| The wireless industry has experienced explosive growth in bothsubscribers and minutes of use in recent years. With the addition of wirelessdata services, the pressure on network resources and quality will continue todramatically increase. In order to provide better communication quality andsufficientresourcesforsuchalargedemand,smartantennasystems arebeingdeployed. Smart antennas can improve the system performance byincreasingchannel capacity and spectrum efficiency, steering multiple beams to trackmany mobiles, extending range coverage, reducing delay spread, multi-pathfading and co-channel interference. Smart antenna systems are usuallycategorized as either switched-beam or adaptive-array systems. Althoughboth systems attempt to increase gain in the direction of the user, only theadaptive-array system offers optimal gain. In a smart antenna system thearrays arenot smart. It is thedigital signal processingthat makes them smart.Using signal processing algorithms, smart antenna systems can locate andtrack signals of both users and interferers, and can dynamically adjust theantenna patterns to enhance reception while minimizing interference. Signalprocessingalgorithmisclassifiedasdirectionofarrival(DOA)algorithmandbeam-forming algorithm. The DOA computes the direction of arrival of allsignals by computing the time delays among the antenna elements, whilebeam-forming algorithm controls the weights according to predefinedobjectives. These dynamic calculations enable the system to change itsradiation pattern for optimized signal reception. Beam-forming algorithm iscorealgorithmofadaptivealgorithms.This thesis is focused on Least Mean Square (LMS) algorithm withvariable step size of beam-forming algorithms for smart antenna system.LMS algorithm is by far the most commonly used adaptive algorithm,because it was proposed at first, its implementation was simple and required relatively little computation. LMS algorithm is an approximation of thesteepest descent algorithm which uses an instantaneous estimate of thegradient vector of a cost function. The estimate of the gradient is based onminimization of the mean square error between the received signal and thereference signal. The parameter of step sizeμplays a very important rolein LMS algorithm. It is usually chosen after experimentation for a givenapplication. The step sizeμhas to be 0<μ<(1/2)max Smallμmay yieldaccurate weight vector and less misadjustment error, but slow convergence;while largeμleads to fast convergence but unstable. We usually use smallμto get less misadjustment and ignore convergence. To solve thiscontradiction, variable step size LMS algorithm was proposed. In the initialstage of convergence step size should be relatively large, so that the speed ofconvergence and tracking speed of time-varying systems can be fast. Aftertheconvergenceofthealgorithm,asmall adjustment shouldmaintainasmallstep size to achieve homeostasis imbalance noise. The studyshows that mathcurve like sigmoid function, probability curve and witch curve suit the rulesofstepsizeaftertransformation.In this paper, a relationship between step sizeμand signal error e(n)is established. In the process of change, e (n)is getting smaller and close tozerogradually,while change ofμis thesameas e (n). When e (n)is zero,μis zero. So we can establish mathematical smoothing curves which pass(0,0) and we can adjustμas e (n)changes. Sigmoid function isY = 1 /(1+e-X).The function can be changed as follows: taking the absolutevalue of x and taking half units translation downwards along y axis. So the relationship betweenμand e (n)can beμ(n)=β(1/(1+e-α|e(n)|)-0.5). Probability curve is Y=e-X2.Taking x axisymmetric and translating one unitupwards along y axis gives the relationship betweenμand e (n):μ(n)=β(1-e-ae2(n)). Witch curve is Y=(8a3)/(X2+4a2). Setting a=0.5 and takingY=a symmetric result the relationship betweenμand e (n):μ(n)=β(1-1/(αe2(n)+1)). In addition of these three mathematical curves, it isfoundthat hyperbolictangent curve is also inaccordwith thechanges ofstepsizeμ. This paper puts forward a step size LMS algorithm based onhyperbolic tangent curve which is Y = tanh(X). Taking the absolute value ofx yields the relationship betweenμand e (n):μ(n)=βtanh(α|e(n)|γ). In thesealgorithms,βis used to control the shape of the function which decides theincreasingspeed of the curve,βis used to control the range of the functionandγin the improved algorithm is also used to control the shape of thefunction, which decides the decreasing speed of the curve. In order todetermine the ranges of these coefficients well, this paper carries out adetailed analysis and shows that there are two circumstances to get optimalvalue. One of them is to set the value ofβnear1/λmax (In sigmoid function,βis set near 2/λmax ) in order to get faster convergence. The other is to setsmallμand bigβin order to adjustμnear 1/λmax in the initial stage ofconvergence, it has less misadjustment error but slow convergence speed.Therefore, when setting coefficients, we can choose them accordingto actualsituation. When improving algorithm, the algorithm is required to be simple enough for hardware to realize. The parameters should be easy to control.The speed of convergence should be fast and the misadjustment error shouldbe small. For completion of a comprehensive iteration, the algorithm isrequired to have small amount of computation and small memory space. Tocompare the performance of all the algorithms, the same initial value of ?is set by adjusting the parameters of each LMS algorithm with variable stepsize. Research shows that when adjusting each algorithm to optimalperformance,thespeedofconvergenceofthesealgorithms basedon differentmathematical curves are similar, but the algorithm based on hyperbolictangent curve has the smallest misadjustment error. This improves theperformanceoftheLMSalgorithmwithvariablestepsize.Since MATLAB provides a powerful data processing and statisticalfunctions and relative simple GUI, a graphical user interface simulationsystem using MATLAB to analysis beam-forming algorithm of LMSalgorithm with variable step size is developed for smart antenna. Thefunctions ofthis system arelistedas follows: adjustingdifferent numbers andspacing of element to study the influence to antenna pattern; adjustingdifferentdirectionofarrivalandinterferencetostudytheinfluencetoantennapattern; adjusting different parameters to compare the impact of the specificalgorithm with different coefficients; adjusting optimal parameters of eachalgorithm and compare the performance of all of them to determine the bestalgorithm. The system provides the performance functions as follows: themean square error learning curve, the weight vector learning curve, the meansquare deviation learning curve, the desired signal and the signal contrastcurve and SNR curve. We can analyze various performances of algorithmsandfacilitatedataanalysisandstatisticsthroughthissystem. |
