Wireless Sensor Network, 2009, 1, 245-256
doi:10.4236/wsn.2009. 14031 Published Online November 2009 (http://www.scirp.org/journal/wsn).
On the Implementation of a Probabilistic Equalizer for
Low-Cost Impulse Radio UWB in High Data Rate
Transmission
Sami MEKKI1, Jean-Luc DANGER1, Benoit MISCOPEIN2
1Institut Telecom/Telecom Paris Tech (ENST), Paris, France
2France Télécom R & D, Meylan Cédex, France
Received April 11, 2009; revised June 9, 2009; accepted June 10, 2009
Abstract
This paper treats the digital design of a probabilistic energy equalizer for impulse radio (IR) UWB receiver in
high data rate (100 Mbps). The aim of this study is to bypass certain complex mathematical function as a
chi-squared distribution and reduce the computational complexity of the equalizer for a low cost hardware
implementation. As in Sub-MAP algorithm, the max* operation is investigated for complexity reduction and
tested by computer simulation with fixed point data types under 802.15.3a channel models. The obtained re-
sults prove that the complexity reduction involves a very slight algorithm deterioration and still meet the
low-cost constraint of the implementation.
Keywords: Impulse Radio Ultra-Wideband, Probabilistic Energy Equalizer, Inter-Sym bol Interference, Chi-Squared
1. Introduction
Ultra-wideband impulse radio is considered as a
promising candidate for indoor communications and
wireless sensor networks, as described in [1]. Despite the
numerous advantages afforded by the ultra-wideband
(UWB) [1], this system faces the technological limits
which brake the development of impulse radio (IR)
UWB. Coherent IR-UWB reception, based on Rake
receiver is limited in number of implementable Rake
fingers [2]. An alternative is given by the transmitter
reference (TR) method [3], however the electronic
architecture is more complex as it needs analog delay
lines and mixers. Non-coherent energy detection receiver
is far less complex as a few components like shottky
diodes and capacitors suffice. Though, the energy
detection is simple to implement, transmitting impulses
at high data rate leads to inter-symbol interference (ISI)
which decreases the performance of the receiver [4–6].
An efficient scheme is necessary to improve the system
performance.
A probabilistic energy equalizer is proposed in [7],
which handles different types of interference. Besides the
ISI, the proposed equalizer could manage the intra-
symbol interference, called also inter-slot interference
(IStI) in an pulse position modulation.
Nevertheless, equalization process is mathematically
complex to implement. The problem is mainly located on
the energy distribution which follows a chi-squared
distribution [8] and on the number of multiplications
required by the equalizer.
arrayM 
In this paper, the probabilistic equalizer defined in [7]
is simplified by applying the Jacobi logarithm [9] where
addition become max* operation (using Viterbi's notation
[10]) and multiplications become additions. In order to
make this possible, an approximation of the chi-squared
distribution is considered and rewritten in the logarithmic
domain as the probabilistic equalizer. The simplified
equalizer is embedded into the iterative loop of a chann el
decoder which applies the Sub-MAP algorithm in the
decodin g process.
This article is organized as follows: Section 2 defines
the system model under consideration, where energy
distribution is established. Equalization principle is
reviewed in Section 3. In Section 4, the energy
distribution is approximated by a simple function for
hardware implementation. Results with the approximated
distribution are compared to the chi-squared distribution
in Section 5. In the same Section, the hardware
implementation results in fixed point precision d ata types
are also depicted and compared to the theoretical results
in floating point precision. In Section 6 we rewrite the
probabilistic equalizer in the logarithmic domain to base
10 with respect to max10* operation and to the
approximated distribution. In Section 7, the complexity

246 S. MEKKI ET AL.
and the performance of the logarithmic equalizer is
studied and compared to the complexity of a linear
equalizer. Finally, conclusion and forthcoming work in
the field are given in Section 8.
2. System Design{TC “1 Transmitter and
Receiver Design.”\f f}
We consider an IR-UWB receiver based on energy
detection. Data transmission is ensured via the
M
-
array pulse position modulation (M-PPM) over a
bandwidth . Transmitting pulses over a high
dispersive channel causes inter-symbol interference (ISI)
and intra-symbol interference denoted as inter-slot
interference (IStI). The received signal over a time
symbol has the following expression
W
s
T
)()(=)(
0=
tztxtynkn
k
n 

 (1)
where is an additive white Gaussian noise with
variance and mean zero, and is the channel
response of the transm itted symbol defi ned by:
)(tzn
2
 )(tx kn
th
kn )( 
)()(=)( thTAtptx slotknkn  (2)
where is the impulse channel response,
)(th

denotes the convolution product, is the pulse
shape, is the time slot duration for an M-PPM
modulation,i.e. , and takes value in
according to transmitted symbol.
)(tp
kn
A,
slot
T
,M
slots MTT =
1}{0,1, 
Let
K
denotes the number of interfering symbol
assumed by the receiver, even though the real number of
interfering symbol is greater. Thus for digital treatment
the received signal (1) becomes a finite sum defined as:
)()(=)( 1
0=
tztxty nkn
K
k
n 

 (3)
The received energy per time slot in the
received symbol is given by slot
Tth n
 dttzts nn
slot
Tm
s
nT
slot
Tm
s
nT
mn2
)(
1)(
,)()(= 


 (4)
where .
)(=)( 1
0=txtskn
K
k
n


Following the approach of Urkowitz [11], it was
shown that the energy of a signal of duration can
be represented as a sum of samples in number
which is know as the degrees of freedom (DoF). Let
stands for the DoF during a time slot . Thus,
the energy in the slot of symbol is given by
slots
T
slot
T
WTslot
2
th
n
L
2
th
m
2
,,
2
1=
,)(= 

mnmn
L
mn zs 

 (5)
where and are respectively the sample
of and in slot of symbol.
mn
s,
)t
mn
z,
)(t
n
th

(snzth mth n
Assuming , then the received energy
follows a non-central chi-squared distribution
0)( 2
,
2
1= 

mn
Ls




















2
,,
1
2
2
)
,,
(
2
1
,
,
2
,, 2
1
=)|(


mnmn
L
mn
B
mn
L
mn
mn
mnmn
B
Ie
B
Bp 



(6)
with DoF and noncentrality parameter defined as
. The function is the
-order modified Bessel function of the first kind
[8]. If the noncentrality parameter is equal to zero; i.e.
; the received energy follows a central
chi-squared distribution de fine d as
L2
2
=


th
1)
0=
2
,
1=
,)( mn
L
mn sB
L(
,mn
B
)(
1uIL
2
2
,
1
,
2
,)(2
1
=0)|(


mn
Lmn
LL
mn e
L
p




 (7)
where )(z

is the gamma function [8].
The energy distribution is studied in next sections and
simplified for hardware implementation.
3. Energy Equalization Principle
To benefit from the iterative process of a communication
system, we consider a probabilistic equalizer that can be
embedded into the iterative loop of a channel decoder
based on SISO (Soft-Input/Soft-Output) decoding.
Thus, the considered equalizer takes the accumulated
energy per slot (i.e. mn,

) and per symbol (i.e.
),,,(= ,,2,1 Mnnnn E



) as reference, in order to
retrieve the transmitted symbol . So the detector
computes a conditioned probability
regarding the interfering symbols on . It has been
shown in [7] that the equalization is performed by
computing
n
x
)|( nn xEp
n
x












 )()|(=)|(1
1=
,,
1=
11
kn
K
k
mnmn
M
m
Kn
x
n
x
nn xBpxEp
 
(8)
where )(kn
x

is the a priori probability provided by
the SISO decoder and is defined in
Section 2. It was also established that the set of all the
)| (,,mnmn Bp 

S. MEKKI ET AL. 247
However the smaller the number of DoF , the larger
the approximation error. Due to the large bandwidth
in UWB-IR, the number DoF could b e big enough [16] to
consider the Gaussian distribution as an approximation to
the chi-squared density. For instance is around
for and . According to the
previous Remark, the Gaussian approximation has the
same mean and variance as the non-central chi-squared
distribution, i.e. , given by [17]:
L2
L2
W
30 GHzW 3=
,n
E
nsTslot 5=
),( 222

 mN
m:
possible values that could take, has a finite
cardinal. Figure 1 summarize the transmission and the
receiver design under consideration.
mn
B,
In order to reduce the complexity and make the
equalizer feasible, we investigate the implementation in
finite precision.
Moreover the probability given by Equation (8) needs
some mathematical simplifications and approximations
of the probability density function (pdf) ,
corresponding either to the central (7) or non-central
chi-squared (6) distribution. This will be investigated in
the following section.
)|( ,, mnmn BpE
mn
BLm,
2
22=


(9)
mn
BL ,
242244=


 (10)
4. Chi-Squared Distribution Approximation
for Hardware Implementation This can be extended to the central chi-squared
distribution by considering . 0=
,mn
B
The chi-squared distribution defined by (7) and (6) is a
three variable function (, and ). Thus,
building a look-up table according to these parameters
would occupy a great memory. For instance, if the energy
distribution is coded in bits and , and
are coded respectively in 14 -bit, -bit and -bit long,
the space memory allocated to this look-up table would
occupy Mbits (or Mbytes). This corresponds to a
costly silicon area in a FPGA or ASIC technology and thus
incompatible with low-cost constraints.
mn,
Emn B,
n
E
6
2

mn,
6
7
56
m,B2

448
Using these results and the aforementioned
assumptions, we obtain the approximation for the energy
distribution (n oticed ) pe r slot,and as

p0
,mn
B2>>2L
22
22
2
2,
,,,, 2
2
)(
exp
=)|()|(

















m
BpBp
mn
mnmnmnmn
E
EE 
(11)
Figure 2(a) shows the error measured by
for ,
and .
|)|()|(| ,,,, mnmnmnmnBpBp EE  
1=
2

0
, mn
E0>
,mn
B
An approximation for the chi-squared distribution is
thus necessary. In the literature, there are some proposals
for the calculation of the non-central chi-squared
distribution [12] and the use of the normal approximation
to the chi-squared distribution [13,14], but those
approximations require high bit precision and are
therefore too complex for digital design.
Table 1. Look-up table Input/Output size with x2
distribution.
An intuitive approximation can be found by considering
the Remark in [15] which stands that when a variable

is used to approximate a variable , it is equivalent to
match the mean and variance of and .

 
Parameters Quantization size
En,m 14 bits
Bn,m 6 bits
σ2 6 bits
p(En,m|Bn,m) 7 bits
x 2 Table size 448 Mbits (56 Mbytes)
It is notably shown in [15], that a chi-squared distribution
can be app roxim ated by a Gaussian distributio n.
Figure 1. Transmitter and receiver design.

248 S. MEKKI ET AL.
(a) Error for (b) Error for 1=
2
 0.5=
2

Figure 2. Error measured by for
|)B|(Ep)B|p(E|mn,mn,mn,mn,

0E mn, 

, . 0>Bmn,
It is noticed that the error tends to zero as
decreases (Figure 2 (b)). According to [7], the energy
equalizer operates at ; i.e. corresponds to
for a pulse energy equals to unity in coded
system. So, the maximum error, considered between the
chi-squared and Gaussian distributions, is
as shown in Figure 2(a). We denote the normal function by
2

3
10
1<
2
 1=
2

dBSNR 3=
5=  
/2
2
2
1
=)( t et 

 (12)
Using (9), (10) and (12), equation (11) can be rewritten
as follows










22
2,
22
,,1
=)|(






m
Bpmn
mnmn
E
E
 (13)
As the energy distribution is simply deduced from the
normal function )(t

, the digital implementation can
only use two look-up tables. The first one contains the
values of the normal function 0,)( tt

. The second
one contains the values of the ratio 0>,1/ xx . The
input/output precision of the look-up tables will be
analyzed in the simulation Section according to the
hardware constraints.
5. Performance of the Approximated Linear
Equalizer
In this section, computer simulations have been run to
assess the performance of the linear energy equalizer with
the approximated Gaussien distribution defined by (13).
The BER computation has been performed via simulations
in both floating point precision and fixed point precision
data types. In the firsts part of simulations, we compare
the performance of the receiver with the Gaussian
approximation (11) and with the exact calculation of the
chi-squared distribution in floating point precision.
Second part of simulations has been run in fixed point data
types with the approximated distribution.
The block fading multipath channel is generated
randomly according to IEEE 802.15.3a UWB channel
models [18]. Channel estimation is out of the scope of
this paper. The channel state information (CSI) is
assumed perfectly known at the receiver side.
Nevertheless, channel parameters can be approached by
the mean of the expectation-maximization (EM)
algorithm as studied in [19] or by a set of a specific
training sequence.
5.1. Chi-Squared Versus Gaussian Approximation
Simulations in Double Precision
We consider an UWB-IR system as defined in Figure 1.
Transmission is ensured by a 4-PPM modulation at
. Thus we get bits per transmitted symbol.
We have implemented a duo-binary turbo code as it is
defined in the standards [20,21]. This channel coder is
chosen because it is suited to QPSK (quadratic phase
shift keying) and 4-PPM modulations. The encoded data,
at the input of the encoder, are -bit long blocks. The
turbo encoder rate is and 10 iterations of the SISO
decoder are performed at the receiver side. The equalizer
is jointly implemented into the iterative loop of the
decoder to benefit from the iterative process of the
decoder. The efficiency of the energy equalizer will not
sMbit/100 2
864
1/2

S. MEKKI ET AL. 249
be treated in this paper, the reader should refer to [7] for
more details concerning the equalizer performances.
The receiver assumes that there are only two
interfering symbols, i.e. 2=
K
and , but the real
number of interfering symbols could be more. The CSI is
assumed over
5=P
P
time slots duration and not otherwise.
In our case, for a data rate of Mbps, the time slot
duration is , so the receiver has a perfect CSI only
over . This duration is sufficient for channel
models as CM1 and CM2, although their respective
maximum excess delay are 80 and as it is
studied in [7]. However for highly dispersive channel as
CM3 and CM4 with maximum excess delay of
and respectively, channel knowledge should be
extended to . Nevertheless, we consider only
simulations with
100
ns
ns
=K
5
ns25
ns200
ns115
ns140
3= 2
K
for the Gaussian approximation
performance comparison.
It is noticed that the results with Gaussian
approximation match the chi-squared performances in
floating point precision even for highly dispersive
channel such CM3 and CM4 with a slight degradation of
performance.
5.2. Fixed Point Precision Simulations
The fixed point precision is subject to hardware
constraints. The duo-binary turbo coder hardware
implementation is out of the scope of this paper. The
digital design of the channel coder is furnished by Turbo-
Concept for an optimum efficiency [22]. The eergy
detector of UWB platform is a logarithmic one [2
guarantee the scalar value of the energy mn, E for
equalization, a look-up table of the function x
10 is
required. Computer simulations in fixed point precision
are achieved by means of the SystemC class sc_fix [24].
thok-up tae functions:
n
3]. To
The Gaussian approximation for energy
are computedrough the lobles of th
equalization


2
=)(/2x
e
x,
2
xxg 1/=)( and x xh 10=)( . Figure 4
sh
fix
he fractional part of the object. Hence each
ob
d point simulations.
Reng the number of bits for each variable involves a
significant perform ance decre a se.
ows the Gaussian approximation computation archi-
tecture for the chi-squared distribution.
According to the class sc_ of SystemC, a signed or
an unsigned object are defined by two parameters: the
total word length noted as wl , i.e. the total number of
bits used in the type, and the integer word length noted
as iwl , i.e. the number of bits that are on the left of the
binary point (.) in a fixed point number. The remaining
bits stand for t
ject is represented by a pair of parameters noted
>,< iwlwl .
Simulations have been carried out with different
parameter sizes. Table 2 shows the word sizes of the
parameters considered for the fixe
duci
Figure 3. Chi-squared vs gaussian approximation in float precision using duo-binar y turbo code at rate 1/2th K=2. wi

250 S. MEKKI ET AL.
Figure 4. Approximated energy distribution architecture for the linear equalizer with x2 approximation.
Table 3 lists the Input/Output size look–up table
ne
Table 2. Parameters size definition.
Param
cessary for density computation.
eters Quantization size >,<iwlwl
mn,
log >6,2<
mn,
 >14,1<
mn
B,
2
>6,2<

>6,1<
2

m
2
> 12,2<
2


>12,2<
)|(,, mnmn Bp 

>13,3<
)|( nnxEp >13,6<
)( k x

>4,1<
We notice that the total memory occupied by the
lo
2 and 3 under the same
co
Table 3. Look-up table input/output size.
Parameters Input size Output size ble size
ok-up tables is lower than the chi-squared look-up
table as it is described in Table 1.
Simulations according to Table
nditions as for double precision lead to the results
depicted in Figure 5.
Ta
(Kbits)


2
=)( /2
2
x
e
x >8,2< >18,0< 4.5
xxg 1/=)( >12,2< >6,4< 24
x
xh10=)( >6,2< >14,1< 0.875
Figure 5. chi-squared float precision versus the Gaussian approximation in fixed point precision for K=2.

S. MEKKI ET AL. 251
esults in
degrade
.3. Complexity of the Linear Equalizer
he linear equalizer with a chi-squared distribution has
4 and equation (8), the amount of
no
n

(14)
According to the decomposition in (14),
re
R fixed point precision data types are close to
those obtained in double precision with chi-squared
distribution. We notice, that even the quantization error
of the energy distribution is around 13 1/2 , which is
lower than the considered maximum err3
105=    ,
the receiver performances are slightlyd
compared to double precision sim ulations.
or
5
T
an expensive lookup table (Table 1). Due to the size of
the ROM and its cost, the chi-squared distribution is
approximated by a simple implementable function (13)
wi th some memor ies.
According to Figure
n trivial multiplications in the linear domain is about
K
MKM 2)(4  in a symbol period. We denote by
on the multiplication by a power of 2
which is equivalent to a shift in hardware impl ementa tio.
The detail of multiplications is as follows1

trivial multiplicati






























MulKM
MulK
kn
K
k
Mul
MulM
mnmn
M
m
cases
K
M
Kn
x
n
x
Mtimes
nn xBpxEp
2)(
2)(
1
1=
1
1)(
,,
1=
1
11
)()|(=)|(  E
(n Ep
e should
)|n x
quires 1 2)(   K MKM multiplication. W
notice thatputed )|( nn xEp is com
M
times per sy mb ol
duration. Tymbol period,e total amount of
multiplication is equal to K MKM 2)(  multiplications.
Once the number of mult
per symbol period, we divide up each terms. The a priori
probability, i.e. )( kn x
hus, in a s
ipli
th
scation is established for (14)

, will not be discussed since it is
provided by th decoder. So our analysis is
focused rather on the energy distribution. In the term
)|( ,,
MBp 
, we calculate
e SISO
mnmn
1=m
M
times )|( ,,mnmnBp .
chitecture in Figure 4,
requires3 non-trivial multiplication. Hence
period there is 1
3K Madditional multiplications on the
back of KM( This leads to K
MKM 2)(4
multiplica by the equalizer.
As example, we consider a 4-PPM modulation at
With respect to the ar
K
M2)
)|( ,, mnmn B
per symbol
p
,
tions calculated per tim e sym bol
Mbps0 and 2= 10
K
at the receiver side, this leads to
attiplic and a total memory ofKbits30.25
r the energy distribution c,
,, mnmn
sion/Gmul12.8
i.e. )|( Bp

. We should notice that the equalizer
computes the energy distribution 1K
M
times per time
sym Mbps for2=
bol. So, at 100
K
with a 4-PPM the
frequency of tab le acce ss is about 3. . If we co ns id er
a hardware that runs atMHz400 ,level of parallelism
to achieve the energy distribution is equal to 8. Thus, the
total amount of mem a factor of 8, i.e.
KbitsKbits242=830.25
GHz2
the
isory

.
The next par of this paper will be focused on
probabilistic equalizer com
m
r
de
the
plexity reduction
ean of S
ity
dware devices, t
max
max=

*
10
max
of the
Reduct
he ex
=),(
*
10 ba
),( ba 
operation is essentially a
by the
robabilistic
plementable in
uired by the
(15)
ub-MAP algorithm known also as Jacobi
algorithm [9].
6. Complexio
q
pe
log
log
n o
nsive
(10
(1
f the
ualizer is im
area
)10ba 
10 || ba 
P
req
m
Energy Equalizer
n though the defined eEve
ha
equalizer could be decreased by simple computational
method. This is made possible by computing in the
logarithmic domain where only additions and
comparisons operations with small memories are
required. The channel decoder should operate also in the
logarithmic domain, in order to get the better
performance of the receiver. The decoder properties are
not discussed in this paper. However, Sub-MAP
algorithm, also called Max-Log-MAP or Dual Viterbi
[21,10] based decoder is a good candidate for joint
decoder equalizer receiver in the logarithmic domain.
As in the Sub-MAP decoding [10,25], we consider the
*
max function which operates in a logarithm to base
10 10
fined as

)
a
x
operation
adjuste factor carried ou a lookup
table read-only memory (ROM); whutputs the
correction term )10(1log || ba  given the input )(ba
d by a correction
; i.e a t by ich o

in hardware implementation. As the m a
x
property, *
is an associative operator (s ee APPENDIX 1 for the proo
]),,([=),,( *
10
*
10
*
10 cbamaxmaxcbamax (16)
10
max
f):
(17)
Using the max10operation defined in (15) ,
of the probabilistic equalizer in the logarithmic domain is
For notation sim
*
plicity we consider
)( i a =),,,( {1,
*
1021
*
10 i Nmaxaaamax 
},N
the output
according to Table 3 foomp ut i ng
1Mul stands for Multi
p
lication in
(
14
)
.

252 S. MEKKI ET AL.
gi
(18
Considering the result of equation (18), it is noticed
that the multiplication operations are replaced
co
ven by:
=)|(log xEp









  )(log)|(log 1
1=
,,
1=
1
,,
1
10 kn
K
k
mnmn
M
m
Kn
x
n
x
nn
xBpmax  

*
)
by
mparators and adders which are costless and easy to
implement. Since the equalizer output should be a
probability, i.e. [0,1])|(

nn xEp , a normalization
process is applied as follow:
)|(
)|(
=)|( nn p xEp (19)
where p is the normalized probability at the output of
the equalizer. In the logarithmic domain normalization
becomes








)|(log)|(log=)|(log nn
n
x
nnnn xEpxEpxEp (20)
)|(log)|(log= *
10 nn
n
x
nn xEpmaxxEp  (21)
Gaussian approximation studied in Section 4 is
assumed for equalization in the logarithmic domain so
that the energy distribution is feasible or im plementable.
Thus, the approximated energy distribution in
logarithmic domain is equal to:
nn
n
x
nnxE
xEp
10ln2
)(
log
2
1
2log
2
1
=)|(log 22
2
2,
22,,




m
Bp mn
mnmn

 
 (22)
One should notice that with normalization process at
e output of the equalizer, the redundant constants are
m
[23] which provides logarithmic energies per time slot
. So, the only available data is
th
reoved. In addition, due to hardware restraint, the
energy detector is a logarithm to base 10 detector as in
slot
Tmn, logE, mn B, log
and )(2log 2
 L. Expending 22

 in (22), we get
10ln)2(24
)(
22,mn mL B 


)](2
2,
22
2
,
2
mn
mn BLL
L 


(23)
[2log
1
)|(log 2
,, mnmn Bp
 

10ln2.10
)(
)2(2log
2
1
)(2log
2
1
))
,
2
2
(2
2
(2log
2
2,
,
22
mn
BLL
mn
mn
mL
BL 






(24)
where the symbol means “proportional to” and
e function log stands for the logarithmic to base 10.

th
Rewriting (24) taking into account max*10and removing
the redundant constant such as 2
)(2log 2 
, leads to

10ln2.10
)(
lo),(2g
2
)
,
2
2
(2log)
2
(2log
2
2,
,
2
10,,
mn
BLL
mn
mnmnmn
mL
L






E
(25)
the devision part in (25) can be transformed into multi-
plicationas follows:
log2glo
1
)|(log *BmaxBp E





 10)(
10ln2
2
2,m mn

(26)
where
  log2log),(2log
2
1
)|(log,
2*
10,,
L
BLmaxBpmnmnmn 


]log2log),(2log[)(2log= ,
2*
10
2mn
BLmaxL 
(27)
and 2

m
,mn


is easily calculated as follows
(28)
It is noticed that the energy distribution in logarithmic
domain is achieved by th e mean of two lookup tables. A
first ROM for the max*10 function and a second ROM for
10x function. The memories size will be treated in
sim
s used for the probabilistic equalizer
complexity study.





 mn
B
L
mn
mn m,
log
)
2
(2log
,
log
2, 101010=




the
ulation Section.
The advantage of working in logarithmic domain is
that the amount of multiplications is confined only on the
energy distribution. Figure 6 depicts the new energy
distribution architecture implemented in digital design.
This architecture i
is defined as

S. MEKKI ET AL. 253
Figure 6. Energy distribution architecture for logarithmic equalizer.
7. Performance of the Logarithmic Equalizer
with the Approximated Distribution
he performances of the logarithmic equalizer with the
d point
prece chi-squared distribution in
ouble precision. Reception is ensured by a logarithmic
mbols could be more.
ained sults for
hi
channel. It has been proven that for channel models
CM3 and CM4, the optimal compromise is to consider
Simulations run withfor CM3 and CM4 in
plex slightly
able 4. Parameters size definition.
7.1. Fixed Point DataTypes Simulation in
Logarithm Domain
T
approximated distribution are simulated in fixe
ision and compared to th
d
energy detector [23]. Simulations in fixed point precision
are carried out by the mean of the class sc_fix of
SystemC as in 5.2. Table 4 shows the word sizes of
parameters considered for the fixed point simulations.
With respect to the equalizer expression (18) and to
the approximated energy distribution in logarithmic
domain (26), we consider two ROM types whose sizes
are defined in Table 5.
Figure 7 shows the results obtained if the receiver
assumes that there are only 2 interfering symbols, i.e.
CSI is known only over 5=P slots, however the real
number of interfering sy
We notice that for less dispersive channel such as
CM1 and CM2, the results in fixed point precision data
types are close to those obt in double precision with
chi-squared distribution. Regarding the re
ghly dispersive channel (CM3 and CM4), we get a
loss of dB1 at 4 10= BER .According to [7], the receiver
could be improved if the suppo sed number of in te rfe r ing
symbols are bigger than 2, especially in hi ghly dispersive
fixed point data types are depicted in Figure 8. Although
the comity iscreased due to cardinal of the
set }{ ,mn B, i.e. 88|=}{|,mn B for 3=K [7], the receiver
is improved of dB2 for CM3 at 4
10=  BER .
3=K [7]. 3=K
in
T
Parameters Quantization size >,< iwlwl
mn,
log <6,2>
mn
B, log,2>
)(2log2  L
<8
<7,4>
<8,4>
2

m
)| B <7,4> (log ,,mnmn p

)|nn x (logEp <6,4>
) (log k x 
<6,4>
Table 5. ROM input/out p
Parameters Input
size Osize Table size (Kbits)
ut size.
utput
=)(xg )10(1log||d
 <6,2> <4,0> 0.25
x
xh 10=)( <9,5> <8,3> 4

254 S. MEKKI ET AL.
Figure 7. Chi-squared float precision versus the logarithmic Gaussian approximation in fixed point precision for K=2.
Figure 8. Simulation in fixed point data types for CM3 and CM4.
7.2. Complexity of the Logarithmic Equalizer
According to the equalizer expression (18) and Figure 9,
the computational complexity of the equalizer in
terms of non-trivial multiplication is equal to 1 2K
M
odulation multiplication per symbol. Thus, with 4-PPM m
at and Mbps100 2=
K
, we get
Regarding the memory size, the logarithmic approximati-
sation/. Gmultiplic6.4

S. MEKKI ET AL. 255
Figure 9. Equalizer architecture in the logarithmic domain.
Table 6. Complexity requirement with 4-PPM and K=2.
Number of Multiplications
Function Linear x2 Linear x2 approximated Logarithmic x2 approximated
p(En|xn) (M+K2)MK 3.20GMultip/s(M+K2)MK 3.2 GMultip/s 0
2MK+1 6.4 GMultip/s
Total Equalizer Multiplications 4 GMultip/s
Total required mory
f parallelism
)|( ,, mnmnBp  0 K+1
3M 9.6 GMultip/s
3. 2 GMultip/s 12.8 GMultip/s 6.
me
per level o448 Mbit 30.25 Kbit 16.25 Kbit
on requ. Comp
equalizer the in the logarithm domain, i.e.
number of m
promiare impor
instanruns at d the
quess is e same
imated
complex for
it allows a compromise
etween the number of multiplications and the size of
h is
a Gaussian distribution instead of a
leads to reduce significantly the
quired memory for distribution computation. The
is to calculate all the probabilities
idomain e
ope , the computa of the
equalizer is highly reduced compared to the linear
equalver, only two ble types are
required for equalizer calculation in logarithmic domain.
on fo
mmunications,” IEEE Conference on Ul-
ystems and Technologies, pp. 265–269,
ires Kbits16.25
complexity
ultiplications
d before, we r
aring to the linear
and the m
sing for a low cost hardwemory size, is lementation. F
ce, if the hardware Mhz an400
Ghz for th
n Table 2
fre
ency of table acce3.2 xample quoteequire 8 level of parallelism
to achieve the energy distribution. So the total required
memory is a factor of 8; i.e. KbitsKbits 130=816.25)
which is lower than the required memory in linear
domain (Kbits242). Moreover, the r equired bits for each
parameters in Table 4 are shorter than i.
7.3. Complexity Summary
Table 6 is the synthesize of the complexity requirement
for the linear and the logarithmic equalizer with the
chi-squared distribution approximation. We notice that
the logarithmic equalizer with the approx
energy distribution is far the less
hardware implementation, since
b
the required memory for equalization calculation.
8. Conclusion
In this paper, we a have shown how a complex and
costly probabilistic equalizer is simplified for digital
design by using the logarithmic domain. A first
simplification concerns the energy distribution whic
approximated by
chi-squared. This
re
second simplification
n the logarithmic by the mean of th *
10
max
ration. Hencetional complexity
izer. Moreolookup ta
Computer simulations demonstrated the performance of
the receiver in finite precision. It showed, that for highly
dispersive channels such as CM3 and CM4, the receiver
is still able to equalize and decode the transmitted
informations with a slight increase in complexity.
As perspective, some operations or memories could
even be simplified or reduced by the mean of polynomial
approximations with a negligible loss on the receiver
performance. This could be a subject of investigatir
future research.
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Appendix
*
10
max
]),,([=),,( *
10
*
10
*
10 cbamaxmaxcbamax
P
From the defi*
10
max
)1010(10log cba m =),,(
*
10 cbaax
in other hand we can write
),(
*
10
)10(10log10=10=1010 bamax
ba
ba 

)10(10log=),,( ),(
*
10
*
10 c
bamax
cbamax 
]),,([= *
10
*
10 cbamaxmax