 Open Access
 Total Downloads : 2
 Authors : Bhavitra.M.B, Ms.M.Kanthimathi
 Paper ID : IJERTCONV2IS12015
 Volume & Issue : NCACCT – 2014 (Volume 2 – Issue 12)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Complexity Reduction of Maximum Likelihood Detector using closed form approximation in Cooperative Networks
Bhavitra.M.B, ME,Communication Systems and Ms.M.Kanthimathi, Assistant Professor,Department of ECE, Sri Sai Ram Engineering College, Chennai.
Email:bhavimadhavan@gmail.com, kanthimathi.ece@sairamedu.in
Abstract The maximum likelihood detection in differential AF and DF cooperative networks using Mary differential phase shift keying with Rayleigh fading, has practical complexity. To reduce the complexity of the maximum likelihood detector, two algorithms were analysed. The first algorithm eliminates number of symbols in ML search. In high signaltonoise ratios, this algorithm determines a single symbol, means without calculating the likelihood function, maximum likelihood estimate of the information symbol, can be obtained. For low to medium signalto noise ratios multiple symbols were determined, thus the closedform approximate expression was derived, which is very accurate throughout the whole signal to noise ratio range. Closed form approximation requires only five sample evaluation per symbol. Combining these algorithms closedform approximate maximum likelihood detector was obtained. Finally, the obtained BER performance of closed form ML detector is compared with bit error performance of diversity combiner.
Index terms Maximum Likelihood Detector (ML), Amplify and Forward (AF), Decode and forward (DF), MDifferential Phase shift keying (MDPSK).
I.INTRODUCTION
In cooperative networks the function of relay is to retransmit received signal from source to destination. Here the information symbol is modulated by differential phase shift keying and corresponding symbol is detected using maximum likelihood detector at the receiver. Objective of the paper is to reduce complexity of maximum likelihood detector to estimate the single information symbol. As the instantaneous channel state information is not available at the receiver Mdifferential phase shift keying modulation is used. To reduce complexity, two algorithms were analysed and denoted as closed form approximate ML detector. In high SNR, the algorithm detects single information symbol without calculating likelihood function. For low to medium SNR, multiple symbols were detected hence, calculating likelihood function becomes inevitable. Thus closed form approximation was made for

RELATED WORKS
In the existing system, non coherent modulations were used in amplify and forward networks. In [2] maximum likely
ctois used for noncoherent frequency shift keying in Rayleigh fading. However ML detection in [2] has no closed form solutions are very complex for implementation. In [3] suboptimum detectors is used, it has lower complexity but some degradation in performance. In [4] simple diversity combiner is used but diversity combiner suffers from strict sub optimality as it does net exploits non Gaussianity of overall noise at destination. This may lead to significant performance loss compared with optimum (ML) detector.

SYSTEM MODEL
Cooperative networks consist of source S, relay R, and destination D shown in Fig 1. Let hsd denote channel coefficient of direct link between S and D, hsr denote channel coefficient of first hop from S and R, hrd denote channel coefficient of second hop from R and D.
Relay R
hs,r hr,d
hs,d
Source Destination
S D
Fig1: General cooperative network
calculating the likelihood function.
Channel coefficients are independent circularly symmetric complex Gaussian random variables with zero mean and variances sd , sr and rd respectively. Here MDPSK modulation is used hence each information symbol is drawn from
Mphase constellation set :={ej2n/M:n=0,….,M1} and is differentially encoded in to two information bearing symbols by
{
( ) ( )
x(k)=x(k1)c(k) ,k=1,2,… is symbol index. Transmission divided into two phases. In phase1, S broadcasts x(k) with power Es, while R and D listen. The received signals at the relay and the
( ) ( ) } (8)
destination are, respectively, given by
( ) ( ) ( ) (1)
( ) ( ) ( ) (2)
In phase 2, R scales its received signal by a factor ( ) and forwards to destination. Then received signal at phase 2 is given by,
( ) Es ( ) ( ) ( ) ( )
( ) (3)
A. Amplify and forward scheme:
For phase 2 relay which amplifies the received signal by a factor
( )
( ) ( ( )) (4)
( ) (9)
( ) (10)
( ) (11)
( ) (12)
V. ML DETECTOR APPROXIMATE METHOD For low to medium SNR number of symbols
estimated was more than one, so likelihood function calculation becomes inevitable. So the analysed simple closed form approximation was given below.

ClosedForm Approximation
Using five sample Gaussian Legendre quadrature the term ( ) can be approximated as follows:

Decode and forward scheme:
( )
( )
(14)
Here the relay first decodes the received signals,
{ ( )}
( )
(15)
( ) ( { ( ) ( )}) (5)
( )( )
The decoded signal is reencoded as follows
( ) ( ) ( ) (6)
usr(k ), usd(k ), urd(k ) are noise components and are
independent CSCG random variable with variance 2. The average received SNR of StoR, StoD, and RtoD are expressed as ,
.
IV. ML DETECTOR FOR MDPSK
In DPSK signalling, information c(k) is carried by the phase difference of two consecutive signals, hence the receiving signals has been considered as vector.

ClosedForm Approximate condition for symbol estimation
For  = 1, the ML estimate of c(k) is immediately obtained from the singleton without solving the LF integral. For > 1, the LF integral is approximately calculated. Therefore, closedform approximate ML detectors are as follows:
{   ( ) (16)
( ) ( ) { }
The parameters involved in this above equation are defined as,
{ )
{
[
) [
) ( )
{
(17)
The received signal can be detected using ML detector and it would be reduced using equation (13) and the parameters involved in reduction equation are as follows:
(18)
( )
(19)
{ ( ) ( )} (7)
{ ( ) ( ) ( )}
(20)
 ( ) ( ) ( )
 ( ) ( ) ( )
(21)
(22)
Thus the estimated value ( ) should be within constellation .

Complexity Comparison
The diversity combiner equation used to estimate the information symbol is as follows:
( ) {[ ( ) ( )
( )
( ) ( )] ( ) } ( )
Here for estimating the single information symbol, the detector needs to check the entire constellation set.
VI. SIMULATION RESULTS
The distance between source and destination is normalized to unity and denoted by d, 0 d 1.The total signal transmission power in the network is E=Es+Er. For various SNR, average bit error rate value can be evaluated and the inference values are tabulated for both amplify and forward networks, as well as decode and forward networks by using ML detector and diversity combiner.

Simulation Graph:
Fig 2:4DPSK for Amplify and Forward network
Fig 3:8DPSK for Amplify and Forward network
Fig 4:4DPSK for Decode and Forward network
Fig 5:8DPSK for Decode and Forward network/p>

Inferences: Average BER
SNR D 
AMPLIFY AND FORWARD 

ML ETECTOR 
DIVERSITY INER 

4D 
8D 
4D 
8D SK 

0 
0.3 
0.3 
0.71 
0.6 

5 
0.3 
0.2 
0.67 
0.4 

10 
0.2 
0.0 
0.63 
0.2 

15 
0.1 
0.0 
0.53 
0.0 

20 
0.0 
0.0 
0.33 
0.0 

25 
0.0 
0.0 
0.12 
0.0 
SNR D 
DECODE AND FORWARD 

ML ETECTOR 
DIVERSITY INER 

4D 
8D 
4D 
8D SK 

0 
0.2 
0.2 
0.49 
0.6 

5 
0.1 
0.1 
0.27 
0.4 

10 
0.0 
0.0 
0.11 
0.1 

15 
0.0 
0.0 
0.04 
0.0 

20 
0.0 
0.0 
0.01 
0.0 

25 
0.0 
0.0 
0.00 
0.0 
Table 1: Average BER for Amplify and Forward
Table 2: Average BER for Decode and Forward VII CONCLUSION
In this paper, closed form approximation for maximum likelihood detector was derived for Amplify and Forward, decode and forward cooperative networks employing M Differential phase shift keying in Rayleigh fading. The algorithm was analysed, which can substantially reduce the complexity of ML detection. Specifically, the first algorithm substantially narrows down the ML search while causing no loss of optimality. The second algorithm provides a very accurate closedform approximation for the LF. Combining these algorithms, a closedform approximate ML detector was obtained. In future closed form approximation can be done for amplify and forward networks using multiple relays.
REFERENCES

Peng Liu, Student Member, IEEE, IlMin Kim, Senior Member, IEEE,and Saeed Gazor, Senior Member, IEEE,MaximumLikelihood Detector for Differential AmplifyandForward Cooperative Networks IEEE Trans. Vehicular Technology, vol. 62, no. 8, October 2013.

R. Annavajjala, P. C. Cosman, and L. B. Milstein, On the performance of optimum noncoherent amplifyand forward reception for cooperative diversity, in Proc. IEEE MILCOM, Oct. 2005, pp. 32803288.

M. R. Souryal, Noncoherent amplifyandforward generalized likelihood ratio test receiver, IEEE Trans. Wireless Commun., vol. 9, no. 7,pp. 23202327, Jul. 2010.

Q. Zhao, H. Li, and P. Wang, Performance of cooperative relay with binary modulation in Nakagamim fading channels, IEEE Trans. Veh.Technol., vol. 57, no. 5, pp. 33103315, Sep. 2008.

G. Farhadi and N. C. Beaulieu, A low complexity receiver for noncoherent amplifyandforward cooperative systems, IEEE Trans. Commun., vol. 58, no. 9, pp. 2499 2504, Sep. 2010.

J. G. Proakis, Digital Communications, 4th ed. New York, NY, USA:McGrawHill, 2000.

Q.Zhao and H.Li, Differential modulation for cooperative wireless systems, IEEE Trans. Signal Process., vol. 55, no. 5, pp. 22732283,May 2007.