BATAILLE, P.

HAMON, F.

LECOMTE, M.

TESTARD, M.

I-Revues

Design and performances of a low cost ltered COFDM
system using -constellations and real channel estimator
P. Bataille, F. Hamon, M. Lecomte and M. Testard,
SACET, Campus de Kerlann, Ecole Louis de Broglie, 35170 Bruz, France
Abstract - In order to dene a multicarriers modem
for High Frequency high rate radio communications
we analyze new digital techniques.
The system originality comes from the using of -
constellations with a low cost soft decision  detec-
tor applied to fractionnally spaced multicarrier mod-
ulation [1]. We also demonstrate that even if the -
detector is more complex than a classical one it keeps
the advantage to easily provide a soft bit information
to a soft decoder.
1 Introduction
Most of radio channels and particularly the HF channel
are more or less time variant and frequency selective.
The frequency selectivity arises from multi-path propaga-
tion and results in InterSymbols Interferences. These ISI
can be counteracted using equalizer (DFE ...), or using
a coding multicarriers modulation system. In the latter
solution, the number of carriers is chosen to get a at
frequency response on each subchannels. Therefore each
subchannel doesn't need to be equalized.
The techniques described in this paper are designed for
multicarrier modem, therefore "equalization" is reduced
to a scalar channel gain compensation on each sub-carrier.
The time selectivity is a consequence of the Doppler eect.
It results in signal fading aecting the communication.
These signal fades can be bridged by applying diversity
techniques. Here, we propose a transmission scheme using
a modulation diversity. Contrary to time diversity, modu-
lation diversity has the advantage to not increase the sig-
nal bandwidth thus preserving the spectral eÆciency. One
disadvantage of -constellations is the complexity of the
optimal receiver. As far as we are concerned the receiver
used is a low cost sequential suboptimal detector. This de-
tector keeps the performance of the universal detector de-
creasing highly the complexity. Furthermore, we suggest
new method to provide a soft bit information from this
detector without increasing sensitively the system cost.
A multicarrier modulation scheme is much more sensitive
to frequency osets than a single-carrier system. Fre-
quency osets (due to Doppler shifts, or oscillators imper-
fections) introduce ICI (interchannel interference). The
use of a square root Nyquist pulse shapping function avoid
overlapping between sub-carriers and allows to reduce this
ICI.
In this paper, we dene a transmission system in a fading
environment using :
 ltered multicarrier modulation.
 -constellations in which order diversity is equal to
the dimension of the constellation.
 a bilinear channel estimator.
 a low cost suboptimal sequential detector which pro-
vides a soft bit decision.
The paper outline is as follows. In section II we describe
the waveform model with a focus on the -constellations,
and on the bilinear estimation. Next, in section III we
describe the soft bit decision detector and we analyze its
complexity. Finally, before concluding, we analyze in sec-
tion IV the simulation results.
2 Waveform model
2.1 Signal and channel model
In an ltered multicarriers modulation system, the trans-
mitted baseband signal is :
x(t) =
+1
X
n= 1
K 1
X
k=0
c
k
n
g(t  nT
s
)e
j2f
k
t
(1)
 M is the number of subchannel.
 T
s
is the symbol time.
 c
nk
is a complex data symbol transmitted during
time interval [n:T
s
; (n+1):T s] on the k
th
subchannel.
 g(t), the shapping function, is the square root of a
Nyquist function with a roll of factor 
 f
k
is the k
th
carrier of the base band signal. f
k
is dened as: f
k
=
kM
T
s
K
where K is the number of
carriers andM is the number of samples in a symbol
duration.
In the case of multicarrier system ltered with a Nyquist
fonction the orthogonality between sub-carrier signals is
guaranteed if the sub-carrier spectral support are disjoint:
f
k
  f
k 1
>
1+
T
s
. So the the ltering allow to reduce the
ICI but in the other hand it reduces the spectral eÆciency
with a factor
1
1+
. In our system the ltering is realise
with the polyphase lters technique.
When the number of carriers is well chosen (e.g multi-path
delay T
d
<< T
s
) each subchannel can be represented by
the Rayleigh channel model without ISI.
Thus we assume that the received complex signal of the
k
th
subchannel is :
~y
i
= ~c
i
: ~
i
+ ~n
i
(2)
where ~
i
and ~n
i
are two independent Gaussian complex
noise.
Let
^
~
i
be the channel estimate so the estimate of the co-
herent received signal is :
~z
i
=
~y
i
^
~
i
=
~c
i
: ~
i
^
~
i
+
~n
i
^
~
i
(3)
The system uses a PSAM (pilot symbol assisted modula-
tion) channel estimation. The channel estimates ^
m
result
from a bilinear interpolation of the pilot in the time and
frequency grid. The frame is so made of data and pilot
periodically inserted in the time and frequency direction.
2.2 Modulation diversity and  constella-
tion
2.2.1 Principle:
The modulation diversity denoted by L is given by the
minimal number of dierent components between two
points of the constellation. If the diversity L is equal to
the dimension N of the constellation, then one unfaded
component is enough to decide on the transmit symbol.
As we have previously noticed, the modulation diversity
has the advantage to not increase the signal power or the
signal bandwidth. This diversity is carried out from the
building of particular N-dimensional signal constellations.
The digital transmission of symbols from N-dimensional
constellation is seen by the channel as
N
2
successive
complex signal transmission. However the ML detection
is realized on N-dimensional signal. So, if we assume
diversity and interleaving on the N-component, then
the detector increases with N its chances to receive an
unfaded component.
2.2.2 The pairwise error probability on the
Rayleigh channel and -constellations:
In this section we introduce a second signicant parameter
(the rst one is the diversity) for the design of constellation
adapted to the Rayleigh channel : the product distance.
If x = (x
1
; ::; x
i
; ::x
N
) and y = (y
1
; ::; y
i
; ::; y
N
) are two
N-dimensional symbols, the product distance d

of these
symbols is dened by :
d

=
N
Y
i=1
jx
i
  y
i
j (4)
In a Rayleigh fading channel a pairwise error probability
majoration has been given by Divsalar and Simon. This
majoration depending on the signal to noise ratio (SNR)
denoted by  . At high SNR this relation become:
p(x! y)  (
4
 
)
N
1
d

(x;y)
2
(5)
where d

(x;y) =
Q
N
i=1
jx
i
  y
i
j is the product distance
between the N -dimensional symbols x and y. The last
relation supposes that x
i
6= y
i
with if1; ::; Ng. As we can
see from 5, two parameters allow to reduce the pairwise
error probability majoration :
 d

(x;y) > 1
 the N -dimension of the constellation which implic-
itly acts as a N -diversity, as the error probability
decreases with  
 N
.
Particular N-dimensional signal constellation have been
designed by [2][3] : the -constellations. In this ap-
proach the authors use algebraic tools to buildR
N
lattices
adapted to Rayleigh fading channel in term of diversity
and product distance. The generating matrix of this R
N
lattice is denoted by G.
2.3 Transmission scheme including -
constellations and polyphase ltering
In the system proposed we design -constellations apply-
ing the generating matrix to N-dimensional 32QAM sym-
bols. The transmission scheme is described on Fig. 1.
ic
iy
cn
0
cn
k-1
D
EM
U
X
M
U
X
n
m-1
x
nx
0
Filtered
OFDM
modulator
PI-constellation
symbol set
Real
Soft
ML detector
decision
α^ i PSAM
components
Real
bits/ piN
Interleaver
Mapper
estimation
Y
deinterleaver
X
components
h(t,  )τ
Filtered
OFDM
M
U
X
D
EM
U
X
demodulator
n
0
n
k-1
y
y
bits
soft bits
bilinear
Fig. 1: System transmission scheme
To take advantage of the modulation diversity, we need
to decorrelate the symbol component from the fading
channel. This independance is achieved by a real compo-
nent interleaver. At the output of the components dein-
terleaver we get the N-dimensional vector Y .
The soft decision ML detector which is described in the
next section, use the weighted metric
D(CjZ; ^) =
N
X
k=1
j^
k
j
2
jz
k
  c
k
j
2
(6)
where Z is a coherent demodulated version of Y and ^ a
N-dimensional vector estimate of the channel state.
In order to dene the numerical structure of the ltered
modulator we take a sampled version of 1 at time (nTs+
mTe).
x(nM +m) =
L 1
X
l=0
C
(nM+m)
N
n l
g(lM +m) (7)
C
(nM+m)
N
n l
=
N 1
X
k=0
c
k
n l
e
j2
k
K
(nM+m)
where C
(nM+m)
N
n l
is the inverse Discrete Fourrier trans-
form of the sequence c
k
n l
over K points. And where the
samples x(nM+m) are provide byM polyphase lter [1].
The m
th
poliphase lter is dened by: g
m
(l) = g(m+lM)
wich is a decimate version of the prototype lter g(m).
The details of the demodulator structure is described in
[1].
3 A low cost soft decision detector
adapted to  constellation
With this kind of modulation, the number M of points in
the constellation increases exponentially with the required
L diversity . Also we assume that the universal ML de-
tector yields to prohibitive complexity.
The proposed detector is based on FastML [2][7] algo-
rithm. This algorithm uses the high diversity constella-
tion following property: without disturbance one compo-
nent of the received point is suÆcient to detect the trans-
mited symbol. Therefore, the decoding criteria is xed
on only one component, the least aected by fading. The
FastML algorithm operates sequentially, by comparing the
least faded received components with the corresponding
reference symbol components arranged beforehand. This
schedule allows us to decrease the number of symbols and
components tested.
Furthermore we demonstrate in the next sections that we
can take advantage of the algorithm properties to provide
easily a soft bit information to a soft decoder.
3.1 Algorithm description
3.1.1 Algorithm principle:
Let Z = (Z
1
; :::; Z
N
) be the coherent received signal, and
fC
i
= (C
i
1
; :::; C
i
N
)g
i=1;::;M
the possible symbols set. The
components of Z are arranged such that Z
1
is the least
faded component, and Z
N
is the most faded.
All of the C
i
are put in order to :
jC
1
1
  Z
1
j < jC
2
1
  Z
1
j < ::: < jC
M
1
  Z
1
j (8)
During the maximum likelihood symbol research, the can-
didates C
i
are evaluated successively from i = 1 to M un-
til a stopping criteria activation. This criteria is activated
when:
D(Z;C
i
) < d(Z
1
; C
i+1
1
) (9)
where D(Z;C
i
) is the weigthed distance between the re-
ceived symbol Z and the reference symbol C
i
, and where
d(Z
1
; C
i+1
1
) is the partial weighted distance according to
the least faded component. Therefore the set of symbols
C
i+2
; :::; C
M
is not evaluated by the algorithm.
It is shown in [7] that the FASTML preserves the universal
detector optimality. Indeed we have :
8 i 2 (1; :::;M) D(Z;C
i
) > d(Z
1
; C
i
1
);
(8)) d(Z
1
; C
i+1
1
) < d(Z
1
; C
i+2
1
) < ::: < d(Z
1
; C
M
1
)
(9)) D(Z;C
i
) < D(Z;C
j
) 8 j 2 (i+ 1; :::;M)
Hence, the untreated symbols would not have been chosen.
3.1.2 Soft bit information supplying methods:
VFSoftML
With the high spectral eÆciency modulation, it is usually
complex to provide soft bit information. Indeed, the de-
cision rule is based on symbols that map several bits. An
exhaustive method could be the following :
Let the vectors 0 = (0
0
; :::;0
m 1
) and 1 =
(1
0
; :::;1
m 1
) where m = log
2
(M) is the number of
bits maps by one symbol. Let Cb
i
= (Cb
i
0
; :::; Cb
i
m 1
)
be the binary symbol associated to C
i
= (C
i
0
; :::; C
i
N 1
).
And (Cb
0;0
; :::;Cb
0;m 1
), (Cb
1;0
; :::;Cb
1;m 1
) two m
subsets such that:
Cb
0;j
= (Cb
i
jCb
i
j
= 0);
Cb
1;j
= (Cb
i
jCb
i
j
= 1):
As we map Cb
i
with the reference symbol C
i
, we
map the subsets (Cb
0;0
; :::;Cb
0;m 1
) with the subsets
(C
0;0
; :::;C
0;m 1
). 0
i
and 1
i
are dene such that:
0
j
= min
i
D(Z;C
i
0;j
);
1
j
= min
i
D(Z;C
i
1;j
):
Then 0
j
represent the minimal weighted distance be-
tween the received symbol and the reference symbols is-
sued from the subset Cb
0;j
. The soft bit value vector
supplied to the binary decoder is dened by:
Soft =
1 0
1 +0
:
The complexity of this exhaustive method is prohibitive
in the case of large constellation, furthermore it is not
adapted to the sequential detection algorithm.
Here after we propose a simplied soft decision detec-
tion algorithms.
The aim is to provide values to the vectors 0 and 1 with
the lower cost. The basic idea of the VFSoftML is the fol-
lowing : In the FASTML algorithm nd (with nd << M)
N-dimensional distances are calculated such that:
D
(1)
> :: > D
(i)
> ::: > D
(nd)
; (10)
where D
(1)
represents the rst calculated distance and
D
(j)
the jth calculated distance.
Furthermore, each distance D
(i)
is mapped into a binary
symbol Cb
j
by the relation (i) = j. Now, at time t =
t
i
, D
(i)
is the minimal distance calculated during the
algorithm process.
Therefore D
(i)
can supply the rst m   1 values of the
vector 0, and the last value of the vector 1.
We iterate this process for each N-dimensional distance
calculation step crushing the previous values :
1((i)) = Cb:D + Cb:1((i  1));
0((i)) = Cb:D + Cb:0((i  1)):
This method can not warrant to supply the vectors 0 and
1 with the m needed values. However, we assume that
if a value have not been supplied, its complement is given
with the maximal ability. Indeed, an indeterminate value
corresponds to a symbol set rejected by the algorithm.
The analysis of the calculation complexity of the system is
given in [7].The complexity depends on the constellation
size. At every step of the detecting process we evaluate
the partial weighted distance : d(z
i
; c
i
) = j^
i
j
2
jz
i
  c
i
j
2
.
Such a calculation is refered as one operation. The Fig. 2
compares the universal hard detector complexity with the
VFSoftML complexity. The complexity unit is the partial
weighted distance cost.
16.0 18.0 20.0 22.0 24.0 26.0
SNR(dB)
1e+00
1e+01
1e+02
N
b 
of
 o
pe
ra
tio
ns
 p
er
 b
it
Perfect CSI
VFSoftML complexity
Universal detector complexity
Fig. 2: Comparative estimation of universal and VFSoftML com-
plexity
4 Simulation results
We give the BER resulting from the transmission scheme
described in section 2.4. The fading HF channel is a
Rice channel characterized by T
d
= 1:10
 3
s and the
normalized Doppler band B
d
= 0:1. The number of
sub-carriers is K = 128, the ro o factor is  = 0:25.
The spectral eÆciency including the loss introduce by
pilots and ltering is  = 3:75bit=s=Hz. For the L = 4
constellation the generating matrix G is computed in
[2]. The initial symbols are issued from a conventional
32QAM . The channel state information (CSI) is provided
by the bilinear channel estimator previously described.
The Fig. 3 compares the BER of a conventional 32QAM
constellation with their rotated version with diversity
L = 2 and L = 4. With a diversity L = 4 we obtain
signicant gains of 3dB at BER = 5:10
 3
16.0 18.0 20.0 22.0 24.0
SNR (dB)
1e-03
1e-02
1e-01
TE
B
Rice HF channel Td=1ms and Bd=0.1
No Diversity 32QAM
Diversity L=2
Diversity L = 4
Fig. 3: BER of a L = 2 and L = 4 constellation, the CSI is given
by a bilinear lter  = 3:75bit=s=Hz.
5 Conclusion
We have proposed new digital techniques for high fre-
quency communication. The transmission scheme is based
on a multicarriers ltered OFDM system, associated to
-constellations and soft decoder.
Modulation diversity allows to ght eÆciently against
the fading environment without increasing the frequency
band.
The design of the receiver allows to pilot any soft
decoders.
Complexity analysis and simulation results realized with
a real channel estimator demonstrate the interest of
such a solution which can be adapted to the material
realization constraints.
akwnoledgement : We would like to aknowledge R. Vallet
for his help and availability concerning the polyphase
ltering part.
References
[1] K. Haj Taieb, Robert Vallet and Daniel Duponteil:
"Performance of Fractionally Spaced Multi-Carrier Modu-
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\Good lattice constellations for both Rayleigh fading and
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[3] Giraud, X., Boutillon, E. and Belore, J.C., \Alge-
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[4] X. Giraud :"Constellations pour le canal a
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[5] M. Leconte, M. Testard : \A model of High Frequen-
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Design and performances of a low cost filtered COFDM system using π-constellations and real channel estimator

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