GPS Signal Joint Acquisition Method of Mean Function and Autocorrelation Function under Multiplicative and Additive Noise

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1 GPS Sgnal Jont Acquton Metho of Mean Functon an Autocorrelaton Functon uner Multplcatve an Atve Noe CHAO WU, LUPING XU, HUA ZHANG School of Aeropace Scence an Technology X an Unverty X an 76 Chna wuchao@6.com Abtract: GPS gnal etecton metho bae on hgh orer cyclotatonarty (DMHOC the tate-of-the-art metho for acqurng the GPS gnal uner multplcatve an atve noe. However, nce the metho only ue autocorrelaton functon to obtan the pea an etmate the receve gnal frequency, the pea may be bure n the noe when power pectrum of the multplcatve noe (PSMN low. To olve the problem, the GPS gnal jont acquton metho of mean functon an autocorrelaton functon uner multplcatve an atve noe ha been propoe. The propoe metho ue the energy rato threhol to etermne whch functon (mean functon or autocorrelaton functon to be ue for etmaton of the receve gnal frequency. The mulaton reult how that the propoe metho more robut to the change of multplcatve noe than DMHOC. Key-Wor: GPS acquton, Cyclotatonarty, Multplcatve an atve noe, Mean functon, Autocorrelaton functon, Energy rato Introucton Global Potonng Sytem (GPS gnal [] the rect equence prea pectrum moulaton gnal, an every atellte tranmt a partcular peuoranom noe (PRN coe. Satellte are acqure by correlatng the receve gnal wth local coe gnal an comparng the reult agant a threhol. In practce the local replca of the tranmtte coe gnal ffer from the receve coe gnal by a coe phae hft an a Doppler hft. Both have to be etermne multaneouly n a two-menonal earch []. The reult of th earch, whch uually calle acquton, are requre for preettng ubequent tage of the GPS recever [3]. To mprove the acquton probablty of atellte navgaton gnal, accumulatng the receve ata[4] coul be ue to ncreae the gnal to noe rato (SNR. For example, the non-coherent etecton metho [5] ha been propoe, whch can reuce the nfluence of the bt-tranton on gnal accumulaton but mae the noe ampltue quare at the tme of etecton. The coherent etecton metho [6] ha been propoe, whch mprove the SNR but entve to bt-tranton. The fferental combnaton metho [7,8] have been propoe, whch can uppre the effect of the Doppler offet an bt-tranton but the correlaton pea wa attenuate ue to the multplcaton of the noy gnal. The moel of the metho above n receve GPS gnal are almot aume to be atve. However, when GPS gnal pae through the onophere [9,], each electron wll caue a phae change n the carrer. The carrer phae fluctuaton caue by the abunant electron can be moele a a multplcatve noe proce. Moreover, the comprehenve carrer phae changng effect n a huge number of path [,] can alo be moele a the multplcatve noe. The problem of etmatng the frequence of harmonc uner multplcatve an atve noe ha been cue n [3,4]. Due to the feature of cyclotatonarty of harmonc, the mean functon calle cyclc mean, an autocorrelaton functon calle cyclc autocorrelaton. Many metho[5-7] only ue the mean functon to etmate the gnal frequency n low power pectrum of the multplcatve noe (PSMN. In other wor, the gnal wth PRN coe trppe off procee by frequency oman metho, uch a Fat Fourer Tranformaton (FFT, whch not proper for hgh PSMN [8,9,]. Snce there lttle cuon on how to acqure GPS gnal egrae by both multplcatve an atve noe, the gnal moel ha been et up n E-ISSN: Volume, 5

2 Ref []. However, the part of term n gnal moel ha been omtte, an the gnal moel wrtten a: K AC ( τ xn + w+ ( co( π f τ w where A the ampltue of -th GPS gnal. ( the bt gn. τ nt τ. τ the coe phae. T amplng nterval. w ( n enote multplcatve noe proce. w+ enote atve Gauan noe proce. The correlaton functon can be wrtten a: + AC( τ C( τ K + ( n ( n rnm (, + σδ + ( m ( + co( π f τ co( πf τ σδ( m + + where τ nt τ. τ nt τ. m m n + n+. n n. If the multplcatve noe large enough, the correlaton reult relate to tme nt. Therefore, uner the conton of large multplcatve noe, the noe procee of the receve GPS gnal cannot be moelle a we ene tatonary an the conventonal GPS gnal acquton metho bae on correlaton may not alway wor. Bae on the gnal moel, GPS gnal etecton metho bae on hgh orer cyclotatonarty (DMHOC ha been propoe. However, the metho only ue the autocorrelaton functon to acqure the GPS gnal. Th may not acqure the gnal n low PSMN, whch analyze n the followng cuon. To acqure the GPS gnal uner the multplcatve an atve noe, th paper ha propoe the GPS gnal jont acquton metho of mean functon an autocorrelaton functon. Frtly, the receve GPS prove to be cyclotatonary. Due to the property, the Fourer ere coeffcent of mean functon an autocorrelaton functon can be ue to etmate the frequency an coe phae by Fourer tranformaton (FT proce. Snce the propoe metho can chooe the etecton cheme bae on mean functon or autocorrelaton functon by the energy rato threhol, the propoe metho more robut to the change of multplcatve noe than DMHOC. The mulaton reult how that the propoe metho can have better etecton performance than DMHOC uner low PSMN crcumtance. The ret of th paper organze a follow: In Secton the receve gnal moel gven an prove to be cyclotatonary. In Secton 3, the conventonal metho (GPS gnal etecton metho bae on hgh orer cyclotatonarty ha been analyze. Bae on the analy, the paper propoe the GPS gnal jont acquton metho of mean functon an autocorrelaton functon n Secton 4. The mulaton reult how the effectvene of the propoe metho compare wth the DMHOC n Secton 5. Fnally, the concluon are rawn n Secton 6. Sgnal Moel The receve gnal from a pecfc atellte vehcle (SV can be wrtten a Ct ( τ t (( A+ w ( t S( t (3 co( ( f + f t + w ( t π IF + where τ the coe phae elay. f Doppler frequency. A the gnal ampltue. c ( the peuo ranom noe (PRN coe. w ( the multplcatve noe. w + ( the atve noe. ( the bt gn. It aume that T the pero of receve gnal S( t. E S( t + T C( t τ ( t Aco( π ft S ( ( E St ( +, τ ( ( τ ( A + σδ( t Γ R t T t E St+ T+ t St+ T ( τ co( π co( π + σδ ( ( (, ft ft + tτ (4 (5 E St+ tτ St RS ttτ where Γ Ct ( τ Ct ( τ t ( t (. f fif + f. t t+ T. t t+ T + t τ. E St the mean functon of S( t. RS ( tt, τ ( ( the autocorrelaton functon of S( t. σ an σ + are multplcatve noe varance an atve noe, repectvely. A the equaton are hown, both mean functon an autocorrelaton functon are peroc. So t prove that receve gnal cyclotatonary [8]. E-ISSN: Volume, 5

3 3 GPS gnal etecton metho bae on hgh orer cyclotatonarty (DMHOC GPS gnal etecton metho bae on hgh orer cyclotatonarty (DMHOC can be rawn a follow []: Sn co( π f nt IF t branch Square proce. Fat Fourer Tranform( FFT Abolute Value proce n( π f nt IF j Local coe L t branch Square proce Fat Fourer Tranform( FFT Abolute Value proce Larger The etmate frequency Smaller Compare wth the et varance Choong the poble frequency, tang the average a etmate frequency, an calculatng the varance of poble frequency Fg The bloc agram of DMHOC Bae on the bloc agram of DMHOC, the gnal through the quare proce can be wrtten a: ' Cn ( τ Cn ( τ n ( A+ ω S exp( j π fnt + W+ (6 ( A + ω exp( j4 π fnt + ( W+ + ' W+ ( ncn ( τ Cn ( τ n ( A+ ω exp( j π fnt where Sn n Fg repreent the gtal gnal of St ( T at amplng nterval.,..., L '. τ the local coe phae. τ the receve coe phae. n the ata bt of receve gnal. Both real part an magnary part of W + are atve σ Gauan noe wth zero mean an varance +, ω the multplcatve noe wth mean an σ varance. Snce the mean functon of S ( n can be wrtten a: ( + σ E S A exp( j4 π f nt + σ + (7 Due to the fferent frequence, we can tnguh the gnal from atve noe an etmate the gnal frequency by FFT proce. Th eal tuaton. A Fg how, n practce S ( n ntea of ( E S n performe on FFT proce. Snce the quare proce performe on the receve gnal, the etmate frequency correponng to the pea tme than the f σ + A frequency. When an mall (.e PSMN low [], the pea may be bure n the atve noe. The Fg below how the ampltue uner fferent PSMN when gnal power -6BW an the power pectrum of the atve noe (PSAN -8 Bm/Hz: E-ISSN: Volume, 5

4 4.5 x Ampltue Ampltue (a PSMN-7 Bm/Hz (b PSMN- Bm/Hz Fg The Ampltue of FFT of S ( n when f 4Hz Comparng Fg (a wth Fg (b, the obvou autocorrelaton functon RS ( t, can be wrtten a, pea appear n Fg (b. Th ue to the fact that PSMN ecreae, the pea bure n the repectvely: atve noe, an the correct pea har to etect. ( ( Th wll lea to the ncreae of varance, an the lm N E StCt τ FE α N N t exp( jαt performance of etmate frequency wll be (8 egrae. Th wll be prove n Secton 5. D ( δα ( ω + δα ( + ω 4 The propoe metho Snce DMHOC only ue autocorrelaton functon to etmate the frequency an har to etect the gnal uner low PSMN crcumtance, th paper propoe the jont acquton metho of mean functon an autocorrelaton functon, an ue the energy rato threhol to etermne whch functon (mean functon or autocorrelaton functon to be ue for etmaton of the receve gnal frequency. To get r of the turbance caue by two pero (the pero of PRN coe an the pero of the frequency, the pero of PRN coe houl be remove. The generalze Fourer ere coeffcent of the mean functon E( StCt ( ( τ an lm N FR α RS ( t, exp( jαt N N t A + σ δα ( ω + A + σ + + σ+ δα 4 δα ( + ω (9 where t aume that D contant regarle of the nfluence of ata pero on the coeffcent. ω π ft F α. T the amplng nterval. E an F ( α can be calculate by Fourer tranform R (FT, an can be ue to etmate the coe phae an frequency of the receve gnal. The pecfc tep of the propoe metho can be ecrbe a follow: E-ISSN: Volume, 5

5 Ye No All of coe phae are earche Sn co( π f nt IF n( π f nt IF j S Local coe Square proce FFT FFT Calculatng energy rato Calculatng energy rato γ Comparng Larger Comparng Smaller Smaller The coe phae an frequency are etmate Acquton fale γ The coe phae an frequency are etmate Bae on the etmate frequency, local coe correlate wth the receve gnal wth the remove frequency, an coe phae etmate Larger Frtly, the S( n : Fg 3 The bloc agram of the propoe metho S t converte to the gtal gnal where n the nex correponng to the pea of C( nt τ ( nt A + w S ( co( π( fif + f ( nt τ + w+ Then, multple by two orthogonal reference S n can be converte to the gnal nuo, S ( n : S C( nt τ ( nt A + w exp( jf nt + W π + S n : There are two tep to proce the Step: S ( n correlate wth the local coe. Snce the PRN coe ha been remove, t aume that the amplng nterval T ( T > T. Then the gnal goe through the fat Fourer tranform (FFT: S n S ncn τ ( FFT ( F c where τ c the local coe phae. Calculatng the energy rato E r: N SF( n SF N n n n E r (3 N SF N n n n F S n. Comparng the E r wth the et threhol γ. If E r larger than the γ, the gnal acqure. If not, the coe phae τ c nee to be upate. Step: f all of coe phae are earche, the energy rato of tep tll maller than the S n goe through the quare proce threhol. an then performe on FFT: ( SF n FFT S n (4 Calculatng the energy rato: N SF( n SF( n N n n n E r N SF N n n n (5 where n the nex correponng to the pea of F S n. Comparng the E r wth the et threhol γ. If E r larger than the γ, the gnal acqure. The etmate frequency can be calculate by n. Then, local coe correlate wth the receve gnal wth the remove frequency: E-ISSN: Volume, 5

6 S C where n IFFT FFT ( S exp( π jfn nt ( FFT ( C (6 S n. f the etmate frequency of repreent the conjugaton operaton. IFFT nvere fat Fourer tranform (IFFT. The coe phae can be etmate by co mparng the energy rato of the pea of S ( n wth the et threhol β. If the energy rato larger than β, the coe phae uccefully etmate. Bae on C the analy above, the frequency an coe phae can be etmate. To evaluate the performance of the propoe metho, the mean quare error (MSE of the frequency an coe phae calculate: N N MSET T T (7 where T f or C. T f repreent the MSE of frequency. T C repreent the MSE of coe phae. T repreent the -th etmaton of T. MSE of frequency (Hz γ γ5 γ5 γ75 γ MSE of frequency (Hz γ γ5 γ5 γ75 γ PSAN(Bm/Hz (a PSMN-Bm/Hz, t aume that the local coe ynchronze wth receve gnal coe PSAN(Bm/Hz (b PSMN-77Bm/Hz, t aume that the local coe ynchronze wth receve gnal coe MSE of phae β β6 β β8 β4 β3 MSE of phae β β6 β β8 β4 β PSAN(Bm/Hz PSAN(Bm/Hz E-ISSN: Volume, 5

7 (c PSMN-Bm/Hz, t aume that the receve gnal frequency nown ( PSMN-8Bm/Hz, t aume that the receve gnal frequency nown Fg 4 MSE when the gnal length m an N To avo the nteracton effect between β an γ, we aume that the receve gnal frequency nown when MSE of phae calculate, an aume that the local coe ynchronze wth receve gnal coe when MSE of frequency calculate. The Fg 4 how the MSE of frequency an coe phae wth fferent γ value an β value, The gnal length The gnal power The threhol γ The threhol β m -6Bw 5 6 repectvely. The parameter correponng to mnmum of MSE houl be electe. So γ 5, an β 6. Samplng frequency / T Hz The PRN rate.3mc/ 5 Smulaton Reult To prove that the propoe metho more robut to the change of multplcatve noe than DMHOC, the mulaton reult have been mae. The parameter are et a follow: Table Smulaton parameter. The varance of DMHOC.3 The number of branche of DMHOC L The number of Monte Carlo mulaton Frequency f Hz 5 x Ampltue 3 Ampltue (admhoc when PSMN-8Bm/Hz, (bthe propoe metho when PSMN- 8Bm/Hz E-ISSN: Volume, 5

8 Ampltue 5 5 Ampltue (cdmhoc when PSMN-Bm/Hz, (The propoe metho when PSMN- Bm/Hz, Fg 4 The ampltue for frequency etmaton obtane by two metho when PSAN-8Bm/Hz The etecton probablty The propoe metho DMHOC The etecton probablty The propoe metho DMHOC PSAN(Bm/Hz (a PSMN-8Bm/Hz PSAN(Bm/Hz (b PSMN-9Bm/Hz E-ISSN: Volume, 5

9 The etecton probablty The propoe metho DMHOC Fg 4(a an Fg 4(c how the one branch gnal ampltue through FFT proce of DMHOC. Fg 4(b an Fg 4( how the gnal ampltue through FFT proce of the propoe metho when the energy rato larger than or equal to the et threhol. Comparng Fg 4(a an Fg 4(b, the obvou pea can be een from both fgure uner the ame noe conton (PSMN-8Bm/Hz an PSAN-8Bm/Hz. However, the pea bure n the noe n Fg 4(c when PSMN-Bm/Hz an PSAN-8Bm/Hz, an clear pea tll can be een n Fg 4(. Th becaue DMHOC only ue autocorrelaton functon to obtan the pea an correct pea may be bure n low PSMN. So the varance calculate by pea etecton ncreae n low PSMN, an the etecton probablty of frequency of DMHOC wll ecreae. To further llutrate the fact, the etecton probablty comparon mae n Fg 5 by Monte Carlo mulaton. The etecton probablty the probablty that the metho etmate the correct frequency unt. A the Fg 5 (a how, the etecton curve of propoe metho almot overlap wth the etecton curve of DMHOC. However, when PSMN ecreae, the etecton performance of DMHOC egrae baly compare wth the etecton performance of the propoe metho n Fg 5(b or Fg 5(c. So t prove that the propoe metho more robut to the change of multplcatve noe than DMHOC. 6 Concluon For GPS gnal acquton uner the multplcatve an atve noe, th paper ha propoe the jont PSAN(Bm/Hz (c PSMN-Bm/Hz Fg 5 The etecton probablty comparon acquton metho bae on mean functon an autocorrelaton functon. The gnal moel et up. The mean functon an autocorrelaton functon utlze to prove that the receve gnal cyclotatonary. Bae on cyclotatonary feature, the energy rato calculate by the Fourer ere coeffcent of the mean functon or autocorrelaton functon, an the propoe metho ue the energy rato threhol to etermne whch functon (mean functon or autocorrelaton functon to be ue for etmaton of the acquton parameter (nclung coe phae an frequency. The mulaton reult how that the propoe metho ha better etecton performance than DMHOC uner low PSMN crcumtance. In other wor, the propoe metho more robut to the change of multplcatve noe than DMHOC. We emontrate that the propoe metho ha better frequency etmaton performance than DMHOC. The propoe metho apple n hgh ynamc envronment[5]. Th a topc of our further reearch. Acnowlegment Th wor wa upporte by the Natonal Natural Scence Founaton of Chna (Grant NO. 6738, NO. 6434, the Natural Scence Bac Reearch Plan n Shaanx Provnce of Chna (Grant No. 3JQ84, the Reearch Fun for the Doctoral Program of Hgher Eucaton of Chna (Grant NO. 334, the Open Reearch Fun of The Acaemy of Satellte Applcaton (Grant NO. 4_CXJJ-DH_, the X'an Scence an technology plan (Grant NO. CXY35(4, the E-ISSN: Volume, 5

10 Funamental Reearch Fun for the Central Unverte (Grant NO. 43B, NO. 4B, JB433, an the Open Fun of Key Laboratory of Precon Navgaton an Tmng Technology, Natonal Tme Servce Center, CAS (Grant NO. 4PNTT, NO. 4PNTT7, NO. 4PNTT8. Reference: [] V. Barrle, G. M. Meurl, G. Blotta, An open GIS for the gnfcance analy of placement arng from GPS networ repeate over tme: an applcaton n the area of Catrovllar, WSEAS Tran on Sgnal Proceng, Vol, 4, pp [] A. Polyoro, C. L. Weber, A unfe approach to eral earch prea-pectrum coe acquton--part I: general theory, IEEE Tran on Communcaton, Vol.3, No.5, 984, pp [3] B. C. Geger, C. Vogel, Influence of Doppler bn wth on GPS acquton probablte, IEEE Tran. on Aeropace an Electronc Sytem, Vol.49, No.4, 3, pp [4] T. H. Ta, S. U. Qaar, A. G. Dempter, Partal fferental potcorrelaton proceng for GPS LC gnal acquton, IEEE Tran on Aeropace an Electronc Sytem, Vol.48, No.,, pp [5] B. Slar, Dgtal communcaton. NJ: Prentce Hall,. [6] A. Polycloro, C.L. Weber, A unfe approach to eral earch prea-pectrum coe acquton part II:A Matche-Flter Recever, IEEE Tran on Communcaton, Vol.3, No.5, 984, pp [7] A. S. Ayaz, Analy of fferental acquton metho by ung Monte-Carlo mulaton, In Proceeng of the 8th nternatonal techncal meetng of the atellte von of The Inttute of Navgaton (ION GNSS 5, Long Beach, CA. 5. [8] W. Yu, B. Zheng, R. Waton, Dfferental combnng for acqurng wea GPS gnal, Sgnal Proceng, Vol.87, No.5, 7, pp [9] K. V. Rey, P. H. Deepa, L. V. Rao, Near pont montorng of the onophere ung ual frequency GPS ata over the hyearaba regon: a alman flter approach, Internatonal Conference on Informaton Communcaton an Embee Sytem (ICICES, 4, pp. -5. [] I. Progr, Inoor Geolocaton Sytem: Theory an Applcaton. Sprnger Scence an Bune Mea, 3. [] A. Gupta, S. Joh, Etmaton of multpath fang channel ung fractal bae VSLMS algorthm, WSEAS Tran on Sgnal Proceng, Vol, 4, pp.3-4 [] K. Hanea, A. Khatun, M. Daht, Meaurement-bae analy of patal egree of freeom n multpath propagaton channel, IEEE Tran on Antenna an Propagaton, Vol 6, No, 3, pp [3] M. Ghogho, A. Swam, B. Garel, Performance analy of cyclc tattc for the etmaton of harmonc n multplcatve an atve noe, IEEE Tran on Sgnal Proceng, Vol.47, No., 999, pp [4] G. Zhou, G. B. Ganna, Harmonc n multplcatve an atve noe: performance analy of cyclc etmator, IEEE Tran on Sgnal Proceng, Vol.43, No.6, 995, pp [5] C. Wu, L. P. Xu, H. Zhang, W. B. Zhao, A bloc zero-pang metho bae on DCFT for L parameter etmaton n wea gnal an hgh ynamc envronment, Fronter of Informaton Technology & Electronc Engneerng, 5. Do:.63/FITEE.558 (n pre [6] S.H. Kong, A etermntc compree GNSS acquton technque, IEEE Tran. Veh. Technol., Vol.6, No., 3, pp. 5 5 [7] B. Km, S.H. Kong, Degn of FFT-bae TDCC for GNSS acquton, IEEE Tran. on Wrele Communcaton, Vol.3, No.5, 4, pp [8] W. A. Garner, A. Napoltano, L. Paura, Cyclotatonarty: Half a century of reearch, Sgnal Proceng, Vol.86, No.4, 6, pp [9] F. Shen, M. Ga, Z. L. Wang, Coe acquton ung the locally optmum tet tattc n both multplcatve an atve noe, In Proceeng of the IEEE Internatonal Conferenee on Mechatronc an Automaton.Pcataway,, pp [] P. Huang, Y. P, I. Progr, GPS Sgnal Detecton uner Multplcatve an Atve Noe, Journal of Navgaton, Vol.66, No.4, 3, pp E-ISSN: Volume, 5