A DECONVOLUTION ALGORITHM FOR ESTIMATING JOINTLY .pdf

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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. RMSE of LOS code delay and carrier phase
offset vs. CNR for 2-path static channel.                                                      offset vs. CNR for 3-path static channel.


funded by the Finnish Funding Agency for Technol-                                                                    multipath performance,” IEEE Aerospace Con-
ogy and Innovation (Tekes). The work of A. H. Sayed                                                                  ference, vol. 3, p. 1686, March 2004.
was supported in part by NSF grants ECS-0601266                                                 [6]                  S. K. Kalyanaraman, M. S. Braasch, and J. M.
and ECS-0725441.                                                                                                     Kelly, “Code tracking architecture influence on
                                                                                                                     GPS carrier multipath,” IEEE Transactions on
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                                LOS code delay for 4−path static channel, BPSK signal                                           LOS code delay for 1−path static channel, BOC signal
                          3                                                                                               3
                     10                                                                                              10
                                                                                                                                                                     LS
                                                                                                                                                                     MMSE
                                                                                                                                                                     POCS1
                                                                                                                                                                     POCS2
                                                                                                                                                                     based on max(Acf)
                     10
                          2
                                                                                                                     10
                                                                                                                          2                                          based on true τ
                                                                                                                                                                                    1




                                                                                                     RMSE [meters]
     RMSE [meters]




                          1                                                                                               1
                     10           LS                                                                                 10
                                  MMSE
                                  POCS1
                                  POCS2
                                  based on max(Acf)
                                  based on true τ
                          0                      1                                                                        0
                     10                                                                                              10
                          30         35        40       45         50         55          60                              30          35            40        45           50            55
                                                     CNR [dBHz]                                                                                      CNR [dBHz]

                               LOS carrier phase for 4−path static channel, BPSK signal                                        LOS carrier phase for 1−path static channel, BOC signal

                                                                                                                                                                     LS
                                                                                                                                                                     MMSE
                                                                                                                                                                     POCS1
                                                                                                                                                                     POCS2
                                                                                                                                                                     based on max(Acf)
                                                                                                                      0                                              based on true τ
                                                                                                                10                                                                  1




                                                                                                RMSE [rads]
 RMSE [rads]




                      0
                 10


                                  LS
                                  MMSE
                                  POCS1
                                  POCS2
                                  based on max(Acf)
                                  based on true τ
                                                 1
                          30         35        40       45         50         55          60                              30        35         40       45         50        55          60
                                                     CNR [dBHz]                                                                                      CNR [dBHz]




Fig. 7. RMSE of LOS code delay and carrier phase                                               Fig. 8. RMSE of LOS code delay and carrier phase
offset vs. CNR for 4-path static channel.                                                      offset vs. CNR for 1-path static channel.


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                             LOS code delay for 2−path static channel, BOC signal                                         LOS code delay for 3−path static channel, BOC signal
                   3                                                                                         3
                  10                                                                                        10




                   2                                                                                         2
                  10                                                                                        10




                                                                                            RMSE [meters]
  RMSE [meters]




                   1                                                                                         1
                  10                                                                                        10
                                                                                                                                  LS
                                LS                                                                                                MMSE
                                MMSE                                                                                              POCS1
                   0                                                                                         0                    POCS2
                  10            POCS1                                                                       10
                                POCS2                                                                                             based on max(Acf)
                                based on max(Acf)                                                                                 based on true τ1
                                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, BOC signal                                       LOS carrier phase for 3−path static channel, BOC signal




                                                                                            RMSE [rads]
  RMSE [rads]




                   0                                                                                             0
                  10                                                                                        10

                                                                                                                                 LS
                                LS
                                                                                                                                 MMSE
                                MMSE
                                POCS1
                                                                                                                                 POCS1
                                POCS2                                                                                            POCS2
                                based on max(Acf)                                                                                based on max(Acf)
                                based on true τ1                                                                                 based on true τ
                                                                                                                                                1


                       30        35        40          45        50        55         60                             30        35         40      45          50        55          60
                                                    CNR [dBHz]                                                                                 CNR [dBHz]




Fig. 9. RMSE of LOS code delay and carrier phase                                           Fig. 10. RMSE of LOS code delay and carrier phase
offset vs. CNR for 2-path static channel.                                                  offset vs. CNR for 3-path static channel.


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