GPS Interference Mitigation Using Derivative-free Kalman Filter-based RNN

Transcription

1 518 WEI-LUNG MAO, GPS INERFERENCE MIIGAION USING DERIVAIVE-FREE KALMAN FILER-BASED RNN GPS Interference Mtgaton Usng Dervatve-free Kalman Flter-based RNN We-Lung MAO Graduate School of Engneerng Scence and echnology and Dept. of Electrcal Engneerng, Natonal Yunln Unversty of Scence and echnology, awan Manuscrpt receved September 12, 215 Abstract. he global postonng system (GPS) wth accurate postonng and tmng propertes has become ntegral part of all applcatons around the world. Rado frequency nterference can sgnfcantly decrease the performance of GPS recevers or even completely prohbt the acquston or tracng of satelltes. he approaches of system performances that can be further enhanced by preprocessng to reject the jammng sgnal wll be nvestgated. A recurrent neural networ (RNN) predctor for the GPS ant-jammng applcatons wll be proposed. he adaptve RNN predctor s utlzed to accurately predct the narrowband waveform based on an unscented Kalman flter (UKF)-based algorthm. he UKF algorthm as a dervatve-free alternatve to the extended Kalman flter (EKF) n the framewor of state-estmaton s adopted to acheve better performance n terms of convergence rate and qualty of soluton. he adaptve UKF-RNN flter can be successfully appled for the suppresson of nterference wth a number of dfferent narrowband formats,.e. contnuous wave nterference (CWI), mult-tone CWI, swept CWI and pulsed CWI, to emulate realstc crcumstances. Smulaton results show that the proposed UKF-based scheme can offer the superor performances to suppress the nterference over the conventonal methods by computng mean squared predcton error (MSPE) and sgnal-to-nose rato (SNR) mprovements. Keywords Global postonng system (GPS) recever, narrowband nterference, drect sequence spread spectrum (DSSS), recurrent neural networ (RNN) predctor, unscented Kalman flter (UKF) algorthm 1. Introducton he global postonng system (GPS) [1] can provde accurate postonng and tmng nformaton useful n many applcatons. here s an ncreasng nterest n postonng technques based on GPS and cellular networ nfrastructure or on the ntegraton of the two technologes for a wde spread of applcatons, such as tracng systems, navgaton, and ntellgent transportaton systems. he satelltes supply servce to consumers around the world by usng drect sequence spread spectrum (DSSS) modulaton method. GPS spreads the bandwdth of transmttng sgnals wth coarse/acquston (C/A) code, whch results n a 43 db processng gan. DSSS technque nherently exhbts a modest ant-jammng property that can cope wth narrowband nterference. herefore, t s necessary to develop addtonal technques for protecton of the GPS n jammng envronments. he nterference cancellaton technques [2] can be categorzed as tme doman technques [3 6], frequency doman technques [12], and tme-frequency doman technques [13]. For the tme doman approaches, lnear adaptve flterng methods usng fnte mpulse response (FIR) or nfnte mpulse response (IIR) flters [11] have attracted great attenton. However, the jammng sources can be nherently statonary/nonstatonary and assocated hgh order statstcs, so nonlnear adaptve flters may be more sutable to the predcton of these nterferng sgnals. he study of narrowband nterference rejecton has attracted consderable attenton n recent years [3 6]. An enhanced nonlnear adaptve (ENA) algorthm [4] that does not need the AR parameters s proposed and can outperform the exstng lnear and nonlnear adaptve flters. herefore, the ENA algorthm wll be used as the bass of comparson n ths project. he artfcal neural networ (ANN) s one of the alternatve methods used to acheve narrowband nterference suppresson n DS-SS [6 8]. he ppelned recurrent neural networ [6] provdes a better SNR mprovement than the nonlnear ACM flter [3], [4] when the statstcs and number of code dvson multple access (CDMA) users are unnown to those recevers. Several algorthms have been proposed for tranng of the recurrent neural networ (RNN). he most wdely nown s the real tme recurrent learnng (RRL) algorthm [7], whch can be used to update the weghts of the RNN n real tme. However, RRL suffers from the slow convergence problem and cannot provde a good soluton durng the learnng process. he extended Kalman flter (EKF)-based method s probably the most wdely used estmaton algorthm for neural networs [9], [1]. he EKF algorthm, whch ln- DOI: /re SIGNALS

2 RADIOENGINEERING, VOL. 25, NO. 3, SEPEMBER earzes all nonlnear transformatons of state equaton and measurement equaton, substtutes Jacoban matrces for the lnear transformatons n the KF equatons. It s recognzed to be nadequate due to the lnearzaton scheme [9]. Recently, Juler and Uhlmann [9] have ntroduced a new flter founded on the ntuton that t s easer to approxmate a Gaussan dstrbuton than t s to approxmate arbtrary nonlnear functons. hey named ths flter as unscented Kalman flter (UKF). he UKF leads to more accurate results than EKF and n partcular t generates much better estmates of the covarance of the state. In ths paper, we study the performance of the UKFbased RNN predctors for statonary and nonstatonary nterference cancellaton n GPS recevers. Snce the nterferng sgnals may not be statonary, a substtute model based on recurrent neural networ (RNN) structure s more sutable for nonstatonary sgnal predcton [8], [1]. he UKF can outperform the EKF n terms of predcton and estmaton error, at an equal computatonal complexty for general state-space problems. Once the predcton of the jammng sgnal s obtaned, an error sgnal can be computed by subtractng the estmate from the receved sgnal. he error sgnal s then fed nto the correlator for despreadng. he remander of ths paper s organzed as follows. System model of receved jammng sgnals s constructed n Sec. 2. Desgn of UKF-based neural networ s descrbed n Sec. 3. Smulaton results are demonstrated n Sec. 4. Some conclusons are stated n the last secton. 2. Proposed System Models GNSS systems are contnuously gong through progressve evoluton n the feld of postonng and navgaton. A recever computes ts poston, velocty and tme soluton by processng receved data from a constellaton of satelltes. he rado frequency nterference becomes the most dsruptve operaton of GPS recevers. Unfortunately, the low level of GNSS sgnal s susceptble to many types of nterference, whch can be ether ntentonal or unntentonal. hs nterference can sgnfcantly degrade the qualty of, or totally dsable some of, the processes n the GPS recevers. he satelltes broadcast rangng codes and navgaton data at two frequences: prmary L1 and secondary L2. Only the L1 sgnal, free for cvlan use, wll be consdered. A smplfed bloc dagram of an ant-jammng GPS model s shown n Fg. 1. he drect sequence spread spectrum sgnal s gven by s() t 2 Pd() t c()cos(2 t f t) (1) where P s the sgnal power, d(t) s the 5-bps navgaton data sequence, c(t) s the C/A code sequence wth a chppng rate of 1.23 MHz, and f L1 s L1 carrer frequency ( MHz). s the coherent ntegraton tme, c s the L1 chp duraton of rangng code, and the nteger PG = / c = 246 (.e. 43 db) s the processng gan of the GPS system. he receved sgnal r(t) can be modeled as rt () st () nt () It () (2) where n(t) s addtve whte Gaussan nose (AWGN) wth varance 2, and the narrowband jammng source I(t) has a bandwdth much smaller than the GPS spreadng bandwdth. he receved sgnal s bandpass fltered, amplfed and down converted. Due to the down-converson, the spectrum of the sgnal s shfted to the baseband frequency. o further smplfy the analyss, the receved sgnal s assumed to pass through a flter matched to the chp waveform and s sampled synchronously once durng each chp nterval. he observaton becomes r ( ) s ( ) n ( ) I ( ) (3) where {s()}, {n()}, and {I()} are dscrete tme sampled waveform of {s(t)}, {n(t)}, and {I(t)}, respectvely. hey are assumed to be mutually ndependent. he n() can be modeled as band-lmted and whte, and the jammng source beng consdered has a bandwdth much narrower than 1/ c. he s() sequence s d()c() tang values of 1. he low power jammng sgnal can be suppressed by GPS recevers wth the 43 db processng gan. However, f strong jammng sgnals are presented, they can result n degradaton of navgaton accuracy or even complete loss of recever tracng. In ths paper, four nds of narrowband nterference wll be consdered: Sngle tone contnuous wave nterference (CWI) I ( ) Icos( ) (4) a where I s ampltude and s ts frequency offset from the center frequency of the spread spectrum sgnal. c s the chp duraton, whch s equal to the samplng nterval. s a random phase unformly dstrbuted over the nterval [, 2). Mult-tone CWI b c N c (5) 1 I () t I cos[ ] where I,, and represent ampltude, random phase, and frequency offset, respectvely, of the -th nterferer from the central frequency of spread spectrum sgnal, and N s the number of narrowband nterferers. (c) Pulsed CWI Ic Icos( c ) ( l1) N ( l1) N N1 ( l1) N N1 ln l 1, 2, 3,... (6)

3 52 WEI-LUNG MAO, GPS INERFERENCE MIIGAION USING DERIVAIVE-FREE KALMAN FILER-BASED RNN cos L1 t c 2cos L1 t where the on-nterval s N 1 c seconds long and off-nterval s (N N 1 ) c seconds long. We consder the case n whch N and N 1 are much greater than unty. (d) Perodcally swept (lnear FM) CWI 2 Id( ) I cos[ ( l 1) c.5 ( c) ], ( l1) K lk, l 1,2,3,... (7) Fg. 1. GPS spread spectrum system: ransmtter. Ant-jammng recever. where I and are the ampltude and random phase of the swept CWI. represents the offset from the GPS carrer frequency, s the sweep bandwdth, = /K s the frequency rate, and K s the sweep perod. In Fg. 1, the narrowband canceller composed of an RNN predctor and an adder s employed to suppress the jammng sgnals. he predctor/subtractor mplementaton essentally produces a replca of the narrowband nterference Î() whch can be subtracted from the receved sgnal to enhance the wdeband components. he {s()} and {n()} sequences are wdeband sgnals wth nearly flat spectra. hus, these two sequences cannot be estmated from ther past values. he predcton of the GPS receved sgnal usng the adaptve nonlnear predctor based on prevously receved values wll, n effect, be an estmate of the nterferng sgnal. he nterferng sgnal {I()} can be predcted because of ts correlated property. he error sgnal z() s obtaned as z ( ) s ( ) n ( ) I ( ) I ˆ( ) s ( ) n ( ) (8) where z() can be vewed as an almost nterference-free sgnal and s fed nto the correlator. However, the jammng sgnals always have statstcally statonary/nonstatonary propertes, and the nonlnear structure sutable for estmaton s the artfcal neural networ. RNNs are the most general type of neural networs. hey have feedbac, a property whch maes them capable of learnng on-lne and adaptng to statstcal varaton of ncomng tme seres. RNNs have been proven [8] to be better than tradtonal sgnal processors n modelng and predctng nonlnear and chaotc tme seres, as well as n a wde varety of applcatons rangng from speech processng to adaptve channel equalzaton. 3.1 Recurrent Neural Networ Dynamcs he detaled structure of an RNN s llustrated n Fg. 2. It has a neural module and a comparator of ts own. Specfcally, the module conssts of a fully connected RNN wth N hdden neurons, P external nput neurons, and one output neuron. In each neuron, one-unt delayed verson outputs of hdden neurons are assumed to be fed bac to the nput. Besdes the P + N nputs, one bas nput whose value s always at +1 s ncluded. Let matrx W a denote the synaptc weghts of the N neurons n the hdden layer that are connected to the feedbac nodes n the nput layer, and matrx W b represent the synaptc weghts of these hdden neurons that are connected to the nput nodes. It s further assumed that the bas term of hdden neurons are absorbed n the weght matrx W b. he weght matrces W a and W b can be expressed as Wa Wa1... Waj... W an, (9) 3. Proposed Recurrent Neural Networ (RNN) Predctor he predcton of a tme seres s synonymous wth modelng of physcal systems responsble for ts generaton. Wb Wb1... Wbj... W bn, (1) W Wa Wb W1... Wj... W N (11) where W a, W b and W are N-by-N, N-by-(P + 1) and (P + N + 1)-by-N matrces, respectvely. he W aj, W bj and

4 RADIOENGINEERING, VOL. 25, NO. 3, SEPEMBER q 1 () w, j ^ q 2 () q 1 ( 1) y( ) r( 1) - e( ) q 2 ( 1) + q N () 1 r( 1) r( P 1) r( 1) z 1 I q N ( 1) r() Input layer Hdden layer Output layer Fg. 2. Proposed recurrent neural networ predctor. W j are defned by W aj wa j,1... wa j, N, (12) Wbj wb,1 j wb, j P 1, (13) Wj W aj W. (14) bj A (P + 1)-by-1 nput vector R() can be constructed n terms of GPS observaton samples r(), r( 1),, r( P + 1), and a bas nput (+1). Let the N-by-1 vector Q() denote the state of vector of an RNN, and the 1-by-1 vector y() denotes the correspondng output of the system. hese vectors can then be descrbed as ( ) [1, r( ), r( 1),..., r( P1)] R, (15) ( ) [ ( ),..., ( )] Q q1 qn. (16) Based on the defnton above, an nput vector consstng of the total (P + N + 1) nput sgnal can be represented as U( ) Q ( ) R ( ). (17) he dynamc behavor of an RNN can be descrbed by the followng par of nonlnear state space equatons: Q( 1) Φ WQ( ) WR( ) a b ( WU 1 ( )) ( WU j ( )) ( WU N ( )), (18) y( ) CQ ( ) (19) where C s a 1-by-N matrx, whch represents the synaptc weghts of the output node connected to the hdden neurons, and : R N R N s a dagonal map, x1 ( x1) x 2 ( x2) Φ (2) xn ( xn) wth ( ) tanh( /2) (1 ax ax x a x e ) (1 e ). he nonlnear functon ( ) represents the sgmod actvaton functon of a hdden neuron, and a s the gan of a neuron. Any EKF-based tranng algorthm s a second order, recursve procedure that s partcularly effectve n tranng both recurrent and feedforward neural networ archtectures for a wde varety of problems. It typcally requres fewer tranng cycles than does ts RRL [7] counterpart and tends to yeld superor nput-output mappng. hs neural networ tranng problem s vewed as a parameter estmaton problem, and the synaptc weghts are the parameters to be estmated. he unscented Kalman flterng s a dervatve free alternatve to EKF method for state estmaton. he UKF method can utlze a determnstc samplng approach to calculate mean and varance terms. he dynamc behavor of an RNN can be modeled as the nonlnear dscrete tme state equatons: x x u, (21) 1 y H x v (22) where x s an N(N + P + 1)-by-1 vector obtaned by rearrangng the weght matrx W() nto a column vector, and y s an N-by-1 observaton vector. he frst equaton states that the state of the neural networ s represented as a statonary process corrupted by process nose u. he second equaton, nown as the observaton equaton, represents the desred response vector y as a nonlnear functon of weght vector x and measurement nose v. he N(N + P + 1)-by-1 process nose vector u, whch has the PDF u ~ N(,R u ), s ndependent from sample to sample, so that E[u m u n ] = for m n (vector WGN). v. s an N-by-1 observaton nose vector wth PDF v ~ N(,R v ), and s ndependent from sample to sample; thus E[v m v n ] = for m n (vector WGN). 3.2 Node Decoupled Kalman Flter (NDEKF) Learnng Algorthm he practcal applcaton of the EKF algorthm s lmted by ts computatonal complexty. It has greater computatonal complexty and storage requrements than the RRL algorthm. As the EKF-based algorthm s domnated by storng and updatng the error covarance matrx M, the NDEKF learnng algorthm s proposed here n order to reduce the computatonal burden of the GEKF nherent n processng large matrces. he NDEKF algorthm dvdes the weghts of the RNN nto several groups of smaller sze, wth the weghts grouped by output node, and the nteracton between weght estmates can be gnored. hs smplfcaton ntroduces many zeros nto matrx M. herefore, the weghts are decoupled so that the weght groups are mutually exclusve of one another; as a result, M can be rearranged nto bloc-dagonal form. hese smaller-weght groups can be processed ndvdually wth a sgnfcant reducton n computaton. he smplfed matrx M s gven as

5 522 WEI-LUNG MAO, GPS INERFERENCE MIIGAION USING DERIVAIVE-FREE KALMAN FILER-BASED RNN M M1, M 2, M g, (23) where g denotes the total number of groups, and matrx M, s the weght condtonal error covarance matrx of the -th group. he global gradent matrx H also needs to be rearranged, so the weght vectors x () correspondng to a gven output node neuron are grouped as a sngle bloc. Hence, the H s composed of ndvdual submatrces H, ; that s H H1, H2, Hg,. (24) Based on the smplfyng assumpton above, the NDEKF algorthm for the -th group can be expressed as 1 v g A R H, M, H, (25), 1 G M H A, 1,2,..., g, (26),,,,,, xˆ wˆ G y h( x ˆ ), 1,2,..., g, (27) M u, 1 IG H,, M R, 1,2,...,, g (28) where A s the global scalng factor requred to compute the Kalman gan matrx for all weght groups. For group, the vector G, s the Kalman gan of neurons, H, s the gradent matrx of each weght wth respect to each output node, and vector x, refers to the estmated weghts. he concatenaton of vector x, forms the vector x. he matrx R v s a dagonal measurement nose covarance matrx, and R u s an addtonal postve process nose matrx. 3.3 UKF Learnng Algorthm he unscented transform s a method for calculatng the statstcs of a random varable whch undergoes a nonlnear transformaton. Consder propagatng a random varable X (dmenson equal to L) through a nonlnear functon Y = f(x). Assume X has mean X and covarance P x. o calculate the statstcs of Y, we form a matrx χ of + 1 sgma vector χ accordng to the followng: χ x, (29) χ x( ( L ) P ), = 1,,L, (3) x χ x( ( L ) P ), = L + 1,, (31) x L where λ = α 2 (L + κ) L s a scalng parameter. he constant α determnes the spread of the sgma ponts around X, and s usually set to a small postve value. he constant κ s a secondary scalng parameter, whch s usually set to (3 L). ( ( L ) P ) s the -th column of the matrx x square root. hese sgma vectors are propagated through the nonlnear functon Υ f( ) (32) where the mean and covarance matrces for Y are approxmated usng a weghted sample mean and covarance of the posteror sgma ponts, wth weghts W gven by y W, (33) (m) (c) Py W ( y)( y ) (34) (m) 1 W ( L ), (35) (c) 1 2 W ( L) 1, (36) W W 2 ( L ), 1,...,. (37) (m) (c) 1 1 he unscented Kalman flter s a straghtforward extenson of the U to the recursve estmaton, where the state RV s redefned as the concatenaton of the orgnal state and nose varables: x a = [x u v ]. he U sgma pont selecton scheme s appled to ths new augmented state random varable to calculate the correspondng sgma matrx χ a. he UKF equatons are descrbed further. Note that no explct calculatons for Jacobans or Hessans matrces are necessary to mplement ths algorthm. Furthermore, the overall number s of the same order as the EKF. he UKF algorthm s represented as follows: (1) Intalze wth xˆ x E ( ˆ )( ˆ ) P E x x x x, (38) a a xˆ ˆ Ex x, (39) P a a a a a u P ˆ ˆ E ( x x)( x x) R (4) v R for 1,...,. (2) Calculate the sgma ponts: χ a a a a a a ˆ ˆ ˆ ( L) 1 1 ( L) 1 x x P x P. (41) (3) he tme-update equatons: Each sgma pont s propagated through the lnear process model of RNN. It s expressed as x x χ χ. (42). / 1 1 he transformed ponts are utlzed to calculate the mean and covarance of the predcton value of x

6 RADIOENGINEERING, VOL. 25, NO. 3, SEPEMBER (m) x W, / 1 xˆ χ, (43) (c) x x u ˆ ˆ W (, / 1 )(, / 1 ) P χ x χ x R. (44) he sgma ponts are propagated through the nonlnear observaton model of RNN. It s gven as y H χ, χ, (45) x x n / 1 / 1 1 (m) x W, / 1 yˆ y. (46) (4) he measurement-update equatons: From the resulted transformed observatons, the mean, covarance, and cross-covarance matrces are computed. P W ( y yˆ )( y yˆ ) R,(47) yy (c) v, / 1, / 1 P W ( χ xˆ )( y y ˆ ), (48) xy (c), / 1, / 1 1 xy yy κ P P. (49) he measure update of state estmate can be appled usng the Kalman flter equatons. xˆ xˆ κ ( y y ˆ ), (5) P P κpyy κ (51) wth x a = [x u v ], χ a = [(χ x ) (χ u ) (χ v ) ], L where λ s the composte scalng parameter, L s the dmenson of the augmented state, R u s the process nose covarance, R v s the measurement nose covarance, and W are the weghts. he UKF method does not requre complcated analytcal dervatves, such as Jacobans, and the computatonal cost of the algorthm s the same order of magntude as the EKF. Both operatons are O(L 3 ) [9]. he U method can provde a more drect and explct mean and covarance nformaton through the nonlnear transformaton. 4. Smulaton Results In ths secton, the computer smulaton results are llustrated to demonstrate the ant-jammng GPS recever system. Fve types of algorthms are compared, namely LMS, RLS, ENA [3], NDEKF and UKF methods. he followng parameters are chosen: (1) Normalzed LMS: he tap number of the normalzed LMS flter s 1, the adaptaton constant s.1, and the forgettng factor s set at.99. (2) RLS: he tap number of the standard RLS flter s 1, the forgettng factor s set at.99, and the dagonal elements of the error covarance matrx are ntalzed on the order of 1 2. (3) ENA: he ENA algorthm was proposed n [3], [4] wth coeffcents beng updated usng the LMS algorthm. he tap number of the ENA flter s 1, the adaptaton constant s.1, and the forgettng factor s set at.99. (4) NDEKF-RNN: he RNN s composed of 5 external nput neurons (P), a hdden layer of 4 recurrent neurons (N), and 1 lnear output neuron. he number of groups s 4, the error covarance matrx s M () = 1I, the dagonal process nose for group s R u = 1 2 I, and the dagonal measurement nose s R v = 1 2 I. (5) UKF-RNN: he RNN archtecture used s same as NDEKF above. he error covarance matrx s P = I, the dagonal process nose for group s R u = 1 2 I, and the dagonal measurement nose s R v = 1 2 I. he state number L s selected as 9, α s set to 1 3, β s set to 2, and κ s set to for state estmaton n the UKF learnng method. In ths smulaton, the receved sgnal s band-pass fltered, amplfed and down-converted to IF and then dgtzed. he ntermedate frequency f IF s fxed at 1.25 MHz, and a samplng frequency f S of 5 MHz s selected. d() s bnomally dstrbuted wth a value of 1, and c() s randomly selected wth unform probablty from 24 PRN codes of GPS system. he varance of bacground thermal nose n() s held constant at. 1 relatve to the sgnal s(), the power of whch was 1.. he smulaton results are ensemble-averaged over 1 ndependent runs, and 11 data ponts are obtaned n each run. 4.1 Performance Indexes he smulaton results of the UKF-based RNP are obtaned to confrm the jammng rejecton characterstcs. he performance s expressed n terms of SNR mprovement and MSPE. (1) SNR mprovement: he metrc adopted to verfy the steady state performance s the SNR mprovement, whch s defned n [4] and gven by SNR 2 Er ( ) s ( ) 1log (db). (52) Ez ( ) s ( ) mprovement 2 (2) Mean squared predcton error (MSPE, V MSPE ): he MSPE s used as an ndex to evaluate the convergence rate of transent responses for varous algorthms. It s defned as, SIM num 1 2 V( ) e ( ), (53) SIM num 1 n1 1 VMSPE( n) log V( ) 1 (54) (( n1) 1) 1

7 524 WEI-LUNG MAO, GPS INERFERENCE MIIGAION USING DERIVAIVE-FREE KALMAN FILER-BASED RNN where SIM num s the total number of smulatons (whch s 5 here), and e () s the predcted error of the -th teraton for the -th run. 4.2 Interference Suppresson Performances (1) Statonary jammng sgnals CWI sgnal: Fgure 3 presents the SNR mprovements and averaged MSPE for sngle-tone CWIs. he contnuous wave nterference consdered here s a sngletone snusodal sgnal. he CWI offset frequency s set as = 1.2 MHz, s a random phase unformly dstrbuted over the nterval [, 2), and the nterference nose rato (INR) s vared from 2 db to 5 db. he UKF-RNN algorthm can acheve faster convergence rates and better SNR mprovement values than the other algorthms. On average, the UKF-RNN method provdes 1.14 db, 5.29 db, 7.2 db and 8.25 db more n terms of SNR mprovements than the NDEKF-RNN, ENA, RLS, and LMS methods, respectvely. In Fg. 3, we compare the performance of the UKF-RNN algorthm wth others by plottng the value of the MSPE versus the number of teratons. As we can see after about 3 teratons, the UKF-RNN has a better convergence rate than the LMS, RLS, ENA, and NDEKF- RNN. Fgure 3 shows that the UKF-RNN scheme s also superor n both convergence speed and predcton error. he MSPE can declne sgnfcantly to 1 4 n 3 teratons, whle the NLMS and RLS methods reach the steady state after 2 teratons and have larger MSPE results. Fg. 3. Sngle tone CWI suppresson performances of SNR mprovement vs. INR, averaged MSPE vs. the number of teratons. Fg. 4. Multple tone CWI suppresson performances of SNR mprovement vs. INR, averaged MSPE vs. the number of teratons. Mult-tone CWI: he nterferng sgnal consdered s a fve-tone sgnal, where the fve offset frequences are ept constant at =.2 MHz,.4 MHz,.6 MHz,.8 MHz and 1 MHz, and s are..d. random phases unformly dstrbuted over the nterval [, 2). It s shown that UKF-RNN method has better convergence rates than all the other algorthms, and t can move rapdly to the steady state at about 3 teratons. As can be seen from Fg. 4 and, the smulaton results are smlar to those n the case of the CWI sgnal. On average, the UKF-RNN method offers 1.94 db, 5.48 db, 7.22 db and 8.66 db more n terms of SNR mprovements than NDEKF-RNN, ENA, RLS, and LMS methods, respectvely. (2) Nonstatonary jammng sgnals Swept CWI: he frst nd of nonstatonary sgnal consdered s the lnear FM. he normalzed sweep rate () s set to 5 MHz/sec, and the relatve sweep bandwdth s set to 1.5 Hz, and the sweepng perod ncludes 75 samples long. From Fg. 5, the SNR mprovements are computed usng the last 1 data ponts. It s shown that UKF-RNN method s sutable for predctng nonstatonary jammng sgnals and outperforms the other algorthms. On average, the UKF-RNN offers 3.34 db, 5.86 db, 1.9 db and 8.3 db more n terms of SNR mprovements over the NDEKF-RNN, ENA, RLS, and LMS methods, respectvely. he MSPE versus the number of teratons s shown n Fg. 5. he frequency ncreases lnearly at the begnnng of each sweepng nterval, resettng at the end of each

8 RADIOENGINEERING, VOL. 25, NO. 3, SEPEMBER Fg. 5. Lnear FM suppresson performances of SNR mprovement vs. INR, averaged MSPE vs. the number of teratons. nterval. In each smulaton, the change n frequency happens at the 75 th teraton pont, so a transent state s presented n each curve. For the RLS algorthm, t s shown that that t performs poorly at estmatng nonstatonary sgnals. he step sze of the LMS flter s constant, so the rate of convergence s lmted. he performance of the LMS-based ENA flter s between those of the LMS/RLS and NDEKF-RNN flters. Because the Kalman gan matrx of the NDEKF-RNN and UKF-RNN methods allows an adjustable learnng rate, we can see that t acheves a faster convergence rate n the transent state. Pulsed CWI: Another nonstatonary sgnal we consdered s the pulsed CWI. In ths experment, the frequency offset s set to.5 MHz, the on-nterval s set at 1 c and the off-nterval s set at 5 c. Smulaton results of SNR mprovements are gven n Fg. 6. It s shown that the UKF-RNN method provdes superor SNR mprovements on both on- and off-ntervals. On average, the UKF-RNN method offers 2.39 db, 5.52 db, 7.52 db and 9.47 db more n the SNR mprovements than the NDEKF-RNN, ENA, RLS, and LMS methods, respectvely. In Fg. 6, the off-nterval s from the 5 th to the 1 th teraton ponts, and others are on-nterval. It s shown that the convergence speed of the MSPE of the UKF-RNN s usually faster than those of the LMS, RLS, ENA and NDEKF-RNN algorthms n both the on- and off-ntervals. 5. Conclusons Rado frequency nterference s one of the foremost concerns n safety and postonng crtcal applcatons that utlze GNSS recevers. A recurrent neural networ usng an UKF algorthm has been proposed to suppress GPS narrowband nterference. he UKF-RNN method, whch s an mproved dervatve-free and powerful nonlnear estmaton approach, can robustly estmate the statonary and nonstatonary jammng sgnals. he proposed RNN flter converges fast and guarantees ts robustness aganst dfferent nds of obstacles. he UKF recursons are conducted and the correspondng cancellaton performances are presented. Smulaton results have also confrmed that the proposed UKF-RNN algorthm mproves the capablty of combatng nterference and of acceleratng the convergence rate. he results llustrate that the UKF-based RNN method s capable of ncreasng the SNR mprovement and reducng the MSPE of the receved sgnals over those of the conventonal methods n varous nterference crcumstances. Fg. 6. Pulsed tone CWI suppresson performances of SNR mprovement vs. INR, averaged MSPE vs. the number of teratons. Acnowledgments he authors would le to than the Mnstry of Scence and echnology of the Republc of Chna, awan, for fnancally supportng ths research under contract No. MOS E

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