IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 2, JANUARY 15, 2014 271
An MMSE Strategy at Relays With Partial CSI for a
Multi-Layer Relay Network
Pannir Selvam Elamvazhuthi, Student Member, IEEE,BikashKumarDey, Member, IEEE,and
Srikrishna Bhashyam, Senior Member, IEEE
Abstract—We consider a relay network with a single s
ource-des-
tination pair and multiple layers of relays between them. We as-
sume that these layers sequentially relay the signal transmitted by
the source to the destination. Unlike exist
ing work, we also assume
that the destination and all the forward layers present between the
transmitting layer and the destination receive signals during every
transmission phase. We optimally combin
e the se signals, say
of
them, using a precoder at each relay layer for onward transmis-
sion. We obtain this precoding matrix by minimizing the m ean-
squared error (MSE) at the relays,and
do not require channel-
state-information (CSI) of the forward channels at the relays unlike
existing system s that minimize MSE at the destination and require
CSI of the forward channels at th
e relays. Our closed-form solution
for this matrix is valid for any
number of layers, whereas mini-
mizing MSE at the destination does not have closed-form solution
for
.For , we enhance an
existing scheme to obtain
a sub-optimal closed-form precoder solution and use it for com-
parison. We show using simulations that our scheme approaches
the bit-error-rate (BER
) performance of this scheme, when
is in-
creased, even with partial C SI.
Index Terms—MMSE, channel-state-information, multi-layer
relay network, relay precoder, amplify-and-forward.
I. INTRODUCTION
J
ING and Hassibi [1] proved that a spatially distributed
network of single-antenna radio nodes can emulate a mul-
tiple-input multiple-output (MIMO) communication system.
They showed that this creates a distributed space-time code
and achieves the same diversity as that of a MIM O system at
high total transmi tted power. These au thors also extended it t o
include multiple-antenn a nodes in [2] and [3].
When multiple radio nodes a re present, the effectiveness
of cooperative communication [4], [ 5] can be increased by
Manuscript received March 17, 2013; revised August 05, 2013; accepted
September 24, 2013. Date of publication October 04, 2013; date of current ver-
sion December 23, 2013. The associate editor coordinating the review of t his
manuscript and approving it for publication was Prof. Samson Lasaulce. The
work of P. Selvam and B. K. Dey was supported in part by Bharti Centre for
Communication.
P. S. Elamvazhuthi is with Delphi Automotive S y ste ms Pvt. Ltd., Technical
Center I ndia, Kalyan i Platina, Whitefield, Bengaluru 560066, India (e-mail:
pannir.selvam@delphi.com).
B. K. Dey is with the Ind ian Institute of Technology Bombay, Powai, Mumbai
400076, India (e-mail: bik ash@ee.iitb.ac.in).
S. Bhashyam is with the Indian Institute of Technology Madras, Chennai
600036, India (e-mail: skrishna@ee.iitm.ac.in).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Dig
ital Object Identifier 10.1109/TSP.2013.2284471
arranging t hese nodes into a layered architecture t
o relay in-
formation. Pottie and Kaiser [6] sho wed how a distr
ibuted and
layered signal processing architecture ca
n overcome the energy
and bandwidth constraints in many wireles
s sensor network
applications.
A relay can use a simple forwarding techniq
ue called
amplify-and-forward (AF) [7], in which it
amplifies what is
received and transm its. Other well-know
n r elaying methods
include decode-and-forward (DF) (
e.g., [8]), coded coopera-
tion (e.g., [9]), c ompress-and-fo
rward(e.g.,[10],[11]),and
partial-decode-and-forward (PD
F) (e.g., [12]). Chiu et al.
[13] designed the precoder for no
t only the relay but also the
source, while using the PDF stra
tegy at the relays. They used
orthogonal space-time block c
oding [14] in b oth the source and
the relay transmissions. Thou
gh other relaying protocols can
perform better, the AF prot
ocol is widely considered interestin g
because of its simplicity
of implementation. In this paper, we
restrict our attention t
o the AF protocol. In an AF multi-layer
relay system, the perfo
rmance of the system depends on the
precoders used at the re
lays, and we consider the problem of
designing such p reco d
ers.
Our system model cons
ists of a set of single-antenna radio
nodes grouped into
layers of relays, , between
asourceandadestin
ation (we will call the source S o r
and
the destination D o
r
in the sequel) as show n in Fig. 1. Ear-
lier work involvi
ng multiple layers of relays use only the signal
reaching at a part
icular lay e r
from the preceding layer
to construct the
signals transmitted from
. On the contrary, in
this paper, we
construct th e transmit signals at
using signals
that have reac
hed from
, although the signal
from
reaches with lower power compared
to that from
. Thus, we take advantage of the broadcast na-
ture of the w
ireless medium by utilizing overheard signals. We
call these
low power overheard signals as leaked signals.
We will cal
l the channel states of the channels from
to
,availab
le at
,tobebackward CSI and the channel states
of the cha
nnels f rom
to as forward CSI.Here
.Forwar
d and backw ard CSI are together called the global
CSI,and
just the backward CSI is referred to as partial CSI.
We note
here that obtainin g forward CSI r equires either sending
the est
imated channels from the receiving nodes over reliable
feedb
ack channels, or direct estimation of these channels from
the b
ackward transmission (in a time-division duplex system).
The f
easibility of this de pend s on the application scenar io.
Ding
et al. [15] employed AF strategy at the r elays, designed
auni
tary precoder, and achieved maxim um diversity gain.
The
closed form expression of the precoder was extended to
1053-587X © 2013 IEEE
272 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 2, JANUARY 15, 2014
Fig. 1. A multi-layer relay network. Here S, ,D,and represent the
source,
th layer, th relay in , the destination, and the length of the corre -
sponding link respectively. The relays are assumed to be half-duplex.
-ary signals of larger constellation size in [16]. Gomadam
and Jafar [17] obtained relay precoders by maximizing the
receive signal-to-noise ratio (SNR) at t he destination. Many
authors [18]–[23] have considered minimizing the MSE at the
destination (MMSED) to obtain relay precoders using global
CSI. This global optimization is challenging, and authors of
[17] and [23] considered two layers of relays and obtained the
precoder by iterative techniques, w hile the authors of [18]–[22]
derived the optimal precoders in closed-form for a single layer
of relays. The disadvantage in an iterative technique is that
at any iteration, only an approximate solution is found. For
real-time co mputation at the relay s, the quality of the solution
then depends on the processing speed of the relays. Further, to
the best of our knowled ge, iterative so lutions are also available
only for upto two layers of relays, and their generalization
to more layers or incorporating leaked signals is challenging
under the MMSED criterion.
In some systems, the relays m ay be able to exchange informa-
tion amongst themselves before transmission. In such a system,
the relays are said to be co opera tive. O therwise, the precoder
matrix would be diagonal and the relays are said to b e non-co-
operative. All the literature discussed in the previous paragraph,
showed the efficacy of the derived precoders when the relays
are non-cooperative except [ 19], in which the authors derived
the p recoders for cooperative relays. For either cooperative and
non-cooperative relays, to our knowledge, existing literatu re
provides minimum MSE (MMSE) d esign of AF precod ers:
• in closed form only when there is a single layer of relays,
and in the form of iterative numerical solution when there
are two layers,
• assuming that global CSI is av ailab le with the relays, and
• without considering leaked signals.
All the above concerns are addressed in this paper. Unlike
[18]–[23], which ado pted MMSED, we m in im ize MSE at the
relays (MMSER) and obtain relay precoder matrices. We show
that the MMSER strategy makes the optimal precoder design
a lay er- wise optimization as opposed to a global optimization.
This yields optimal precoders in closed form for arbitrary
number of relay l ayers even when leaked signals are used,
and it r equires partial CSI, i.e., only the backward CSI. In
the absence of optimum MMSED precoder solution for more
than two layers of relays, we have proposed som e suboptimal
MMSED precoding schemes, and we show using simulatio n
that despite the lack of forward CSI, our MMSER precoding
strategy outperform s/approaches the perfo rm ance of these
MMSED strategies by using more leaked signals.
A. Contribution
Our contributions in this paper are:
• For the multi -layer relay network shown i n Fig. 1, we pro-
pose a novel MMSER relaying strategy, which do es not
require forward CSI in th e transm itting nodes.
• We obtain closed form solutions for the optimum MMSER
relay precoders for both cooperative and no n-coop erative
relays for arbit rar y number of layers of relays while also
including leaked signals.
• We enhance the MMSED s trateg ies (though these enhan ce-
ments may not be optimal) proposed in [18], [19], and [21]
to work in thi s mu lti-layer network for meani ngf ul com-
parisonwithMMSER.
• We show using simulations that combining more number
of leaked signals improves t he performance of MMSER,
which outperforms/approaches that of MMSED schemes
that use global CSI.
B. Notation and Organization
denotes the identity matrix and is the vector of
elements .For denotes zero if
and if . represents a circularly sym-
metric complex G aussian random variable with real and imagi-
nary parts having mean 0 and variance
.
is a d iagonal matrix with diagonal elements .
The remainder of this paper is organized as follows. In
Section II, we state th e problem a nd introduce the MMS E R
strategy. Thereafter, in Section III, the MMSER strategy is
presented in detail and the precoders for the relay layers are de-
rived. Section IV gives details on how we extend and enhance
the MMSED strategy, so that the BER performance of MMSER
can be m eaningfully compared. In Section V, we describe
the equalizer that is used at de stin ation D for M M S ER an d
MMSED schemes to decode the received vector. In S ection VI,
we present simulation results. Finally, in Section VII, we
summarize and conclude the paper.
II. S
YSTEM MODEL AND THE PROPOSED SCHEM E
In our system model shown in Fig. 1, denotes the th
relayinthe
th layer and we assume that it is half-duplex,
.Weuse and to represen t the links an d the channel
coefficients respectively, with the first two subscripts denoting
the transmitter and the next two the receiver. Therefore, the
channel coefficients of the links,
from
from ,and from
are denoted as ,and respectively. Let
and represent
ELAMVAZHUTHI et al.: MMSE STRATEGY AT RELAYS WITH PARTIAL CSI 273
TABLE I
R
ECEIVED AND TRANSMITTED VECTORS—RELAY L AY ERS
the vectors/matrices of channel coefficients from S to to
,and to D respectively. We assume that the channels are
Rayleigh fading and quasi static.
Let
be the set of all links and be the set
of links from
to .Asanexample,for
,thelinkset is given by
Let us define the length of the link as ,or
simply by
when the l ayers are understood from the context.
All the lin ks in link set
havethesamelength .
Now, let L
be a class of subsets of with .
Clearly, L
L ,if .Asanexample,linkclassL is
given by
L
The Proposed MMSER- Strategy
Let us consider the
hopnetworkshowninFig.1.Inthe
MMSER-
strategy, S transm its in phase 0,
and
receive and store for later use; transmits
in phase 1 , and
receive and store for later use and
so on till phase
, when D receives from . To be specific, in
MMSER-
, the relays and D store all signals received through
the link s ets in L
for later use. For , MMSER-
would use more number of leaked signals, and thus is expected
to perform better than MMSER-
.
We assume synchronous reception and transmission at the
relay nodes and all noise signals added at the receiver front-ends
are complex zero-mean independent and identically distributed
(i.i.d.) Gaussian random variables with variance
.
Let us denote the transmitted, received, and noise signals by
,and respectively with subscript and superscript on them
denoting layer and phase respectively. Le t the signal transmitted
by S be
with an average pow er of Watts, where
is a unit variance constellation point. I n v arious phases,
the layer
would have r eceived vectors each of size
from previous transmissions, starting from phase till phase
,where . All these vectors are stacked together
as given in Table I to form the overall received vector
.
For example, let us take
and . Here,
for and for .
Hence,
and would have the overall received vectors
respectively. Also and .
The stacked received vector is transmitted by
relays, after
precoding with
in phase as shown in Table I. The precoder
matrix
at is given by
(1)
where
with and .The
precoder submatrices
can be selected to be non-diagonal or
diagonal depending upon whether th e relays would cooperate or
not respectively.
In phases
to , D receives
.If , then it receives
from S directly in phase 0.
Now, we define the cost function at the relay layer
to be
the MSE
(2)
where
asgiveninTableI
and
is the expectation operator. The relays then transmit
a scaled version of the solution to meet the layer-wise power
constraint. The cost function given in (2) i s motivated by the
following facts:
• Somewhat similar to the principle behind regenerative re-
laying (DF), the relays are desired to tr ansmi t a signal that
is close t o the source symbol or its scaled version.
• The otherwise complex optim ization probl em (with
MMSED criterion) is replaced by a smaller layer-wise
optimization problem which, as we will see, yields a
solution in closed form.
• The requirement of only backw ard CSI gives an added
practical advantage.
Now, our aim is to minimi ze
under the power constraint
and find the precoder matrices .
III. MMSER P
RECODER
Khajehnouri and Sayed [18] minim ized the MSE
at the destination and found the precoder matr ix for when
with no power constraint. Here, and
.Krishnaet al. [19] derived a non-diagonal precoder
matrix for cooperative relays with average p ower constraint
.
To compare with their results, these authors modified the
th
diagonal element of the precoder matrix to restrain power in the
Khajehnouri-Sayed scheme [18] as
(3)
for non-cooperative relays in Khajehnouri–Sayed equation (24).
Lee et al. [20] obtained
using constrained optimization [24]
for non-cooperativ e relays as
(4)
274 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 2, JANUARY 15, 2014
where we have removed the uncertainty channel term s to match
the scope of this paper. In both (3) and (4), we have chang e d
symbol notation to be consisten t with this paper.
Let us call these non-cooperative systems, which use (3)
and (4), as MMSED-Khajehnouri/Krishna (MMSED-KK)
and MMSED-Lee (MMSED-L) respectively. We also call
the cooperative system proposed by Krishna et al. [19]
as MMSED-Krishna (MMSED-K). We note th at, both
MMSED-KK and MM SED-L require
or forward CSI at
the relay
from (3) and (4) respectively.
In our strategy, we minimize the MSE at the relays resulting
in precoders that do not depend on forward CSI, an d use leaked
signals to improve the performance. Let us now derive the pre-
coders for MMSER-
for any .
A. MMSER Precoder Matrix at
In our proposed MMSER scheme, each layer obtains an es-
timate of the signal vector
. This estimate or a scaled version
of this estimate can be transmitted, subject to the sum transmit
power constraint for each layer of relays. Expanding the expres-
sion of the M SE given in (2), we get
(5)
Now,theestimateisobtainedbyfinding the optimum
given by , subject to the constraint
. We write the constraint function as
(6)
and let
. This problem is an
MMSE estimation pro blem with a convex constraint on the esti-
mate. It is a convex optimization problem with a unique solution
as discussed in [25].
Now, given the optimization variable
,
the c ost function
and th e inequality
constraint function
,wedefine the La-
grangian
as
(7)
where
is the Lagrange multiplier and the do main of
.
Claim 1: (1) When the relays cooperate, the optimum pre-
coder matrix
is given by
(8)
where
(2) When the relays do not cooperate, the o ptimum precoder
vector
is given by
(9)
where
(10)
(11)
and
.
.
.
.
.
.
.
.
.
(12)
Here
and represent the th diagonal elements
of
and re-
spectively.
Proof: See Appendix A.
We notice the similarity of (8) and (9), where and
are ana logous to and respectively, except for the extra
summing operator in the denominator in (9).
For the non-cooperative case, the optimum precoder
,is
made from the optimum vectors,
, by noting that these vec-
tors give the
th diagonal elements of all submatrices,
, that make up the p reco der.
The sum transm it power constraint at each layer allows for
optimal allocation o f power among the relays depending on the
quality of the estimate at each relay. Specializing (8) to layer 1,
we get
(13)
as
and from
Tab le I. F ro m (13), we can observe that the relays with better
backward channels are allocated more power.
B. Information Required for MMSER Precoders
From (8) and (9), we can see that the precoders of
MMSER depend on two co rrelation matrices
and
. From Table I, we see that, these matrices depend on
(if ,thenon ), and the
precoder matrices
for .Since (see (13 ))
depends only on backward CSI at
, the information req uir e d
to construct
at is also the backward CSI at . Therefore,
MMSER-
does n ot require forward CSI.
IV. E
XTENSION OF MMSED SCHEMES
Derivation of precoders for MMSED schemes, MMSED-KK
and MMSED-L ((3) and (4) give the diago nal e lem ents of t he
precoders of these schemes for
), is complicated when
. The sy stem in [23] has two layers, i.e., ,andthe
precoders are obtained iteratively. In this S ection, we enhance
the MMSED strategies (though these enhancements may not be
optimal) proposed in [ 18], [19], and [21] to obtain closed-form
solutions to precoders in a multi-lay er network for meaningful
comparison with the MMSER-
system proposed in this paper.
We call these systems E-MMSED, for Enhanced-MMSED sys-
tems, and find their precoders to be dependent on global CSI as
shownin(17)and(18).
ELAMVAZHUTHI et al.: MMSE STRATEGY AT RELAYS WITH PARTIAL CSI 275
A. E-MMSED Strategy
For a meaningful comparison, we consider the total number
of phases as
, and the total power transmitted as to
bethesameasthatusedbyMMSER.Weassume:Stransmits
times in as many phases; all the relays average their received
signals and transmit
timesinasmanyphases.
Thus,Dwouldhaveavectorof
received signals.
Stransmits
repeatedly times from phase zero
to phase
, and the relays follow suit transmitting a signal
vector
, which is explained later, from p hase to . For each
of these transmissions, wh en S transmits or the relay s transm it,
the channel does not va ry as we assume a slow varying channel.
Hence, the average power
can be equally divided in various
phases when S transmits and
when the relays transmit. Thus,
Watts and Watts respectively, with
, t he total power available.
The relays in all the layers would receive in phase
, a vector given by
(14)
.
.
.
.
.
.
with .
We assume that all the relays transmit together in their trans-
mission phases, as though they are in a single layer. We also
assume that the noise is uncorrelated, i.e.,
where when and when is
the Kronecker delta function.
The relays average all the signals received and repeat trans-
mission of the sign al
in phases to ,where
(15)
from (14). As
is a di-
agonal precoder matrix, let us define it as
where is the multiplying factor of the relay .
From (15),
becomes
(16)
which is transmitted
times, so that the total number of phases
would be
, the same as that of MMSER.
Now, we will derive precoders for enhanced MMSED-KK
(E-MMSED-KK) and enhanced MMSED-L (E-MMSED-L)
schemes.
1) Precoder of E-MMSED-KK: We take (3) and replace
,
the power t ransmitted by S , with
and , the power trans-
mitted by the relays, with
, as we allocate fractions of
powers to them due to their multiple transmissions. Further, the
noise variance
is replaced by ,sincethenoisevari-
ance at each of the relays
after
averaging over the
phases (in (15)) is . Therefore,
we get
and a diagonal element of
theprecoderofE-MMSED-KKas
(17)
2) Precoder of E-MMSED-L: Similarly, to get the diagonal
elements of the precoder of E-MMSED-L, we replace the signal
power transmitted and noise varian ce as in E-MMSED-KK into
(4) to get its diagonal element
as
(18)
We note that in both (17 ) and (18), we have a double summation
inthedenominatorinsteadofasinglesummation,when
is
greater than 1. Also, w e see that in both cases, the relay
needs forward CSI .
B. Selection of
Let us now select the best and use it while comparing
its performance with MMSER-
system. As it is hard to derive
BER, we obtain SNR at D for any
and attempt to select the
value of
that maximizes it.
In phase
, D receives a scalar
(19)
where
.Substi-
tuting (16) into ( 19), we get
where
(20)
are respectively, the signal and noise components of the received
signal in phase
, without considering the noise that is add ed
at D . We do not take the noise added at D with these compo -
nents, as it is not consid ered while derivin g optimum precoder in
MMSED. Now, we concatenate these components in D into vec-
tors as
and .
The signal and noise powers from these vectors a re d e
fined
as
(21)