On the design of uplink and downlink grouporthogonal multicarrier wireless systems
ABSTRACT Grouporthogonal multicarrier codedivision multiple access (GOMCCDMA) 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 grouporthogonal 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.

Article: A unified view of diversity in multiantennamulticarrier systems: analysis and adaptation strategies
EURASIP Journal on Wireless Communications and Networking 2012(1). · 0.54 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Optimum detection based on maximum likelihood (ML) has been recently explored within the context of linearly precoded subcarrier cooperative MIMOOFDM systems, showing that large performance gains can be obtained in comparison to an alllinear 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 MLbased 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  SourceAvailable from: Guillem Femenias[Show abstract] [Hide abstract]
ABSTRACT: This paper proposes a novel receiver structure based on soft information for linearly precoded MIMOOFDM systems. The architecture combines an MMSEbased front end with an iterative technique based on maximum likelihood detection (MLD) in a structure that exhibits two very attractive features. Firstly, it can fully exploit the diversity benefits of spreading the information symbols in the space and frequency domains by optimally estimating them. Secondly, and under the realistic assumption of the presence of a cyclic redundancy check (CRC) mechanism, the far more computationally demanding MLD component needs only be used when the MMSE front end has failed. Simulation results reveal that the MLD iterative mechanism adds only a negligible amount of computations to the simple MMSE detector while significantly improving its performance.
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1656 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 10, OCTOBER 2008
On the Design of Uplink and Downlink
GroupOrthogonal Multicarrier Wireless Systems
Felip RieraPalou, Guillem Femenias, and Jaume Ramis
Abstract—Grouporthogonal multicarrier codedivision multi
ple access (GOMCCDMA) 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 grouporthogonal 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—Grouporthogonal multicarrier codedivision
multipleaccess(GOMCCDMA),
(CDM), maximum likelihood (ML), diversity, multiuser inter
ference (MUI).
codedivisionmultiplex
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 userspecific 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, serialtoparallel (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 MCCDMA a firm candidate for
the next generation of wireless systems [3], [4], [5], [6].
Grouporthogonal MCCDMA (GOMCCDMA) has been
recently proposed in [7] as a combination of MCCDMA
and orthogonal frequency division multiple access (OFDMA)
ULTICARRIER 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 (TEC200500997/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 (email: {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 GOMCCDMA 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 (MLMUD)
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 MLMUD in combination with group
orthogonal MCCDMA since most of the detected information
would be discarded. To avoid such a waste of processing, a
mobile terminal could use one of the lowcomplexity 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 grouporthogonal modulation and ML detection in the
downlink, grouporthogonal multicarrier codedivision multi
plexing (GOMCCDM) has been recently introduced in [8]
as an extension to the MCCDM scheme proposed in [9].
In GOMCCDM 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 (MLMSD) to estimate all
the symbols simultaneously. In comparison with conventional
OFDM, GOMCCDM has the potential advantage offered by
the frequency diversity which can be fully exploited when
using ML detection.
Given the group independence within a grouporthogonal
multicarrier (GOMC) system, any performanceanalysis needs
only to consider a single group and, to all effects, the re
sults of conventional MCCDMA [5] or MCCDM [9] apply.
Nevertheless, the additional aspect that needs to be taken into
account when designing a GOMCCDMA 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,
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RIERAPALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUPORTHOGONAL 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 nearoptimum BER
performance.
The objective of this paper is to present a new unified
analysis of grouporthogonal multicarrier systems covering
both link directions, mobiletobase (uplink) and baseto
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 nonbold, 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 singlecell 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 quasisynchronously. 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 frequencyflat 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 MCCDMA system with
N subcarriers with capacity to transmit a maximum of N
codemultiplexed 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 GOMC an attractive option for multi
service multirate 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 energynormalized 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, serialtoparallel 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
quasisynchronicity 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 CDMAbased 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 nearfar 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 timevaryingand frequencyselective
with an scenariodependent 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
j2?
= 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
lth 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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .×
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
c1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .×
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
cK−1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 GOMC 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 kth 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 elementwise 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
unitpower transmitted symbols and normalized power delay
profile, the operating signaltonoise 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 MLMUD 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 leastsquares 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
RIERAPALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUPORTHOGONAL 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→ ajA) =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?
(spacedfrequencycorrelation 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) =
closedform 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
upperbound 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 GOMC 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 matrixdefining
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
WalshHadamard 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 WalshHadamard 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 GOMCCDM 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 WalshHadamard
spreading. In order to find the BER upperbound, 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 WalshHadamard 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 chipspecific 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
RIERAPALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUPORTHOGONAL 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 WalshHadamard spreading and its
rotated version for a group size of four BPSK symbols (K =
N = 4). In contrast to conventional WalshHadamard which
attains˜Dmin= 1, for rotated WalshHadamard spreading, all
classes are characterized by Dc =˜Dmin = 4. Taking into
account the asymptotic BER expression in (11), this indicates
that while using conventional WalshHadamard 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. Singleerror performance
It is proved in [7] that, in the uplink case, the classes
defined by a singleerror 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 GOMCCDM 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
ij2for 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 logscale, 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 nonzero 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.
WalshHadamard
Dc
N
14
22
22
41
43
44
Rotated WalshHadamard
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 18path profile corresponding to a typical indoor
environment with no lineofsight (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 ETSIBRAN
A, ETSIBRAN 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
ETSIBRAN 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 quasisynchronicity constraint of the system
can be relaxed permitting the interuser 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
classdefining matrix Gc has rank 1 (1st entry in the table).
In contrast, when using rotated WalshHadamard 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 MCCDMA/CDM, such a
system could be used to codemultiplex up to 64 indepen
dent symbols in either uplink or downlink. WalshHadamard
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
RIERAPALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUPORTHOGONAL 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 48 subcarriers in the case of channel
A and 816 subcarriers for channel E. These results agree
with the predictions anticipated in Section IIID (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, 4QAM and 16QAM). Again, it can
be observed that simulations match very well the theoretical
results proving that the derived BER expressions are valid for
highorder 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 grouporthogonal
transmission in the downlink when using nonrotated 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 WalshHadamard codes,
the asympotic BER curves for all group sizes asymptotically
decay with slope 1 (i.e. no diversity), when using rotated
WalshHadamard 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. WalshHadamard (nonrotated) 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 WalshHadamard 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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RIERAPALOU et al.: ON THE DESIGN OF UPLINK AND DOWNLINK GROUPORTHOGONAL 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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SIAM,
Felip RieraPalou 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 crosslayer 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 corecipient 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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