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A DECONVOLUTION ALGORITHM FOR ESTIMATING JOINTLY THE LINE-OF-SIGHT CODE DELAY AND CARRIER PHASE OF GNSS SIGNALS Danai Skournetou1, Ali H. Sayed2 and Elena Simona Lohan1 1 Department of Communications Engineering, Tampere University of Technology P.O.Box 553, FIN-33101, Finland; {danai.skournetou, elena-simona.lohan}@tut.ﬁ 2 Department of Electrical Engineering, University of California, Los Angeles, CA 90095, U.S. sayed@ee.ucla.edu concerned. ABSTRACT BIOGRAPHIES An important task of a Global Navigation Satellite Danai Skournetou received the M.Sc. degree in System (GNSS) receiver is to achieve ﬁne synchro- Information Technology from Tampere University of nization between the received Line-of-Sight (LOS) Technology in 2007. Currently, she is a PhD stu- signal and the reference code, which would allow dent in the Dept. of Communications Engineering the computation of the satellite-receiver distance. of TUT where she also works as a researcher in This synchronization process, known also as tracking the Digital Transmission Group (http://www.cs.tut.ﬁ/ stage, requires the Doppler shift to be successfully ∼simona/Positioning Group Webpage.htm). removed from the received signal (or that the residual Ali H. Sayed is Professor and Chairman of error is kept within allowable limits) and typically Electrical Engineering at the University of Califor- involves the estimation of signal parameters such as nia, Los Angeles. He is also the Principal Inves- the code delay, the carrier frequency and/or carrier tigator of the UCLA Adaptive Systems Laboratory phase. (www.ee.ucla.edu/asl). A challenging issue in the estimation of the syn- Elena Simona Lohan is an Adjunct Professor chronization parameters is the mitigation of multipath at Tampere University of Technology, where she effects that appear due to the wireless propagation got her PhD degree in 2003. Her research interests channel characteristics. In this paper, we deal with include satellite positioning techniques, CDMA sig- the problem of joint LOS code delay and carrier nal processing, and wireless channel modeling and phase estimation of GNSS signals in a multipath estimation. environment. The problem is formulated into a linear system of equations in which the unknowns are the I. INTRODUCTION channel complex coefﬁcients corresponding to each Knowledge of the Line-of-Sight (LOS) carrier observed signal sample. We introduce a modiﬁed phase is of great value in the context of Global Projection Onto Convex Sets (POCS) algorithm that Navigation Satellite Systems (GNSSs). Carrier-phase- we optimize for both the new L1 Open Service and based positioning is a representative example wherein Coarse/Acquisition (C/A) signals employed by the fu- phase information is fully utilized for pseudorange ture European Galileo and the Global Position System calculation at a sub-centimeter level [1]. Phase acqui- (GPS), respectively. We compare the performance of sition is also required for smoothing the ranging code the algorithm with other state-of-art deconvolution measurements; it is normally performed in differential algorithms. The simulation results indicate that our approaches and results in improved accuracy. More- modiﬁed POCS algorithm is the most resistant in over, carrier phase estimation loops can be incorpo- closely-spaced multipath static channels both when rated in the receivers for aiding the code tracking LOS code delay and carrier phase estimation are loops and improving their overall performance. Although accurate carrier phase estimates are desir- LOS and carrier phase estimation in multipath envi- able, the presence of multiple channel paths causes ronments, for BOC-modulated signals, by trying to distortions to the received signal, which cannot be overcome some of the above-mentioned limitations treated via differential approaches due to the very (namely, scarcity of methods able to estimate jointly localized nature of multipath [2]. Moreover, the char- phase and delay of LOS in multipath channels, limi- acterization of the carrier phase multipath from ﬁeld tation of most existing methods in the presence of data is a difﬁcult problem since the exact sources multiple correlation peaks as in BOC modulation, of these errors are not easily recognizable. In [3], high complexity of Maximum Likelihood-based al- the above-mentioned problem is approached from a gorithms). geometric perspective that involves different conﬁg- We ﬁrst formulate the multipath estimation prob- urations of the antenna-reﬂector(s) geometry. Carrier lem in terms of a linear system of equations, where phase multipath is also commonly studied using a the unknowns are the channel complex coefﬁcients phasor diagram that illustrates the relation between corresponding to each observed signal sample. Then the phase of the LOS signal and the multipath [4]– we apply deconvolution methods to solve this system [6]. and to ﬁnd the correct code delay of each multipath One can ﬁnd in the literature other methods that component. The output is further used to form the car- aim to estimate carrier phase offsets assuming multi- rier estimates. We also introduce a modiﬁed Projec- path environment. Such methods include the Ashtech tion Onto Convex Sets algorithm (i.e., a constrained Enhanced Strobe Correlator [7] and the Multipath deconvolution approach previously used for various Estimating Delay Lock Loop (MEDLL); the latter CDMA systems [12]–[16]), which we optimize for jointly estimates the delay, relative amplitude, and both the new L1 Open Service and GPS signals. phase parameters of the multipath signals based on Deconvolution approaches for LOS estimation the maximum likelihood theory [8]. Both are ad- have been previously used in [12], [17]. Our POCS- vanced techniques with improved performance in based proposed algorithm is different from the pre- long delay multipath errors, however, they are heavily viously proposed deconvolution approaches in two patented. main ways: ﬁrst, it incorporates some knowledge A new modulation technique, called Binary Offset about the static multipath channel via estimated level Carrier (BOC), has been introduced for future GNSS crossing rates of receiver correlation function; second, signals, including several Galileo signals as well as it uses an adaptive threshold to reduce the various the GPS L5 signal [9]. Compared to the Binary sources of interference (noise, multipath, sidelobes Phase Shift Keying (BPSK) modulation used in GPS, in the autocorrelation function of BOC-modulated with only one triangular-shaped peak in the envelope signals). of the AutoCorrelation Function (Acf) (if unlimited The remainder of this manuscript is organized as bandwidth is assumed), BOC modulation introduces follows. Section II introduces the signal and the more challenges, both in the code delay and carrier channel model. Section III includes the formulation of phase tracking stages, due to the presence of multiple the problem whereas Section IV contains state-of-art peaks (e.g., the possibility to track a wrong peak is methods, a step-by-step description of the modiﬁed higher). While the majority of the existing research POCS algorithm and how the LOS code delay and work is done for GPS signals, only few studies can carrier phase are computed. Section V includes the be found for the modernized GPS and new Galileo simulation results and a discussion on the perfor- signals. In [10] the ability to track the phase of the mance of the algorithms in various multipath channel GPS L5 signal in the presence of Gaussian noise was proﬁles. Lastly, Section VI concludes the highlights studied (both pilot and data channels were consid- of this research work and describes future plans. ered). Also, in [11] the performance of different phase discriminators was tested for the phase tracking of II. SIGNAL MODEL dataless channels, which have been included in the The signals of interest are GPS and Galileo civil new BOC modulated signals. However, the ability to signals. Both use a Direct-Sequence Code Division track the LOS phase of these new signal types in Multiple Access (DS-CDMA) technique, and either multipath environment is still to be investigated. BPSK for C/A code of GPS, or sine Binary-Offset- In this paper, we deal with the problem of joint Carrier (BOC) modulation, for Galileo Open Services Examples of time−domain waveforms (OS) [1], [18]. We notice that recent standardization documents specify Multiplexed BOC (MBOC) as Code sequence 1 the modulation type to be employed for OS signals PRN sequence and which is a combination of sine-BOC(1, 1) and 0 sine-BOC(6, 1) [19]. However, since both BOC and −1 0 1 2 3 4 5 MBOC modulation types have been incorporated in Chips the standards, and since MBOC signals are supposed sine−BOC(1,1), i.e., NB=2 sine−BOC code also to work with sine-BOC receivers, the focus here 1 is on sine-BOC, and our algorithm requires only 0 few (and straightforward) modiﬁcations to work with MBOC as well. −1 0 1 2 3 4 5 Therefore, the signal to be transmitted, x(t), can BOC samples be written as the convolution between the modulat- ing waveform sBOC (t), the PseudoRaNdom (PRN) Fig. 1. Example of sine-BOC(1, 1) modulation. CDMA code and the modulated data, [20]: +∞ SF x(t) = sBOC (t) bn ck,n δ(t − nT − kTc ) sine-BOC(1,1) modulations, we have N B = 1 and n=−∞ k=1 NB = 2, respectively. An example of how a PRN (1) waveform is modulated via sine-BOC(1, 1) (i.e. the where is the convolution operator, b n is the n-th signal sBOC (t) for NB = 2) is shown in Fig. 1. complex data symbol (in case of a pilot channel, it is Then, the signal x(t) is modulated onto the carrier equal to 1), T is the symbol period, c k,n is the k−th frequency fc and its passband form is given by chip corresponding to the n−th symbol, T c is the chip period, SF is the spreading factor (SF = Tsym /Tc ) g(t) = Eb x(t)e−j2πfc t , (3) and δ(t) is the Dirac pulse. The signal s BOC (t) stands for both BPSK and sine-BOC modulated signals, and where Eb is the data bit energy and {·} represents it can be expressed as in Eq. (2) [20] (for cosine-BOC the real part. The signal g(t) is typically transmitted modulation, the expression of s BOC (t) is also given over a multipath static or multipath fading channel in [20]): where all interference sources (except for the mul- tipaths) are lumped into a single additive Gaussian NB πt sBOC (t) = ± sin , 0 ≤ t ≤ Tc noise term v(t): Tc NB −1 L = pTB (t) (−1)i δ(t − iTB ) (2) rx (t) = αl g(t − τl )e−j(2πfD t+φl ) + v(t), (4) i=0 l=1 where pTB (t) is the pulse shaping ﬁlter applied to where rx (t) is the received signal, α l , τl and φl are pulses of duration TB = Tc /NB . For instance, the amplitude, the code delay and the phase offset if inﬁnite bandwidth is assumed, p TB (t) will be a of the l-th path, respectively, L is the number of rectangular pulse of unit amplitude if 0 ≤ t ≤ TB channel paths, f D is the Doppler shift introduced by and 0 otherwise. Here, N B is a modulation-related the channel, and v(t) is the additive Gaussian noise parameter, also called the BOC-modulation order of zero mean and double-sided power spectral density [20]. For example, for the most encountered GNSS N0 . modulations, namely BPSK and sine-BOC(1,1) mod- The received signal r x (t) is then down-converted ulations, we have N B = 1 and NB = 2, respec- to a low IF frequency or to baseband using a conven- tively. For instance, if inﬁnite bandwidth is assumed, tional I/Q demodulator, followed by low-pass ﬁltering pTB (t) will be a rectangular pulse of unit amplitude for rejecting the higher frequency components. In if 0 ≤ t ≤ TB and 0 otherwise. Here, N B is a our paper, we assume without loss of generality that modulation-related parameter, also called the BOC- the signal has been down-converted to baseband (the modulation order [20]. For example, for the most IF frequency can be then modeled as an additional encountered GNSS modulations, namely BPSK and Doppler shift). The baseband signal, r B (t), is now the 1 combination of the ﬁltered in-phase (I) and quadra- BPSK 0.9 sine BOC(1,1) ture (Q) demodulator branches: Averaged squared correlation 0.8 rB (t) = rIB (t) + jrQB (t) + vI,Q (t) 0.7 0.6 L = Eb αl x(t − τl )[cos(2πfD t + φl ) 0.5 l=1 0.4 + j sin(2πfD t + φl )] + vI,Q (t) 0.3 L 0.2 j(2πfD t+φl ) = Eb αl x(t − τl )e 0.1 l=1 0 −3 −2 −1 0 1 2 3 + vI,Q (t) (5) Delay error [chips] where fD is the Doppler shift to be removed during Fig. 2. Example of averaged squared correlation func- the acquisition stage and the new noise term v I,Q (t) tion with pulse shaping for BPSK and sine-BOC(1,1) can be shown to have the same power as v(t). modulation in single-path channel. After the signal is acquired, ﬁne code synchro- nization is required for successfully despreading the signal. Both acquisition and delay tracking stages correlation function can be written as (i.e., code synchronization) are usually based on 1 c N the code epoch correlation between the incoming R(τk ) = Rm (τk ) (8) down-converted signal and the reference modulated Nc m=1 PRN code (xref ), with a certain candidate Doppler 1 2 R(τk ) = R(τk ) (9) frequency fD , delay τ and phase offset φ given by Nnc N nc where Nc is the coherent integration time (expressed +∞ SF in code epochs or ms for GPS/Galileo signals) and xref (τ , fD , φ) = sBOC (t − τ ) bn Nnc is the non-coherent integration time, expressed n=−∞ k=1 in blocks of length Nc ms (note that the subscript m has been dropped in Eqs. (8) and (9) for clarity rea- × ck,n δ(t − nT − kTc − τ ) sons). Examples of the averaged squared correlation function for BPSK and sine-BOC(1,1) modulation for × ej(2πfD t+φ) (6) one- and two- path channel, no noise, zero residual Doppler error, no phase offset and unit bit energy are In order to generate xref (·) we ﬁrst assume a certain shown in Fig. 2. Doppler shift estimate fD (produced in the acquisi- tion stage), then the code delay ( τ ) and the phase offset (φ) of the LOS signal are estimated by cross- III. PROBLEM FORMULATION correlating rB (t)e−j(2πfD t) with x(t − τk ) where τk The target is to estimate jointly the code delay belongs to a candidate region of delays. The cross- and the carrier phase offset of the LOS signal. If we correlation function is given by substitute rB (t) into Eq. (7), after some manipula- tions, we can write the output of the crosscorrelation 1 mT function yR as Rm (τk ) = E r(t)e−j2πfD t x(t − τk )dt T (m−1)T L (7) yR = Eb αl Rm (τi , τj )e−j(2πΔfD t+φl ) + v(t), where m is the code epoch index and E(·) is the l=1 expectation operation, with respect to the PRN code. (10) In order to reduce the noise level, both coherent where ΔfD is a small residual error that is equal to (R(τk )) and non-coherent (R(τ k )) integration may fD − fD and resulted from the Doppler estimation be used. The averaged coherent and non-coherent during the acquisition stage and R m (τi , τj ) is the ideal auto-correlation function of the modulated code solution of the so-called Least Squares (LS) problem given by is 1 mT wLS = (HT H)−1 HT y (14) Rm (τi , τj ) = E x(t − τi )x(t − τj ) , T (m−1)T (11) Another approach is to minimize the mean square where τi ,τj belong in the range of possible code error (MMSE) and its solution is given by delays. It is possible to rewrite Eq. (10) into a system −1 of linear equations [21]: wM M SE = σ 2 I + HT H HT y, (15) ⎡ ⎤ ⎡ ⎤⎡ ⎤ y0 h0,0 ··· h0,dmax w0 ⎢ . ⎥ = ⎢ . .. . ⎥⎢ . ⎥ where σ 2 is the estimated noise variance and letter T ⎣ . ⎦ . ⎣ . . . . . ⎦⎣ . ⎦ . is used to denote matrix transposition. ydmax hdmax ,0 · · · hdmax ,dmax wdmax ⎡ ⎤ A third approach, namely Projection Onto Convex v0 Sets (POCS), is an iterative constrained deconvolution ⎢ . ⎥ + ⎣ . ⎦, . (12) approach which was originally proposed in [14], vdmax [16] for delay estimation in Rake receivers. More precisely, POCS ﬁnds a solution that complies with a where yi is the complex correlation output at code de- predeﬁned set of constraints. Each constraint is used lay τi computed by Eq. (7) for i = 0 till the maximum to form a closed convex set and the estimate that delay spread of the channel, d max = τmax Ts , where satisﬁes them all is the POCS solution. For example, Ts is the sampling period. The d max × dmax matrix an estimate for w is found via H is the pulse shape deconvolution matrix, each √ element, hi,j , of which is equal to Eb Rm (τi , τj ) wP OCS ∗ = Pw, (16) (note that H is a real Toeplitz matrix) and the vector v contains the complex AWGN noise terms of the where P is the operation of projecting the solution w, despreaded signal. The elements of the unknown onto the convex set C . When the constraint is applied vector w have the following interpretation: ideally, on the variance of the estimation error (i.e., on the if a path is present at delay τ i , then wi should be variance of y − wP OCS ∗ ) then the convex set can be ai ejφi , else wi = 0. Now, the target is to estimate deﬁned as [13], [14] the non-zero elements of w; the positions of which indicate the path delays. Thus, we have the problem C = {f : y − Hf 2 ≤ β}, (17) of solving a linear system of equations, where the path positions, amplitudes and phases are changing where β is a scalar bound that is a function of the in time. Moreover, the phase offsets can be computed noise variance after integration. The traditional POCS by using an appropriate phase discriminator function. solution at the (k)-th iteration can be written as [22]: In the following section, we describe some methods that can be applied for solving the above-mentioned 1 −1 (k) (k−1) linear problem and for resolving the multiple paths. wP OCS = wP OCS + I + HT H HT λ (k−1) IV. MULTIPATH DECOMPOSITION × y − HwP OCS (18) Now that we have formulated our problem into a where λ is the Lagrange multiplier associated with system of linear equations, we can consider some the variance of the residual constraint and I is the methods for estimating the unknown vector w. Let unity matrix. The optimal λ is found when the noise us ﬁrst write Eq. (12) into compact form as variance is known or when it is a-priori estimated. y = Hw + v (13) Starting with an initial estimate, the algorithm con- verges to a feasible solution by cyclically projecting One well-known approach for dealing with such onto the constraint sets [13]. In [21], a modiﬁed linear models is to minimize the squared difference POCS was used for path acquisition in WCDMA between the data (known vector y) and Hw. The systems with promising results. IV-A. Proposed POCS with R(τk ) denoted in Eq. (9), are denoted by R k We now describe how POCS can be adjusted to and that they are taken at sampling instants τ k , ﬁt our problem. As it was mentioned earlier, POCS k = 1, 2, . . . , then:ł utilizes a set of constraints with the aim of reducing the estimation error. The ﬁrst constraint employed by LCR(lj ) = card k|(Rk ≤ lj ∧ Rk+1 > lj ) POCS is the one related to the range of possible delays. In particular, in each iteration k, only the (Rk+1 ≤ lj ∧ Rk > lj ) , (22) delay estimates that are located within a window of ±γ chips around the position of the estimate with the where card is the cardinal of a set and ∧ is ’and’ maximum magnitude are considered. The delay range operator. The potential of LCR information in GNSS constraint P1 is deﬁned as context was noted in [23], where it was found that zi , if i ∈ [˘ − γ τ ˘ τ + γ] it can be used as a reliable indoor/outdoor CNR P1 : zi = (19) 0, else identiﬁer. However, the LCR information can be also used within the code tracking context for thresholding where zi is the i-th element of a vector z, γ is a pre- the Acf [24]. Here, we also apply the LCR function deﬁned constant (here it depends on the modulation on the normalized non-coherent Acf in order to ˘ index NB ) and τ is given by compute ξ ; however, the threshold is applied on the τ = arg max z ˘ (20) vector w(k,nc) at each iteration. The reason for using norm τ the LCR information to deﬁne the second POCS Imposing such a constraint ensures that POCS is constraint is that we can discard the noisy estimates. applied on the area where most of the signal power More precisely, we have found that at lower levels, is present under the assumption that the signal dom- where the majority of noise ”spikes” are present, the inates noise. The second constraint, P 2 , has to do LCR is larger than in the higher Acf levels (e.g. above with the magnitude of the estimates. More precisely, noise ﬂoor) [23], [24]. Consequently, the level that we would like to keep only the estimates z i which corresponds to the maximum LCR will reveal also are larger than a threshold ξ : the level in which Acf is the most noisy and thus we (k,nc) zi , if zi ≥ ξ can use it to reject the noisy estimates in wP OCS . For P2 : zi = (21) 0, else brevity, we have omitted the more detailed description of the LCR concept; interested readers may refer We notice that P1 and P2 differ from the P constraint in that they are applied after the adaptation step has taken place, i.e., they are not used for forming the Algorithm 1 POCS Algorithm for multipath decom- iterative POCS solution as it is done in the original position POCS. Now, we can describe the steps of the proposed 1. For nc = 1 to Nnc (0,nc) algorithm. First, we see that POCS is applied for each 2. Set wP OCS = y(nc) non-coherent block separately and that for each non- 3. For k = 1 to Kiter −1 coherent block we use y (nc) as the initial state for (k,nc) (k−1,nc) wP OCS = wP OCS + σ 2 I + HT H wP OCS , where nc is the nc-th non-coherent block. (k−1,nc) Then for each iteration, the adaptation rule is applied. × HT y(nc) − HwP OCS (k,nc) (k,nc) Notice that the constant term 1/λ which appears in w(k,nc) = |wP OCS |2 / max |wP OCS |2 norm the original form of POCS algorithm has been now 4. Apply P1 and P2 constraints substituted by the dynamically estimated noise vari- (k,nc) wnorm (i) of w(k,nc) norm ance. Another important element of the algorithm is (k,nc) the choice of the threshold ξ . Its computation is based 5. If wnorm = 0 k,nc k,nc on the Level Crossing Rate (LCR) of the normalized wP OCS (i) = wP OCS (i) non-coherent Acf and which is deﬁned as the number Else k,nc of crossings (both from below and from above) at wP OCS (i) = 0 level lj . Assuming that the time samples of the normalized correlation function R(τk )/ max(R(τk )) ˘ to [23], [24]. Given the Acf level q that results in After the LOS delay has been determined, the LOS maximum LCR, the POCS threshold is computed as phase offset for each algorithm ( φalgo ) is computed as ξ = min {[˘ + 0.5 q 0.99]} , for NB = 1, 2 (23) φalgo = F −1 {walgo (τalgo )}, {walgo (τalgo )} (30) ˘ q = arg max LCR(q) (24) q where F is a properly chosen phase discriminator The constant of 0.5 has not been added to q arbitrar- ˘ function. When the data bits have been already ily. In particular, we performed a set of simulations in removed or when we deal with dataless channels which we tested the performance of POCS algorithm (e.g., Galileo OS signals), the inverse double arctan for different values, starting from 0.1 and up to 0.8 discriminator function (F = tan−1 ) is the most 2 with a step of 0.1 and we found the when we add appropriate due to its wide linear range of operation 0.5 to q we get the best performance. The results of ˘ (i.e. it can detect phases that vary between −π to π ) our experiments are not included due to the limited and better noise resistance [10], [11], [25]. space. V. SIMULATION RESULTS IV-B. LOS delay and phase estimation The main target of the simulations is to compare the performance of the modiﬁed POCS algorithm Here, we describe how the LOS signal is detected (POCS1 and POCS2 versions) with the LS and and also how the code delay and phase offset are MMSE methods. Regarding POCS algorithm, we computed. In all algorithms, we apply non-coherent Kiter ,nc empirically found via simulations that three iterations integration on the returned complex vector wP OCS (Kiter = 3) were adequate for satisfactory perfor- in order to further reduce the noise. Then, the LOS mance, but we omit here the experimental results for signal, for both LS and MMSE, is the estimate the sake of clarity. Moreover, γ was set to 2 chips with the maximum magnitude. The LOS code delay when NB = 1 and to 1 chip when N B = 1. Moreover, estimate for LS and MMSE algorithms are given by the noise variance σ 2 utilized by MMSE and POCS 1 (K ,nc) 2 was estimated from the non-coherent Acf by using τLS = arg max wLSiter (25) τ Nnc Nnc ﬁrst order statistics. 1 In addition, we have included two more cases (K ,nc) 2 τM M SE = arg max wM M SE iter (26) for reference purposes. In the ﬁrst one, the LOS τ Nnc Nnc code delay is simply estimated by the position of Concerning the POCS algorithm, we tested two dif- max(Acf ) in the time axis and then the LOS phase is ferent cases. In the ﬁrst case, the LOS signal is computed by applying the inverse double arctangent detected as in LS and MMSE (i.e., based on the discriminator function on the coherent Acf at the maximum magnitude). In the second case, the LOS estimated LOS delay. In the second scenario, we signal is the one that corresponds to the ﬁrst estimate would like to compare the estimators performance of the non-coherent estimated vector ( wnew ) that with the ideal code delay estimator, meaning when P OCS is above a threshold ξ . The estimated LOS delays for the estimated LOS delay is the true one and the LOS the ﬁrst (P OCS1) and second (P OCS2) case are phase is computed using the same phase discriminator given by as in the rest of the algorithms (i.e. double arctan). Of course for the case of true delay the error is zero 1 (K ,nc) 2 τP OCS1 = arg max wP OCS iter (27) as expected, but we are also interested in looking at τ Nnc Nnc the estimated phase when it is computed (based on τP OCS2 = wnew (1), P OCS (28) the coherent Acf as well) using the true delay. We performed simulations for both sine-BOC(1, 1) where wnew is given by P OCS and BPSK modulated signals. First, we give a detailed ⎧ ⎫ ⎨ ⎬ description of the channel proﬁles and simulation 1 Kiter ,nc 2 wnew = wP OCS (i) : Kiter wP OCS (i) ≥ξ parameters we used. Second, we present the results P OCS ⎩ Nnc ⎭ and analyze the estimators performance in terms of Nnc (29) Root Mean Square Error (RMSE). V-A. Simulation parameters being 0, is not visible. When LOS phase is concerned, Regarding the channel setup, we considered only the RMSE derived based on max(Acf ) and LOS true the case of static multipath channel because we delay are the same, POCS1 and POCS2 also perform wanted to see the maximum achievable performance, similarly, and LS follows last (see bottom of Fig. 3). and because modeling the phases in fading chan- One may wonder why, for example at CN R = nels introduces additional errors. More precisely, the 30 dBHz, when we estimate LOS delay based on model we used employs a decaying Power Delay max(Acf ) we have RM SE ≈ 8 meters but when Proﬁle (PDP), meaning that αl = α1 e−ζP DP (τl −τ1 ) , we estimate the LOS phase at same CNR, RMSE where α1 is the average amplitude of the 1-st path is the same with the case where LOS phase is and ζP DP is the power decaying proﬁle coefﬁcient, computed based on true delay. In order to get a better (assumed in the simulations to be equal to 0.09 when understanding on this observation, we can look at the the path delays are expressed in samples). top of Fig. 4 where an instance of the LOS absolute The number of channel paths varied between 1 and carrier phase error versus the code delay error is 4, while the carrier phase offset introduced in the l-th depicted for CN R = 30 dBHz. From the ﬁgure we was assumed to be uniformly distributed between −π notice that the phase error is not the minimum when and π . The time separation between successive paths the delay error is but it rather ﬂuctuates around the τl+1,t − τl,t , at any time instance t, was assumed to zero delay error. We have also encountered similar be uniformly distributed between 0 and 0.35 chips behavior of the phase error at very high CNR values, for BOC modulated signals or between 0 and 1 (i.e. thus the presence of the noise cannot fully explain between 0 and half the width of the main lobe of the above mentioned situation. An in-depth research the Acf). The latter parameter values indicate closely- on the relation between the delay and carrier phase spaced paths which are typical in indoor and densely error is not in the scope of this paper and belongs to populated urban scenarios. The oversampling factor our future plans. was equal to 10 for both BOC(1, 1) (i.e., the number When the number of paths is increased to 2, we see of samples per BOC interval) and BPSK modulation. the overall performance of the estimators deteriorates The processing of Acf is done in a window (S ) of as expected (see Fig. 5). However, now POCS2 8 chips length with a resolution of 0.1 and 0.05 has the best performance for CN R ≥ 40 dBHz chips in the case of BPSK and BOC(1, 1) modulation, both for delay and phase estimation. Also, we see respectively (in BOC signals we need ﬁner resolution that although the RMSE of LOS phase decreases as due to the more complicated shape of Acf). The CNR increases for LS, MMSE and POCS, it remains coherent integration time (Nc ) was set to 20 ms, while almost constant when the phase is estimated based on the non-coherent integration (Nnc ) was performed in max(Acf ) or on the true delay. This saturated behav- 2 blocks of Nc length. The simulations are based ior also implies the need for sophisticated algorithms on N rand = 5000 random realizations (of channel even when the CNR conditions are very good. When and signals), each realization having an observation we have 3 or 4 paths present, the performance of interval of Nc Nnc ms. POCS2 remains the best for both phase and delay estimation (see Figs. 6 and 7). In particular, when CNR is very high (e.g. above 55 dBHz) the RMSE V-B. Discussion for POCS2 reaches the zero level. We also notice, that We start our discussion ﬁrst on the results where when the number of paths is greater than 2, looking BPSK-modulated signal is used and then when BOC at the delay that corresponds to max(Acf ) leads to modulation is used. signiﬁcant errors even at very good CNR conditions. In the top of Fig. 3, we see the performance of the When BOC(1, 1) modulated signals are used, we estimators in single-path proﬁle. At lower CNR val- see that in the single path scenario, there is some ues (30 − 35 dBHz), the time delay that corresponds slight deterioration in the performance of the algo- to max(Acf ) seems to best represent the LOS delay. rithms in respect to RMSE delay when compared POCS1, POCS2 and MMSE reach RM SE = 0 for with the case of BPSK modulation (see Fig. 8). CN R ≥ 35 dBHz while LS seems to perform the When the number of paths is increased, we see that worst. We note, that due to the logarithmic scale used the performance of POCS2 is signiﬁcantly better for for plotting, the curve for the RMSE of true delay, middle to higher CNRs when compared to the rest LOS code delay for 1−path static channel, BPSK signal Absolute LOS carrier phase error vs. code delay error at CNR=30 dBHz 3 1 10 10 LS MMSE POCS1 POCS2 2 0 Absolute phase error [rads] 10 based on max(Acf) 10 based on true τ1 RMSE [meters] 1 −1 10 10 0 −2 10 10 −1 −3 10 10 30 35 40 45 −20 −10 0 10 20 30 CNR [dBHz] Delay error [samples] LOS carrier phase for 1−path static channel, BPSK signal Real and imaginary part of coherent Acf at CNR=30 dBHz 1 10 0.1 0 −0.1 0 −0.2 10 Coherent Acf RMSE [rads] −0.3 −0.4 real LS −0.5 imag −1 10 MMSE POCS1 −0.6 POCS2 −0.7 based on max(Acf) based on true τ1 −0.8 −2 10 −0.9 30 35 40 45 50 55 60 −20 −10 0 10 20 30 CNR [dBHz] Delay error [samples] Fig. 3. RMSE of LOS code delay and carrier phase Fig. 4. Absolute LOS phase error vs. delay error offset vs. CNR for 1-path static channel, BPSK (top). Real and imaginary parts of coherent Acf, for modulation and inﬁnite bandwidth. single path static channel at CN R = 30 dBHz. and for both delay and phase estimation (see Figs. is the most resistant in closely-spaced path environ- 9 and 10). We also notice that we have obtained ments when good CNR conditions occur. In the case similar results for a 4 path static channel but due of single-path scenario, the proposed algorithm seems to the limited space they are not included here. to be affected more by the noise when it is used for LOS carrier phase estimation. To overcome this VI. CONCLUSIONS AND FUTURE WORK limitation, our future plans include the incorporation In this paper, we proposed a modiﬁed POCS al- of a mechanism that would allow to differentiate the gorithm for estimating jointly the code delay and the algorithm depending on whether single or multipath carrier phase of the received LOS signal. The pro- channel is detected. In particular, the method de- posed algorithm has been optimized for both the new scribed in [17] has been shown to have very good L1 Open Service and GPS signals and it is compared performance in terms of detection probability, there- with other state-of-art methods (namely Least Square fore it is an excellent choice for further optimizing and Minimum Mean Square approaches) that are ap- the proposed algorithm. Other future plans include plicable in problems where multipath decomposition the assessment of the modiﬁed POCS in the case is required. where fading channel is assumed and when MBOC We tested the performance of the proposed algo- modulation is used. rithm in terms of Root Mean Square Error versus CNR for various multipath proﬁles and static channel. VII. ACKNOWLEDGMENTS The results were based on Monte Carlo simulations This work was carried out in the project ”Fu- and showed that the proposed alternation of POCS ture GNSS Applications and Techniques (FUGAT)” LOS code delay for 2−path static channel, BPSK signal LOS code delay for 3−path static channel, BPSK signal 3 3 10 10 LS MMSE POCS1 POCS2 2 2 10 based on max(Acf) 10 based on true τ 1 RMSE [meters] RMSE [meters] 1 1 10 10 LS MMSE 0 10 0 10 POCS1 POCS2 based on max(Acf) based on true τ 1 −1 −1 10 10 30 35 40 45 50 55 60 30 35 40 45 50 55 60 CNR [dBHz] CNR [dBHz] LOS carrier phase for 2−path static channel, BPSK signal LOS carrier phase for 3−path static channel, BPSK signal RMSE [rads] RMSE [rads] 0 10 0 10 LS LS MMSE MMSE POCS1 POCS1 POCS2 POCS2 based on max(Acf) based on max(Acf) based on true τ1 based on true τ1 30 35 40 45 50 55 60 30 35 40 45 50 55 60 CNR [dBHz] CNR [dBHz] Fig. 5. RMSE of LOS code delay and carrier phase Fig. 6. 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