# On the design of uplink and downlink group-orthogonal multicarrier wireless systems

**ABSTRACT** Group-orthogonal multicarrier code-division multiple access (GO-MC-CDMA) has been proposed as an attractive multiplexing technique for the uplink segment of wireless systems. More recently, a variant of this scheme has also been proposed for the downlink. This paper presents a unified bit error rate (BER) performance analysis of group-orthogonal wireless systems when using maximum likelihood (ML) multiuser/multisymbol detection covering both link directions. Valuable design rules regarding the number of subcarriers per group and the selection of spreading codes are derived. Simulations results using realistic system parameters and ETSI BRAN channel models are also presented which serve to validate the analytical results.

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**ABSTRACT:**This paper considers the use of precoding in MIMO-OFDM systems based on deterministic channel state information at the transmitter (CSIT). The proposed technique combats subcarrier fading using a linear spreading of the source information in the spatial and frequency domains. Unlike previous works, it is assumed that the receiver does not necessarily has to rely on linear processing, allowing in this way more powerful receiver strategies. In particular, it is shown that a detector based on the maximum likelihood (ML) criterion brings along important performance benefits. Different spreading strategies are evaluated in terms of their performance and ML detection complexity. It can be observed that spreading the information and optimising the system over a group of subcarriers (rather than on a per subcarrier-basis or all subcarriers together) extracts virtually all the diversity the wireless channel can provide while remaining computationally feasible. Simulation results are presented within the context of the new WLAN standard IEEE 802.11n in order to illustrate the practicality of the proposed technique.Wireless Days (WD), 2010 IFIP; 11/2010 - [Show abstract] [Hide abstract]

**ABSTRACT:**Optimum detection based on maximum likelihood (ML) has been recently explored within the context of linearly precoded subcarrier cooperative MIMO-OFDM systems, showing that large performance gains can be obtained in comparison to an all-linear setup (i.e., linear transmitter/receiver) at the cost of increased computational complexity. Nonetheless, low complexity alternatives, yet retaining optimality, have been also proposed. Most of these previous works have assumed the availability of perfect channel state information at the transmitter (CSIT) despite this is rarely the case since feedback delay and/or quantisation effects will usually render imperfect CSIT. This paper studies how these imperfections affect the performance of the ML-based receiver. Particularly, it focuses on fully loaded setups, that is, those configurations where the number of independent streams transmitted equals the number of available spatial modes. Remarkably, for these configurations, an interesting interplay arises between the presence of CSIT imperfections, the spreading of the information symbols in the frequency and spatial domains and the diversity order achieved.Information Sciences and Systems (CISS), 2012 46th Annual Conference on; 01/2012 - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper considers novel receiver structures for linearly precoded MIMO-OFDM systems, making special em- phasis on soft detection principles. After reviewing the operation of MMSE detectors, an optimal iterative soft receiver based on list sphere detection (LSD) is proposed. This scheme provides the best possible performance from an error rate point of view at the expense of a rather high computational cost. To address this downside, a lower complexity yet optimal detector that combines linear and non-linear processing is proposed for a class of precoders that are particularly important from a practical point of view. I. I NTRODUCTIONIEEE Transactions on Wireless Communications 01/2011; 12(12):326-330. · 2.42 Impact Factor

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1656 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008

On the Design of Uplink and Downlink

Group-Orthogonal Multicarrier Wireless Systems

Felip Riera-Palou, Guillem Femenias, and Jaume Ramis

Abstract—Group-orthogonal multicarrier code-division multi-

ple access (GO-MC-CDMA) has been proposed as an attractive

mutiplexing technique for the uplink segment of wireless systems.

More recently, a variant of this scheme has also been proposed for

the downlink. This paper presents a unified bit error rate (BER)

performance analysis of group-orthogonal wireless systems when

using maximum likelihood (ML) multiuser/multisymbol detection

covering both link directions. Valuable design rules regarding the

number of subcarriers per group and the selection of spreading

codes are derived. Simulations results using realistic system

parameters and ETSI BRAN channel models are also presented

which serve to validate the analytical results.

Index Terms—Group-orthogonal multicarrier code-division

multipleaccess(GO-MC-CDMA),

(CDM), maximum likelihood (ML), diversity, multiuser inter-

ference (MUI).

code-divisionmultiplex

I. INTRODUCTION

M

bilities) of the combination of two complementary techniques:

code division multiple access (CDMA) and orthogonal fre-

quency division multiplexing (OFDM). On one hand, CDMA

multiplexes users by means of a user-specific spreading code

allowing them to simultaneously use the same frequency

spectrum. The properties of these codes (e.g. orthogonality)

make user separation at the receiver possible. On the other

hand, OFDM is a block transmission scheme where the

incoming user symbols are grouped, serial-to-parallel (S/P)

converted and modulated onto different subcarriers. Choosing

the subcarriers to be orthogonal allows the group of symbols

to be transmitted in parallel and, assuming perfect synchroni-

sation, without interference. The S/P conversion permits the

transmission rate to be reduced to a fraction of the original

user rate combating in this way the frequency selectivity of

the channel. The attractive features derived from the CDMA-

OFDM combination makes MC-CDMA a firm candidate for

the next generation of wireless systems [3], [4], [5], [6].

Group-orthogonal MC-CDMA (GO-MC-CDMA) has been

recently proposed in [7] as a combination of MC-CDMA

and orthogonal frequency division multiple access (OFDMA)

ULTI-CARRIER code division multiple access (MC-

CDMA) [1] is one specific case (see [2] for other possi-

Paper approved by J. Wang, the Editor for Wireless Spread Spectrum of

the IEEE Communications Society. Manuscript received September 26, 2006;

revised February 19, 2007. This work has been supported by the Spanish

Ministry of Education and FEDER (Fondo Europeo de Desarrollo Regional)

through project MARIMBA (TEC2005-00997/TCM), Govern de les Illes

Balears through project XISPES, and a Ram´ on y Cajal fellowship.

The authors are with the Mobile Communications Group, Dept. of

Mathematics and Informatics, University of the Balearic Islands (UIB),

Crta. Valldemossa km 7.5, Palma de Mallorca, Spain (e-mail: {felip.riera,

guillem.femenias, jaume.ramis}@uib.es.

Digital Object Identifier 10.1109/TCOMM.2008.060497

suitable for the uplink segment of a wireless system. The

basic idea behind GO-MC-CDMA is to partition the avail-

able (orthogonal) subcarriers into (orthogonal) groups and

distribute users among the groups. The main advantage of this

system is that each group functions as an independent MC-

CDMA system with a smaller number of users making the

use of maximum likelihood multiuser detection (ML-MUD)

within each group feasible. This idea is mainly applicable in

the context of an uplink transmission where the base station

has to detect all incoming users. In the downlink, and since

a given user is only interested in his own data, it is not

very sensible to use ML-MUD in combination with group-

orthogonal MC-CDMA since most of the detected information

would be discarded. To avoid such a waste of processing, a

mobile terminal could use one of the low-complexity single-

user detection techniques such as maximal ratio combining

(MRC) or equal gain combining (EGC) but this would come

at the cost of a severely degraded performance in terms of

bit error rate (BER). In order to exploit the combination

of group-orthogonal modulation and ML detection in the

downlink, group-orthogonal multicarrier code-division multi-

plexing (GO-MC-CDM) has been recently introduced in [8]

as an extension to the MC-CDM scheme proposed in [9].

In GO-MC-CDM a given group of (orthogonal) subcarriers

are used to multiplex symbols (using orthogonal codes) from

the same user. The mobile terminal can then use maximum

likelihood multisymbol detection (ML-MSD) to estimate all

the symbols simultaneously. In comparison with conventional

OFDM, GO-MC-CDM has the potential advantage offered by

the frequency diversity which can be fully exploited when

using ML detection.

Given the group independence within a group-orthogonal

multicarrier (GO-MC) system, any performanceanalysis needs

only to consider a single group and, to all effects, the re-

sults of conventional MC-CDMA [5] or MC-CDM [9] apply.

Nevertheless, the additional aspect that needs to be taken into

account when designing a GO-MC-CDMA system is the parti-

tioning of the total number of available subcarriers into groups.

This important parameter determines in great manner the

overall performance, capacity and complexity of the system. It

should be noted that using an efficient implementation of the

ML detector such as the sphere decoder [10] whose average

complexity has been shown to be cubic [11], the group size can

be chosen more in accordance with performance criteria rather

than driven by complexitylimitations. Basically, the group size

should be carefully chosen as a tradeoff between diversity gain

and interference enhancement. It was already shown in [7] that

BER performance is maximized by choosing the subcarriers in

a group equispaced across the available bandwidth. Moreover,

0090-6778/08$25.00 c ? 2008 IEEE

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RIERA-PALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUP-ORTHOGONAL MULTICARRIER WIRELESS SYSTEMS1657

it was proved that the maximum diversity order for a user

is given by the rank of the channel correlation matrix. This

already provides a foundation to fix the number of subcarriers

per group. However, it is easy to see experimentally that in

most realistic scenarios, fewer subcarriers than those needed

to achieve full diversity already provide near-optimum BER

performance.

The objective of this paper is to present a new unified

analysis of group-orthogonal multicarrier systems covering

both link directions, mobile-to-base (uplink) and base-to-

mobile (downlink) when using ML detection. This analysis

highlights which are the important factors affecting the BER

and provides a suitable base onto which to choose the number

of subcarriers per group. Simulation results using realistic

system parameters are used to illustrate the design guidelines.

This paper is organized as follows: the system model used

in this study is described in Section II. Section III presents a

unified analysis covering both, uplink and downlink scenarios.

Asymptotic results are also derived which provide valuable in-

sight into the relevant factors affecting the BER performance.

Simulation results supporting the analytical derivations are

provided in Section IV. Finally, the main conclusions of the

paper are stated in Section V.

A brief note on notation: scalars, vectors and matrices are

denoted by lower case non-bold, lower case bold and upper

case bold symbols, respectively. Vectors are assumed to be

column. Hermitian transpose and transpose are denoted by

(·)Hand (·)T, respectively. The expression D(x) is used to

denote a diagonal matrix with vector x at its main diagonal.

II. SYSTEM MODEL

We consider a single-cell scenario where all resources are

available to the users connected to the base station. The

downlink operates synchronously whereas the uplink, as in

[7], is assumed to operate quasi-synchronously. This implies

that the time offset among users is upper bounded. The

system is able to simultaneously serve up to Kmax active

users using the total available bandwidth of W Hz. Without

loss of generality, the total number of subcarriers (Ntotal)

is partitioned into Ng groups each with N = Ntotal/Ng

subcarriers. Note that Ntotalshould be chosen large enough so

that each subcarrier experiences frequency-flat fading. Since

groups are independent of each other, all the modeling and

analysis can be performed on a single group which, to all

effects, behaves like a conventional MC-CDMA system with

N subcarriers with capacity to transmit a maximum of N

code-multiplexed symbols. Note that the group size N does

not necessarily impose a limit on the user’s information rate

since any user can participate (or indeed fully use) more than

one group making GO-MC an attractive option for multi-

service multi-rate networks where users have different traffic

requirements.

Figure 1 depicts the transmission/reception scheme for

group j in which K distinct (and independent) data symbols

are jointly transmitted. Notice that K ≤ N with equal-

ity holding when a group is fully loaded. At this point it

is necessary to distinguish between the uplink and down-

link scenarios. In the uplink, we will denote the block of

jointly transmitted symbols in group j at discrete time in-

stant n by aj

aj

each element aj

complex MQAM or MPSK constellation. It follows that K

denotes the number of active users in the group. For the

downlink, the block of transmitted symbols is denoted by

aj

aj

all K symbols in the block belong to the same user. For

both scenarios we assume energy-normalized symbols such

that E

|aj

in the downlink or uplink, is then multiplied by a different

spreading code of the form ci= [ci

E?|ci

convenience, we define now the N × K spreading matrix as

C = [c0c1...cK−1]. Since groups are separated by the fact

of using different subcarriers, the same spreading matrix can

be used in all groups. To simplify notation, and given that

each group is independent of the others, allowing the analysis

to be conducted on just one individual group, from now on

the group index will be dropped (i.e. the transmitted symbols

vector will be denoted by aup(n) and adw(n)).

After spreading, the resulting chips from each data symbol

are added up together, serial-to-parallel converted and fed,

jointly with the symbols from other groups, to a bank of

orthogonal carrier modulators usually implemented using the

inverse Fourier transform (IFFT). A cyclic prefix is then

added to minimise the effects of the dispersion introduced

by the channel. It is assumed that the CP length exceeds

the maximum channel delay spread and therefore, there is

no interference among successively transmitted symbol blocks

(i.e. there is no interblock interference (IBI)). Furthermore, the

quasi-synchronicity assumed in the uplink refers to the fact

that the delay between the first and last users in the group to

reach the base station is less than the duration of the cyclic

prefix. This condition implies that the relative delays of the

users are absorbed in the random phases of the subcarriers. It

is assumed, as usually done in CDMA-based networks, that

a slow power control mechanism ensures that all users in the

uplink are received with the same average power at the base

station avoiding in this way the near-far effect. Raising this

assumption of perfect slow power control leads to a different

analysis which is currently a topic of further research.

The channel linking the base station and a particular mobile

terminal is assumed to be time-varyingand frequency-selective

with an scenario-dependent power delay profile S(τ) given by

up(n) =

i(n) generated by user i is drawn from a

?

0(n) aj

1(n) ... aj

K−1(n)

?T

, where

dw(n) =

?

i(n)|2?

0(n) aj

1(n) ... aj

K−1(n)

?T

where in this case

?

= 1. Each incoming symbol aj

i(n), either

0ci

1...ci

N−1]Twith

j|2?

= 1/N for 0 ≤ j ≤ N − 1. Orthogonal

spreading codes are assumed throughout this work. For later

S(τ) =

P−1

?

l=0

φlδ(τ − τl)

where P denotes the number of independent paths of the

channel and φl and τl denote the power and delay of the

l-th path. It is assumed that the power delay profile is the

same for all users in a group and normalized to unity (i.e.

?P−1

l=0φl = 1). A single realization of the channel impulse

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1658IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008

a0

a1

...

aK−1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .×

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

c0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

c1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .×

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

cK−1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

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Σ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S/P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IFFT

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P/S

+

Cyclic

prefix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cyclic

prefix

removal

+

S/P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FFT

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ML

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

ˆ a0

ˆ a1

...

ˆ aK−1

Fig. 1. Individual group GO-MC transmission architecture.

response at time instant t will then have the form

h(t;τ) =

P−1

?

l=0

hl(t)δ(t − τl)

where it will hold that E?| hl(t) |2?= φl. The corresponding

P−1

?

which when evaluated at the N equispaced subcarrier fre-

quencies assigned to a specific group, yields the N × 1

vector¯h(t) = [¯h(t;f0)¯h(t;f1)...¯h(t;fN−1)]T. Assuming

that the channel is static over the duration of an OFDM

symbol and to simplify the notation, we will express the

subcarrier frequency response during the discrete time instant

n as¯h(n) = [¯h0(n)¯h1(n)...¯hN−1(n)]T.

Now, important differences arise between the uplink and

downlink cases. In the uplink, each symbol in the transmission

vector aup(n), comes from a different user and hence is

transmitted through a different channel. Let us denote by

¯hk(n), with 0 ≤ k ≤ K − 1, the channel frequency response

for the k-th user evaluated at the subcarrier frequencies of

the group to whom user k belongs at instant n and let

H(n) =?¯h0(n)¯h1(n) ...¯hK−1(n)?

to be operating in a common environment so that the different

channel frequency responses are all independent realizations

derived from the common power delay profile. The uplink sys-

tem matrix characterizing the channel and spreading processes

for a group when K users (K ≤ N) are active can then be

expressed as

frequency response will be given by

¯h(t;f) =

l=0

hl(t)exp(−j2πfτl)

represent the channel

frequency response for the whole group. All users are assumed

Aup(n) = H(n) ◦ C

=

⎛

⎝

⎜

¯h0

0(n)c0

...

0

...

...

...

¯hK−1

0

(n)cK−1

...

0

¯h0

N−1(n)c0

N−1

¯hK−1

N−1(n)cK−1

N−1

⎞

⎠,

⎟

(1)

where ◦ denotes the element-wise product between two equal-

sized matrices. Notice that columns of Aup(n) are independent

since users can be safely assumed to be sufficiently apart.

In the downlink, all symbols in the transmission vector,

adw(n), belong to the same user and therefore they all prop-

agate through the same channel. In this case, the (downlink)

system matrix accounting for the spreading and channel effects

for an arbitrary group wholly used by user k when transmitting

K data symbols is defined as

Adw(n) = D?¯hk(n)?C

=

⎝

Notice that the columns of Adw(n) are now potentially linearly

dependent and the task of separating the transmitted symbols

lies solely on the spreading codes.

The reception process for a group is illustrated on the left

handside of Fig. 1. After removing the cyclic prefix and S/P

conversion and assuming perfect subcarrier synchronisation,

the received signal, either in the uplink or the downlink, is

sampled and demodulated (tipically using the FFT) yielding

the N × 1 received signal vectors given by

r(n) = A(n)a(n) + υ(n)

⎛

⎜

¯hk

0(n)c0

...

N−1(n)c0

0

...

...

...

¯hk

0(n)cK−1

...

N−1(n)cK−1

0

¯hk

N−1

¯hk

N−1

⎞

⎠.

⎟

(2)

where the N × 1 complex vector υ(n) is made of zero-

mean complex Gaussian random variables with variance

E{|υq(n)|2} = σ2

A(n) and a(n) are chosen accordingly to the uplink or down-

link scenario. Notice that with the definition of normalized

unit-power transmitted symbols and normalized power delay

profile, the operating signal-to-noise ratio can be expressed

as Es/N0 = 1/2σ2

since successive symbols are independent from one another

due to the CP, the explicit time index n will be dropped from

subsequent equations.

υ, for all q = 0,...,N −1. In this equation,

v. In order to simplify the notation, and

III. GENERAL MAXIMUM LIKELIHOOD DETECTION

ANALYSIS

A. Maximum Likelihood detection. Unified BER analysis.

The symbols estimates calculated using ML-MUD or ML-

MSD satisfy [12]

ˆ a = argmin

a

?A a − r?2.

This procedure amounts to evaluate all the possible transmitted

vectors and choosing the closest one (in a least-squares sense)

to the received vector. Sphere detection [10], [13] has been

Authorized licensed use limited to: IEEE Xplore. Downloaded on October 17, 2008 at 04:45 from IEEE Xplore. Restrictions apply.

Page 4

RIERA-PALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUP-ORTHOGONAL MULTICARRIER WIRELESS SYSTEMS 1659

proposed for efficiently performing this search. The BER anal-

ysis which follows is general enough so as to accommodate

the uplink and downlink cases. It will be seen, however, that

the BER minimisation requires of different actions depending

on the link direction.

We consider the general case where a group, formed by N

subcarriers, is used to transmit K independent symbols, either

uplink1or downlink, all drawn from a constellation of size M.

The bit error probability for an arbitrary user/symbol in the

group can be upper bounded using the union bound as [14]

Pb≤

1

KMKlog2M

MK

?

i=1

MK

?

j?=i

j=1

P (ai→ aj)Nb(ai,aj).

(3)

where Pr{ai→ aj}, usually called the pairwise error proba-

bility (PEP), represents the probability of erroneously detect-

ing symbol vector ajwhen aiwas transmitted and Nb(ai,aj)

is the number of differing bits between symbols ai and aj.

To proceed further in the analysis, the PEP conditioned on a

given system matrix A can be shown to be [15]

??

=1

π

0

P (ai→ aj|A) =1

2erfc

?π/2

?A(ai− aj)?2

4σ2

?

ij? ?A(ai− aj)?2, the

v

?

exp

−?A(ai− aj)?2

4σ2

vsin2φ

?

dφ.

Now, defining the random variable d2

unconditional PEP can be expressed as

P (ai→ aj) =1

π

?π/2

?π/2

0

?+∞

Md2

−∞

e−x/4σ2

vsin2φpd2

ij(x)dxdφ

=1

π

0

ij

?

−

1

4σ2

vsin2φ

?

dφ

(4)

where px(·) and Mx(·) denote the probability density function

(pdf) and moment generating function (MGF) of a random

variable x, respectively.

As was done by Cai et al. in [7], we define the error vector

eij = ai− aj = [e0

specific matrix Sk

N × N diagonal matrix

⎛

k=0

and the N×N channel correlation matrix R = E?¯hk(¯hk)H?

(spaced-frequencycorrelation function)evaluated at the group-

assigned subcarrier frequencies. Note that R does not depend

on the specific user since we assume that all of them conform

to the same power delay profile. We further define the NK ×

1 vector¯h =(¯h0)T(¯h1)T

NK matrix Sij= [S0

ije1

ij... eK−1

ijD(ck). We additionally define the

ij

]Tand the N × N user-

ij= ek

Tij= D

⎝

?K−1

?

ek

ijck

0···

K−1

?

k=0

ek

ijck

N−1

?T⎞

⎠

(5)

which can be determined from the Fourier transform of S(τ)

?

... (¯hK−1)T?T

ij

and the N ×

ijS1

ij... SK−1

]. By definition, d2

ij?

1We emphasise that, in the uplink, slow power control ensures that all

users in the group are received with equal average power and therefore we

can perform the analysis for an arbitrary user.

?Aeij?2= eH

uplink case, that is, when A is given by (1), it is found that

ijAHAeij. Expanding this expression for the

d2

ij=

K−1

?

l=0

el

ij

∗

N−1

?

q=0

h

l

q

∗cl

q

∗

K−1

?

p=0

ep

ijh

p

qcp

q=¯hHSH

ijSij¯h.

Similarly, in the downlink, when A is given by (2), the

expansion of d2

ijresults in

d2

ij=

N−1

?

q=0

h

k

q

∗h

k

q

K−1

?

l=0

el

ij

∗cl

q

∗

K−1

?

p=0

ep

ijcp

q= h

kHTH

ijTijh

k.

For both cases, uplink and downlink, d2

in complex variables¯hkwith MGF

ijis a quadratic form

Md2

ij(s) = |IN− sGij|−1,

where IN denotes the identity matrix of dimension N and

??K−1

Let λij = {λij,1,λij,2,..., λij,Dij} denote the set of

Dij distinct positive eigenvalues of Gij with corresponding

multiplicities αij =

results in [16], the MGF of d2

Gij=

k=0Sk

TijRTH

ijRSk

ij

H

, uplink case

, downlink case

ij

(6)

?αij,1,αij,2,..., αij,Dij

1

(1 − sλij,d)αij,d=

?. Using the

ijcan also be expressed as

Md2

ij(s) =

Dij

?

d=1

Dij

?

d=1

αij,d

?

p=1

κij,d,p

(1 − sλij,d)p

(7)

where, using [17, Theorem 1], it can be proved that

λp−αij,d

ij,d

(αij,d− p)!

⎡

⎣

?

d??=d

with

Φ

beingthe

?n1,...,nd−1,nd+1,...,nDij

P (ai→ aj) =1

π

d=1

p=1

κij,d,p=

×∂αij,d−p

∂sαij,d−p

⎢⎢

Dij

?

d??=d

d?=1

1

(1 − sλij,d?)αij,d?

⎤

⎦

?

⎥⎥

????????

s=

1

λij,d

= λp−αij,d

ij,d

Φ

Dij

?

d?=1

λnd?

ij,d?

?

set

?

?αij,d?+nd?−1

λij,d

nd?

?αij,d?+nd?

nonnegative

such that

1 −

λij,d?

of integers

?

sin2φ

d??=dnd?

=

αij,d− p, which allows (4) to be written as

Dij

?

Dij

?

p−1

?

αij,d

?

κij,d,p

?π/2

⎝1 − Ω

?⎛

0

⎛

?λij,d

2

⎝

sin2φ +λij,d

?

?λij,d

2

4σ2

v

⎞

⎠

p

dφ

=

d=1

αij,d

?

p=1

κij,d,p

⎛

4σ2

v

⎞

⎠

4σ2

p

×

g=0

?p − 1 + g

g

⎝1 + Ω

v

?

⎞

⎠

g

(8)

with Ω(c) =

closed-form BER upper bound for an arbitrary power delay

profile is obtained. It is later shown that this bound is tight,

accurately matching the simulation results.

?c/(1 + c). By substituting (8) into (3), a

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1660 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008

B. BER analysis based on PEP classes

Since there are many pairs (ai,aj) giving exactly the

same PEP, it is possible to define a pairwise error class

C(Dc,λc,αc) as the set of all pairs (ai,aj) characterized with

a common matrix Gij = Gc with Dc distinct eigenvalues

λc = {λc,1,...,λc,Dc} with corresponding multiplicities

αc= {αc,1,αc,2, ... ,αc,Dc} and therefore, a common PEP

denoted by P(Dc,λc,αc). A more insightful BER expression

can then be obtained by using the PEP class notation, avoiding

the exhaustive computation of all the PEPs. Instead, the BER

upper-bound can be found by computing the PEP for each

class and weighing it using the number of elements in the

class and the number of erroneous bits this class may induce.

The BER upper bound can then be rewritten as

1

KMKlog2M

?

Pb≤

×

∀C(Dc,λc,αc)

K log2M

?

N=1

N W(Dc,λc,αc,N)P(Dc,λc,αc)

(9)

where W(Dc,λc,αc,N) corresponds to the number of ele-

ments in the class C(Dc,λc,αc) inducing N erroneous bits.

C. Asymptotic performance

Now, in order to gain further insight on the parameters

affecting the BER performance in GO-MC systems we focus

on the asymptotic case of large SNR. When Es/N0 =

1/2σ2

to infinity. It is easy to see that when s → ∞ the MGF from

(7) can be approximated by

v→ ∞, the argument of the MGF in (4) also tends

Md2

ij(s) ?

1

??Dij

d=1λαij,d

ij,d

?

(−s)

?Dij

d=1αij,d

allowing the asymptotic PEP of the different classes to be

expressed as

?π/2

=(2˜Dc)!

2˜Dc!2

Pasym(Dc,λc,αc) =1

π

0

(4σ2

?Dc

?˜ Dc

vsin2φ)˜ Dc

d=1λαd

(Es/N0)−˜ Dc

d=1λαc,d

c,d

dφ

c,d

(10)

where˜Dc =

class Gc. From (10) it is clear that the probability of error

will be mainly determined by the classes whose matrix Gc

have the smallest rank which we denote by˜Dmin, allowing

the asymptotic BER to be written as

?Dc

d=1αc,d is the rank of the matrix-defining

Pb≤

?

∀C(˜ Dmin,λc,αc)

×W(˜Dmin,λc,αc,N)(Es/N0)−˜ Dmin

K MKlog2M?˜ Dmin

In light of (11), the asymptotic BER minimisation is

achieved by maximising the minimum class rank˜Dmin and

the eigenvalue product of all the classes with rank˜Dmin.

K log2M

?

N=1

N

(2˜Dmin)!

2(˜Dmin!)2

d=1λαc,d

c,d

.

(11)

In the following, we pursue only the maximization of˜Dmin

(i.e. maximisation of the diversity order) since the maxi-

mization of the product of eigenvalues {λ˜ Dmin,d}Dmin

more difficult as it involves the simultaneous optimization of

all the eigenvalue products in the classes with˜Dmin rank.

The maximisation of the diversity order requires of different

actions depending on the link direction.

1) Uplink minimum rank maximisation: It is stated in [7]

that, assuming that channels of different users are statistically

independent, the maximum diversity order is achieved by

using equispaced group subcarriers and spreading codes which

are linearly independent of each other. Typical codes like

Walsh-Hadamard or Gold satisfy this requirement. Since this

maximisation holds for all classes, the class with minimum

rank in particular will also have maximum rank.

2) Downlink minimum rank maximisation: As it happens

in the uplink, choosing the subcarriers for a group equispaced

across the whole bandwidth minimizes subcarrier correlation

allowing us to optimize the system performance if an ad-

equate family of codes is properly selected. To this end,

rotated spreading transforms have been proposed for downlink

multicarrier systems by Bury et al in [18]. It is shown in

[18] that the often used Walsh-Hadamard codes lead to poor

diversity gains when employed to perform the frequency

spreading in the downlink of multicarrier systems. This can

be explained by the fact that for certain symbol blocks, the

energy is concentrated on one single subcarrier. A deep fade

on this subcarrier dramatically raises the probability of error

in the detection process, irrespective of the state of all other

subcarriers, limiting in this way the achievable diversity order

(asymptotic BER slope) to one. A similar effect is observed

in the GO-MC-CDM framework under study which is best

illustrated through an example: suppose 4 subcarriers are

used to transmit 4 symbols multiplexed by code using binary

modulation with alphabet {+1,−1} and Walsh-Hadamard

spreading. In order to find the BER upper-bound, the different

pairwise error classes need to be computed as indicated by (9).

One of these classes will comprise the PEP between blocks

which differ in all symbols such as ai = [1 1 1 1]Tand

aj= [−1 −1 −1 −1]T. In this situation, using (5) and (6),

it can easily be seen that, for this particular class,

⎛

⎜

where Rx,y denotes the (x,y)th entry of the channel cor-

relation matrix. Obviously in this case˜Dmin = 1 and this

will become the dominant term in the BER expression given

by (9) leading to a diversity order of one. For the particular

case of Walsh-Hadamard sequences, this behavior is due to

the fact that all columns (or rows) add up to zero except

for the first entry. A similar effect can be observed in other

spreading sequences such as those based on the discrete

Fourier transform (DFT). As pointed out in [18], a spreading

that maximizes the diversity gain can be found by applying a

rotation to the columns of the conventional spreading matrix

C as Crot= CD(θ), where θ = [θ0 θ1...θN−1] with each

θi denoting the chip-specific rotation which in the scheme

d=1is far

Tij=

⎜

⎝

8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

⎞

⎟

⎟

⎠,

Gij=

⎛

⎜

⎜

⎝

8R0,0

8R1,0

8R2,0

8R3,0

0

0

0

0

0

0

0

0

0

0

0

0

⎞

⎟

⎟

⎠

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Page 6

RIERA-PALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUP-ORTHOGONAL MULTICARRIER WIRELESS SYSTEMS1661

proposed is given by

θi= exp

?2πjcs

i

NΔ

?

,s = 0, 1,...,N − 1

with Δ being constellation dependent and is selected so as

to make 2π/Δ the minimum angle which rotates the transmit

symbol alphabet onto itself (e.g. Δ = 2 for BPSK, Δ = 4

for MQAM). Table I lists the characteristics of the different

pairwise error classes for Walsh-Hadamard spreading and its

rotated version for a group size of four BPSK symbols (K =

N = 4). In contrast to conventional Walsh-Hadamard which

attains˜Dmin= 1, for rotated Walsh-Hadamard spreading, all

classes are characterized by Dc =˜Dmin = 4. Taking into

account the asymptotic BER expression in (11), this indicates

that while using conventional Walsh-Hadamard spreading no

diversity gain will be achieved, the rotated spreading has the

potential (depending on the channel correlation matrix R)

to attain a diversity gain equal to the number of employed

subcarriers. Notice that, logically, in both cases there are the

same number of different pairwise errors (240 = 42×42−42).

By using an algebraic result [19] stating that given a

full rank diagonal matrix X and an arbitrary matrix Y ,

rank

= rank(Y ), the choice of an appropriate set

of rotated codes guarantees that for all PEPs Tij(see (5)) has

full rank, which in turn ensures that, for all classes, the rank

of Gcis given by the rank of the channel correlation matrix.

?

XY XH?

D. Single-error performance

It is proved in [7] that, in the uplink case, the classes

defined by a single-error PEP (SEP) are one of the dominant

terms when computing the bit error probability since their

corresponding Gc matrix rank is minimum. Moreover, it is

shown that the BER can indeed be approximated by means

of SEP terms. In the downlink, assuming rotated codes are

used and since all PEP classes have a common rank, the SEP

classes play also a very important role in the BER. Conse-

quently, it is interesting to examine more carefully the factors

influencing (asymptotically)the SEP classes. In this subsection

it is shown how the performance of a GO-MC-CDM system

depends on the rank of the channel correlation matrix and

also on the product of its eigenvalues. This provides us with

design rules to choose a reasonable number of subcarriers per

group balancing performance and detection complexity. For

the classes induced by SEPs, CSEP(Dc,λc,αc), the class-

defining matrices are given by GSEP

(error assumed to be in symbol l) where E denotes the error

power, that is, E = |el

belonging to classes CSEP(Dc,λc,αc). We note that E is

solely determined by the constellation of the modulation in

use. The matrix GSEP

c

has the same eigenvalues as ER [7]

allowing the asymptotic SEP to be expressed as

c

= ED(cl)RD(cl)H

ij|2for any pair ij inducing a PEP

PSEP

asym(D,ρ,α) =(2˜D)!

2˜D!2

1

E

?D

d=1αd?D

d=1ραd

d

?Es

N0

?−˜ D

(12)

where ρ = {ρ1, ρ2, ..., ρD} denotes the set of D non-

zero distinct eigenvalues of R with corresponding multiplicity

α = {α1, α2, ..., αD} and˜D =?D

d=1αd is the rank of

the channel correlation matrix.

Equation (12) expresses one of the dominant PEPs solely as

a function of the environment characteristics. Noticeably, this

PEP performance(and hence the BER performance)is affected

not only by the rank of the channel correlation matrix R (i.e.

diversity order˜D) but it is also a function of the product of its

eigenvalues. A somewhat clearer interpretation of the effect of

the eigenvalues is obtained by representing (12) in logarithmic

form

?PSEP

= log10

2˜D!2

log10

asym(D,ρ,α)?

?

(2˜D)!

?

−

D

?

d=1

αdlog10E

− log10

D

?

d=1

(ρd)αd−˜Dlog10

?Es

N0

?

.

(13)

Clearly, (13) indicates that the SEP terms will be decaying

with respect to Es/N0with slope˜D (i.e. diversity order˜D) on

a log-scale, however it also shows that the Es/N0point from

which such diversity is achieved is affected by the eigenvalue

product of R.

The maximum attainable diversity order is given by the

number of independent paths in the channel delay profile.

If error performance is to be optimized, enough subcarriers

per group need to be allocated to ensure that rank(R)=P.

Defining the sampled channel order L as the channel delay

spread in terms of chip (sampling) periods (TcW), it is shown

in [7] that the maximum rank of R is attained by setting the

number of subcarriers per group to N = L+ 1. While this is

a valuable design rule in channels with short delay spread, in

most practical scenarios where L can be in the order of tens

or even hundreds of taps, the number of subcarriers required

to achieve full diversity would make the use of ML detection

difficult even if using efficient search strategies (i.e. sphere

decoding). Moreover, very often maximum diversity would

only be attained at unreasonably large Es/N0levels.

In order to determine the number of subcarriers worth using

in a given environment (i.e. a particular channel power delay

profile), it is useful to use as reference the characteristics of

the ideal case where all subcarriers in the group are totally

uncorrelated (iid channel). It is straightforward to see that for

the iid channel (with each subcarrier following a Rayleigh

distribution), R = IN with˜D = rank(R) = N and in this

case the correlation matrix has only one non-zero eigenvalue

ρ1= 1 with multiplicity α1= N. Moreover, it is easy to show

that, for a given rank˜D, the iid case maximizes?˜ D

results in the maximum diversity order (N) and will also lead

to the largest eigenvalue product?D

Since setting the group size to guarantee full diversity

(N = L + 1) is unfeasible for most realistic scenarios we

need to be able to measure what each additional subcarrier is

contributing in terms of diversity gain. Ideally, each additional

subcarrier should bring along an extra diversity order, that

is, an increase in rank by one as it is indeed the case for

uncorrelated channels. For correlated channels, however, this

is often not the case and therefore to choose the group size it

is useful to have some form of measure. A widely used tool

i=1ραd

i.

Therefore, for a fixed number of subcarriers, the iid channel

d=1(ρd)αdminimizing in

this way the probability of error.

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Page 7

1662 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008

TABLE I

RANKS AND NUMBER OF ENTRIES OF THE DIFFERENT PEP CLASSES FOR K = N = 4 AND BPSK MODULATION.

Walsh-Hadamard

Dc

N

14

22

22

41

43

44

Rotated Walsh-Hadamard

Dc

N

41

42

42

43

44

W(Dc,λc,αc,N)

8

32

64

64

64

8

Σ = 240

λc

[64]

W(Dc,λc,αc,N)

64

32

64

64

16

λc

[3.29 3.39 4.25 5.07]

[10.13 8.51 6.58 6.78]

[2.17 2.36 14.5 12.92]

[23.78 13.14 10.42 0.66]

[30.74 24.8 5.11 4.06]

[15.09 16.91]

[14.06 17.93]

[3.29 3.39 4.25 5.07]

[3.33 3.81 4.7 36.14]

[13.16 13.55 17.01 20.24]

Σ = 240

in principal component analysis [20] to assess the ”practical”

dimensionality of a correlation matrix is the cumulative sum

of eigenvalues (CSE) defined as

?i

For the iid channel, Φ(i) is always a discrete linearly-

increasing function with respect to the number of subcarriers

employed and it can serve as a reference against which

to measure the contribution of each subcarrier in arbitrary

channels.

As an example, suppose we are trying to determine the

appropriate group size for the models ETSI BRAN A and E.

Both models were measured across a total bandwidth of 20

MHz with a channel sampling chip period of 10 ns. Model

A is an 18-path profile corresponding to a typical indoor

environment with no line-of-sight (NLOS). This profile has

a root mean square (rms) delay of 50 ns and a maximum

delay spread of 390 ns. Model E, representative of large open

environments, has also 18 paths (also NLOS) and rms and

maximum delay spreads of 250 ns and 1760 ns, respectively.

Table II shows the CSE for channel models ETSI-BRAN

A, ETSI-BRAN E and the iid model for different number

of subcarriers chosen equispaced across a bandwidth of 20

MHz. It can be inferred from the table that when only a

few subcarriers are used per group (e.g. N = 2, 4) both

ETSI-BRAN channels behave similarly to the iid model with

each of the subcarriers contributing almost equally. When

increasing the number of subcarriers (e.g. N = 8, 16), this

no longer holds: notice how the CSE values for channel

A quickly saturate and gets farther apart from those of the

iid channel indicating that the additional subcarriers do not

contribute substantially in increasing the diversity order and

therefore, as later confirmed in the simulations, reducing the

probability of error. The same holds for channel E but to

a much lesser extent, with the departure with respect to

the iid model being more evident for N = 16 subcarriers.

These results seem to indicate that, for channel A, N=4

would provide a good compromise between performance and

detection complexity. In contrast, for channel E, N=8 would

seem a more appropriate choice to fully exploit the channel

characteristics. Notice that despite both channels having the

same number of paths and consequently equal maximum

potential diversity, the appropriate group size differs.

We conclude this section by noting another important fact to

be observed in (12): the spreading codes do not appear in the

Φ(i) =

j=1ρj

?N

j=1ρj

.

(14)

asymptotic SEP expression. Consequently, in the uplink, the

asymptotic BER performance is independent of the employed

spreading codes. Moreover, and as a consequence of the code

independence, the quasi-synchronicity constraint of the system

can be relaxed permitting the inter-user delay in a group to

exceed the CP duration without (asymptotically) performance

degradation. In the downlink scenario, spreading codes do

indeed play a role as the SEP classes are only among the

dominant BER classes when rotated codes are used. Notice

for example in Table I how when using conventional Walsh-

Hadamard codes, the SEP class (4th entry in the table) has

rank 4 whereas the dominant class will be the one whose

class-defining matrix Gc has rank 1 (1st entry in the table).

In contrast, when using rotated Walsh-Hadamard codes, all

classes have the same rank for their respective Gcmatrices.

IV. SIMULATION RESULTS

Without loss of generality we assume system parameters

similar to those in use in current wireless networks such

as HIPERLAN/2 or IEEE 802.11a, that is, carrier frequency

around 5 GHz and a total bandwidth of 20 MHz. The

chip sampling period is set to Tc = 10 ns which is the

period with which the ETSI BRAN channel profiles were

sampled/modelled. The number of total subcarriers, Ntotal, is

fixed to 64 ensuring that each subcarrier undergoes frequency

flat fading. If using coventional MC-CDMA/CDM, such a

system could be used to code-multiplex up to 64 indepen-

dent symbols in either uplink or downlink. Walsh-Hadamard

spreading codes (with or without rotation) are employed for

the symbol multiplexing. The results shown in this section

are for fully loaded groups, that is, the number of symbols

transmitted in a group equals the number of subcarriers in

the group. Perfect channel state information is assumed at

the receiver. We have used channel profiles ETSI BRAN A

and E to check the analytical results and design rules derived

in previous sections. These two channels, both having 18

paths correspond to very different environments and serve to

illustrate the different design choices. Since profiles A and

E have maximum delay spreads of 390 ns and 1760 ns,

respectively, 40 and 177 subcarriers would in principle be

required to guarantee full diversity [7].

Figures 2 and 3 show the analytical and simulated BERs in

the uplink scenario for channels A and E, respectively, when

using different number of subcarriers per group. From both

figures it can be appreciated that the theoretical curves match

very accurately the simulation results except for scenarios with

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Page 8

RIERA-PALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUP-ORTHOGONAL MULTICARRIER WIRELESS SYSTEMS1663

TABLE II

CUMULATIVE EIGENVALUE SUM OF THE CHANNEL CORRELATION MATRIX FOR iid CHANNEL, CHANNEL A AND CHANNEL E FOR DIFFERENT NUMBER

OF SUBCARRIERS.

Channel iid

N = 4

0.2500

0.5000

0.7500

1.0000

Channel A

N = 4

0.5427

0.8115

0.9412

1.0000

Channel E

N = 4

0.3341

0.6195

0.8264

1.0000

N = 2

0.5000

1.0000

N = 8

0.1250

0.2500

0.3750

0.5000

0.6250

0.8750

1.0000

N = 16

0.0625

0.1250

0.1875

0.2500

0.3125

0.3750

0.4375

0.5000

0.5625

0.6250

0.6875

0.7500

0.8125

0.8750

0.9375

1.0000

N = 2

0.6563

1.0000

N = 8

0.5256

0.7809

0.9021

0.9575

0.9828

0.9938

0.9982

1.0000

N = 16

0.5241

0.7780

0.8986

0.9542

0.9800

0.9912

0.9961

0.9985

0.9995

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

1.0000

N = 2

0.6128

1.0000

N = 8

0.2483

0.4354

0.5968

0.7500

0.8648

0.9443

0.9904

1.0000

N = 16

0.1730

0.3321

0.4665

0.5890

0.6830

0.7638

0.8392

0.9060

0.9555

0.9868

0.9939

0.9999

0.9999

0.9999

0.9999

1.0000

05101520

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Pb

N=1

N=2

N=4

N=8

N=16

Lines: analytical

Circles: simulation

Fig. 2.

using different group sizes.

Analytical and simulated BER for uplink ETSI BRAN A model

a large number of users at very low SNRs, where it is well

known that the union bound tends to overestimate the true

probabiliy of error. Another important point to note is the

clear benefit of multicarrier transmission as it can be seen

how increasing the number of subcarriers per group greatly

reduces the probability of error. Nevertheless, notice that in

the case of channel A, while enlarging the group size up to 4

subcarriers per group brings along clear benefits from a BER

point of view, further increasing the group size would be far

more questionable since the gain achieved is greatly reduced

whereas decoding complexity increases substantially (even

when using sphere decoding techniques). Certainly, groups of

16 subcarriers would not make any sense since the gain with

respect to 8 subcarrier groups is negligible. The situation is

different for channel E where it can be seen how in this case,

there are clear gains up to 8 subcarrier groups and even for

groups with 16 subcarriers the gain is significant. In view of

these results, and in spite of both channels having the same

05101520

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Pb

N=16

N=2

N=4

N=8

N=1

Lines:analytical

Circles: simulation

Fig. 3. Analytical and simulated BER for uplink ETSI BRAN E model using

different group sizes.

number of paths and therefore the same maximum potential

diversity, a good balance between performance and complexity

would result in groups of 4-8 subcarriers in the case of channel

A and 8-16 subcarriers for channel E. These results agree

with the predictions anticipated in Section III-D (see Table II)

confirming the usefulness of the CSE in dimensioning group-

orthogonal multicarrier systems.

Figure 4 shows results for channel A (uplink) for a fixed

group size of 4 subcarriers/group when using different modu-

lation schemes (BPSK, 4-QAM and 16-QAM). Again, it can

be observed that simulations match very well the theoretical

results proving that the derived BER expressions are valid for

high-order constellations.

The asymptotic indepedence with respect to the spreading

codes in the case of the uplink is illustrated in Fig. 5 (Channel

A). This figure presents results for groups of 2, 4 and 8

subcarriers and it can be appreciated how as the SNR becomes

large, the BERs corresponding to systems with and without

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Page 9

1664IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008

05 10 1520 25 30

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Es/N0 (dB)

Pb

16−QAM

4−QAM

BPSK

Lines: analytical

Circles: simulation

Fig. 4.

using different modulations. N=4 with 4 users/group.

Analytical and simulated BER for uplink ETSI BRAN A model

05101520

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Pb

Lines: analytical (no spreading)

Stars: simulation (no spreading)

Circles: simulation (spreading)

N=2

N=4

N=8

Fig. 5. Analytical and simulated BER for uplink ETSI BRAN A model with

and without spreading.

spreading code become closer. Notice how even at low SNRs

the spreading code does not seem to have much influence.

Figures 6 and 7 show the performance of group-orthogonal

transmission in the downlink when using non-rotated and

rotated codes, respectively. The good match between analytical

and simulated results confirms that the derived PEP/BER

equations are also applicable to the downlink scenario. As in

the uplink, the benefits of multicarrier transmission in terms

of BER reduction are evident. Comparing the results in Figs. 6

and 7 the improvement due to the use of rotated codes can

be appreciated: whereas when using Walsh-Hadamard codes,

the asympotic BER curves for all group sizes asymptotically

decay with slope 1 (i.e. no diversity), when using rotated

Walsh-Hadamard codes the diversity is in line with the number

of subcarriers employed and approaches the one attained

in the uplink scenario. In the case of rotated codes it can

be appreciated how, for practical error rates, increasing the

05 101520

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Pb

N=1

N=2

N=4

N=8

N=16

Lines: analyical

Circles: simulation

Fig. 6.

using different group sizes. Walsh-Hadamard (non-rotated) codes.

Analytical and simulated BER for downlink ETSI BRAN E model

05101520

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/N0 (dB)

Pb

N=1

N=2

N=4

N=8

N=16

Lines: analytical

Circles: simulation N=1,2,4,8

Stars: simulation N=16

Fig. 7.

using different group sizes. Rotated Walsh-Hadamard codes.

Analytical and simulated BER for downlink ETSI BRAN E model

number of subcarriers from 8 to 16 does not improve the

BER in any significant way (for clarity, simulation results for

N = 16 are depicted by stars in this figure). Recall that to

ensure full diversity in this channel (order 18), 40 subcarriers

are required, however, it is clear that for the practical range

of BER, 8 subcarriers are enough to minimise the probability

of error. In fact, further increasing the number of subcarriers

may induce a larger probability of error (in this BER range)

due to the increased level of interference.

V. CONCLUSIONS

We have presented a unified BER analysis of group-

orthogonal multicarrier systems using ML detection encom-

passing uplink and downlink transmission. By focusing on

the asymptotic case of large SNR, the relevant parameters

affecting the BER have been identified. It has been shown that

an adequate group size balancing performance and detection

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Page 10

RIERA-PALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUP-ORTHOGONAL MULTICARRIER WIRELESS SYSTEMS 1665

complexity can be made based on the eigenvalues of the chan-

nel correlation matrix. The spreading codes have been shown

to have very different influence in the two link directions:

whereas in the uplink their effect is almost negligible, a good

code choice (i.e rotated codes) in the downlink is crucial to

maximise the diversity order. Simulation results using realistic

system parameters and channel models have shown the derived

analytical expressions to be accurate.

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[16] G. Femenias, “BER performance of linear STBC from orthogonal

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[17] S. Amari and R. Misra, “Closed-form expressions for distribution of

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SIAM,

Felip Riera-Palou received the MS degree in

Computer Engineering from the University of the

Balearic Islands (UIB), (Palma de Mallorca, Spain)

in 1997, the MSc and PhD degrees in Communi-

cation Engineering from the University of Bradford

(Bradford, UK) in 1998 and 2002, respectively, and

the MSc degree in Statistics from the University

of Sheffield (Sheffield, UK) in 2006. From May

2002 to March 2005, he was with Philips Research

Laboratories (Eindhoven, The Netherlands) first as a

postdoctoral fellow (Marie Curie program, European

Union) and later as a member of technical staff. In April 2005 he became a

research associate (Ramon y Cajal program, Spanish Ministry of Science and

Education) in the Mobile Communications Group of the Dept. of Mathematics

and Informatics at UIB where he is researching techniques suitable for future

wireless systems.

Guillem Femenias was born in Petra, Spain, in

1963. He received both the Telecommunication En-

gineer degree and the Ph.D. degree in Telecom-

munications from the Universitat Polit` ecnica de

Catalunya (UPC), Spain, in 1987 and 1991, respec-

tively.

From 1987 to 1994, he was a Researcher at UPC,

where he became an Associate Professor in 1990.

Since 1995 he has been in an Associate Professor

position at the Department of Mathematics and

Informatics of the Universitat de les Illes Balears

(UIB), Spain. His current research interests and activities span the fields of

digital communications theory and wireless personal communication systems,

with particular emphasis on MIMO cross-layer design in radio resource

management strategies applied to fourth generation systems.

Dr. Femenias has been the Project Manager of projects ARAMIS,

DREAMS, DARWIN and MARIMBA, funded by the Spanish and Balearic

Islands goverments. In the past, he was also involved in some european

projects (ATDMA, CODIT, COST). Dr. Femenias was a co-recipient of the

Best Paper Award at the IFIP International Conference on Personal Wireless

Communications 2007.

Jaume Ramis Bibiloni received the Engineer of

Telecommunication degree from the Polytechnic

University of Catalonia (UPC) in 1997. Since 2002

he has held several positions at the Department

of Mathematics and Informatics of the University

of the Balearic Islands (UIB) and at this moment

he is an Associate Professor of this Department

where he is currently pursuing the PhD degree.

His current research interests are wireless personal

communication systems, with particular emphasis on

third generation systems and beyond. He is author

of articles in international journals on this topic as well as of communica-

tions to international conferences. He is collaborating on projects concerned

with Wireless LAN (WLAN) funded by the Spanish and Balearic Islands

governments.

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