Radial Basis Function Neural Network Receiver for Wireless

Channels

Pedro Henrique Gouvêa Coelho and Fabiana Mendes Cesario

State Univ. of Rio de Janeiro, FEN/DETEL, R. S. Francisco Xavier, 524/Sala 5027E, Maracanã, RJ, 20550-900, Brazil

Keywords: Neural Networks, Artificial Intelligence Applications, Channel Equalization, Wireless Systems.

Abstract: Artificial Neural Networks have been widely used in several decision devices systems and typical signal

processing applications. This paper proposes an equalizer for wireless channels using radial basis function

neural networks. An equalizer is a device used in communication systems for compensating the non-ideal

characteristics of the channel. The main motivation for such an application is their capability to form

complex decision regions which are of paramount importance for estimating the transmitted symbols

efficiently. The proposed equalizer is trained by means of an extended Kalman filter guaranteeing a fast

training for the radio basis function neural network. Simulation results are presented comparing the

proposed equalizer with traditional ones indicating the efficiency of the scheme.

1 INTRODUCTION

Channel equalization purpose is to remove the

effects of the channel on the transmitted symbol

sequence, namely the inter-symbol interference

(ISI). Typically, this task can be done either by

inverse filtering, Decision-Feedback-Equalization

(DFE) or by means of sequential detection usually

using Viterbi algorithm. Wireless channels can

exhibit delay dispersion, in other words, Multi Path

Components (MPCs) can have different runtimes

from the transmitter (TX) to the receiver (RX).

Delay dispersion causes ISI, which can greatly

degrade the transmission of digital signals. Even a

delay spread that is smaller than the symbol duration

can cause a considerable Bit Error Rate (BER)

degradation. If the delay spread becomes

comparable with or larger than the symbol duration,

as occurs often in second and third generation

cellular systems, then the BER becomes

unacceptably large if no countermeasures are taken.

Also when a signal is transmitted through wireless

medium then due to multipath effect there is

fluctuation in signal amplitude, phase, and time

delay. This effect is often known as fading (Proakis,

2001). Coding and diversity can decrease, but not

completely eliminate, errors due to ISI. On the other

hand, delay dispersion can also be a positive effect.

Since fading of the different MPCs is statistically

independent, resolvable MPCs can be interpreted as

diversity paths. Delay dispersion thus gives the

possibility of delay diversity, if the RX can separate,

and exploit, the resolvable MPCs. Equalizers are RX

structures that work both ways - they reduce or

eliminate ISI, and at the same time exploit the delay

diversity inherent in the channel. The operational

principle of an equalizer can be visualized either in

the time domain or the frequency domain. In this

paper the time-domain approach is pursued. For an

interpretation in the frequency domain, remember

that delay dispersion corresponds to frequency

selectivity. In other words, ISI arises from the fact

that the transfer function is not constant over the

considered system bandwidth. The objective of an

equalizer is thus to reverse distortions by the

channel. That is, the product of the transfer functions

of channel and equalizer should be constant

(Proakis, 2001). The channel dynamics may not be

known at startup. Moreover the channel may vary

with time, so an adaptive implementation of the

equalizer is essential. The following different modes

of adaptation can be listed:

• Adaptation using a training signal;

• Decision directed adaptation - An error signal

is generated by comparing input and output of the

decision device;

• Blind adaptation: Exploiting signal properties

instead of using an error signal for adaptation;

In this paper a training signal is used for the

658

Henrique Gouvêa Coelho P. and Mendes Cesario F..

Radial Basis Function Neural Network Receiver for Wireless Channels.

DOI: 10.5220/0005470106580663

In Proceedings of the 17th International Conference on Enterprise Information Systems (ICEIS-2015), pages 658-663

ISBN: 978-989-758-096-3

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

equalizer adaptation. Summing up, digital

communication systems operates on time varying

dispersive channels which often employ a signaling

format in which customer data are organized in

blocks preceded by a known training sequence. The

training sequence at the beginning of each block is

used to estimate channel or train an adaptive

equalizer. Depending on the rate at which the

channel changes with time, there may not be a need

to further track the channel variations during the

customer data sequence. This paper proposes a

channel equalizer for wireless channels using Radial

Basis Function (RBF) neural networks as the

equalizer structure on a symbol by symbol decision

basis. RBFs (Mulgrew, 1996) have been used in the

area of neural networks where they are applied as a

replacement for the sigmoidal transfer function.

Such networks have three layers: the input layer, the

hidden layer with the RBF nonlinearity, and a linear

output layer, as shown in Fig. 1(Burse et al, 2010).

Due to obvious reasons, the most popular choice for

the nonlinearity is the Gaussian function. The RBF

equalizer classifies the received signal according to

the class of the center closest to the received vector

(Assaf et al, 2005). The output of the RBF equalizer

supplies an attractive alternative to the Multi-Layer

Perceptron (MLP) type of Neural Network for

channel equalization problems because the structure

of the RBF network has a close relationship to

Bayesian schemes for channel equalization and

interference exclusion problems. This paper is

divided into four sections. Section 2 does a brief

discussion of RBF artificial neural networks. Section

3 presents the application of RBF neural networks to

the equalization problem and section 4 ends the

paper by presenting conclusions.

2 RBF NEURAL NETWORKS

RBF neural networks are the second more used

architecture after feedforward neural networks.

Denoting the input (vector) as x and the output as

y(x) (scalar), the architecture of a RBF neural

network is given by





















||)(||

expy(x)



(1)

using Gaussian function as basis functions. Note

that, c

are called centers and  is called the width.

There are M basis functions centered at c

, and w

are named weights.

RBF neural networks are very popular for function

approximation, curve fitting, time series prediction,

control and classification problems. The radial basis

function network differs from other neural networks,

showing many distinctive features. Due to their

universal approximation, more concise topology and

quicker learning speed, RBF networks have attracted

considerable attention and they have been widely

used in many science and engineering fields (Oyang

et al., 2005), (Fu et al., 2005), (Devaraj et al., 2002),

(Du et al., 2008), (Han et al., 2004). The

determination of the number of neurons in the

hidden layer in RBF networks is somewhat

important because it affects the network complexity

and the generalizing capability of the network. In

case the number of the neurons in the hidden layer is

insufficient, the RBF network cannot learn the data

adequately. On the other hand, if the number of

neurons is too high, poor generalization or an

overlearning situation may take place (Liu et al.,

2004). The position of the centers in the hidden layer

also influences the network performance

significantly (Simon, 2002), so determination of the

optimal locations of centers is an important job.

Each neuron has an activation function in the hidden

layer. The Gaussian function, which has a spread

parameter that controls the behavior of the function,

is the most preferred activation function. The

training method of RBF networks also includes the

optimization of spread parameters of each neuron.

Later on, the weights between the hidden layer and

the output layer must be selected suitably. Finally,

the bias values which are added with each output are

determined in the RBF network training procedure.

In the literature, several algorithms were proposed

for training RBF networks, such as the gradient

descent (GD) algorithm (Karayiannis, 1999) and

Extended Kalman filtering (EKF) (Simon, 2002).

Several global optimization methods have been used

for training RBF networks for different science and

engineering problems such as genetic algorithms

(GA) (Barreto et al., 2002), the particle swarm

optimization (PSO) algorithm (Liu et al., 2004), the

artificial immune system (AIS) algorithm (De Castro

et al., 2001) and the differential evolution (DE)

algorithm (Yu et al., 2006). The Artificial Bee

Colony (ABC) algorithm is a population based

evolutional optimization algorithm that can be used

to various types of problems. The ABC algorithm

has been used for training feed forward multi-layer

perceptron neural networks by using test problems

such as XOR, 3-bit parity and 4-bit encoder/decoder

problems (Karaboga et al., 2007). Due to the need of

fast convergence, EKF training was chosen for the

RBF equalizer reported in this paper, details on the

RadialBasisFunctionNeuralNetworkReceiverforWirelessChannels

659

training process can be found on (Simon, 2002).

3 RBF NEURAL EQUALIZER

Radial Basis Function Neural Networks have been

used for channel equalization purposes (Lee et al.,

1999), (Gan et al., 1999), (Kumar et al. 2000), (Xie

and Leung, 2005). Typically, such networks have

three layers: the input layer, the hidden layer with

the RBF nonlinearity, and a linear output layer, as

shown in Fig. 1 (Burse et al., 2010). The RBF

equalizer classifies the received signal according to

the class of the center closest to the received vector.

The output of the RBF NNs gives an attractive

alternative to traditional equalization methods for

channel equalization problems because the structure

of the RBF network has a close relationship to

Bayesian methods for channel equalization and

interference rejection problems. Simulations carried

out on time-varying channels using a Rayleigh

fading channel model to compare the performance of

RBF with an adaptive maximum likelihood

sequence estimator (MLSE) show that the RBF

equalizer produces superior performance with less

computational complexity (Mulgrew, 1996). Several

techniques have been developed in literature to solve

the problem of blind equalization using RBF (Tan et

al., 2001), (Uncini et al., 2003) and others. RBF

equalizers require less computing demands than

other equalizers (Burse et al., 2010).

Figure 1: RBF neural network (from Burse et al., 2010).

A comprehensive review on channel equalization

can be found in (Qureshi, 1985). A recent review on

Neural Equalizers can be found in (Burse et al.,

2010). The equalization scheme can be seen in Fig. 2

(taken from (Molisch, 2011)). The adaptive

equalizer in the figure is the RBF Neural equalizer

trained by EKF according to (Simon, 2002). The

considered channel uses the Rayleigh model

(Molisch, 2011) using QPSK modulation.

Figure 2: Equalization procedure (from Molisch, 2011).

The QPSK ideal constellation symbols are shown in

figure 3. In other words when the communications

channel is ideal, there is no distortion or noise so

that the symbols are always received with no error.

For a real channel the received symbols will show

some dispersion as shown in figure 4.

Figure 3: QPSK ideal constellation.

Figure 4: QPSK real scenario constellation.

Several simulations were performed for realistic

channel characteristics. Two case studies were

carried out.For the first case study, a flat fading

channel was considered. Flat fading channels have

amplitude varying channel characteristics and are

narrowband (Molisch, 2011). A transmission of an

ICEIS2015-17thInternationalConferenceonEnterpriseInformationSystems

660

image was included in both case studies. The

transmitted image is depicted in figure 5.

Figure 5: Original transmitted image in case studies.

The simulations also made possible to plot results

for comparing the performance in terms of Bit Error

Rate (BER) against Signal to Noise Ratio (SNR) and

Symbol Error Rate (SER) against SNR. The

received image for the RBF – EKF equalizer and the

Decision Feedback Equalizer (DFE) which is a quite

popular traditional equalizer is shown in figures 6

and 7.

Figure 6: RBF-EKF received image for flat fading.

Figure 7: DFB received image for flat fading.

In a qualitative way, one can see that the EKF-EBF

equalizes better. For a quantitative description figure

8 shows the BER x SNR and SER x SNR for

comparing the two equalizers. The theoretical curve

is also shown for comparative purposes. One can see

that the RBF-EKF equalizer performs better as the

comparison of the received images indicated. It can

be also seen that for low SNRs the performance of

the EKF-RBF equalizer is very close the theoretical

performance. As SNR values increase the equalizer

begins to get away from the theoretical model.

Figure 8: BER x SNR for case study 1.

Figure 9 shows a constellation diagram for the

equalizers in case study 1, and it can be seen a

cluster formation around the original symbols for

both equalizers, indicating that errors might occur in

the receiver output.

Figure 9: BER x SNR for case study 1.

In case study 2, a frequency selective fading was

considered which is a more severe type of fading

(Molisch, 2011). Figures 10 and 11 show the

received images corresponding to EKF-RFB and

DFB equalizers.

Figure 10: DFB received image for case study 2.

One can see a more intensive degradation in the

image for both equalizers, although the DFB is still

worse. The performance curves are depicted in

RadialBasisFunctionNeuralNetworkReceiverforWirelessChannels

661

figure 12 which shows clearly the degradation in

performance for both equalizers as far as frequency

selective fading is concerned.

Figure 11: EKF-RBF received image for case study 2.

Figure 12: BER x SNR for case study 2.

4 CONCLUSIONS

This paper proposed a radial basis function (RBF)

equalizer trained by an extended Kalman filter

(EKF). The advantages of using a Kalman filter for

training the RBF neural equalizer are that it provides

the same performance as gradient descent training,

but with only a fraction of the computational effort.

Moreover if the decoupled Kalman filter is used, the

same performance is guaranteed with further

decrease on the computational effort for large

problems. The equalizer was simulated and two case

studies were reported where its performance was

compared with the popular Decision feedback

equalizer and the results indicated the proposed

equalizer performed better. For future work the

authors intend to consider improvements on the RBF

equalizer as far as the tracking of time-variatons is

concerned.

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