Noise Performance of a Matched Filter PN Code Synchroniser u

7/27/2019 Noise Performance of a Matched Filter PN Code Synchroniser u

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Noise Performance of a Matched Filter PN Code Synchroniser Using aMaximum Likelihood DetectorR F OrmondroydSchool of Electronic and Electrical EngineeringUniversity of Bath, BathB A2 7A Y , UK

A B S T R A C TUsing a computer simulation, this paper compares theacquisition performance of a digital matched filter PNcode synchroniser, based on a simple threshold detector,with that using a maximum likelihood detector underconditions of additive Gaussian noise. Both types ofsynchroniser have a hard-limiter at their input. It isfound that the conventional digital matched filtersynchroniser has excellent acquisition performance inrelatively good to moderately poor input SNRs, but failsas the input SNR falls below a threshold SNR,whereas the maximum likelihood detector system hasgood acquisition performance over a much wider nnge ofinput SNR, even with data modulation.

I N T R O D U C T I O NSerial-search synchronisation systems are widely used toobtain coarse synchronisation between the locallygenerated replica pseudo-noise sequence and the noise-corrupted received sequence in direct-sequence spread-spectrum receivers. These systems use the measure ofcorrelation between the two sequences to indicatewhether or not they are in coarse synchronisation. Thecorrelator can be based on active correlation or passivecorrelation using a digital matched filter. In this paper,passive correlation is assumed. A baseband model of theserial search strategy is shown in figure 1.If the correlation value is low, it is probable thatsynchronisation has not been achieved. The code phaseof the locally generated code is then updated according tosome search strategy and the new correlation value isthen obtained and checked. If the correlation value is

values [11. Here, a priori knowledge of the operationofthe system is used to search those code epochs which aremost likely to lead to synchronisation first and then toexpand the area of search only if these have not lead tosynchronisation.Because of the very poor SIR at the input to thesynchroniser, the correlator output is corrupted by noiseand the lock-detector may indicate lock prematurely afalse alarm) or may miss a correct indication of lock.Both consequences worsen the search time, particularlythe occurrence of the false alarm which may take a longtime to be verified. If an active correlator is being usedit is possible to reduce the effects of noise on theprobability of correctly detecting lock ( P d ) and theprobability of a false alarm Pf,) y correlating the codesfor a longer time (the dwell-time) before updating thecode epoch. This can reduce the acquisition time in poorSIR conditions: however, it can also result in a linearincrease in search time with dwell-time if the SIR isrelatively good. For the case of the digital matched filterof figure 1, signal averaging is over the fixed codelength.

D I G IT A L M A T C H E D F I L T E RS Y N C H R O N I S E RThe advantage of using a digital matched filter for manysystems is that the passive correlation is formed within asingle chip period, rather than over the dwell-time, zd,which is many chip periods. For a code sequence oflength, L , with a chip rate, fc = l / T c , the passivecorrelator signal consists of a train of impulsescorresponding to the in-lock condition, which occur at a

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Figure I Schema tic block diagram of a seriai-senrch PN code acquisition systembased on a mtchedfi l terhigh however, this indicates that coarse synchronisationis likely and the search is stopped. During the initialsearch, a number of strategies can be followed. Thesimplest is to linearly increment the phase of the localcode by one cell (typically 0.5 chip). Thus, in noise-free conditions, a code of L chips would have 2L cellsand, on average, it would take L cell phase updates toachieve coarse lock to within 0.5 chip. Other strategiesbased on binary search patterns and expanding windowsearches can also be used to try and skew the likelihoodof acquisition away from the average value, L , to lower

period of L / j c .Thus, in noise free conditions, with twocells per chip, lock can be obtained, on average, afterL/fc seconds for the matched filter structure comparedwith Lzd seconds for the serial synchroniser.It would appear, therefore, that since l l f


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detections of the in-lock signal. In this case, because ofthe fixed structure of the digital matched filter, it is notpossible to improve the reliability of the sampleestimates from the passive correlator except by.increasing the length of the code, which effectivelyincreases the dwell-time. Indeed, when a simplethreshold decision is made on the output of the matchedfilter at low SNRs, Pd can be lower and Pfa can be muchhigher than for the fixed-dwell detector and thiscounteracts any benefits resulting from the higherdecision rate at the detector. Consequently, in highlevels of noise, it is by no means certain that thematched filter will outperform the active correlator serialsearch system.An analysis of the mean acquisition time of a digitalmatched filter PN code synchroniser has been carried outby Pandit [21 for the case where the codes are at basebandand there is no data modulated onto the PN code. Let Tvbe the time interval in which a false alarm occumng canaffect the rh impulse at an instant t and let Pav be theprobability of detecting this Correlation impulse. If Pd isthe probability of detection of a correlation impulse, nfais the falseam rate of the detector with T , as the falsealarm verification time, then the mean acquisition timeT,,,, is given by [2]:

results by showing the effect of the noise and hardlimiting on the mean value of the wanted sample outputand on the mean value of the spurious values. Twocurves are shown for the spurious output. One assumesthat only the mean of the peak positive spurii are ofinterest, whereas the other curve assumes that anabsolute value detector is being used. The differencebetween peak positive and absolute peak spurii isrelatively small.

30 -25 -20 -15 -10 -5 0 5 10input SNR dB)

Thus, as for all types of serial synchroniser, theacquisition performance of the system depends verycritically on the probabilities of detection and false alarmunder the prevailing noise and interferenceconditions andthese in tum depend on the threshold setting of thedetector which follows the matched filter. In mostcases, it is extremely difficult to get these probabilitiesin a convenient closed form. In this paper, a computersimulation of the digital matched filter has been carriedout in order to obtain the Pd and nfu characteristicsas afunction of the key system parameters.Simulation ResultsIn the simulation of the matched filter and the detectorsystem, additive Gaussian noise was added to a 2047chip maximal length sequence prior to hard limiting.The hard limited signal was fed to the matched filter. Nodata was added to the PN sequence. The clock frequencyassumed for the sequence was 3Mchip/s. The time seriesof outputs from the matched filter represents a singlewanted sample value, indicating the in-lock condition,and 2046 sample values indicating an out-of-lockcondition. All have random amplitudes due to the effectof the additive noise and hard-limiting. In thesimulation 1OOO complete code sequences were passedthrough the filter and the amplitude distributions of the

. wanted and spurious matched filter outputs were obtainedat each signal to noise ratio. Figure 2summarises these

Figure 2 Effect of the input SNR on the mean valuesof the wanted and peak spurious output sam ples f romthe digitalfilter in AWGN and hard-limiter inputThe hard limiter at the input to the digital matched filtereffectively acts as an AGC operating on the totalreceived signal. Consequently, as the variance of thenoise at the filter input is increased, the mean wantedoutput from the matched filter, representing the in-locksignal reduces in value. It is apparent that at SNRsworse than -19 dB, where the characteristic of the meanvalue of the wanted signal crosses the characteristic ofthe spurii, it would appear to be very difficult to achievelock because of the large increase in nfu. In practice,however, the situation is more complicated to assess,and it is necessary to examine the amplitudedistributions of the wanted and spurious signals at thematched filter output. These are shown in figure 3, forthree representative input SNRs.At relatively good input SNRs (figure 3 6 , there is aclear distinction between the distributions of the wantedand spurious samples and it is a relativelystraightforward matter to choose a threshold level whichoptimisesPd and minimises PfU.At relatively poor input SNRs (figure 3b) where themean value of the wanted samples is of the Same order asthe mean value of the peak spurii, there is virtuallytotal overlap of the two distributions. However, becausethe standard deviation of the distribution of the spurii isso much smaller than that of the wanted signal, it is stillpossible to set a threshold level (typically around anormalised threshold level of 0.10) such that P d .although small, is still useful and larger than the falsealarm probability, Pfa . Figure 3c shows t h e situation atvery low input SNRs. Here, the mean of thedistribution of the wanted samples is below the mean ofthe distribution of spurii.

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0.0 0.05 0.10 0.15Normalised ilter o w tFigure 3 Distributions of the wanted and spurious correlations fo r the digital matched filter

f o r AWGN and har d-limited inputAgain however, because of the difference in the variancesof the two distributions, it is still possible to set athreshold level which will ensure that Pd >> P f u .However, in this case, P d is extremely low and this hasa very large effect on the mean acquisition time set byequations 1 ) - 3).In order to be able to use the mean acquisition timeequations, it is necessary to obtain the P d vs nfucharacteristics of the digital matched filter/detectorsystem as a function of the threshold level of thedetector. From the simulations carried out above, theeffect of threshold level on P d and nfa are shown infigures4 nd 5 respectively.

0.02 0.04 0.06 0.08 0.1 0.12 0.14normalised t h r es ho ld l ev e lFigure4 Effect of the threshold level on the probabilityof detecting the wanted sample in AWGN and hard-limited inputIt is of interest to note that although the threshold levelhas a large effect on the false alarm rate, input S N R hasa very small effect. These characteristics were then usedwith equations 1 ) - 3) to obtain the mean acquisitiontime as a function of threshold level at each value ofS N R This is shown in figure 6

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0.02 0.04 0.06 0.08normalised threshold levelFigure 5 Effect of the threshold level on the averagenumber of fals e alarms.for a 2047 chip search in AWGNand hard-limited input

10-4) . . . I . . . . . . I . . . . . L0.04 0.06 0.08 0.1 0.12 0.14normalised t hreshold l v l

Fi ure 6 Effect of threshold level and input SNR on thmean acquisition time of a digital matched filt er subjectto AGWN and a hard-limited filter input.


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To obtain this result it was assumed that the verificationtime was vr = 10ms. It is seen that the meanacquisition time is sharply optimised with regard tothreshold level. The explanation for this is as follows.If the threshold level is set too high, the false alarm ratewill be negligible, but the probability of detection willalso be low and it is this term which is dominant insetting the mean acquisition time. As the thresholdlevel is lowered P d increases and since this is still thedominant term, Tma reduces. At low threshold settings,nfu increases significantly whilst Pd starts to saturate toa value close to 1 and nf now becomes the dominantterm causing T m a to increase. Note also how thesensitivity of the mean acquisition system to thresholdlevel: a) gets more critical as the input SNR worsens,(b) the sensitivity to threshold is lower if the thresholdis set slightly high (because the effect of an increase innfa is worse than the effect of a slightly reduced P d .From this graph, it is possible to plot a curve of theoptimum value of mean access time vs. S M , as shownin figure 7. For this result, the threshold level is re-optimisedat each value of SNR.

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gain afforded by the matched filter. The system shownin figure 8 overcomes these restrictionsby takingadvantage of the periodic nature of PN codes toaccumulate the correlation values from the output of thematched filter which are each separated by one entiresequence period.

the register in order to implement a rake type detectorand this is particularly important in applications wherethe received signal undergoes multipath fading. In allcases,a maximum likelihood detector is then used on thecombined output to select the largest 'bin' after anappropriate number of accumulations.Simulation ResultsIn this case, maximal length sequences of length 127chips were used to illustrate the method. As for the caseabove, Gaussian noise was added to the incoming PNsequence. This was hard limited prior to the matchedfilter. Two sets of simulations were carried out:- a) nodata modulated onto the code, (b) random data modulatedonto the code. In the latter case, it was assumed that the

In this system, the output samples from the digitalmatched filter are accumulated within the register stack.For a digital matched filter of length L there are Laccumulator 'bins'. Each matched filter output sampleis fed to the appropriate address bin using modulo-Lcounting. Output samples separated by sampleperiods are thus accumulated. If the PN code is notmodulated with data, the samples in each bin are simplyadded, whereas if the PN code is ita modulated thesamples are added as absolute values For the case of adata modulated sequence, it is assumed that the data bitperiod is precisely one sequence period and is clocked byan appropriate code epoch, such as the all 1 s state.Nevertheless, since the precise startpoint of the sequenceis unknown, the matched filter output is degraded due toeven and odd uartial correlation uroblems. It is also

Figure 7 Effect of the input SNR on the optimum meanacquisition timefor a digital matche dfilter subject toAWGN and hard-limited input

DIGITAL MATCHED FILTER WITHMAXIMUM LIK ELIH O O D DETECTO RThe simple digital matched filter synchroniser fails atlow input S N R because of the very low Pd andcorrespondingly high nf due to the fixed processing

Figure 9shows a typical result for the case of a datamodulated code. In this figure, the probability that themaximum value detected is the correct value is plottedagainst the number of accumulations (i.e. sequenceperiods). Consequently by accumulating for asufficiently large number of times, the probability offalse alarm becomes negligible.Although not shown, substantially improved character-istics are obtained for the case where there is no datamodulated onto the code due to the fact that the digital

Figure 8 Schematic block diagram of a digital matchedfilter PN synchronisation systembased on = mum likelihood detection1149


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filter now provides a full correlation rather than a p,artialcorrelation, and this is reflected in a much improvedmean acquisition time characteristic,shown in figure 10.H oo

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0 50 100 150 200 250 300 350 400 450 500Number of AccumulationsFigure 9 Effert of he number of accumulations on theprobability that the maximum value in the accumulatoris the correc t in-lock condition, as a function of inputS N RIn this graph are plotted the mean acquisitioncharacteristics for 90 probability of acquiring lock forthe three cases considered. For the case of the digitalmatched filter with threshold detector, the results are re-simulated for a maximal code length of 127 chips and achip rate of 100kHz.

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digital matched filter threshold

Log-likelihoodsequent ialdetector w ith data)

m a tc hed fi lt er w i t h \maximum ikel ihood . \detec tor with data) \

matched filter with \m ax imum l i k e l i hod *4 \ \detector no data) e \

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Figure 10 Effect of SNR on the mean arquisition timej > r : a )digital nzatchedfilter and threshold detector, b )matched ilter with maximuni likelihood detector , c )matr hedfilte r w ith maximum likelihood detector anddata modulated p.n . code, d) sequential detector.

rate) [3,4]. This system also has data modulation and ahard-limi ed input.C O N C L U S I O N S

It is clear that the digital matched filter with thresholddetector provides excellent performance in relatively goodsignal to noise ratio conditions, but the performance issignificantly degnded as the S N R worsens. The increasedeffective dwell time afforded by the accumulator andmaximum likelihood detector provide significantperformance benefits, even when the code is datamodulated. However, the effect of data modulation oncode acquisition is severe and the performance of thematched filter with maximum likelihood detectionbecomes only marginally better than the sequentialdetector [3 41. However, at very poor SN Rs there is anindication that the performance of the sequential detectorand the digital matched filter with maximum likelihooddetector are very similar.

A C K N O W L E D G E M E N T SThe author wishes to thank Dr K Ravi for many fruitfuldiscussions and for his help with the simulation resultsof the sequential detector.

R E F E R E N C E Sl ] for example Holmes, J K and Woo, K T A noptimum P N code search technique fo r a given apriori s ignal location density, NTC 78 Conferencerecord, 1978, pp. 18.6.1 -18.6.6.

[2] Pandit, M: Mean acquisition time of active andpassive correlation systems f o r spread-spectrumcommunications, IEE Proceedings, Vol. 128PartF, NO. 4 pp 100-109 , 1981.[3] Ravi, K V and Ormondroyd, R F: Comparison ofthe acqu isition performance of biased-square-law andquantized log-likelihood sequential detectors or PNacquisition, Proceedings of the IEEE IntemationalSymposium on Spread-spectrum Techniques andApplications, London, U.K, 1990, pp 53-58 .[4] Ravi, K.V. and Ormondroyd, R F: Performanceofsequential detectors fo r the acquisition of datamodulated spread-spectrum pseudo noise signals,IEEE International Conference on Communications,ICC91, Conference Record, Vol. 2, 1991, pp19.7.1-19.7.5,

Also shown in this figure is the corresponding resultobtained for serial search using a log-likelihoodsequential detector under similar conditions to thematched filter (i.e. 127 chip PN code and lOOkHz chip

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Noise Performance of a Matched Filter PN Code Synchroniser u