^{*}

Galileo is the Global Navigation Satellite System that Europe is building and it is planned to be operational in the next 3-5 years. Several Galileo signals use split-spectrum modulations, such as Composite Binary Offset Carrier (CBOC) modulation, which create correlation ambiguities when processed with large or infinite front-end bandwidths (i.e., in wideband receivers). The correlation ambiguities refer to the notches in the correlation shape (i.e., in the envelope of the correlation between incoming signal and reference modulated code) which happen within +/– 1 chip from the main peak. These correlation ambiguities affect adversely the detection probabilities in the code acquisition process and are usually dealt with by using some form of unambiguous processing (e.g., BPSK-like techniques, sideband processing, etc.). In some applications, such as mass-market applications, a narrowband Galileo receiver (i.e., with considerable front-end bandwidth limitation) is likely to be employed. The question addressed in this paper, which has not been answered before, is whether or not this bandwidth limitation can cope inherently with the ambiguities of the correlation function, to which extent, and which the best design options are in the acquisition process (e.g., in terms of time-bin step and ambiguity mitigation mechanisms).

New advances in the field of satellite positioning and the design of new satellite systems to be used for location purposes in the years to come created the need for new modulation and signal types at the transmitter side. The upcoming satellite systems, such as the European Galileo and Chinese Compass systems, need to preserve compatibility with the existing Navstar GPS system, while keeping the interference levels at minimum. This motivated the introduction of a w modulation family, namely the Binary Offset Carrier (BOC) family, which currently have several variants, such as sine BOC [1-3], cosine BOC [3,4], alternate BOC (AltBOC) [5,6], Composite BOC (CBOC) [7,8], and Time Multiplexed BOC (TMBOC) [_{sc} used in these split-spectrum modulations. As a consequence of the spectrum splitting, the Auto-Correlation Envelope (ACE) of these signals has also different properties compared with the BPSK case. Two main consequences have been noticed in the literature with respect to the new ACE: on one hand, notches in the ACE shape appear within +/–1 chip interval, which introduces some challenges in the acquisition process [

• A thorough analysis of the ambiguities in the BOC/ CBOC correlation functions, in the presence of limited receiver bandwidth, ranging from 3 MHz (severe limitation, narrowband receiver) to 24.552 MHz (bandwidth used in the Galileo specifications [

• An explanation based on semi-analytical model regarding the fact that a limited front-end receiver bandwidth acts as an ‘unambiguous method’, by reducing the ambiguities in the correlation function, and thus removing the need for a supplementary unambiguous processing.

Additionally, an overview of unambiguous methods and a generic block diagram for them is offered in the context of split-spectrum modulations, and design recommendations for Galileo E1 receiver acquisition architecture are done.

The CBOC and AltBOC modulations used in Galileo split the signal spectrum into two symmetrical components around the carrier frequency, by multiplying the pseudorandom (PRN) code with a rectangular sub-carrier. This spectrum splitting is seen also as a splitting in the correlation domain, meaning that the correlation function envelope will exhibit additional sidepeaks (besides the main correlation peak) and additional low values within +/–1 chip interval from the main peak. These low values (or notches) in the correlation function are typically referred to as ambiguities, as illustrated in

These ambiguities create problems in the acquisition process, in the sense that, if a too high time-bin step is used, we might lose the main correlation peak, as illustrated in

One solution is to diminish the time-bin step (to values lower than half of the main correlation lobe width), at the expense of a larger acquisition time, since more time bins need to be tested. Another solution is to modify the ambiguous correlation into an ‘unambiguous’ one, via frequency-domain or time-domain processing, as explained for example in [13-15] and illustrated in the generic block diagram of

An overview of the state-of-art unambiguous acquisition methods in Galileo is as follows:

1) B&F methods, denoted as such after the initial of the first authors who introduced them in [1,10]; in here, only the upper or the lower sideband of the received signal is filtered and correlated with a similarly filtered reference modulated code. Both upper and lower sidebands can be then combined non-coherently (dual-band processing) or only one of the sidebands can be used (single sideband processing). The single-sideband block diagram is illustrated in _{1}(f) = H_{3}(f) = upper/lower lobe selection filters, applied both on incoming signal and reference modulated code, and H_{2}(f) = 1.

2) mM\&H methods, denoted as such after the initial of the first authors who introduced them in [11,12]; in here, both upper and lower lobes of the signal spectrum are selected, plus everything between them. The incoming signal is shifted with a shift parameter a, which is modulation dependent, e.g., a = 1 for CBOC (details on optimum a parameters for other split-spectrum BOC modulations are given in [_{1}(f) is a filter selecting both main frequency lobes of the signal, plus everything between them, H_{2}(f) is a hold filternamely, = chip interval, = BOC interval (e.g., for CBOC), and H_{3}(f) = 1.

3) Unambiguous Adjacent Lobe (UAL) methods, proposed by the Author in [13-15]. In here, a is the same modulation-dependent factor as used in mM\&H methods, H_{1}(f) = H_{3}(f) = 1 (absent filters)and is a hold filter.

After the correlation of the incoming signal with the reference code, coherent and non-coherent integration may be applied, as shown in

Then a decision variable is formed. For example, for a serial search approach, the decision variable, denoted in what follows via Z, is the non-coherent correlation output.

An example of the normalized unambiguous correlation functions after single sideband processing of a CBOC signal is shown in

Regarding the complexity of various unambiguous methods, a comparison has been provided in [

The focus here is on Galileo E1 Open Service signals, which employ CBOC modulation. Two CBOC variants, i.e. CBOC (+) and CBOC (-) are used currently [

1) CBOC (+) signal at transmitter and CBOC (+) modulated reference code (e.g., full processing of Galileo E1 data channels)

2) CBOC (+) signal at transmitter and sine BOC (1,1) -modulated reference code (e.g., low complexity one-bit processing of Galileo E1 data channels)

3) CBOC (-) signal at transmitter and CBOC (-) -modulated reference code (e.g., full processing of Galileo E1 pilot channels)

4) CBOC (-) signal at transmitter and sine BOC (1,1) -modulated reference code (e.g., low complexity one-bit processing of Galileo E1 pilot channels)

The detection probabilities P_{d} are computed within +/–

0.35 chips error, based on the fact that the main correlation lobe for a CBOC-modulated signal is about 0.7 chips (i.e., if we acquire the signal with an error less than half of the main lobe, we consider that acquisition was done ‘correctly’ and we can move to the tracking stage; if the error is higher than half of the width of the main lobe, then the acquisition was unsuccessful).

Under additive white Gaussian channel assumption, it is straightforward to show, following the model of [

Above, is the detection probability per bin, computed as the probability that the decision variable is higher than a decision threshold, provided that we are in a correct bin (hypothesis H_{1}):

where is the decision statistic corresponding to the estimated code phase, is the acquisition threshold, is the true Line Of Sight (LOS) delay of the channel (expressed in chips), and. The acquisition threshold is computed based on a pre-defined target false alarm probability P_{fa} (here, 10^{-3}). It was assumed that the residual Doppler error was 0, thus,.

In Equation (1) and are the Cumulative Distribution Function (CDF) under correct and incorrect-bin hypotheses, respectively, and they can be written as [

with and being the noise variance, number of degrees of freedom, and non-centrality parameter (dependent on signal power), and being the generalized Marcum-Q function. For dual sideband approaches, and for single sideband approaches, , with being the non-coherent integration length. The parameters and are obtained via simulations, according to the carrier-tonoise ratio level and according to the acquisition method (ambiguous or unambiguous).

There can be several correct bins (denoted here as N_{t}), and therefore the total detection probability P_{d} is given by:

that is, the sum of probabilities of detecting the signal in the i-th bin, provided that all the previous tested hypotheses for the prior correct bins gave a misdetection. In Equation (4), is the delay error associated with the first sampling point in the two-chip interval where we have the N_{t} correct bins. Equation (4) is valid only for fixed sampling points. However, due to the random nature of the channels, the sampling point (with respect to the channel delay) is randomly fluctuating, hence, the global P_{d} will be computed as the expectation operator over all possible initial delay errors (under uniform distribution, we simply take the temporal mean):

_{d}. Here, only three possible sampling sequences are shown for illustration purposes. The total number of sampling sequences depends on a discrete step, chosen sufficiently small. The step of searching the time bins in this figure is chips.

The analysis presented here has been done semi-analytically, for serial search and single-path channel in order to find out the relative performance of various acquisition algorithms. Since the channel LOS delay is unknown, we can have several possible sequences of samples of the correlation function, as illustrated in _{1} corresponds to the ‘correct acquisition’ case (i.e., samples within the main lobe of the correlation

envelope), and Hypothesis H_{0} corresponds to the ‘incorrect acquisition’ cases (i.e., samples outside the main lobe).

For a severe receiver front-end bandwidth limitation, for example 3 MHz, the correlation shapes with ambiguous and unambiguous processing are highly modified (see

If we compare

_{W} = 3 MHz double-sided bandwidth. As seen in both figures, ambiguous average detection probability is always better than unambiguous approaches in this case. These figures are for a CBOC (-) tx with sine BOC (1,1) rx. Similar plots were obtained for the other 3 combinations, and are not reproduced here due to overlapping findings.

_{W} = 3 MHz double-sided bandwidth. UAL and mM\&H algorithms are perfectly overlapping. The differences between worst-case and average case detection probabilities for ambiguous, mM\&H and UAL cases are very small (less than 0.01 dB) and therefore they are not distinguishable in _{1} hypothesis). As already mentionedin the discussion related to

In

among the unambiguous methods, at low receiver bandwidths. DSB stands for dual sideband processing, and SSB stands for single sideband processing.

The dual-sideband unambiguous methods are compared with ambiguous acquisition for various time-bin steps in

The main conclusion based on the plots presented in this section is that unambiguous approaches bring no benefit whatsoever compared with ambiguous approaches for low receiver bandwidths, no matter on the time-bin

step to be used. Thus, a significant bandwidth limitation already acts as an ‘unambiguous’ method.

Figures 11 and 12 are for a large time-bin step of chips and for B_{W}_{ }= 24.552 MHz doublesided bandwidth. For large time-bin steps (e.g., 0.5 chips), there is a clear gap between worst-case and average-case detection probabilities, and this gap is the highest for the ambiguous case. In fact, in the ambiguous case we can fail to detect the signal completely if the combination between sampling sequence and channel delay is a ‘bad’ combination. The unambiguous approaches for high front-end bandwidths and high time-bin steps bring indeed a significant enhancement over ambiguous ones, especially if we consider the worst-case detection probabilities. In

The performance at a small time-bin step of chips is shown in

The dual-sideband unambiguous methods are compared with ambiguous acquisition for various time-bin steps in

The novel finding in this paper is that, a small receiver bandwidth (e.g., 3-4 MHz double sideband, as typically used in mass-market receiver) has an inherent robustness towards the correlation ambiguities of a BOCCBOCmodulated signal and there is no need for additional unambiguous processing in such low receiver bandwidths. Therefore, for a mass-market or narrowband Galileo receiver, the recommendation is to employ the classical ambiguous correlation method in the acquisition process (no supplementary filtering or unambiguous processing) and time-bin steps of the order of 0.5 chips (in order to achieve a good tradeoff between performance and complexity).

For wideband receivers (e.g., bandwidth as specified in Galileo SIS-ICD [

The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 227890 (GRAMMAR project). This research work has also been supported by the Academy of Finland.