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International Conference On Information Technology And Business ISSN 2460-7223
Improving Relay Matrices for MIMO Multi-Relay
Communication Using Gradient Projection
Apriana Toding
Dept. Electrical Engineering.
Universitas Kristen Indonesia Paulus, Makassar, Sul-Sel 90245, Indonesia
Email: aprianatoding@ukipaulus.ac.id
Abstract—In this paper, we design the optimal relay matrices In this paper, we propose the optimal relay matrices for
for multiple-input multiple-output (MIMO) relay communication MIMOrelaycommunication systems with parallel relay nodes
systems with parallel relay nodes using the projected gradient using the projected gradient (PG) approach which significantly
(PG) approach. We show that the optimal relay amplifying reduces the computational complexity of the optimal design.
matrices have a beamforming structure. Using the optimal
structure, the relay power loading algorithm is developed to We show that the optimal relay amplifying matrices have a
minimize the mean-squared error (MSE) of the signal waveform beamforming structure. In addition to the PG approach, we
estimation at the destination node. Simulation result demonstrate constrain the power at each relay node which is more practical
the effectiveness of the proposed relay amplifying matrix with compared to the constraints in [9] as that constraint may
multiple parallel relay nodes using the PG approach in the system exceed the available power budget at the relay nodes. Simula-
bit-error-rate performance.
Index Terms—MIMOrelay,parallel relay network, beamform- tion result demonstrate the effectiveness of the proposed relay
ing, non-regenerative relay, projected gradient. amplifying matrix with multiple parallel relay nodes using the
PG approach in the system bit-error-rate performance.
I. INTRODUCTION The rest of this paper is organized as follows. In Section II,
weintroducethesystemmodelofMIMOrelaycommunication
In order to establish a reliable wireless communication link, system with parallel relay nodes. The relay matrices design
one needs to compensate for the effects of signal fading algorithm is developed in Section III. In Section IV, we show
and shadowing. An efficient way to address this issue is somenumerical simulations. Conclusions are drawn in Section
to transmit signals through one or more relays [1]. This V.
can be accomplished via a wireless network consisting of
geographically separated nodes. II. SYSTEM MODEL
When nodes in the relay system are installed with multiple Fig. 1 illustrates a two-hop MIMO relay communication
antennas, we call such system multiple-input multiple-output system consisting of one source node, K parallel relay nodes,
(MIMO) relay communication system. Recently, MIMO relay and one destination node. We assume that the source and
communication systems have attracted much research interest destination nodes have N and N antennas, respectively, and
and provided significant improvement in terms of both spectral s d
efficiency and link reliability. In [3]-[6], the authors have each relay node has Nr antennas. The generalization to the
studied the optimal relay amplifying matrix design for the system with different number of antennas at each node is
source-relay-destination channel. In [3] and [4], the optimal straightforward. To efficiently exploit the system hardware,
relay amplifying matrix maximizing the mutual information each relay node uses the same antennas to transmit and receive
(MI) between the source and destination nodes was derived signals. Due to its merit of simplicity, we consider the amplify-
assuming that the source covariance matrix is an identity and-forward scheme at each relay.
matrix. In [5] and [6], the relay amplifying matrix was The communication process between the source and desti-
designed to minimize the mean-squared error (MSE) of the nation nodes is completed in two time slots. In the first time
signal waveform estimation at the destination. In [7], the slot, the Ns×1 source signal vector s is transmitted to relay
author investigated the joint source and relay optimization for nodes. The received signal at the ith relay node can be written
MIMOrelaynetworksusingprojectedgradient (PG) approach. as
However, in [2]-[7], the authors investigated the optimal relay yr;i = Hsr;is + vr;i; i = 1;··· ;K (1)
amplifying matrix design for two-hop MIMO relay networks
with a single relay node. In [8], some linear relaying strategies where H is the N × N MIMO channel matrix between
sr;i r s
are presented for multiple relays in MIMO relay networks by the source and the ith relay node, yr;i and vr;i are the received
making use of local CSI. In [9], the authors investigated the signal and the additive Gaussian noise vectors at the ith relay
optimal relay amplifying matrices for two-hop MIMO relay node, respectively.
networks with multiple parallel relay nodes with sum relay In the second time slot, the source node is silent, while
power constraints at the output of the second hop channel. each relay node transmits the amplified signal vector to the
150|International Conferences on Information Technology and Business (ICITB), 20th -21th August 2015
International Conference On Information Technology And Business ISSN 2460-7223
destination node as where(·)−1 denotes the matrix inversion. Substituting (7) back
into (6), it can be seen that the MSE is a function of F can
x =Fy ; i = 1;··· ;K (2)
r;i i r;i be written as
h i
where F is the N ×N amplifying matrix at the ith relay ˜H˜−1˜ −1
i r r MSE=tr IN +H C H (8)
node. Thus the received signal vector at the destination node s
can be written as III. MINIMAL MSE RELAY DESIGN
K In this section, we address the relay amplifying matrices
yd = XHrd;ixr;i +vd (3) optimization problem for systems with a linear receiver at the
i=1 destination node. In particular, we show that the optimal relay
where H is the N ×N MIMO channel matrix between matrices has a general beamforming structure. Base on (5) and
rd;i d r
the ith relay and the destination node, yd and vd are the (8), the relay amplifying matrices optimization problem can be
received signal and the additive Gaussian noise vectors at the formulated as
destination node, respectively. Substituting (1)-(2) into (3), we h ˜H˜−1˜i−1
min tr I +H C H (9)
have Ns
{Fi}
K s:t: tr F H HH +I FH≤P ;i=1;··· ;K(10)
yd = X(Hrd;iFiHsr;is+Hrd;iFivr;i)+vd i sr;i sr;i Nr i r;i
i=1 where(10)isthepowerconstraint at the relay node, and Pr;i >
˜ ˜ 0 is the corresponding power budget availabe at the ith relay.
= H FH s+H Fv +v =Hs+v (4)
rd sr rd r d
T T T T A. Optimal Relay Design Using Projected Gradient (PG)
where H , [H ; H ; · · · ; H ] is a KN × N
sr sr;1 sr;2 sr;K r s Approach
channel matrix between the source node and all relay nodes, Let us introduce the following singular value decomposi-
H ,[H ;H ;···;H ] is an N × KN channel
rd rd;1 rd;2 rd;K d r tions (SVD)
matrix between all relay nodes and the destination node,
F , bd[F ;F ;··· ;F ] is the KN × KN block diag- H =U Λ VH; H =U Λ VH (11)
1 2 K r r sr;i s;i s;i s;i rd;i r;i r;i r;i
onal equivalent relay matrix, vr , vT ;vT ;··· ;vT T
r;1 r;2 r;K where Λ and Λ are R ×R andR ×R diagonalmatrix.
is obtained by stacking the noise vectors at all the relays, s;i r;i s s r r
Here R , rank(H ), R , rank(H ), rank(·) denotes
˜ s sr;i r rd;i
H,HrdFHsr as the effective MIMO channel matrix of the the rank of a matrix. The following theorem states the structure
˜
source-relay-destination link, and v , H Fv + v as the
rd r d of the optimal F .
equivalent noise vector. Here (·)T denotes the matrix (vector) i
transpose, and bd[·] stands for a block-diagonal matrix. We as- THEOREM 1: The optimal structure of Fi as the solution to
the problem (9)-(10) is given by
sumethat all noises are independent and identically distributed
(i.i.d.) Gaussian noise with zero mean and unit variance. The F =V AUH; i=1;···;K (12)
i r;1 i s;1
transmission power consumed by each relay node (2) can be where A is an R×R diagonal matrix and R , min(R ;R )..
expressed as i s r
PROOF:
H H H Without loss of generality, Fi can be written as
E[tr(x x )] = tr F H H +I F ;i=1;···;K
r;i r;i i sr;i sr;i Nr i
(5) Ai Xi UH
F = Vr;1 V⊥ s;1
where E[·] denotes statistical expectation, tr(·) stands for the i r;1 Y Z (U⊥ )H
i i s;1
matrix trace, and (·)H denotes the matrix (vector) Hermitian i = 1;··· ;K (13)
transpose.
where V⊥ (V⊥ )H=I −V VH, U⊥ (U⊥ )H=I −
Using a linear receiver, the estimated signal waveform r;1 r;1 Nr r;1 r;1 s;1 s;1 Nr
ˆ H Us;1UH , such that [Vr;1;V⊥ ] and [Us;1;U⊥ ] are unitary
vector at the destination node is given by s = W yd, where s;1 r;1 s;1
WisanN ×N weightmatrix. The minimal MSE (MMSE) matrices. The matrices Ai;Xi;Yi;Zi are arbitrary matrices
d s
with dimensions of R × R, R × (N − R), (N − R) × R,
approach tries to find a weight matrix W that minimizes the r r
(N −R)×(N −R),respectively. Substituting (13) back into
statistical expectation of the signal waveform estimation given r r
(9), we obtain that H FH =U Λ AΛ VH and
by rd;i i sr;i r;i r;i i s;i s;i
P
H H K H H H H
h i Hrd;iFiF H = Ur;iΛr;i(AiA +XiX )Λ U .
H i rd;i i=1 i i r;i r;i
ˆ ˆ
MSE = tr E s−s s−s Thus we can rewrite equation (9) as
"
H H H H K K
˜ ˜ ˜ X X
= tr W H−I W H−I +W CW (6)
Ns Ns H H H H H
MSE=tr I + V Λ A Λ U U Λ (AA +
Ns s;i s;i i r;i r;i r;i r;i i i
˜ i=1 i=1
where C is the equivalent noise covariance matrix given by #−1
H H H K
˜ ˜˜ X
C=E vv =H FF H +I .TheweightmatrixW −1
rd rd Nd H H H H
XX )Λ U +I U Λ AΛ V
which minimizes (6) is the Wiener filter and can be written as i i r;i r;i Nd r;i r;i i s;i s;i
i=1
˜ ˜H ˜ −1˜ i = 1;··· ;K: (14)
W=(HH +C) H (7)
151|International Conferences on Information Technology and Business (ICITB), 20th -21th August 2015
International Conference On Information Technology And Business ISSN 2460-7223
TABLE I problem
PROCEDUREOFAPPLYINGTHEPROJECTEDGRADIENT
ALGORITHMTOSOLVETHEPROBLEM(15)-(16) ¯ ˜ ¯ ˜ H
min tr (A −A )(A −A ) (18)
¯ i i i i
Ai
(0) ¯ 2 ¯H
1) Initialize the algorithm at a feasible A for i = 1;··· ;K; Set s:t: tr A (Λ +I )A ≤P : (19)
i i s;i Nr i r;i
n=0.
(n) By using the Lagrange multiplier method, the solution to the
2) Compute the gradient of (15) ∇f(A );
i
˜(n) (n) (n) ¯(n) problem (18)-(19) is given by
Project A =A −sn∇f(A )toobtainA .
i i i i
(n+1) (n) ¯(n) (n)
Update A with A =A +δn(A −A )
i i i i i ¯ ˜ 2 −1
(n+1) (n) A =A[(λ+1)I +λΛ ]
3) if max abs kA −A k≤ε,thenend. i i Nr s;i
i i
Otherwise, let n := n + 1 and go to step 2). where λ > 0 is the solution to the nonlinear equation
˜ 2 −1 2
tr A [(λ+1)I +λΛ ] (Λ +I )
i Nr s;i s;i Nr
Substituting (11) back into the left-hand-side of 2 −1 H
˜
[(λ+1)I +λΛ ] A =P : (20)
the transmission power constraint (10), we have Nr s;i i r;i
2 H 2 H H
tr A (Λ +I )A +Y(Λ +I )Y+XX +ZZ :
i s;i Nr i i s;i Nr i i i i i Equation (20) can be efficiently solved by the bisection method
From (13), we find that Xi= 0R×(Nr−R);Yi= 0(Nr−R)×R; [11]. The step size parameters δ and s are determined by
and Z =0 ; minimize the power consumption. n n
i (Nr−R)×(Nr−R) the Armijo rule [11], i.e., sn = s is a constant through
Thus we have F = V A UH .
i r;i i s;i all iterations, while at the nth iteration, δn is set to be
The remaining task is to find the optimal Ai;i = 1;··· ;K. γmn. Here m is the terminal nonnegative integer that sat-
n
From (14), we can write the optimization problem as isfies the following inequality MSE(A(n+1))−MSE(A(n))≤
i i
" K K mn (n) H ¯(n) (n)
αγ realtr (∇f(Ai) ) (A −A ) , where α and γ
X H H H H X i i
min tr I + V Λ A Λ U U Λ A are constants. According to [11], usually α is chosen close
Ns s;i s;i i r;i r;i r;i r;i i
Ai −5 −1
i=1 i=1 to 0, for example αε[10 ; 10 ], while a proper choice of γ
K #−1 is normally from 0:1 to 0:5.
H H H −1X H
A Λ U +I U Λ AΛ V
i r;i r;i Nd r;i r;i i s;i s;i B. Simplified Design
i=1
(15) By introducing
¯
F,H F: (21)
s:t: tr A (Λ2 +I )AH ≤P ; rd
i s;i Nr i r;i
i = 1;··· ;K: (16) The received signal vector at the destination can be equiv-
¯ ¯ ¯ ¯
alently written as yd = Hs + v, where H , FHsr, and
¯ ¯
Both the problem (9)-(10) and the problem (15)-(16) have v , Fvr + vd. Considering (2) and (21), the transmission
matrix optimization variable. However, in the former problem, power consumed at the output of Hrd can be expressed as
the optimization variable F is an N ×N matrix. In general, H H H
i r r ¯ ¯
E[tr((H x )(H x ) )] = tr F H H +I F
the problem (15) - (16) is nonconvex and globally optimal rd r rd r sr sr KNr
H H H
≤tr(H H )tr F H H +I F : (22)
solution is difficult to obtain with a reasonable computational rd;i rd;i i sr;i sr;i Nr i
complexity. Fortunately, we can resort to numerical methods, Substituting (10) into (22) we have
such as the projected gradient algorithm [11] to find (at least)
K K
a locally optimal solution of (15) - (16). The procedure of the ¯ H ¯H X X H
tr F H H +I F ≤ P tr(H H ):(23)
projected gradient algorithm is listed in Table I, where δ and sr sr KNr r;i rd;i rd;i
n i=1 i=1
sn denote the step size parameters at the nth iteration. max
Here PK P ,P ; is the total transmission power budget
absk·k denote the maximum among the absolute value of all i=1 r;i r
elements in a matrix, and ε is a positive constant close to 0. available to all K relay nodes. Using (23), the relaxed relay
h i
−1 optimization problem can be written as
˜H˜−1˜
THEOREM 2: If f(Ai) = tr IN +H C H is
s ¯H¯−1¯−1
min tr I +H C H (24)
chosen as the objective function, then its gradient ∇f(Ai) with ¯ Ns
F
respect to Ai can be calculated by using results on derivatives ¯ H ¯H ¯
s:t: tr F H H +I F ≤P;i=1;···;K(25)
of matrices in [13] as sr sr KNr r
¯ H H
where P , P tr(H H ). Let H = U Λ V denote
r r rd rd sr s s s
∇f(A) = 2 [MR]T[SC]T +[MR]T[D]T the singular value decomposition (SVD) of H , where the
i i i i i i i i sr
H −1 T T ∗ dimensions of U , Λ , V are KN ×KN , KN ×N , N ×
−[Ei Gi Ri] [Si] s s s r r r s s
i = 1;··· ;K: (17) Ns, respectively. We assume that the main diagonal elements
of Λs is arranged in a decreasing order. Using Theorem 1 in
¯
PROOF: See Appendix A. [10], the optimal structure of F as the solution to the problem
˜ ¯ (24)-(25) is given by
The projection of Ai onto the feasible set of Ai given
¯ H
by (16) is performed by solving the following optimization F=QΛfUs;1 (26)
152|International Conferences on Information Technology and Business (ICITB), 20th -21th August 2015
International Conference On Information Technology And Business ISSN 2460-7223
where Q is any N ×N semi-unitary matrix with QHQ= variance. We define SNR = σ2P KN =N and SNR =
d s s s s r s r
I , U contain the leftmost N columns of U , and Λ is σ2P N =(KN ) as the signal-to-noise ration (SNR) for the
Ns s;1 b s f r r d r
an N ×N diagonal matrix. The proof of (26) is similar to source-relay link and the relay-destination link, respectively.
s s
the proof of Theorem 1 in [10]. From (26), we see that the We transmit Ns × 1000 randomly generated bits in each
¯
optimal F has a beamforming structure. In fact, the optimal channel realization, and all simulation results are averaged
¯ ¯
Fdiagonalizes the source-relay-destination channel H up to a over 200 channel realizations. In all simulations, the MMSE
rotation matrix Q. Using (26), the relay optimization problem linear receiver in (7) is employed at the destination for symbol
(24)-(25) becomes detection.
!
h i−1−1 In our example, a parallel MIMO relay system with K = 2
2 2 relay nodes, N = N = 5, and N = 4 are simulated. We
min tr IN + ΛfΛs Λ +IN (27) s d r
Λ s f s
f compare the BER performance of the propose optimal relay
2 2 ¯ matrices using Projected Gradient (ORP) algorithm in (12)
s:t: tr Λf Λs+INs ≤Pr: (28)
with ZF algorithm in [8], MMSE algorithm in [8], and the
Let us denote λf;i;λs;i, i = 1;··· ;Ns, as the main diagonal naive amplify-and-forward (NAF) Algorithm. While Fig. 2
elements of Λ , Λ , respectively, and introduce
f s demonstrates BER versus SNR for SNR fixed at 20 dB.
s r
ai , λ2 ; yi , λ2 λ2 +1; i = 1;··· ;Ns:(29) It can be seen that the propose algorithm outperforms all
s;i f;i s;i competing algorithms in the whole SNR range.
s
Theoptimization problem (27)-(28) can be equivalently rewrit-
ten as V. CONCLUSIONS
Ns In this paper, we have derived the general structure of the
min X aixi+yi+1 (30) optimal relay amplifying matrices for parallel MIMO relay
y a x y +a x +y +1
i=1 i i i i i i communication systems using the projected gradient approach.
Ns The proposed algorithm has less computational complexity
X ¯
s:t: y ≤P y ≥0; i = 1;··· ;N (31)
i r i s compared to the existing techniques. Simulation result shows
i=1 the effectiveness of the proposed algorithm.
where y , [y ;y ;··· ;y ]T. The problem (30)-(31) can be
1 2 Ns VI. APPENDIX
solved by an iterative method developed in [10], where in
iteration, y is updated alternatingly by fixing the other vector. Base on (11) and (12), we have H = U Λ VH,
sr;i s;i s;i s;i
After the optimal y is found, λf;i can be obtained from (29) H =U Λ VH,F=V AUH,PK H FH =
rd;i r;i r;i r;i i r;i i s;i i=1 rd;i i sr;i
as PK U Λ AΛ VH, and PK H FFHHH =
i=1 r;i r;i i s;i s;i i=1 rd;i i i rd;i
r yi PK H H H
U Λ AA Λ U :Thusf(A)canbewrittenas
λf;i = ; i = 1;··· ;Ns: (32) i=1 r;i r;i i i r;i r;i i
λ2 x +1
s;i i "
K K
¯ X H H H H X
Using (21) and the optimal structure of F in (26), we have f(A )= tr I + V Λ A Λ U U Λ A
i Ns s;i s;i i r;i r;i r;i r;i i
H F =QΛ Φ,wherematrixΦ containsthe(i−1)N + i=1 i=1
rd;i i f i i r
1 to iN columns of UH . Then we obtain K #−1
r
s;1 X
H H H −1 H
A Λ U +I U Λ AΛ V
F =H† QΛ Φ; i = 1;··· ;K (33) i r;i r;i Nd r;i r;i i s;i s;i
i rd;i f i i=1
where (·)† denotes matrix pseudo-inverse. Finally, we scale F (35)
i
P
in (33) to satisfy the power constraint (10) at each relay node H K H H H H
Let us define Z , Vs;jΛ A Λ U ,andYi,
i j=1;j6=i s;j j r;j r;j
as PK U Λ AAHΛHUH+I :Thenf(A)canbe
j=1;j6=i r;j r;j j j r;j r;j Nd i
˜ written as
Fi = αiFi; i = 1;··· ;K (34)
where the scaling factor α is given by α = H H H H H
i i f(A )= tr I +(Z +V Λ A Λ U )(Y +U Λ
i Ns i s;i s;i i r;i r;i i r;i r;i
q H H −1
P =tr(F [H H +I ]F ); i=1;··· ;K. H H H −1 H
r;i i sr;i sr;i Nr i AiA Λ U ) (Ur;iΛr;iAiΛs;iV +Zi)
i r;i r;i s;i
IV. SIMULATIONS (36)
In this section, we study the performance of the proposed Applying I +AHC−1A−1=I −AH(AAH+C)−1A.
optimal relay beamforming algorithms for parallel MIMO Ns Ns
Then, (36) can be written as
relay systems with linear MMSE receiver. All simulations
are conducted in a flat Rayleigh fading environment where H H H H H
f(A )= tr I −(Z +V Λ A Λ U )((U Λ
i Ns i s;i s;i i r;i r;i r;i r;i
the channel matrices have zero-mean entries with variance AΛ VH +Z)(ZH+V ΛHAHΛHUH)
σ2=N and σ2=(KN ) for H and H , respectively. The i s;i s;i i i s;i s;i i r;i r;i
s s r r sr rd +(Y +U Λ AAHΛHUH))−1
BPSKconstellations are used to modulate the source symbols, i r;i r;i i i r;i r;i
and all noise are i.i.d Gaussian with zero mean and unit (U Λ AΛ VH +Z): (37)
r;i r;i i s;i s;i i
153|International Conferences on Information Technology and Business (ICITB), 20th -21th August 2015
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