO broadcasting; thus in AF, DF and CF MIMO relaying. We also present comparative, simulation-based performance analysis results to justify GSVD-based beamforming.

The paper is organized as follows: Section II presents the mathematical framework, highlighting how and under which constraints GSVD can be used for beamforming. Section III examines how GSVD-based beamforming can be applied in certain simple MIMO and MIMO relaying configurations. Performance analysis is conducted in section IV on one of these applications. Section V concludes with some final remarks.

Notations: Given a matrix A and a vector v, (i) A(i, j)

gives the ith element on the jth column of A; (ii) v(i)

{ˆy1 }R(r+1,r+s) = ˜Σ{x }R(r+1,r+s) +

_

UHn1

_

R(r+1,r+s) ,

{ˆy2 }R(p−t+r+1,p−t+r+s) = ˜Λ{x }R(r+1,r+s) +

_

VHn2

_

R(p−t+r+1,p−t+r+s) ,

{ˆy1 }R(1,r) = {x }R(1,r) +

_

UHn1

_

R(1,r) ,

{ˆy2 }R(p−t+r+s+1,p) = {x }R(r+s+1,t) +

_

VHn2

_

R(p−t+r+s+1,p) . (1)

gives the element of v at the ith position. {A}R(n) and

{A}C(n) denote the sub-matrices consisting respectively of the

first n rows, and the first n columns of A. Let {A}R(m,n)

denote the sub-matrix consisting of the rows m through n

of A. The expression A = diag (a1, . . . , an) indicates that

A is rectangular diagonal; and that first n elements on its

main diagonal are a1, . . . , an. rank (A) gives the rank of

A. The operators ( ・ )H, and ( ・)−1 denote respectively the

conjugate transpose and the matrix inversion. C mÃ-n is the

space spanned by mÃ-n matrices containing possibly complex

elements. The channel between the wireless terminals T1 and

T2 in a MIMO system is designated T1 →T2.

II. MATHEMATICAL FRAMEWORK

Let us examine GSVD to see how it can be used for

beamforming. There are two major variants of GSVD in the

literature (e.g. [20] vs. [21]). We use them both here to

elaborate the notion of GSVD-based beamforming.

A. GSVD – Van Loan definition

Let us first look at GSVD as initially proposed by Van Loan

[20, Thm. 2].

Definition 1: Consider two matrices, H ∈C mÃ-n with

m ≥n, and G ∈C pÃ-n, having the same number n of

columns. Let q = min (p, n). H and G can be jointly

decomposed as

H = UΣQ, G = VΛQ (2)

where (i) U ∈C mÃ-m,V ∈C pÃ-p are unitary, (ii) Q ∈

C nÃ-n non-singular, and (iii) Σ= diag (σ1, . . . , σn) ∈

C mÃ-n, σi ≥0; Λ= diag (λ1, . . . , λq) ∈C pÃ-n, λi ≥0.

As a crude example, suppose that G and H above represent

channel matrices of MIMO subsystems S →D1 and S →D2

having a common source S. Assume perfect channel-stateinformation

(CSI) on G and H at all S,D1, and D2. With

a transmit precoding matrix Q−1, and receiver reconstruction

matrices UH,VH we get q non-interfering virtual broadcast

channels. The invertible factor Q in (2) facilitates jointprecoding

for the MIMO subsystems; while the factors U,V

allow receiver reconstruction without noise enhancement. Diagonal

elements 1 through q of Σ,Λrepresent the gains

of these virtual channels. Since Q is non-unitary, precoding

would cause the instantaneous transmit power to fluctuate.

This is a drawback not present in SVD-based beamforming.

Transmit signal should be normalized to maintain the average

total transmit power at the desired level.

This is the essence of ‘GSVD-based beamforming’ for

a single source and two destinations. As would be shown

in Section III, this three-terminal configuration appears in

various MIMO subsystems making GSVD-based beamforming

applicable.

B. GSVD – Paige and Saunders definition

Before moving on to applications, let us appreciate GSVDbased

beamforming in a more general sense, through another

form of GSVD proposed by Paige and Saunders [21, (3.1)].

This version of GSVD relaxes the constraint m ≥n present

in (2).

Definition 2: Consider two matrices, H ∈C mÃ-n and

G ∈C pÃ-n, having the same number n of columns. Let

CH =

_

HH,GH

_

∈C nÃ-(m+p), t = rank(C), r =

t −rank (G) and s = rank(H) + rank (G) −t.

H and G can be jointly decomposed as

H = U (Σ 01 )Q = UΣ{Q}R(t) ,

G = V (Λ 02 )Q = VΛ{Q}R(t) , (3)

where (i) U ∈C mÃ-m,V ∈C pÃ-p are unitary, (ii)

Q ∈C nÃ-n non-singular, (iii) 1 ∈C mÃ-(n−t), 2 ∈

C pÃ-(n−t) zero matrices, and (iv) Σ∈C mÃ-t,Λ∈

C pÃ-t have structures

Σ_

⎛

⎝

IH

˜Σ

H

⎞

⎠

and

Λ_

⎛

⎝

G

˜Λ

IG

⎞

⎠.

IH ∈C rÃ-r and IG ∈C (t−r−s)Ã-(t−r−s) are identity

matrices. H ∈C (m−r−s)Ã-(t−r−s), and G ∈

C (p−t+r)Ã-r are zero matrices possibly having no

rows or no columns. ˜Σ= diag (σ1, . . . , σs) ,˜Λ=

diag (λ1, . . . , λs) ∈C sÃ-s such that 1 > σ1 ≥. . . ≥

σs > 0, and σ2

i + λ2i

= 1 for i ∈ {1, . . . , s}.

Let us examine (3) in the MIMO context. It is not difficult

to see that a common transmit precoding matrix

_

Q−1

_

C(t)

and receiver reconstruction matrices UH,VH would jointly

diagonalize the channels represented by H and G.

For broadcasting, only the columns (r+1) through (r +s)

of Σand Λare of interest. Nevertheless, other (t −s)

columns, when they are present, may be used by the source

S to privately communicate with the destinations D1 and

configuration # common channels # private channels

S → {D1,D2} S →D1 S →D2

m > n,p ≤n p n −p

m ≤n, p > n m 0 n −m

m ≥n, p ≥n n 0 0

m < n, p < n, m + p −n n −p n −m

(m + p) > n

n ≥(m + p) 0 m p

TABLE I

NUMBERS OF COMMON CHANNELS AND PRIVATE CHANNELS FOR

DIFFERENT CONFIGURATIONS

D2. It is worthwhile to compare this fact with [22], and

appreciate the similarity and the conflicting objectives GSVDbased

beamforming for broadcasting has with MIMO secrecy

communication.

Thus we can get ˆy1 ∈C mÃ-1, ˆy2 ∈C pÃ-1 as in (1) at

the detector input, when x ∈C tÃ-1 is the symbol vector

transmitted. It can also be observed from (1) that the private

channels always have unit gains; while the gains of common

channels are smaller.

Since, σis are in descending order, while the λis ascend

with i, selecting a subset of the available s broadcast channels

(say k ≤s channels) is somewhat challenging. This highlights

the need to further our intuition on GSVD.

C. GSVD-based beamforming

Any two MIMO subsystems having a common source

and channel matrices H and G can be effectively reduced,

depending on their ranks, to a set of common (broadcast) and

private (unicast) virtual channels. The requirement for having

common channels is rank (H) + rank (G) > rank (C)

where C =

_

HH,GH

_

H.

When the matrices have full rank, which is the case with

most MIMO channels (key-hole channels being an exception),

this requirement boils down to having m +p > n . Table I

indicates how the numbers of common channels and private

channels vary in full-rank MIMO channels. It can be noted

that the cases (m > n,p ≤n) and (m ≥n, p ≥n)

correspond to the form of GSVD discussed in the Subsection

II-A. Further, the case n ≥(m + p) which produces only

private channels with unit gains, can be seen identical to zero

forcing at the transmitter. Thus, GSVD-based beamforming is

also a generalization of zero-forcing.

Based on Table I, it can be concluded that the full-rank

min (n,m + p) of the combined channel always gets split

between the common and private channels.

D. MATLAB implementation

A general discussion on the computation of GSVD is found

in [23]. Let us focus here on what it needs for simulation:

namely its implementation in the MATLAB computational

environment, which extends [14, Thm. 8.7.4] and appears as

less restrictive as [21].

The command [V, U, X, Lambda, Sigma] = gsvd(G, H);

gives1 a decomposition similar to (3). Its main deviations

from (3) are,

1Reverse order of arguments in and out of ‘gsvd’ function should be noted.

)

)

D1

y1 , r1

S

x ,w

(

(

)

)

D2

y2 , r2

_

H1 __

n1

_

__

H2

n2

Fig. 1. Source-to-2 destination MIMO broadcast system

• QH = X ∈C nÃ-t is not square when t < n. Precoding

for such cases would require the use of the pseudo-inverse

operator.

• Σhas the same block structure as in (3). But the structure

of Λhas the block G shifted to its bottom as follows:

Λ_

⎛

⎝

˜Λ

IG

G

⎞

⎠.

This can be remedied by appropriately interchanging the

rows of Λand the columns of V. However, restructuring

Λis not a necessity, since the column position of the

block ˜Λwithin Λis what matters in joint precoding.

Following MATLAB code snippet for example jointly

diagonalizes H,G to obtain the s common channels (3)

would have given.

MATLAB code

% channel matrices

H = (randn(m,n)+i*randn(m,n))/sqrt(2);

G = (randn(p,n)+i*randn(p,n))/sqrt(2);

% D1, D2: diagonalized channels

[V,U,X,Lambda,Sigma] = gsvd(G,H);

w = X*inv(X’*X); C = [H’ G’]’; t = rank(C);

r = t – rank(G); s = rank(H)+rank(G)-t;

D1 = U(:,r+1:r+s)’*H*w(:,r+1:r+s);

D2 = V(:,1:s)’*G*w(:,r+1:r+s);

III. APPLICATIONS

Let us look at some of the possible applications of GSVDbased

beamforming. We assume the Van Loan form of GSVD

for simplicity, having taken for granted that the dimensions

are such that the constraints hold true. Nevertheless, the Paige

and Saunders form should be usable as well.

A. Source-to-2 destination MIMO broadcast system

Consider the MIMO broadcast system shown in Fig. 1,

where the source S broadcasts to destinations D1 and D2.

MIMO subsystems S →D1 and S →D2 are modeled

to have channel matrices H1 ,H2 and additive complex

Gaussian noise vectors n1 , n2. Let x = [x1, . . . , xn]T

)

)

R1

y1 , F1

(

(

S

x ,w

(

(

)

)

D

y3 ,r1

y4 ,r2

)

)

R2

y2 , F2

(

(

_

___

H3

_ n3

H1 ___

n1

_

___

H2

n2 _

H4 ___

n4

Fig. 2. MIMO relay system with two 2-hop-branches

be the signal vector desired to be transmitted over n ≤

min (rank (H1 ) , rank (H2 )) virtual-channels. The source

employs a precoding matrix w.

The input y1 , y2 and output ˆy1 , ˆy2 at the receiver filters

r1 , r2 at D1 and D2 are given by

y1 = H1wx + n1 ; ˆy1 = r1 y1 ,

y2 = H2wx + n2 ; ˆy2 = r2 y2 .

Applying GSVD we get H1 = U1 Σ1 V and H2 =

U2 Σ2V. Choose the precoding matrix w = α

_

V−1

_

C(n)

;

and receiver reconstruction matrices r1 =

_

U1

H

_

R(n)

_ , r2 =

U2

H

_

R(n)

. The constant α normalizes the total average

transmit power.

Then we get,

ˆy1(i) = αΣ1(i, i) x(i) + ˜n1(i) ,

ˆy2(i) = αΣ2(i, i) x(i) + ˜n2(i), i∈ {1 . . . n},

where Ëœn1 , Ëœn2 have the same noise distributions as n1 , n2 .

B. MIMO relay system with two 2-hop-branches (3 time-slots)

Fig. 2 shows a simple MIMO AF relay system where a

source S communicates a symbol vector x with a destination

D via two relays R1 and R2. MIMO channels S →R1, S →

R2, R1 →D and R2 →D are denoted: Hi , i ∈ {1, 2, 3, 4}.

Corresponding channel outputs and additive complex Gaussian

noise vectors are yi , ni for i ∈ {1, 2, 3, 4}. Assume relay

operations to be linear, and modeled as matrices F1 and F2 .

Assume orthogonal time-slots for transmission. The source

S uses w as the precoding matrix. Destination D uses

different reconstruction matrices r1 , r2 during the time slots

2 and 3. Then we have:

Time slot 1: y1 = H1wx + n1 , y2 = H2wx + n2

Time slot 2: y3 = H3 F1 y1 + n3

Time slot 3: y4 = H4 F2 y2 + n4

Let ˆy = r1 y3 +r2 y4 be the input to the detector. Suppose

n ≤min

i

(rank (Hi )) virtual-channels are in use.

)

)

R

y1 , F

(

(

S

x ,w

(

(

)

)

D

y2 ,r1

y3 ,r2

_

___

H3

_ n3

H1 ___

n1

H2 _

n2

Fig. 3. MIMO relay system having a direct path and a relayed path

Applying GSVD on the broadcast channel matrices we get

H1 = U1 Σ1 Q and H2 = U2 Σ2 Q. Through SVD we

obtain H3 = V1 Λ1 R1

H and H4 = V2 Λ2 R2

H. Choose

w = α

_

Q−1

_

C(n)

; F1 = R1U1

H; F2 = R2U2

H; r1 = _

V1

H

_

R(n)

; r2 =

_

V2

H

_

R(n)

. The constant α normalizes

the total average transmit power. Then we get ˆy to be

α{(Λ1Σ1 +Λ2Σ2 )}C(n)x+Λ1˜n1+Λ2˜n2+˜n3 + ˜n4

_

R(n)

,

where each Ëœni has the same noise distribution as ni .

Remarks:

• The matrices U1

H and U2

H can be used by the relays

to extract (and if necessary decode) each channel passing

through them. Hence, the same beamforming matrices

can be used with the DF and CF schemes.

• The relay operations Fi can be modeled more generally

as Ri PiUi

H, i ∈ {1, 2} with the diagonal matrices Pi

governing power allocation among the virtual-channels.

C. MIMO relay system having a direct path and a relayed

path (2 time-slots)

Fig. 3 depicts a MIMO relay system having 3 nodes: source

S, relay R and destination D. The S →R, S →D and

R →D MIMO channels are H1 ,H2 and H3 . Corresponding

channel outputs are y1 , y2 and y3 ; additive complex

Gaussian noise vectors are n1 , n2 and n3 . Relay operation is

linear and represented by a matrix F. For a transmit symbol

vector x we get:

Time slot 1: y1 = H1wx + n1 , y2 = H2wx + n2

Time slot 2: y3 = H3 Fy1 + n3

Let ˆy = r1 y2 + r2 y3 be the input to the detector.

Assume n ≤min

i

(rank (Hi )) virtual-channels to be in

use. Applying GSVD on the broadcast-phase channel matrices

we get H1 = U1 Σ1 Q and H2 = U2 Σ2 Q. Applying

SVD we obtain H3 = VΛRH. Choose w = α

_

Q−1

_

C(n)

;

F = RU1

H; r1 =

_

U2

H

_

R(n)

; and r2 =

_

VH

_

R(n)

. The

constant α normalizes the total average transmit power. Then

we get ˆy to be

α {(ΛΣ1 + Σ2 )}C(n) x +Λ˜n1 + ˜n2 + ˜n3

_

R(n)

, (4)

where each Ëœni has the same noise distribution as ni .

S1

x1 ,w1

(

(

)

)

D1

y5 ,r5

y3 ,r3

)

)

R

y1 ,r1

y2 ,r2

x3 ,w3

(

(

S2

x2 ,w2

(

(

)

)

D2

y6 ,r6

y4 ,r4

_

__

H1

n1

H5 _

n5

_

H3 __

n3

_

__

H4

_ n4

H2 __

n2

H6 _

n6

Fig. 4. CF relaying with Network Coding

D. CF relaying with Network Coding (3 time-slots)

Network coding schemes that code 2-messages at a time

(e.g. those based on XOR operation), require broadcasting

information to 2 destinations. The simplest network to support

CF relaying (see Fig. 4), for instance, has three such broadcast

phases, each of which can be exploited via GSVD-based

beamforming.

Let S1, S2 be the sources; D1,D2 the destinations; and R

the CF relay. MIMO channels S1 →R, S2 →R, R →D1,

R →D2, S1 →D1, and S2 →D2 are denoted respectively

Hi , i ∈ {1, . . . , 6}. Corresponding channel outputs

and additive complex Gaussian noise vectors are yi , and

ni , i ∈ {1, . . . , 6}.

Output of receiver filters ˆyi = ri yi , i ∈ {1, . . . , 6}, are

used to decode the signals at R, D1 and D2. The sources

S1, S2 transmit the codewords x1 , x2 respectively in the 1st

and 2nd time-slots. The relay XORs what it decodes from ˆy1

and ˆy3 to form x3 , and transmits it in the 3rd time-slot.

Applying GSVD, separately for each time-slot, provides the

transmit precoding and receiver reconstruction matrices for

diagonalizing all 6 channels.

IV. PERFORMANCE ANALYSIS

This section evaluates the performance of GSVD-based

beamforming, comparing it with that of SVD-based beamforming

for a specific example: the MIMO AF relay system

outlined in Subsection III-C.

SVD-based beamforming is also possible for this case since

(i) AF relaying is used; and (ii) the system has a single sourcedestination

pair. Define y =

_

y2

H, y3

H

_

H. Then we have,

y =

_

H3FH1

H2

_ __ _

ˆH

wx +

_

H3 Fn1

+

_

n3

n2

(5)

Suppose ˆH = ˆU

ˆΣ

ˆVH is the SVD. The channel can be

diagonalized by choosing the transmit precoding matrix w =

ˆV

_

C(n)

and receiver reconstruction matrix r =

ˆUH

_

R(n)

.

0 2 4 6 8 10 12 14 16

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

average normalized transmit SNR (dB)

outage probability

k = 3

k = 2

k = 1

(solid) GSVD−based b/f

(dash) SVD−based b/f

Fig. 5. Outage performance of GSVD-based beamforming vs. SVD-based

beamforming, for MIMO AF relaying with Ns = 4,Nr = 3,Nd = 5, for

n = 3 common channels.

However, with this approach, the choice of F is not straightforward.

• An apparent choice is selecting F to invert H3 ; which

is essentially zero-forcing in the forward direction.

• Another is to choose F = V3U1

H, where Hi =

Ui ΣiVi

H, i ∈ {1, 3} as governed by the SVD. It’s

optimality, reasoned out for slightly different configurations

in [11, Eqn. (22)], and [24, Eqn. (7)], may be

appreciated in the light that aligning the eigemodes of

input and output channels is almost the best the relay can

do towards improving the signal-to-noise ratio (SNR) at

the destination. This form of F is assumed here.

Fig. 5 compares the outage performance of the three common

channels2 of a MIMO AF relay configuration having

Ns = 4,Nr = 3,Nd = 5 source, relay and destination

antennas, for both GSVD-based beamforming and SVD-based

beamforming. Fig. 6 shows the average symbol error rate

(SER) of quadrature phase shift keying (QPSK) modulation

for the same configuration. Monte-Carlo simulation based on

(4) and (5), has been employed with 107 simulation points.

The constant α too was found through simulation.

As expected, the first channel (k = 1) shows better outage

and SER performance than the other two. GSVD-based

beamforming fares within 3dB of SVD-based beamforming

for moderate SNR. Interestingly, GSVD-based beamforming

appears to have higher diversity order for this case. This

observation is yet to be established theoretically.

Incidentally, the instantaneous per channel received SNR

γ(gsvd)

i , i ∈ {1, . . . , n} for GSVD-beamforming can be written

from (4) as

γ(gsvd)

i =

(Λ(i, i) Σ1(i, i) +Σ2(i, i) )2

Λ(i, i) 2 + 2 α2P, (6)

2GSVD-based beamforming over this MIMO configuration yields 3 common

channels and a single source-to-destination private channel. The private

channel is not considered here. Performance over the others are compared

against the best 3 of the 4 channels SVD-based beamforming produces.

0 2 4 6 8 10 12 14 16 18

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

average transmit SNR (dB)

average SER

k = 3

k = 2

k = 1

(solid) GSVD−based b/f

(dash) SVD−based b/f

Fig. 6. The average SER of GSVD-based beamforming vs. SVD-based

beamforming, for MIMO AF relaying with Ns = 4,Nr = 3,Nd = 5, for

n = 3 common channels.

where P is the transmit SNR. SVD-based beamforming would

not give an as concise form since r and U3 are not generally

orthogonal.

More interestingly GSVD-beamforming allows the symbols

to be decoded at the relay; and the received SNR over the ith

virtual channel would be α2P (Σ1(i, i) )2. Perfect decoupling

of the virtual channels at both the relay and the destination

makes GSVD-based beamforming usable with all AF, DF, and

CF relay processing schemes. This is a feat not achievable with

SVD-based beamforming.

V. CONCLUSION

The use of generalize singular value decomposition (GSVD)

for coordinated beamforming in MIMO systems has been

examined. Several applications of GSVD-based beamforming

have been summarized. Performance of one of them was

evaluated; and seen to perform within 3dB of SVD-based

beamforming. This, combined with the applicability with DF

and CF relay processing schemes makes GSVD-beamforming

promising. However, further analysis on different MIMO

configurations is required to assess its usefulness. From a

theoretical point-of-view, incorporating GSVD into random

matrix theory is vital to accurately characterize GSVD-based

beamforming. An interesting design problem would be seeking

the ways of utilizing the common and private virtual channels

in hybrid.

ACKNOWLEDGMENT

This work is supported in part by the Alberta Ingenuity

Fund through the iCORE ICT Graduate Student Award.

REFERENCES

[1] B. Farhang-Boroujeny, Q. Spencer, and L. Swindlehurst, “Layering

techniques for space-time communication in multi-user networks,” in

Proc. Vehicular Technology Conference. VTC Fall. IEEE 58th, vol. 2,

Orlando, FL, Oct. 2003, pp. 1339-1343.

[2] L.-U. Choi and R. Murch, “A transmit preprocessing technique for

multiuser MIMO systems using a decomposition approach,” IEEE Trans.

Wireless Commun., vol. 3, no. 1, pp. 20-24, Jan. 2004.

[3] Z. Pan, K.-K. Wong, and T.-S. Ng, “Generalized multiuser orthogonal

space-division multiplexing,” IEEE Trans. Wireless Commun., vol. 3,

no. 6, pp. 1969-1973, Nov. 2004.

[4] D. Hoang and R. Iltis, “Generalized eigencoding for MIMO ad-hoc

networks,” in Proc. Information Sciences and Systems, 40th Annual

Conference on, Princeton, NJ, Mar. 2006, pp. 723-728.

[5] F. Boccardi and H. Huang, “A near-optimum technique using linear

precoding for the MIMO broadcast channel,” in Proc. Acoustics, Speech

and Signal Processing, IEEE International Conference on, vol. 3,

Honolulu, HI, Apr. 2007, pp. 17-20.

[6] H. Lee, K. Lee, B. M. Hochwald, and I. Lee, “Regularized channel

inversion for multiple-antenna users in multiuser MIMO downlink,”

in Proc. Communications, IEEE International Conference on, Beijing,

China, May 2008, pp. 3501-3505.

[7] C.-B. Chae, D. Mazzarese, N. Jindal, and R. Heath, “Coordinated

beamforming with limited feedback in the MIMO broadcast channel,”

IEEE J. Sel. Areas Commun., vol. 26, no. 8, pp. 1505-1515, Oct. 2008.

[8] K.-H. Park, Y.-C. Ko, M.-S. Alouini, and J. Kim, “Low complexity

coordinated beamforming in 2-user MIMO systems,” in Proc. Communications,

IEEE International Conference on, Dresden, Germany, Jun.

2009.

[9] C.-B. Chae, S. hyun Kim, and R. Heath, “Linear network coordinated

beamforming for cell-boundary users,” in Proc. Signal Processing Advances

in Wireless Communications, IEEE 10th Workshop on, Perugia,

Italy, Jun. 2009, pp. 534-538.

[10] X. Tang and Y. Hua, “Optimal design of non-regenerative MIMO

wireless relays,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1398-

1407, Apr. 2007.

[11] O. Munoz-Medina, J. Vidal, and A. Agustin, “Linear transceiver design

in nonregenerative relays with channel state information,” IEEE Trans.

Signal Process., vol. 55, no. 6, pp. 2593-2604, Jun. 2007.

[12] R. U. Nabar, O. Oyman, H. B¨olcskei, and A. J. Paulraj, “Capacity scaling

laws in MIMO wireless networks,” in P