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  1. 1. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710705 | P a g eAnalysis of Adaptive Direct sequence Spread Spectrum usingLeast mean squares AlgorithmMainak Chakraborty*, Moloy Kumar Chowdhury***(Department of Electronics & Communication Engg, Heritage Institute of Technology, Kolkata)** (Department of Electronics & Communication Engg, Heritage Institute of Technology, Kolkata)ABSTRACTSpread spectrum forms an importantaspect of digital communication technology wherethe message signal is transmitted over a channelhaving bandwidth much greater than required.Direct Sequence Spread Spectrum (DSSS) belongsto the category of spread spectrum methods andforms an important part in mobilecommunications. The objective of this paper is toinvestigate the FIR optimal, Wiener and leastmean square (LMS) algorithms in the design oftraversal tap delay line filters for the purpose ofcompensating the effect of the quasi-staticcommunication channel in mobile communicationusing DSSS. The designed equalizers remove thedistortion caused by the channel from thetransmitted signal with the help of an adaptive,recursive Weiner filtering technique using theconcept of Weiner-Hopf equation using LMSalgorithm and to aid in the equalization process, atraining sequence is first transmitted to adapt thefilter coefficients. The performance metrics arestudied for DSSS system with and withoutadaptive equalization. Further, the convergence ofLMS algorithm is also verified as well as theimpact of different step-sizes on the speed of theconversion and the accuracy of the overallalgorithm. Exhaustive Monte Carlo simulativeanalysis have been performed to investigate thescheme under varying distortion levels and Signalto noise ratio (SNR) values via impulse response,frequency response and uncoded Bit Error Rate(BER) simulations.Keywords - Direct Sequence Spread Spectrum,Least Mean Square, Weiner-Hopf equation, ChannelEqualization, Adaptive filter, Transversal tap delayline filters, Bit error rate.I. INTRODUCTIONIn digital communication, spread spectrumtechnique is used to increase the transmitted signalbandwidth to a higher value than required fortransmitting the information digits. Spread spectrumtechnology was developed for military andintelligence purpose, and radar engineers employedthe method of spectrum spreading for jamming. Mostrecent spread spectrum techniques (2.4GHz) areassociated with IEEE 802.11 standard. There are afew number of spread spectrum methods amongwhich two are approved by FCC, namely Directsequence spread spectrum and Frequency Hoppingspread spectrum. IEEE 802.11 direct sequence spreadspectrum technology encompasses differential binaryphase shift keying for 1Mbps and differentialquadrature phase shift keying at 2Mbps. Both thesedigital modulation techniques used are based onphase shift keying modulation.Sometimes signals are at a low probabilityto intercept, and as a result they are difficult to bedetected or to be demodulated after detection. Toensure the security, a signal is needed to bedeveloped which is very difficult to interfere with orto “jam”. Employment of spread spectrum techniqueseliminates the problems and also fulfils the questionof security[1].Direct sequence spread spectrum techniquesoffers several advantages like easy implementationand high data rate. At present almost all WirelessLANs use this technology. Here both the transmitterand receiver are set with frequency at 22 MHz. InDSSS, chipping code is used to combine with datasignal. The RF carrier is then modulated by thecombined signal so that the transmitted signal isspread over a wide bandwidth. Another name ofchipping code is processing gain which is implied toprotect high signal from interference. Sum of chips inthe code will determine how big the spreadingoccurs, and sum of chip per bit and code rate willdetermine the data rate.[2]We may consider frequency translation of amessage signal of B Hz by DSBSC modulationtechnique. The modulated signal (of bandwidth 2BHz) .It would typically override the noise occupyingthe same part of the spectrum. So it would be easy todetect by spectrum analyzer giving very highprobability of intercept. Moreover, to achievesynchronous demodulation, a carrier synchronizedwith transmitter is required at the receiver [19].Therecovered signal to noise ratio comes as 3 dB betterthan measured one and it occurs due to coherentaddition from each of sidebands where noise is notadded. This 3 dB improvement is called asprocessing gain. In practice, transmission bandwidthand message bandwidth are in the ratio of 2:1 andthousand of different carriers are implemented in
  2. 2. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710706 | P a g eorder to generate thousands of DSBSC signals fromsame message. As the carriers are spread over alarge bandwidth (much greater than 2B Hz) all ofthese DSBSC signals are spread over the samebandwidth. Here the power of an individual DSBSCin spread spectrum phenomenon is thousand timesless as compared to the single DSBSC case. It wouldbe hard to find out by spectrum analyzer, as over theentire bandwidth occupied by one of these DSBSCsignals, it would be in literally „buried in the noise‟state. Instead of the total transmitted power beingconcentrated in a band of width 2B Hz, the multiplecarriers have spread it over a very wide bandwidth.We observe that the signal-to-noise ratio for eachDSBSC is very low (well below 0 dB). At thereceiver in order. to recover the message from thetransmitted spread spectrum signal , thousands oflocal carriers are needed, at the same frequency andof the same relative phase, as all those at thetransmitter. All these carriers come from a pseudorandom binary sequence (PRBS) generator. [9]With the help of a stable clock, and a long sequence,it can be proved that the spectrum of a pseudorandom binary sequence generator is a good sourceof these carriers. Similarly, a second PRBSgenerator, at the same rate, and correctly aligned, isenough to regenerate all the local carriers at thereceiver demodulator for perfect synchronization.Alternatively the PRBS signal is termed as PN –pseudo noise - signal, since its spectrum approachesthat of random noise [7].In other words, correct sequence at the receivermeans that the message received from each of thethousands of minute DSBSC signals are combined inphase, coherently and are added up to produce afinite message output. Otherwise they add withrandom phases, which in turn results weak powerlevel noise-like output [14].II. ADAPTIVE FILTERINGIn direct sequence code division multipleaccess(DS/CDMA) system, to recover thetransmitted information, the received signal shouldfirst despread using a locally generated pseudo noise(PN) code sequence. The code acquisition performsinitial code timing alignment between the locallygenerated and received signals. Code acquisitionmethod is usually conducted using a correlator toserially search the code phase, which is related to thechannel propagation delay. This approach performswell for the adaptive filtering approaches have highacquisition-based capacity and drawback of theseschemes is high computational complexity [8]. Dueto down-sampling operations, the computationalcomplexity of the adaptive filters can be effectivelyreduced. Code synchronization is a very essentialand important part of any spread spectrum (SS)system in order to remove the spreading effectinduced by the transmitter and to exploit theprocessing gain of the spread signal [12].Thereceiver must be able to estimate the delay offsetbetween the spreading code in the received signaland the locally generated replica of the code beforedata demodulation is started. Code synchronizationis usually developed over two steps, namelyacquisition and tracking [9].In this application ofmultirate processing, not only the computationalcomplexity, but also the mean acquisition time canbe effectively reduced. Here the basic idea of thescheme used in multirate adaptive filtering forDS/CDMA code acquisition is discussed [10].Figure 1 shows a generic adaptive signal processingsystem. The system consists of three parts processor,adaptation algorithm, performance function. Theprocessor is a part of the system that is actualresponsible for processing of input signal x andgenerating output signal y . The processor ishowever to be a FIR digital filter. The performancefunction however takes the signals x and y asinputs along with other data d that may affect theperformance of the system from x to y .Theperformance function is the quality measure of theadaptive system. To the objective function and incontrol theory it corresponds to the cost function.The output  is the performance function of thequality signal illustrating the processor at its presentstate and indicating whether it is performing welltaking into account the input signal, output signaland other relevant parameters[18]. The quality signal is finally fed to the adaptation algorithm. Thetask of adaptation algorithm is to change of theparameters of the processor in such a wayperformance is maximized. In the example ofadaptive filters this would require change in tapweights. Close loop adaptation has advantage ofbeing usable in many situations where no analyticsynthesis procedure is either exist or is knownand/or when the input signal characteristics variesconsiderably. Further in cases where physical systemcomponents values are variable or inaccuratelyknown closed loop adaptation will be the best choiceof parameters. In the event of partial system failurethe adaptation mechanism may even be able toreadjust the processor [21].In a way that the system will still achievean acceptable performance. System reliability canoften be improved using performance feedback.There are however some inherent problems aswell.The processor model and the way the adjustableparameters of the processor model is chosen affectsthe possibilities of building a good adaptivesystem[6]. We must be sure that the desired responseof the processor can really be achieved by adjustingthe parameters within the allowed range.Using toocomplicated a processor model would on the otherhandmake analytic analysis of the systemcumbersome or even impossible.The performancefunction also has to be chosen carefully [17]. If the
  3. 3. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710707 | P a g eFigure 1 Closed Loop Generic Adaptive Systemperformance function does not have a uniqueextremum, the behaviour of the adaptation processmay be uncertain. Further, in many cases it isdesirable(but not necessary)that the performancefunction is differentiable. Choosing “other data”carefully may also afffect the usefulness of theperformance function of the system [20].Finally theadaptation algorithm must be able to adjust theparametersof the processorin a way to improve theperformance as fast as possible and to eventuallyconvergeto the optimum solution(in the sense of theperformance function).Too fast an adatationalgorithm may, however, resultsin instability orundesiredoscillatory behaviour. These problems arewell known from optimization and controltheory[5].The processor and the performancefunction:- In this section a common processormodel, th eadaptive linear combiner and a commonperformance function, the mean square error wull bepresented.Throughout the text,vectornotation will beused where all the vectors are assumed to be columnvectors. And are usually indicated as a transpose ofa row vector unless stated otherwise [13].The adptive linear combiner:- One of the mostcommon processor model is the so called adaptivelinear combiner shown in figure 3.2. In the mostgeneral form the combiner is assumed to have L+1inputs denoted0 1[ ..... ]Tk k k LkX x x x……………………………(1)Where the subscript k is used as the time index .Similarly the gainsor weights of the differentbranches are denoted0 1[ ..... ]Tk k k LkW w w w……………………………(2)Since we are now discussing with an adaptivesystem the weights will also vary in time and hencehave a time subscript k. The output y of the adaptivemultiple input linear combiner can be expressed as0Lk lk lkly w x …………………………………(3)As can be seen from equation (3.3) this is nothingbut a dot product between the input vector and theweight vector ,henceT Tk k k K Ky X W W X ……………………….(4)III. LMS ALGORITHMThe After Least mean squares (LMS)algorithms are a class of adaptive filter used tomimic a desired filter by finding the filtercoefficients that relate to producing the least meansquares of the error signal (difference between thedesired and the actual signal). It is a stochasticgradient descent method in that the filter is onlyadapted based on the error at the current time [4].The LMS algorithm uses a special estimate of thegradient that is valid for the adaptive linearcombiner. Thus, the LMS algorithm is morerestricted in its use than the other methods. On theother hand, LMS algorithm is important because ofits ease of computation and because of it does notrequire the repetitive gradient estimation[16]. TheLMS algorithm is quite often the best choice formany different adaptive signal processingapplications in processing domain. The LMSalgorithm is a widely used algorithm for adaptivefiltering. The algorithm is described by the followingequations [11].10( ) ( )* ( )Miiy n W n x n i  ……………(5)( ) ( ) ( )e n d n y n  ………………………(6)( 1) ( ) 2 ( )* ( )i iw n w n ue n x n i    ….(7)In these equations, the tap inputs x(n),x(n-1),……,x(n-M+1) form the elements of the referencesignal x(n), where M-1 is the number of delayelements. d(n) denotes the primary input signal, e(n)denotes the error signal and constitutes the overallsystem output. ( )iw n denotes the tap weight at thenth iteration. In equation (3), the tap weights updatein accordance with the estimation error. And thescaling factor u is the step-size parameter u controlsthe stability and convergence speed of the LMS
  4. 4. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710708 | P a g ealgorithm. The LMS algorithm is convergent in themean square if and only if u satisfies the condition: 0< u < 2 / tap-input power M where tap input poweris:120{ ( ) }MkE u n k . Here max is the largestEigen value of the correlation matrix of the inputdata. The stability would be confirmed if thefollowing relation is satisfied:20inputsignalpower  …………….(8)If step size increases, it would increase adaptationrate as well as residual mean squared errorThe LMSalgorithm is most commonly used adaptive algorithmbecause of its simplicity and a reasonableperformance. Since it is an iterative algorithm it canbe used in a highly time-varying signal environment.It has a stable and robust performance againstdifferent signal conditions. However it may not havea really fast convergence speed compared othercomplicated algorithms like the Recursive LeastSquare (RLS). It converges with slow speeds whenthe environment yields a correlation matrix Rpossessing a large Eigen spread. Usually trafficconditions are not static, the user and interfererlocations and the signal environment are varying withtime, in which case the weights will not have enoughtime toconverge when adapted at an identical rate.The,μ isthe step-size needs to be varied in accordance withthe varying traffic conditions. There are severalvariants of the LMS algorithm that deal with theshortcomings of its basic form. The Normalized LMS(NLMS) introduces a variable adaptation rate [15]. Itimproves the convergence speed in a non- staticenvironment. In another version, the Newton LMS,the weight update equation includes whitening inorder to achieve a single mode of convergence [3].For long adaptation processes the Block LMS is usedto make the LMS faster. In block LMS, the inputsignal is divided into blocks and weights are updatedblock wise. A simple version of LMS is called theSign LMS. It uses the sign of the error to update theweights. Also, LMS is not a blind algorithm i.e. itrequires a priori information for the reference signal[7].In the block diagram (Fig. 2), the source containsdigital information. The PNS is pseudo randomsequence which is itself semi random in nature. It ismultiplied and applied to BPSK modulation input.The modulated output is given to a time varyingchannel. The channel is quasi static in nature. At thereceiver adaptive equalizer is adjusted depending oncoefficients that are determined by LMS algorithm.The equalized output is applied to the BPSKdemodulator where it is further multiplied by PNS toobtain binary output which is an estimate of theoriginal source signal.Table:- Various Parameters and ValuesParameter ValuesTotal Iteration 700Step Size 0.125Taps 7SNR 30dBFigure 2 Block Diagram of DSSS-LMS
  5. 5. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710709 | P a g eIV. RESULTS OF SIMULATIONV.Fig. 3 Training Sequence and Equalized OutputFig.4 Noise Value for different IterationFig.5 PNS Bit SequenceFig.6 Original Message Bit SequenceFig.7 Channel and Equalizer Impulse ResponseFig.9 LMS Adaptation Learning CurveFig.8 Dsss Frequency Spectrum
  6. 6. Mainak Chakraborty, Moloy Kumar Chowdhury / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.705-710710 | P a g eFig 3 shows four discrete time signals The firstsequence is training sequence for equalizer training.The training sequence is convolved to the channel.This is shwon in output after channel. Final output ofequalizer shows non zero values at only two timeindices. The next picture shows equalized binaryfinal output and equals the training sequence. In Fig4, we observe the noise amplitude for differentiterations. The total no. of iteration are set at k=700.In Fig. 5, PNS sequence is shown which ispseudorandom bit sequence only for 25 iteration. InFig. 6, original message bit sequence generated bythe source for 10 iteration only. In Fig. 7, the imulseresponse of channel and equalizer is observed. Herewe see that equalizer impulse response is inverse ofchannel response and cascade combination ofchannel and equalizer againstno. Of taps. In Fig.8 itexhibits the direct sequence spread spectrumfrequencey spectrum. In Fig. 9, we observe LMSadaptation curve for step size0.125and outputestimation error (in dB) is plotted against total no. ofiteration. Here we see that untill 100 iteration, LMScurve exhibits fluctuations and after it becomesconstant at -0.1dB..VI. CONCLUSIONHere we implemented the adaptive directsequence spread spectrum communication systembased on LMS algorithm. Before the actual data istransmitted training sequence is initially sent toadjust the equalizer to nullify the channel distortion.Then the data is transmitted at the end of each packettransmission and the channel profile changes. Thusthe channel is modeled as quasi static in nature. Theoutput sequence at DSSS demodulator equals theoriginal digital information sent by source. We alsoobserved that LMS adaptation learning curvegradually approaches a constant value. Based onLMS algorithm, the equalizer impulse response isobserved to be inverse of channel response.REFERENCES[1] G. R. Ansari and B. Liu, "Multirate signalprocessing," in Sanjit. K. Mitra and JamesF. Kaiser (ed.), Handbook for DigitalSignal Processin, 981-1084, John Wiley,1993.[2] M.G. Bellanger, G. Bonnerot, and M.Coudreuse, "Digital filtering by polyphasenetwork: application to sample-ratealteration and filter banks," IEEETransactions on Acoustics, Speech, andSignal Processing, vol. 24, pp. 109-114,April, 1976.[3] M. Bellanger, Digital processing of signals:Theory and practice, New York: JohnWiley & Sons, 1989.[4] Crochiere, R.E., and Rabiner, L.R.,"Interpolation and decimation of digitalsignals - A Tutorial Review," Proceedingsof the IEEE, vol.78, pp. 56-93, March1981.[5] R.E., Crochiere and L.R. Rabiner, Multiratedigital signal processing Englewood Clifs,Prentice-Hall, 1983.[6] A.G. Dempster and M.D. Macleod,"General algorithms for reduced adderinteger multiplier design," Electron. Letters,vol. 31, pp. 261-264, Oct. 1995.[7] W. Drews and L. Gaszi, "A new desigmethod for polyphase filters using all-passsections," IEEE Transactions on Circuitsand Systems, vol. 33, pp. 346-348, March1986.[8] N. J. Fliege, Multirate digital signalprocessing, New York: John Wiley & Sons,1994.[9] F. J. Harris, Multirate Signal Processing,Upper Saddle River: Prentice Hall, 2004.[10] T. Hentchel, Sample Rate Conversion inSoftware Configurable Radios,Morwood,MA: Artech House, 2002.[11] A Antonieu, Digital Filter Analysis andApplication,McGraw Hill New York[12] J W Adams FIR Digital Filters ,IEEE,1991[13] J W Adams and A N Wilson,A newApproach to Digital FIRfilter,IEEE,1983.[14] J Babts, Digital Signal ProcessingApplications, Prentice Hall 1995.[15] S. K. Mitra, Digital Signal Processing,McGrawHill, 1999.[16] Andrea Goldsmith, WirelessCommunications. Cambridge UniversityPress,2004.[17] Bernard Sklar, Digital Communications.Pearson Education Press,2005[18] Chanpreet Kaur et al. / International Journalof Engineering Science and Technology(IJEST) “High Speed and Cost EfficientAdaptive Equalizer for ISI Removal”Vol 3,No 1, Jan 2011.[19] J.G. Proakis, “Adaptive equalization fortdma digital mobile radio,”[20] IEEE Trans. Vehic. Technol., vol. 40, no. 2,pp. 333–341, 1991.[21] B. Widrow and S. D. Stearns, AdaptiveSignal Processing. Prentice Hall SignalProcessing Series, 1985.