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.

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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) 01 ∈C mÃ-(n−t), 02 ∈

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

C pÃ-t have structures

Σ_

⎛

⎝

IH

˜Σ

0H

⎞

⎠

and

Λ_

⎛

⎝

0G

˜Λ

IG

⎞

⎠.

IH ∈C rÃ-r and IG ∈C (t−r−s)Ã-(t−r−s) are identity matrices. 0H ∈C (m−r−s)Ã-(t−r−s), and 0G ∈

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 0

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 â€¢ 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 0G shifted to its bottom as follows:

Λ_

⎛

⎝

˜Λ

IG

0G

⎞

⎠.

This can be remedied by appropriately interchanging the rows of Λand the columns of V. However, restructuring Î›is 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