School of Aerospace Science and Technology,Xidian University,Xi’an 710071,China

Code acquisition method based on wavelet transform filte ing

Chao Wu*,Luping Xu,Hua Zhang,and Wenbo Zhao

School of Aerospace Science and Technology,Xidian University,Xi’an 710071,China

In order to improve the acquisition probability of satellite navigation signals,this paper proposes a novel code acquisition method based on wavelet transform filte ing.Firstly,the signal vector based on the signal passing through a set of partial matched filter(PMFs)is built.Then,wavelet domain filte ing is performed on the signal vector value.Since the correlation signal is low in frequency and narrow in bandwidth,the noise out-of-band can be filtere out and the most of the useful signal energy is retained. Thus this process greatly improves the signal to noise ratio(SNR). Finally,the detection variable when the filtere signal goes through the combination process is constructed and the detection based on signal energy is made.Moreover,for the better retaining useful signal energy,the rule of selection of wavelet function has been made.Simulation results show the proposed method has a better detection performance than the normal code acquisition methods under the same false alarm probability.

code acquisition,wavelet transform,filte ing,detection probability.

1.Introduction

The globalnavigationsatellite system(GNSS)signal is the direct sequence spread spectrum modulation signal,and every satellite is transmitting a particular pseudorandom noise(PRN)code.The position parameters are acquired by correlating the received signal with local code signals and comparing the results with a threshold.In practice the local replica of the transmitted code signal differs from the received code signal by a code phase shift and a Doppler shift.Both have to be determined simultaneously in a twodimensional search[1].The search is usually called acquisition,a very important stage for subsequent processing [2–4].

To reduce the acquisition time,the full parallel correlators method based on matched filte(MF)was proposed in[5],which greatly reduces the acquisition time at the cost of hardware.To improve the acquisition probability of satellite navigation signals,accumulating the receiveddata[6]canincreasethe signalto noise ratio(SNR). For example,the non-coherent detection method[7]has been proposed,which can reduce the influenc of the bittransition on signal accumulation but make the noise amplitude squared at the time of detection.The coherent detection method was proposed in[2].The method improves the SNR but is sensitive to bit-transition.The differential combinationmethodwas proposedin[8,9],whichcansuppress the effect of the Doppler offset and bit-transition but the correlation peak is attenuated due to the multiplication of the noisy signal.The partial differential postcorrelation method for signal acquisition[6]has been proposed, which improves the differential combination method but the correlation peak is still attenuated due to the multiplication of the noisy signal.The methods which include the non-coherentmethod,the coherentmethod and the differential combination method[10,11]have been analyzed. The methods were combined in[12,13].In this way,the longerintegrationtime can be used for overcomingthe bittransition and improvingthe SNR.All these works just develop the long integration time,lacking the consideration of the received signal’s bandwidth and frequency which contain the signal inner information.The discrete cosine transform(DCT)domain filterin method[14]has been proposed(we call the method DF below),which performs DCT domain filterin on the received signal.However,in the DCT domain,we could not confir the frequency per point so that the noise out of useful band cannot be selectivelyfiltere out.Nevertheless,to filte outthenoise,three points of all points in the DCT domain[14]is chosen to reconstruct the signal.Although the large noise energy has been filtere out,this keeps the noise variance unchanged andmakes the useful signal energyloss increase,whichinfluence the further improvement of detection probability.

In this work,we firstl build a signal model when the signal goes through the partial matched filte(PMF)and analyze the bandwidth and frequency of the signal afterPMF,which provides the theory basis for wavelet transform filtering Then the signal passing through PMF is fil tered in the wavelet domain.The bandwidth of the filtere signal can be changed by adjusting the level of wavelet decomposition.Since different wavelet functions can be used to divide the frequency band,the rule of selecting the wavelet function based on the energy ratio has been proposed,and the wavelet function retaining more useful signal energy than any other wavelet functions has been selected.In addition,based on the analysis of the noise energyandvariancebeforeandafterthefiltering boththeenergy and the variance of the noise are reduced.Finally,the signal goes through the combination block which includes non-coherent integration and coherent integration,and the detection variable based on wavelet transform(WT)filteing is built.Since the method adopts the wavelet filterin process with the selected wavelet function and combination process,the detection probability has been greatly improved.In the paper,we call the method WT.The simulation results show this method has a better detection performance than the normal code acquisition method under the same false alarm probability.

This paper is organized as follows.Section 2 describes the PMF signal model.In Section 3,the method based on WT is proposed.In Section 4,the detection performance under different channels is analyzed.Section 5 proposes theruleofselectingthewaveletfunctionandcomparesWT with the normal methods.Finally,the conclusion is drawn in Section 6.

2.PMF signal model

The received signal can be written as

where n represents the nth sampling point of the signal,L is the number of multipath,and h(l)is the plural fading factor of the lth path.τlis the initial code phase of the lth path.Suppose?0≤l1,l2≤ L,|τl1?τl2|≥nc,and ncis sampling points per chip.Without loss of generality,we only consider the largest path which is supposed to be the lth path for acquisition below.P(n)is the additive complex zero-mean Gaussian noise with variance σ2.c(n) is the spread spectrum code of the global position system (GPS).f1is the frequency of the received signal.

Then,let r(n)pass Q matched filters

where sr(q)represents the results of the qth PMF output. Co is the number of correlation points.q=1,...,Q.τ is the initial phase of c(n),Δf is the search step of the frequency,and i is the search number of times.

Hypothesis H1:when τ=τl,the local code is synchronized with the received signal and sr(q)can be simplifie as

Hypothesis H0:when τ/=τl,the local code is unsynchronized with the received signal and sr(q)can be simplifie as

where W(q)is the additive complex white Gaussian noise. Both real part and imaginary part of W(q)are additive complex white Gaussian noise with zero mean and variance Co×σ2,f0=f1?i×Δf,when πf0is small, sin(πf0)=πf0.

3.Code acquisition method based on WT

3.1 Process of WT

As can be seen from(3),the center frequency and bandwidth of the useful signal in signal sr(q)is mainly determinedbyf0andh(l).Due to the fact that channelcoherent time being dozens to hundredsof milliseconds,h(l)can be regardedas slowly changing signals.When h(l)is omitted and the received signal is synchronized,f0can be made fall in a small range by adjusting variables i and Δf.

Fig.1 shows the signal spectrum when the received signalgoesthrougha set ofmatchedfilters InFig.1,different curves represent the signal spectrum in different correlation time,and the peaks of different curves represent the most useful signal energy.Under the low frequency offset (i.e.f0is small),the most of the useful signal energy is concentrated in the low frequency part.The larger the correlation time is,and the better the energy would concentrate.Therefore,the received signal can be decomposed in the wavelet domain,and then low frequency band with the informationof useful signals can be extractedto reducethe noise.However,in the DF method,the reduction of signal points after DCT filterin does not have influenc on the noise variance[14].This is not good for further improvement on the detection probability.This will be proved in the followingsimulation.Finally andthe most importantly,WT proposedin this paperreduces bothnoise’s energyand variance,and differentwavelet functionscan be selected to express the useful signal.The block diagram of WT is presented in Fig.2.

Fig.1 The signal spectrum

Fig.2 Block diagram of WT

The specifi steps of WT can be demonstrated as follows:

(i)Let the signals(n)go throughQPMFs,and we can obtain signal vectorsr=[sr(1),...,sr(q),...,sr(Q)].

(ii)The wavelet transform filtering

Performing wavelet transform on signal vectorsr:

whereq=0,1,...,Q?1,Jis thelevelof waveletdecomposition,AJ,i(q)andDk,i(q)are the wavelet basis functions,iis the shift factor,andsc()represents wavelet coefficient

According to the analysis above,sc(1,i)is the low frequency coefficien and contains the most useful signal energy.Thus,sc(1,i)is chosen to reconstruct the signal:

wheren=0,1,...,Q?1.Since the noise obeys the uniformdistribution in the whole frequencydomain,the noise energy after filterin is reduced by a factor of 2J.Thus we have:

whereW′(q)isthenoisesignalamplitudeafterthewavelet domain filtering andW′is the mean ofW′(q);W(q)is the noise signal amplitude before the wavelet domain fil tering,andWis the mean ofW(q).

The noise variance after filterin can be written as

The noise variance before filterin can be written as

Since the average is unchanged before and after filteing,W=W′=0.Based on the equations from(7)to(9), we obtain:

From the analysis above,not only the noise energy after filterin isof that before filtering but also the noise variance after filterin isof the variance before filteing.

(iii)Putsf(n)into the energy detector for searching the largest energy,the detection variable can be written as

whereQ1is coherent time,Q2is non-coherent time. SupposeQ=Q1Q2,we search for the biggestZasZmaxin every code phase.Successful acquisition is made whenZmaxis bigger than the thresholdγ.In(13),represents the useful signal energy which is supposed to be constant or is slightly reduced compared to the noise,butstd1is reduced compared withstd2.Thus,the detection probability will be improved under the same threshold.

3.2Threshold of WT

When the local code is unsynchronized with the received signal,thethresholdγcanbeset bythedemandofthefalse alarm probability.Different code phases have differentZvalues,which are independent.Zobeys the chi-squared distribution:

whereΓ()is theGammafunction,andkis thedegree.Due to(13),the degreeis 2Q2.The total false alarm probability can be written as

wherePis the number of possible code phases.Sincecan be simplifie as

Thus,γcan be calculated whenPfaandQ2are fi ed.

4.Performance analysis

4.1Detection probability

When the local code is synchronizedwith the receivedsignal,the detection probabilityPDcan be written as

4.2Signal detection performance under additive white Gaussian noise(AWGN)channel

Under H1,the detection variableZobeys the non chi-squared distribution of degree 2Q2[16],the probability density function can be written as

whereλ=2E/Q1×std1.Thefina acquisitionprobability can be obtained based on(17)and(18).

Based on the analysis above,the useful signal energy before filterin is almost the same as the amount of energy after filtering and the useful signal energy can be written as

4.3Signal detection performance under fading channel

Whenthesignalchannelis thefadingchannel[15],thedistribution of the detection variableZobeys conditional non chi-squareddistribution of degree2Q2,and the probability density function can be written as

whereβis the non-central parameter which obeys the exponential distribution,Pβ(β)=e?β.The statistical average of(20)based onβcan be written as

The average detection probability can be obtained by utilizing(17),(20),and(21).The detection probabilitycan be solved by the numerical integration.

5.Simulation analysis

5.1Simulation condition

In this section,computersimulation is carriedout based on the GPS signal.We choose Matlab as the platform for our simulation.

To select the wavelet function,the energy ratio(ER) based on(5)and(6)can be define aswheresr(q)is the signal before wavelet filteringsf(q)is the signal after wavelet filteringEris calculated underthe condition that the noise variance is 0.Fig.3 shows the differentErvalues when the received signal is processed by the proposedmethodWT using differentwavelet functions (J=1)and the method based on DCT filterin(DF)[14].

Fig.3 Energy ratio comparison

As Fig.3 shows,different wavelet functions harr,db4, coif3 and sym2 lead to different ERs,and ER of the wavelet function increases with the digital frequency offset(DFO).Similar to the standard of selecting the wavelet function[17],ER is chosen as the standard of selecting the wavelet function.Moreover,the ER of db4 and coif3 is lower than that of DF.Since a large ER value leads to a large useful energyloss,DF leads to more energyloss than WT of db4 and coif3,which has a negative influenc on the detection probability.Thus the wavelet function db4 is selected.We use Monte Carlo simulation to calculate the actual detection probability.In the following simulation,J=1 andJ=2 represent the different levels of wavelet decomposition of the WT method.T1represents the theoretical detection probability of the WT method whenJ=1,andT2represents the theoretical detection probability of the WT method whenJ=2.WT1n is the method accumulating the received signal directly without filterinwhenQ1=1,Q2=Q.WT2n is the method accumulating the received signal directly without filterin whenQ1=Q,Q2=1.The simulation parameters are as follows:

Table 1 Simulation parameters

5.2Simulation results and analysis

5.2.1 AWGN channel

(i)The comparison of detection probability curves

As can be seen from Fig.4(a)and Fig.4(b),the actual detection probabilities ofJ=1 andJ=2 are higher than other detection probabilities.This is due to the fact that the filterin in the wavelet domain can be sectional for retaining the most of useful energy and filterin out a large amount of noise in the wavelet domain.In other word,WT greatly improves the SNR.In Fig.4(a),we can observe that the detection probability forJ=2 is higher than that forJ=1.However,in Fig.4(b),we have the opposite observation.This can be explained by the bandwidth and frequency offset of the useful signal.

Fig.4 Detection probabilities under AWGN channel

The detection is performed on signalsr(q)of(3),in which the signal digital center frequency isCo×f0.By observing Fig.1 and(3),we fin that the useful signalbandwidthisbroadenedduetosignaltruncationandslowly changing signalh(l).WT is used to divide the signal frequency band,and to reconstruct the signal from the low frequency part.The parameterJis the number of frequency bands.The value ofJshows that the more the number of frequency bands is,the narrower the low part frequency and the larger the reduced noise are.However, the frequency band that is reconstructed is decreased.This in turn leads to reduction of useful energy and the reduction of detection probability.In the following simulation, we give the detailed explanation in the following experiment.

(ii)The influenc ofJon the detection probability

It can be observed in Fig.5(a)and Fig.5(b)that the actual curve of the detection probability does not overlap with the curve of the theoretical detection probability.Besides the influenc of the simulationerror,this is caused by thefrequencyoffsetandthe bandwidthofthe usefulsignal. This leads to the reduction of useful signal energy after fil tering.The parameterEof the actual detection probability reflect all of useful signal energy:

However,the detection variableZcan be represented as

Fig.5 Detection probability of WT under AWGN channel

Because the frequency band of the useful signal is broadened,notallusefulenergyis containedinfiltere signalsf(n).

The fittin degree of the curve of the actual detection probability in comparison with its theoretical detection probability reflect the reduction of the useful energy reconstructed.Even so,since the reconstructed signal contains most useful energy and the most energy of the noise is filtere out,the proposed method still realizes high detection probability compared with other methods.Hence, we just need to adjust the range off0to make most useful signal pass the filtering

5.2.2 Fading channel

(i)The comparison of detection probability curves

From Fig.6,we can observe that the proposed method consistently produces a higher detection probability than the probabilities of other methods under the fading channel.Although how to choose the parameterJis influence byf0,the proposed method can successfully filte out the amount of noise,and keep the high detection probability.

(ii)The influenc ofJon the detection probability

In Fig.7(a),the curve of actual detection probability does not overlap with the curve of theoretical detection probability under the fading channel.This is because the actual detection probability is still affected by the bandwidth and frequency offset of the useful signal.Although the curves of actual detection probabilityand of theoretical detection probability do not coincide completely,the proposed method still generates the highest detection probability due to the reduction of noise.Similar observations can be made for Fig.7(b).

Fig.6 Detection probabilities under fading channel

Fig.7 Detection probability of WT under fading channel

5.2.3 Acquisition sensitivity analysis

To test acquisition sensitivity of the WT method and normal methods(WT1n,WT2n and DF),Table 2 and Table 3 show the average of the SNR values when the detection probability is 0.9 and the frequency offset is given the uniform distribution over the range of–100 Hz to 100 Hz.

Similar to the definitio of sensitivity[14],the method has a high acquisition sensitivity when the average of the SNR values shown in the tables is low.It can be observed from Table 2 that the acquisition sensitivity of WT is at least 2.5 dB higher than the acquisition sensitivity of the normal methods under the fading channel.Table 3 shows that,the acquisition sensitivity of WT is at least 1.2 dB higher than the acquisition sensitivity of the normal methods under the AWGN channel.Above all,WT has the higher acquisition sensitivity than the normal methods.

Table 2 Acquisition sensitivity under fading channel

Table 3 Acquisition sensitivity under AWGN channel

6.Conclusions

In this paper,a novel method for improvement of GNSS signal acquisition probability using wavelet domain filteing is proposed.Based on the PMF signal model,the bandwidth and frequency of useful signals are analyzed,which provides the theoretical basis of the proposed method. Then,the wavelet transformis usedto dividethe frequency band of the received signal,and low frequency band is chosen to reconstruct the signal.This process improvesthe SNR.Furthermore,the combination process can also improve the detection probability.Since different wavelet functionscanbeusedto dividethefrequencyband,therule of selecting the wavelet function based on the energy ratio has been proposed.In simulation,the detection probability and the acquisition sensitivity are analyzed compared with the normal methods.The analysis proves that the WT methodhas a better detection performancethan the normal methods.Since the usefulsignal bandis broadened,the parametersQ1,Q2andJneed to be chosen properly based on the range of the frequencyoffset.In this way,the higher detection probability can be obtained.

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Biographies

Chao Wuwas born in 1988.He received his M.S. degree in School of Electronic Engineering from Xidian University.He is currently pursuing his Ph.D.degree at Xidian University.His research interest lies in GNSS weak signal detection.

E-mail:wuchaoid@126.com

Luping Xuwas born in 1961.He received his Ph.D.degree in signal and information processing from Xidian University,in 1996.Since 2000, he has been a professor at Xidian University.His main research interests are target detection,spread spectrum,satellite and mobile communication,and SAR image processing.

E-mail:lpxu@mail.xidian.edu.cn

Hua Zhangwas born in 1982.He received his B.A. and M.S.degrees in telecommunication engineering from Xi’an University of Technology,in 2005 and 2008,respectively.From 2008 to 2011,he was studying in Xidian University for his Ph.D.degree. Now he is a lecturer in Xidian University.His main research interests include X-ray pulsar weak signals processing and its navigation mechanism.

E-mail:zhanghua@mail.xidian.edu.cn

Wenbo Zhaoreceived her B.S.degree from the Department of Computer Science,Nanjing University of Astronautics and Aeronautics,China in 2006.She is currently working toward her Ph.D. degree in the Department of Computer Science at the Nanyang Technological University.Her research interests include the design of energy effi cient MAC and routing protocols for wireless,ad hoc and sensor networks.

E-mail:wenbozhao2007@gmail.com

10.1109/JSEE.2015.00127

Manuscript received August 27,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61172138;61401340),and the Fundamental Research Funds for the Central Universities(K5051302015).