# System Level Evaluation of Innovative Coded MIMO-OFDM Systems for Broadcasting Digital TV

Authors: Y. Nasser; J.-F. Hélard; M. Crussière
Journal: International Journal of Digital Multimedia Broadcasting
 (2008)

---

## Abstract

Single-frequency networks (SFNs) for broadcasting digital TV is a topic of theoretical and practical interest for future broadcasting systems. Although progress has been made in the characterization of its description, there are still considerable gaps in its deployment with MIMO technique. The contribution of this paper is multifold. First, we investigate the possibility of applying a space-time (ST) encoder between the antennas of two sites in SFN. Then, we introduce a 3D space-time-space block code for future terrestrial digital TV in SFN architecture. The proposed 3D code is based on a double-layer structure designed for intercell and intracell space time-coded transmissions. Eventually, we propose to adapt a technique called effective exponential signal-to-noise ratio (SNR) mapping (EESM) to predict the bit error rate (BER) at the output of the channel decoder in the MIMO systems. The EESM technique as well as the simulations results will be used to doubly check the efficiency of our 3D code. This efficiency is obtained for equal and unequal received powers whatever
is the location of the receiver by adequately combining ST codes. The 3D code is then a very promising candidate for SFN architecture with MIMO transmission.


---

## Body

## 1. Introduction

Broadcasting digital
TV  is currently an area  of intensive development and standardisation
activities. The terrestrial broadcasting is the most challenging transmission
system among the existing radio diffusion systems due to the presence of strong
echoes.Technically, single-frequency
networks (SFNs) [1] present great advantages by transmitting lower power
at various sites throughout the coverage area. In an SFN, the different
antennas transmit the same signal at the same moment on the same carrier
frequency. The existing SFN architectures are achieved in a single-input single-output
(SISO) system since their deployment is very simple due to the use of one
transmitting antenna by site. However, due to the increase of client services
demand, it is desirable to deploy SFN with new MIMO techniques which ensure high
spectrum efficiency as well as high diversity gain. The MIMO technique combined
with the orthogonal frequency division multiplexing (OFDM) technique is pursued
as a potential candidate for future generations of terrestrial portable and
mobile digital video broadcasting related to DVB-T2 and DVB-NGH proposals. Actually,
one of the main research topics concerns the optimisation of the MIMO-OFDM schemes
in order to obtain high-spectrum efficiency for high definition television
(HDTV) services. In the literature, there are few studies on the SFN with MIMO
transmission. The authors of [2] propose a new SFN model to increase the diversity
gain in MIMO SFN architecture. In [3], an array antenna receiver using a maximum ratio
combining technique is proposed to improve the system performance of the SFN
transmission. This lack of studies on this original idea motivates our work to
extend the application of the MIMO-OFDM transmission to the SFN architecture.The
optimisation of the MIMO-OFDM schemes in the SFN is highly desirable to be led in
terms of the bit error rate (BER) after channel decoding. However, the
optimisation of the MIMO-OFDM systems by simulations is time consuming. Thus,
it is very important to accurately abstract the system level BER performance
into analytical expression. Moreover, the system level performance abstraction
should take into account the different transmission conditions, that is,
modulation and coding scheme (MCS), synchronization errors, channel fading, and
so forth.This paper
presents a complete study on the optimisation of the MIMO-OFDM schemes for SFN
architectures. The optimisation is double checked analytically and by
simulations. This work has been carried out within the framework of a new
European CELTIC project calledBroadcast
for 21st Century (B21C) project [4] and constitutes an extension of some previous works [5–7]. In this paper, we propose a 3D MIMO-OFDM scheme
taking advantage of the particular characteristics of an SFN. More precisely, the
contribution of this work is multifold. First of all, we investigate the
possibility of applying a space-time block code (STBC) encoder between the antennas
of two sites in SFN architecture. Secondly, using an iterative receiver, a
generalized framework is proposed for modelling the effect of unbalanced powers
received from different transmitting antennas in MIMO-OFDM systems. This is a
critical problem in SFN with mobile and portable reception. Another
contribution of this work is the proposal of a new 3D space-time-space (STS) block
code for SFN environment. The use of a second space dimension in the STS code will
be justified as being particularly adapted and efficient in the case of SFN
transmission. The proposed code is based on a double-level construction of ST
coding resulting from the combination of two coding schemes: the intercell ST
coding and the intracell ST coding. Eventually, we propose in this paper to adapt
a technique, initially used for OFDM systems, to predict the BER at the output
of the channel decoders of the MIMO-OFDM systems. This technique, called
exponential effective SNR mapping (EESM) [7], is empiric but has been validated within the 3GPP
project for the OFDM study item [8]. It provides a reliable link between link
level simulations and system level simulations. In our contribution, we show that
the EESM technique is independent of the MIMO scheme and of the power imbalance
at the receiving side. It depends only on the modulation and coding scheme
(MCS).This paper is
structured as follows. Section3 describes the architecture of an SFN in which
MIMO techniques are used. In Section 4, we present the transmission system
model. Section 5 presents the receiving model with iterative receiver. In
Section
6, we discuss the construction of different STBC schemes considered in this
paper and we describe our proposed 3D code for MIMO-OFDM transmission. In Section
7, we adapt an accurate abstraction of the system level BER performance,
initially proposed for the OFDM systems, to the MIMO-OFDM systems. In
Section 8,
we double check the efficiency of the proposed 3D code using the BER
abstraction method described in Section 7. Conclusions are drawn in
Section 8.


## 2. MIMO Systems in SFN

In this paper,
we propose to apply a MIMO communication scheme between the antennas located in
the different sites of an SFN architecture. Such a system could be implemented
usingM
T transmit antennas (Tx) by site as shown in
Figure 1. Without loss of generality, we will consider in our
study the transmission behaviour of two neighbouring cells using a total of (
2
×
M
T
)
Tx and M
R receive antennas (Rx). The extension of our
study to more sites could be adequately adapted.Figure 1
SFN with unequal received powers.Classically,
in SFN architectures, the different antennas transmit at the same moment the
same signal on the same frequency. For the SFN to work properly,  the resulting delay
spreadτ
max of the different received signals must be less
than the duration of the guard interval (GI) time inserted at the beginning of
each OFDM symbol.As a starting
point, let us assume that each site holds one antenna and that the receiver
receives signals from both antennas. In the case of an SFN, the time offset
between the signals received from each site antennas could be seen as a
superposition of the time offset between transmitters’ signals (the signal time
delay between the transmitting antennas) and the signal time offset between each
transmitter and the receiver. The first offset is generally negligible since
the transmitters are synchronized with an ultrastable reference like the global
positioning system (GPS). The second offset could be seen as follows. When the
mobile terminal (MT) moves within one cell, it receives signal from its own
cell antenna but also from the neighbouring cell antenna. Since the MT is not
equidistant to both antennas, the signal received from each one will be delayed
according to the position of the MT. This results into a delayΔ
τ between the two signals received from both
antennas or equivalently between the channel impulse responses (CIRs) between the
transmitters and the receiver. The delays are directly related to the distances
between the transmitters and the receiver and thus to the signal strength ratio
at the receiver. Assuming an equal transmitted power P
0 at each antenna, the received power from the ith antenna is(1)
P
i
=
P
0
d
i
α
, where d
i is the distance between the receiver and
the ith transmitter and α is the
propagation constant which depends on the transmission environment.The delay of
each CIR between theith transmitter
and the receiver is(2)
τ
i
=
d
i
c
, where c is the light velocity.Without loss
of generality, let us assume that the first transmitter site is the reference
site. Substitutingd
i from (2) in (1), the CIR delay of the
ith link (i.e., between the ith
transmitter and the receiver) with respect to the reference antenna can be
expressed by(3)
Δ
τ
i
=
τ
i
−
τ
1
=
(
10
−
β
i
/
10
α
−
1
)
d
1
c
,
where d
1 is the distance between the reference transmitter (first one) and
the receiver; and β
i is the received power difference (expressed in
dB) between the signal received from the reference site and the signal received
from the ith transmitter. It is given
by(4)
β
i
[
dB
]
=
−
10
⋅
α
⋅
log
10
(
d
i
d
1
)
.In the sequel,
we will assume that the power received from the reference antenna is equal to 0 dB and the distanced
i is greater than d
1 whatever i
>
1.
It is a real situation where the MT is closer to its own cell antenna than to
the other antennas. In this  case,  β
i is neither than the power attenuation
factor between the ith transmitter
and the MT. As a consequence, the transmission model becomes equivalent to a
system with unbalanced powers received from each site antennas.
Figure 2 shows an example of the relation between the power
attenuation factor β and the CIR
delay of the ith link with respect to the reference antenna signal.Figure 2
SFN with unequal received powers.If we now
consider that the number of Tx in one site is greater than one (i.e.,M
T
>
1), the choice of an adequate MIMO scheme
should then be based on this imbalance. Moreover, it should be adequate for
intercell environment, (i.e., between antennas signals of each site) and
intracell environment (i.e., between antennas signals in each site). Furthermore,
it should be chosen adequately to cope with equal and unequal received powers. This
will be the subject of Section 6 where we propose a 3D STS code adapted to such
situations.We note that
in this paper we consider independent CIRs with the dominant problem of the SFN
architecture, that is, the problem of the CIR delays and the power loss. However,
in real situations, other problems like CIRs correlations should be considered
also. The reader may refer to [9] and the references therein for more details.


## 3. Transmission Model

In this
section, we describe the transmission model of the double-layer STBC constructed
between the antennas of the different sites. The double layer proposed here has
to cope with the equal and unequal received powers. The first layer in our
proposed code corresponds to the intercell ST coding while the second
corresponds to the intracell ST coding.Figure3 depicts the transmitter modules at each site. Information
bits b
k are first channel encoded, randomly
interleaved, and fed to a quadrature amplitude modulation (QAM) module. We
recall that we restrict our study to two sites only and the generalisation
could be done in different forms. Therefore, the SFN transmission system
involving the two sites (described in Figure 1) could be seen as a double-layer scheme in the
space domain. The first layer is seen between the 2 sites separated by D km. The
second layer is seen between the antennas separated by
d meters
within one site. For the first layer, a space time block code (STBC) scheme is
applied between the two signals transmitted by each site antennas. In the
second layer, we use a second STBC encoder for each subset of M
T signals transmitted from
the same site. For the first layer (resp., the second layer), the STBC encoder
takes L (resp., M) sets of data complex symbols and transforms them into a (2,
U) (resp., (
M
T
,
V
)) output matrix according to the STBC scheme.
This output is then fed to 2
×
M
T OFDM modulators, each using N
c subcarriers. In order to have a fair analysis
and comparison between different STBC codes, the signal power at the output of
the ST encoder is normalized by
2
×
M
T.Figure 3
MIMO-OFDM transmitter.The double-layer
encoding matrix of the proposed code is described by(5)
X
(
1
)
=
(
X
11
(
2
)
⋯
X
1
U
(
2
)
X
21
(
2
)
⋯
X
2
U
(
2
)
)
,
X
p
q
(
2
)
=
(
f
p
q
,
11
(
s
1
,
…
s
M
)
…
f
p
q
,
1
V
(
s
1
,
…
s
M
)
⋮
⋱
⋮
f
p
q
,
M
T
1
(
s
1
,
…
s
M
)
⋯
f
p
q
,
M
T
V
(
s
1
,
…
s
M
)
)
.   In (5), the superscript indicates the layer, f
p
q
,
i
t
(
s
1
,
…
s
M
) is a function of the input complex symbols s
m and depends on the STBC encoder scheme. The
subscripts p and q are such that p
=
1
,
2 and  q
=
1
,
…
,
U.
They reflect, respectively, the STBC encoder input size at each layer. The time dimension of the resulting 3D code
is equal to U
×
V and the resulting coding rate is R
=
(
L
×
M
)
/
(
U
×
V
).In order to
simplify the transmission model, the double-layer encoding matrix given in (5) will be represented by X
=
[
x
i
,
t
]
, where x
i
,
t
(
i
=
1
,
…
,
2
×
M
T
;
t
=
1
,
…
,
U
×
V
) is the output of the double-layer STBC encoder
on a given subcarrier  n.
In other words, the layers construction is transparent from the transmission
model viewpoint. Moreover, we set Q
=
L
×
M vas the number of the complex symbols at the
input of the double-layer STBC encoder and we set T
=
U
×
V as the number of the corresponding output
symbols. The ST coding rate is then R
=
Q
/
T.


## 4. Iterative STBC Receiver

### 4.1. Receiving Model

We assume that
the transmitter and the receiver are perfectly synchronised. Moreover, we
assume perfect channel state information (CSI) at the receiver. In this paper,
the transmission is described in frequency domain for simplicity reasons.
However, in real scenario, the signal is transferred to the time domain and
cyclic prefix (CP) insertion operations are achieved at the transmitting side.
Reciprocal operations are done at the receiving side. The signal  received on
the subcarriern by the antenna j is a superposition of the transmitted
signal by the different antennas  multiplied by the channel coefficients h
j
,
i
[
n
] (“i”
is the index of the transmitting antenna) to which additive white Gaussian
noise (AWGN) is added. It is given by (6)
y
j
,
t
[
n
]
=
∑
i
=
1
2
M
T
P
i
h
j
,
i
[
n
]
x
i
,
t
[
n
]
+
w
j
,
t
[
n
]
, where y
j
,
t
[
n
] is the signal received on the nth subcarrier by the
jth receiving antenna during the tth OFDM symbol period. h
j
,
i
[
n
] is the frequency channel coefficient assumed
to be constant during T symbol
durations, x
i
,
t
[
n
] is the signal transmitted by the ith antenna, and w
j
,
t
[
n
] is the additive AWGN with zero mean and
variance N
0
/
2.
In the sequel, we will drop the subcarrier index n for simplicity. By introducing an equivalent receive matrix Y
∈
ℂ
M
R
×
T whose elements are the complex received
symbols expressed in (6), we can write the received signal on the
nth subcarrier on all receiving antennas
as(7)
Y
=
H
P
X
+
W
, where H is the (
M
R
,
2
M
T
) channel matrix whose components are the
coefficients h
j
,
i
,
P is a (
2
M
T
,
2
M
T
) diagonal matrix containing the signal
magnitudes P
i
,
X is a (
2
M
T
,
T
) complex matrix containing the transmitted
symbols x
i
[
t
]. W is a (
M
R
,
,
T
) complex matrix corresponding to the AWGN.Let us now
describe the transmission link with a general model independently of the ST
coding scheme. We separate the real and imaginary parts of the complex symbols input
vectors
{
s
q
:
q
=
1
,
…
,
Q
} of the outputs X of the double-layer ST encoder as well as
those of the channel matrix 						H, and the received signal
Y.
Let s
q
,
R and s
q
,
𝒥 be the real and imaginary parts of
s
q.
The main parameters of the double code are given by its dispersion matrices U
q and V
q corresponding (not equal) to the real and
imaginary parts of X, respectively. With these notations, X is given by(8)
X
=
∑
q
=
1
Q
(
s
q
,
ℜ
U
q
+
j
s
q
,
ℑ
V
q
)
,
where U
q is the dispersion matrix having the same
dimensions of X such that(9)
U
q
(
m
,
n
)
=
{
1
,
if
X
(
m
,
n
)
=
s
q
,
ℜ
,
−
1
,
if
X
(
m
,
n
)
=
−
s
q
,
ℜ
,
0
,
elsewhere,
where V
q could be deduced from (9) by replacing the real part by the imaginary part.In the sequel,
we separate the real and imaginary parts ofS
,
Y, and
X, and stack them row-wise in vectors
of dimensions (
2
Q
,
1
)
,
(
2
M
R
T
,
1
),
and (
4
M
T
T
,
1
),
respectively. We obtain(10)
s
=
[
s
1
,
ℜ
,
s
1
,
ℑ
,
…
,
s
Q
,
ℜ
,
s
Q
,
ℑ
]
T
,
y
=
[
y
1
,
ℜ
,
y
1
,
ℑ
,
…
,
y
T
,
ℜ
,
y
T
,
ℑ
,
…
,
y
M
R
T
,
ℜ
,
y
M
R
T
,
ℑ
]
T
,
x
=
[
x
(
1
,
1
)
,
ℜ
,
x
(
1
,
1
)
,
ℑ
,
…
,
x
(
2
M
T
,
T
)
,
ℜ
,
x
(
2
M
T
,
T
)
,
ℑ
]
T
, where [
⋅
]
T holds for matrix transpose.Since we use
linear ST coding, the vectorx can be written as(11)
x
=
F
⋅
s
, where F has the dimensions (
4
M
T
T
,
2
Q
) and is obtained through the dispersion
matrices of the real and imaginary parts of
s.
It is given by (12)
F
=
[
F
1
(
1
,
1
)
⋯
⋯
F
Q
(
1
,
1
)
⋮
⋮
⋮
⋮
F
1
(
1
,
T
)
⋯
⋯
F
Q
(
1
,
T
)
⋮
⋱
⋮
⋮
F
1
(
2
M
T
,
T
)
⋯
⋯
F
Q
(
2
M
T
,
T
)
]
,
where F is composed of 2
M
T blocks of 2
T rows each, that is, the data transmitted on each antenna is gathered in one
block having 2
T rows and  2
Q columns according to the ST coding scheme. The
different components of F are given by (13)
F
q
(
m
,
t
)
=
[
U
q
,
ℜ
(
m
,
t
)
−
V
q
,
ℑ
(
m
,
t
)
U
q
,
ℑ
(
m
,
t
)
V
q
,
ℜ
(
m
,
t
)
]
. As we change
the formulation of
S
,
Y, and X in (10), it can be shown that vectors x and y are related through the matrix G of dimensions (
2
M
R
T
,
4
M
T
T
) such that (14)
y
=
G
B
x
+
w
. The matrix B is a (
4
M
T
T
,
4
M
T
T
) diagonal matrix whose components are given by(15)
B
i
,
i
=
P
i
2
⋅
T
(
p
−
1
)
+
1
≤
i
≤
2
T
⋅
p
,
p
=
1
,
…
,
2
M
T
. Matrix G is composed of blocks G
j
,
i
(
j
=
1
,
…
,
M
R
;
i
=
1
,
…
,
2
M
T
) each having (
2
T
,
2
T
) elements given by
(16)
G
j
,
i
=
(
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
0
⋯
0
0
0
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
0
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
0
⋯
0
0
⋯
0
⋱
0
0
0
⋯
0
⋱
0
0
0
⋯
0
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
)
(
2
T
,
2
T
)
. Now,
substituting x from (11) in
(14), the relation between y and
s becomes(17)
y
=
G
B
F
s
+
w
=
G
e
q
s
+
w
,
where G
e
q is the equivalent channel matrix between s and y. It is assumed to be known perfectly at the receiving side.


### 4.2. STBC Detector

The detection
problem is to find the transmitted datas given the vector y. In the case of
orthogonal STBC (OSTBC), the optimal receiver is made of a concatenation of ST
decoder and channel decoder modules. In nonorthogonal STBC (NO-STBC) schemes,
there is an interelement interference (IEI) at the receiving side. The optimal
receiver in this case is based on joint ST and channel-decoding operations.
However, such receiver is extremely complex to implement and requires large
memory to store the different points of the trellis. Moreover, it could not be
implemented reasonably in one chip. Thus, the suboptimal solution proposed here
consists of an iterative receiver where the ST detector and channel decoder
exchange extrinsic information in an iterative way until the algorithm
converges. The iterative detector shown in
Figure 4 is composed of a parallel interference canceller
(PIC), a demapper which consists in computing the soft information of the
transmitted bits, that is, a log likelihood ratio (LLR) computation [10], a soft-input soft-output (SISO) decoder [11], and a soft mapper.Figure 4
Iterative receiver structure.At the first
iteration, the demapper takes the estimated symbolss
^,
the knowledge of the channel G
e
q
, and of the noise variance, and computes the
LLR values of each of the coded bits transmitted per channel use. The estimated
symbols s
^ are obtained via minimum mean square error
(MMSE) filtering according to(18)
s
^
p
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
y
, where g
p
t
r of dimension (
2
M
R
T
,
1
) is the pth
column of G
e
q
(
1
≤
p
≤
2
Q
). s
^
p
(
1
) is the estimation of the real part (p odd) or imaginary part
(p even) of s
q
(
1
≤
q
≤
Q
)
. Once the estimation of the different symbols s
q is achieved by the soft mapper at the first
iteration, we use this estimation for the next iterations process.From the
second iteration, we perform PIC operation followed by a simple inverse filtering (instead of MMSE filtering at the
first iteration):(19)
y
^
p
=
y
−
G
e
q
,
p
s
˜
p
(
1
)
,
s
^
p
(
2
)
=
1
g
p
t
r
g
p
g
p
t
r
y
^
p
, where G
e
q
,
p of dimension (
2
M
R
T
,
2
Q
−
1
) is the matrix G
e
q with its pth
column removed, s
˜
p
(
1
) of dimension (
2
Q
−
1
,
1
) is the vector s
˜ estimated by the soft mapper with its pth entry removed.


## 4.1. Receiving Model

We assume that
the transmitter and the receiver are perfectly synchronised. Moreover, we
assume perfect channel state information (CSI) at the receiver. In this paper,
the transmission is described in frequency domain for simplicity reasons.
However, in real scenario, the signal is transferred to the time domain and
cyclic prefix (CP) insertion operations are achieved at the transmitting side.
Reciprocal operations are done at the receiving side. The signal  received on
the subcarriern by the antenna j is a superposition of the transmitted
signal by the different antennas  multiplied by the channel coefficients h
j
,
i
[
n
] (“i”
is the index of the transmitting antenna) to which additive white Gaussian
noise (AWGN) is added. It is given by (6)
y
j
,
t
[
n
]
=
∑
i
=
1
2
M
T
P
i
h
j
,
i
[
n
]
x
i
,
t
[
n
]
+
w
j
,
t
[
n
]
, where y
j
,
t
[
n
] is the signal received on the nth subcarrier by the
jth receiving antenna during the tth OFDM symbol period. h
j
,
i
[
n
] is the frequency channel coefficient assumed
to be constant during T symbol
durations, x
i
,
t
[
n
] is the signal transmitted by the ith antenna, and w
j
,
t
[
n
] is the additive AWGN with zero mean and
variance N
0
/
2.
In the sequel, we will drop the subcarrier index n for simplicity. By introducing an equivalent receive matrix Y
∈
ℂ
M
R
×
T whose elements are the complex received
symbols expressed in (6), we can write the received signal on the
nth subcarrier on all receiving antennas
as(7)
Y
=
H
P
X
+
W
, where H is the (
M
R
,
2
M
T
) channel matrix whose components are the
coefficients h
j
,
i
,
P is a (
2
M
T
,
2
M
T
) diagonal matrix containing the signal
magnitudes P
i
,
X is a (
2
M
T
,
T
) complex matrix containing the transmitted
symbols x
i
[
t
]. W is a (
M
R
,
,
T
) complex matrix corresponding to the AWGN.Let us now
describe the transmission link with a general model independently of the ST
coding scheme. We separate the real and imaginary parts of the complex symbols input
vectors
{
s
q
:
q
=
1
,
…
,
Q
} of the outputs X of the double-layer ST encoder as well as
those of the channel matrix 						H, and the received signal
Y.
Let s
q
,
R and s
q
,
𝒥 be the real and imaginary parts of
s
q.
The main parameters of the double code are given by its dispersion matrices U
q and V
q corresponding (not equal) to the real and
imaginary parts of X, respectively. With these notations, X is given by(8)
X
=
∑
q
=
1
Q
(
s
q
,
ℜ
U
q
+
j
s
q
,
ℑ
V
q
)
,
where U
q is the dispersion matrix having the same
dimensions of X such that(9)
U
q
(
m
,
n
)
=
{
1
,
if
X
(
m
,
n
)
=
s
q
,
ℜ
,
−
1
,
if
X
(
m
,
n
)
=
−
s
q
,
ℜ
,
0
,
elsewhere,
where V
q could be deduced from (9) by replacing the real part by the imaginary part.In the sequel,
we separate the real and imaginary parts ofS
,
Y, and
X, and stack them row-wise in vectors
of dimensions (
2
Q
,
1
)
,
(
2
M
R
T
,
1
),
and (
4
M
T
T
,
1
),
respectively. We obtain(10)
s
=
[
s
1
,
ℜ
,
s
1
,
ℑ
,
…
,
s
Q
,
ℜ
,
s
Q
,
ℑ
]
T
,
y
=
[
y
1
,
ℜ
,
y
1
,
ℑ
,
…
,
y
T
,
ℜ
,
y
T
,
ℑ
,
…
,
y
M
R
T
,
ℜ
,
y
M
R
T
,
ℑ
]
T
,
x
=
[
x
(
1
,
1
)
,
ℜ
,
x
(
1
,
1
)
,
ℑ
,
…
,
x
(
2
M
T
,
T
)
,
ℜ
,
x
(
2
M
T
,
T
)
,
ℑ
]
T
, where [
⋅
]
T holds for matrix transpose.Since we use
linear ST coding, the vectorx can be written as(11)
x
=
F
⋅
s
, where F has the dimensions (
4
M
T
T
,
2
Q
) and is obtained through the dispersion
matrices of the real and imaginary parts of
s.
It is given by (12)
F
=
[
F
1
(
1
,
1
)
⋯
⋯
F
Q
(
1
,
1
)
⋮
⋮
⋮
⋮
F
1
(
1
,
T
)
⋯
⋯
F
Q
(
1
,
T
)
⋮
⋱
⋮
⋮
F
1
(
2
M
T
,
T
)
⋯
⋯
F
Q
(
2
M
T
,
T
)
]
,
where F is composed of 2
M
T blocks of 2
T rows each, that is, the data transmitted on each antenna is gathered in one
block having 2
T rows and  2
Q columns according to the ST coding scheme. The
different components of F are given by (13)
F
q
(
m
,
t
)
=
[
U
q
,
ℜ
(
m
,
t
)
−
V
q
,
ℑ
(
m
,
t
)
U
q
,
ℑ
(
m
,
t
)
V
q
,
ℜ
(
m
,
t
)
]
. As we change
the formulation of
S
,
Y, and X in (10), it can be shown that vectors x and y are related through the matrix G of dimensions (
2
M
R
T
,
4
M
T
T
) such that (14)
y
=
G
B
x
+
w
. The matrix B is a (
4
M
T
T
,
4
M
T
T
) diagonal matrix whose components are given by(15)
B
i
,
i
=
P
i
2
⋅
T
(
p
−
1
)
+
1
≤
i
≤
2
T
⋅
p
,
p
=
1
,
…
,
2
M
T
. Matrix G is composed of blocks G
j
,
i
(
j
=
1
,
…
,
M
R
;
i
=
1
,
…
,
2
M
T
) each having (
2
T
,
2
T
) elements given by
(16)
G
j
,
i
=
(
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
0
⋯
0
0
0
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
0
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
0
⋯
0
0
⋯
0
⋱
0
0
0
⋯
0
⋱
0
0
0
⋯
0
h
(
j
,
i
)
,
ℜ
−
h
(
j
,
i
)
,
ℑ
0
⋯
0
h
(
j
,
i
)
,
ℑ
h
(
j
,
i
)
,
ℜ
)
(
2
T
,
2
T
)
. Now,
substituting x from (11) in
(14), the relation between y and
s becomes(17)
y
=
G
B
F
s
+
w
=
G
e
q
s
+
w
,
where G
e
q is the equivalent channel matrix between s and y. It is assumed to be known perfectly at the receiving side.


## 4.2. STBC Detector

The detection
problem is to find the transmitted datas given the vector y. In the case of
orthogonal STBC (OSTBC), the optimal receiver is made of a concatenation of ST
decoder and channel decoder modules. In nonorthogonal STBC (NO-STBC) schemes,
there is an interelement interference (IEI) at the receiving side. The optimal
receiver in this case is based on joint ST and channel-decoding operations.
However, such receiver is extremely complex to implement and requires large
memory to store the different points of the trellis. Moreover, it could not be
implemented reasonably in one chip. Thus, the suboptimal solution proposed here
consists of an iterative receiver where the ST detector and channel decoder
exchange extrinsic information in an iterative way until the algorithm
converges. The iterative detector shown in
Figure 4 is composed of a parallel interference canceller
(PIC), a demapper which consists in computing the soft information of the
transmitted bits, that is, a log likelihood ratio (LLR) computation [10], a soft-input soft-output (SISO) decoder [11], and a soft mapper.Figure 4
Iterative receiver structure.At the first
iteration, the demapper takes the estimated symbolss
^,
the knowledge of the channel G
e
q
, and of the noise variance, and computes the
LLR values of each of the coded bits transmitted per channel use. The estimated
symbols s
^ are obtained via minimum mean square error
(MMSE) filtering according to(18)
s
^
p
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
y
, where g
p
t
r of dimension (
2
M
R
T
,
1
) is the pth
column of G
e
q
(
1
≤
p
≤
2
Q
). s
^
p
(
1
) is the estimation of the real part (p odd) or imaginary part
(p even) of s
q
(
1
≤
q
≤
Q
)
. Once the estimation of the different symbols s
q is achieved by the soft mapper at the first
iteration, we use this estimation for the next iterations process.From the
second iteration, we perform PIC operation followed by a simple inverse filtering (instead of MMSE filtering at the
first iteration):(19)
y
^
p
=
y
−
G
e
q
,
p
s
˜
p
(
1
)
,
s
^
p
(
2
)
=
1
g
p
t
r
g
p
g
p
t
r
y
^
p
, where G
e
q
,
p of dimension (
2
M
R
T
,
2
Q
−
1
) is the matrix G
e
q with its pth
column removed, s
˜
p
(
1
) of dimension (
2
Q
−
1
,
1
) is the vector s
˜ estimated by the soft mapper with its pth entry removed.


## 5. 3D STSBC Construction

The aim of
this section is to judiciously build the proposed double-layer 3D STS code so
that the resulting MIMO scheme behaves efficiently in an SFN context. We then
need to choose the adequate ST coding scheme to apply to each layer of our 3D
code. In the sequel, we will consider different coding schemes to apply to the
different layers. First, we will consider the well-known orthogonal Alamouti ST
coding scheme [12] for its robustness and its simplicity. In this case,
the maximum likelihood (ML) receiver is simply implemented. This code is described
by its dispersion matrix given by (20)
X
=
[
s
1
s
2
−
s
2
*
s
1
*
]
. For NO
schemes, we consider in this work the well-known space multiplexing (SM) scheme
[13]. SM is designed to maximize the rate by transmitting
symbols sequentially on  different antennas. Its coding scheme is given by (21)
X
=
[
s
1
s
2
]
tr
. Finally, we
consider the full rate and the fully diverse Golden code [14]. The Golden code is designed to maximize the rate
such that the diversity gain is preserved for an increased signal constellation
size. It is defined by (22)
X
=
1
5
[
α
(
s
1
+
θ
s
2
)
α
(
s
3
+
θ
s
4
)
μ
α
¯
(
s
3
+
θ
¯
s
4
)
α
¯
(
s
1
+
θ
¯
s
2
)
]
, where θ
=
(
1
+
5
)
/
2
,
θ
¯
=
1
−
θ
,
α
=
1
+
j
(
1
−
θ
)
,
α
¯
=
1
+
j
(
1
−
θ
¯
).
To identify the most efficient ST code, the OFDM parameters are derived from
those of a DVB-T system (see Table 1). Moreover, we have considered the possibility
to extend the size of the constellation size up to 256-QAM. The spectral
efficiencies 4 and 6 b/s/Hz are obtained for different ST schemes as shown in
Table 2. In all simulations, we assume that two Rx are
used by the MT.Table 1
Simulations parameters.
FFT size
8 K
Sampling frequency(
f
s
=
1
/
T
s
)
9.14 MHz
Guard interval (GI) duration
1
0
24
×
T
s
=
112
μ
s
RateR
c of convolutional code
1/2, 2/3, 3/4
Polynomial code generator
(
133
,
171
)
o
Channel estimation
perfect
Constellation
16-QAM, 64-QAM, 256-QAM
Spectral Efficiencies
η
=
4 and 6
b/s/HzTable 2
Different MIMO schemes and
efficiencies.
Spectral efficiency
ST scheme
ST rateR
Constellation
R
c
η
=
4
[
bit/Sec/Hz
]
Alamouti
1
64-QAM
2/3
SM
2
16-QAM
1/2
Golden
2
16-QAM
1/2
3D code
2
16-QAM
1/2
η
=
6
[
bit/Sec/Hz
]
Alamouti
1
256-QAM
3/4
SM
2
64-QAM
1/2
Golden
2
64-QAM
1/2
3D code
2
64-QAM
1/2In the simulations results given hereafter, we separate the single-layer
case and the double-layer case. For NO schemes, we show in [7] that the receiver converges after 3 iterations. This
implies an acceptable complexity as compared to the ML detection. This can be
observed with Golden code, but also with SM scheme. That is, for NO-STBC schemes,
we will present in the sequel the performances after 3 iterations only.


### 5.1. Single-Layer Case:
Inter-Cell ST Coding

In the case of
single-layer reception, we have one antenna by site. Then, the second-layer
matrixX
(
2
) in (5) resumes to one element. The multiple-input component
of the MIMO scheme is then only obtained by the single antenna in each site.
Due to the mobility, the MT is assumed to occupy different locations and the
first-layer ST scheme must be efficient face to unequal received powers. For
equal received powers, we assume that the powers of matrix B in (14) are equal to 0 dB.
Figure 5 presents a simple case of MIMO transmission in SFN
environment using Alamouti scheme. As it is shown in this figure, the STBC
symbols are transmitted through the set of one antenna in each site using
Alamouti coding.Figure 5
Alamouti scheme in SFN environment.Figure6 shows
the required E
b
/
N
0 to obtain a BER equal to 10
−
4 for a
spectral efficiency η
=
4
b/s/Hz with a Rayleigh channel model. Since we have
one Tx antenna by site, we set β
1
=
0
dB and we change β
=
β
2.
As expected, this figure shows that the Golden code presents the best
performance when the Rx receives the same power from both sites (
i
.
e
.
,
β
1
=
β
2
=
0
dB
).
When  β
2
decreases, Alamouti scheme is very
efficient and presents a maximum loss of only 3 dB in terms of required E
b
/
N
0 with
respect to equal received powers case.
Indeed, for very small values of β,
the transmission scenario  becomes equivalent to a scenario with one transmitting
antenna. In this figure, the value β
lim
=
−
6.2
dB presents the power imbalance limit where the
Alamouti and the Golden code schemes have the same performance at a BER
=
1
0
−
4.
It is also straightforward to note that
the SISO transmission in existing SFN presents the worst results when it is
compared to the distributed MIMO technique.Figure 6
RequiredE
b
/
N
0 to obtain a BER
=
1
0
−
4,
single-layer case, η
=
4
b/s/Hz.


### 5.2. Double-Layer Case

Considering
the whole double-layer space domain construction, one ST coding scheme has to
be assigned to each layer of the proposed system. The resulting 3D STS code
should be efficient for both environments in SFN architectures. In this paper,
we restrict our study toM
T
=
2
Tx by site. We propose to construct the first
layer with Alamouti scheme, since it is the most resistant for the unequal
received powers case. In a complementary way, we propose to construct the
second layer with the Golden code since it offers the best results in the case
of equal received powers. After combination of the two space layers with time
dimension, (5) yields
(23)
X
=
1
/
5
(
α
(
s
1
+
θ
s
2
)
α
(
s
3
+
θ
s
4
)
α
(
s
5
+
θ
s
6
)
α
(
s
7
+
θ
s
8
)
j
α
¯
(
s
3
+
θ
¯
s
4
)
α
¯
(
s
1
+
θ
¯
s
2
)
j
α
¯
(
s
7
+
θ
¯
s
8
)
α
¯
(
s
5
+
θ
¯
s
6
)
−
α
*
(
s
5
*
+
θ
*
s
6
*
)
−
α
*
(
s
7
*
+
θ
*
s
8
*
)
α
*
(
s
1
*
+
θ
*
s
2
*
)
α
*
(
s
3
*
+
θ
*
s
4
*
)
j
α
*
¯
(
s
7
*
+
θ
*
¯
s
8
*
)
−
α
*
¯
(
s
5
*
+
θ
*
¯
s
6
*
)
−
j
α
*
¯
(
s
3
*
+
θ
*
¯
s
4
*
)
α
*
¯
(
s
1
*
+
θ
*
¯
s
2
*
)
)
, where  θ
=
(
1
+
5
)
/
2
,
θ
¯
=
1
−
θ
,
α
=
1
+
j
(
1
−
θ
)
,
α
¯
=
1
+
j
(
1
−
θ
¯
).Since the
distanced between the transmitting
antennas in one site is negligible with respect to the distance D
(Figure 1), the power attenuation factors in the case of our 3D
code are such that β
1
=
β
2
=
0
dB and β
=
β
3
=
β
4. Figure 7
presents an overview of the proposed 3D STSBC. Since
the received powers from each antenna in the same site are equal, we apply the
Golden code between the two signals transmitted in a given site. However, we
apply the Alamouti code between the signals transmitted by the different sites’
antennas.Figure 7
3D STS scheme in SFN environment.Figure8 shows the results in terms of required E
b
/
N
0 to obtain a BER equal to
1
0
−
4
for different values of β and 3 STBC schemes, that is, our proposed 3D code scheme, the
single-layer Alamouti scheme, and the single-layer Golden scheme. The results
obtained in this figure assume that the transmission is achieved through the
COST 207 TU-6 channel model [15]. The value β in this
figure corresponds to
β
2
for the single-layer case and to β
=
β
3
=
β
4 for our 3D code. We assume that the MT is
moving with a velocity of 10 km/h and the distance d
1 of the reference antenna is equal to 5 km. The CIRs between
different transmitters and the MT are delayed according to (3). Figure 8
shows that the proposed scheme presents the best
performance whatever
the spectral efficiency and the factor β are.
Indeed, it is optimized for SFN systems and unbalanced received powers. For β
=
−
12 dB, the proposed 3D code offers a gain equal
to 1.5 dB (resp., 3.1 dB) with respect to the Alamouti scheme for a spectral
efficiency η
=
4
b/s/Hz (resp. η
=
6
b/s/Hz). This gain is greater when it is compared to
the Golden code. The maximum loss of our code due to unbalanced received powers
is equal to 3 dB in terms of
E
b
/
N
0.
This means that it leads to a powerful code for SFN systems.Figure 8
RequiredE
b
/
N
0 to obtain a
BER
=
1
0
−
4,
double-layer case, η
=
4
b/s/Hz
,
η
=
6
b/s/Hz, TU-6 channel.Figure9 evaluates the robustness of the different schemes to the two values of MT
velocity, that is, 10 km/h and 60 km/h.
We show in this figure that the Alamouti scheme is very robust to the MT
velocity. The degradation of the Golden code might reach 1 dB in terms of
required E
b
/
N
0 at a
BER
=
1
0
−
4. Our 3D code presents an intermediate behaviour. Its degradation due to
the MT velocity, that is, to Doppler effect is about 0.2 dB only.Figure 9
RequiredE
b
/
N
0 to obtain a BER
=
1
0
−
4
,
η
=
6
b/s/Hz,
TU-6 channel, different values of MT velocity.


### 5.3. Complexity Considerations

As we have shown, the 3D code outperforms the other MIMO schemes in
different reception scenarios. Let us now compare the different MIMO schemes in
terms of complexity implementation. At this stage, different complexity points
could be evaluated. First, at the transmission side, the implementation of the Alamouti
and the Golden code schemes between different sites in SFN architecture does
not increase the complexity when it is compared to that of the SISO case.
Indeed, we just need to synchronize the transmission from both sites as it
should be already done with SFN in the SISO case. This task can be ensured by
an ultrastable reference like the GPS. However, for the 3D code, an additional
front-end RF should be used at each site. At the receiving side, the iterative receiver
used for NO schemes like the SM scheme or the Golden code is the same of that
used for the 3D code. Moreover, when compared to the ML detection, we had
showed in [7]
that the iterative receiver converges after 3
iterations only. However, the ML detection complexity increases with the MIMO
architecture size and the modulation order [16]. Thus, the proposed 3D code complexity is of the same
order of the NO codes complexity.


## 5.1. Single-Layer Case:
Inter-Cell ST Coding

In the case of
single-layer reception, we have one antenna by site. Then, the second-layer
matrixX
(
2
) in (5) resumes to one element. The multiple-input component
of the MIMO scheme is then only obtained by the single antenna in each site.
Due to the mobility, the MT is assumed to occupy different locations and the
first-layer ST scheme must be efficient face to unequal received powers. For
equal received powers, we assume that the powers of matrix B in (14) are equal to 0 dB.
Figure 5 presents a simple case of MIMO transmission in SFN
environment using Alamouti scheme. As it is shown in this figure, the STBC
symbols are transmitted through the set of one antenna in each site using
Alamouti coding.Figure 5
Alamouti scheme in SFN environment.Figure6 shows
the required E
b
/
N
0 to obtain a BER equal to 10
−
4 for a
spectral efficiency η
=
4
b/s/Hz with a Rayleigh channel model. Since we have
one Tx antenna by site, we set β
1
=
0
dB and we change β
=
β
2.
As expected, this figure shows that the Golden code presents the best
performance when the Rx receives the same power from both sites (
i
.
e
.
,
β
1
=
β
2
=
0
dB
).
When  β
2
decreases, Alamouti scheme is very
efficient and presents a maximum loss of only 3 dB in terms of required E
b
/
N
0 with
respect to equal received powers case.
Indeed, for very small values of β,
the transmission scenario  becomes equivalent to a scenario with one transmitting
antenna. In this figure, the value β
lim
=
−
6.2
dB presents the power imbalance limit where the
Alamouti and the Golden code schemes have the same performance at a BER
=
1
0
−
4.
It is also straightforward to note that
the SISO transmission in existing SFN presents the worst results when it is
compared to the distributed MIMO technique.Figure 6
RequiredE
b
/
N
0 to obtain a BER
=
1
0
−
4,
single-layer case, η
=
4
b/s/Hz.


## 5.2. Double-Layer Case

Considering
the whole double-layer space domain construction, one ST coding scheme has to
be assigned to each layer of the proposed system. The resulting 3D STS code
should be efficient for both environments in SFN architectures. In this paper,
we restrict our study toM
T
=
2
Tx by site. We propose to construct the first
layer with Alamouti scheme, since it is the most resistant for the unequal
received powers case. In a complementary way, we propose to construct the
second layer with the Golden code since it offers the best results in the case
of equal received powers. After combination of the two space layers with time
dimension, (5) yields
(23)
X
=
1
/
5
(
α
(
s
1
+
θ
s
2
)
α
(
s
3
+
θ
s
4
)
α
(
s
5
+
θ
s
6
)
α
(
s
7
+
θ
s
8
)
j
α
¯
(
s
3
+
θ
¯
s
4
)
α
¯
(
s
1
+
θ
¯
s
2
)
j
α
¯
(
s
7
+
θ
¯
s
8
)
α
¯
(
s
5
+
θ
¯
s
6
)
−
α
*
(
s
5
*
+
θ
*
s
6
*
)
−
α
*
(
s
7
*
+
θ
*
s
8
*
)
α
*
(
s
1
*
+
θ
*
s
2
*
)
α
*
(
s
3
*
+
θ
*
s
4
*
)
j
α
*
¯
(
s
7
*
+
θ
*
¯
s
8
*
)
−
α
*
¯
(
s
5
*
+
θ
*
¯
s
6
*
)
−
j
α
*
¯
(
s
3
*
+
θ
*
¯
s
4
*
)
α
*
¯
(
s
1
*
+
θ
*
¯
s
2
*
)
)
, where  θ
=
(
1
+
5
)
/
2
,
θ
¯
=
1
−
θ
,
α
=
1
+
j
(
1
−
θ
)
,
α
¯
=
1
+
j
(
1
−
θ
¯
).Since the
distanced between the transmitting
antennas in one site is negligible with respect to the distance D
(Figure 1), the power attenuation factors in the case of our 3D
code are such that β
1
=
β
2
=
0
dB and β
=
β
3
=
β
4. Figure 7
presents an overview of the proposed 3D STSBC. Since
the received powers from each antenna in the same site are equal, we apply the
Golden code between the two signals transmitted in a given site. However, we
apply the Alamouti code between the signals transmitted by the different sites’
antennas.Figure 7
3D STS scheme in SFN environment.Figure8 shows the results in terms of required E
b
/
N
0 to obtain a BER equal to
1
0
−
4
for different values of β and 3 STBC schemes, that is, our proposed 3D code scheme, the
single-layer Alamouti scheme, and the single-layer Golden scheme. The results
obtained in this figure assume that the transmission is achieved through the
COST 207 TU-6 channel model [15]. The value β in this
figure corresponds to
β
2
for the single-layer case and to β
=
β
3
=
β
4 for our 3D code. We assume that the MT is
moving with a velocity of 10 km/h and the distance d
1 of the reference antenna is equal to 5 km. The CIRs between
different transmitters and the MT are delayed according to (3). Figure 8
shows that the proposed scheme presents the best
performance whatever
the spectral efficiency and the factor β are.
Indeed, it is optimized for SFN systems and unbalanced received powers. For β
=
−
12 dB, the proposed 3D code offers a gain equal
to 1.5 dB (resp., 3.1 dB) with respect to the Alamouti scheme for a spectral
efficiency η
=
4
b/s/Hz (resp. η
=
6
b/s/Hz). This gain is greater when it is compared to
the Golden code. The maximum loss of our code due to unbalanced received powers
is equal to 3 dB in terms of
E
b
/
N
0.
This means that it leads to a powerful code for SFN systems.Figure 8
RequiredE
b
/
N
0 to obtain a
BER
=
1
0
−
4,
double-layer case, η
=
4
b/s/Hz
,
η
=
6
b/s/Hz, TU-6 channel.Figure9 evaluates the robustness of the different schemes to the two values of MT
velocity, that is, 10 km/h and 60 km/h.
We show in this figure that the Alamouti scheme is very robust to the MT
velocity. The degradation of the Golden code might reach 1 dB in terms of
required E
b
/
N
0 at a
BER
=
1
0
−
4. Our 3D code presents an intermediate behaviour. Its degradation due to
the MT velocity, that is, to Doppler effect is about 0.2 dB only.Figure 9
RequiredE
b
/
N
0 to obtain a BER
=
1
0
−
4
,
η
=
6
b/s/Hz,
TU-6 channel, different values of MT velocity.


## 5.3. Complexity Considerations

As we have shown, the 3D code outperforms the other MIMO schemes in
different reception scenarios. Let us now compare the different MIMO schemes in
terms of complexity implementation. At this stage, different complexity points
could be evaluated. First, at the transmission side, the implementation of the Alamouti
and the Golden code schemes between different sites in SFN architecture does
not increase the complexity when it is compared to that of the SISO case.
Indeed, we just need to synchronize the transmission from both sites as it
should be already done with SFN in the SISO case. This task can be ensured by
an ultrastable reference like the GPS. However, for the 3D code, an additional
front-end RF should be used at each site. At the receiving side, the iterative receiver
used for NO schemes like the SM scheme or the Golden code is the same of that
used for the 3D code. Moreover, when compared to the ML detection, we had
showed in [7]
that the iterative receiver converges after 3
iterations only. However, the ML detection complexity increases with the MIMO
architecture size and the modulation order [16]. Thus, the proposed 3D code complexity is of the same
order of the NO codes complexity.


## 6. Analytical System Level Evaluation

In the
previous section, we have proposed a new 3D STSBC for MIMO-OFDM systems in SFN
architecture. Using the system level simulations, we have showed that this new
ST code is very efficient to cope with equal and unequal received powers.
However, explicit bit level simulation of each MT in every cell of the SFN
would be forbidding time consuming. The problem becomes more noticeable when
MIMO-OFDM techniques are used in SFN architectures. As a consequence, it is
desirable to evaluate the system level performance in terms of BER without
achieving system simulation. Thus, the practical need of
an accurate abstraction of the system level simulation into analytical
evaluation highly motivates our work to achieve an analytical BER expression of
the MIMO-OFDM systems using iterative receiver.In
some studies, it has been shown that the BER at the output of the channel
decoder is directly related to the SINR at the output of the detector [8, 17, 18]. In the OFDM system, [8]
proposes a new technique called effective exponential
SINR mapping (EESM) to evaluate the BER. The technique is based on the
computation of an effective SINR derived from the different values of the
estimated SINRs on each subcarrier. The authors of [19]
propose a new method to adapt the EESM technique to
the SM transmission and an ML receiver. Their method, however, could not be
suited for our work since an iterative receiver is used and, as it will be
shown, the SNR expression is not computed for each layer.In
this paper, we propose to adapt the EESM technique to the MIMO-OFDM systems
using the iterative receiver. The first step in our work consists in computing
the SINRs expressions at the output of the detector. The
second step is to establish an accurate relationship between the different
SINRs and the coded BER through the adapted EESM technique. We note that the
SINR expressions and the predicted BER given hereafter are not specified for a
given antenna, that is, we do not separate between the different antennas like
in [19]. However, it is possible to apply our
methods for each antenna received signal.


### 6.1. SINR Evaluation

Without loss
of generality, we assume in the sequel that we are interested by thepth symbol. Using the vector-matrix
notation of Sections 4 and 5, the estimated received symbol at the first
iteration in (18) could be written in an equivalent form as(24)
s
^
p
=
I
0
(
1
)
+
I
1
(
1
)
+
I
2
(
1
)
,
I
0
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
p
s
p
,
I
1
(
1
)
=
∑
q
=
1
q
≠
p
2
Q
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
s
q
,
I
2
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
w
. In (24), I
0
(
1
) is the useful received signal, I
1
(
1
) is the IEI due to the nonorthogonality of the
considered STBC. We can easily verify that it is equal to zero for O-STBC
schemes. I
2
(
1
) is the coloured noise. The superscript ℓ in the signal expressions I
(
ℓ
) indicates the number of iteration in the
iterative process.The complex
transmitted data symbols are assumed i.i.d. having zero mean and unit variance
(the variance of the real and imaginary parts is equal to1
/
2). Due to this distribution, the SINR
expression can be deduced from (24) by(25)
SINR
=
E
{
|
I
0
(
1
)
|
2
}
E
{
|
I
1
(
1
)
|
2
}
+
E
{
|
I
2
(
1
)
|
2
}
. The
expectations values in (25) over the random data symbols are given by(26)
E
{
|
I
0
(
1
)
|
2
}
=
1
2
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
p
|
2
,
E
{
|
I
1
(
1
)
|
2
}
=
1
2
∑
q
=
1
q
≠
p
2
Q
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
|
2
,
E
{
|
I
2
(
1
)
|
2
}
=
N
0
2
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
|
2
. At the second
iteration, the estimated symbol expressed in (24) becomes more complex. It is obtained using (17)
and
(18) in (19) by(27)
s
^
p
(
2
)
=
I
0
(
2
)
+
I
1
(
2
)
+
I
2
(
2
)
,where
(28)
I
0
(
2
)
=
s
p
,
I
1
(
2
)
=
∑
q
=
1
q
≠
p
2
Q
1
g
p
t
r
g
p
g
p
t
r
[
g
q
s
q
−
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
s
˜
q
(
1
)
]
,
I
2
(
2
)
=
1
g
p
t
r
g
p
g
p
t
r
w
. For next
iterations, it is clear from (27)
that the expressions of the estimated received symbol
as well as the estimated SINR become more complex. Therefore, some
manipulations should be considered to give an analytical expression of the
SINR.Based on the
structure of the iterative receiver, we already know that the outputs of the
soft Gray mapper are complex symbols which belong to the set of constellation
points. LetI
t
(
2
)
=
I
1
(
2
)
+
I
2
(
2
)
=
s
p
(
2
)
−
I
0
(
2
) be the total interference power at the second
iteration. Then, two cases can be presented at this stage.(I) If the estimated
symbols
˜
p
(
1
) at the output of the Gray mapper is equal to
the transmitted symbol s
p,
the useful signal I
0
(
2
) in (28) is such that I
0
(
2
)
=
s
p
=
s
˜
p
(
1
) and the total interference signal at the
second iteration becomes(29)
I
t
(
2
)
=
I
1
(
2
)
+
I
2
(
2
)
=
s
p
(
2
)
−
s
p
(
1
)
. Since I
1
(
2
) and I
2
(
2
) are independent and the complex outputs of the
Gray mapper are zero mean with unit variance, the estimated SINR at the second
iteration is(30)
SINR
(
2
)
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
|
2
}
=
1
2
⋅
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
, where s
^
p
(
2
) is estimated through (19) and s
˜
p
(
1
) is the output of the soft Gray mapper at the
first iteration.(II) If the
estimated symbols
˜
p
(
1
) at the first iteration is different from the
transmitted symbol s
p,
the difference between the received signals at the first two successive
iterations yields by substituting I
0
(
2
) from (28) in (27):(31)
s
^
p
(
2
)
−
s
˜
p
(
1
)
=
s
p
+
I
1
(
2
)
+
I
2
(
2
)
−
s
˜
p
(
1
)
. Since s
˜
p
(
1
) is different from s
p in this case, and the different transmitted
symbols are i.i.d., we can verify due to the expectation operation that (32)
SINR
(
2
)
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
|
2
}
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
+
s
p
−
s
˜
p
(
1
)
|
2
}
=
E
{
|
I
0
|
2
}
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
=
1
2
⋅
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
. It is clear
from the last term of (30) and (32)
that the SINR expression at the second iteration is
simpler than that of (28). In this case, only the estimated symbols at each
iteration are used for SINR estimation, that is, we do not have to compute
complex expressions. Also, we can show that (30) and (32)
could be generalized for successive iterations. In
the next section, we will exploit our theoretical SINR model through BER
measurements at the output of the channel decoder.


### 6.2. BER Evaluation with EESM
Technique

In order to
evaluate the BER at the output of the channel decoder, we propose in this
section to adapt the EESM technique to the MIMO-OFDM context. At the first
step, we will develop analytically the EESM technique. At the second step, we
will present its application in the OFDM system. Then, we will adapt it to the
MIMO-OFDM context using the SINR expressions computed at the previous subsection.LetJ denote the packet size
in complex data symbols. In general, the data symbols in the packets are
transmitted over different resource elements (e.g., subcarriers) and, therefore,
they may experience different propagation and interference conditions. Thus,
the data symbols may have different SINR values. Let SINR be the vector of J instantaneous SINR received at the output of the
detector. The problem of determining an accurate BER prediction method comes
back to looking for a relationship such that(33)
P
e
=
f
(
SINR
)
, where  P
e   denotes the bit
error probability (BEP) and f is
the prediction
function, which should be invariant with respect to the
fading realization and
to the multipath channel model, and should be
applicable to different MCSs in a soft way, that is, by changing the values of
some generic parameters [18]. In the context of
AWGN channel, the SINR becomes SNR and it remains
constant over the packet. In this context, a direct relationship ξ exists between the
SNR and  the error probability: (34)
P
e
,
WGN
=
ξ
(
SNR
)
. The function ξ is called the mapping function. It is obtained through theoretical analysis or system level
simulation with AWGN
channel. In the general context of a fading channel, where the SINR varies, the
function f in
(33)
can be written exactly as a compound function of the AWGN
function ξ and a compression function r [8]: (35)
P
e
=
ξ
o
r
(
SINR
)
=
ξ
(
SINR
eff
)
with
SINR
eff
=
r
(
SINR
)
. The function r is referred to as the compression functionsince its role is to compress the vector
SINR of J components
into one scalar SINR
eff. The scalar  SINR
eff  is called the
effective SINR and it is
defined as the SINR which would yield the same error probability in an
equivalent AWGN channel as the associated vector SINR in a fading channel. By writing (35)
, we have merely turned the problem of determining the
evaluation function f into
the problem of determining the compression function r.In an OFDM
system, it was concluded that the key issue to accurately determine the
appropriate BER after channel decoding is to use the effective SINR in
combination with AWGN curves. The work in [8]
proposes the EESM technique which is based on the
Chernoff Union bound [18]
to find the effective SINR. The key EESM technique
expression relevant to an OFDM system is given by (36)
SINR
eff
=
−
λ
ln
(
1
N
∑
n
=
0
N
−
1
exp
(
−
SINR
[
n
]
λ
)
)
.
SINR
[
n
] is the SINR obtained over the
nth sub-carrier and λ is a
unique parameter which must be estimated from the system level simulations for
each MCS. It is estimated once by preliminary simulation for each MCS. When the  SINR
eff   is computed, it will
be used for BER prediction at the output of the channel decoder with a simple
lookup table (LUT) as shown in Figure 10. This LUT gives the BER at
the output of the channel decoder as a function of the SNR for a Gaussian
channel. It is computed analytically or by simulations. The uniqueness of λ for each
MCS is derived from the fact that the effective SINR must fulfil the
approximate relation(37)
BER
(
SINR
)
=
BER
AWGN
(
SINR
eff
)
, where BER
AWGN is the BEP for the AWGN
channel which depends only on the MCSs.Figure 10
BER prediction through EESM.In our study,
the EESM  technique must be adapted to the MIMO-OFDM system. Indeed, the
estimated received symbol at each subcarrier is a superposition of different
symbols transmitted by the different antennas on that subcarrier. Therefore,
the EESM technique will be applied on the set of Q symbols transmitted on theM
T antennas during T OFDM symbols. The effective SINR is, therefore, computed through(38)
SINR
eff
=
−
λ
ln
(
1
N
Q
∑
q
=
1
Q
∑
n
=
1
N
exp
(
−
SINR
q
[
n
]
λ
)
)
. Using the
effective SINR of (38), we are now able to evaluate the BER using the LUT as
shown in Figure 10.


## 6.1. SINR Evaluation

Without loss
of generality, we assume in the sequel that we are interested by thepth symbol. Using the vector-matrix
notation of Sections 4 and 5, the estimated received symbol at the first
iteration in (18) could be written in an equivalent form as(24)
s
^
p
=
I
0
(
1
)
+
I
1
(
1
)
+
I
2
(
1
)
,
I
0
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
p
s
p
,
I
1
(
1
)
=
∑
q
=
1
q
≠
p
2
Q
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
s
q
,
I
2
(
1
)
=
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
w
. In (24), I
0
(
1
) is the useful received signal, I
1
(
1
) is the IEI due to the nonorthogonality of the
considered STBC. We can easily verify that it is equal to zero for O-STBC
schemes. I
2
(
1
) is the coloured noise. The superscript ℓ in the signal expressions I
(
ℓ
) indicates the number of iteration in the
iterative process.The complex
transmitted data symbols are assumed i.i.d. having zero mean and unit variance
(the variance of the real and imaginary parts is equal to1
/
2). Due to this distribution, the SINR
expression can be deduced from (24) by(25)
SINR
=
E
{
|
I
0
(
1
)
|
2
}
E
{
|
I
1
(
1
)
|
2
}
+
E
{
|
I
2
(
1
)
|
2
}
. The
expectations values in (25) over the random data symbols are given by(26)
E
{
|
I
0
(
1
)
|
2
}
=
1
2
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
p
|
2
,
E
{
|
I
1
(
1
)
|
2
}
=
1
2
∑
q
=
1
q
≠
p
2
Q
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
|
2
,
E
{
|
I
2
(
1
)
|
2
}
=
N
0
2
|
g
p
t
r
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
|
2
. At the second
iteration, the estimated symbol expressed in (24) becomes more complex. It is obtained using (17)
and
(18) in (19) by(27)
s
^
p
(
2
)
=
I
0
(
2
)
+
I
1
(
2
)
+
I
2
(
2
)
,where
(28)
I
0
(
2
)
=
s
p
,
I
1
(
2
)
=
∑
q
=
1
q
≠
p
2
Q
1
g
p
t
r
g
p
g
p
t
r
[
g
q
s
q
−
(
G
e
q
⋅
G
e
q
t
r
+
σ
w
2
I
)
−
1
g
q
s
˜
q
(
1
)
]
,
I
2
(
2
)
=
1
g
p
t
r
g
p
g
p
t
r
w
. For next
iterations, it is clear from (27)
that the expressions of the estimated received symbol
as well as the estimated SINR become more complex. Therefore, some
manipulations should be considered to give an analytical expression of the
SINR.Based on the
structure of the iterative receiver, we already know that the outputs of the
soft Gray mapper are complex symbols which belong to the set of constellation
points. LetI
t
(
2
)
=
I
1
(
2
)
+
I
2
(
2
)
=
s
p
(
2
)
−
I
0
(
2
) be the total interference power at the second
iteration. Then, two cases can be presented at this stage.(I) If the estimated
symbols
˜
p
(
1
) at the output of the Gray mapper is equal to
the transmitted symbol s
p,
the useful signal I
0
(
2
) in (28) is such that I
0
(
2
)
=
s
p
=
s
˜
p
(
1
) and the total interference signal at the
second iteration becomes(29)
I
t
(
2
)
=
I
1
(
2
)
+
I
2
(
2
)
=
s
p
(
2
)
−
s
p
(
1
)
. Since I
1
(
2
) and I
2
(
2
) are independent and the complex outputs of the
Gray mapper are zero mean with unit variance, the estimated SINR at the second
iteration is(30)
SINR
(
2
)
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
|
2
}
=
1
2
⋅
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
, where s
^
p
(
2
) is estimated through (19) and s
˜
p
(
1
) is the output of the soft Gray mapper at the
first iteration.(II) If the
estimated symbols
˜
p
(
1
) at the first iteration is different from the
transmitted symbol s
p,
the difference between the received signals at the first two successive
iterations yields by substituting I
0
(
2
) from (28) in (27):(31)
s
^
p
(
2
)
−
s
˜
p
(
1
)
=
s
p
+
I
1
(
2
)
+
I
2
(
2
)
−
s
˜
p
(
1
)
. Since s
˜
p
(
1
) is different from s
p in this case, and the different transmitted
symbols are i.i.d., we can verify due to the expectation operation that (32)
SINR
(
2
)
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
|
2
}
=
E
{
|
I
0
(
2
)
|
2
}
E
{
|
I
1
(
2
)
+
I
2
(
2
)
+
s
p
−
s
˜
p
(
1
)
|
2
}
=
E
{
|
I
0
|
2
}
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
=
1
2
⋅
E
{
|
s
^
p
(
2
)
−
s
˜
p
(
1
)
|
2
}
. It is clear
from the last term of (30) and (32)
that the SINR expression at the second iteration is
simpler than that of (28). In this case, only the estimated symbols at each
iteration are used for SINR estimation, that is, we do not have to compute
complex expressions. Also, we can show that (30) and (32)
could be generalized for successive iterations. In
the next section, we will exploit our theoretical SINR model through BER
measurements at the output of the channel decoder.


## 6.2. BER Evaluation with EESM
Technique

In order to
evaluate the BER at the output of the channel decoder, we propose in this
section to adapt the EESM technique to the MIMO-OFDM context. At the first
step, we will develop analytically the EESM technique. At the second step, we
will present its application in the OFDM system. Then, we will adapt it to the
MIMO-OFDM context using the SINR expressions computed at the previous subsection.LetJ denote the packet size
in complex data symbols. In general, the data symbols in the packets are
transmitted over different resource elements (e.g., subcarriers) and, therefore,
they may experience different propagation and interference conditions. Thus,
the data symbols may have different SINR values. Let SINR be the vector of J instantaneous SINR received at the output of the
detector. The problem of determining an accurate BER prediction method comes
back to looking for a relationship such that(33)
P
e
=
f
(
SINR
)
, where  P
e   denotes the bit
error probability (BEP) and f is
the prediction
function, which should be invariant with respect to the
fading realization and
to the multipath channel model, and should be
applicable to different MCSs in a soft way, that is, by changing the values of
some generic parameters [18]. In the context of
AWGN channel, the SINR becomes SNR and it remains
constant over the packet. In this context, a direct relationship ξ exists between the
SNR and  the error probability: (34)
P
e
,
WGN
=
ξ
(
SNR
)
. The function ξ is called the mapping function. It is obtained through theoretical analysis or system level
simulation with AWGN
channel. In the general context of a fading channel, where the SINR varies, the
function f in
(33)
can be written exactly as a compound function of the AWGN
function ξ and a compression function r [8]: (35)
P
e
=
ξ
o
r
(
SINR
)
=
ξ
(
SINR
eff
)
with
SINR
eff
=
r
(
SINR
)
. The function r is referred to as the compression functionsince its role is to compress the vector
SINR of J components
into one scalar SINR
eff. The scalar  SINR
eff  is called the
effective SINR and it is
defined as the SINR which would yield the same error probability in an
equivalent AWGN channel as the associated vector SINR in a fading channel. By writing (35)
, we have merely turned the problem of determining the
evaluation function f into
the problem of determining the compression function r.In an OFDM
system, it was concluded that the key issue to accurately determine the
appropriate BER after channel decoding is to use the effective SINR in
combination with AWGN curves. The work in [8]
proposes the EESM technique which is based on the
Chernoff Union bound [18]
to find the effective SINR. The key EESM technique
expression relevant to an OFDM system is given by (36)
SINR
eff
=
−
λ
ln
(
1
N
∑
n
=
0
N
−
1
exp
(
−
SINR
[
n
]
λ
)
)
.
SINR
[
n
] is the SINR obtained over the
nth sub-carrier and λ is a
unique parameter which must be estimated from the system level simulations for
each MCS. It is estimated once by preliminary simulation for each MCS. When the  SINR
eff   is computed, it will
be used for BER prediction at the output of the channel decoder with a simple
lookup table (LUT) as shown in Figure 10. This LUT gives the BER at
the output of the channel decoder as a function of the SNR for a Gaussian
channel. It is computed analytically or by simulations. The uniqueness of λ for each
MCS is derived from the fact that the effective SINR must fulfil the
approximate relation(37)
BER
(
SINR
)
=
BER
AWGN
(
SINR
eff
)
, where BER
AWGN is the BEP for the AWGN
channel which depends only on the MCSs.Figure 10
BER prediction through EESM.In our study,
the EESM  technique must be adapted to the MIMO-OFDM system. Indeed, the
estimated received symbol at each subcarrier is a superposition of different
symbols transmitted by the different antennas on that subcarrier. Therefore,
the EESM technique will be applied on the set of Q symbols transmitted on theM
T antennas during T OFDM symbols. The effective SINR is, therefore, computed through(38)
SINR
eff
=
−
λ
ln
(
1
N
Q
∑
q
=
1
Q
∑
n
=
1
N
exp
(
−
SINR
q
[
n
]
λ
)
)
. Using the
effective SINR of (38), we are now able to evaluate the BER using the LUT as
shown in Figure 10.


## 7. Application of the EESM Technique to the Proposed 3D STS Code

In this
section, we validate through the EESM technique and the SINR analysis the
efficiency of the proposed 3D STS code. The considered simulation parameters
are the same of those given in Table1. The parameter λ is
estimated using the AWGN channel model. Its estimation is done as follows. For
a given channel model and a given MCS, we estimate the different SINRs at the
output of the detector. Therefore, we select a value of λ and we
compute the BER at the output of the channel decoder according to (38)
and Figure 10. The accurate value of λ is that
which ensures minimum error between the predicted and the simulated BERs. Once the
value of λ is decided, it does not change with the
channel or the MIMO scheme. We will show by simulations that this value depends
only on the spectral efficiency.The results
given in this section are obtained with the COST 207 TU-6 channel model. The AWGN
results used to estimate the parameterλ are obtained using Alamouti scheme since NO
schemes are not efficient with AWGN channel. They will be plotted in the same
figure of those obtained by EESM technique or by simulations, with TU-6
channel. First of all, we will validate our SINR analysis and EESM technique on
the Alamouti and the Golden code schemes. Then, it will be suited by the
validation of the efficiency of our proposed code. In all figures, the results
are given with mobile velocity equal to 10 km/h.Figure11 compares the BER obtained by simulations and the BER obtained
with the EESM technique for the Alamouti scheme, considering a spectral
efficiency η
=
6 b/s/Hz and different values of transmitted
powers. These figures show the accuracy of the proposed technique based on the
SINR analytical expression. Moreover, they show that the parameter λ is
constant (
λ
=
22
.
6 for η
=
6) and it is independent of the power imbalance
but depends on the MCSs or equivalently on the spectral efficiency. The
parameter λ is obtained by simulations. It is computed
once for a given MCS.Figure 11
Validation
of EESM technique, Alamouti scheme,η
=
6
b/s/Hz
,
λ
=
22
.
6,
TU-6 channel.Figure12 compares the BER obtained
by simulations and the BER evaluated with the EESM technique for the Golden
code scheme, a spectral efficiency η
=
6
b/s/Hz and different values of transmitted powers.
Since, compared with the parameters used in Figure 11, the spectral efficiency does not change, the
parameter λ
=
22
.
6 gives an accurate BER prediction and validates
our analytical expressions and prediction method. Again, we show that this
parameter is independent of the power imbalance.Figure 12
Validation of EESM technique,
Golden code scheme,η
=
6
b/s/Hz
,
λ
=
22.6, TU-6 channel.In Figure13, we give the same kind of results of those given in
Figure 8 by using the EESM technique. Figure 13 validates our proposed prediction method for all
considered STBC schemes in the SFN architecture. Again, it shows that parameter
λ is independent of the power imbalance and of
the STBC scheme. Moreover, it shows again the superiority of the proposed 3D
STSBC whatever the power imbalance factor β.Figure 13
Validation of EESM technique,η
=
6
b/s/Hz
,
λ
=
22.6, TU-6 channel.


## 8. Conclusion

In this paper,
a new 3D STSBC has been presented for MIMO transmission in SFN architecture
including two transmitting sites. The proposed 3D STSBC is based on a double-layer
structure defined for intercell and intracell situations by adequately
combining the Alamouti code and the Golden code schemes. We showed that our
proposed 3D STS scheme is very efficient to cope with equal and unequal received
powers in SFN scenarios whatever  the receiver position is.Moreover, we
have proposed an analytical SINR evaluation of MIMO-OFDM systems using an
iterative receiver as well as an adaptation of the EESM technique to efficiently
evaluate the BER at the output of the channel decoder. Using the EESM technique
and the analytical evaluation, we have showed again the superiority of the
proposed 3D code. It is then a very promising candidate for the broadcasting of
the future terrestrial digital TV in SFN architectures.



---

*Source: 359206-2008-10-27.xml*