Cross-correlation codeless processing of BOC modulated signals

: Encrypted global navigation satellite system signals (GNSS) can be tracked using codeless techniques that do not require the knowledge of the spreading code used for signal generation. These techniques can also be applied to binary offset carrier (BOC) modulated signals whose unknown code sequence can be removed through squaring. In this study, an alternative codeless approach based on the cross-correlation principle is considered. Cross-correlation codeless processing is commonly used for tracking signal components on different frequencies, such as the global positioning system (GPS) L1 and L2 P(Y) signals, and it is adapted here to BOC modulations broadcast on a single frequency. A cross-correlation codeless framework is proposed where the BOC signal is split into two data streams that are cross-multiplied to remove the unknown code sequence. Two architectures, open-loop and closed-loop processing, are proposed and analysed. The codeless cross-correlation function is introduced and open-loop processing is used to reconstruct it from the received samples. Closed-loop processing based on cross-correlation phase lock loop (PLL) and delay lock loop (DLL) are used to estimate the signal parameters. The proposed cross-correlation framework is thoroughly analysed theoretically, through simulations and using real data. The analysis shows the effectiveness of the cross-correlation framework.


Introduction
Codeless and semi-codeless techniques have been traditionally used to generate measurements from the global positioning system (GPS) L2 P(Y) signal [1,2] that is encrypted and requires approaches able to operate without prior knowledge of the signal pseudo-random noise (PRN) code. In codeless and semi-codeless techniques, a non-linear operation, such as squaring, is used to remove the impact of the unknown code that is treated as a nuisance component.
More recently, codeless techniques have been adapted to binary offset carrier (BOC) signals [3][4][5][6], where the structure of these modulations is exploited for codeless tracking. Codeless techniques can be used to monitor the signal correlation function [3,4], to assess the code/subcarrier divergence [5,6] and to generate signal measurements. Moreover, codeless techniques can be used to introduce additional layers of security [7,8]. In this respect, short data segments of encrypted global navigation satellite system (GNSS) signals can be recorded and used to sign GNSS observables. Techniques based on codeless processing can then be used to verify the authenticity of the signed signals.
Codeless techniques developed for BOC modulations usually track the subcarrier component [9] and integrate it coherently over the duration of a code chip. The impact of the unknown PRN code is then removed through squaring.
In this paper, an alternative approach based on the crosscorrelation principle is considered. Cross-correlation codeless processing has been widely used for jointly processing the GPS L1 and L2 P(Y) signals [2], on the 1575.42 and 1227.60 MHz frequencies. In this respect, the L1 P(Y) signal is used to remove the unknown code on the L2 P(Y) component. This principle can also be used for BOC modulated signals broadcast on a single frequency. More specifically, BOC signals are characterised by two symmetric spectral lobes and can be decomposed in two side-band components. This principle is used by different types of side-band BOC processing techniques [10][11][12], which split BOC signals into two separate data streams. These techniques assume the knowledge of the signal code and use it to despread the received samples. In this way, long coherent integrations can be implemented. This approach cannot be adopted here since the code is unknown and consequently coherent processing cannot be implemented. In this work, side-band processing is used to obtain two BOC side-band components that are treated as the GPS L1 and L2 P(Y) signal pair. In particular, these side-band components are cross-multiplied to remove the impact of the unknown code. Using this principle, a cross-correlation codeless phase lock loop (PLL) is designed along with two code tracking architectures. The first architecture is openloop and allows one to reconstruct and monitor the signal codeless cross-correlation function (CCCF), which is a function of the received code chips. The second approach is closed-loop and consists of a cross-correlation delay lock loop (DLL) able track, in a codeless way, the code component.
In addition to adapt cross-correlation codeless technique to BOC signals, the paper provides a thorough characterisation of the BOC cross-correlation codeless framework. Open-loop processing is characterised from a theoretical point of view and a closed-form expression for the CCCF is derived. The cross-correlation codeless PLL is also analysed and a closed-form expression for the PLL tracking jitter is provided.
Simulations and real data processing support theoretical findings and show the feasibility of cross-correlation codeless processing. In this respect, a fully software MATLAB receiver has been developed and used to process encrypted Galileo E1a signals with centre frequency equal to 1575.42 MHz. The receiver processes the open Galileo E1c component in a standard way and uses Doppler measurements to aid codeless processing. In this way, long integrations can be implemented in the cross-correlation codeless PLL and DLL.
Real data processing demonstrates the feasibility of crosscorrelation codeless techniques that can be effectively used as an alternative to squaring PLL approaches.
The reminder of this paper is organised as follows: the BOC signal model is introduced in Section 2 while cross-correlation codeless processing is described in Section 3. The tracking jitter of the cross-correlation codeless PLL is analysed in Section 4 and simulation results are provided in Section 5. Section 6 describes the experimental results obtained processing real Galileo E1a signals and Section 7 finally concludes the paper.

BOC signals
After amplification, filtering and down-conversion, the analogue BOC modulated signal can be modelled as [10, (2)] where C is the received signal power, c( ⋅ ) is the signal spreading code and s b ( ⋅ ) is the BOC subcarrier. η(t) is a noise term usually modelled as an additive white Gaussian noise (AWGN). All the signal components in (1) are functions of the time variable, t. The communication channel delays the useful signal component and introduces a Doppler effect. In (1), the delay introduced by the communication channel is modelled by τ 0 , which shifts the signal code and subcarrier, and by φ 0 , the additional phase experienced by the carrier component. In this context, the Doppler effect is the signal frequency shift perceived by the receiver and caused by the relative motion between the receiver and the transmitting satellite [13]. Its impact is neglected on the code component whereas it is modelled by f d, 0 and Δ 0 , on the carrier and subcarrier, respectively. f d, 0 is in Hz while Δ 0 is the normalised Doppler and is unitless. The BOC subcarrier is a square wave running at the subcarrier frequency, f sub . In the following, sine BOC modulations are considered. Similar results can be obtained for cosine-phased BOC modulations. For sine BOC signals If only the first term in the Fourier series expansion in (2) is retained, the subcarrier can be effectively approximated as a pure sinusoid of fundamental frequency, f sub Thus, the subcarrier term in (1) can be written as where f d, s = Δ 0 f sub is the Doppler frequency on the subcarrier expressed in Hz and φ s = − 2π f sub + f d, s τ 0 − π/2 is the additional phase caused by the propagation delay on the subcarrier. Using (4), it is possible to express (1) as Equation (5) implies that the received baseband BOC modulated signal is made of two symmetric lobes shifted in frequency with respect to the subcarrier fundamental frequency, f sub , and the Doppler shifts on the carrier and on the subcarrier. A schematic spectral representation of (5) is provided in Fig. 1.
The analogue signal, y(t), is sampled with a frequency f s = 1 T s and the digital sequence, y[n] = y(nT s ), is obtained where A = 2 C π and c τ 0 [n] = c(nT s − τ 0 ).
The two components in (6) are separated in frequency and can be isolated using, for example, two complementary filters as shown in Fig. 1. In the following, the two BOC side lobes will be isolated through an integration process and the use of complementary filters is not required. The transfer functions of complementary filters are depicted in Fig. 1 only to better highlight the fact that the two BOC side lobes are considered as separate signals.
Two separate data streams are obtained by multiplying y[n] in (6) by two complex carriers at the nominal subcarrier frequency Signal y 0 [n] is obtained by shifting the upper lobe of the BOC modulation close to baseband. Similarly, y 1 [n] is obtained by upconverting the lower BOC side lobe close to baseband. In the following, it will be assumed that the code values, c τ 0 [n] are unknown at the receiver side. For this reason, codeless techniques are adopted. The signals, y 0 [n] and y 1 [n], form a pair similar to that formed by the L1 and L2 P(Y) signals and thus can be processed using cross-correlation codeless processing [2].

Cross-correlation codeless processing
In order to implement cross-correlation codeless techniques, it is necessary, at first, to integrate data streams (7) over the code chip duration. It is assumed that the code component in (6) is binary phase shifting keying (BPSK) modulated and that c τ 0 [n] takes values in {-1, 1}. Moreover, c τ 0 [n], is assumed approximately constant over a code chip.
A BOC modulation is defined by two parameters [14]: the subcarrier frequency, f sub , introduced in Section 2, and the code frequency, f sc , that defines the duration of an individual element of the spreading code usually denoted as chip. For example, in the BOC(10, 5) adopted by the M-code modulation [6,15], the subcarrier frequency is f sub = 10 ⋅ 1.023 MHz, whereas the code rate is f sc = 5 ⋅ 1.023 MHz. When the BOC signal is split into two separate streams, each stream can be approximated as a BPSK modulation with code rate equal to f sc . The chip duration is T sc = 1/ f sc and c τ 0 [n] can be assumed to be approximately constant over this time period. It is assumed that the start of a code chip is effectively recovered using the cross-correlation DLL described in Section 3.3. In addition to this, it is assumed that K, the number of samples integrated over each code chip is approximately constant. In reality, K varies over time oscillating between T sc /T s and T sc /T s and, for each code chip, the DLL described in Section 3.3 is able to determine the exact value of K. The assumption of a constant K is made only to simplify derivations and does not compromise the generality of the results.
Using these assumptions, it is finally possible to compute the integrated data streams The integration/averaging over K samples acts as a low-pass filter that preserves only the signal components close to baseband. This is the integration process mentioned in Section 2 that isolates the two BOC side lobes. In particular The second signal component from y[n] is shifted to high frequencies by (7) and it is averaged out by the summation in (8).
The noise term, η 0 [m] is given by and defines an AWGN with variance reduced by a factor K with respect to η(nT s ). The properties of η 0 [m] are better analysed in Section 4. Similarly Under the assumption that c τ 0 [n] is constant over the integration interval, it is possible to show that ỹ 0 [m] and ỹ 1 [m] are given by where and The product between the chip integration time, KT s , times the maximum Doppler frequencies f d, 0 ± f d, s is small and closed to zero. Thus, the Dirichlet kernels in (13) are approximately equal to 1 For the same reason, the additional phase introduced during the integration process can be neglected as in (14). Cross-correlation codeless processing is obtained by where the BPSK assumption, A schematic representation of cross-correlation codeless processing for tracking the carrier parameters is shown in Fig. 2. The residual carrier error is removed from d[m] through the multiplication by the local carrier generated by the numerically controlled oscillator (NCO). The local carrier is generated using the previous estimates of the Doppler frequency and carrier phase. After carrier removal, d[m] is integrated over M epochs and a cross-correlation correlator is obtained.
In particular, the prompt cross-correlation codeless correlator is obtained as where subscript k denotes different time epochs. Each epoch has duration KMT s . P k is used to compute the discriminator output The error signal, φ d, k , is filtered and used to update the NCO that produces a new estimate of the signal carrier frequency and generates a new local carrier for the next processing epoch. In Fig. 2, H(z) denotes the transfer function of the filter used to process the discriminator output (18).

Frequency aiding from unencrypted channels
Cross-correlation codeless processing requires long integration times, MKT s , and thus the PLL used for the frequency and phase recovery needs a small loop bandwidth [13,16]. While this requirement ensures loop stability, it also reduces the ability of the In this way, the dynamics sensed by the codeless loop is significantly reduced. Frequency aiding is commonly used in codeless loops [2,3,5] and it is adopted here for cross-correlation processing.

Open-loop processing
In the previous sections, it was assumed that the start of the code chips was known or recovered using a DLL. When the start of the code chip is unknown, it is possible to use open-loop processing where a bank of correlators is adopted to test several delay values in parallel. Open-loop processing is used for monitoring the correlation function of encrypted signals [4][5][6] and it is adapted here to cross-correlation processing. Open-loop cross-correlation processing of BOC modulated signals is depicted in Fig. 3. Several branches are present in Fig. 3: for each branch a different delay is tested and the coherent accumulation process on the code chips of y 0 [n] and y 1 [n] is delayed by a different value. The outputs of the two coherent accumulation blocks are then multiplied and further integrated after Doppler removal. The Doppler frequency can be estimated using the PLL in Fig. 2. In this way, the empirical CCCF is sampled at different delay instants. The delays should be selected in the In Appendix, the ideal CCCF, determined in the presence of perfect carrier recovery, is derived. The CCCF is a periodic function of period T sc and it is given by (Appendix) where Under ideal conditions, i.e. when the code chip is modelled as a rectangular pulse, R d (τ) becomes The block processing scheme depicted in Fig. 3 allows one to estimate R d (τ) and monitor the properties of the filtered code chip, c(t).

Cross-correlation DLL
The CCCF obtained using real data is a noisy and delayed version of (19). It has a maximum for τ = τ 0 and exhibits a symmetric peak that can be maximised using a DLL [18]. The principle is the same as that used in standard tracking loops: early and late correlators can be used to maximise the CCCF. A schematic representation of the DLL adopted to recover the start of a code chip in cross-correlation processing is shown in Fig. 4. A nominal delay, τ, is used to compute the prompt correlator. More specifically, τ determines the start of the prompt accumulation process. The early and late correlators are computed in a similar way considering τ ± d s /2 as start of their accumulation processes. d s is the early-minus-late code spacing in seconds. The early and late correlators are obtained according to the same processing described above for the prompt correlator and are denoted as E k and L k .
The discriminator output is computed from the early and late correlators as where Fig. 2 Cross-correlation codeless processing for tracking the carrier parameters is a normalisation constant selected to have a discriminator function with unit gain. G(d s , T sc ) has been computed assuming an ideal rectangular chip and a CCCF equal to (21). The discriminator output passes through the loop filter, denoted in Fig. 4 by the transfer function H(z), and a new estimate of the code rate, f^d , sc , is obtained. Finally, the NCO updates, τ, which indicates the start of the accumulation process and determines the number of samples used in the coherent integration process, K. In particular, τ is updated as where the index m has been added to indicate the different integration epochs and 'mod' denotes the modulus operator with respect to the code chip duration, T sc . T¯s is given by and represents the duration of a sampling interval scaled in order to take into account the effect of the code Doppler, f^d , sc , obtained at the output of the loop filter. Finally, K m is the number of samples accumulated during the mth epoch. While in the previous sections, K, was considered constant to simplify mathematical derivations, in a real implementation, it is updated at each accumulation epoch and it is given by

Tracking jitter
In this section, the tracking jitter obtained for cross-correlation codeless processing is provided. Only the codeless PLL described in Section 3 is considered. The tracking jitter determines the amount of noise transferred from the input signal, y[n], to the final phase estimate and it is defined as [18] where σ ϕ is the tracking jitter, σ d is the standard deviation of the discriminator output (18) and G d is the discriminator gain. T c = KMT s is the loop update rate and B eq is the loop equivalent bandwidth. B eq is a loop design parameter and is used to determine the coefficients of the loop filter [13]. G d is defined as where S( ⋅ ) is the loop discriminator input-output function. The correlator outputs, P k , are functions of the residual carrier phase error, which is denoted here as ϕ. Discriminator (18) is used to extract an error signal from the correlator output. In particular, φ d, k in (18) is, in general, a non-linear function of the input residual phase, ϕ, and it is affected by noise where η ϕ is the noise at the input of the discriminator function propagated from y[n] through the prompt correlator and η S is the discriminator output noise. G d is used to linearise the discriminator function and approximate the PLL as a linear system. For the derivations, it is assumed that the input noise, η[n] = η(nT s ) in (6) is an AWGN with independent real and imaginary parts each with variance [3] where N 0 is the power spectral density of the input noise and B RX is the one-sided receiver equivalent bandwidth. In the following where The variance of η d [m] has been computed taking into account the fact that the three terms in (32) are uncorrelated since their products lead to monomials of odd order of zero mean Gaussian random variables. Moreover, the variance in (34) Using the results in (35), it is finally possible to compute the postcorrelation signal-to-noise power ratio (SNR) The tracking jitter can be directly computed using (36) and the results obtained in [2,3]. In particular, in [2] it is shown that for a four-quadrant discriminator as in (18), the normalised standard deviation of the discriminator output is given by In this way, the tracking jitter of the cross-correlation codeless PLL is given by (39)

Simulation results
In this section, simulations are used to support the theoretical results developed in Section 4. Two types of simulations are used: full Monte Carlo simulations have been adopted to verify the validity of (36) whereas semi-analytic simulations [19] have been used to analyse the tracking jitter and (39).

Post-correlation SNR
Full Monte Carlo simulations have been used to estimate the postcorrelation SNR and support the validity of (36). A synthetic GNSS signal was generated using the parameters reported in Table 1. The signal was modulated by a cosine BOC (15, 2.5) that is the modulation adopted by the Galileo E1a component. The signal was modulated with a random code and a sampling frequency, f s = 40 MHz, was considered. The simulations were performed as a function of the input carrier-to-noise power spectral density ratio (C/N 0 ). For each C/N 0 value, AWGN was added to the simulated GNSS signal samples. The noise variance was determined from the input C/N 0 and (30). For each C/N 0 value, 10 4 simulation runs were performed. For each simulation run, a prompt correlator, P k , was evaluated. The mean and standard deviation of the correlators were finally estimated from the P k obtained through Monte Carlo simulations. The correlators were computed using the processing schemes described in Section 3 and assuming perfect carrier, subcarrier and code synchronisation. Indeed the goal of these type of simulations was to verify the validity of (36) and not to test the proper functioning of the cross-correlation codeless loops. These loops are tested through simulations in Section 5.2 and by processing real GNSS data in Section 6. Simulation results related to the post-correlation SNR are provided in Fig. 5, which compares simulation and theoretical findings. A good agreement between theoretical and simulation results is obtained supporting the validity of (36): the postintegration SNR increases linearly as a function of the input C/N 0 . The slope of the SNR curve in Fig. 5 is equal to 2 and it is determined by the squaring factor in (36).

Tracking jitter
Semi-analytic simulations [19] have been used to support the validity of the theory developed in Section 4. In this framework, analytic results on the statistical properties of the prompt correlators (35) were used to reduce the computational load that full Monte Carlo would have required. The statistical properties of P k were assessed in Section 5.1 and used to simulate the correlator outputs that are a function of the input C/N 0 , of the assumed signal Doppler frequency and phase, and of the NCO output. Analytic results were used only for the evaluation of P k , whereas all the other elements of the cross-correlation codeless PLL described in Section 3 were implemented and used to determine the tracking jitter through simulations. A second-order PLL with different loop bandwidths was considered along with a total integration time, T c , equal to 400 ms. Perfect code alignment was assumed for the analysis of the PLL tracking jitter. For the simulation, a cosine BOC(15, 2.5) was considered since this modulation is adopted by the Galileo E1a PRS. Simulations were conducted as a function of the input C/N 0 and for each C/N 0 value, 10 4 simulation runs were conducted. During the simulations, a constant Doppler frequency was assumed and a linearly varying carrier phase was generated. For each epoch, the phase estimated by the loop was compared with that generated by the simulation scenario and used to compute the residual phase error that, in turn, was used to estimate the tracking jitter.
Results are provided in Fig. 6, which compares the tracking jitter obtained through simulations with (39). A good agreement between theoretical and simulation results is obtained for high/ moderate C/N 0 values. This fact is expected since theoretical results have been obtained assuming signal lock conditions. Using this assumption, it is possible to linearise the effects of the loop discriminator (see (29)) and obtain a closed-form expression for the tracking jitter. Thus, (39) is unable to predict non-linear effects, such as loss of lock, which occur at low C/N 0 values. The practically vertical dashed lines observable in Fig. 6 represent the tracking thresholds of the cross-correlation codeless PLL, i.e. the lowest C/N 0 values for which the PLL is able to maintain signal lock. The tracking threshold is a function of the integration time and of the loop bandwidth. Lower loop bandwidths reduce the tracking jitter and lead to lower tracking threshold. Similar results were obtained by considering different combinations of integration times and loop bandwidths. They are not reported here to avoid the repetition of similar results. In all cases and under lock conditions, a good agreement between simulations and theoretical results was obtained. The semi-analytic framework used here supports the validity of the theoretical findings provided in Section 4 and is able to predict the non-linear behaviour of the cross-correlation tracking PLL.

Experimental results
In order to demonstrate the feasibility of cross-correlation codeless processing, real Galileo E1a data have been collected and processed using the approaches detailed in Section 3. A fully software MATLAB receiver able to process Galileo E1 signals has been developed. Standard processing has been used to acquire and track the E1c pilot signal that has been used to aid codeless processing.
Data have been collected using a National Instruments (NI) PXIe-5663 Radio Frequency Signal Analyser configured according to the parameters listed in Table 2. A sampling frequency equal to 40 MHz was selected in order to be able to collect the full cosine BOC(15, 2.5) adopted by the E1a signal [9]. The characteristics of the BOC signal used for the analysis are summarised in Table 3 along with the different processing parameters adopted in the software receiver.    Fig. 7 that shows the CCCF computed using a real Galileo E1a signal from Satellite Vehicle (SV) 12. In this case, 21 correlators were used to span a code chip period. The reference delay, corresponding to the zero in the code delay axis in Fig. 7, is the delay estimated from the E1c channel that is used for frequency/delay aiding. The total integration time was equal to 400 ms.
In this case, the CCCF is S-shaped and its amplitude oscillates between a minimum value of about 700 and a maximum of about 1400. Note that the amplitude scaling of the CCCF depends on the quantisation applied to the input samples and on the specific implementation of the accumulation/correlation process. In this respect, the CCCF amplitudes are numerical values without a specific unit of measurement. It was however important to remark that the CCCF maximum and minimum values are approximately related by a factor 2. This result was predicted by the theory developed in Section 3.2 and in Appendix.
The CCCF in Fig. 7 is less sharp than the ideal CCCF in (21). This fact was expected and is due to filtering effects at both transmission and reception side. Under real conditions, the code chips cannot be approximated as perfect rectangular pulses and the CCCF is less sharp than (21). In the limit case, when only the first harmonic term in the Fourier series expansion of a rectangular pulse is retained, the CCCF assumes the following form: The results in Fig. 7 are in agreement with (40) and the theoretical findings presented in Section 3.2.
The MATLAB software receiver used to demonstrate crosscorrelation codeless techniques also implements closed-loop processing with the PLL and DLL detailed in Section 3. The parameters used for the tracking loops are provided in Table 3. Since Doppler aiding from the E1c signal was implemented, it was possible to adopt first-order loops with narrow bandwidths.
Results related to cross-correlation codeless tracking are provided in Fig. 8, which shows the cross-correlation correlator outputs as a function of time. Fig. 8a shows the absolute values of the early, prompt and late correlators. After an initial transient, the DLL is able to balance the early and late components that assume similar amplitudes. The prompt correlator has an amplitude greater than that of the other two correlators showing the proper functioning of the codeless DLL.
The prompt correlator is further analysed in Fig. 8b, which shows the in-phase and quadrature components of the of prompt correlators. After an initial transient, the codeless PLL is able to recover the residual phase of the E1a signal carrier and concentrate the signal energy on the in-phase component of the correlator. The real part of the prompt correlator always assumes positive values. This fact is expected and is the results of the multiplication in (16) that removes the impact of the spreading code and of the navigation message.
These results have been obtained using real cosine BOC(15, 2.5) data and show the feasibility of cross-correlation codeless processing that can be used to monitor encrypted BOC signals.

Conclusions
In this paper, cross-correlation codeless processing was considered as an alternative to the squaring PLL usually adopted for monitoring encrypted BOC signals. While cross-correlation codeless processing is commonly used for tracking signal components on different frequencies, such as the GPS L1 and L2 P(Y) signals, it was adapted here to BOC modulations broadcast on a single frequency. It was shown that BOC signals can be decomposed into two independent data streams that can be used in a way similar to that commonly adopted for the GPS L1 and L2 P(Y) signal pair.
The cross-correlation codeless PLL was detailed along with open-and closed-loop processing. Open-loop processing can be adopted to monitor the CCCF of encrypted signals, which is a function of the received code chips. Open-loop processing was also characterised from a theoretical point of view and a closed-form expression for the CCCF was derived. Closed-loop processing was obtained by implementing a cross-correlation codeless DLL based on early/late processing.
The cross-correlation codeless PLL was analysed from a theoretical point of view and a closed-form expression for the PLL tracking jitter was provided.
Monte Carlo and semi-analytic simulations supported theoretical findings and provided additional insights on the characteristics of cross-correlation codeless processing.
Finally, real Galileo E1 data were used to demonstrate the feasibility of cross-correlation processing that can be effectively used to track encrypted BOC modulations.

Codeless cross-correlation function
In this appendix, the ideal CCCF is derived. The CCCF is evaluated as a function of the delay between the start of a code chip and the accumulation interval over the chip duration. In particular, ỹ 0 [m] and ỹ 1 [m] in (8) have been computed assuming that the integration interval is aligned with the start of a code chip. In the following, a delay, τ is assumed. For the derivation, it is assumed that the carrier Doppler is perfectly recovered.
The ideal CCCF is defined as  (9) and (11) have been used taking into account that the carrier/subcarrier Doppler frequencies and phases are perfectly recovered. The noise terms in (9) and (11) are zero mean and thus are removed when computing the expected value in (41). The accumulation process is depicted in Fig. 9: continuous lines represent the code chips whereas the dashed rectangle represents the accumulation interval. The signal code, c τ 0 [n], is unknown and can be considered as a sequence of pulses modulated by independent random variables taking value in {-1, 1}. τ models the total delay between the start of a code chip and the start of the accumulation interval. When the accumulation over a chip duration is considered, two possibilities can occur: either the two chips over which the accumulation is performed have the same sign or a chip transition occurs. The first case, which occurs with a 0.5 probability, is depicted in Fig. 9a. In this case, the accumulation process is independent from τ and In the absence of filtering, when a chip is approximated as a perfect square pulse The delay, τ, should be always considered in the [0, T sc ] range. This is due to the fact that when τ exceeds T sc , the subsequent chip should be considered to measure the delay. Since the spreading code is unknown, it is possible to measure delays only modulo T sc .