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arXiv:cs/0507005v1 [cs.IT] 4 Jul 2005

A Genetic Algorithm Based Finger Selection

Scheme for UWB MMSE Rake Receivers

1

Sinan Gezici, Mung Chiang, H. Vincent Poor and Hisashi Kobayashi

Department of Electrical Engineering

Princeton University, Princeton, NJ 08544

{sgezici,chiangm,poor,hisashi}@princeton.edu

Abstract—Due to a large number of multipath components in

a typical ultra wideband (UWB) system, selective Rake (SRake)

receivers, which combine energy from a subset of multipath com-

ponents, are commonly employed. In order to optimize system

performance, an optimal selection of multipath components to be

employed at fingers of an SRake receiver needs to be considered.

In this paper, this finger selection problem is investigated for

a minimum mean square error (MMSE) UWB SRake receiver.

Since the optimal solution is NP hard, a genetic algorithm (GA)

based iterative scheme is proposed, which can achieve near-

optimal performance after a reasonable number of iterations.

Simulation results are presented to compare the performance

of the proposed finger selection algorithm with those of the

conventional and optimal schemes.

Index Terms—Ultra-wideband (UWB), impulse radio (IR),

MMSE Rake receiver, optimization, genetic algorithm (GA).

I. INTRODUCTION

Recently impulse radio (IR) ultra wideband (UWB) sys-

tems ([1]-[5]) have drawn considerable attention due to their

suitability for short-range high-speed data transmission and

precise location estimation. In an IR-UWB system, very short

pulses with a low duty cycle are transmitted, and each in-

formation symbol is represented by positions or polarities of

a number of pulses. Each pulse resides in an interval called

“frame”, and positions of pulses in frames are determined

by time-hopping (TH) sequences specific to each user, which

prevents catastrophic collisions among pulses of different users

[1].

Commonly, Rake receivers are employed in an IR-UWB

system to collect energy from different multipath components.

A Rake receiver combining all the paths of the incoming

signal is called an all-Rake (ARake) receiver. Since a UWB

signal has a very wide bandwidth, the number of resolvable

multipath components is usually very large. Hence, an ARake

receiver is not implemented in practice due to its complexity.

However, it serves as a benchmark for the performance of

more practical Rake receivers. A feasible implementation of

multipath diversity combining can be obtained by a selective-

Rake (SRake) receiver, which combines the M best, out of

L, multipath components [6]. Those M best components are

determined by a finger selection algorithm. For a maximal

ratio combining (MRC) Rake receiver, the paths with highest

1This research is supported in part by the National Science Foundation

under grants ANI-03-38807, CNS-0417603, and CCR-0440443, and in part

by the New Jersey Center for Wireless Telecommunications.

signal-to-noise ratios (SNRs) are selected, which is an optimal

scheme in the absence of interfering users and inter-symbol

interference (ISI). For a minimum mean square error (MMSE)

Rake receiver, the “conventional” finger selection algorithm

is to choose the paths with highest signal-to-interference-

plus-noise ratios (SINRs). This conventional scheme is not

necessarily optimal since it ignores the correlation of the

noise terms at different multipath components. In other words,

choosing the paths with highest SINRs does not necessarily

maximizes the overall SINR of the system. In [7], the optimal

finger selection problem is shown to be an NP-hard problem,

and two suboptimal algorithms are proposed based on an

approximate objective function and constraint relaxations. In

this paper, we propose a genetic algorithm (GA) based scheme,

which performs finger selection by iteratively evaluating the

exact objective function without the need for any constraint

relaxations. Using this technique, near-optimal solutions can

be obtained in many cases with a degree of complexity that

is much lower than that of the optimal exhaustive search

algorithm.

The remainder of this paper is organized as follows. Section

II describes the transmitted and received signal models in

a multiuser frequency-selective environment. The finger se-

lection problem is formulated and the optimal algorithm is

described in Section III, followed by a brief description of the

conventional algorithm in Section IV. In Section V, the GA-

based finger selection scheme is presented. Simulation results

are presented in Section VI, and concluding remarks are made

in the last section.

II. SIGNAL MODEL

We consider a K-user IR-UWB system, in which the

transmitted signal from user k is represented by:

s(k)

tx(t) =

?

Ek

Nf

∞

?

j=−∞

d(k)

j

b(k)

⌊j/Nf⌋ptx(t − jTf− c(k)

jTc),

(1)

where ptx(t) is the transmitted UWB pulse, Ek is the bit

energy of user k, Tf is the “frame” time, Nf is the number

of pulses representing one information symbol, and b(k)

{+1,−1} is the binary information symbol transmitted by user

k. In order to allow the channel to be shared by many users

and avoid catastrophic collisions, a TH sequence {c(k)

⌊j/Nf⌋∈

j}, where

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c(k)

j

sequence provides an additional time shift of c(k)

to the jth pulse of the kth user where Tcis the chip interval and

is chosen to satisfy Tc≤ Tf/Ncin order to prevent the pulses

from overlapping. We assume Tf = NcTc without loss of

generality. The random polarity codes d(k)

variables taking values ±1 with equal probability [8]-[10].

We assume a synchronous system and a tapped delay line

channel with tap spacing Tc. Note that this channel model can

represent any channel of the form?ˆL

represent the discrete channel for user k, where L is assumed

to be the number of multipath components for each user. Then,

the received signal can be expressed as

∈ {0,1,...,Nc− 1}, is assigned to each user. This TH

jTcseconds

j

are binary random

l=1ˆ α(k)

l

δ(t−ˆ τ(k)

l

) if the

···α(k)

channel is bandlimited to 1/Tc[11]. Let α(k)= [α(k)

1

L]

r(t) =

K

?

k=1

?

Ek

Nf

∞

?

j=−∞

L

?

jTc− (l − 1)Tc) + σnn(t),

l=1

α(k)

l

d(k)

j

b(k)

⌊j/Nf⌋

× prx(t − jTf− c(k)

where prx(t) is the received unit-energy UWB pulse, and n(t)

is zero mean white Gaussian noise with unit spectral density.

We assume that the TH sequence is constrained to the

set {0,1,...,NT − 1}, where NT

there is no inter-frame interference (IFI). However, the pro-

posed algorithm is valid for scenarios with IFI as well, and

this assumption is made merely to simplify the expressions

throughout the paper. From the analysis in [12], the results of

this paper can easily be extended to the IFI case as well.

Because of the high resolution of UWB signals, it is

desirable to employ symbol-rate sampling instead of chip-

rate or frame-rate sampling at the receiver. In order to enable

symbol-rate sampling, the received signal is correlated with

a symbol-length template signal, and the correlator output is

sampled once per symbol [13]. The template signal for the lth

path of the incoming signal is given by

(2)

≤ Nc − L, so that

s(1)

temp,l(t) =

(i+1)Nf−1

?

j=iNf

d(1)

j

prx(t − jTf− c(1)

jTc− (l − 1)Tc),

(3)

for the ith information symbol, where user 1 is considered as

the desired user, without loss of generality. Note that the use of

such template signals results in equal gain combining(EGC) of

different frame components, which may not be optimal under

some conditions [12]. However, it is very practical since it

facilitates symbol-rate sampling. Since we consider a system

that employs template signals of the form (3), i.e. EGC of

frame components, it is sufficient to consider the problem of

selection of the optimal paths for just one frame. Hence, we

assume Nf= 1 without loss of generality.

Figure 1 shows the receiver structure, which uses one

correlator for each multipath component. The outputs of

the correlators are sampled at the symbol rate. Let L =

{l1,...,lM} denote the set of multipath components that the

receiver collects. From (2) and (3), the discrete signal for the

Fig. 1.

combined by the MMSE combiner.

The receiver structure. There are M multipath components that are

lth path can be expressed, for the ith information symbol, as2

rl= sT

lAbi+ nl,

(4)

for l = l1,...,lM, where A = diag{√E1,...,√EK}, bi=

[b(1)

i

···b(K)

which can be expressed as a sum of the desired signal part

(SP) and multiple-access interference (MAI) terms:

i

]Tand nl ∼ N(0, σ2

n). sl is a K × 1 vector,

sl= s(SP)

l

+ s(MAI)

l

,

(5)

where the kth elements can be expressed as

?

s(SP)

l

?

k=

?

α(1)

l,

0,

k = 1

k = 2,...,K

(6)

and

?

s(MAI)

l

?

k=

?

0,

d(1)

k = 1

k = 2,...,K

1d(k)

1

?L

m=1α(k)

m I(k)

l,m,

,

(7)

with I(k)

mth path of user k collides with the lth path of user 1, and 0

otherwise.

l,mbeing the indicator function that is equal to 1 if the

III. OPTIMAL FINGER SELECTION

We aim to find the optimal set of multipath components,

L = {l1,...,lM}, that maximizes the overall SINR of the

system. In other words, we need to choose the best samples

from the L received samples rl, l = 1,...,L, in (4).

In order to reformulate this combinatorial problem, we first

define an “assignment vector” x, the ith element of which

is equal to 1 if the ith multipath component is selected, and

0 otherwise. Since M multipath components are selected by

the Rake receiver, x satisfies

denotes the ith element of x. Also let px denote a length

M vector, the elements of which are the indices of the non-

zero elements of x. For example, if the second and the third

multipath components are selected for a system with L = 4

and M = 2, then x = [0 1 1 0] and px= [2 3].

?L

i=1[x]i = M, where [x]i

2Note that the dependence of rlon the index of the information symbol,

i, is not shown explicitly.

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From the assignment vector x, we define an M × L

“selection matrix” X as follows:

X =?e[px]1···e[px]M

?T,

(8)

where eiis an L×1 unit vector having a 1 at its ith position

and zero elements for all other entries, and [px]i represents

the ith element of px.

Using the selection matrix X, we can express the vector of

received samples from any M multipath components as

r = XSAbi+ Xn,

(9)

where n is the vector of thermal noise components n =

[n1···nL]T, and S is the signature matrix given by S =

[s1···sL]T, with slas in (5).

From (5)-(7), (9) can be expressed as

r = b(1)

i

?

E1Xα(1)+ XS(MAI)Abi+ Xn,

(10)

where S(MAI)is the MAI part of the signature matrix S.

Then, the linear MMSE receiver can be expressed as

ˆbi= sign{θTr},

(11)

where the MMSE weight vector is given by [14]

θ = R−1Xα(1),

(12)

with R being the correlation matrix of the noise term:

R = XS(MAI)A2(S(MAI))TXT+ σ2

nI.

(13)

The overall SINR of the system can be expressed as [7]

SINR(X) =E1

σ2

n

(α(1))TXT

?

I +

1

σ2

n

XS(MAI)A2(S(MAI))TXT

?−1

Xα(1). (14)

Hence, the optimal finger selection problem can be formulated

as finding X that maximizes the SINR expression in (14),

subject to the constraint that X has the previously defined

structure. Note that the objective function to be maximized is

not concave and the optimization variable X takes binary val-

ues, with the previously defined structure. Hence, the problem

is NP-hard.

IV. CONVENTIONAL ALGORITHM

Instead of the solving the optimal finger selection problem,

the “conventional” finger selection algorithm chooses the M

paths with largest individual SINRs, where the SINR for the

lth path can be expressed as

SINRl=

E1(α(1)

)TA2s(MAI)

l)2

(s(MAI)

ll

+ σ2

n

,

(15)

for l = 1,...,L.

This algorithm is not optimal since it ignores the correlation

of the noise components of different paths, which is due to

the MAI from the interfering users in the system. Therefore,

it does not always maximize the overall SINR of the system

given in (14).

V. FINGER SELECTION USING GENETIC ALGORITHMS

In this section, we propose a GA based finger selection

approach, which directly uses the SINR expression in (14),

and tries to achieve the optimal performance in an iterative

fashion.

A. Genetic Algorithm

The GA is an iterative technique for searching for the global

optimum of an objective function [15]. The name comes from

the fact that the algorithm models the natural selection and

survival of the fittest [16].

The GA starts with a population of chromosomes, where

each chromosome is represented by a binary string3. Let Nipop

denote the number of chromosomes in this population. Then,

the fittest Npopof these chromosomes are selected, according

to a fitness function. After that, the fittest Ngoodchromosomes,

which are also called the “parents”, are selected and paired

among themselves (pairing step). From each chromosome

pair, two new chromosomes are generated, which is called

the mating step. In other words, the new population consists

of Ngoodparent chromosomes and Ngoodchildren generated

from the parents by mating. After the mating step, the mutation

stage follows, where some chromosomes (the fittest one in

the population can be excluded) are chosen randomly and are

slightly modified; that is, some bits in the selected binary string

are flipped. After that, the pairing, mating and mutation steps

are repeated until a threshold criterion is met.

The GA has been applied to a variety of problems in

different areas [15]-[17]. Also, it has recently been employed

in the multiuser detection problem [18]-[20]. The main char-

acteristics of the GA algorithm is that it can get close to

the optimal solution with low complexity, if the steps of the

algorithm are designed appropriately.

B. Finger Selection via the GA

In order to be able to employ the GA for the finger

selection problem we need to consider how to represent the

chromosomes, and how to implement the steps of the iterative

optimization scheme.

A natural way to represent a chromosome is to consider the

assignment vector x defined in Section III, which denotes the

assignments of the multipath components to the M fingers of

the RAKE receiver. In other words, [x]i= 1 if the ith path is

selected, and [x]i= 0 otherwise; and?L

the SINR expression given by (14). Note that, given a value

of x, SINR(X) can be uniquely evaluated. By choosing this

fitness function, the fittest chromosomes of the population

correspond to the assignment vectors with the largest SINR

values.

Now the pairing, mating and mutation steps need to be

designed for the finger selection problem:

i=1[x]i= M.

Also, the fitness function that should be maximized can be

3Although we consider only the binary GA, continuous parameter GAs are

also available [15].

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1) Pairing: The assignments to be paired among them-

selves are chosen according to a weighted random pairing

scheme [15], where each assignment is chosen with a proba-

bility that is proportional to its SINR value. In this way, the

assignments with large SINR values have a greater chance of

being chosen as the parents for the new assignments.

2) Mating: From each assignment pair, two new pairs are

generated in the following manner: Let x1and x2denote two

finger assignments, and let px1and px2consist of the indices

of the multipath components chosen as the Rake fingers. Then,

the indices of the new assignments are chosen randomly from

the vector p = [px1px2]. If the new assignment is the same

as x1or x2, then the procedure is repeated for that assignment.

For example, consider a case where L = 10 and M = 4. If

x1 = [1 0 0 1 0 0 1 1 0 0] and x2 = [0 1 0 1 0 1 0 0 1 0];

that is, px1 = [1 4 7 8] and px2 = [2 4 6 9], then the

new assignments are chosen randomly from the set p =

[1 4 7 8 2 4 6 9]. For example, the new assignments (chil-

dren) could be x3

=[1 1 0 1 0 0 0 0 1 0] and x4

[0 0 0 1 0 1 1 0 1 0] (corresponding to px3= [1 2 4 9] and

px4= [4 6 7 9], respectively).

Note that by designing such a mating algorithm, we make

sure that a multipath component that is selected by both

parents has a larger probability of being selected by the new

assignment than a multipath component that is selected by

only one parent does.

3) Mutation: In the mutation step, an assignment, except

the best one (the one with the highest SINR), is randomly

selected, and one 1 and one 0 of that assignment are randomly

chosen and flipped. This mutation operation can be repeated a

number of times for each iteration. The number of mutations

can be determined beforehand, or it might be defined as a

random variable.

Now, we can summarize our GA based finger selection

scheme as follows:

• Generate Nipopdifferent assignments randomly.

• Select Npopof them with the largest SINR values.

• Pairing: Pair Ngoodof the finger assignments according

to the weighted random scheme.

• Mating: Generate two new assignments from each pair.

• Mutation: Change the finger locations of some assign-

ments randomly except for the best assignment.

• Choose the assignment with the highest SINR if the

threshold criterion is met; go to the pairing step other-

wise.

In the simulations, we stop the algorithm after a certain

number of iterations. In other words, the threshold criterion

is that the number of iterations exceeds a given value. As the

number of iterations increases, the performance of the algo-

rithm increases, as well. The other parameters that determine

the tradeoff between complexity and performance are Nipop,

Npop, Ngood, and the number of mutations at each iteration.

In terms of the computational complexity, the algorithm

needs at most Nipop+ Niter(Ngood+ Nmut) calculations of

the SINR expression in (14), where Niter is the number of

iterations, and Nmutis the number of mutations. On the other

=

101520 2530

5

10

15

20

25

30

Eb/N0 (dB)

Average SINR (dB)

Optimal

Conventional

Genetic Algorithm, 1 Iteration

Genetic Algorithm, 5 Iteration

Genetic Algorithm, 10 Iteration

Fig. 2.

is the bit energy. The channel has L = 15 multipath components

and the taps are exponentially decaying. The IR-UWB system has

Nc = 20 chips per frame and Nf = 1 frame per symbol. There

are 5 equal energy users in the system and random TH and polarity

codes are used.

Average SINR versus Eb/N0 for M = 5 fingers, where Eb

hand, the exhaustive search for the optimal solution requires

?L

VI. SIMULATION RESULTS

Simulations have been performed to evaluate the perfor-

mance of various finger selection algorithms for an IR-UWB

system with Nc = 20 and Nf = 1. In these simulations,

there are five users in the system (K = 5) and the users’

TH and polarity codes are randomly generated. We model the

channel coefficients as αl = sign(αl)|αl| for l = 1,...,L,

where sign(αl) is ±1 with equal probability and |αl| is

distributed lognormally as LN(µl,σ2). Also the energy of

the taps is exponentially decaying as E{|αl|2} = Ω0e−λ(l−1),

where λ is the decay factor and

Ω0 = (1 − e−λ)/(1 − e−λL)). For the channel parameters,

we choose λ = 0.1, σ2= 0.5 and µlcan be calculated from

µl = 0.5

?

We average the overall SINR of the system over different

realizations of channel coefficients, TH and polarity codes of

the users.

In Figure 2, we plot the average SINR of the system for

different noise variances when M = 5 fingers are to be

chosen out of L = 15 multipath components, and all the users

have equal energy (Ek = 1 ∀k). For the GA, Nipop = 32,

Npop = 16, and Ngood = 8 are used, and 8 mutations

are performed at each iteration. As is observed from the

figure, the GA based scheme performs considerably better than

the conventional scheme, and gets very close to the optimal

exhaustive search scheme after 10 iterations. The GA scheme

needs to evaluate the SINR expression less than 200 times for

the 10 iterations case, whereas the optimal algorithms needs

SINR calculations for

M

?

different assignments.

?L

l=1E{|αl|2} = 1 (so

ln(1−e−λ

1−e−λL) − λ(l − 1) − 2σ2?

, for l = 1,...,L.

Page 5

5 10152025

16

16.5

17

17.5

18

18.5

19

19.5

Number of Fingers

Average SINR (dB)

Conventional

Genetic Algorithm, 1 Iteration

Genetic Algorithm, 5 Iterations

Genetic Algorithm, 10 Iterations

Genetic Algorithm, 20 Iterations

Genetic Algorithm, 50 Iterations

Fig. 3.

20dB, Nc = 75 and L = 50. All the other parameters are the same

as those for Figure 2.

Average SINR versus number of fingers M, for Eb/N0 =

3003 evaluations. Note that the gain achieved by using the

proposed algorithm over the conventional one increases as the

thermal noise decreases. This is because when the thermal

noise becomes less significant, the MAI becomes dominant,

and the conventional technique gets worse since it ignores the

correlation between the MAI noise terms when choosing the

fingers.

Next, we plot the SINR of the proposed and conventional

techniques for different numbers of fingers in Figure 3, where

there are 50 multipath components and Eb/N0= 20dB. The

number of chips per frame, Nc, is set to 75, and all other

parameters are kept the same as before. In this case, the

optimal algorithm takes a very long time to simulate since

it needs to perform exhaustive search over many different

finger combinations and therefore it was not implemented.

The improvement using the GA based scheme over the con-

ventional one decreases as M increases since the channel is

exponentially decaying and most of the significant multipath

components are already combined by both of the algorithms.

The GA based scheme results in about a 1dB improvement

for M = 5 after 10 iterations with Nipop= 128, Npop= 64,

Ngood = 32, and 32 mutations. The improvement is not

significant since the MAI is not very strong in this case.

Finally, we consider an MAI-limited scenario, in which

there are 5 users with E1= 1 and Ek= 10 ∀k ?= 1, and all

the parameters are as in the previous case. Then, as shown in

Figure 4, the improvement by using the proposed algorithm

increases significantly. The main reason for this is that the

GA based scheme considers the correlations caused by MAI

whereas the conventional scheme simply ignores it.

VII. CONCLUDING REMARKS

Since UWB systems have a large numbers of multipath

components, only a subset of those components can be used

5 10 1520 25

13

14

15

16

17

18

19

20

Number of Fingers

Average SINR (dB)

Conventional

Genetic Algorithm, 1 Iteration

Genetic Algorithm, 5 Iterations

Genetic Algorithm, 10 Iterations

Genetic Algorithm, 20 Iterations

Genetic Algorithm, 50 Iterations

Fig. 4.

users with each interferer having 10dB more power than the desired

user. All the other parameters are the same as those for Figure 3.

Average SINR versus number of fingers M. There are 5

due to complexity constraints. Therefore, the selection of the

optimal subset of multipath components is important for the

performance of the receiver. The optimal solution to this finger

selection problem requires exhaustive search which would

become prohibitive for UWB systems. Therefore, we have

proposed a GA based iterative finger selection scheme, which

depends on the direct evaluation of the objective function. In

each iteration, the set of possible finger assignments is updated

in search of the best assignment according to the proposed GA

stages.

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