Maximum Likelihood Delay Estimation in a Navigation ...

Maximum Likelihood Delay Estimation in a Navigation Receiver for Aeronautical Applications Maximum Likelihood Verz¨ogerungssch¨atzung in einem Navigationsempf¨anger f¨ur Luftfahrtanwendungen a German

F. Antreich a,∗ , J. A. Nossek b , W. Utschick c

Aerospace Center (DLR), Institute for Communications and Navigation, 82234 Wessling, Germany for Circuit Theory and Signal Processing, Munich University of Technology (TUM), 80333 Munich, Germany c Associate Institute for Signal Processing, Munich University of Technology (TUM), 80333 Munich, Germany

b Institute

Abstract The potential of the SAGE (Space Alternating Generalized Expectation Maximization) algorithm for navigation systems in order to distinguish the line-of-sight signal (LOSS) is to be considered. The SAGE algorithm is a low-complexity generalization of the EM (Expectation-Maximization) algorithm, which iteratively approximates the maximum likelihood estimator (MLE) and has been successfully applied for parameter estimation (relative delay, incident azimuth, incident elevation, Doppler frequency, and complex amplitude of impinging waves) in mobile radio environments. This study discusses receivers using a single antenna or an antenna array. Whereas we estimate the complex amplitudes, relative delays, and Doppler frequencies of the impinging waves, in the latter additionally the spatial signature of the impinging waves (incident azimuth and elevation) is estimated. The results of the performed computer simulations and discussion indicate that the SAGE algorithm has the potential to be a very powerful high resolution method to successfully estimate parameters of impinging waves for navigation systems. SAGE has proven to be a promising method to combat multipath for aircraft navigation applications due to its good performance, fast convergence, and low complexity. Zusammenfassung In dieser Arbeit wird der SAGE (Space Alternating Generalized Expectation Maximization) Algorithmus im Hinblick auf Verz¨ ogerungssch¨ atzung des direkten Ausbreitungspfades (Line-of-sight signal) f¨ ur Satellitennavigationssysteme untersucht. Der SAGE Algorithmus ist eine komplexit¨ atsreduzierte Generalisierung des EM (Expectation Maximization) Algorithmus, der den Maximum Likelihood Sch¨ atzer (MLE) zu approximieren sucht und der schon erfolgreich dazu eingesetzt worden ist Signalparameter (relative Verzgerung, Azimut, Elevation, Dopplerfrequenz und komplexe Amplitude der einfallenden Wellenfronten) in typischen Szenarien f¨ ur Mobilfunkanwendungen zu sch¨ atzen. Diese Studie untersucht Empf¨ anger mit einer Einzelantenne und mit einer Gruppenantenne. Hierbei werden f¨ ur den Einzelantennenfall die relativen Verz¨ ogerungen, die komplexen Amplituden und die Dopplerfrequenzen der einfallenden Wellenfronten gesch¨ atzt, im Falle einer Gruppenantenne wird noch zus¨ atzlich die r¨ aumliche Signatur (einfallender Azimut und Elevation) der einzelnen Wellenfronten gesch¨ atzt. Die Ergebnisse der durchgef¨ uhrten Computersimulationen und der Untersuchung lassen darauf schließen, dass der SAGE Algorithmus ein leistungsstarkes und hochaufl¨ osendes Verfahren ist, das in der Lage ist erfolgreich Signalparameter von einfallenden Wellenfronten f¨ ur die Anwendung in der Satellitennavigation zu sch¨ atzen. SAGE hat sich aufgrund seiner sehr hohen Leistungsf¨ ahigkeit, guten Konvergenzeigenschaften und niedrigen Rechenaufwandes, als viel versprechende Methode zur Unterdr¨ uckung von Mehrwegesignalen in Satellitennavigationsempf¨ angern f¨ ur Luftfahrtanwendungen erwiesen. Key words: Delay estimation, multipath mitigation, satellite navigation, maximum likelihood

Preprint submitted to Elsevier

12 February 2009

Schl¨ usselw¨ orter : Verz¨ ogerungssch¨ atzung, Mehrwegeunterdr¨ uckung, Satellitenavigation, Maximum Likelihood

tipath, where the scatterers are located close to the receiver and the extra path covered by the reflection is shorter than one chip period (about 300 m for a chip rate of 1.023 Mcps). This situation is referred to as coherent multipath. Thus, in order to perform accurate synchronization in the presence of multipath corrupted signals and especially when coherent multipath is present we follow the approach of obtaining the MLE for estimation problems of higher order. Therefore, signal parameters of a number of superimposed delayed replicas have to be estimated jointly. As this leads to a multi-dimensional non-linear optimization problem the reduction of the complexity of this problem is the most important issue to be solved in order to perform precise positioning in a navigation receiver. Several techniques have been proposed in the literature to solve the multipath problem in GNSS receivers, (see for instance [33] and [28]). Lately, interesting approaches like [23] and [26] have appeared. The first method applies the maximum likelihood principle to the delay estimation in the presence of multipath and unintentional interference in an antenna array receiver, whereas multipath and interference are modeled as colored noise. The latter develops efficient multipath mitigation techniques (with low-complexity) based on a Newton-type method in single antenna and array antenna navigation receivers. In both works, a connection is made between the multipath estimation problem in navigation systems and the same problem in communication systems; further work of the same authors has appeared more recently in [24], [27], and [7] respectively. In this paper, in order to cope with the synchronization problem induced by multipath signals in GNSS receivers, we explore the potential application of an iterative method estimating the parameters of all impinging wavefronts in white Gaussian noise which already was assessed for communication systems: the SAGE algorithm (Space Alternating Generalized Expectation Maximization algorithm). However, a detailed study on the capabilities of SAGE for precise time-delay estimation has not been performed yet. The SAGE algorithm [9] is based on the Expectation Maximization (EM) algorithm [5], [17], which facilitates optimizing maximum likelihood cost functions that arise in statistical estimation problems, but with improvements regarding the trade-off between convergence rate and complexity. The potential of the SAGE algorithm in communication systems has been proven in [10] where it is shown that SAGE is a promising candidate for channel estimation in direct-sequence code division multiple access systems (DS-CDMA). The intention of this paper is to serve as a preliminary study on the application of SAGE for aircraft navigation based on [2]. The performance of the algorithm is assessed by computer simulations under severe multipath conditions.

1. Introduction Nowadays, both the increasing sizes and velocities of aircrafts and the increase of air traffic lead to stricter requirements for any aircraft navigation aid, such as DME (Distance Measuring Equipment) or ILS (Instrument Landing System), with regards to safety and traffic flow (see [11] and [15] for details on aircraft navigation systems). The complexity of the infrastructure and its costs would rise significantly in order to still meet the requirements for aviation navigation concerning accuracy, integrity, availability, and continuity and therefore to guarantee safety and efficiency of air traffic. Thus, Global Navigation Satellite Systems (GNSS) as GPS or the upcoming European Galileo will play a very important role in future aviation navigation; not only in critical situations like approach and landing where very precise positioning is required, but also in the airport ground control of taxiways, holding areas, and transition areas, which can affect the efficiency of the airport operation. The quality of the data provided by a GNSS receiver depends largely on the synchronization error with the signal transmitted by the GNSS satellite (navigation signal), that is, on the accuracy in the propagation delay estimation of the direct signal (line-of-sight signal, LOSS). The synchronization of a navigation signal is usually performed by a Delay Lock Loop (DLL), which basically implements an approximation of a maximum likelihood estimator (MLE). The problem which arises is that the order of this estimator (the DLL), which is given by the number of impinging wavefronts is chosen according to the assumption that only the LOSS is present. This means that this estimator tries to estimate the relative propagation delay of only one signal replica. In case the LOSS is corrupted by several superimposed delayed replicas this estimator becomes biased, because of the change of the order of the estimation problem at hand. Thus, multipath can introduce a bias up to a hundred meters of range error when using a standard 1-chip wide DLL [19]. In order to reduce this bias, several methods have been proposed such as the Narrow Correlator [32] or the Double Delta Correlator [12]. Although they achieve much better results than a conventional 1-chip wide DLL in terms of multipath-caused timing bias, their performance is severely degraded in case short delayed multipath is present as for single-antenna receivers their ability to distinguish the LOSS from the reflections is limited by the intrinsic time resolution the signal bandwidth imposes [8]. Relatively short delays are just the case of real-life mul∗ Corresponding author. phone: +49 8153-28-2119 fax: +49 815328-2328 Email addresses: [email protected] (F. Antreich), [email protected] (J. A. Nossek), [email protected] (W. Utschick).

2

z

This paper is organized as follows. The signal model is outlined in Section 2. A short introduction to maximum likelihood estimation of superimposed signals is given in Section 3. The SAGE algorithm is described in Section 4, and its initialization is discussed in Section 5. In Section 6 we show the results of our simulations. Finally, in Section 7 we discuss our conclusions.

ϑ` y Array

2. Data Model

φ`

We assume that L narrowband planar wavefronts of wavelength λ, complex amplitude γ` , delay τ` , Doppler frequency ν` , incident azimuth φ` , and incident elevation ϑ` , 1 ≤ ` ≤ L are impinging on an array of M , 1 ≤ m ≤ M isotropic sensor elements. The transmission medium is considered linear such that the noise corrupted baseband signal at the antenna output y(t) ∈ CM ×1 can be modelled as a superposition of L wavefronts generated by L point sources and additional complex white Gaussian noise n(t) ∈ CM ×1 , N (0, σn2 ). The L point sources are located far from the array such that the direction of propagation is nearly equal at each sensor and the wavefronts are approximately planar (far-field approximation). Thus, the propagation field within the array aperture consists of plane waves and we can write y(t) =

L X

s` (t) + n(t)

x Fig. 1. Definitions of Azimuth (−180◦ < φ` ≤ 180◦ ) and Elevation (0◦ ≤ ϑ` ≤ 90◦ ).

fF ym (t)

(1)

(2)

Here, a` (φ` (t), ϑ` (t)) denotes the steering vector of an uniform rectangular array (URA) of M = Mx · My isotropic sensors with Mx elements being displaced by ∆x along the x-axis and My elements displaced by ∆y along the y-axis as depicted in Fig. 1. Assuming ∆x = ∆y = λ2 (λ is the wavelength corresponding to the carrier frequency), i. e. a` (φ` (t), ϑ` (t)) = [1, . . . , e jπ (mx u` + my v` ) ,

0 ≤ mx ≤ Mx − 1 and 0 ≤ my ≤ My − 1

(3) (4)

where u` = cos φ` (t) cos ϑ` (t) and v` = sin φ` (t) cos ϑ` (t),

FR (z)

% ↓

ym [n]

The GPS L1 (Link 1) signal which also contains the GPS C/A code has a total one-sided bandwidth of approximately 15 MHz to accommodate the military P code signal of GPS with 10.23 MHz chip rate which is also modulated on the same carrier (1575.42 MHz) as the C/A code. Most of the signal power of the GPS C/A code is contained only in the one-sided bandwidth of B =1.023 MHz as the null-tonull bandwidth of the main lobe of the spectrum of the C/A code is equal to 2 · B [19]. In order to just use the socalled C/A band B, sufficient filtering of the received signal ym (t) is necessary. Here, ym (t) denotes the signal received by antenna m of the URA. Analog filters with a sharp cutoff can not be realized (or even closely approximated) with sufficient accuracy at reasonable complexity [16]. Thus, we apply the realization [16] and [18], which is shown in Fig. 2 in order to get a sufficient statistic ym [n] using a realizable analog filter F(f ) with bandwidth BF , while BF ≥ B as the analog filter has no sharp cutoff. After this pre-filtering with F(f ) the signal is sampled with a sampling frequency of fF = T%s = 2 % B ≥ 2 BF . Here, % denotes the oversampling ratio and Ts is the sampling period with respect to the Nyquist rate 1 Ts = 2 B. Then the sampled signal is filtered by a digital lowpass filter HL (z) with bandwidth BL = B and by a digital filter

where s` (t) is given by

. . . , e jπ ((Mx −1) u` + (My −1) v` ) ]T

HL (z)

Fig. 2. Receive Filtering

`=1

s` (t) = a` (φ` (t), ϑ` (t)) γ` (t) e j2πν` (t)t c(t − τ` (t)).

F(f )


FR e j2πf Ts =

(5)

and mx ∈ N and my ∈ N. Fig. 1 gives the definitions of azimuth and elevation for this URA. In Eq. (2) c(t − τ` (t)) denotes the pseudo-random-noise (PN) sequence with delay τ` (t). We apply Gold codes [19] as used for the GPS C/A code with code period T = 1 ms, 1023 chips per code period each with a time duration Tc = 977.52 ns and a rectangular chip shape.

1 |f | < B F(f )

(6)

which reverses the effect of F(f ) in the passband B. Finally, downsampling by the ratio % is performed. Thus, we get a sufficient statistic y[n] where the spatial observations of the URA are collected at N time instances, as y[n] = y(n · Ts ) with n = 1, 2, . . . , N . Generally, there is a trade-off between oversampling rate and analog pre-filtering complexity. Oversampling relaxes 3



the requirements on the analog pre-filter regarding a sharpcutoff. However, downsampling to the Nyquist rate has to be done afterwards in order to obtain a sufficient statistic with uncorrelated noise samples. Thus, all sharp-cutoff filtering is done by a discrete-time system, and only nominal continuous-time filtering is required. Since discrete-time FIR filters can have an exactly linear phase, it is possible using this oversampling and reverse filtering approach to implement antialiasing pre-filtering with virtually no phase distortion. The channel parameters are assumed constant during the observation interval TN = N ·Ts since the coherence time of the assumed channel is Tcoh ≥ TN . Collecting the samples of the observation interval leads to Y = [y[1], y[2], . . . , y[N ]] ∈ CM ×N , M ×N

N = [n[1], n[2], . . . , n[N ]] ∈ C

S` = [s` [1], s` [2], . . . , s` [N ]] ∈ C iT h T T , θ = θT 1 , θ2 , . . . , θL

(8) ,

T

θ ` = [γ` , τ` , ν` , φ` , ϑ` ] .

(9)

L X

(11)

S` (θ ` ) + N =

T

The problem at hand is to estimate θ ` = [γ` , τ` , ν` , φ` , ϑ` ] , ` = 1, 2, . . . , L. The estimation of L is not taken care of in this work. In the following we presume L is given. As we assume spatially and temporally uncorrelated elements in N, the covariance of the noise is R = σn2 ·I and the variance σn2 is considered to be known. Thus, the likelihood function for our signal model is given by the conditional probability density function (pdf)   1 kY − S(θ)k2F p(Y; θ) = . (18) exp − (πσn2 )M ·N σn2

(12)

Here, the matrix S(θ) ∈ CM ×N contains the superimposed PL impinging wavefronts `=1 S` (θ ` ), A ∈ CM ×L denotes the array steering matrix which includes the array steering vector a(φ` , ϑ` ) for each impinging wavefront, Γ ∈ CL×L is a diagonal matrix which comprises the complex amplitudes of the impinging wavefronts γ` in its diagonal, C ∈ RL×N contains the sampled PN sequence for each impinging wavefront c(τ` ) delayed by τ` , and D ∈ CL×N contains the sampled Doppler frequencies for each impinging wavefront  d(ν` ) = e j2πν` Ts , e j2πν` 2·Ts , . . . , T . . . e j2πν` n·Ts , . . . , e j2πν` N ·Ts .

Here, k · kF denotes the Frobenius norm of a matrix. The maximum likelihood estimator (MLE) is given by ˆ = arg max {p(Y; θ)} . θ θ

· · · a(φL , ϑL )] ∈ CM ×L ,

(19)

Estimation of θ is a computationally prohibitive task since there is no analytical solution for the global maximum, p(Y; θ) generally is not a concave function of θ, and θ and L usually have high dimension. Since the values for maximization of the complex amplitude γ` can be given in closed form as a function of the other parameters [10], the T computation of the MLE for θ ` = [γ` , τ` , ν` , φ` , ϑ` ] is a 4 · L-dimensional non-linear optimization procedure.

(13)

Thus, we get

A = [a(φ1 , ϑ1 ) a(φ2 , ϑ2 ) · · · a(φ` , ϑ` ) · · ·

(17)

3. Maximum Likelihood Estimation of Superimposed Signals

`=1

= A Γ (C D) + N.

(16)

whereas diag{γ` }L `=1 represents a diagonal matrix containing the values γ` , ` = 1, 2, . . . , L in its diagonal. In Eq. (12) denotes the Hadamard-Schur product (element-byelement multiplication).

(10)

Thus, the signal model can be written in matrix notation

Y = S(θ) + N =



   cT (τ )  2     ..     . , C=   T  c (τ` )      ..   .   T c (τL )   dT (ν1 )    dT (ν )  2     ..     .  , D=   dT (ν` )      ..   .   T d (νL )

(7)

,

M ×N

cT (τ1 )

(14)

4. The SAGE Algorithm Γ = diag{γ` }L `=1 ,

(15)

Whereas no closed form expression can be found for the MLE, a numerical approach employs either a grid search or 4

an iterative maximization of the likelihood function. In this work we try an iterative approach, the SAGE algorithm [9,10,6] . Instead of performing the above mentioned high dimensional non-linear optimization procedure directly, the SAGE algorithm gives a sequential approximation of the MLE given in Eq. (19) [6] as it performs a sequence of maximization steps in spaces of lower dimension and thereby reducing the complexity considerably. In fact the SAGE algorithm can be considered as a generalization of the well known Expectation-Maximization (EM) algorithm [5,17]. In the following we sketch the fundamental ideas of the SAGE algorithm. The basic concept of the SAGE algorithm is the hidden data space [9]. Instead of estimating the parameters of all waves θ in parallel in one iteration step as done by the EM algorithm, the SAGE algorithm estimates the parameters of each wave θ ` sequentially. Also, instead of estimating the complete θ ` , SAGE breaks down the multi-dimensional optimization problem into several smaller problems by conditioning sequentially on a subset of parameters θ S while keeping the parameters of the complement subset θ S¯ fixed. We choose the hidden data space as one noisy wave X` = S` + N` , where N` is white Gaussian noise with variance β` σn2 . This choice of the hidden data space leads to a fast convergence rate and low complexity due to sequential updating and one-dimensional optimization procedures. The stochastic mapping of thePhidden data space to the L observable signal is Y = X` + `0 =1 S`0 + N`0 . We choose

maximum spatial correlation. Various choices of the hidden data space could be made which would individually influence the convergence rate. Less informative hidden data spaces yield faster convergence, but more informative hidden data spaces yield an easier M-step [9]. Although the sequence of log-likelihood values form a monotonically nondecreasing sequence of parameter estimates, convergence to even a local maximum is not guaranteed. Therefore, the SAGE algorithm has to be initialized in a region which is close enough to a local (at best the global) maximum. Then the sequence of estimates will converge in norm to it [9].

5. Initialization of the SAGE Algorithm Several methods are proposed in [10] and [30] in order to perform an initialization of the SAGE algorithm . Since this work should preliminary assess the potential of the SAGE algorithm for navigation systems, we use a technique which commonly is referred to as successive interference cancelˆ` = lation. This method starts with the pre-initial setting θ T [0, 0, 0, 0, 0] for each ` = 1, 2, . . . , L. It successively performs a maximum search of correlation processes for the delay and Doppler frequency, and for azimuth and elevation:

`0 6=`

the sequence of parameter vector θ S (µ) (µ is the parameter update index) as θ S (1) = [τ1 ], θ S (2) = [φ1 ], θ S (3) = [ϑ1 ], θ S (4) = [ν1 ], θ S (5) = [γ1 ], θ S (6) = [τ2 ], θ S (7) = [φ2 ], . . . . For fast convergence we set β` = 1 [10]. Thus, after estimating the hidden data space in the socalled expectation step (E-step) with ˆ` = Y − X

L X

ˆ `0 ), S`0 (θ

τ` ,ν`

ˆ ` (c(ˆ τ` ) d(ˆ ν` ))∗ |2 }. (φˆ` , ϑˆ` ) = arg max{|a(φ` ,ϑ` )H X φ` ϑ`

ν`

(20)

ˆ` ,ϑ ˆ` )H X ˆ ` (c(ˆ a(φ τ` ) d(ˆ ν` ))∗ , MN

n

(26) (27)

In order to perform the E-step, it subtracts signal estimates of waves whose parameter estimates already have been initialized based on the observed data Y. Since in Eq. (26) φ` ˆ ` is summed incoherently over all and ϑ` are unknown, X sensors to provide the initial delay and Doppler estimates. The estimates of the complex amplitudes γ` can be derived from Eq. 25. In order to ensure fast convergence and good estimation results, the initialization of the SAGE algorithm in a GNSS receiver could be augmented by appropriate a priori knowledge of the parameters of the impinging waves. For example, we could consider the estimates of the delay and Doppler frequency provided through acquisition and additionally one could use almanac data to get initial estimates for the spatial signature of the LOSS. Furthermore, a good initialization or a priori knowledge of the parameters which are to be estimated will also reduce the search space of the parameters θ and therefore reduce complexity of the optimization procedures. We assume that there is no knowledge of almanac data and only rough acquisition estimates available which indicate the delay of the LOSS with an accuracy of two chips. This facilitates the initialization procedure considerably.

`0 =1 `0 6=`

the so-called maximization step (M-step) with o n ˆ ˆ H ˆ ν` ))∗ |2 ` (c(τ` ) d(ˆ , τˆ` = arg max |a(φ` ,ϑ` ) X 2 M N β ` σn τ` o n ˆ H ˆ ` (c(ˆ τ` ) d(ˆ ν` ))∗ |2 , φˆ` = arg max |a(φ` ,ϑ` ) X 2 M N β σ ` n φ` o n ˆ H ˆ τ` ) d(ˆ ν` ))∗ |2 ` (c(ˆ , ϑˆ` = arg max |a(φ` ,ϑ` ) X 2 M N β σ ` n ϑ` o n ˆ ˆ H ˆ τ` ) d(ν` ))∗ |2 ` (c(ˆ νˆ` = arg max |a(φ` ,ϑ` ) X , M N β` σ 2 γˆ` =

ˆ ` (c(τ` ) d(ν` ))∗ |2 }, (ˆ τ` , νˆ` ) = arg max{|X

(21) (22) (23) (24) (25)

is performed. Here, c(τ` ) denotes the sampled reference PN sequence delayed by τ` (generated in the receiver). The Estep and the M-step are performed iteratively until the algorithm converges. Eq. (21) is often called rake searcher, as delay estimates τˆ` correspond to the maximum crosscorrelation values. Equivalently, the estimates of azimuth φˆ` of Eq. (22) and elevation ϑˆ` of Eq. (23) are given by the 5

6. Simulation Results

phase arg γ1 = arg γ2 , which corresponds to the worst possible case [28,23,26]. The signal-to-multipath ratio (SMR) is 5 dB for all reflections (cf. [28,24,23] ). The parameter estimates are quantized to a precision of 0.5 ns for τˆ` , 0.5 Hz for νˆ` , and 0.1◦ for φˆ` and ϑˆ` . Convergence of the SAGE algorithm is reached if for all impinging waves ` = 1, 2, . . . , L (k) (k−1) all the following constraints are fulfilled: |ˆ τ` − τˆ` |≤ (k) (k−1) (k) (k−1) ˆ ˆ 0.5 ns, |ˆ ν` − νˆ` | ≤ 0.5 Hz, |φ` − φ` | ≤ 0.1◦ , (k) (k−1) (k) (k−1) (k) |ϑˆ` − ϑˆ` | ≤ 0.1◦ , ||ˆ γ` | − |ˆ γ` || ≤ 0.01, |arg γˆ` − (k−1) arg γˆ` | ≤ 0.01. First we will assess the performance of the SAGE algorithm in a GNSS receiver using a single antenna. We assume an isotropic sensor element with a` = 1 and the paT rameter vector reduces to θ ` = [γ` , τ` , ν` ] . Fig. 3 shows the RMSE of τˆ1 for SAGE versus the relative delay of the multipath ∆τ = |τ1 − τ2 |/Tc and the effective SNR given by Eq. (28). The absolute relative Doppler difference is assumed ∆ν = |ν1 − ν2 | · TN = 0.5. Fig. 4 illustrates the result for the Doppler frequency νˆ1 . Fig. 5 depicts the mean number of iteration cycles k¯ until convergence versus ∆τ and the SNR.

We tested the SAGE algorithm under severe multipath conditions via Monte-Carlo simulations. Receivers using a single antenna (M =1) and an array antenna with M antenna elements are considered. An URA is simulated with M = Mx · My , and ∆x = ∆y = λ2 . The channel parameters are assumed constant during the observation interval TN and the one-sided bandwidth is B =1.023 MHz. The filtering approach of Section 2 is applied. Thus, the rectangular chip pulse shape after filtering and downsampling can be assumed ideal bandlimited as described in [35] and the sampling rate after filtering and downsampling is fs = 1/Ts = 2 · B. We apply the initialization as discussed in Section 5. In order to describe the behavior and to assess the performance of the SAGE algorithm we adopt the root mean square error (RMSE) and the Cramer-Rao lower bound (CRLB) which is derived in Appendix A. The number of iteration cycles performed by the SAGE algorithm is denoted by k ∈ N and the mean number of iteration cycles until convergence of the algorithm is given by k¯ ∈ R. Signal-to-noise ratio (SNR) denotes the LOSS-to-noise ratio. The effective SNR in dB can be obtained following [4] as

CRLB(ˆ τ1 )

(28) CRLB, RMSE / Tc

SNR = C/N0 − 10 · log10 (2 · B) + 10 · log10 (Nc ),

p

RMSE τˆ1

√

whereas C/N0 in dB-Hz denotes the carrier-to-noise density ratio and Nc ∈ N is the number of code periods within TN . Here, SNR means pre-correlation SNR but we take into account the observation time TN given by the number of code periods Nc for which correlation is averaged. Typical values of C/N0 for GPS range from 35 - 55 dB-Hz. Please notice that C/N0 is dependent on the receiver and system noise figure, noise temperature, and on elevation of the satellite. Following [21,20] an aviation navigation receiver being interoperable with a standard GPS antenna without preamplifier, shall be capable of tracking GPS satellites with a minimum input signal power of - 166 dBW at the receiver port in the presence of background thermal noise density of -206.6 dBW/Hz. This leads to a minimum C/N0 of 40.3 dB-Hz for a GPS aviation receiver. In order to achieve SNR ranging from -15 dB to -5 dB for C/N0 = 40.3 dB-Hz and B = 1.023 MHz, the observation time TN is ranging from 5 ms to 50 ms . In order to assess the performance of the SAGE algorithm with respect to delay estimation of the LOSS for a GNSS receiver, we analyze the behavior of our estimator for a single reflective multipath as a function of its relative delay to the LOSS (L = 2). This procedure is commonly used in GNSS literature in order to characterize multipath mitigation capabilities. In the following parameters with the subscript 1 refer to the LOSS and parameters with the subscript 2 refer to the reflection. Thus, the signal parameters T of the LOSS are θ1 = [γ1 , τ1 , ν1 , φ1 , ϑ1 ] and the signal paT rameters of the multipath are θ2 = [γ2 , τ2 , ν2 , φ2 , ϑ2 ] . The reflected multipath and the LOSS are considered to be in-

−1

10

−2

10 −15

−10 0 0,2 0,4 0,6 0,8

SNR

−5

1 1,2

∆τ

Fig. 3. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ and the SNR. Parameters: L = 2, M = 1, ∆ν = 0.5.

Fig. 6, Fig. 7, and Fig. 8 show the results for τˆ1 , νˆ1 , and k¯ versus ∆τ and ∆ν where SNR = -10 dB. In Fig. 6 and Fig. 7 it can be observed that the estimator is biased for short relative delays ∆τ and small ∆ν. Here, the ability of SAGE to distinguish the two waves is slowly dwindling away towards shorter relative delays and smaller absolute relative Doppler differences, while the CRLB is tending to infinity towards small ∆τ and small ∆ν. This behavior is also observed in [28,26,34] for a MLE in a single antenna GNSS receiver. Simulation results have shown that for small ∆τ SAGE more and more often converges to either two equal delay estimates (ˆ τ1 = τˆ2 ) or to the true LOSS delay estimate and a noisy multipath delay estimate, which mostly has a large difference to the true value. It seems as if the algorithm takes advantage of the a priori knowledge that two impinging waves are present (L = 2) in order to obtain better estimates. 6

p

RMSE νˆ1

CRLB(ˆ ν1 )

p CRLB(ˆ ν1 )

RMSE νˆ1

3

√ CRLB, RMSE Hz

10

2

√ CRLB, RMSE Hz

10

1

10

0

10

2

10

1

10

0

10 0

−15

0,5

−1

10 1,2

−10 1

0,8

0,6

0,4

0,2

SNR

−5

0

∆ν

1

0,8

1

1,2

0,4

0,2

0

∆τ

∆τ

Fig. 4. RMSE of the Doppler frequency of the LOSS in Hz versus the relative delay ∆τ and the SNR. Parameters: L = 2, M = 1, ∆ν = 0.5.

0,6

Fig. 7. RMSE of the Doppler frequency of the LOSS in Hz versus the relative delay ∆τ and the relative Doppler frequency ∆ν. Parameters: L = 2, M = 1, SNR = -10 dB.

30 30 25 25 20

15

k¯

k¯

20

15 10

10

5

5

0 0 −15

0

−10

0,5

0

0 0,2

0,2 0,4

0,4

0,6

0,6

0,8 −5

SNR

0,8

1

∆τ

1,2

∆ν

¯ versus the relative delay Fig. 5. Mean number of iteration cycles k ∆τ and the SNR. Parameters: L = 2, M = 1, ∆ν = 0.5.

1

CRLB, RMSE / Tc √

0

10

−1

10

−2

10 0

0,5 0 0,2 0,4 0,6 0,8 1

1 1,2

∆τ

with the RMSE of the optimal estimator for the single path case, which also is a MLE (MLE-L1). The MLE-L1 gives the performance of a conventional GNSS receiver as discussed in Section 1. One can observe that even if SAGE nearly reaches the CRLB as illustrated in Fig. 3 and Fig. 6, it only provides better performance compared to the MLEL1 for higher SNR and for bigger ∆τ . This behavior is also observed for the estimators in [28,26] for a one antenna approach. Thus, this approach can only provide limited improvement in performance and may only be applicable for stationary GNSS receivers where a longer observation time TN could be considered [3], for example in reference stations for Ground Based Augmentation Systems (GBAS). We will now apply the SAGE algorithm with an antenna array in order to further reduce the estimation error of the delay of the LOSS under presence of severe multipath and to even resolve impinging waves with small ∆τ and small ∆ν. In the following we will assess the performance of a GNSS

10

∆ν

1 1,2

¯ versus the relative delay ∆τ Fig. 8. Mean number of iteration cycles k and the relative Doppler frequency ∆ν. Parameters: L = 2, M = 1, SNR = -10 dB.

p CRLB(ˆ τ1 )

RMSE τˆ1

1

∆τ

Fig. 6. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ and the relative Doppler frequency ∆ν. Parameters: L = 2, M = 1, SNR = -10dB.

In Fig. 9 and Fig. 10 the RMSE of τˆ1 of SAGE is compared 7

duces the multi-dimensional non-linear optimization procedure of the MLE into 4 · L one-dimensional problems per iteration cycle. Compared to the MLE and the hybrid beamformer presented in [23,24] and the proposed subspace methods presented in [25,1], the SAGE algorithm is able to precisely estimate the delay of the LOSS even if multipath with small relative delay ∆τ and small absolute relative Doppler difference ∆ν is present. This is mainly due to the fact that the MLE which is given in Eq. (19) additionally estimates the complex amplitudes γ` , the Doppler frequencies ν` , the azimuths φ` , and the elevations ϑ` of all impinging waves explicitly, which is exploited to enhance the resolution abilities especially for small ∆τ and small ∆ν, but increases the complexity of the estimator. If the complex amplitudes γ` , the Doppler frequencies ν` , the azimuths φ` , and the elevations ϑ` are not explicitly estimated the CRLB for τˆ1 of such a MLE will tend to infinity towards small relative delays ∆τ and small ∆ν. This is the case for the MLE given in [23,24]. Thus, the MLE as given in Eq. (19) is very much suitable for the estimation of synchronization parameters for navigation applications and can be solved with reasonable complexity by the SAGE algorithm. The results have shown that SAGE can be considered as a high resolution method for precise synchronization.

0.2 SAGE MLE-L1

0.18 0.16

RMSE / Tc

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

∆τ

Fig. 9. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ for SAGE and the MLE-L1. Parameters: SNR = - 10 dB, L = 2, M = 1, ∆ν = 0.

0.2

SAGE MLE-L1

0.18 0.16

RMSE / Tc

0.14 0.12

p

RMSE τˆ1

0.1

CRLB(ˆ τ1 )

0.08 0.06 0.04 −1

10

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

CRLB / Tc

0.02

1.2

−2

10

RMSE,

√

∆τ

Fig. 10. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ for SAGE and the MLE-L1. Parameters: SNR = 5 dB, L = 2, M = 1, ∆ν = 0.

−15

−3

10 1.2

receiver using an URA with M = 9. This size of the aperture still seems to be reasonable for aeronautical applications as ∆x = ∆y = λ2 = 0.0951 m for the GPS L1 signal. Fig. 11, Fig. 12, Fig. 13, and Fig. 14 demonstrate the estimation performance of the SAGE algorithm for the delay τˆ1 , the Doppler νˆ1 , the azimuth φˆ1 , and the elevation ϑˆ1 of the LOSS for a scenario with φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , and ∆ν = 0.5. Fig. 16, Fig. 17, Fig. 18, and Fig. 19 depict the estimation performance for a scenario with different absolute relative Doppler differences ∆ν with φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , and SNR = -10 dB. Fig. 11 and Fig. 16 illustrate that the proposed estimator attains the CRLB for spatially non-coherent wavefronts. It significantly reduces the RMSE of τˆ1 under presence of severe multipath compared to the case where only one antenna is used which is depicted in Fig. 3 and Fig. 6. The complexity of the estimator and its implementation increases due to the use of an antenna array, but SAGE re-

−10 1.0

0.8

0.6

0.4

0.2

0

−5

SNR

∆τ

Fig. 11. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, ∆ν = 0.5.

Concerning the estimation of the incident azimuth φˆ1 and elevation ϑˆ1 of the LOSS the proposed estimator is statistically efficient. In Fig. 15 and Fig. 20 the mean number of iteration cycles k¯ versus the SNR or ∆ν respectively and ∆τ is depicted. Fewer iteration cycles are needed for SAGE to converge for using an antenna array than for a single antenna approach whereas taking into account the same quantization of the parameters. As we have shown that the SAGE algorithm is capable of resolving spatially non-coherent signals even with small relative delay of the multipath to the LOSS ∆τ and small absolute relative Doppler difference ∆ν, we now want to assess the behavior of SAGE when the impinging wavefronts have small relative azimuth ∆φ = |φ1 −φ2 | or small relative 8

p

RMSE νˆ1

CRLB(ˆ ν1 ) 8 7 6

k¯

5 4 1

3 2 0

10

−15 −15

RMSE,

√

CRLB Hz

10

−1

10

−10

−10 −2

10

1,2

1

0,8

0,6

0,4

SNR

0,2

−5

0

SNR

∆τ

−15

RMSE,

RMSE,

√ CRLB [◦ ]

0

−1

−10

CRLB(ˆ τ1 )

−2

10

0

0,5

1,2 1 0,8

1

0,8

0,6 0,6

0,4 0,4

0,2

0

−5

0,2

SNR

0

Fig. 16. RMSE of the time-delay of the LOSS / Tc versus the relative delay ∆τ and the relative Doppler frequency∆ν. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, SNR = -10 dB.

Fig. 13. RMSE of the the azimuth of the LOSS in degrees versus the relative delay ∆τ and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, ∆ν = 0.5. q CRLB(ϑˆ1 )

p

RMSE νˆ1

CRLB Hz

RMSE ϑˆ1

∆ν

1

∆τ

∆τ

0

10

−15

CRLB(ˆ ν1 )

0

0

10

1,2

RMSE,

√

RMSE,

√ CRLB [◦ ]

p

RMSE τˆ1

10

10 1,2

∆τ

CRLB(φˆ1 )

√ CRLB / Tc

q

0,8

1

1,2

0

¯ versus the relative delay Fig. 15. Mean number of iteration cycles k ∆τ and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, ∆ν = 0.5.

Fig. 12. RMSE of the Doppler frequency of the LOSS in Hz versus the relative delay ∆τ and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, ∆ν = 0.5. RMSE φˆ1

−5

0,2

0,4

0,6

−10 1

0,8

0,6

0,5

1,2 1

0,4

0,8

0,2

0

−5

∆ν

0,6

SNR

0,4 0,2 0

∆τ

1

∆τ

Fig. 14. RMSE of the elevation of the LOSS in degrees versus the relative delay ∆τ and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, ∆ν = 0.5.

Fig. 17. RMSE of the Doppler frequency of the LOSS in Hz versus the relative time-delay / Tc and the SNR. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, SNR = -10 dB.

9

q

RMSE φˆ1

CRLB(φˆ1 )

7 6

k¯

5 4 3 2 0

RMSE,

√ CRLB [◦ ]

0

10

0

0,5 1,2

1

0,8

0,5 0,6

0,4

0,2

1

0

∆ν

0 0,2 0,4

∆τ

0,6 0,8

∆ν

Fig. 18. RMSE of the the azimuth of the LOSS in degrees versus the relative delay ∆τ and the relative Doppler frequency ∆ν. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, SNR = -10 dB. q

1,2

∆τ

¯ versus the relative delay Fig. 20. Mean number of iteration cycles k ∆τ and the relative Doppler frequency ∆ν. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, SNR = -10 dB.

CRLB(ϑˆ1 )

burden of the different algorithms. Thus, in order to really compare different methods concerning delay estimation for navigation applications a more detailed analysis with respect to performance in terms of accuracy and complexity of SAGE and other high resolution methods as MUSIC and ESPRIT or one of their numerous variations needs to be performed in the future. Fig. 25 depicts the mean number of iteration cycles k¯ needed until convergence.

0

10

0

RMSE,

√ CRLB [◦ ]

RMSE ϑˆ1

1

1

0,5 1,2

p CRLB(ˆ τ1 )

RMSE τˆ1

1 0,8 0,6

∆ν

0,4 0,2

∆τ

0

1

RMSE,

√ CRLB / Tc

Fig. 19. RMSE of the elevation of the LOSS in degrees versus the relative delay ∆τ and the relative Doppler frequency∆ν. Parameters: φ1 = 50◦ , φ2 = 102◦ , ϑ1 = 40◦ , ϑ2 = 26◦ , L = 2, M = 9, SNR = -10 dB.

elevation ∆ϑ = |ϑ1 − ϑ2 |. Fig. 21, Fig. 22, Fig. 23, and Fig. 24 depict the RMSE of the delay τˆ1 , the Doppler frequency νˆ1 , the azimuth φˆ1 , and the elevation ϑˆ1 for a scenario with a relative delay of the multipath to the LOSS of ∆τ = 0.5, SNR = -10 dB, and an absolute relative Doppler difference of ∆ν = 1. The relative azimuth ∆φ and the relative elevation ∆ϑ range from 0◦ to 20◦ . The results show that SAGE is capable of resolving the two wavefronts even for small ∆φ and ∆ϑ. Thus, the SAGE algorithm can be considered as a high resolution method as other methods like MUSIC [22] or ESPRIT and Unitary ESPRIT [13]. However, MUSIC and ESPRIT need preprocessing steps like spatial smoothing [13] in order to decorrelate spatially coherent wavefronts. As these preprocessing steps decrease the effective antenna aperture, the SAGE algorithm performs slightly better in the case where spatially coherent wavefronts are received [30,31]. On the other hand one also has to consider memory requirements and computational

−2

10

20 15

20 10

15 10

5 5

∆ϑ [◦ ]

0

0

∆φ [◦ ]

Fig. 21. RMSE of the time-delay of the LOSS / Tc versus the relative azimuth ∆φ and the relative elevation ∆ϑ in degrees. Parameters: SNR = -10 dB, ∆τ = 0.5, ∆ν = 1, L = 2.

7. CONCLUSION In this work we have addressed the problem of estimating the propagation time-delay of the LOSS in a GNSS receiver under the presence of severe multipath (SMR = 5 dB) for aeronautical applications. We tested the SAGE algorithm in order to reduce the complexity in solving the 10

p CRLB(ˆ ν1 )

RMSE νˆ1

4.2

k¯

4

RMSE,

√

CRLB Hz

4.4 0

10

3.8 3.6 3.4

20

3.2 0

20

0

15 15 10

5

10 5

10

0

15

∆φ [◦ ]

0

∆ϑ [◦ ]

Fig. 22. RMSE of the Doppler frequency of the LOSS in Hz versus the relative azimuth ∆φ and the relative elevation ∆ϑ in degrees. Parameters: SNR = -10 dB, ∆τ = 0.5, ∆ν = 1, L = 2.

20

RMSE,

√

CRLB [◦ ]

0

15 20 10 15 5

∆φ [◦ ]

5

◦

∆ϑ [ ]

0

0

Fig. 23. RMSE of the azimuth of the LOSS in degrees versus the relative azimuth ∆φ and the relative elevation ∆ϑ in degrees. Parameters: SNR = -10 dB, ∆τ = 0.5, ∆ν = 1, L = 2. q CRLB(ϑˆ1 )

RMSE ϑˆ1

20

20

∆φ [◦ ]

MLE given in Eq. (19). The SAGE algorithm iteratively approximates the MLE and significantly reduces the complexity by breaking down the multi-dimensional non-linear optimization problem of the MLE as given in Eq. (19) into a number of one-dimensional ones. Using a single antenna (M = 1) this approach is only suitable for higher SNR and even then only provides limited improvement compared to a conventional GNSS receiver which can be approximated by the MLE-L1. However, using an array antenna (M = 9) SAGE is capable of precisely estimating the delay of the LOSS under presence of severe reflective multipath, where the relative time-delay ∆τ and the absolute relative Doppler difference ∆ν is small. Also for spatially coherent wavefronts the SAGE algorithm is capable of resolving the signal parameters of the LOSS. This work clearly shows that the proposed method using an antenna array significantly reduces the estimation error of synchronization parameters compared to a conventional GNSS receiver. Thus, SAGE has proven to be a very promising method to combat multipath for navigation applications, especially for safety critical applications like aviation due to its good performance and low complexity.

10

10

15

Fig. 25. Mean number of iteration cycles versus the relative azimuth ∆φ and the relative elevation ∆ϑ in degrees. Parameters: SNR = -10 dB, ∆τ = 0.5, ∆ν = 1, L = 2.

q CRLB(φˆ1 )

RMSE φˆ1

√ CRLB [◦ ]

10

5

∆ϑ [◦ ]

RMSE,

5

Acknowledgement

0

10

The authors would like to thank the reviewers for their very useful suggestions and comments, which have enhanced the quality of this work. Furthermore, we like to thank Gonzalo Seco Granados from the University of Barcelona (UAB) for the valuable suggestions to this work.

20

15 20 10 15 10

5

∆φ [◦ ]

5

∆ϑ [◦ ]

0

0

Appendix A. Cramer-Rao Lower Bound (CRLB) Fig. 24. RMSE of the elevation of the LOSS in degrees versus the relative azimuth ∆φ and the relative elevation ∆ϑ in degrees. Parameters: SNR = -10 dB, ∆τ = 0.5, ∆ν = 1, L = 2.

The likelihood function for the defined signal model (c.f. Section 2) is given by the conditional probability density function (pdf) 11

p(Y;θ)=

exp(−vec{Y−S(θ)}H R−1 (θ)vec{Y−S(θ)}) π

N

det(R(θ))

(A.1)

,

whereas R(θ) denotes the covariance matrix of the noise. We define γ = [γ1 γ2 . . . γ` . . . γL ]T as the vector of amplitudes for the L impinging wavefronts, φ = [φ1 φ2 . . . φ` . . . φL ]T as the vector of incident azimuths, ϑ = [ϑ1 ϑ2 . . . ϑ` . . . ϑL ]T as the vector containing the incident elevations, τ = [τ1 τ2 . . . τ` . . . τL ]T as the vector of delays, ν = [ν1 ν2 . . . ν` . . . νL ]T as the vector of Doppler frequencies, η = [Re{γ}T Im{γ}T ]T ,ψ = [φT ϑT ]T , and ζ = [τ T ν T ]T . The parameter θ has complex and real components. In order to avoid confusion with the case that θ is purely complex we denote the vector of real parameters ξ = [η T ψ T ζ T ]T . The CRLB is found as the [i, i] element of the inverse of the so called Fisher information matrix (FIM) I(ξ) or   var(ξˆi ) ≥ I−1 (ξ) i i , (A.2) h

∂ ln p(Y,ξ) ∂ ln p(Y,ξ) ∂ 2 ln p(Y,ξ) =E . ∂ξi ∂ξj ∂ξi ∂ξj

i

h

i

n

+2·Re

∂R(ξ) ∂ξi

∂vec{S(ξ)}H ∂ξi

R−1 (ξ)

R−1 (ξ)

∂R(ξ) ∂ξj

(A.3)

∂vec{S(ξ)} ∂ξj

+

o

(A.4)

.

The derivatives are defined as the matrix or vector of partial derivatives (see [14] p. 47-48). Each complex element g = gRe + j gIm is differentiated as ∂g ∂gRe ∂gIm = +j . ∂ξi ∂ξi ∂ξi

(A.5)

As we assume spatially and temporal uncorrelated elements in N the covariance of the noise is R = σn2 · I. Thus, we get o n ∂vec{S(ξ)} ∂vec{S(ξ)}H R−1 = (A.6) [I(ξ)]i j =2·Re ∂ξi ∂ξj n  H o ∂S (ξ) ∂S(ξ) . (A.7) = 22 Re tr ∂ξi ∂ξj σn

The FIM can be block partitioned and is symmetric. Thus, we can write   Iηη Iηψ Iηζ

   I(ξ)= IT  ηψ 

Iψψ

   Iψζ ,  

(A.8)

T IT ηζ Iψζ Iζζ

where

[Iηη ]i j = [Iηψ ]i j

=

[Iηζ ]i j

=

  2 ·Re tr (C D)H 2 σn   2 ·Re tr (C D)H 2 σn   2 ·Re tr (C D)H 2 σn

 ∂ΓH AH A ∂Γ (C D) , ∂ηi ∂ηj

(A.9)

∂ΓH AH ∂A Γ(C D) , ∂ηi ∂ψj  H ∂(C D) ∂Γ AH AΓ , ∂ηi ∂ζj

(A.10)



[Iψζ ]i j

=

[Iζζ ]i j

=

   2 ·Re tr (C D)H ΓH ∂AH ∂A Γ(C D) , 2 ∂ψi ∂ψj σn    2 ·Re tr (C D)H ΓH ∂AH AΓ ∂(C D) , 2 ∂ψi ∂ζj σn !) ( H 2 ·Re tr ∂(C D) ΓH AH AΓ ∂(C D) . 2 ∂ζi ∂ζj σn

(A.12) (A.13) (A.14)

[1] F. Antreich, H. Cao, and J. A. Nossek. TST-MUSIC for Delay Estimation in a GNSS Receiver. In Proceedings of the 2nd Workshop on GNSS Signals & Signal Processing, GNSS Signals 2007, Noordwijk, The Netherlands, April 2007. [2] F. Antreich, O. Esbri-Rodriguez, J. A. Nossek, and W. Utschick. Estimation of Synchronization Parameters using SAGE in a GNSS Receiver. In Proceedings of the 16th Annual International Technical Meeting of the Satellite Division of the Institute of Navigation (ION) - Global Satellite Navigation System (GNSS), Long Beach, CA, USA, September 2005. [3] F. Antreich and J. A. Nossek. Multipath Mitigation in a Galileo SoL Receiver. In 3rd ESA Workshop on Satellite Navigation User Equipment Technologies NAVITEC ’2006, Noordwijk, The Netherlands, December 2006. [4] M. S. Braasch and A. J. van Dierendonck. GPS Receiver Architecture and Measurements. Proceedings of the IEEE, 87(1), January 1999. [5] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum Likelihood from Incomplete Data via the EM Algorithm. J. Royal Statistical Soc. B., 39(1), 1977. [6] F. A. Dietrich. A Tutorial on Channel Estimation with SAGE. Technical Report TUM-LNS-TR-06-03, 2006. [7] C. Fern´ andes-Prades. Advanced Signal Processing Techniques for Global Navigation Satellite Systems Receivers. PhD thesis, Department of Signal Signal Theory and Communications, Universitat Polit` ecnica de Catalunya, 2005. [8] C. Fern´ andes-Prades, J. A. Fern´ andez-Rubio, and G. SecoGranados. On the Equivalence of Joint Maximum Likelihood Approach and the Multiple Hybrid Beamforming in GNSS Synchronization. In Proceedings of Sixth Baiona Workshop on Signal Processing in Communications, 2003. [9] J. A. Fessler and A. O. Hero. Space-Alternating Generalized Expectation-Maximization Algorithm. IEEE Transactions on Signal Processing, 42(10), October 1994. [10] B. H. Fleury, M. Tschudin, R. Heddergott, D. Dahlhaus, and K. I. Pedersen. Channel Parameter Estimation in Mobile Radio Environments Using the SAGE Algorithm. IEEE Journal on Selected Areas in Communications, 17(3), March 1999. [11] B. Forssell. Radionavigation Systems. Prentice Hall International (UK) Ltd., 1991. [12] G. A. Mc Graw and M. S. Braasch. GNSS Multipath Mitigation Using Gated and High Resolution Correlator Concepts. In Proceedings of the National Technical Meeting of the Institute of Navigation ION NTM99, 1999. [13] M. Haardt. Efficient One-, Two-, and Multidimensional HighResolution Array Signal Processing. PhD thesis, Department Electrical Engineering of Munich University of Technology, 1996. [14] S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory, volume 1. Prentice Hall PTR, 1993. [15] M. Kayton and W. R. Fried. Avionics Navigation Systems. John Willey & Sons, Inc., 1997. [16] H. Meyr, M. Moeneclaey, and S. A. Fechtel. Digital Communication ReceiversSynchronization, Channel Estimation, and Signal Processing. John Wiley & Sons,Inc., 1998. [17] T. K. Moon. The Expectation-Maximization Algorithm. IEEE Signal Processing Magazine, November 1996. [18] A. V. Oppenheim and R. W. Schafer with J. R. Buck. DiscreteTime Signal Processing. Prentice-Hall Inc., 1999.

The FIM for a complex multivariate Gaussian pdf is [14], [29], [23] i h [I(ξ)]i j = tr R−1 (ξ)

=

References

where

[I(ξ)]i j =−E

[Iψψ ]i j

(A.11) 12

[19] B. W. Parkinson and J. J. Spilker. Global Positioning System: Theory and Applications, volume 1. Progress in Astronautics and Aeronautics, 1996. [20] RTCA. Minimum Operational Performance Standards for Global Navigation Satellite System (GNSS) Airborne Antenna Equipment. DO-228, 1995. [21] RTCA. Minimum Operational Performance Standards (MOPS) for Global Positioning (GPS)/Wide Area Augmentation System (WAAS) Airborne Equipment. DO-229C, 2001. [22] R. O. Schmidt. Multiple Emitter Location and Signal Parameter Estimation. IEEE Transactions on Antennas and Propagation, 34(3), March 1986. [23] G. Seco-Granados. Antenna Arrays for Multipath and Interference Mitigation in GNSS Receivers. PhD thesis, Department of Signal Signal Theory and Communications, Universitat Polit` ecnica de Catalunya, 2000. [24] G. Seco-Granados, J. A. Fern´ andez-Rubio, and C. Fern´ andezPrades. ML Estimator and Hybrid Beamformer for Multipath and Interference Mitigation in GNSS Receivers. IEEE Transactions on Signal Processing, 53(3), March 2005. [25] J. Selva-Vera. Subspace Methods to Multipath Mitigation in a Navigation Receiver. In Proceedings of the IEEE Vehicular Technology Conference (VTC), 1999. [26] J. Selva-Vera. Efficient Multipath Mitigation in Navigation Systems. PhD thesis, Department of Signal Signal Theory and Communications, Universitat Polit` ecnica de Catalunya, 2004. [27] J. Selva-Vera. An Efficient Newton-Type Method for the Computation of ML Estimators in a Uniform Linear Array. IEEE Transactions on Signal Processing, 53(6), June 2005. [28] J. Soubielle, I. Fijalkow, P. Duvaut, and A. Bibaut. GPS Positioning in a Multipath Environment. IEEE Transactions on Signal Processing, 50(1), January 2002. [29] P. Stoica and A. Nehoria. MUSIC, Maximum Likelihood, and Cramer-Rao Bound. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(5), May 1989. [30] M. Tschudin. Analyse und Vergleich von hochaufloesenden Verfahren zur Funkkanalparameterschaetzung . PhD thesis, ETH Zuerich, 1999. [31] M. Tschudin, C. Brunner, T. Krupjuhn, M. Haardt, and J. A. Nossek. Comparison between Unitary ESPRIT and SAGE for 3-D Channel Sounding. In Proceedings of the IEEE Vehicular Technology Conference (VTC), 1999. [32] A. J. van Dierendonck and A. J. Fenton. Theory and Performance of Narrow Correlator Spacing in a GPS Receiver. Navigation, 39(3), 1992. [33] R. D. J. van Nee and P. C. Fenton. The Multipath Estimating Delay Lock Loop Approaching Theoretical Accuracy Limits . In Proceedings of IEEE Position Location and Navigation Symposium, Las Vegas, NV, USA, April 1994. [34] L. Weill. Achieving Theoretical Accuracy Limits for Pseudoranging in the Presence of Multipath. In Proceedings of Institute of Navigation GPS, Palms Springs, CA, USA, 1995. [35] J. O. Winkel. Modeling and Simulating GNSS Signal Structures and Receivers. PhD thesis, Department of Civil Engineering and Surveying of University FAF Munich, 2000.

13