Research Article A Novel Adaptive Algorithm Addresses...

Research ArticleA Novel Adaptive Algorithm Addresses Potential Problems ofBlind Algorithm

Muhammad Yasin and Muhammad Junaid Hussain

National University of Sciences and Technology H-12 Islamabad Pakistan

Correspondence should be addressed to Muhammad Yasin myasinkhattakgmailcom

Received 1 February 2016 Revised 17 April 2016 Accepted 3 May 2016

Academic Editor Yu Jian Cheng

Copyright copy 2016 M Yasin and M J Hussain This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A hybrid algorithm called constant modulus least mean square (CMLMS) algorithm is proposed in order to address the potentialproblems existing with constant modulus algorithm (CMA) about its convergence It is a two-stage adaptive filtering algorithmand based on least mean square (LMS) algorithm followed by CMA A hybrid algorithm is theoretically developed and the sameis verified through MatLab Software Theoretical model is verified through simulation and its performance is evaluated in smartantenna in presence of a cochannel interfering signal and additive white Gaussian noise (AWGN) of zero mean This is also testedin Rayleigh fading channel using digital modulation technique for Bit Error Rate (BER) Finally a few computer simulations arepresented in order to substantiate the theoretical findings with respect to proposed model Corresponding results obtained withthe use of only CMA and LMS algorithms are also presented for further comparison

1 Introduction

Smart antenna is an active area of research Most of theantenna manufactures and digital signal processing devel-opers are in opinion that smart antenna provides a solesolution for signal quality and capacity improvement eitherto direct a beam towards a desired user or to minimizea mean squared error (MSE) to fulfill today requirementsIn this regard various researchers are striving to excel inthese prospects In [1] LLMS algorithm is developed usingLMS-LMS algorithms for stable convergence to get optimumresults whereas the same author has proposed anotheralgorithm for beamforming known as RLMS [2 3] using acombined RLS-LMS algorithm to provide a robust perfor-mance Similarly in [4] Feng et al proposed a fast recursivetotal least squares (TLS) algorithm for adaptive FIR filteringwhereas in [5] a simple variable step size LMS (VSSLMS)adaptive algorithm is presented for simple robust efficientfast convergence and low steady state MSE In [6] live modelof Bessel beamformer is developed alongwith its convergenceanalysis [7] whereas the modified Bessel beamformer ispresented in [8 9] with automatic gain control (AGC) for

beamforming to avoid the operator involvement for adjustingthe step size parameter that controls the convergence rate ofthe algorithm In [10] generalized Sato algorithm (GSA) andthe multimodulus algorithm (MMA) are modified by addingconstellation information in their cost functions to improveequalizer performance that is the dynamic convergenceprocess In [11] a variable step size least mean fourth (LMF)algorithm is proposed that achieves a better performancethan the LMF algorithm in different noise environmentsand is shown to achieve a lower steady state error than thetraditional LMF algorithm Dominique Godard [12] was thefirst to introduce a family of blind equalization algorithmlike CMA that has potential problems about its convergence[13] First problem is that its convergence is not guaranteedbecause the cost functionMSE is not convex and may havefalse minima Second potential problem is that if there ismore than one strong signal the algorithm may acquire anundesired signalThis problem can be overcome if additionalinformation about the desired signal is available [13] whereaspractical application of the CMAhas demonstrated a numberof circumstances in which it fails to converge or equallybad from a practical standpoint converges to a solution

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 5983924 8 pageshttpdxdoiorg10115520165983924

2 International Journal of Antennas and Propagation

Xk

1

2

3

N

W1

W1

W2

W2

W3

W3

Wk

Wk

LMSalgorithm

ek(LMS)

dk

yk(LMS)

CMAek(CMA)

yk(CMA)

Overalloutput

Ad

sum

sum

sum

Figure 1 Proposed model

which fails to equalize the input signal [14] Those who haveproposed various forms of CMA include supervised CMA (S-CMA) [15] normalized CMA (N-CMA) [15] and modifiedCMA (M-CMA) [16] to redress the convergence problems ofCMA

In this paper the effectiveness of a hybrid algorithmcalledCMLMS algorithm is proposed in order to evade the use ofpotential problems existing with CMA algorithm regardingits convergence [13ndash19] Its performance is evaluated in anadaptive linear array having multiple inputs including thepresence of a cochannel interfering signal AWGN of zeromean and Rayleigh fading channel Finally a few computersimulations are presented in order to validate the theoreticalfindings with respect to the proposed model

The paper is planned as follows Section 2 explainsthe proposed model for the adaptive antenna array systemSection 3 presents mathematical model of the CMLMSalgorithm Computer simulations are provided in Section 4Results are presented in Section 5 Section 6 concludes thepaper

2 Proposed Model

In this section we model the adaptive antenna array systemwith a hybrid algorithm that consists of LMS algorithmfollowed by CMA as shown in Figure 1 We explain the saidproposed model that involves the use of LMS (nonblind) andCMA (blind) algorithms Both of them are alienated by anarray image factor LMS is a nonblind algorithm that requiresa reference signal also known as training signal to updateits complex weight vector During the training period thetraining signal is sent by the transmitter to the receiver andreceiver uses this information to compute new weight forconvergence to form a beam in the desired direction whereasCMA is a blind algorithm that does not require a reference

signal to train the adaptive weights but beamformer outputis used as feedback to train the beamformer for optimumconvergence based on the idea of reducing system overheadandmaintaining gain on the signal whileminimizing the totaloutput energy As a result a number of bits for transmittinginformation are increased which leads to enhancing capacity

This arrangement as shown in Figure 1 is made in orderto address potential problems existing with CMA algorithmwith respect to its convergence

Section 1 of the hybrid algorithm produces an output119910119896(LMS) that is calculated by using (1) and updates its weightsusing (6) This output 119910119896(LMS) is estimated by LMS algorithmwhich is then fed into Section 2 (ie CMA) after it has beenmultiplied by the image of the desired signal array factor Itis pertinent to mention that error signal used for adjustmentof adaptive system by optimizing the weight vector of LMSalgorithm is sourced from external reference signal (119889119896) thatis estimated by using (7) whereas for updating the CMAweights reference signal is obtained from its self-referencedversion (ie output of CMA is used as feedback to train theCMA for optimum convergence) and this is estimated byusing (11)

Thus in the proposed scheme as shown in Figure 1 theimmediate output 119910119896(LMS) yielded from Section 1 is multipliedby image of the desired signal array factor (119860119889) that resultsin a filtered signal (119860119889119910119896(LMS)) This filtered signal is furtherprocessed byCMAsection using (8) and eventually we get anoptimum output using (14) through adaptation process using(15) by proposed beamformer with input signal array vectorX119896

The number and types of parameters within proposedmodel depend on the computational structure chosen forthe smart antenna system However this structure may bereduced to general adaptive filter structure that has beenproven useful for adaptive filtering tasks as shown in Figure 2

International Journal of Antennas and Propagation 3

Xk

CMLMS filter

Wk+1(CMLMS) = Wk(CMLMS) + 2120583ek(CMLMS)Xk

yk(CMLMS)

ek(CMLMS)

Overall outputsum

Figure 2 CMLMS adaptive filter structure

3 Mathematical Model

Considering a linear beamformer having a hybrid algorithmthat combines the use of LMS algorithm followed by CMA inan arrangement using multiple inputs at its arrayrsquos elementsas shown in Figure 1 then output of the LMS section at 119896thiteration can be defined as

119910119896(LMS) =WLMSX119896 (1)

where 119896 is the iteration numberThe signal array vector received on the elements of

antenna is written by

X119896 = [1199091 1199092 119909119873]119879 (2)

where 119879 signifies the transpose of the vector within thebrackets and linear array having 119873-element composed ofisotropic radiating antenna elements

As signal array vector consists of desired and otherinterfering signals [20 21] therefore it can also be writtenas

X119896 = 119904119889 (119896) 119886 (120579119889) +119871

sum

119894=1

119904119894 (119896) 119886 (120579119894) + 119899 (119896) (3)

where 119904119889 and 119904119894 are the desired and interfering signals arrivingat the array at angles 120579119889 and 120579119894 respectively 119871 is the numberof interfering signals and 119899 is a white and zero mean complexGaussian noise at the array elements 119886(120579119889) and 119886(120579119894) arethe steering vectors for the desired and interfering signalsrespectively which is also known as image of the desiredand interfering signals array factor However when LMSalgorithm converges its output tends to approach desiredsignal (119904119889)with both interfering signal (119904119894) andGaussian noise(119899) being suppressed Therefore steering vector or image ofthe desired signal array factor (119860119889) is described as

119860119889 (120579) = 119886 (120579119889) = [1 119890minus119895120601 119890

minus119895(119873minus1)120601] (4)

where 120601 = (2120587119889120582) sin 120579 is the phase shift observed at eachsensor due to the angle of arrival of the wavefront and assume119889 is the uniform distance between array elements 120582 = 119888119891

where 119891 is in Hertz Therefore the steering vector or imageof the desired signal array factor (119860119889) can be written as

119860119889 (120579) = 119886 (120579119889)

= [1 119890minus119895(2120587120582)119889sin(120579)

119890minus119895(2120587120582)119889(119873minus1) sin(120579)

]

(5)

The input stage of the CMLMS scheme is based on theLMS algorithm as shown in arrangement in Figure 1 with itsweight vector at (119896 + 1)th iteration updated accordingly andis given by

W119896+1(LMS) =W119896(LMS) + 2120583119890119896(LMS)X119896 (6)

where120583 is the step size and its error signal used for adjustmentof adaptive system by optimizing the weight vector is given by

119890119896(LMS) = 119889119896 minus 119910119896(LMS) (7)

where 119889119896 is the reference signal also known as pilot signalThis reference signal is used as desired response from theadaptive processor connected with the antenna array ele-ments which guide the beamformer to map the main beamtowards a specified direction only Output of the LMS sectionis denoted by 119910119896(LMS) at 119896th iteration as defined in (1) Withthis filtered signal (output of the LMS section) forming theinput to the following CMA section the input signal vectorof the CMA section becomes

X119896(CMA) = 119860119889119910119896(LMS) (8)

where 119860119889 is the image array factor of the desired signalPutting value of (1) into (8) then

X119896(CMA) = 119860119889WLMSX119896 (9)

For the CMA stage its weight vector is updated accordingto

W119896+1(CMA) =W119896(CMA) + 2120583119890119896(CMA)X119896(CMA) (10)

where 119890119896(CMA) is the error signal and is given by

119890119896(CMA) = (119910119896(CMA) minus119910119896(CMA)1003816100381610038161003816119910119896(CMA)

1003816100381610038161003816

) (11)


where 119910119896(CMA) is the output of the CMA section and is givenby

119910119896(CMA) =WCMAX119896(CMA) (12)

Putting value of (9) into (12) then

119910119896(CMA) =WCMA119860119889WLMSX119896 (13)

Equation (13) finally becomes the output of the CMLMSbeamformer and is given by

119910119896(CMLMS) =WCMLMSX119896 (14)

whereWCMLMS is the required optimum solution or optimalweight vector for proposed beamformer with input signalarray vector X119896 and is given by

W119896+1(CMLMS) =W119896(CMLMS) + 2120583119890119896(CMLMS)X119896 (15)

where 119890119896(CMLMS) is overall error signal and is given by

119890119896(119862MLMS) = (119910119896(CMLMS) minus119910119896(CMLMS)1003816100381610038161003816119910119896(CMLMS)

1003816100381610038161003816

) (16)

and 120583 is the step size which is given by [22]

0 lt 120583 lt1

120582max (17)

where 120582max is the largest eigenvalue of autocorrelationmatrixwhich is denoted by R This autocorrelation matrix describescorrelation between various elements of signal array vectorX119896 The stability of proposed algorithm is maintained bychoosing the step size parameter by trial-and-error methodwithin specified range as stated above in (17)

Equation (15) implies that the adaptive process willfinally converge to mean square error as the adaptationprogresses In summary the proposed beamformer performsthe following steps

Step 1 Obtain X119896 in (2) by signal array vector

Step 2 Get output 119910119896(CMLMS) of the CMLMS beamformerin (14) concluding both parts of LMS and CMA

Step 3 Calculate overall error signal 119890119896(CMLMS) for optimiz-ing the weight vector in (16)

Step 4 Calculate the robust adaptive beamformer weightsW119896+1(CMLMS) in (15)

Step 5 Repeat the above steps in closed loop to get optimumresults

4 Simulation Results

In this section performance analysis is carried out forproposed algorithm and the same is also compared with LMSandCMA algorithms for further assessmentWe are adaptinga strategy that an incoming signal coming in at a certain

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 1000

1

2

3

4

5

6

7

8Proposed CMLMS algorithm

Angle of arrival (degree)

Gai

n (d

B)

AOA = 40

AOA = minus20

Figure 3 Beam pattern achieved by CMLMS algorithm

angle (here 40∘ amp minus20∘) using119873 (here119873 = 8) antennas andsuppressing all other incoming signals at the same time Thisassumption is met while discussing our results

We consider the following parameters for the simulationspurpose

(i) A linear array model with 8 elements and spacingbetween two elements is 05120582

(ii) The channels are AWGN and Rayleigh fading withinterference signal included

(iii) All weight vectors for algorithms under analysis areinitially set at zero

(iv) Number of samples is fifty for all simulations

AOA for desired signals are set at minus20 and 40 degrees asshown in Figure 3 for proposed algorithm and subsequentMSE is obtained Optimum gain towards desired users withminimum sidelobe level (SLL) is obtained Small SLL indi-cates that proposed algorithm will get less interference thatwill enhance security and provide quality signal to desiredusers

Substantial reduction in MSE is also observed whicheliminates the potential problems existing with CMA regard-ing its convergence Convergence is the process of minimiz-ing the power of the error signal The performance curve asshown in Figure 4 indicates that CMLMS has minimumMSEand starts to converge from the iteration number 30 whenmeasured after 50 iterations

In this case step size is set at 00001 by trial-and-errormethod but within range as specified in (17) It is to benoted that step size has significant effect on convergence andstability of the proposed beamformer The step size withinbounded range gives marked improvement in reductionof SLL and in error minimization Obtained results aresummarized in Table 1


Table 1 Input and output estimates

Input parameter Output parameterAOA (degree) Number of elements Element spacing SLL (dB) Gain (dB)40∘ amp minus20∘ 8 05120582 1818 8001

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Iteration number

Mea

n sq

uare

erro

r

Proposed CMLMS algorithm

CMLMS

Figure 4 Convergence behavior of CMLMS algorithm


1

2

3

4

5

6

7



Gai

n (d

B)

CMA = 30

LMS = 0

CMLMS = minus30

Figure 5 Beam patterns achieved by CMLMS LMS and CMAalgorithms

5 Performance Comparison

51 Performance with respect to Array Gain Proposed algo-rithm is also compared with CMA and LMS algorithms asshown in Figure 5 for its improved performance in termsof array gain with different AOA set for desired users

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Iteration number

Mea

n sq

uare

erro

r


CMLMSLMSCMA

Figure 6 Convergence behaviors of CMLMS LMS and CMAalgorithms

Table 2 Performance analysis

Algorithms Input parameter Output parameterAOA (degree) SLL (dB) Gain (dB)

CMA 30∘ 2034 7859LMS 0∘ 1596 6864CMLMS minus30∘ 1818 8001

You see the optimum array gain with respect to CMA andLMS algorithms The SLL of proposed algorithm is slightlylessgreater than CMA and LMS algorithms respectively butwith constant amplitude It means that CMLMS algorithmsaves power by reduction in SLL as compared to CMAHowever at the same time it has more array gain incomparison with LMS with slightly large SLL but this can becompromised keeping large gain and robustness in MSE asshown in Figure 6

Because of this it may be more useful where signalstatistics vary rapidly with time In this case step size is setat 00008 by trial-and-error method for algorithms underconsideration and all other parameters remain the same asdescribed above for better comparison The performanceanalysis of study algorithms are summarized in Table 2

52 Performance with respect to Convergence If we compareand observe the MSE curves as shown in Figure 6 then weconclude that CMLMS algorithm has minimum MSE and


0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS

BER CMATheoretical BER

100

10minus1

10minus2

10minus3

EbNo (dB)

Figure 7 BER performance of CMLMS LMS and CMA algo-rithms

follows steady path The learning curves of the understudyalgorithms exhibit that CMA and LMS algorithms have fluc-tuation and their performance is worst than the performanceof the proposed scheme

It means that the proposed algorithm illustrates clear per-formance advantages over other CMA and LMS techniquessimulated for comparisons in both areas of convergencespeeds and MSE floor Therefore proposed algorithm canachieve lower MSE and faster robust convergence than CMAand typically LMS algorithm for the same adaptation size oriterations which is extremely important in the application ofwireless cellular communication where signal statistics varyrapidly with time

53 Performance with respect to Bit Error Rate Proposedalgorithm is also operated in digital domain in order to assessits performance and compared with CMA and LMS algo-rithms as shown in Figure 7 All these algorithms are tested inRayleigh fading channel using digital modulation techniqueto measure their Bit Error Rates (BERs) The simulationsare designed for quadrature amplitude modulation (QAM)signal with 0 to 10 dB SNR and step size is fixed at 85616119890 minus04 Values for BERs are obtained through the semianalytictechnique and the same values are also compared with thetheoretical BERs at different SNR

The BER performance curves are shown in Figure 7which indicates that the computed values of BER are smallerthan theoretical values of BER means BER performanceis greatly improved It is realized that the CMLMS is thebest in performance followed by CMA and LMS algorithmsFigure 7 shows the BER performance of our proposed schemeand achieves a BER of 10minus2 at an SNR of 8 dB The dataobtained from Figure 7 is provided in Table 3 which indicatesthat proposed scheme outperforms both CMA and LMSalgorithms

Table 3 BER performance analysis

119864119887119873119900(dB) BER theory BER LMS BER CMA BER CMLMS

0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057

If we take one computed BER value of LMS CMA andCMLMS algorithms say at 5 dB SNR with respect to BERreduction capabilities then we have computed BER value ofLMS CMA and CMLMS algorithms at 5 dB SNR which are00613 00253 and 00204 respectively

The BER values of CMLMS are 3327 (00204 is 3327of 00613) as compared to LMS algorithm whereas the BERvalues of CMLMS are 8063 (00204 is 8063 of 00253)as compared to CMA Then BER reduction capability ofCMLMS at 5 dB SNR is 6673 as compared to LMS algo-rithm whereas reduction of the BER for CMLMS is 1937compared to CMA Therefore CMLMS algorithm is morecost effective for wireless cellular communication system ascompared to LMS and CMA algorithms in this respect

6 Discussion on Results

The following outcomes are deducted from the given results

(i) The proposed algorithm does not always require anexternal reference signal for its operation but adaptsitself entirely through self-referencing (ie outputof proposed algorithm is used as feedback to trainthe beamformer for its optimum convergence) tothe desired signal using the correct reference signalduring the initial a few iterations only for LMS Theconfiguration (as shown in Figure 1) uses a (nonblind)LMS trained equalizer first to open the intersymbolinterference (ISI) communication eye and when theeye was open the training finishedLMS switched outand the system reverted to blind decision feedback

(ii) The convergence of proposed algorithm is robust inpresence of noise and follows steady path Thereforeproposed algorithmmay correctly address the poten-tial problems existing with CMA algorithm regardingits convergence

(iii) The steady state MSE of proposed algorithm is foundto be the most favorable as compared to CMA andLMS An adaptive system with small minimum MSEindicates that this system has accurately modeledpredicted adapted andor converged to a solution forthe given system


(iv) The proposed algorithm has achieved similar beampatterns as obtained with CMA and LMS but withoptimum array gain as tabulated in Table 1 and shownin Figures 3 and 5

(v) With optimum array gain proposed algorithmenhances range

(vi) The proposed algorithm directs its energy towardsdesired users only therefore it saves energy due towhich battery life installed at Base Transceiver Station(BTS) increases

(vii) As there is no leakage of energy towards interfererstherefore minimum BTS is required to cover theservice area and infrastructures cost may be reduced

(viii) With optimum array gain and no leakage of energytowards interferers security of subscribers may beenhanced

(ix) Leakage of classified informationtapping may bereducedrestricted

(x) Complexity of proposed algorithm is slightly more ascompared to CMA and LMS when treated as singleentity However at the same time it has more arraygain and steady state MSE in comparison with CMAand LMS as shown in Figure 6 Further the proposedalgorithm illustrates a clear performance advantagesover other CMA and LMS techniques simulated forcomparisons in both areas of convergence speeds andMSE floor Because of this it may be more usefulwhere signal statistics vary rapidly with time There-fore its application is extremely important in wirelesscellular communication where signal statistics varyrapidly

(xi) Further as powerful low cost digital signal proces-sors (DSPs) are commercially available nowadaystherefore algorithm complexity or computational costwith respect to execution time would not make muchdifference Besides all other requirements are met bythe proposed technique So it is better to use proposedtechnique for getting aforesaid advantages for smartantenna

(xii) BER reduction capability of CMLMS say at 5 dB SNRis 6673 as compared to LMS algorithm whereasreduction of the BER for CMLMS is 1937 comparedto CMA Hence CMLMS algorithm provides costeffective solution for wireless cellular communicationsystem as compared to LMS and CMA algorithms inthis respect If SNR value increases then BER reduc-tion capability of CMLMS algorithm also increases astabulated in Table 3

7 Conclusions

In this paper we have introduced a new promising tech-nique for adaptive beamforming called CMLMS algorithmbeing a breakthrough design delivering high forward gainand unmatched interference rejection The performance ofCMLMS with CMA and LMS algorithms is discussed in the

noise and Rayleigh fading channels models Relevant resultsdepict that smart antenna equipped with proposed algorithmcan increase the number of active users (means increasein capacity) in 3G and beyond system significantly withoutlosing of performance quality

Competing Interests

The authors declare that they have no competing interests

References

[1] J A Srar K-S Chung and A Mansour ldquoAdaptive arraybeamforming using a combined LMS-LMS algorithmrdquo IEEETransactions on Antennas and Propagation vol 58 no 11 pp3545ndash3557 2010

[2] J A Srar andK S Chung ldquoAdaptive array beamforming using acombined RLS-LMS algorithmrdquo in Proceedings of the 14th Asia-Pacific Conference on Communications pp 1ndash5 Tokyo JapanOctober 2008

[3] J A Srar and K-S Chung ldquoPerformance of RLMS algorithmin adaptive array beam formingrdquo in Proceedings of the 11th IEEESingapore International Conference on Communication Systems(ICCS rsquo08) pp 493ndash498 Guangzhou China November 2008

[4] D Feng X Zhang D Chang and W Zheng ldquoA fast recursivetotal least squares algorithm for adaptive FIR filteringrdquo IEEETransactions on Signal Processing vol 52 no 10 pp 2729ndash27372004

[5] T I Haweel ldquoA simple variable step size LMS adaptive algo-rithmrdquo International Journal of Circuit Theory and Applicationsvol 32 no 6 pp 523ndash536 2004

[6] M Yasin and P Akhtar ldquoDesign and performance analysisof live model of Bessel beamformer for adaptive array sys-temrdquoCOMPELmdashThe International Journal for Computation andMathematics in Electrical and Electronic Engineering vol 33 no4 pp 1434ndash1447 2014

[7] M Yasin and P Akhtar ldquoConvergence analysis of Besselbeamformer and its comparison with LMS in adaptive arraysystemrdquo COMPELmdashThe International Journal for Computationand Mathematics in Electrical and Electronic Engineering vol34 no 3 pp 952ndash961 2015

[8] M Yasin and P Akhtar ldquoMathematical model of bessel beam-former with automatic gain control for smart antenna arraysystemrdquoArabian Journal for Science and Engineering vol 39 no6 pp 4837ndash4844 2014

[9] M Yasin P Akhtar and A H Pathan ldquoMathematical modelof bessel beamformer with automatic gain control for smartantenna array system in Rayleigh fading channelrdquo IEEJ Trans-actions on Electrical and Electronic Engineering vol 9 no 3 pp229ndash234 2014

[10] L He and S A Kassam ldquoConvergence analysis of blindequalization algorithms using constellation-matchingrdquo IEEETransactions on Communications vol 56 no 11 pp 1765ndash17682008

[11] S M Asad A Zerguine and M Moinuddin ldquoOn the con-vergence analysis of a variable step-size LMF algorithm ofthe quotient formrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo10) pp 3722ndash3725 Dallas Tex USA March 2010

[12] F B Gross Smart Antennas for Wireless Communications withMATLAB McGraw-Hill New York NY USA 2005


[13] C B Dietrich Jr Adaptive arrays and diversity antenna config-urations for handheld wireless communication terminals [PhDthesis] Virginia Polytechnic Institute State University 2000chapter 3

[14] J R Treichler V Wolff and C R Johnson Jr ldquoObservedmisconvergence in the constant modulus adaptive algorithmrdquoin Proceedings of the 25th Asilomar Conference on SignalsSystems and Computers pp 663ndash667 Pacific Grove Calif USA1991

[15] A Ozen I Kaya and B Soysal ldquoA supervised constantmodulusalgorithm for blind equalizationrdquo Wireless Personal Communi-cations vol 62 no 1 pp 151ndash166 2012

[16] I Chahed J Belzile and A B Kouki ldquoBlind decision feedbackequalizer based on high order MCMArdquo in Proceedings ofthe IEEE Canadian Conference on Electrical and ComputerEngineering (CCECE rsquo04) vol 4 pp 2111ndash2114 Niagara FallsCanada May 2004

[17] D L Jones ldquoA normalized constant modulus algorithmrdquo inProceedings of the 29th IEEE Asilomar Conference on SignalsSystems and Computers vol 1 pp 694ndash697 PacificGrove CalifUSA 1996

[18] S S Moghaddam and M S Moghaddam ldquoA comprehensivesurvey on antenna array signal processingrdquo Trends in AppliedSciences Research vol 6 no 6 pp 507ndash536 2011

[19] A-J van der Veen and A Paulraj ldquoAn analytical constantmodulus algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 5 pp 1136ndash1155 1996

[20] L Zhang H C So L Ping and G Liao ldquoEffective beamformerfor coherent signal receptionrdquo Electronics Letters vol 39 no 13pp 949ndash951 2003

[21] L Zhang H C So L Ping and G Liao ldquoAdaptive multiple-beamformers for reception of coherent signals with knowndirections in the presence of uncorrelated interferencesrdquo SignalProcessing vol 84 no 10 pp 1861ndash1873 2004

[22] BWidrow and SD StearnsAdaptive Signal Processing PearsonEducation Upper Saddle River NJ USA 1985

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Xk

1

2

3

N

W1

W1

W2

W2

W3

W3

Wk

Wk

LMSalgorithm

ek(LMS)

dk

yk(LMS)

CMAek(CMA)

yk(CMA)

Overalloutput

Ad

sum

sum

sum

Figure 1 Proposed model

which fails to equalize the input signal [14] Those who haveproposed various forms of CMA include supervised CMA (S-CMA) [15] normalized CMA (N-CMA) [15] and modifiedCMA (M-CMA) [16] to redress the convergence problems ofCMA

In this paper the effectiveness of a hybrid algorithmcalledCMLMS algorithm is proposed in order to evade the use ofpotential problems existing with CMA algorithm regardingits convergence [13ndash19] Its performance is evaluated in anadaptive linear array having multiple inputs including thepresence of a cochannel interfering signal AWGN of zeromean and Rayleigh fading channel Finally a few computersimulations are presented in order to validate the theoreticalfindings with respect to the proposed model

The paper is planned as follows Section 2 explainsthe proposed model for the adaptive antenna array systemSection 3 presents mathematical model of the CMLMSalgorithm Computer simulations are provided in Section 4Results are presented in Section 5 Section 6 concludes thepaper

2 Proposed Model

In this section we model the adaptive antenna array systemwith a hybrid algorithm that consists of LMS algorithmfollowed by CMA as shown in Figure 1 We explain the saidproposed model that involves the use of LMS (nonblind) andCMA (blind) algorithms Both of them are alienated by anarray image factor LMS is a nonblind algorithm that requiresa reference signal also known as training signal to updateits complex weight vector During the training period thetraining signal is sent by the transmitter to the receiver andreceiver uses this information to compute new weight forconvergence to form a beam in the desired direction whereasCMA is a blind algorithm that does not require a reference

signal to train the adaptive weights but beamformer outputis used as feedback to train the beamformer for optimumconvergence based on the idea of reducing system overheadandmaintaining gain on the signal whileminimizing the totaloutput energy As a result a number of bits for transmittinginformation are increased which leads to enhancing capacity

This arrangement as shown in Figure 1 is made in orderto address potential problems existing with CMA algorithmwith respect to its convergence

Section 1 of the hybrid algorithm produces an output119910119896(LMS) that is calculated by using (1) and updates its weightsusing (6) This output 119910119896(LMS) is estimated by LMS algorithmwhich is then fed into Section 2 (ie CMA) after it has beenmultiplied by the image of the desired signal array factor Itis pertinent to mention that error signal used for adjustmentof adaptive system by optimizing the weight vector of LMSalgorithm is sourced from external reference signal (119889119896) thatis estimated by using (7) whereas for updating the CMAweights reference signal is obtained from its self-referencedversion (ie output of CMA is used as feedback to train theCMA for optimum convergence) and this is estimated byusing (11)

Thus in the proposed scheme as shown in Figure 1 theimmediate output 119910119896(LMS) yielded from Section 1 is multipliedby image of the desired signal array factor (119860119889) that resultsin a filtered signal (119860119889119910119896(LMS)) This filtered signal is furtherprocessed byCMAsection using (8) and eventually we get anoptimum output using (14) through adaptation process using(15) by proposed beamformer with input signal array vectorX119896

The number and types of parameters within proposedmodel depend on the computational structure chosen forthe smart antenna system However this structure may bereduced to general adaptive filter structure that has beenproven useful for adaptive filtering tasks as shown in Figure 2


Xk

CMLMS filter


yk(CMLMS)

ek(CMLMS)

Overall outputsum




119910119896(LMS) =WLMSX119896 (1)



X119896 = [1199091 1199092 119909119873]119879 (2)



X119896 = 119904119889 (119896) 119886 (120579119889) +119871

sum

119894=1

119904119894 (119896) 119886 (120579119894) + 119899 (119896) (3)


119860119889 (120579) = 119886 (120579119889) = [1 119890minus119895120601 119890

minus119895(119873minus1)120601] (4)



119860119889 (120579) = 119886 (120579119889)

= [1 119890minus119895(2120587120582)119889sin(120579)

119890minus119895(2120587120582)119889(119873minus1) sin(120579)

]

(5)


W119896+1(LMS) =W119896(LMS) + 2120583119890119896(LMS)X119896 (6)


119890119896(LMS) = 119889119896 minus 119910119896(LMS) (7)


X119896(CMA) = 119860119889119910119896(LMS) (8)


X119896(CMA) = 119860119889WLMSX119896 (9)





1003816100381610038161003816

) (11)



119910119896(CMA) =WCMAX119896(CMA) (12)


119910119896(CMA) =WCMA119860119889WLMSX119896 (13)


119910119896(CMLMS) =WCMLMSX119896 (14)





1003816100381610038161003816

) (16)


0 lt 120583 lt1

120582max (17)











1

2

3

4

5

6

7



Gai

n (d

B)

AOA = 40

AOA = minus20














0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Iteration number

Mea

n sq

uare

erro

r


CMLMS



1

2

3

4

5

6

7



Gai

n (d

B)

CMA = 30

LMS = 0

CMLMS = minus30




0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Iteration number

Mea

n sq

uare

erro

r


CMLMSLMSCMA









0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS


100

10minus1

10minus2

10minus3

EbNo (dB)








0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057


















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Xk

CMLMS filter


yk(CMLMS)

ek(CMLMS)

Overall outputsum




119910119896(LMS) =WLMSX119896 (1)



X119896 = [1199091 1199092 119909119873]119879 (2)



X119896 = 119904119889 (119896) 119886 (120579119889) +119871

sum

119894=1

119904119894 (119896) 119886 (120579119894) + 119899 (119896) (3)


119860119889 (120579) = 119886 (120579119889) = [1 119890minus119895120601 119890

minus119895(119873minus1)120601] (4)



119860119889 (120579) = 119886 (120579119889)

= [1 119890minus119895(2120587120582)119889sin(120579)

119890minus119895(2120587120582)119889(119873minus1) sin(120579)

]

(5)


W119896+1(LMS) =W119896(LMS) + 2120583119890119896(LMS)X119896 (6)


119890119896(LMS) = 119889119896 minus 119910119896(LMS) (7)


X119896(CMA) = 119860119889119910119896(LMS) (8)


X119896(CMA) = 119860119889WLMSX119896 (9)





1003816100381610038161003816

) (11)



119910119896(CMA) =WCMAX119896(CMA) (12)


119910119896(CMA) =WCMA119860119889WLMSX119896 (13)


119910119896(CMLMS) =WCMLMSX119896 (14)





1003816100381610038161003816

) (16)


0 lt 120583 lt1

120582max (17)











1

2

3

4

5

6

7



Gai

n (d

B)

AOA = 40

AOA = minus20














0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Iteration number

Mea

n sq

uare

erro

r


CMLMS



1

2

3

4

5

6

7



Gai

n (d

B)

CMA = 30

LMS = 0

CMLMS = minus30




0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Iteration number

Mea

n sq

uare

erro

r


CMLMSLMSCMA









0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS


100

10minus1

10minus2

10minus3

EbNo (dB)








0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057


















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910119896(CMA) =WCMAX119896(CMA) (12)


119910119896(CMA) =WCMA119860119889WLMSX119896 (13)


119910119896(CMLMS) =WCMLMSX119896 (14)





1003816100381610038161003816

) (16)


0 lt 120583 lt1

120582max (17)











1

2

3

4

5

6

7



Gai

n (d

B)

AOA = 40

AOA = minus20














0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Iteration number

Mea

n sq

uare

erro

r


CMLMS



1

2

3

4

5

6

7



Gai

n (d

B)

CMA = 30

LMS = 0

CMLMS = minus30




0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Iteration number

Mea

n sq

uare

erro

r


CMLMSLMSCMA









0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS


100

10minus1

10minus2

10minus3

EbNo (dB)








0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057


















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Iteration number

Mea

n sq

uare

erro

r


CMLMS



1

2

3

4

5

6

7



Gai

n (d

B)

CMA = 30

LMS = 0

CMLMS = minus30




0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Iteration number

Mea

n sq

uare

erro

r


CMLMSLMSCMA









0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS


100

10minus1

10minus2

10minus3

EbNo (dB)








0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057


















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8 9 10

BER

BER CMLMSBER LMS


100

10minus1

10minus2

10minus3

EbNo (dB)








0 01889 01041 00589 005861 01682 00949 00492 004832 01482 00859 00410 003943 01292 00772 00343 003184 01114 00690 00292 002555 00951 00613 00253 002046 00804 00542 00225 001637 00673 00478 00204 001298 00559 00421 00185 001019 00461 00371 00168 0007710 00378 00328 00151 00057


















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















7 Conclusions



Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A Novel Adaptive Algorithm Addresses...

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