ORIGINAL ARTICLE

Neural network-based adaptive multiuser detection schemesin SDMA–OFDM system for wireless application

Kala Praveen Bagadi • Susmita Das

Received: 9 February 2012 / Accepted: 21 June 2012

� Springer-Verlag London Limited 2012

Abstract Neural network applications in adaptive mul-

tiuser detection (MUD) schemes are suggested here in the

context of space division multiple access–orthogonal fre-

quency division multiplexing system. In this paper, various

neural network (NN) models like feed forward network

(FFN), recurrent neural network (RNN) and radial basis

function network (RBFN) are adopted for MUD. MUD

using NN models outperforms other existing schemes like

genetic algorithm–assisted minimum bit error rate (MBER)

and minimum mean square error MUDs in terms of BER

performance and convergence speed. Among these NN

models, the FNN MUD performs efficiently as RNN in full

load scenario, where the number of users is equal to

number of receiving antennas. In overload scenario, where

the number of users is more than the number of receiving

antennas, the FNN MUD performs better than RNN MUD.

Further, the RBFN MUD provides a significant enhance-

ment in performance over FNN and RNN MUDs under

both overload and full load scenarios because of its better

classification ability due to Gaussian nonlinearity. Exten-

sive simulation analysis considering Stanford University

Interim channel models applied for fixed wireless appli-

cations shows improvement in convergence speed and BER

performance of the proposed NN-based MUD algorithms.

Keywords SDMA � Multiuser detection � Maximum

likelihood � Genetic algorithm � Feed forward network �Recurrent neural network � Radial basis function network

1 Introduction

Orthogonal frequency division multiplexing (OFDM) is a

parallel transmission scheme, where a serial data stream

with high data rate is split into a set of low data rate sub-

streams by modulating it with orthogonal subcarriers. This

technique provides protection against inter symbol inter-

ference (ISI) and inter carrier interference (ICI) [1]. Space

division multiple access (SDMA) is a form of multiple

input–multiple output (MIMO) communication, which is

currently a major research focus. It allows many sub-

scribers to share a frequency band simultaneously, which

makes efficient use of the spectral band width. In the

SDMA uplink scheme, each user is equipped with a single

antenna, while the base station receiver possesses an array

of antennas. The multiple users in the SDMA system are

differentiated by the unique user’s specific channel impulse

response (CIR) at the receiver antenna [2]. Thus, both the

OFDM and SDMA technologies will be in great demand for

the next-generation broadband wireless communications

(4G) to solve the capacity problem. So the combination of

both stands as a major breakthrough for the next-generation

fixed and mobile wireless applications.

At the receiver’s end of the SDMA–OFDM system, the

multiuser detection (MUD) process plays a vital role where

a single receiver jointly detects multiple simultaneous users

over MIMO channel. MUD could be carried out by sepa-

rating the user signals transmitted on the same frequency

band, provided that they are separated in spatial domain

with their CIRs [3, 4]. However, the ISI and co-channel

interference (CCI) make the MUD as a complex task.

Hence, an efficient MUD technique is necessary for sepa-

rating the users appropriately in SDMA system. The most

popular MMSE MUD with very low complexity detects the

user’s signals at the cost of poor performance. On the other

K. P. Bagadi (&) � S. Das

Department of Electrical Engineering,

National Institute of Technology, Rourkela, India

e-mail: kalapraveen.bagadi@gmail.com

S. Das

e-mail: sdas@nitrkl.ac.in

123

Neural Comput & Applic

DOI 10.1007/s00521-012-1033-z

hand, the highly complex maximum likelihood (ML)

detector is capable of achieving the optimal performance

through an exhaustive search, which restricts its usage in

practical systems. Thus, the trade-off between complexity

and the BER performance draws considerable attention of

researchers [5–11]. Minimizing mean square error may not

give a guarantee that the BER of the system is also mini-

mized. Hence, Alias and Chen et al. [12, 13] proposed

minimum bit error rate (MBER) MUD, which minimizes

BER directly rather than mean square error unlike the

commonly used MMSE MUD. The MBER MUD with

conjugate gradient (CG) algorithm updates the linear MUD

weight vectors as proposed in [14], and it requires initial

decision of weight vector condition. Later, Alias et al. [15]

suggested genetic algorithm (GA)–assisted MBER. It is

preferred due to its faster convergence speed, and it does

not require any initial condition of weight vector. Subse-

quently, the MBER MUD algorithm was modified using

other advised evolutionary techniques like particle swarm

optimization (PSO) [16] and differential evolutionary (DE)

[17] algorithm. However, all these schemes exhibit almost

equal suboptimal performance, are complex and are limited

by their slow rate of convergence. Design of multiuser

detectors that are minimizing the error probability of

detection and are realistic from a computational complexity

point of view has been a major research focus.

The deployment of neural networks (NN) having highly

nonlinear parallel structures can become a good alternative

to the above discussed techniques [18]. There are wide

ranges of NN applications in communication system for

signal detection at receiver’s end [19, 20]. In SDMA–

OFDM system, at the receiver’s end, the signal gets cor-

rupted with both noise and multiuser interference. Hence, a

better classifier is necessary to distinguish users appropri-

ately from the combined received signal. Being adaptive

nonlinear classifiers, the neural network models can do

multiuser signal detection by mitigating multiuser inter-

ference. Earlier, several NN model–based MUD schemes

were utilized for CDMA system [21–33]. Application of

NN-based MUD in SDMA–OFDM communication system

is not reported prior in research literatures. In this research

article, we have tried to explore the possibility of using

NN-based MUD schemes in this system to enhance its

performance and adaptability in a multiuser environment.

In this article, the neural network models used for MUD are

feed forward neural network (FNN), recurrent neural net-

work and radial basis function network (RBFN). The

simulation results show that the NN model–based MUD

techniques perform better than GA MBER MUD, and

among NN models, the RBFN-aided MUD consistently

outperforms the FNN, RNN and other existing techniques

by providing near optimal BER performance and faster

convergence.

The rest of the paper is organized as follows. The gen-

eralized SDMA–OFDM system model along with the

mathematical representation of received signal is presented

in Sect. 2. Section 3 describes some of the existing MUD

techniques. The proposed NN model–based MUD tech-

niques are mentioned in Sect. 4. Simulation analysis with

results is discussed in Sect. 5. Finally, the conclusion is

provided in Sect. 6.

2 SDMA–OFDM system model

Figure 1 demonstrates the uplink transmission of SDMA–

OFDM system model [15]. In this figure, each of the L

simultaneous users is equipped with a single transmitting

antenna, and the base station is equipped with a P element

antenna array. This scenario can improve capacity of the

system. The received signal ‘y[n, k]’ at the kth subcarrier of

the nth OFDM block can be characterized by the super-

position of L independently transmitted user signals. Thus,

the received signal corrupted with AWGN at each fre-

quency bin can be expressed in vector form as:

y ¼ Hxþ n ð1ÞHere, the indices [n, k] are omitted for the sake of notational

convenience. In the above equation y ¼ y1; y2; . . .; yP½ �T , x ¼x1; x2; . . .; xL½ �T and n ¼ n1; n2; . . .; nP½ �T are the received

vector, the transmitted vector and the noise vector with zero

mean and variance r2n respectively. H is the frequency domain

channel matrix given as follows:

H ¼

H1;1 H1;2 � � � H1;L

H2;1 H2;2 � � � H2;L

..

. ... . .

. ...

HP;1 HP;2 � � � HP;L

26664

37775 ð2Þ

where HP,L is the channel gain between the Pth receiver

antenna and Lth user link. The lth (l = 1, 2, …, L) column

of channel matrix H is often referred to as the spatial sig-

nature of the lth user across the receiving antenna array.

Further, in the SDMA–OFDM system each user’s signal

separately undergoes OFDM modulation, which is descri-

bed as follows.

2.1 OFDM modulation and demodulation

The OFDM block diagram that includes a guard interval

(GI) to mitigate the impairment of multipath radio channels

is given in Fig. 2 [1]. In OFDM modulation, a large

number of closely spaced orthogonal subcarriers signals are

used to carry data. The orthogonality between subcarriers

must be carefully maintained in OFDM system to avoid

ICI. Initially, the input binary serial data is encoded using

any one of the conventional modulation techniques such as,

Neural Comput & Applic

123

BPSK, QPSK or QAM. While this binary data is trans-

formed into a multilevel signal, the symbol rate is reduced

to, D ¼ R=log2 M symbols/s, where R refers to the bit rate

of the data stream in bits/s and M denotes the modulation

order. When this serial data is converted to parallel, the

data rate gets further reduced by N, where N is the number

of parallel channels. So the parallel symbols are essentially

low data rate symbols, and since they are narrowband, they

experience flat fading. This parallel data stream is then

subjected to an IFFT to produce a time domain symbol

‘Xl(n)’. In OFDM, the time domain symbols appear as

frequency spectrum because these symbols are modulated

with multiple carrier frequencies. The OFDM-modulated

symbol ‘Xl(n)’ is represented mathematically as:

XlðnÞ ¼ IDFTfxlðkÞg; k ¼ 1; 2; . . .;N

¼XN�1

k¼0

xlðkÞejð2pkn=NÞ ð3Þ

After OFDM modulation, GI is inserted to suppress ISI

caused by multipath distortion. A GI is a copy of the last

part of the OFDM symbol, which is larger than the maxi-

mum delay spread and is ‘‘prepended’’ to the OFDM

symbol. This makes the symbol periodic, helps in identi-

fying frames correctly and avoids ISI. Then, this parallel

data is converted to serial data and transmitted through the

wireless channel.

At the receiver’s end, the serial data is converted back to

parallel form and GI is removed. Finally, the time domain–

received symbol ‘Yp(n)’ is passed through the FFT block

for extracting the frequency spectrum as follows:

ypðkÞ ¼ DFTfYpðnÞg; n ¼ 1; 2; . . .;N ð4Þ

The recovered binary data is obtained back through

‘‘signal demapper’’ block.

3 Existing multiuser detection techniques

Multiuser detection (MUD) is one of the receiver design

technologies for detecting desired user signal by eliminat-

ing noise and interference from neighborhood user’s signal.

Generally, multiuser system’s receiver suffers from the

inter user interference, where a strong user signal source

may influence the reception of weak user signal. The effect

of interference is more pronounced in SDMA like wireless

multiuser communication systems. MUD techniques are

used to overcome this problem. In the detection process,

the estimated signal vector ‘x’ can be expressed as:

x ¼ WHy ð5Þ

where ‘W’ is the (P 9 L) dimension weight matrix and ‘y’

is the received signal vector.

1x

H

11H21H

1PH

12H22H

2PH

1LH

2LH

PLH

User 1Signal

Mapper

User 2

User L

Signal

Mapper

Signal

Mapper

2x

Lx

1b

2b

Lb

OFDM

Modulator

OFDM

Modulator

OFDM

Modulator

1n

1n

Pn

OFDM

Demodulator

OFDM

Demodulator

OFDM

Demodulator

1y

2y

Py

Channel

Estimator

Mul

tius

er D

etec

tion

1x

2x

ˆLx

Signal

Demapper

Signal

Demapper

Signal

Demapper

1b

2b

ˆLb

Fig. 1 Block diagram of standard SDMA–OFDM system with L users and P receiving antennas

S/P

Con

vert

or

Mul

ti C

arri

er

Mod

ulat

or (

IFFT

)

Add

Cyc

lic P

refi

x

P/S

Con

vert

or

D/A

Con

vert

or

Input Data

Symbols

Output to

Transmitter

( )lx k ( )lX n

A/D

Con

vert

or

Rem

ove

Cyc

lic P

refi

x

S/

P C

onve

rtor

Mul

ti C

arri

er

Dem

odul

ator

(F

FT

)

P/S

Con

vert

or

Recovered

Symbols

Received

Data

( )PY n ( )py k

(a) (b)

Fig. 2 Schematic diagram of OFDM Block. a OFDM modulator, b OFDM demodulator

Neural Comput & Applic

123

3.1 Minimum mean square error (MMSE) detection

The most popular linear MMSE MUD scheme assumes

a priori knowledge of noise variance and channel covari-

ance. In this MMSE MUD, the weight matrix ‘W’ can be

expressed by minimizing mean square error, that is, MSE ¼E xl � xlj j2h i

, where xl is lth user signal xl is lth user esti-

mated signal. Thus, the weight vector wl is expressed as [15]:

wl ¼ ðHHH þ 2r2nIPÞ�1Hl ð6Þ

where (.)H indicates the Hermitian transpose and IP is P

dimensional identity matrix, wl is lth column of weight

matrix W and Hl is lth column of channel matrix H. In

general, the received signal contains residual interference

which is not Gaussian distributed due to multiuser inter-

ference. But these linear detectors assume that the received

signal is corrupted only by AWGN. Hence, a nonlinear

detector is essential to mitigate this residual interference.

3.2 Maximum likelihood (ML) detection

The high complex, nonlinear and optimal ML detector uses

an exhaustive search for finding the most likely transmitted

user vector. For a ML detector supporting L simultaneous

transmitting users, a total of 2mL metric evaluations have to

be invoked in order to detect the transmitted symbol vector

x, where m denotes the number of bits per symbol. ML

detector uses the maximum a posterior (MAP) detection

when all the transmitted vectors are equally likely. The ML

detector calculates the Euclidean distance between the

received signal vectors and the product of all possible

transmitted signal vectors with the given channel and finds

the minimum distance. The solution of ML detector can be

expressed as follows [2].

x ¼ arg minu

y� H~xuk kn o

; u ¼ 1; 2; . . .; 2mL ð7Þ

where ~x is most likely transmitted symbol vector and u is

the set of total metric evaluations.

3.3 Genetic algorithm (GA)–assisted minimum bit

error rate (MBER) detection

In MBER MUD, the detection of user ‘l’ can be described

by minimizing probability of error PE encountered at the

receiver’s end of the SDMA–OFDM system, which is a

function of weight vector wl, where wl is the lth column of

weight matrix W. For a BPSK modulation scheme, the

probability of error can be expressed as [15]:

PEðwlÞ ¼1

Nb

XNb

j¼1

QsgnðbðjÞl ÞwH

l �yj

rffiffiffiffiffiffiffiffiffiffiwH

l wp

" #ð8Þ

where Nb is the number of equiprobable combinations of

the binary vectors of the L users, i.e. Nb = 2L, bðjÞl ; j 2

1; . . .;Nb is the transmitted bit of user l, and �yj; j 2 1; . . .;Nb

constitutes a possible value of the noiseless received signal

vector y. The MBER solution is defined as:

wlðMBERÞ ¼ arg minwl

PEðwlÞ ð9Þ

Relatively, Eq. (9) can be used as the cost function to

optimize weights in the other evolutionary techniques such

as PSO [16] and DEA [17].

4 Proposed neural network (NN)-based detection

techniques

The complex nonlinear characteristics of the NN models

are led to use it widely in several equalization problems. In

the NN-based MUD process, NN model will be designed

according to the SDMA structure, and then the model will

be trained using training symbol vectors. Generally, the

training algorithms like back propagation (BP) [18], real-

time recurrent learning (RTRL) [34] and least mean square

(LMS) [18] are applied to FNN, RNN and RBF structures,

respectively. The well-trained network is used as multiuser

detector in the testing phase, and the received signal vector

of the SDMA–OFDM system is fed to the trained networks

to detect transmitted signal vector. The wireless channel

response should be constant during training and testing

period. In the NN models, the system parameters are

considered according to:

xt is a (L 9 1) training symbol vector fed to SDMA–

OFDM system shown in Fig. 1, which is a possible

transmitted vector from the set ~xu.

yt is a (P 9 1) response vector of xt from SDMA–

OFDM system as given in Eq. (1).

d is a (2L 9 1) desired vector in training phase response

vector corresponding to real and imaginary parts of xt.

yI is a (2P 9 1) input vector fed to NN models

corresponding to real and imaginary parts of y in testing

phage and corresponding to real and imaginary parts of

yt in training phage.

xK is a (2L 9 1) output vector from the NN models

corresponding to real and imaginary parts of x in testing

phase and corresponding to real and imaginary parts of

xt in training phase.

4.1 Feed forward neural network (FNN) model–based

MUD

The FNN with a hidden layer does better complex map-

ping tasks and has high computational efficiency. The

Neural Comput & Applic

123

architecture shown in Fig. 3 is a simple FNN structure in

testing phase consisting of an input layer of ‘2P’ units, one

hidden layer of ‘HN’ nodes (neurons) and an output layer

of ‘2L’ nodes. In FNN each neuron in the hidden layer

consists of a nonlinear activation such as sigmoid function

and summation operation. And each neuron in the output

layer has a simple linear input–output relationship, so that

they perform simple summations. Hence, the resultant

output at jth node in the hidden layer ‘J’ can express at a

time instant ‘n’ as:

zjðnÞ ¼ uX2P

i¼1

VjiðnÞyIi ðnÞ þ bJ

j ðnÞ !

; j ¼ 1; 2; . . .;HN

ð10ÞThe resultant output at kth node in the output layer ‘K’ is

expressed at a time intent ‘n’ as:

xKk ðnÞ ¼

XHN

j¼1

UkjðnÞzjðnÞ þ bKk ðnÞ; k ¼ 1; 2; . . .; 2L ð11Þ

where, Vji denote a weight associated with the connection

between hidden node j and input node i, Ukj denote a

weight associated with the connection between output node

k and hidden node j, bJj denote bias of the hidden node, and

bKk denote bias of the output node.

uð:Þ denote a nonlinear function having sigmoid

nonlinearity.

In the FNN training process, an iterative gradient des-

cent algorithm that minimizes an empirical error function

such as BP algorithm can be used efficiently to update

connection weights and thresholds to minimize error as

summarized below [18].

1. Initialize randomly all connection weights and thresh-

olds such as VjiðnÞ;UkjðnÞ; bJj ðnÞ and bK

k ðnÞ at iteration

‘n’ (=1).

2. Compute the hidden vector ‘zj(n)’ and output vector

‘xKk ðnÞ’ from Eqs. (10) and (11), respectively.

3. Compute the error term ‘ek(n)’ of each output node as:

ekðnÞ ¼ dðnÞ � xKk ðnÞ; k ¼ 1; 2; . . .; 2L ð12Þ

4. However, the BP algorithm requires the error gradient

d at each layer. Thus, the error gradients at kth node of

output layer and jth node of hidden layer are given,

respectively:

dkðnÞ ¼ ekðnÞ ð13Þ

djðnÞ ¼X

k

dkðnÞUkjðnÞf0 ðzjðnÞÞ ð14Þ

Here f0 ð:Þ represents first derivative function.

5. Update the weights and biases of hidden nodes and

output nodes are from:

Vjiðnþ 1Þ ¼ VjiðnÞ þ gdjðnÞyIi ðnÞ ð15Þ

bJj ðnþ 1Þ ¼ bJ

j ðnÞ þ gdjðnÞ ð16Þ

Ukjðnþ 1Þ ¼ UkjðnÞ þ gdkðnÞzjðnÞ ð17Þ

bKk ðnþ 1Þ ¼ bK

k ðnÞ þ gdkðnÞ ð18Þ

where g is the learning rate parameter, which should be

in between zero and one.

6. Compute the total error dðnÞ � xKðnÞk k2and proceed

the computation to next iteration (n ? 1) from Step 2

until this error is less than the specified value.

4.2 Recurrent neural network (RNN) model–based

MUD

Unlike FNN, RNN can use their internal memory to pro-

cess arbitrary sequences of inputs, and also it has low

structural complexity by eliminating extra hidden layer

}1 1Re Iy y→ jiV

2

Jb

2 L

Jb

2 1−L

Jb

1

Jb

2

Kb

2 L

Kb

2 1−L

Kb

1

Kb

∑

∑

∑

∑

kjU1z

HNz

1HNz −

2z

} 2Im IP Py y→

} 2 1Re IP Py y −→

}1 2Im Iy y→

Layer I Layer J Layer K

1Kx

2KLx

2 1KLx −

2Kx

∫

∫

∫

∫

1y1n

PyPn

Rx1

Rx P

1x

i

ˆLx

i

OFDM

De mod ulator

OFDM

De mod ulator

+

+

+

+

{

{

{

{

Fig. 3 Schematic diagram of proposed FNN MUD at the receiver’s end of SDMA–OFDM system

Neural Comput & Applic

123

used in FNN without much performance degradation.

A RNN is a class of neural network, which has an input

layer ‘2P’ external input units and an output layer of ‘2L’

neurons with each neuron feeding its output signal back to

the inputs of all neurons as illustrated in the architecture

shown in Fig. 4. This creates an internal state of the net-

work which allows it to exhibit dynamic temporal

behavior.

The external input vector yIf ðnÞ; f ¼ 1; 2; . . .; 2P and one

step delayed output vector xKk ðnÞ; k ¼ 1; 2; . . .; 2L with an

initial value zero (xKk ð1Þ ¼ 0) are connected to form an

input vector u(n) to the RNN module, whose hth element at

time index n is denoted by uhðnÞ, which is given by:

uhðnÞ ¼xK

k ðnÞ if h 2 ½1; 2; . . .; 2L�yI

f ðnÞ if h 2 ½2Lþ 1; 2Lþ 2; . . .; 2Lþ 2P�

�

ð19Þ

Let Wkh is a 2L 9 (2L ? 2P) weight matrix, then the

output vector can be expressed as:

xKk ðnÞ ¼ u

X2Lþ2P

h¼1

WkhðnÞuhðnÞ þ bkðnÞ !

;

k ¼ 1; 2; . . .; 2L

ð20Þ

In the network training mechanism, RNN uses the

powerful algorithm called as RTRL to update connection

weights [34].

1. Initialize randomly connection weights ‘WkhðnÞ’ and

bias ‘bkðnÞ’ at iteration ‘n’.

2. Compute the output vector ‘xKk ðnÞ’ by the use of Eq.

(20).

3. Compute the error term ek(n) of each output neuron is

ekðnÞ ¼ dðnÞ � xKk ðnÞ; k ¼ 1; 2; . . .; 2L ð21Þ

4. Application of RTRL algorithm involves primarily

the evaluation of sensitivity parameter; a triply

indexed set of variables pkjh

n o, which is evaluated as

follows:

khW

2b

2Lb

2 1Lb −

1b}1 1Re Iy y→

}1 2Im Iy y→

} 2 1Re IP Py y −→

} 2Im IP Py y→

1Kx

2Kx

2 1KLx −

2KLx

1y

Py

1n

Pn

i

i

1x

ˆLx

∫

∫

∫

∫

Rx 1

Rx P

OFDM

Demodulator

OFDM

Demodulator

+

+

+

+

1Z−

1Z−

1Z−

1Z−

{

{

{

{

Fig. 4 Schematic diagram of proposed RNN MUD at the receiver’s end of SDMA–OFDM system

Neural Comput & Applic

123

pkjhðnþ 1Þ ¼ f

0 ðxKk ðnÞÞ

X2P

i¼1

WkiðnÞpkjhðnÞ þ djkuhðnÞ

" #;

j ¼ 1; 2; . . .; 2L; h ¼ 1; 2; . . .; 2Lþ 2P

ð22Þ

with an initial condition pkjh ¼ 0 and djk is termed as

Kronecker delta as given by, djk ¼ 1 for j = k, zero

otherwise.

5. Compute the incremental change and adjust the

connection weights according to:

Wkhðnþ 1Þ ¼ WkhðnÞ þ gX2L

k¼1

ekðnÞpkjhðnÞ ð23Þ

where g is learning rate parameter that lies between

zero and one.

6. Compute the total error dðnÞ � xKðnÞk k2, and iterate

the computation by returning to Step 2 until this error

is less than a specified value.

4.3 Radial basis function network (RBFN)

model–based MUD

The architecture of RBFN is a three-layered feed forward

network, which consists an input layer of ‘2P’ number of

input units and an output layer of ‘2L’ number of neurons

and also with the hidden layer of ‘HN’ number of neurons

existing between input and output layers as shown in Fig. 5.

The inter connection between input layer and hidden layer

forms hypothetical connection and between the hidden and

output layer forms weighted connections. The sigmoid type

of activation function used in FNN does not yield the

approximation capabilities of RBFN, which uses Gaussian

activation function. The response of such function is non-

negative for all values of input vector, which is defined as:

f ðrÞ ¼ exp�r2

r2

� �ð24Þ

Here, r is spreading parameter (width), and ‘r’ is the

distance between input vector and center vector. Distance

is usually measured by Euclidean norm. If sufficient

}1 1Re Iy y→

∑

1z

HNz

1HNz −

2z

} 2Im IP Py y→

} 2 1Re IP Py y −→

}1 2Im Iy y→

1Kx

2KLx

2 1KLx −

2Kx

Ω

Ω

Ω

∑

∑

∑

Layer I Layer J Layer K

Ω

kjW

1y1n

PyPn

1x

ˆLx

i

i

Rx1

Rx P

OFDM

Demodulator

OFDM

Demodulator

+

+

+

+

{

{

{

{

Fig. 5 Schematic diagram of proposed RBFN MUD at the receiver’s end of SDMA–OFDM system

Table 1 Simulation parameters

Value

Parameters

Number of subcarrier 128

Length of guard band 32

Number of OFDM frames 1,000

Number of receiving antennas 4

Number of users 4 in full load scenario

1–10 in overload

scenario

Modulation scheme BPSK

Channel condition As specified in

Tables 2 and 3

Neural network models parameters

Number of input elements 8 (2P)

Number of output elements 8 (2L)

Number of hidden layers in FNN model 1

Number of hidden neurons in FNN and

RBFN models

16 (2L)

Activation function used in FNN and

RNN models

Bipolar sigmoid

Activation function used in RBFN models Gaussian

GA parameters As specified in [15]

Neural Comput & Applic

123

number of hidden neurons (around HN = 2L) are taken and

also center vectors and connection weights are appropri-

ately tuned, then the RBFN with Gaussian neurons can able

to approximate wide range of pattern classification and

curve fitting problems. Let the input vector be denoted by

yIi and let the center vector of each hidden neuron be

denoted by Cjðj ¼ 1; 2; . . .;HNÞ of size (2P 9 1), then the

output of each neuron in the hidden layer is

zj ¼ exp �X2P

i¼1

ðyIi � CjÞ2

r2j

!; j ¼ 1; 2; . . .;HN ð25Þ

The neurons in the output layer are simple summing

elements. Hence, the output of each neuron of output layer

is

xKk ¼

XHN

j¼1

Wkjzj; k ¼ 1; 2; . . .; 2L ð26Þ

where, Wkj denote a weight associated with the connection

between output neuron k and hidden neuron j.

However, the modeling of RBFN mainly depends on

selection of centers and proper approximation of connection

weights. In the training mechanism, it is essential to fix the

centers of hidden neurons before updating connection

weights. The k-means clustering algorithm is one of effi-

cient techniques used to fix centers [18]. The recursive

gradient descent algorithms such as LMS algorithm used to

update connection weights followed by fixing center are

given as follows:

1. Initialize randomly all connection weights at iteration

‘n’.

2. Compute the output vector xKk ðnÞ by the use of

Eq. (24).

3. Compute the error term ek(n) of each output neuron,

which is

ekðnÞ ¼ dðnÞ � xKk ðnÞ; k ¼ 1; 2; . . .; 2L ð27Þ

4. Adjust the connection weights according to:

Wkjðnþ 1Þ ¼ WkjðnÞ þ gekðnÞzjðnÞ ð28Þ

where g is learning rate parameter that lies between

zero and one.

5. Compute the total error dðnÞ � xKðnÞk k2, and iterate

the computation by returning to Step 2 until this error

is less than the specified value.

5 Simulation analysis

In this section, simulation results are provided which

illustrate the improved BER performance of proposed

MUD schemes over existing ones in SDMA–OFDM sys-

tem. In full load scenario, four receiving antennas and four

users are considered. The bit error rate (BER) for each user

Table 2 SUI-A channel model parameters

Tap 1 Tap 2 Tap 3 Units

Delay 0 14 20 ls

Power 0 -10 -14 dB

k factor 1 0 0

Doppler shift 0.4 0.3 0.5 Hz

Antenna correlation 0.3

Antenna type Omni directional antenna

Table 3 CIR of the different

users at the different antennas

for the P = 4, L = 4 system

User Antenna CIR

User 1 1 (0.5017 ? 0.0304i) ? (0.1894 ? 0.0739i)z-1 ? (0.1505 - 0.0214i)z-2

2 (0.5363 ? 0.1908i) ? (0.1457 - 0.4261i)z-1 ? (0.0090 - 0.0840i)z-2

3 (0.4032 - 0.0003i) ? (0.1395 ? 0.0966i)z-1 ? (0.0939 - 0.0119i)z-2

4 (0.2867 - 0.0170i) ? (0.0902 ? 0.0911i)z-1 ? (0.0439 - 0.0028i)z-2

User 2 1 (0.1705 - 0.0197i) ? (0.0485 ? 0.0660i)z-1 ? (0.0111 ? 0.0024i)z-2

2 (0.0711 - 0.0121i) ? (0.0182 ? 0.0321i)z-1 - (0.0021 - 0.0030i)z-2

3 (0.5591 ? 0.1815i) ? (0.2133 - 0.3020i)z-1 ? (0.1216 ? 0.0355i)z-2

4 (0.5856 ? 0.1551i) ? (0.2493 - 0.1779i)z-1 ? (0.1930 - 0.0006i)z-2

User 3 1 (0.5935 ? 0.1160i) ? (0.2535 - 0.0673i)z-1 ? (0.2165 - 0.0209i)z-2

2 (0.5671 ? 0.0716i) ? (0.2307 ? 0.0189i)z-1 ? (0.1975 - 0.0264i)z-2

3 (0.9184 ? 0.0835i) - (0.3334 ? 0.6678i)z-1 - (0.4045 - 0.1738i)z-2

4 (0.7950 ? 0.1069i) - (0.2622 ? 0.6887i)z-1 - (0.4124 - 0.2070i)z-2

User 4 1 (0.6747 ? 0.1343i) - (0.1659 ? 0.6735i)z-1 - (0.3611 - 0.2085i)z-2

2 (0.5851 ? 0.1619i) - (0.0563 ? 0.6215i)z-1 - (0.2590 - 0.1820i)z-2

3 (0.5397 ? 0.1829i) ? (0.0522 - 0.5364i)z-1 - (0.1266 - 0.1366i)z-2

4 (0.6631 - 0.1386i) - (0.2650 ? 0.2894i)z-1 - (0.0077 ? 0.1156i)z-2

Neural Comput & Applic

123

is calculated by varying signal-to-noise ratio (SNR). The

second case is overload scenario, in which we have taken

four receiving antennas and keeping the SNR fixed at 5 dB,

BER of user 1 using various MUD techniques is calculated

by increasing the number of users to verify the efficiency of

the proposed MUD schemes in a multiuser scenario by

suppressing interference from the neighborhood users. The

remaining simulation parameters are summarized in

Table 1.

The Stanford University Interim (SUI) channel models

for fixed wireless applications are considered for the

simulations. The characterization of these models depends

on the territorial structures in propagation environment and

classified as A-, B- and C-category. The maximum path

loss category is the hilly terrain with moderate-to-heavy

tree densities (A-category). Intermediate path loss condi-

tions are captured in B-category, which is either high tree

density and flat area or low tree density and hilly area. The

minimum path loss category is mostly the flat terrain with

light tree densities (C-category) [35]. With a view to

maximum path loss conditions, SUI-A category channel

model is chosen here and the parameters of the model are

0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

10 0

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

0 2 4 6 8 10 1210-5

10-4

10-3

10-2

10-1

10 0

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

(a) (b)

0 2 4 6 8 10 12 14 16 18 20 2210-5

10-4

10-3

10-2

10-1

100

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

0 2 4 6 8 10 12 1410-5

10-4

10-3

10-2

10-1

100

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

(c) (d)

Fig. 6 The BER performance of the four different users in an SDMA system employing four receiver antennas under SUI channel conditions

given in Table 1. a User 1, b user 2, c user 3, d user 4

Neural Comput & Applic

123

summarized in Table 2. This channel is modeled as mul-

tipath, frequency selective with nonuniform delays, and the

number of taps used is three. The gain associated with first

tap is characterized by a Rician distribution, and the gain

associated with remaining two taps is characterized by a

Raleigh distribution. This channel is assumed to be con-

stant during training and testing phases in NN model–based

detection schemes. Further study over the Gaussian chan-

nel model given in Table 3 [17] is done to ensure the

capability of the proposed MUD techniques under different

channel conditions.

Figure 6a–d illustrates a comparative BER performance

of discussed MUD techniques for four different users in

SDMA–OFDM system provided with four receiving

antennas under SUI-A channel condition as given in

Table 1. From these curves, it is observed that the BER

performance level of 10-4 achieved by different users is at

unequal SNR values because the CIR of each user varies

0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

100

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

0 5 10 15 20 25 30 3510-6

10-5

10-4

10-3

10-2

10-1

100

SNR in dB

Bit

Err

or R

ate

MMSEGA MBERRNN

FNNRBFML

(a) (b)

Fig. 7 The average BER performance of the four different users in an SDMA system employing four receiver antennas over (a) SUI channel

model given in Table 1. b Gaussian channels [17]

1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

10-1

100

Number of Users

Bit

Err

or R

ate

MMSEGA MBERRNNFNNRBFML

Fig. 8 BER performance of user 1 employing various MUD

techniques in an SDMA–OFDM system with four receiving antennas

at 5 dB SNR while increasing number of users

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010-5

10-4

10-3

10-2

10-1

100

Complexity

Bit

Err

or R

ate

GA MBERFNNRNNRBF

Fig. 9 Learning curves with respect to bit error rate of user 1 at SNR

10 dB in an SDMA–OFDM system using GA MBER and NN base

MUD schemes

Neural Comput & Applic

123

due to different multipath propagations. For example, SNR

values at 10-4 BER level for MMSE MUD technique are

22, 11, 21 and 13 dB for user 1, 2, 3 and 4, respectively.

The average BER performance of four different users in

an SDMA–OFDM system equipped with four receiving

antennas for SUI-A channel condition is shown in Fig. 7a,

and the same for Gaussian channel condition is shown in

Fig. 7b, which indicates the efficacy of suggested MUD

schemes for various channel conditions. It is inferred from

Fig. 7 that being a linear detector, the MMSE MUD cannot

mitigate multiuser interference adequately; hence, it results

in poor BER performance, while ML MUD gives the

optimal performance. It is also observed that the BER

performance of FNN and RNN MUDs is almost the same

and better than GA MBER MUD. Further, the BER per-

formance of RBFN MUD shows near optimal performance

and close to ML MUD. More explicitly in Fig. 7a, at 10-4

BER level, the FNN and RBFN MUDs have 0.5 dB and

2 dB SNR gains, respectively, with respect to GA MBER.

Similarly in Fig. 7b, at 10-4 BER level, the FNN and

RBFN MUDs have 1.5 dB and 6 dB SNR gains, respec-

tively, with respect to GA MBER. So it is observed that the

RBFN MUD exhibits near optimal performance regardless

of the channel conditions.

Robustness of suggested NN MUD schemes is further

analyzed through simulation of an SDMA–OFDM system

with overload condition (L [ P). Figure 8 shows the

resultant BER of user 1 for fixed SNR of value 5 dB, when

supporting different number of users in an SDMA system

with four receiving antennas over uncorrelated MIMO flat

Rayleigh fading channel. Up to full load scenario (L B P),

the BER performance of all the suggested NN MUDs

tolerable and the RBFN one is close to ML MUD because

the multiuser interference does not affect much. As the

number of users increases, it is observed that BER perfor-

mance is affected owing to the increased multiuser inter-

ference imposed. Furthermore, it is found that RBFN MUD

being a nonlinear classifier can support MUD, whereas

others are in capable of differentiating the users in the

overload scenario.

Figure 9 shows the BER of suggested MUD techniques

of user 1 with increasing the complexity at SNR value of

10 dB. The complexity of GA MBER MUD can be eval-

uated through the number objective function evaluations.

In each iteration, the cost function of Eq. (9) will have to be

calculated, and the SDMA–MUD weight values will be

updated accordingly. Similarly, the complexity of the NN

MUDs is proportional to the number of training sample

vectors needed to update the connection weights to mini-

mize the error. Therefore, the complexity of the NN MUD

schemes may be estimated in terms of the number of

iterations. In this figure, the rate of convergence of the

RBFN detector is faster, and its complexity is less as it

consumes less number of training symbol vectors to attain

minimum BER level.

Further, the mean square error versus number of training

sample vector in Fig. 10 illustrates the convergence of the

NN models used for SDMA–OFDM schemes. From this

figure, it is observed that the MSE of the RBFN is faster

compared to RNN and FNN models. In this figure, the

RBFN MUD reaches the minimum MSE level at around

120 training sample vectors, whereas the RNN and FNN

models reach minimum MSE level at around 900 training

sample vectors.

6 Conclusion

In this investigation, we have suggested efficient NN-based

MUD schemes for SDMA–OFDM wireless system. It is

observed that the conventional MMSE MUD shows poor

performance, ML MUD is highly complex, and GA MBER

gives suboptimal performance at the cost of complexity.

The nonlinear classification capability offered by NN

models is highly beneficial for MUD by mitigating multi-

user interference. Further, the NN models are adaptable

structures that can be reconfigured according to the number

of users in the SDMA system. The NN-based MUD tech-

niques are able to outperform GA MBER MUD in both full

load and overload scenarios and possess low complexity.

The simulation results demonstrated that the RBFN MUD

has a substantial improvement in SNR compared to FNN,

RNN and GA MBER MUD techniques, and it has capa-

bility of showing performance close to optimal ML

detector in all cases. The RBFN MUD requires less number

of training samples to converge compared to the other NN

models. Thus, the application of NN models at the receiver

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

Number of Training Sample Vectors

Mea

n S

quar

e E

rror

RNNFNNRBFN

Fig. 10 Learning curves with respect to mean square error of user 1

at SNR 10 dB in an SDMA–OFDM system using NN base MUD

schemes

Neural Comput & Applic

123

end of SDMA–OFDM system for MUD gives a promising

solution for wireless communication.

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