Maximum Likelihood Multipath Estimation in Comparison with Conventional Delay Lock Loops

Transcription

1 Maximum Likelihood Multipath Estimation in Comparison with Conventional Delay Lok Loops Mihael Lentmaier and Bernhard Krah, German Aerospae Center (DLR) BIOGRAPY Mihael Lentmaier reeived the Dipl.-Ing. degree in eletrial engineering from University of Ulm, Germany, in 998, and the Ph.D. degree in teleommuniation theory from Lund University, Sweden, in 3. As a Postdotoral Researh Assoiate he spent 5 months at University of otre Dame, Indiana, and four months at University of Ulm. Sine January 5, he has been with the Institute of Communiations and avigation at the German Aerospae Center (DLR). Bernhard Krah reeived the Dipl.-Ing. degree in eletrial engineering from University of Erlangen- uremberg, Germany, in 5. Sine that he has been with the Institute of Communiations and avigation at the German Aerospae Center (DLR). ABSTRACT The implementation of the maximum likelihood estimator for the time-delay estimation problem is pratially intratable for navigation signals due to its omplexity, espeially when due to multipath reeption several superimposed replia are taken into aount. Reently it has been shown that signal ompression tehniques an overome this problem, as the maximum likelihood estimator an be formulated effiiently upon a redued data set of muh smaller size ompared to the original data, where the redued data set forms a suffiient statisti for the estimated signal parameters. This paper fouses on the formulation of suh a signal ompression based estimator. Furthermore the integration of the estimator into navigation reeivers is addressed, in partiular by onsidering the delay lok loop arhiteture that is employed within onventional navigation reeivers. A novel approah for integrating the effiient maximum likelihood estimator into a generi traking loop is proposed. The performane of the proposed method is assessed by omputer simulations. The results show that the onventional delay lok loop is outperformed with respet to noise performane as well as with respet to the multipath bias. ITRODUCTIO In the absene of multipath, the delay lok loop (DLL) of a onventional navigation reeiver implements an approximation of a maximum likelihood (ML) time-delay estimator. owever, in reality the reeiver typially has to ope with a superposition of the line-of-sight signal (LOSS) and some additional replia that are due to refletions. In this ase a bias is introdued into the estimate of the DLL, resulting in a positioning error even if no noise is present. If the refleted signals are taken into aount, it is still possible to formulate an ML estimator (see e.g. []), now having several delays and amplitudes as parameters. Unfortunately, the resulting system of equations does no longer suggest a straightforward exat solution without dramati inrease in omplexity. There are several pratial diffiulties in an implementation of the optimal estimator, given that the optimization problem is not only nonlinear but also multi-dimensional. One approah to redue omplexity is to break down the problem into a one-dimensional one and approximate the ML solution iteratively. An example is the MEDLL introdued in [] and the SAGE algorithm onsidered in []. One of the latest introdued approahes to the address the multipath estimation problem is the reently introdued Vision Correlator [3]. A general framework for effiient implementation of the optimal multi-dimensional ML time-delay estimator has been given in [4]. The purpose of this work is to assess the performane of the ML estimator proposed in [4] when it is integrated into onventional navigation reeiver arhiteture. Previous studies of the estimator have onsidered its open-loop performane. In omputer simulations, the delays and amplitudes have been estimated for speifi integration times without taking into aount the dynamis of the traking loop. The openloop senario has the advantage that it simplifies the

2 omparison with theoretial limits, suh as given by the Cramer Rao lower bound (CRLB). And in the stati ase, when parameters do not hange during the observation time, the performane is equivalent to that of a losedloop senario. In a dynami situation, on the other hand, a omparison between open-loop and losed-loop performane is less straightforward. To take into aount suh senarios, the delay estimator is put diretly into the traking loop as a replaement of the disriminator. In addition to the delay estimate, a phase estimate an be obtained from the omplex amplitude of the ML solution. The paper is strutured as follows: At first the multipath estimation problem is treated theoretially. After the introdution of the signal model the onept of data size redution is desribed. Then the effiient alulation and optimization of the ost funtion in the redued spae is addressed. Seondly, pratial implementation aspets are overed. An approah for integration of the estimator into a generi traking loop is proposed. Its performane is assessed by omputer simulations whose results are shown. The simulated senarios omprise a multipath-free and a multipath senario. The results for eah senario are disussed respetively. Results, outomes and findings are summarized to onlude the paper. PROBLEM FORMULATIO Assume that the omplex valued baseband-equivalent reeived signal is equal to m y( t) = aks( t τ k ) + n( t) () k = where s(t) is the navigation signal transmitted by the satellite, m is the total number of paths reahing the reeiver, and a k and τ k are their individual omplex amplitudes and time delays, respetively. The signal is disturbed by additive white Gaussian noise, n(t). After sampling this an be rewritten as m y = a s( + n= S( a+ n () k = k k In the ompat form on the right hand side the samples of the delayed signals are staked together as olumns of the matrix S(, τ=(τ,,τ ), and the amplitudes are olleted in the vetor a=(a,,a ). Based on this signal model we an use tehniques from standard estimation theory to attak the multipath problem. The ML estimation is given by the set of delays and amplitudes that minimize the quadrati error: τˆ = arg min L C τ a C (, L ( = min y S( a ote that, although our interest is the delay of the first path, all delays and amplitudes are parameters in the optimization problem. As we will see, for a given set of delays, the optimal amplitudes an be derived expliitly, sine their ontribution to the ost funtion is linear. The problem that remains is to find an effiient method to determine the minimizing vetor τ. There are several pratial diffiulties in an implementation of the ML estimator as desribed above. Firstly, the optimisation problem given by (3) is not only nonlinear but also multidimensional. Suh problems usually require iterative methods. Seondly, the data size in a typial navigation system is huge. To redue the influene of noise, the reeived signal typially has to be observed over several odeword lengths, whih an result in vetors y ontaining several millions of samples. This means that even a single numerial evaluation of the ost funtion L ( requires a large omputational burden, making suh an approah infeasible in a real-time appliation. These problems an be approahed by the redued omplexity tehniques suggested in [4] [5]: Data size redution: The large vetor ontaining the reeived signal samples is transformed into a vetor y of muh smaller size before the atual optimization takes plae. The goal is a systemati approah to ahieve suh a redution with a negligible performane loss. ewton-type optimization: Compat symboli expressions for the gradients and essians, in ombination with the redued data size result in a both effiient and robust tehnique. Interpolation methods allow arbitrary delay resolution independent of sampling rate. DATA SIZE REDUCTIO As disussed above, the reeived vetor y is a linear ombination of signatures s(τ k ) plus some additional noise. From a geometrial point of view, the signal term in y is inside the span of the set of signal replia {s(τ ), s(τ ),, s(τ )}. The goal in (3) is to find that vetor within the signal spae spanned by S(, whih is losest to the reeived vetor. For a fixed τ, the best approximation of y is given by an orthogonal projetion onto that signal spae, as illustrated in Figure. (3)

3 P ( S ( S ( ) S ( τ ( S ( ) = () All alulations an now be performed on this redued size model. For a given estimate of delays ˆτ, the orresponding omplex amplitudes â follow diretly from the linear projetion given in (4), Figure : Projetion onto the signal spae This linear projetion is well-known in estimation theory and an be expressed expliitly by the projetion matrix ( S( S( ) S( P( = S( (4) Substituting the projeted vetor P(y into the ost funtion we obtain L C ( S( S( ) ( = y S( S( y (5) In this expression the dependene on the omplex amplitudes a has been eliminated. The ost funtion now depends only on the delay vetor τ. The key to redue the data size is to find a suitable subspae of low dimension that still ontains all possible signal terms. More speifi, if we find a matrix Q that satisfies Q Q = I and Q Q s( = s( (6) then it follows from the eyman-fisher fatorization [6] that Q y is a suffiient statisti for estimating the delays. In other words, there is no information loss if the delays τ are estimated after orrelation with the matrix Q. This means that the original system model an be replaed by Q y = Q S( a Q n (7) + and the minimization of the ost funtion an be performed using the orresponding ost funtion, i.e., τˆ = arg min y P y (8) τ where y = Q y, S ( = Q S(, (9) ( S (ˆ) τ S (ˆ) S (ˆ τ y a ˆ = ) () If Q is a square matrix, then the transformation above is simply a rotation. In order to redue omplexity one needs to find a retangular matrix that has a small number of olumns and, hene, ompresses the data size. ereby, the onditions above should be satisfied as losely as possible if loss in performane is to be avoided The fundamental idea is to find a subspae of small dimension in whih the signal term S(a is onentrated for any value of the parameters. Sine Q ompresses all olumns of the signal matrix S( equally and all these olumns have the same struture, it suffies in the seletion of the subspae to onsider a single signal replia s(a. Furthermore, as the orrelation with Q is a linear operation, the seletion riterion is invariant in the amplitude a, whih allows onsidering a= without loss of generality. The problem an now be desribed as finding a ompression matrix Q in suh a way, that the error of reonstruting s( from its ompressed version Q s( is small. For a given τ the loss an be measured by the relative energy error s( sˆ( s( () The reason why ompression is possible is the fat that we only need to onsider a limited range of possible delays. Sine the most ritial multipaths have delays around one hip duration or less, we may onsider only τ values in a limited interval I τ that is entered at zero. In pratie the tolerated error in the hoie of Q allows a trade-off between performane and omplexity. To selet the ompression matrix Q, a two-fold data ompression has been onsidered in [4][5], as illustrated in Figure. Canonial Components Bank of Correlators Prinipal Components Further Redution Figure : Two-fold data redution

4 The prinipal omponents (PC) method minimizes the average reonstrution error in the desired delay range. This riterion is diretly applied in the prinipal omponents method in order to selet a subspae spanned by a matrix Q p with orthonormal olumns. Q p = arg min s( QQ s( dτ (3) Q Iτ This minimizing matrix Q p an be formed by the eigenvetors orresponding to the greater eigenvalues of the matrix R = s ( s( s Iτ dτ (4) The number PC of olumns in Q p will determine the quality of the signal approximation. The more of the eigenvalues are lose to zero, the better the ahievable ompression. The drawbak of the PC method is that the resulting omplex orrelators do not posses any partiular struture that might simplify a hardware implementation. Suh a struture is maintained in the anonial omponents (CC) method, whih atually also forms the theoretial foundation behind the mathed orrelator tehniques like the ones employed in existing navigation reeivers. The CC method uses the onvolutional fatorization of the navigation signal into ode sequene and pulse, s( t) = ( t) g( t) = iδ ( t it ) g( t) (5) i= where i are the elements of the periodi ode sequene and g(t) is typially a band-limited retangular pulse. After sampling this an be expressed by means of an s x g onvolution matrix C s, s= C g (6) s where s and g are the lengths of the sampled signal vetor s and the pulse vetor g, respetively. The olumns of the matrix C s are irularly shifted against eah other aording to the sample spaing T s. With this the sampled delayed signal an be approximated by b s( C( g ( (7) where the vetor τ b of length defines a delay grid with spaing T s (the yquist sampling period of the signal), overing the area where most energy of the pulse g(t) is loated. From (7) we an dedue that s( is within the span of C(τ b ) and, hene, an be reonstruted after orrelation with the latter. From this we now want to obtain a ompression matrix Q that satisfies the onditions in (6) and has the same span as C(τ b ), Q (8) b = C( R where R is a whitening matrix that follows from a QR or SVD deomposition of C(τ b ). This proedure has the advantage that the resulting orrelators are now mathed to the ode (t), and the orrelation proedure an thus be performed with simple integer values (-,+) and at the hip rate. The outputs of the orrelator bank an be expressed as y b = C( τ y (9) or ) In an implementation, be it in software or hardware, the sparseness of the orrelation matrix an be taken into aount to redue omplexity. It is also possible to use a bank of orrelators that are mathed to the navigation signal s(t) diretly. In this ase the olumns of the orrelation matrix are shifted versions of the sampled navigation signal s=s(τ=), i.e., and y b = S( τ y () or ) Q = S( τ b ) R () The implementation omplexity of this orrelator bank may be larger, sine a multipliation has to be arried out for eah sample of the reeived vetor. The differene between the two orrelation operations and the resulting outputs of the orrelator bank are illustrated in Figure 3 and Figure 4, respetively Signal Mathed Correlator Time [µs] Code Mathed Correlator Time [µs] Figure 3: Illustration of signal-mathed and odemathed orrelation with reeived signal (blue).

5 Amplitude Amplitude Delay [µs] Delay [µs] Figure 4: Output of the bank of signal-mathed (top) and ode-mathed (bottom) orrelators. In order to ahieve a better data ompression, the PC method an be applied to the output of the bank of CC orrelators. While both CC methods an be used equivalently, throughout this paper we will onsider the ode mathed orrelators only. Then the x p PC ompression matrix Q p is alulated from the signal s Q s R C τ s () ( ) ( ) ( ) ( b ) τ = τ = ( The output after the overall ompression then has the struture b y = QpQy = Qp( R) C( y (3) Q COST FUCTIO MIIMIZATIO τ ( k + ) = τ ( k ) µ L( (4) where =ess L( denotes the essian of L( and µ> defines the step size. Alternatively, an approximation of the exat essian an be used, resulting in the so-alled modified variable projetion (MVP) method. Beause of the quadrati onvergene to loal minima, only a small number of iterations are required. Another advantage ompared to other methods is that no speial struture (e.g., Vandermonde) is required in the system model. In the onsidered navigation system, as shown in Figure 5, the ewton-type optimizer is applied to the redued size vetor y at the output of the data redution. Sine the number of operations per iteration is proportional to the data size, the data ompression tehniques desribed above result in a muh smaller omplexity in the optimization implementation. The main drawbak of this optimization tehnique is that the gradient and essian of the ost funtion have to be evaluated in eah iteration. In general this an be a omputational problem and approximations may have to be used. owever, for the struture of the system model onsidered here, ompat symboli expressions for the exat gradients and essians have been derived in [4]. In ombination with the data ompression the ewton-type methods beome thus an attrative solution for our mitigation problem. The ost funtion desribing the minimization problem an be written as (ompare to (8)) {( I P y y } L = y P y = tr ) (5) Let us introdue the notation R = y y, M ( S S ), S = M S (6) y = Cost Funtion Evaluation Figure 5: The ewton-type optimizer ewton Iteration ewton-type methods are regarded as being among the most robust and effiient tehniques in unonstrained optimization. In the ewton-raphson method, for a given ost funtion L(, the estimate in iteration k is given by d d D = s ( τ), s ( τ ),, dτ dτ D d d = s ( ), ( ), τ s dτ dτ τ (7) Using the symboli method introdued in [4], the gradient of this ost funtion an be derived as { { y }} L = R diag SR ( I P) D (8)

6 For the essian one obtains = R { T ( ( ) ( ) ) T -( SR y( I P) D) ( SM ) T -( SD ) ( SR y( I P) D) T -( D ( I P) D) SRy( S) T + I ( SRy( I P) D ) } D I P R I P D M y ( ) (9) erer is the real-part operator and the symbol denotes the element-by-element (adamard) produt of two matries or vetors. If the signal-to-noise ratio is large, whih means that the variane σ is small ompared to the omponents of R y, then an approximate essian is often used in pratie. Under this assumption we have ( I P ) R y (3) and the only term remaining in the essian results in the approximate version appr R = T {( D ( I P ) ( ( ) )} ) D SR y S (3) This matrix is used in the modified variable projetion (MVP) method. During the optimization, the gradient and essian of the ost funtion have to be evaluated in every iteration for another set of possible delays τ. Sine very severe multipath errors are aused by relative delays (between diret and refleted paths) of only a fration of the hip duration T, the grid of the vetor τ in the optimization proedure (see (4)) needs to be suffiiently fine. This means that in the expressions (8) and (9) those matries depending on τ need be available in the desired resolution. All these matries are dedued from delayed versions s( of the signal s and its derivatives. On the other hand it is desirable to ahieve improved delay shift resolution without large oversampling, whih again would inrease omplexity. A way to ahieve arbitrary resolution in τ independent of the sampling period is the use of Fourier interpolation tehniques. Sine the navigation signal is omposed of elementary pulses g(t), we an make use the relation (7) and apply the interpolation on that pulse. Assume that the band-limited pulse g(t) is approximately zero outside a given time interval, and that a vetor g of length g ontains its sampled values with a uniform grid of separation Ts smaller than the yquist period. Then it is possible to alulate the samples of g(t- for a delay τ with the interpolation formula g( = Fdiag( g ) Φ( (3) where g F is the disrete Fourier transform (DFT) of g and j π ( r )( q )/ g [ F] = e, q, r =,, qr, g g (33) j π( r ) τ /( gts) Φ( = e, r =,, [ ] r F are the g x g inverse DFT matrix and a length g Vandermonde vetor, respetively. Conventional fast Fourier transform (FFT) algorithms an be used to ompute g F and F, F, F = fft( I ) g (34) g g = fft( g) where the FFT of a matrix is given by the transforms of its individual olumns. Depending on the implementation of the transform, it may be neessary to rotate the Vandermonde vetor aordingly. Using suffiiently large g together with zero padding, the error in (3) beomes negligible. Sine the interval g should also ontain the non-zero part of the delayed signal g(t-, the omplexity in this interpolation is inreased aordingly with the size of desired delay range. It also depends on the signal length and bandwidth, but not on the delay resolution: the Vandermonde vetor allows the appliation of a ontinuous delay. Another benefit of the Vandermonde struture in (3) is that it allows simple alulation of the derivatives of g( and, hene, s(, whih are required in the omputation of the gradient and essian. To illustrate this, an example of a band limited retangular pulse and its shift by half a sample is given in Figure 6. The pulse has a bandwidth of 5 Mz (one sided) and is sampled with samples/hip = Mz to satisfy the yquist riterion. The samples of the urve appear at the same time instanes but orrespond to a shift of τ=t s /. The Fourier interpolation an be ombined with the data ompression tehniques. The signal matrix and its derivatives at the desired point are omputed with the signal interpolation fatorization of the ompressed signal s ( τ S ( = M Φ( ),, Φ( ) (35) s m g

7 Amplitude Time [samples] Figure 6: Example of a pulse and its shift by /T s with the interpolation matrix M = Q C Fdiag( g ) (36) s s F ene, the interpolation matrix is simply multiplied with a matrix having the orrespondingly shifted Vandermonde vetors as olumns. The derivatives are omputed aordingly, based on the easily determined derivatives of the elements of Ф(. With this the omputations of the ost funtion, gradient and essian an be performed within the redued signal model. The signal interpolation also simplifies the omputation of the PC matrix Q p [5], whih is omposed by eigenvetors of the orrelation matrix with R s = E = M ( s ( s ( ) E( Φ( Φ( ) M s ( ) s (37) b M = R C( CFdiag( g ) (38) s s F Assuming a uniform distribution of the delay τ, the mathematial expetation is alulated by integration over the orresponding delay range I τ =(τ a,,τ b ), see (4). It an be seen in (37) that the Fourier interpolation of the signal may be used to alulate R s from the orrelation matrix R Ф of the Vandermonde vetor Ф(, R Φ = E ( Φ( Φ( ) = τ τ b τ b a τ a Φ( Φ( dτ (39) This has the advantage that the elements in R Ф follow expliitly from τ b aτ b aτ a aτ ( e e ) / a, a e dτ = τ τ b τ a, a = a (4) where a is a salar value that has to be substituted aordingly for the different matrix elements. It should be noted that R Ф depends only on the number of samples in the pulse vetor g, their spaing T s, and the delay interval I τ. The olumns of Q p are then hosen as those eigenvetors of R s, whih orrespond to the p largest eigenvalues of that matrix. Sine the orrelation matrix, by definition, is ermitian, these eigenvetors are orthonormal. ITEGRATIO ITO TE TRACKIG LOOP A onventional DLL uses two orrelators, mathed to the navigation signal, for traking the maximum of the autoorrelation funtion. If the DLL is in lok, the orrelation peak is exatly in the enter between the two orrelators and their outputs are equal. Otherwise, the differene between the early and the late orrelator indiates the diretion and distane to the maximum. The disriminator urve shows this differene as a funtion of the urrent signal delay relative to the lok point, as illustrated in Figure 7 (right). It is used in the feedbak of the loop in order to adjust the urrent loal referene value of the delay and move bak to the stable point at the origin. ene, the two orrelators of a DLL slide along the autoorrelation funtion (see left hand side in Figure 7) until their values beome equal. It an be seen in Figure 7 that the disriminator urve depends on the spaing between the two orrelators. If the spaing is redued from a standard value of hip (red) to. hips (green), the linear region around the origin is redued. Being outside this region orresponds to the ase when both orrelators are on the same side of the peak of the autoorrelation funtion. If this is the ase, their output differene no longer adequately measures the distane to the origin and the performane is redued. On the other hand, a smaller orrelator spaing (ommonly referred to as narrow orrelator ) is known to redue the error due to multipath. Autoorrelation Funtion Early-Late Disriminator Figure 7: Disriminator with different orrelator spaings

8 For omparison, onsider now the ML ost funtion L (, shown in Figure 8, whih an be evaluated from the redued data vetor y with arbitrarily fine resolution in τ. The ewton optimizer estimates the delay of the inoming signal by searhing the minimum of the funtion L ( (marked red in the figure). The urrent referene value τ= (marked green) an be used as initial value for the searh. When no multipath is present, a minimization of the ost funtion is equivalent to a maximization of the autoorrelation funtion. To trak the maximum, the disriminator of a onventional DLL provides an error estimate that is used for the orretion of the urrent delay referene. In its linear region it produes a saled approximation of the delay value that minimizes the funtion L (. Consequently, the disriminator an be interpreted as an approximation of a single-path ML estimator. Figure 8: Cost funtion example evertheless, as the disriminator in the DLL an be regarded as a sub-estimator element within the loop, it is the entire loop itself that finally implements the overall estimator. Thus it is diffiult to ompare the forward ML estimator with the DLL, whih in fat implements a fully sequential estimator. Compared to a sequene of independent ML estimates the sequentially estimating DLL offers robustness against noise and transients when exposed to parameter dynamis. For this reason the inorporation of the ML estimator into the reeiver beomes not straightforward, given that the ML estimator is designed to operate only on time intervals, during whih the signal parameters do not hange. Beside data modulation this fat restrits the possible interval for oherent integration. ene the DLL is sometimes able to outperform the ML estimator in pratial senarios, depending on the integration interval, the noise level and the parameter variations, even when averaging is applied to the sequene of ML estimates with a filter harateristi equal to that of the DLL. In order to overome these shortomings a hybrid solution with an ML estimator inorporated in a generi loop is proposed (in-the-loop MLE approah) as depited in Figure 9. The ML estimator operates in plae of the generi disriminator and provides also a phase estimate to the phase lok loop (PLL) based on (). Figure 9: Traking loop with ML estimator (in-theloop-mle). PERFORMACE WITOUT MULTIPAT For the performane assessment of the proposed approah in a multipath-free senario several simulations have been arried out. The generi inoherent DLL is ompared to the proposed in-the-loop MLE arhiteture and the performane bound that is given by the CRLB by means of the root-mean-square (RMS) traking error in dependene on the C/. The signal used within all simulations was a GPS C/A ode signal of Mz onesided bandwidth. The time of oherent integration was set to ms for the DLL simulations, whih have been arried out for an early/late orrelator spaing of hip,.5 hips,.3 hips and. hips respetively. The MLE simulations have been arried out for a oherent integration time of ms and ms respetively, whereas the MLE assumes a single path being present, i.e., m =. Code mathed orrelators were used for the CC ompression method with = 4, followed by a PC ompression with p = 3. The loop bandwidth for all simulations was equal to z. The CRLB was alulated aording to [6] based upon the onsidered reeived signal and the loop bandwidth. The results, whih are depited in Figure, show for the DLL simulations the well-known effet of improved noise performane as the orrelator spaing gets dereased, as it is overed within [7] for instane. In the figure it an be observed that the in-the-loop MLE attains the CRLB for high C/, but it does diverge from the bound for low C/, whereas the point and amount of divergene depends on the oherent integration time. The phenomenon of divergene may be traed bak to the fat that the ML estimator is non-linear unlike the generi early/late disriminator. For the DLL it is equivalent whether the linear gradient operation (early minus late operation) is applied before or after the linear filtering.

9 RMS error [s] DLL Chip DLL.5 Chip DLL.3 Chip DLL. Chip MLE ms MLE ms CRLB ML estimator behaves linear in a region of ± hip around the in-phase orrelator. This is an advantage ompared to the DLL where the linear region is signifiantly redued for orrelator spaings below hip. For the ML estimator the width of the linear region an be traded against the values and p. If the deviations from the stable lok point are expeted to be small, a muh smaller number of orrelators and PC omponents should be suffiient. - PERFORMACE WIT MULTIPAT C/ [db-z] Figure : Simulated performane of -path in-theloop MLE ompared to DLL and CRLB. The non-linear in-the-loop MLE, on the other hand, has to operate before the loop filter, sine otherwise the linear harater of the loop is lost. This leads to the divergene for low C/, as the ML gradient operates on the data from the oherent interval only, unlike the generi linear disriminator, whih operates on filtered data effetively. ene longer oherent integration times are preferable for the in-the-loop-mle as it is shown by the simulation results. RMSE [m] PC Figure : Performane of -path ML estimator as funtion of subspae dimension p, =4. To determine the influene of the subspae size on the performane, the MLE has been simulated as forward estimator for different p values. The signal and MLE settings were the same as for the previous simulations, whereas the forward MLE operates on a snapshot of ms data at 45 db-z and the resulting RMS error is obtained from a statisti of, independent snapshot estimates for eah p. The results are depited in Figure and show that for the simulated senario of =4 and C/ =45 db-z a number of approximately 3 PC omponents is needed in order to avoid an observable loss due to the PC ompression. It should be noted that this number strongly depends on the number of CC orrelators. The generous hoie of 4 orrelators ensures that the The diret relation between ost funtion and orrelator outputs is no longer given, if the number of paths m in the reeived signal is larger than one. The disriminator of the DLL gets distorted by multipath, and the loop loks to a value that no longer orresponds to the ML solution of the diret path. multipath error [T ] delay [T ] Figure : Multipath error envelope: advantage of narrow orrelator (green). For a single additional path, i.e., m =, the error envelope shows the resulting noise-less multipath error as a funtion of the delay τ = τ -τ between the diret and the seond path. The error depends not only on the relative multipath amplitude and phase, but also on the spaing between the orrelators of the DLL. Figure shows the error for equal phase, amplitude ratio /, and spaing of hip (red) and. hips (green) onsidering a C/A ode signal generated from the band limited example pulse shown in Figure 6. It an be seen that the narrow orrelator is muh less disturbed by the seond path [8]. The effet of multipath on the output of the orrelator bank is shown in Figure 3. The MEDLL [] and the SAGE algorithm [] both operate on the signal-mathed orrelator outputs for searhing the ML solution. More reently, it was suggested to perform the estimation on filtered hip transitions instead [3][9]. Interestingly, there appears to be a lose onnetion between the signal ompression theorem in [9] and the suffiient statistis ondition given in (6) for the output of the ode mathed orrelator bank.

10 Delay [µs] multipath error [T ] Delay [µs] Figure 3: Effet of multipath on the signal-mathed (top) and ode-mathed (bottom) orrelator outputs. The atual ost funtion of an ML estimator for the ase m = is a funtion of two dimensions, one for eah delay. An example is shown in Figure 4. Due to the linearity of the transformation into the subspae the omputation of this funtion is independent of the data redution method. Analogously to the one-dimensional ase, the proposed multipath mitigation algorithm searhes the minimum (marked red) of this ost funtion, starting from some given initial estimate (marked green). As before the urrent delay referene of the DLL an serve for the seletion of the start value. For seleting the relative multipath distane of the initial value, previous estimations an be taken into aount. ote, that a seond minimum exists (marked blue), whih orresponds to an equivalent solution after sorting. Its existene follows from the symmetry of the ost funtion delay [T ] Figure 5: Multipath error envelope: narrow orrelator (green) vs. ML estimator with m = (blue) and m = (red). Figure 5 shows the multipath error of the narrow orrelator in omparison with the ML estimation algorithm. The red urve shows that the error beomes negligible if the true ost funtion is used in the ML estimation. The blue urve results if the ML estimator wrongly assumes a single path, i.e., m =. It an be seen that this urve is still slightly better than that of the narrow orrelator. If the spaing of the narrow orrelator is dereased further it will atually onverge to the blue urve of the -path ML estimator. The orresponding error ould be further redued by inreasing the bandwidth at the reeiver input. The multipath performane of the in-the-loop MLE in presene of noise has been simulated for the same signal and parameter settings as in the previous setion. For a fixed C/ =5 db-z and a oherent integration time of ms the traking error is shown as a funtion of time for a relative multipath delay of τ = -7 s (3 m) and τ = s ( m) in Figure 6 and Figure 7, respetively. The results show again that the -path MLE suessfully mitigates the bias aused by the multipath, even for delays below / of the hip duration T. The figures also show that for smaller τ the variane of the ML estimator is inreased. This is a well-known phenomenon that is also refleted by the orresponding CRLB, whih diverges in the region of small τ. Lower noise levels allow mitigation of multipath with smaller delays. Figure 4: Multipath ost funtion

11 Traking error [m] DLL. Chip MLE Path MLE Path Time [s] Figure 6: Simulated multipath performane of in-theloop MLE with m = and m = ompared to DLL for τ = -7 s (3 m). Traking error [m] DLL. Chip MLE Path MLE Path Time [s] Figure 7: Simulated multipath performane of in-theloop MLE with m = and m = ompared to DLL for τ = s ( m). COCLUSIOS REFERECES [] van ee, D.J.R., J. Siereveld, P. Fenton, and B. Townsend, "The Multipath Estimating Delay Lok Loop: Approahing Theoretial Auray Limits", Proeedings of the IEEE Position, Loation and avigation Symposium, Las Vegas, V, USA, 994. [] Felix Antreih, Oriol Esbri-Rodriguez, Josef A. ossek, Wolfgang Utshik, "Estimation of Synhronization Parameters Using SAGE in a GSS-Reeiver", Proeedings of the IO GSS 5, Long Beah, California, USA, 5. [3] P. Fenton, "The Theory and Performane of ovatel In. s Vision Correlator," Proeedings of the IO GSS 5, Long Beah, CA, September 5, pp [4] Jesus Selva Vera, "Effiient Multipath Mitigation in avigation Systems", Ph.D. thesis, DLR/ Polytehnial University of Catalunya, 4. [5] Jesus Selva, "Complexity redution in the parametri estimation of superimposed signal replias", Signal Proessing, Elsevier Siene Volume 84, Issue, Deember 4, Pages [6] Kay, Steven M., Fundamentals of Statistial Signal Proessing Estimation Theory, Prentie all Signal Proessing Series, Prentie all, ew Jersey, 993 [7] Weill, L., "C/A Code Pseudoranging Auray - ow Good Can It Get?", IO GPS-94, Salt Lake City, pp. 33-4, [8] van Dierendonk. A. J., Fenton P., Ford T., "Theory and Performane of arrow Correlator Spaing in a GPS Reeiver", Journal of The Institute of avigation, Vol. 39, o.3 Fall 99 [9] Weill, L.R., "Ahieving Theoretial Bounds for Reeiver-Based Multipath Mitigation Using Galileo OS Signals," Proeedings of the IO GSS 6, Fort Worth, TX, September 6. The performane of an in-the-loop MLE has been investigated and ompared to a onventional DLL. Simulation results in absene of multipath show that at high C/ the MLE attains the CRLB. At medium to low C/ the MLE is still apable of outperforming the narrow orrelator if the oherent integration time is hosen suffiiently high. While the linear region of a DLL dereases with the orrelator spaing, the one of the MLE an be adjusted by seleting the range overed by the bank of orrelators. In the presene of multipath the simulation results show that the MLE is apable of mitigating the multipath bias even for multipath delays smaller than a tenth of the hip duration. The suggested data ompression and interpolation tehniques are not restrited to the effiient omputation of the ML solution by ewton methods but an be used in a muh wider range of appliations where the signal parameter likelihoods an be of interest.