Research Article On Modelling and Comparative Study of LMS

Research ArticleOn Modelling and Comparative Study of LMS and RLSAlgorithms for Synthesis of MSA

Ahmad Kamal Hassan12 and Adnan Affandi1

1Department of Electrical and Computer Engineering King Abdulaziz University PO Box 80204 Jeddah 21589 Saudi Arabia2Center of Excellence in Intelligent Engineering Systems (CEIES) King Abdulaziz University PO Box 80204Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Ahmad Kamal Hassan kamal393yahoocom

Received 22 August 2016 Revised 18 October 2016 Accepted 19 October 2016

Academic Editor Dimitrios E Manolakos

Copyright copy 2016 A K Hassan and A AffandiThis 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

This paper deals with analytical modelling of microstrip patch antenna (MSA) by means of artificial neural network (ANN) usingleast mean square (LMS) and recursive least square (RLS) algorithms Our contribution in this work is twofoldWe initially providea tutorial-like exposition for the design aspects ofMSA and for the analytical framework of the two algorithmswhile our second aimis to take advantage of high nonlinearity of MSA to compare the effectiveness of LMS and that of RLS algorithms We investigatethe two algorithms by using gradient decent optimization in the context of radial basis function (RBF) of ANN The proposedanalysis is based on both static and adaptive spread factor We model the forward side or synthesis of MSA by means of workedexamples and simulations Contour plots 3D depictions and Tableau presentations provide a comprehensive comparison of thetwo algorithms Our findings point to higher accuracies in approximation for synthesis of MSA using RLS algorithm as comparedwith that of LMS approach however the computational complexity increases in the former case

1 Introduction

Microstrip patch antenna (MSA) has been extensively used inwireless transmission links because of its simplicity in designand fabrication less weight low cost and small size Majorparameters in the design consideration ofMSA include band-width gain directivity polarization and center frequency Itis well understood that there is a tradeoff in selection of theseparameters and design engineers have to assign appropriateweights based on their work objectives [1ndash3] Usually MSAfinds applications in areas where the bandwidth requirementis narrow and with multiband resonance frequencies toaccount for diversity issues [4 5] There exist many otherapplications [6ndash8] where MSA can be integrated with a givenautomation system to better the existing results

Terminology of patch is based on the aspects of radiatingelement ldquophotoetchenrdquo on dielectric constant Patch canbe of different geometries both deterministic and randomDeterministic geometry points to triangular elliptical circu-lar square and last but not least rectangular (used in this

work) shapes of patch Choice of patch dimension such aslength width permeability and thickness of the substrateplays significant role in obtaining the resonance frequencyThere exist numerous models in literature that account forthe determination of dimension of patch based on artificialneural networks (ANN) for modelling purposes [9 10] butan algorithmic comparison of such models is unavailableNevertheless ANN in general and radial basis function (RBF)in particular have been extensively used with excellent resultsin modelling and simulation techniques for other nonlinearmechanical systems [11 12] and signal processing routines[13] and in the design aspects of MSA [14ndash17] The designof MSA using ANN is subdivided into forward modellingand backward modelling Forward modelling accounts forthe synthesis of MSA and hence it is useful in obtainingboth length (119871) and width (119882) of the patch In backwardmodelling the primary task is the extraction of resonancefrequency (119891) Both forward side of the problem (synthesis)and reverse side of the problem (analysis) are basic buildingblocks for commercially available simulation software such

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 9742483 8 pageshttpdxdoiorg10115520169742483

2 Modelling and Simulation in Engineering

as advanced design system (ADS) and Ansoft high frequencysimulation system (HFSS) [18] In this work the analysis forthe synthesis of MSA using RBF is performed We presentherein an exposition of significant algorithms such as leastmean square (LMS) and recursive least square (RLS) andfurther investigate their deviants based on the gradient decentapproach (GDA) The choice of the LMS and RLS algorithmis because they are considered fundamental in many subdis-ciplines of engineering such as adaptive filtering and signalprocessingTheparameters that are fed toANNare resonancefrequency (119891) permeability constant (120576119903) and height of thesubstrate (ℎ) Consequently width (119882) and length (119871) areextracted from ANN The rationale behind the proposedwork is that the MSA design and its pertinent equations areof high nonlinearity therefore choosing an algorithm thatresults in minimal error between the estimated data and thedesired data can be useful in the development of the afore-mentioned commercial software Moreover computationalcomplexity of algorithms is another area which needs to belooked into before choosing an algorithm hence we outlinethe complexity difference in terms of complexmultiplicationsand additions that are performed in an algorithm To the bestof the authorsrsquo knowledge such investigation and comparativeanalysis have not been done previously for the synthesis ofMSA

Organization of this paper is such that after introductionin Section 1 an overview of the design aspects of MSAis given in Section 2 Section 3 introduces RBF LMS andRLS approaches and mathematical formulation of adaptivespread factor Complexity analysis is presented in Section 4Simulation results and comparative analysis of the fouralgorithms are given in Section 5 Subsequent section dealswith concluding remarks of this work and references

2 Design of the Rectangular MicrostripPatch Antenna

Patch antennas are multilayered components with conduc-tors that is patch and ground plane separated by dielectricsubstrate Rectangular microstrip patch antenna (R-MSA)consists of width (119882) and length (119871) as presented in Figure 1The other significant parameters required in designing andfabrication of MSA include substrate thickness (ℎ) and itspermeability (120576119903) Choice of substrate determines the sizeand its relevant application base Most common substrateson the market are RTduroid with dielectric constant of22 and liquid crystal polymer (LCP) with the range ofdielectric constant between 29 and 32 [19] Generally thereare numerous substrates available on the market for usabilityin MSA and in practice the permeability of dielectric withMSA in perspective is in the range of 22 le 120576119903 le 12 Feed isanother crucial aspect in design of patch antenna and thereexist multiple methods by which a signal can be fed Thesimplest method is direct contact method shown in Figure 1and thereby used in this work Other methods can be probefeed microstrip proximity coupling and microstrip aperturecoupling [20]

Ground plane

Feed

W

L

ℎ r

Figure 1 Rectangular microstrip antenna

In MSA physical length (119871) and electrical (effective)length (119871eff ) are different because of fringing fields at theedges of the patch The effective electrical length of the patchantenna is given by [2 19ndash22]

119871eff = 119871 + 2Δ119871 (1)

where Δ119871 is the change in length given by

Δ119871 = 0412ℎ(120576119903eff + 03) (119882ℎ + 0264)(120576119903eff minus 0258) (119882ℎ + 08) (2)

Effective dielectric constant of the dielectric material isexpressed as [19]

120576119903eff = 120576119903 + 12 + 120576119903 minus 12 (1 + 12 ℎ119882)minus12 (3)

The actual length of the patch under specified conditions is

119871eff = 1205822 997904rArr119871 = 1198882119891radic120576119903eff minus 2Δ119871

(4)

Hence the practical width is determined as follows

119882 = 1198882119891radic 2120576119903 + 1 (5)

Based on the above mathematical modelling both syn-thesis and analysis can be designed using soft computingtools We however focus on the synthesis of MSA wherethe inputs to ANN are resonance frequency (119891) substratethickness (ℎ) and dielectric constant (120576119903) while length (119871)and width (119882) are extracted from the RBF designs Synthesisaspect of this work is presented in Figure 2 We utilize thismodel as a test bench to compare LMS and RLS algorithmsin the forthcoming sections

3 Synthesis Using RBF-ANN

In this sectionwe examine the feedforward architecture of theANNbased onRBF Layout of RBF is simplistic and it consistsof hidden layer and output layer neurons The function ofhidden layer is to perform a nonlinear operation on the set ofinputs For the purposes of performing a nonlinear operationone can resort to plethora of nonlinear functions such as

Modelling and Simulation in Engineering 3

RBFLMSRLS

amp W

L

ℎ

r

-GDA

f

Figure 2 Synthesis of ANN-MSA

multiquadrics inverse multiquadrics and Gaussian function[21ndash23] Moreover Kernels of the RBF have been extended inmultiple ways with notable work in [24ndash26] In this work wehowever use the conventional approach expressed in [21 22]for our hypothesis testing The nonlinear function pertinentto 119895th neuron (Φ119895) is considered as Gaussian function whichcan be expressed by means of the following expressions

Φ119895 (119899) = exp(minussum119872119897=1 (119909119897 minus 119888119896)221205902119895 ) (6a)

120590119895 = 119889maxradic2119898 (6b)

where 119899 is the time index 120590119895 is the spread factor of 119895thGaussian function and it is determined empirically 119898 isthe total number of basis functions employed 119889max is themaximum distance between any two bases 119909119897 is the 119897th inputdata and 119888119896 is the 119896th center of basis function

Once the nonlinear activation function is performed onthe input set of data (119909119894) the output can be extracted fromoutput layer neuron (119910119895) as

119910119894 (119899) = 119898sum119895=1

119908119895 (119899)Φ119895 (119899) (7)

where119898 is the total number of basis functions and119908119895 are theweighted interconnection between the hidden neurons andthe output neurons

Weight update is processed by using recursion principleof RBF-ANN and its optimality is an active research areawith the synthesis of MSA in perspective An effort is madeherein to model algorithms with better convergence andstability properties of the objective functionwhich is basicallyerror minimization The following subsections provide anoverview of algorithms utilized in approximating MSA met-rics and recursively update weights and the correspondingactivation function Figure 3 presents an RBF neural networkwhere input data is fed to Gaussian function each nonlinearactivation function has a weighted interconnection with theoutput neuron Our primary focus here is the minimizationof objective function which accounts for the mean squareerror (MSE) between the desired output (119889) and the estimatedoutput (119910) We use two benchmark algorithms namely theLMS and the RLS independently Lastly spread can alsobe recursively updated based on gradient decent approach(GDA) which has the potential to further minimize theMSE

31 Least Mean Square (LMS) Least mean square (LMS)algorithm is used widely in the domain of adaptive filtering[27] and it is alsomore often than not utilized in RBF forMSAdesign [14] The LMS utilizes GDA approach to recursivelyupdate the weights of neurons based on the instantaneousMSE Consider an objective function which is simply basedon instantaneous error of all the output neurons given by

119864 (119899) = 12119868sum119894=1

(119910119894 (119899) minus 119889)2 (8)

where119889 corresponds to the desired output which is comparedwith the approximated result of the output neuron 119910119894 Gradi-ent of the cost function is thus calculated as

120597119864 (119899)120597119908119895 (119899) =119868sum119894=1

(119910119894 (119899) minus 119889)Φ119895 (119899) (9)

Hence the corresponding weights119908119895 update equation can becomputed and updated as

119908119895 (119899 + 1) = 119908119895 (119899) minus 1205781 120597119864 (119899)120597119908119895 (119899) (10)

where 1205781 is the learning rate or step size and it plays animportant role in the convergence properties of the givenobjective function

32 Recursive Least Square (RLS) Recursive least square(RLS) is another powerful adaptive algorithm where the costfunction is minimized by recursively updating the weightsThere is a plethora of literature available on the design ofRLS algorithm therefore [28] and references therein can beinsightful for interested readers The main aim herein is notto redrive the RLS algorithm but to briefly overview its coreprinciples The linear least square objective function is givenby

119864 (119899) = 119899sum119894=1

119891 (119899 119894) (119910119894 (119899) minus 119889)2 (11a)

119891 (119899 119894) = 120582119899minus119894 (11b)

where 120582 is the forgetting factor of RLS and has values in therange of 0 lt 120582 le 1


Output neuron

Desired output

Gradient decent approach for spread update

LMSRLS algorithm

x1

x2

w1

w2

wm

d

y

Φ1

Φ2

Φm

Figure 3 LMS RLS and adaptive spread factor (120590) modelled using RBF-ANN

The order of RLS denoted as 119875(119899) and the correspondingweights update (119908119894) expressions are given by

119875 (119899)= 120582minus1 [119875 (119899 minus 1) minus 119875 (119899 minus 1) 119909 (119899) 119909119879 (119899) 119875 (119899 minus 1)120582 + 119909119879 (119899) 119875 (119899 minus 1) 119909 (119899) ]

(12)

119908 (119899 + 1) = 119908 (119899) + 119875 (119899) 119909 (119899) 119890 (119899) (13)

Convergence of RLS is much faster as compared withLMS though computational complexity of RLS substantiallyincreases and this aspect is shown in the subsequent section

33 Gradient Decent Approach for Adaptive Spread In thissubsection the spread factor has also been made adaptiveusing gradient decent approach for both LMS and RLSalgorithms The relevant gradient of objective function andrecursive update of spread factor are given by [21 22]

120597119864 (119899)120597120590119895 (119899)= minus119908119895 (119899) 119871sum

119897=1

(119910119894 (119899) minus 119889)2 1198661015840 (1003817100381710038171003817x119897 minus 1198881198961003817100381710038171003817120590119895)Q119897119896 (119899)(14a)

Q119897119896 (119899) = [x119897 minus 119888119896 (119899)] [x119897 minus 119888119896 (119899)]119879 (14b)

120590119895 (119899 + 1) = 120590119895 (119899) minus 1205782 120597119864 (119899)120597120590119895 (119899) (14c)

where the term 1198661015840(sdot) in (14a) is the first derivative of Greenrsquosfunction 119866(sdot) with respect to its argument and 1205782 is thelearning rate for adaptive spread

Depiction of GDA based spread update is shownusing dashed-lines of Figure 3 Recursive update of spreadexpressed in (14a) (14b) and (14c) can greatly enhance theapproximation capabilities of the neural networks since it is

tuning themost significant part of RBF namely the activationfunction defined in (6a) It is also to be noted herein thata further extension of the aforementioned model is alsopossible in which one can update centers of the Gaussianfunction as well [21ndash23] however subtractive clusteringmethod is used in this work

4 Computational Complexity

Computation complexity of the four algorithms based onmatrix and vector dimension denoted as 119863 is summarizedin Table 1 It can be seen that algorithm 4 has the high-est computational complexity and it increases rapidly withincreasing the dimension of the system for example numberof neurons of RBF In order to calculate the complexity ofoverall system input vectors are needed to be consideredalongside the inherited routines of a particular algorithmFor the detailed formation of computational complexity wesuggest that the reader follows mechanics of [29] It is shownin the next section that RLS algorithm which needs morecomputational power for the present problem benefits interms of minimizing the MSE which drops from 152 to justover 1 for the testing data There are certainly some emergingalgorithms [30ndash34] and conventional techniques of algorithmmanipulation [35 36] with varying degree of complexitywhich can lead to the potential expansions of this work

5 Experimental Results

In this section four methods have been investigated andanalyzed for synthesis of MSA namely LMS gradient decentapproach for updating spread fused with LMS (120590-GDA-LMS) RLS and lastly gradient decent approach for updatingspread integrated with RLS (120590-GDA-RLS) Four algorithmsmentioned are tested based on the frequency range between22GHz and 5GHzwith substrate thickness variable between02175mm and 05175mm and for a scalar value of dielectricconstant set as 233 which is Rogers RTduroid and similar


Table 1 Comparison of computational complexity

Algorithms Number of complexmultiplications

Number ofcomplex additions

Algorithm 1(9) and (10) 1198632 + 119863 1198632 + 119863Algorithm 2(12) and (13) 41198632 + 2119863 31198632 + 119863Algorithm 3(9) (10) (14a) (14b)(14c)

31198632 + 119863 31198632 + 3119863Algorithm 4(12) (13) (14a) (14b)(14c)

61198632 + 2119863 51198632 + 3119863

Table 2 Mean square error (MSE) in testing phase

Algorithm Initial 120590 Final 120590 MSELMS 0151 0151 15626119890 + 02LMS with adaptive 120590 0151 0149 15322119890 + 02RLS 0151 0151 12916119890 + 02RLS with adaptive 120590 0151 1396 13365119890 + 00

frequency band has been designed using rectangular slotantenna in [37] and therefore adopted herein Normalizedspread used for LMS algorithm is 0151 The number ofepochs used in training is 100 and subtractive clusteringapproach is utilized as in the previous work of [13] Trainingresults for LMS is shown in Figure 4 It can be seen thatduring the training phase the estimated values of length(119871) and width (119882) superimpose the actual output valuesThe mean square error (MSE) for this algorithm is 156 forthe testing batched data By making the spread adaptivefor the LMS algorithm based on gradient decent approachMSE decreases to 153 which is not a substantive decreasein overall error for this approach On the other hand withthe similar set of data as in LMS case RLS is employedwith forgetting factor (120582) set at 093 and training of ANNis done On testing it is found that the MSE drops to 129which is better than both LMS and LMS with adaptivespread Further using the adaptive spread technique in RLSalgorithm performance of ANN is enhanced tremendouslyand the MSE is reduced to 1396 Training results with ANNbased on fourth approach is shown Figure 5 whereas Table 2represents the comparativeMSE analysis for the 4 algorithmsdiscussed Please note that the difference between the initialspread and final spread for the second and fourth algorithminTable 2 is disproportionatewhich is indicative of the overallperformance metric of the two algorithms

In Figure 6 we present a 3D depiction for the variablesinvolved in the synthesis design of MSA namely resonancefrequency (119891) substrate thickness (ℎ) and length (119871) andwidth (119882) using adaptive spread based RLS algorithmThereis excellent match of the algorithm output and the desiredoutput The flouring (contour) of Figure 6 indicates thedegree of match Building on the contour of Figure 6 andto better analyze the four algorithms contour plots for the

10 20 30 40 50 60

Output for LMS training

20

25

30

35

40

45

50

Width (W)Length (L)

W-LMS-trainingL-LMS-training

Figure 4 Training results for LMS algorithm

5 10 15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50

Output for RLS training

Width (W)Length (L)

W-AL4-trainingL-AL4-training

Figure 5 Training results for RLS algorithm with the adaptivespread

23

45

24

68

Testing-RLS with adaptive spread

times10minus4

times109

10

20

30

40

50

60

Wid

th (W

) and

leng

th (L

) of p

atch

Substrate thickness (ℎ) Frequency (f)

Figure 6 Testing results for adaptive spread RLS algorithm


times10minus4

25 3 35 4 45

Frequency (f) times109

25 3 35 4 45

Frequency (f) times10925 3 35 4 45


Output for LMS testing

25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

Testing-LMS with adaptive spread

Testing-RLS Testing-RLS with adaptive spread

Figure 7 Contour plots of the LMS RLS and adaptive spread algorithms in MSA synthesis

testing phase of all four algorithms are provided in Figure 7It can be seen in the contour plots that approximated results of119882 and 119871 are represented by ldquoredrdquo and ldquogreenrdquo lines while theactual or target for both119882 and 119871 is shown using ldquobluerdquo linesIn algorithm 4 results almost superimpose desired ldquobluerdquolines the least matching contours are for LMS The degreeof match between approximated and desired output is shownto improve from algorithm 1 to algorithm 4 in respectiveorder Table 3 represents the desired and approximated widthand length for the four algorithms it can be seen that mostsignificant approximation using RBF is 119871 of the patch whichcontributes significantly toMSE for LMS however it is betterapproximated using RLS algorithms From Table 3 it canalso be seen that algorithms 1 and 3 are almost similar in

approximation of 119882 while in all four test cases 119871 is betterapproximated using algorithm 3 as compared with LMSapproach of algorithm 1 Mean square error (MSE) among allfour cases is minimal while using adaptive spread based RLSalgorithmThe accuracy of our fourth algorithm is 9989 ascompared with 9909 achieved by authors of [14] which istheir highest in the synthesis work of MSA

6 Concluding Remarks

This paper presented the comparative analysis of four dif-ferent algorithms for synthesis aspect of MSA using RBFEstimation based on LMS LMS with adaptive spread RLSand RLS with adaptive spread algorithms is utilized for


Table 3 Comparison of the desired and approximated width and length for the four algorithms

Synthesis parameters119891 = 45GHz 119891 = 33GHz 119891 = 233GHz 119891 = 23GHzℎ = 0238mm ℎ = 0387mm ℎ = 0338mm ℎ = 0588mm120576119903 = 233 120576119903 = 233 120576119903 = 233 120576119903 = 233

Desired width 25832808609737622 35226557195096753 43054681016229374 50542451627747518Algorithm 1 25503126915201797 34727797095017110 42546963629337398 53123805223190246Algorithm 2 25399012168041605 34595677176141677 42368210233612288 52787288550277708Algorithm 3 25503126711966839 34727794498976266 42546963026145313 53123800213238567Algorithm 4 25844082198839743 35219397833997661 43044311230214660 50575068292414450Desired length 21589078002479315 29529980036520211 36147398398220965 42477102918108628Algorithm 1 21180298009632367 28977178217363399 35587003693004334 44507734041334601Algorithm 2 21096419382083074 28866795189303346 35436986349108103 44225886126622122Algorithm 3 21142603463604360 29537218733244114 37292493792678805 41295986097468969Algorithm 4 21513894367659727 29466986874035420 36059519035942024 42453611546508384

approximating length and width of the MSA based on theinput of resonance frequency permeability constant andsubstrate thickness It is shown that by augmenting theadaptive spread schemewith LMS andRLS algorithm there ispossibility of significantly improving the performance of RBFKernel It is also shown in this work that the most significantmetric that accounts for maximum error is length of patchantenna and hence its estimation is of utmost importancewhich is accounted in RLS algorithmwith goodmeasureThecomputational power that is desired by machines has beenexplicitly expressed in terms of complex multiplication andadditions Lastly some potential extension of this work isdiscussed

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] D M Pozar and H D Schaubert Eds Microstrip AntennasTheAnalysis andDesign ofMicrostripAntennas andArrays JohnWiley amp Sons 1995

[3] DM Pozar ldquoMicrostrip antennasrdquo Proceedings of the IEEE vol80 no 1 pp 79ndash91 1992

[4] J Huang F Shan J She and Z Feng ldquoA novel multibandand broadband fractal patch antennardquo Microwave and OpticalTechnology Letters vol 48 no 4 pp 814ndash817 2006

[5] S E El-Khamy O A Abdel-Alim A M Rushdi and AH Banah ldquoCode-fed omnidirectional arraysrdquo IEEE Journal ofOceanic Engineering vol 14 no 4 pp 384ndash395 1989

[6] Y C Huang and C E Lin ldquoFlying platform tracking formicrowave air-bridging in sky-net telecom signal relayingoperationrdquo Journal of Communication and Computer vol 11 pp323ndash331 2014

[7] A K Hassan A Hoque and A Moldsvor ldquoAutomated Micro-Wave(MW) antenna alignment of Base Transceiver Stations

time optimal link alignmentrdquo in Proceedings of the Aus-tralasian Telecommunication Networks and Applications Con-ference (ATNAC rsquo11) pp 1ndash5 Melbourne Australia November2011

[8] M A Hoque and A K Hassan ldquoModeling and performanceoptimization of automated antenna alignment for telecommu-nication transceiversrdquo Engineering Science and Technology vol18 no 3 pp 351ndash360 2015

[9] F Gozasht G R Dadashzadeh and S Nikmehr ldquoA comprehen-sive performance study of circular and hexagonal array geome-tries in the LMS algorithm for smart antenna applicationsrdquoProgress in Electromagnetics Research vol 68 pp 281ndash296 2007

[10] C I Kourogiorgas M Kvicera D Skraparlis et al ldquoModelingof first-order statistics of LMS channel under tree shadowingfor various elevation angles at L-bandrdquo in Proceedings of the 8thEuropeanConference onAntennas andPropagation (EuCAP rsquo14)pp 2273ndash2277 IEEE The Hague The Netherlands April 2014

[11] N I Galanis and D E Manolakos ldquoFinite element analysisof the cutting forces in turning of femoral heads from AISI316l stainless steelrdquo in Proceedings of the World Congress onEngineering (WCE rsquo14) pp 1232ndash1237 July 2014

[12] A PMarkopoulos S Georgiopoulos andD EManolakos ldquoOnthe use of back propagation and radial basis function neuralnetworks in surface roughness predictionrdquo Journal of IndustrialEngineering International vol 12 no 3 pp 389ndash400 2016

[13] W Aftab M Moinuddin and M S Shaikh ldquoA novel kernel forRBF based neural networksrdquoAbstract and Applied Analysis vol2014 Article ID 176253 10 pages 2014

[14] N Turker F Gunes and T Yildirim ldquoArtificial neural networksapplied to the design of microstrip antennasrdquo MikrotalasnaRevija vol 12 no 1 pp 10ndash14 2006

[15] K Guney S Sagiroglu and M Erler ldquoGeneralized neuralmethod to determine resonant frequencies of variousmicrostrip antennasrdquo International Journal of RF andMicrowave Computer-Aided Engineering vol 12 no 1 pp131ndash139 2002

[16] J E Rayas-Sanchez ldquoEM-based optimization of microwavecircuits using artificial neural networks the state-of-the-artrdquoIEEE Transactions onMicrowaveTheory and Techniques vol 52no 1 pp 420ndash435 2004

[17] Q-J Zhang K C Gupta and V K Devabhaktuni ldquoArtificialneural networks for RF and microwave designmdashfrom theory


to practicerdquo IEEE Transactions on Microwave Theory andTechniques vol 51 no 4 pp 1339ndash1350 2003

[18] J Virtanen L Ukkonen A Z Elsherbeni V Demir and LSydanheimo ldquoComparison of different electromagnetic solversfor modeling of inkjet printed RFID humidity sensorrdquo inProceedings of the ACES Conference The 26th Annual Review ofProgress in Applied Computational Electromagnetics TampereFinland April 2010

[19] D C Thompson O Tantot H Jallageas G E Ponchak MM Tentzeris and J Papapolymerou ldquoCharacterization of liquidcrystal polymer (LCP) material and transmission lines on LCPsubstrates from 30 to 110GHzrdquo IEEETransactions onMicrowaveTheory and Techniques vol 52 no 4 pp 1343ndash1352 2004

[20] T C Edwards and M B Steer Foundations of Interconnect andMicrostrip Design vol 3 John Wiley amp Sons 2000

[21] S Haykin Neural Networks A Comprehensive FoundationPrentice Hall Englewood Cliffs NJ USA 2nd edition 1999

[22] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons 2012

[23] S Haykin Neural Networks and Learning Machines vol 3Pearson Education Upper Saddle River NJ USA 2009

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[25] L Zhou G Yan and J Ou ldquoResponse surface method basedon radial basis functions for modeling large-scale structuresin model updatingrdquo Computer-Aided Civil and InfrastructureEngineering vol 28 no 3 pp 210ndash226 2013

[26] R Martolia A Jain and L Singla ldquoAnalysis survey on fault tol-erance in radial basis function networksrdquo in Proceedings of theIEEE International Conference on Computing Communicationamp Automation (ICCCA rsquo15) pp 469ndash473 2015

[27] S Chen B Mulgrew and P M Grant ldquoA clustering techniquefor digital communications channel equalization using radialbasis function networksrdquo IEEE Transactions on Neural Net-works vol 4 no 4 pp 570ndash579 1993

[28] L Zhang and P N Suganthan ldquoA survey of randomizedalgorithms for training neural networksrdquo Information Sciencesvol 364ndash365 pp 146ndash155 2016

[29] S Ljung and L Ljung ldquoError propagation properties of recur-sive least-squares adaptation algorithmsrdquo Automatica vol 21no 2 pp 157ndash167 1985

[30] M Moinuddin and A Zerguine ldquoA unified performance analy-sis of the family of normalized least mean algorithmsrdquo ArabianJournal for Science and Engineering vol 39 no 10 pp 7145ndash71572014

[31] A Timesli B Braikat H Lahmam and H Zahrouni ldquoAnimplicit algorithm based on continuous moving least squareto simulate material mixing in friction stir welding processrdquoModelling and Simulation in Engineering vol 2013 Article ID716383 14 pages 2013

[32] UM Al-Saggaf MMoinuddinM Arif and A Zerguine ldquoTheq-Least Mean Squares algorithmrdquo Signal Processing vol 111 pp50ndash60 2015

[33] H Ait Abdelali F Essannouni L Essannouni and D Abouta-jdine ldquoAn adaptive object tracking using Kalman filter andprobability product kernelrdquo Modelling and Simulation in Engi-neering vol 2016 Article ID 2592368 8 pages 2016

[34] M T Akhtar M Abe and M Kawamata ldquoA new variablestep size LMS algorithm-based method for improved online

secondary path modeling in active noise control systemsrdquo IEEETransactions on Audio Speech and Language Processing vol 14no 2 pp 720ndash726 2006

[35] A M Rushdi ldquoImproved variable-entered Karnaugh mapproceduresrdquo Computers amp Electrical Engineering vol 13 no 1pp 41ndash52 1987

[36] A M Rushdi ldquoPerformance indexes of a telecommunicationnetworkrdquo IEEE Transactions on Reliability vol 37 no 1 pp 57ndash64 1988

[37] J-W Wu H-M Hsiao J-H Lu and S-H Chang ldquoDualbroadband design of rectangular slot antenna for 24 and 5GHzwireless communicationrdquo Electronics Letters vol 40 no 23 pp1461ndash1463 2004

International Journal of

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VLSI Design



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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



as advanced design system (ADS) and Ansoft high frequencysimulation system (HFSS) [18] In this work the analysis forthe synthesis of MSA using RBF is performed We presentherein an exposition of significant algorithms such as leastmean square (LMS) and recursive least square (RLS) andfurther investigate their deviants based on the gradient decentapproach (GDA) The choice of the LMS and RLS algorithmis because they are considered fundamental in many subdis-ciplines of engineering such as adaptive filtering and signalprocessingTheparameters that are fed toANNare resonancefrequency (119891) permeability constant (120576119903) and height of thesubstrate (ℎ) Consequently width (119882) and length (119871) areextracted from ANN The rationale behind the proposedwork is that the MSA design and its pertinent equations areof high nonlinearity therefore choosing an algorithm thatresults in minimal error between the estimated data and thedesired data can be useful in the development of the afore-mentioned commercial software Moreover computationalcomplexity of algorithms is another area which needs to belooked into before choosing an algorithm hence we outlinethe complexity difference in terms of complexmultiplicationsand additions that are performed in an algorithm To the bestof the authorsrsquo knowledge such investigation and comparativeanalysis have not been done previously for the synthesis ofMSA

Organization of this paper is such that after introductionin Section 1 an overview of the design aspects of MSAis given in Section 2 Section 3 introduces RBF LMS andRLS approaches and mathematical formulation of adaptivespread factor Complexity analysis is presented in Section 4Simulation results and comparative analysis of the fouralgorithms are given in Section 5 Subsequent section dealswith concluding remarks of this work and references

2 Design of the Rectangular MicrostripPatch Antenna

Patch antennas are multilayered components with conduc-tors that is patch and ground plane separated by dielectricsubstrate Rectangular microstrip patch antenna (R-MSA)consists of width (119882) and length (119871) as presented in Figure 1The other significant parameters required in designing andfabrication of MSA include substrate thickness (ℎ) and itspermeability (120576119903) Choice of substrate determines the sizeand its relevant application base Most common substrateson the market are RTduroid with dielectric constant of22 and liquid crystal polymer (LCP) with the range ofdielectric constant between 29 and 32 [19] Generally thereare numerous substrates available on the market for usabilityin MSA and in practice the permeability of dielectric withMSA in perspective is in the range of 22 le 120576119903 le 12 Feed isanother crucial aspect in design of patch antenna and thereexist multiple methods by which a signal can be fed Thesimplest method is direct contact method shown in Figure 1and thereby used in this work Other methods can be probefeed microstrip proximity coupling and microstrip aperturecoupling [20]

Ground plane

Feed

W

L

ℎ r

Figure 1 Rectangular microstrip antenna

In MSA physical length (119871) and electrical (effective)length (119871eff ) are different because of fringing fields at theedges of the patch The effective electrical length of the patchantenna is given by [2 19ndash22]

119871eff = 119871 + 2Δ119871 (1)

where Δ119871 is the change in length given by

Δ119871 = 0412ℎ(120576119903eff + 03) (119882ℎ + 0264)(120576119903eff minus 0258) (119882ℎ + 08) (2)

Effective dielectric constant of the dielectric material isexpressed as [19]

120576119903eff = 120576119903 + 12 + 120576119903 minus 12 (1 + 12 ℎ119882)minus12 (3)

The actual length of the patch under specified conditions is

119871eff = 1205822 997904rArr119871 = 1198882119891radic120576119903eff minus 2Δ119871

(4)

Hence the practical width is determined as follows

119882 = 1198882119891radic 2120576119903 + 1 (5)

Based on the above mathematical modelling both syn-thesis and analysis can be designed using soft computingtools We however focus on the synthesis of MSA wherethe inputs to ANN are resonance frequency (119891) substratethickness (ℎ) and dielectric constant (120576119903) while length (119871)and width (119882) are extracted from the RBF designs Synthesisaspect of this work is presented in Figure 2 We utilize thismodel as a test bench to compare LMS and RLS algorithmsin the forthcoming sections

3 Synthesis Using RBF-ANN

In this sectionwe examine the feedforward architecture of theANNbased onRBF Layout of RBF is simplistic and it consistsof hidden layer and output layer neurons The function ofhidden layer is to perform a nonlinear operation on the set ofinputs For the purposes of performing a nonlinear operationone can resort to plethora of nonlinear functions such as


RBFLMSRLS

amp W

L

ℎ

r

-GDA

f




120590119895 = 119889maxradic2119898 (6b)



119910119894 (119899) = 119898sum119895=1

119908119895 (119899)Φ119895 (119899) (7)




119864 (119899) = 12119868sum119894=1

(119910119894 (119899) minus 119889)2 (8)


120597119864 (119899)120597119908119895 (119899) =119868sum119894=1

(119910119894 (119899) minus 119889)Φ119895 (119899) (9)


119908119895 (119899 + 1) = 119908119895 (119899) minus 1205781 120597119864 (119899)120597119908119895 (119899) (10)



119864 (119899) = 119899sum119894=1

119891 (119899 119894) (119910119894 (119899) minus 119889)2 (11a)

119891 (119899 119894) = 120582119899minus119894 (11b)



Output neuron

Desired output


LMSRLS algorithm

x1

x2

w1

w2

wm

d

y

Φ1

Φ2

Φm




(12)

119908 (119899 + 1) = 119908 (119899) + 119875 (119899) 119909 (119899) 119890 (119899) (13)



120597119864 (119899)120597120590119895 (119899)= minus119908119895 (119899) 119871sum

119897=1

(119910119894 (119899) minus 119889)2 1198661015840 (1003817100381710038171003817x119897 minus 1198881198961003817100381710038171003817120590119895)Q119897119896 (119899)(14a)

Q119897119896 (119899) = [x119897 minus 119888119896 (119899)] [x119897 minus 119888119896 (119899)]119879 (14b)

120590119895 (119899 + 1) = 120590119895 (119899) minus 1205782 120597119864 (119899)120597120590119895 (119899) (14c)













31198632 + 119863 31198632 + 3119863Algorithm 4(12) (13) (14a) (14b)(14c)

61198632 + 2119863 51198632 + 3119863





10 20 30 40 50 60


20

25

30

35

40

45

50

Width (W)Length (L)



5 10 15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50


Width (W)Length (L)



23

45

24

68


times10minus4

times109

10

20

30

40

50

60

Wid

th (W

) and

leng

th (L

) of p

atch




times10minus4

25 3 35 4 45


25 3 35 4 45




25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)













Competing Interests


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










RBFLMSRLS

amp W

L

ℎ

r

-GDA

f




120590119895 = 119889maxradic2119898 (6b)



119910119894 (119899) = 119898sum119895=1

119908119895 (119899)Φ119895 (119899) (7)




119864 (119899) = 12119868sum119894=1

(119910119894 (119899) minus 119889)2 (8)


120597119864 (119899)120597119908119895 (119899) =119868sum119894=1

(119910119894 (119899) minus 119889)Φ119895 (119899) (9)


119908119895 (119899 + 1) = 119908119895 (119899) minus 1205781 120597119864 (119899)120597119908119895 (119899) (10)



119864 (119899) = 119899sum119894=1

119891 (119899 119894) (119910119894 (119899) minus 119889)2 (11a)

119891 (119899 119894) = 120582119899minus119894 (11b)



Output neuron

Desired output


LMSRLS algorithm

x1

x2

w1

w2

wm

d

y

Φ1

Φ2

Φm




(12)

119908 (119899 + 1) = 119908 (119899) + 119875 (119899) 119909 (119899) 119890 (119899) (13)



120597119864 (119899)120597120590119895 (119899)= minus119908119895 (119899) 119871sum

119897=1

(119910119894 (119899) minus 119889)2 1198661015840 (1003817100381710038171003817x119897 minus 1198881198961003817100381710038171003817120590119895)Q119897119896 (119899)(14a)

Q119897119896 (119899) = [x119897 minus 119888119896 (119899)] [x119897 minus 119888119896 (119899)]119879 (14b)

120590119895 (119899 + 1) = 120590119895 (119899) minus 1205782 120597119864 (119899)120597120590119895 (119899) (14c)













31198632 + 119863 31198632 + 3119863Algorithm 4(12) (13) (14a) (14b)(14c)

61198632 + 2119863 51198632 + 3119863





10 20 30 40 50 60


20

25

30

35

40

45

50

Width (W)Length (L)



5 10 15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50


Width (W)Length (L)



23

45

24

68


times10minus4

times109

10

20

30

40

50

60

Wid

th (W

) and

leng

th (L

) of p

atch




times10minus4

25 3 35 4 45


25 3 35 4 45




25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)













Competing Interests


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Output neuron

Desired output


LMSRLS algorithm

x1

x2

w1

w2

wm

d

y

Φ1

Φ2

Φm




(12)

119908 (119899 + 1) = 119908 (119899) + 119875 (119899) 119909 (119899) 119890 (119899) (13)



120597119864 (119899)120597120590119895 (119899)= minus119908119895 (119899) 119871sum

119897=1

(119910119894 (119899) minus 119889)2 1198661015840 (1003817100381710038171003817x119897 minus 1198881198961003817100381710038171003817120590119895)Q119897119896 (119899)(14a)

Q119897119896 (119899) = [x119897 minus 119888119896 (119899)] [x119897 minus 119888119896 (119899)]119879 (14b)

120590119895 (119899 + 1) = 120590119895 (119899) minus 1205782 120597119864 (119899)120597120590119895 (119899) (14c)













31198632 + 119863 31198632 + 3119863Algorithm 4(12) (13) (14a) (14b)(14c)

61198632 + 2119863 51198632 + 3119863





10 20 30 40 50 60


20

25

30

35

40

45

50

Width (W)Length (L)



5 10 15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50


Width (W)Length (L)



23

45

24

68


times10minus4

times109

10

20

30

40

50

60

Wid

th (W

) and

leng

th (L

) of p

atch




times10minus4

25 3 35 4 45


25 3 35 4 45




25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)













Competing Interests


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














31198632 + 119863 31198632 + 3119863Algorithm 4(12) (13) (14a) (14b)(14c)

61198632 + 2119863 51198632 + 3119863





10 20 30 40 50 60


20

25

30

35

40

45

50

Width (W)Length (L)



5 10 15 20 25 30 35 40 45 50 5515

20

25

30

35

40

45

50


Width (W)Length (L)



23

45

24

68


times10minus4

times109

10

20

30

40

50

60

Wid

th (W

) and

leng

th (L

) of p

atch




times10minus4

25 3 35 4 45


25 3 35 4 45




25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)













Competing Interests


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus4

25 3 35 4 45


25 3 35 4 45




25

3

35

4

45

5

55Su

bstr

ate t

hick

ness

(ℎ)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)

25 3 35 4 45


times10minus4

25

3

35

4

45

5

55

Subs

trat

e thi

ckne

ss (ℎ

)













Competing Interests


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article On Modelling and Comparative Study of LMS