Optimizations of a MIMO Relay Network

In this paper, we extend recent optimal minimum-mean-square-error MMSE and signal-to-noise ratio SNR designs of relay networks to the cor-responding multiple-input–multiple-output MIMO

Optimizations of a MIMO Relay Network

Alireza Shahan Behbahani, Student Member, IEEE, Ricardo Merched, Senior Member, IEEE, and

Ahmed M Eltawil, Member, IEEE

Abstract—Relay networks have received considerable attention

recently, especially when limited size and power resources impose

constraints on the number of antennas within a wireless sensor

network In this context, signal processing techniques play a

fundamental role, and optimality within a given relay architecture

can be achieved under several design criteria In this paper, we

extend recent optimal minimum-mean-square-error (MMSE) and

signal-to-noise ratio (SNR) designs of relay networks to the

cor-responding multiple-input–multiple-output (MIMO) scenarios,

whereby the source, relays and destination comprise multiple

an-tennas We investigate maximum SNR solutions subject to power

constraints and zero-forcing (ZF) criteria, as well as approximate

MMSE equalizers with specified target SNR and power constraint

at the receiver We also maximize the transmission rate between

the source and destination subject to power constraint at the

receiver.

Index Terms—Minimum-mean-square-error (MMSE),

mul-tiple-input–multiple-output (MIMO), relay networks, relay

optimization, zero-forcing (ZF).

I INTRODUCTION

N EXT-generation wireless networks are demanding high

data rate services to accommodate requests from various

applications In order to provide a reliable transmission, one

needs to compensate for the effect of signal fading due to

mul-tipath propagation and strong shadowing One way to address

these issues is to transmit the signal through one or more relays

[1] This can be accomplished via a wireless network consisting

of geographically separated nodes

The interest in relay networks has recently increased from

several different perspectives, ranging from the modeling of

link abstractions at higher layers in a communication system, to

coding, synchronization, and signal processing designs within a

physical layer The basic motivation behind the use of

coopera-tive communications lies in the exploitation of spatial diversity

provided by the network nodes, as well as the efficient use of

power resources, which can be achieved by a scheme that simply

receives and forwards a given information, yet designed under

certain optimality criterion A recent review on several aspects

of cooperative communications can be found, for instance in [2],

and in the references therein

Manuscript received August 23, 2007; revised June 25, 2008 First published

August 1, 2008; current version published September 17, 2008 The associate

editor coordinating the review of this paper and approving it for publication was

Dr Subhrakanti Dey.

A S Behbahani and A M Eltawil are with the Department of Electrical

Engineering and Computer Science, University of California, Irvine, CA 92697

USA (e-mail: sshahanb@uci.edu; aeltawil@uci.edu).

R Merched is with the Department of Electrical Engineering, Federal

Univer-sity of Rio de Janeiro, Rio de Janeiro 21945–970, Brazil (e-mail: merched@lps.

ufrj.br).

Digital Object Identifier 10.1109/TSP.2008.929120

In a relay framework there are two major issues that influ-ence network performance First is the network topology and the second is the strategy by which each individual relay node relays the information In a general relay network topology there could be multiple sources, multiple destinations as well as direct line-of-sight links between the source and destination, in addi-tion to the relay paths The capacity and performance of relay networks under different topology assumptions is an active re-search topic and there are many references in literature that elab-orate upon possible network scenarios, for instance [3] and the references therein

Regarding the second issue which is relaying strategies, there has been several proposals of which the most dominant are am-plify-and-forward (AF), demodulate-and-forward, decode-and-forward, and compress-and-forward In amplify-and-decode-and-forward, the relays amplify the received signal and forward the scaled signals to the destination [4]–[6] In demodulate-and-forward, the relay demodulates each received symbol individually, re-modulates, and retransmits them to the destination [4] In de-code-and-forward, each relay decodes the entire received mes-sage, re-encodes it and sends the resulting sequence to the des-tination [5], [7], [8] In compress-and-forward, the relay sends

a quantized version of the received signal to the destination [3], [9]

A Prior Work

Coupling relay networks with multiple-input–mul-tiple-output (MIMO) techniques is a natural extension to the state of the art due to the fact that MIMO networks signif-icantly improve spectral efficiency and link reliability through spatial multiplexing [10]–[12] and space-time coding [13], [14] In [15] the authors focus on MIMO wireless relay net-works where each relay is equipped with antennas assisting communication They consider two different sce-narios, coherent and noncoherent MIMO relay networks, where the transmitter spatially multiplexes data (i.e., transmitting statistically independent data streams from different antennas) and each transmit and receive antenna form a pair which is served by some of the relays In the coherent scenario, each relay performs matched filtering for its backward (i.e., the channel between the source and the relay) and forward (i.e., the channel between the relay and the destination) channels and can achieve, asymptotically in number of relays , the upper bound capacity In the noncoherent scenario, the relays

do not have channel state information (CSI) and a simple AF relaying is provided, turning the wireless relay network into

a point-to-point MIMO link achieving spatial multiplexing gain of In [16], the authors propose a zero-forcing (ZF) relaying scheme where each relay performs ZF on backward and forward channel without the need of CSI at the source and

1053-587X/$25.00 © 2008 IEEE

destination This ZF scheme achieves a spatial multiplexing

gain of In [17] a relaying scheme is proposed which

achieves both distributed array gain and maximum multiplexing

gain The proposed scheme relies on a QR decomposition for

the backward and forward channels, and performs successive

interference cancelation (SIC) at the destination

As is evident from the previous literature summary, in

ad-dition to information-theoretic approaches to relay networks

[15], there has been significant interest in analyzing relay

net-works (specifically amplify-and-forward schemes due to their

relative simpler design) rigorously within signal processing

frameworks [18], [19] More specifically, while [15] provides

capacity scaling laws for certain MIMO relay protocols, [18]

studies optimal power distribution strategies under minimum

mean-square-error (MMSE) and signal-to-noise ratio (SNR)

criteria, focusing on a single-input–single-output (SISO) relay

scheme, and [19] targets maximizing mutual information

be-tween source and destination under joint power distribution

at the source and relay, considering a single relay structure

constituted by multiple antennas

B Power Constraints

Traditionally, constraints on power were placed separately at

the transmission devices due to their limited power capability

(source or relay) or due to regulations specifying the maximum

power per transmitter This is a limitation placed at the

“de-vice” level However, due to significant activity in ad hoc

net-works and netnet-works that utilize spectrum sharing and dynamic

spectrum access there has been a recent trend to evaluate

“net-work“ level power constrains where the limitation is no longer

on the ability of a specific transmitter to emit power but rather on

the ability that a set of transmitters broadcasting correlated data

could meet a power or interference constraint at the receiver site

[20], [21] This is due to the fact that a set of transmitters can

co-operate to achieve a much higher power level that was

previ-ously allowed under regulatory enforcement In this approach,

regulations place a limit on the maximum received power or

in-terference at the receiver site versus the transmitter site To

fur-ther illustrate this scenario, consider two cells that are

geograph-ically close to each other Typgeograph-ically with one transmitter per

cell, regulations specify the ”transmission mask” which

spec-ifies how much power can be sent out from the base station as

a function of frequency The main purpose of this mask is to

control power levels in neighboring cells using the same

fre-quency However, for a scenario where several relays (mobile or

fixed) are transmitting within the cell, each with its own

trans-mission mask, it becomes difficult to control how much

inter-ference power affects neighboring cell sites An alternative is

therefore to specify the maximum power that can be received

at the cell edge by all allowable transmitting devices and use

that constraint as well as the physical locations of the relays

to derive power constraints at intermediate radii Another

ap-proach is to have the receiver specify a desired power level (to

achieve a target performance) to be supplied by the network via

a set of relays and transmitters If the requested power creates a

condition that violates interference regulations then the network

would clip or scale the source and/or relay output power

C Contributions

In this paper, we focus on a MIMO amplify-and-forward relay network where each relay is itself equipped with multiple transmit/receive antennas, and intrarelay cooperation is further allowed There is one source with multiple antennas and one destination with multiple antennas Furthermore, we assume

a worst case condition, where there is no direct line of sight link between the source and the destination We discuss both SNR and MMSE designs, considering zero-forcing and power constrained scenarios, including a global power constraint placed at the relay network and finally a power target level imposed at the receiver site We also find the corresponding structure such that the transmission rate is maximized under

a power constraint at the receiver This is in contrast to [22] where we only focused on designing ZF and MMSE without power constraints

The main results of the paper are as follows

1) For a single-antenna case, we obtain a closed-form solution for the optimal relay amplification matrix by minimizing MMSE subject to a global power constraint at the output

of relays The results are shown to be superior to previously published results

2) For multiple antenna systems, we initially consider the case with no power constraints imposed at the relays or receiver and provide two alternative solutions In the first solution,

we minimize MMSE given a target SNR While in the second approach, we maximize SNR subject to ZF Both schemes provide very similar results under these condi-tions The results are shown to be superior to previous re-sults in literature albeit at higher complexity

3) We extend our previous results to include a power con-straint (requirement) at the receiver and provide two alter-native solutions In the first solution, we derive the relay matrix that minimizes the overall MMSE (joint MMSE) subject to the preceding condition In the second approach,

we maximize SNR subject to ZF at the output of equal-izer With no power limitations at the relays, both schemes achieve comparable performance However in reality, im-posing a power constraint at the receiver might result in unbounded power at the relays We provide simulation re-sults that illustrate the effect of power clipping applied at the relays, where depending on the channel condition ei-ther SNR/ZF or MMSE perform better We also provide the cumulative distribution function (CDF) of the total output power of the relay nodes to illustrate that the power is in

an acceptable range for a large percentage of operating sce-narios

4) Finally, we find the relay matrix that maximizes the trans-mission rate under a power constraint at the receiver The remainder of the paper is organized as follows Section II describes the problem formulation In Section III, we minimize MMSE with and without power constraint in order to find relay matrix Section IV, SNR maximization subject to ZF with and without power constraint is introduced and the results are com-pared to MMSE solutions In Section V, we find the relay matrix subject to power constraint at the receiver to maximize transmis-sion rate Simulation results are provided in Section VI and we conclude in Section VII

Fig 1 MIMO relay architecture with intrarelay cooperation.

Notation

We shall use lower case for vectors, while capital letters for

matrices The complex transposition operator is defined as ,

while the conjugate of the elements of a matrix , is given by

Also the , and are the trace and determinant of

matrix respectively The operation stacks the columns

of into a single column vector, and denotes the Kronecker

diag-onal matrix with block elements given by Also, denotes

the pseudoinverse

II PROBLEMFORMULATION

Fig 1 illustrates a MIMO wireless sensor network consisting

of transmit antennas at the source and receive antennas

at the destination We consider an amplify-and-forward relay

scheme consisting of relays, each relay is equipped with

transmit/receive antennas In the subsequent discussions we

de-rive relations between the number of transmit and receive

an-tennas for each condition to have a unique solution As long

as these relations are honored, having unequal numbers of

re-ceive and transmit antennas are possible, and would affect the

diversity gain of the system For the purposes of this paper

we focus on the case where the number of transmit and

re-ceive antennas are equal The relay matrix is represented by

a block diagonal matrix , where each block is given by a

matrix between the source and the relay nodes, while

is the channel matrix between the relay sensors and the destination The channel matrices are

memoryless and a quasi-static fading condition is assumed The

received signal is modeled as

(1) where and are additive Gaussian noise (AGN) with

co-variance matrixes and respectively and is the

trans-mitted signal with covariance matrix A two-phase

(two-hop) protocol is used to transmit data from the source to the receiver In the first phase (hop) the source broadcasts a signal vector towards the relay sensors In the second phase, the relay sensors retransmit the information to the destination At the re-ceiver, depending on the design criterion, one can further em-ploy a MIMO equalizer, which we shall denote later by , in order to compensate for the effect of the overall MIMO channel

In the latter, the goal is to design both separately and jointly, possibly considering a power constraint We investigate the performance of a MIMO relay network under the following scenarios

1) First, the relay matrix and equalizer pair are de-signed under a MMSE criterion, considering two distinct cases: i) without power constraint, where are designed in two independent steps; ii) under an output power constraint, where the relays are selected such that the overall MMSE is minimized Finally, we present a closed-form solution for the case of global power con-straint at the relays albeit for the special case of minimizing the MSE of a SISO system A similar approach was con-sidered in [18], where the authors maximize the received signal power However in the latter, a closed form solution was found only for high SNR situations

2) Second, without considering any predetermined MIMO decoder (equalizer), we design the optimal relay matrix such that it maximizes the output SNR without post-equal-ization (see in Fig 1) This is achieved under a ZF criterion, first with a specified target output SNR Then, by relaxing the target SNR condition, a total power constraint for the relays is enforced Note that in such cases, the role

of the relays is to provide equalization for the underlying forward and backward channels Observe also that in [16],

a similar ZF technique is considered, where each relay performs ZF on its local backward and forward channels, albeit without considering the effect of noise at the relays The proposed ZF approach outperforms the one of [16], however with higher complexity since each relay requires knowledge on the entire backward channel as well as its

Fig 2 MIMO relay equalization.

local forward channel Finally, we maximize the SNR

at the output of the equalizer subject to ZF with power

constraint at the input of the receiver

Finally, in Section V, we design the optimal relay matrix that

maximizes transmission rate and compare it with the capacity

upper bound which is provided in [15] The optimal relay matrix

is found iteratively

A Channel State Information Availability

The assumptions made regarding the availability of CSI entail

certain implementations issues such as the mechanism required

to update channel information at the relays as well as at the

destination The problem of channel estimation and updating

can be decomposed into a training phase and a feedback phase

and can be performed within the context of either a coherent or

a noncoherent network which will be defined as follows in the

context of this paper

1) Coherent MIMO Relay Network: In a coherent MIMO

relay network, the source has no CSI, the destination has perfect

knowledge of all channels, and each relay has perfect knowledge

of all backward channels and its forward channel In this case the

optimizations can be performed at each relay in order to find the

optimum relay matrix Each relay can acquire its local backward

channel through standard training methods, but in order to

ob-tain the backward channels of the other relays each relay needs

to broadcast its backward channel which will require significant

overhead Obtaining each relay’s forward channel is also

chal-lenging and is equivalent to obtaining transmit CSI in

point-to-point wireless systems Finally, acquiring complete knowledge

of channels at the destination can be performed through training

2) Noncoherent MIMO Relay Network: In this case, the

re-lays have no CSI and the optimization process is performed at

the destination Each relay acquires its local backward channel

and sends it to the destination The destination can acquire each

relay’s forward channel through a training procedure initiated

by each individual relay Finally, the receiver sends the optimum

relay matrix via feedback to the individual relays In contrast to

the coherent case, there is no need to send the forward channels

to the relays through feedback which reduces overhead

3) CSI at the Transmitter: In the paper, we assumed that the

source does not know the channel, where the best transmission

strategy in order to improve spectral efficiency gain is to use

spa-tial multiplexing which yields a linear increase in capacity in the

minimum number of antennas at the transmitter and receiver If

the transmitter knows the channels perfectly or partially,

per-formance or capacity could be improved through beamforming

[19], [23] Furthermore, if the transmitter has perfect channel

information, one can design lower complexity receivers [24]

However, obtaining accurate CSI at the source incurs additional overhead due to feedback

III MMSE APPROACH

In this section, we shall first consider an MMSE design given

a certain target SNR One possible approach is to proceed in two steps, where first we design , which along with equalizes for the effect of and the noise Then, given we proceed

by equalizing for the overall channel and the output noise This approach has been considered in [18] for a single-antenna scheme Also we design and (see Fig 2) jointly such that

we minimize the MMSE under a power constraint at the input

of the equalizer

A MMSE With Target SNR—Two-Step Equalization

Let us define

(2) and as

(3)

where , is the target SNR at the receiver per each antenna Also and are signal and noise powers at the receiver The choice of would ensure a certain target SNR at the desti-nation.1Thus, the MMSE solution to

(4)

is given by

(5)

We assume that , so that the system of (2) always has

a solution The choice of a particular solution can be accommo-dated depending on some extra criteria For instance, since one

is normally interested in a low power consumption by the relay sensors, we can pick as the one with minimum norm, i.e.,

, or

(6)

1 The choice of  cannot improve the SNR beyond a limit and the reason is the amplification of noise at the relays This is part of problem with amplify-and-forward relay strategies.

Thus, given , we can now easily express the MMSE equalizer

as

(7) where we have defined the effective noise variance due to at

the output of as

(8)

B Joint MMSE Solution for

Given the optimal equalizer solution in (7), the resulting

MMSE cost is given by [25]

(9) The goal in this section is to compute such that it minimizes

(9) To this end, we shall first consider a SISO relay scheme

Later, we obtain optimality conditions for the MIMO case

1) Analytical SISO Relay Expressions: Let and be the

and backward and forward channels in a SISO

relay network respectively and is the element of diagonal

relay matrix Also we assume that the total power

usage by the network is limited to Our problem is then to

solve

(10)

where is the received signal power at relay , and is

the mean-square error (MSE) which is given by

(11) Now, the optimization problem can be expressed as

(12)

de-note the diagonal elements of and define

, so that

(13) Now by defining and substituting (13) in (12), we can express

the maximization problem in the form of a Rayleigh–Ritz ratio

(14)

Defining , this ratio can be equivalently expressed as

(15)

so that (14) becomes

(16)

In general, for positive definite, we can decompose it into Cholesky factors as ,2[25], and the solution to (16)

is given by

eigenvector of

where the diagonal matrix is defined as

Here, because is rank 1, we can express the elements of explicitly as

(18)

The norm of is then adjusted so that [see (16)], which implies that

(19) or

(20)

Finally, by substituting (20) into (18), the relay coefficients can

be expressed as

(21)

2 We have considered a general approach for decomposing matrix B B which is Cholesky factorization As matrix B B is a diagonal matrix with positive entries, the Cholesky factorization is simply square root factorization where entries of matrixes  L L and  L L are positive square roots of entries of matrix B B B

In [18] the authors maximize the received signal power at the

destination, which includes the input relay noise, subject to a

global power constraint However a closed form solution could

only be obtained for high SNR and after some approximations

and is given by

(22)

Simulations show that the (21) outperforms (22) with a large

margin for different number of relays and global power

2) Joint MIMO MMSE Expressions: From (9), we may note

that it can be further minimized w.r.t Observe that has a

predefined block diagonal structure, and solving for each block

in separate is not straightforward Instead, consider the

defini-tion of in (2), so that (9) can be expressed as

(23)

We can now pose the problem of minimizing (23) subject to a

power constraint at the output of ,

(24) where is the total output power of relays Let us

introduce the SVD of

(25) where , and is a square diagonal matrix whose its

diagonal elements, , , are the singular values

arranged in decreasing order Then it follows that

Now we can rewrite (23) as (26), shown

at the bottom of the page Applying the matrix inversion lemma

to (26), and defining

(27) allows us to write (26) as

(28)

subject to

(29)

We assume that the transmit symbols are spatially white, i.e.,

Also without loss of generality, we assume

is The cost in (28) can thus be expressed as

(30) where is a diagonal matrix with the elements

(31) The cost in (30) is minimized if and only if the expression inside the trace is diagonal [26], meaning that must be diagonal [27] We construct such that all elements are zeros except for

(32)

Using the Lagrange multipliers, the objective function can be written as

(33)

so that differentiating it over gives

(34) and

(35)

By substituting (35) back into (34) we obtain (36), shown at the bottom of the next page, where After setting , we can pick , for instance, as the one with minimum norm

IV SNR APPROACH

In [18], the authors have provided, for the special case of SISO antennas scheme, optimal relay coefficients that

maxi-(26)

mize the signal power at the receiver input, subject to both local

and global power constraints In particular, SNR maximization

subject to local power constraints leads to the traditional power

normalization and phase compensation method employed in the

SISO amplified-and-forward scheme Note that such schemes

assume that each relay retransmits information using the

max-imum available power for each relay, which is not necessarily

optimal In fact, it has been recently shown in [23] that the

optimal SNR maximization subject to local power constraints

scheme is one that can make use of less power at each relay, by

relying on the information of all forward and backward channels

in the network For a global power optimization, [18] provides

an approximate expression for the relay gains In this paper, we

find the optimal relay matrices in the MIMO case under the SNR

approach with and without power constraint In the approach

considered herein, the two noise sources appearing in Fig 1 are

taken into consideration, even though the SNR can be defined

in different ways, depending on the optimality criteria at hand

In this section, we define the output SNR as the ratio between

the power of the input signal and the overall contribution of the

noise sources, whose effect is transferred to the output node

That is

(37)

Thus, without any equalization criteria enforced at this point,

one may seek the optimal relay matrix that solves

(38)

This problem can be shown to be equivalent to the well known

generalized eigenvalue problem (without any constraint), stated

in the form of a Rayleigh–Ritz ratio Note that (38) can be

ex-pressed as

(39) Now, since is block diagonal, we have that

(40) (41)

defined in (41) has dimension In the same fashion,

we have that

(42)

where represents permutation matrix that reorganizes the en-tries of accordingly, and where we have further defined

Thus, the Rayleigh–Ritz ratio in (39) can be expressed as

(43) where in the latter, we have defined

(44) and used the fact that

A Maximum Output SNR Subject to Zero-Forcing Constraint

Under a ZF constraint, we are required to solve (38) subject to

The gain can be defined, for instance, based

on the desired target introduced in (3) In this case, the numerator in (43) becomes Moreover, using (41) and defining , our problem is now

where here incorporates the effect of noise at the relays Note that in order for to have a solution, must at least be a square matrix, which requires that , where denotes truncation Thus, assume is positive definite and consider the

(45) can be expressed as

The solution to this problem is the minimum norm vector [25] that satisfies the linear constraint above, which is given by

(36)

This results in3

(48) where is defined in (44) Note that

is positive definite if is positive definite, which

requires that the matrix is full column rank

This implies that Now, we may remark

that, in case , becomes non-negative definite, and the

above expression no longer holds However, consider instead

the spectral decomposition

(49) and define

(50)

Because we have freedom to choose , many solutions exist

Moreover, one possible solution is obtained by setting ,

so that it corresponds to a minimum norm vector In this case,

the problem becomes

(51)

and the solution in this case can be verified to have a form similar

to (48)

B Maximum Received SNR Subject to Global Power

Constraint

When a global power constraint is enforced, it may not be

possible to achieve a predefined target SNR That is, let be the

total power to be distributed among the relays This implies that

(52)

the input signal powers to each individual relay, and

(53) represents the weighting for the norm of Now, if the

min-imum norm solution in (48) is such that (52) is not satisfied, one

may need to readjust the target SNR in order to meet the power

3 As an alternative procedure, one can define two separate zero forcing criteria,

for the backward and forward channels respectively More specifically, in [16],

the relays matrixes are given by

FF

where H = (H H H H H ) H H H and H = H H H (H H H H H H ) The

coefficient adjusts the output power of each relay to p and is given by

p M=p [tr(H H H H )] + M [tr((H H H H H H ) (H H H H H H ))] Note that

from a complexity point of view, while for the ZF in [16] only knowledge

of local backward and forward channels for each relay is required, in the ZF

provided here each relay needs to know the entire backward channels and only

its local forward channel.

specification That is, the power constrained solution is given by (48), where is

C Maximum SNR (at the Output of ) Subject to ZF and Power Constraint (at the Input of )

In this section, we maximize the SNR at the output of the equalizer, , which is defined as

(54)

subject to the ZF constraint , and power con-straint at the input of

(55)

In view of the ZF condition, the SNR in (54) becomes

(56) where is defined in (8) and is defined as

(57)

By substituting (57) into (56), after some manipulations we can express the SNR maximization as

(58)

subject to the power constraint in (55) Comparing the cost in (58) with (23), we can see that the only difference is the presence

of in the MMSE cost function The problem is similar to (32), where here we simply replace the elements of in (31) by

(59) This allows us to write (58) as

(60) and we find

(61) Under the same power constraint at the input to the equalizer and assuming that the power at the transmitter is fixed at a pre-defined value, simulations have shown that these two scheme have a close performance in the BER sense

V RATEMAXIMIZATION

Here we find in order to maximize the achievable rate

sub-ject to power constraint at the destination The transmission rate

between the source and the destination can be expressed as [28]

(62)

so that we need to solve the following optimization problem:

(63) where in (63) we have used the fact that

By using the same approach as in Section III-B-2), and

using the SVD decomposition in (25), we can express (63) as

(64) where , and is defined in (27) From the Hadamard

inequality [28], the product of the diagonal entries of a positive

definite matrix is always greater or equal than the matrix

deter-minant where equality holds if and only if the matrix is diagonal

Thus, must be diagonal and again we can construct such

that all elements are zeros except for ,

Then (64) can be expressed as

(65)

Using Lagrangian multipliers, the argument in (65) is equivalent

to

(66) Differentiating (66) over gives

(67) which yields

(68) The above is a function of which should be

calcu-lated first By inserting (68) in the power constraint (65), we

were able to find numerically Again, after constructing as

Fig 3 BER performance of MMSE scheme (21) versus maximization of re-ceived signal (22) subject to global power constraint of 10 dB Here, M = 1,

N = 1, and K = 10.

, we can pick for instance as the one with minimum norm In the simulations section, we compare the maximum achievable rate provided here with the upper bound capacity in [15] given by

(69)

where is the total power at the source The capacity in (69)

is the capacity upper bound of a MIMO relay network which is derived by using the cut-set theorem [15]

VI SIMULATIONRESULTS

In this section, we provide some numerical results to verify our analytical calculations We assume that all relays are at equal distance from the source and destination so that the forward and backward channels have the same statistics, which are gen-erated as zero-mean and unit-variance independent and iden-tically distributed (i.i.d.) complex Gaussian random variables The transmission signaling is in spatial multiplexing mode (i.e., the source transmits independent data streams from different antennas) with total transmit power level of 0 dB, which is uniformly distributed among the transmit antennas ( for each antenna at the source) Also all simulations are conducted using a QPSK constellation, and the noise variances are as-sumed to be the same for all antennas We plot bit error rate (BER) curves versus SNR, which is defined per bit per antenna

at each relay antenna The BER provided here is averaged over different channel realizations unless otherwise mentioned Fig 3 shows the BER for the MMSE criterion compared to the received signal maximization in [18] for a SISO relay net-work with 10 relays and global power constraint of 10 dB It can be seen that the MMSE solution here outperforms the one

of (22), specially at high SNR

Fig 4 shows the BER performance of the ZF and MMSE in two steps for the case when , , for 1, 2, and

5 We remark that for the sake of comparison, we have chosen

Fig 4 BER performance of MMSE in two steps and ZF Here M = 3 and

N = 3.

Fig 5 Comparison of BER performance of ZF in (48) and (47) Here, we have

fixed number of relays to K = 5 and M and N are 2 and 4 Also the total output

power of relays are fixed to 7 dB.

, since in the ZF criterion, it is required that

, while for the MMSE, we must satisfy Increasing

the number of relays improves the system performance, and we

may note that both MMSE and ZF criteria behave similarly

From Fig 4, for BER , we achieve 10-dB gain when the

number of relays is increased from 1 to 2 This gain is mainly

due to the distributed diversity gain that we obtain by adding

one more relay It should be mentioned that since we have not

imposed any power constraint at this point, the ZF and MMSE

criteria choose the best power (relay output power) in order to

maximize the SNR or minimize the MMSE at the input of the

receiver

Fig 5 compares the performance of ZF in (48) and (47) for

five relays and two different values for and In order to

have a fair comparison, we have fixed the total output power of

the relays to 7 dB for both cases As Fig 5 shows, our ZF

outper-forms the one in [16] In the latter, each relay peroutper-forms ZF for

Fig 6 BER performance of joint MMSE subject to power constraint of 0 and

3 dB for M = 3, N = 3, and different K.

Fig 7 CDF of the total output power of relays for joint MMSE, where M = 3,

N = 3, and K = 1, 3 Here, the power constraint at the receiver is set to 0 dB.

the backward and forward channels locally without considering the effect of noise at the relays Here in addition to considering the effect of noise, each relay requires knowledge of the entire backward channels

Fig 6 compares the joint MMSE scheme for 3 antennas at the source and destination and 3 antennas at each relay under two different received target powers, 0 and 3 dB It can be seen that

by increasing the target power, improved BER performance can

be obtained Since the power constraint is at the receiver and the total output power of relays is not constrained, there is a possibility of having channels for which is large To elaborate

on this issue we provide CDF of the total output power of relays for joint MMSE as shown in Fig 7 where the power constraint

at the receiver, , is set to 0 dB and SNR 18 dB with , , and 1, 3 The Figure illustrates that when , 96.6% of total output power is less than 15 dB which will be further distributed among 3 antennas and when , 98%

of the total output power is less than 15 dB which again will