Genetic Algorithm Based Equalization for Direct Sequence Ultra-Wideband Communications Systems

Abstract

We propose a genetic algorithm (GA) based equalization approach for direct sequence ultra-wideband (DS-UWB) wireless communications, where GA is combined with a RAKE receiver to combat the inter-symbol interference (ISI) due to the frequency selective nature of UWB channels for high data rate transmission. Simulation results show that the proposed GA based structure significantly outperforms the RAKE receiver. It also provides a close bit error rate (BER) performance to the optimal maximum likelihood detection (MLD) approach, while requiring a much lower computational complexity.

Full text

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Genetic Algorithm based Equalization for Direct
Sequence Ultra-Wideband Communications Systems
Nazmat Surajudeen-Bakinde, Xu Zhu, Jingbo Gao and Asoke K. Nandi
Department of Electrical Engineering and Electronics
University of Liverpool, Liverpool, UK.
{nazmat.surajudeen-bakinde, xuzhu, jgao and a.nandi}@liverpool.ac.uk
Abstract—We propose a genetic algorithm (GA) based equal-
ization approach for direct sequence Ultra-wideband (DS-UWB)
wireless communications, where GA is combined with a RAKE
receiver to combat the inter-symbol interference (ISI) due to
the frequency selective nature of UWB channels for high data
rate transmission. Simulation results show that the proposed GA
based structure significantly outperforms the RAKE receiver. It
also provides a close bit error rate (BER) performance to the
optimal maximum likelihood detection (MLD) approach, while
requiring a much lower computational complexity.
Index Terms—Genetic algorithm, Ultra-wideband, Maximum
likelihood detection, RAKE receivers.
I. INTRODUCTION
Ultra-wideband (UWB) is a promising radio technology for
networks delivering extremely high data rates of 110Mbps
to 480Mbps at distance of 2m to 10m respectively in an
unlicensed frequency spectrum of 3.1GHz to 10.6GHz [1].
UWB systems transmit signals that demonstrate extremely
low-power-spectral-density and occupy very wide bandwidths
(greater than 25% of their center frequency) [2].
In an impulse-based DS-UWB system, the transmitted data
bit is spread over multiple consecutive pulses of very low
power density and ultra-short duration. This introduces resolv-
able multipath components having differential delays in the
order of nanoseconds. Thus, the performance of a DS-UWB
system is significantly degraded by the inter-chip interference
(ICI) and inter-symbol interference (ISI) due to multipath
propagation [3].
In a frequency-selective fading channel, a RAKE receiver
can be used to exploit multipath diversity by combining con-
structively the monocycles received from the resolvable paths
[4]. Maximum ratio combining (MRC)-RAKE is optimum
when the disturbance to the desired signal is sourced only from
additive white Gaussian noise (AWGN), therefore it has low
computational complexity. However the presence of multipath
fading, ISI, and/or narrowband interference (NBI) degrade the
system performance severely [5]. The maximum likelihood
detection (MLD) is optimal in such a frequency selective
channel environment as UWB channel but its computational
complexity grows exponentially with the constellation size and
the number of RAKE fingers. The genetic algorithm (GA) is of
much lower computational complexity compared to the MLD
approach.
This work was supported by the Commonwealth Scholarship Commission,
the University of Liverpool,UK and the University of Ilorin,Nigeria.
The high computational complexity of MLD motivates
research into suboptimal receivers with reduced complexity
such as linear and non-linear equalizers. In [6], performance
of non-linear frequency domain equalization schemes viz.
decision feedback equalization (DFE) and iterative DFE for
DS-UWB systems were studied. Eslami et al in [7] presented
the performance of joint rake and MMSE equalizer receiver for
UWB communications systems. The performance of the GA
based equalization algorithm at small and moderate number
of RAKE fingers is better than the performance of these
suboptimal receivers at the same number of RAKE fingers
and moderate number of equalizer taps. The comparison of
the GA and linear MMSE equalizer is presented in our next
paper.
GA is a well studied and effective search technique used
in lots of work in CDMA communications systems as can be
found in [8] and [9] . GA has also been applied to UWB
communications systems in [10] and [11]. In [10], a GA-
based iterative finger selection scheme which depends on the
direct evaluation of the objective function was proposed. UWB
pulse design method was carried out in [11] using the GA
optimization. However, to the best of our knowledge, no work
has been done on using GA for channel equalization in DS-
UWB communications systems.
In this paper, we propose an approach using GA for channel
equalization in DS-UWB wireless communications, where GA
is combined with a RAKE receiver to combat the ISI due to the
frequency selective nature of UWB channels for high data rate
transmission. Simulation results show that the proposed GA
based structure significantly outperforms the RAKE receiver.
It also provides a close bit error rate (BER) performance to
the optimal MLD approach, while requiring a much lower
computational complexity. The impact of the number of RAKE
fingers on the algorithm and the speed of convergence in
terms of the BER against the number of generations are also
presented.
Section II is the system model. The RAKE receiver is
in section III. The proposed GA-based channel equalization
approach and the computational complexity of the RAKE-GA
in comparison to the RAKE-MLD are presented in Section
IV. Section V presents the simulation results and section VI
draws the conclusion.
978-1-4244-2948-6/09/$25.00 ©2009 Crown
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II. SYSTEM MODEL
A. Transmit Signal
The transmit pulse vTR(t), is generated by using the ternary
orthogonal code sequence of the form given by
vTR(t) =
Nc−1
?
i=0
big (t − iTc)
(1)
where bi is the ith component of the spreading code, Tc is
the chip width, g (t) represents the transmitted monocycle
waveform which is normalized to have unit energy and Nc
is the length of the spreading code sequence.
The transmit signal for the DS-UWB can be expressed as
x(t) =
?
Ec
∞
?
k=−∞
dkvTR(t − kTf)
(2)
where Ec is the energy per transmitted pulse, dk ∈ {±1}
is the kthtransmit symbol, Tf is the interval of one symbol
or frame time and each frame is subdivided into Ncequally
spaced chips giving Tf= NcTc.
B. Channel Model
The UWB channel model derived from the Saleh-Valenzuela
model with a couple of slight modifications is used in this
paper. A log-normal distribution rather than a Rayleigh distri-
bution for the multipath gain magnitude is used because the
log-normal distribution fits the measurement data better. In
addition, independent fading is assumed for each cluster as
well as each ray within the cluster. Therefore, the multipath
model which consists of the discrete time impulse responses
can be expressed in a simpler form as
h(t) =
Ltot−1
?
l=0
hlδ (t − τl)
(3)
where Ltotis the total number of paths, τl(= lTc) is the delay
of the lthpath component and hlis the lthpath gain [12].
C. Receive Signal
The receive signal r(t), which is the convolution of the
transmit signal in (2) with the discrete time channel impulse
responses given in (3) and the addition of noise is shown in
(4) as
r(t)=
x(t) ∗ h(t) + n(t)
=
?
Ec
∞
?
k=−∞
dkvTR
Ltot−1
?
l=0
hl(t − kTf− τl) + n(t)
(4)
where n(t) is the additive white Gaussian noise (AWGN) with
zero mean and a variance of σ2, ∗ denotes the convolution
operator.
III. RAKE RECEIVER
A typical RAKE receiver is composed of several correlators
followed by a linear combiner, as shown in Fig. 1. The signal
received at the RAKE receiver is correlated with delayed
versions of the reference pulse multiplied by the tap weights,
the output signals are then combined linearly. The performance
of a RAKE receiver depends on the path selection approach
and the combining method [4].
MRC, which was used for the finger weight estimation and
the selective RAKE (SRAKE) combiner in [13], is employed
in this paper. In the path selection part, L fingers select
the received signal at the delay times τfl(l = 0,...,L − 1)
corresponding to L strongest paths. Each correlator correlates
the received signal with the reference waveform at the delay
times τfl, and integrates over symbol duration Tf.
We assume perfect channel state information (CSI) is avail-
able at the receiver. Also perfect chip synchronization between
the transmitter and the receiver is assumed. The correlator’s
output for the kthdesired symbol is given by
yfl
k
=
ˆ(k+1)Tf+τfl
kTf+τfl
r(t)vTR(t − kTf− τfl)dt
(5)
(5) in vector notation is expressed as shown in (6)
yk=
?
Esdkh + ik+ nk
?T
(6)
where yk=
?
yf0
k,...yfL−1
k
, Es= NcEcwhich is the en-
?
?T
ergy per symbol, h = [hf0,...hfL−1]T, ik=
with ifL
k
denoting ISI of the kth symbol for the lth corre-
lator and nk =
nf0
k
component of the kthsymbol for the lthcorrelator. L is the
number of RAKE fingers. The SRAKE receiver output with
MRC technique is expressed as
if0
k,...ifL−1
k
?T
?
k,...nfL−1
with nfL
k
being the noise
˜dk= ˜ γTyk
(7)
where ˜ γ = [˜ γ0,..., ˜ γL−1]Tis the finger weights of the RAKE
receiver estimated from the channel taps, ˜ γl=ˆhfl where
ˆhfl=. The hard estimates in the MRC-RAKE
receiver is obtained from the decision device which is given
as
?
ˆ
hf0,...ˆhfL−1
?T
ˆdk= sign(˜dk)
(8)
IV. GENETIC ALGORITHMS IN DS-UWB COMMUNICATIONS
SYSTEMS
A. Genetic Algorithm
The GA works on the Darwinian principle of natural selec-
tion called "survival of the fittest". GA possesses an intrinsic
flexibility and freedom to choose desirable optima according to
design specifications. GA presumes that the potential solution
of any problem is an individual and can be represented by a
set of parameters. These parameters are regarded as the genes
of a chromosome and can be structured by a string of values
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in binary form. A fitness value is used to reflect the degree of
goodness of the chromosome for the problem which would be
highly related with its objective value.
Throughout a genetic evolution, a fitter chromosome has a
tendency to yield good quality offspring which means a better
solution to any problem. In a practical GA application, a pop-
ulation pool of chromosomes has to be installed and these can
be randomly set initially. In each cycle of genetic operation,
termed as evolving process, a subsequent generation, is created
from the chromosomes in the current population. This can only
succeed if a group of these chromosomes, generally called
"parents" or a collection term "mating pool" is selected [14] .
B. Fitness Function Evaluation
The GA searches only a specified number of the possible
solutions and refine them through the selection, reproduction,
crossover and mutation operations. The GA minimizes the
fitness function in terms of the distance measure criteria.
The size of the available possible solutions is?M,2M?but
size being refined by the GA operators G times, that is the
number of generations such that P × G ≤ 2M. P is spread
by the same ternary code sequence of length Nc used in
spreading the transmitted signal to obtain sk= (1,M × Nc)
which is then convolved with the discrete time channel impulse
response,ˆh already defined in (7).
The correlator’s output expressed in vector notation is given
by
only (M,P) is utilized by the GA where P is the population
yGAk=
?
Esskˆh.
(9)
A good initial guess is critical in GA optimization. The
GA performance without RAKE provides a very bad initial
guess and so the initial guess of the GA is obtained from the
soft estimates of the RAKE receiver output˜dk, from (7). The
fitness function for the GA is now expressed as given in 10.
J =
P
?
k=1
??˜ γT(yk− yGAk)??.
(10)
When the whole search space of possible solutions are
utilized, we have the MLD scheme. The output,˜dk of the
RAKE receiver in (7) is also used as the input to the MLD.
The MLD detector searches through all the possible solutions
of data bits,
and the one close in distance to the
transmitted data based on the distance measure criteria is
chosen. The cost function for the RAKE-MLD is also (10)
but with P = 2Mthereby making the MLD to spend longer
simulation time and be more computationally complex.
?M,2M?
C. RAKE-GA in DS-UWB
The proposed GA based equalization approach is carried out
as follows. After evaluating the fitness of individuals within
the population (M,P) specified from?M,2M?according to
sizes of P = 50 and 100 were considered. The number of
variables for the fitness function is the number of symbols
(10), the bitstring population type was used and population
per block which is M and it is used in generating the initial
population such that (P × G) ≤ 2Mwhere G is the number
of generations used as the stopping criteria for the algorithm.
• Proportional fitness scaling was used to convert the
raw fitness score returned by the objective function to
values in a range that is suitable for the selection function.
It makes the expectation proportional to the raw fitness
scores. This is advantageous when the raw scores are
in good range. When the objective values vary a little,
all individuals have approximately the same chance of
reproduction.
• Stochastic selection now chooses parents for the next
generation based on their scaled values from the fitness
scaling function. It lays out a line in which each parent
corresponds to a section of the line of length proportional
to its expectation. A certain elites are now chosen which
are guaranteed to survive to the next generation.
• Scattered crossover combines two parents to form a
child for the next generation. It creates a random binary
vector, then selects the genes where the vector is a 1 from
the first parent, and the genes where the vector is a 0 from
the second parent, and combines the genes to form the
child.
• Gaussian mutation was used and it adds a random
number from a Gaussian distribution with mean zero to
each vector entry of an individual. The variance of this
distribution can be controlled with the scale and shrink
parameters. The scale parameter determines the variance
at the first generation, that is it controls the standard
deviation of the mutation and it is given by scale ∗
(1 − 0), where scale = 0 ∼ 10. The shrink parameter
controls how the variance shrinks as generations go by,
that is it controls the rate at which the average amount of
mutation decreases and the variance at the kthgeneration
G is given by vark = vark−1
shrink = −1 ∼ 3,G = number of generations. If the
shrink parameter is 0, the variance is constant, if the
shrink is 1, the variance shrinks to 0 linearly as the
last generation is reached and a negative value of shrink
causes the variance to grow.
• Hybrid function specifies another minimization function
that runs after the GA terminates in order to improve
the value of the fitness function. It uses the final point
from the genetic algorithm as its initial point. The uncon-
strained minimization function was used in this work.
• Stopping criteria determines what causes the algorithm
to terminate. Generations G was used in terminating the
algorithm .
• Vectorize makes the algorithm to run faster if the fitness
function is vectorized. The fitness function is called once
and it computes the fitness function for all individuals in
the current population at once [15]. Fig. 1 is the block
diagram of the RAKE-GA for DS-UWB systems.
?1 − shrink.k
G
?
where
D. Computational Complexity
The symbolic computational complexities in terms of com-
plex valued floating point multiplication and addition are car-
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1
~
?
?
fL

yk
f0
r(t)

vTR(t-? ?fL-1)
k dˆ

?

?




GA

vTR(t-? ?f0)




? ?

0
~
f?

yk
fL-1
. . . . . .
kd~
Fig. 1.RAKE-GA for DS-UWB
TABLE I
COMPUTATIONAL COMPLEXITY
(M = 10,P = 100,G = 10)
Receiver
RAKE-GA
L
5
10
15
20
10
Normalized
1
1.2
1.3
1.5
4.2RAKE-MLD
ried out for the RAKE, RAKE-GA and RAKE-MLD receivers.
The order of complexities of the RAKE receiver is O(LM), it
is O?[GP (LM + logP)] + LM2?for the RAKE-GA and for
L is the number of RAKE fingers, M is the number of
symbols per block, Ncis the length of the ternary orthogonal
code sequence. G is the number of generations and P is the
population size for the RAKE-GA only. The computational
complexities of the receivers depend on the derivation of
the finger weights, the fitness function evaluation and the
demodulation of the signal.
Table I shows the complexities of the RAKE-GA at L =
10,15,20 and the RAKE-MLD at L = 10 which were
normalized to the RAKE-GA complexity at L = 5. The
RAKE-MLD at L = 10 is more than four times more complex
than RAKE-GA at L = 5 and almost four times more complex
with the same number of fingers when L = 10. The RAKE-
MLD is still more complex even at L = 10 than RAKE-GA
at L = 15,20 being more than three times more complex at
L = 15 and almost three times more complex at L = 20. The
RAKE-GA at L = 10,15,20 are not even twice as complex
as at L = 5. The RAKE-GA is less complex than the RAKE-
MLD with performance very close.
the RAKE-MLD we have O??M2M(LM + log2)??
where
V. SIMULATION RESULTS
A. Simulation Setup
The simulation for the RAKE, RAKE-GA and RAKE-
MLD receivers were carried out using BPSK modulation at
a transmission rate of Rb= 250Mbps with symbol duration
or frame length of Tf = 4ns. Each packet consists of 1000
symbols. A ternary code length of Nc = 24 is used for
spreading with the result of the chip width, Tc = 0.167ns.
The simulated IEEE 802.15.3a UWB multipath channel model
[12] with perfect CSI for a single user scenario is employed for
simulation. The channel model 3 (CM3) which is a non-line-
of-sight (NLOS) environment with a distance of 4 ∼ 10m,
mean excess delay of 14.18ns and RMS delay spread of
14.28ns is considered in this work. The number of RAKE
fingers used are L = 5,10,15,20.
For the proposed RAKE-GA approach, the population size
P = 50 and 100 while the number of generations is G =
1 ∼ 20. The proportional scaling is employed for the scaling
of the fitness values before selection. The crossover of 0.85
is used with elite count of 0.05. The Gaussian mutation
values are shrink = 1.0 and scale = 0.75. In addition, the
unconstrained minimization hybrid function was employed to
improve the value of the fitness function.
B. Performance Evaluation
Fig. 2 shows the RAKE-GA at a population of P = 100
and generation of G = 10 compared with the RAKE-MLD
and RAKE receivers all at L = 10. The RAKE receiver
encountered error floor because the number of RAKE fingers
was too small for it to capture a large signal energy and also
RAKE receiver with MRC weight estimation cannot remove
ISI. The proposed RAKE-GA and RAKE-MLD were able to
remove the ISI symbol by symbol using the distance measure
criteria since the RAKE soft estimates output was used as
the input to the two receivers. The RAKE, RAKE-GA and
RAKE-MLD were of the same BER at very low SNR values.
The RAKE-GA and RAKE-MLD performed better than the
RAKE receiver at the same RAKE fingers at moderate to high
SNR values though at higher complexities. The RAKE-MLD
was of a bit lower BER at moderate to high SNR values and
the difference in performance was highly insignificant.
Fig. 3 is a plot of BER against SNR for RAKE-GA at P =
100, G = 10 at values of L = 5,10,15,20. This shows the
impact of the number of RAKE fingers on the performance of
the scheme. This BER performance improvement is as a result
of increase in the number of RAKE fingers. The RAKE-GA at
L = 5 was of higher BER to the system when L = 10,15,20
where they were almost of the same BER at all SNR values.
Fig 4 shows the impact of the number of generations in
the BER performance, where G = 1 ∼ 20 for P = 100
and G = 2 ∼ 20 for P = 50 both at L = 10 to show
the speed of convergence of the algorithm. The algorithm at
G = 1 ∼ 10 for P = 100 gave better BER generally than
at G = 2 ∼ 20 for P = 50. However, at the same values
of PG for the two scenarios, it was discovered that the case
with lower P value gives a lower BER but at a little higher
complexity. For example at PG = 500 for the two cases, the
BER was 3.2 × 10−3at G = 5 and P = 100 with a little
lower complexity than at G = 10 and P = 50 having a BER
of 1.98×10−4both with L = 10. It can thus be concluded that
the GA with a relatively large population size achieves a lower
steady state BER than the case with only half the population
size at a cost of slightly more generations. The decrease in
BER was gradual from G = 1 ∼ 8 until when G = 8 and
there was no significant difference in the BER up to when
G = 10 for P = 100. When P = 50, the BER decreases
gradually from G = 2 ∼ 8 until when G = 10 ∼ 20 when
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Page 5

there was no difference in BER values anymore The RAKE-
GA at P = 50 is of a little lower complexity than the one
with P = 100 both at the same values of G and L.
VI. CONCLUSION
We have proposed a GA based channel equalization scheme
in DS-UWB wireless communications using proportional scal-
ing, Gaussian mutation and hybrid function. Our simulation
results have shown that the proposed GA based scheme
significantly outperforms the RAKE receiver and also gives
a very close BER performance to the optimal MLD approach
at a much lower computational complexity. With a moderate
number of RAKE fingers, RAKE-GA can achieve a good BER
performance. A further increase in the number of RAKE fin-
gers has little effect on the performance. GA with a relatively
large population size achieves a lower steady state BER than
the case with only half the population size at a cost of slightly
more generations.
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05 10 15 2025 30
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
RAKE,RAKE−GA and RAKE−MLD for CM3 with L=10
Eb/No(dB)
BER


RAKE
RAKE−GA,P=100,G=10
RAKE−MLD
Fig. 2.BER vs. SNR for all receivers
05 1015 20 2530
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
RAKE−GA for CM3 with P=100 and G=10
Eb/No(dB)
BER


RAKE−GA,L=5
RAKE−GA,L=10
RAKE−GA,L=15
RAKE−GA,L=20
Fig. 3.BER vs. SNR for RAKE-GA
05 10 1520
10
−5
10
−4
10
−3
10
−2
10
−1
RAKE−GA for CM3 with L=10,SNR=20dB
No of Generations
BER


P=50
P=100
Fig. 4.Convergence speed of RAKE-GA
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.