Code and Carrier Phase Tracking Performance of a Future Galileo RTK Receiver

Transcription

1 Code and Carrier Phase Tracking Performance of a Future Galileo RTK Receiver Thomas Pany, Markus Irsigler, Bernd Eissfeller Institute of Geodesy and Navigation, University FAF Munich, Germany Jón Winkel IfEN GmbH, Munich, Germany INTRODUCTION The Institute of Geodesy and Navigation of the University of the Federal Armed Forces (FAF) Munich takes part in a project to develop a future real-time kinematic (RTK) receiver capable of tracking navigation signals broadcasted from the modernized GPS and Galileo. One important point in this project is the assessment of the pseudorange code and phase accuracy of the new navigation signals. It determines the achievable positioning accuracy, a very high code accuracy can potentially improve the success-rate of resolving the integer ambiguities of the phase measurements [6], and it is of importance for the stability of the signal tracking process. Tracking of GPS signals has been extensively discussed in the literature but the results are often not directly applicable for the new signals. Especially binary offset carrier (BOC) signals, which might be part of Galileo as well as of the modernized GPS, need to be investigated in more detail. We implement in Matlab/Simulink and C a simulation of parts of a global navigation satellite system (GNSS) receiver, consisting of the radio frequency (RF) frontend and of the tracking channel. The frontend simulation yields bandlimited signals at the intermediate frequency (IF) and includes the bandpass filter in the satellite and receiver. The simulation of the tracking channel tries to be close to a software correlation (SWC) receiver. We implement standard tracking techniques which have been developed for GPS signals and assess their performance in the new signal environment. No new techniques, (e.g. new multipath mitigation techniques) are discussed here. The simulation gives us the possibility to directly assess the accuracy of the new signals. However, to gain more insight and a deeper understanding we also use and develop analytical formulas and methods. We investigate the pseudorange code and phase measurement accuracy due to thermal noise and due to multipath, both being the two most important accuracy limiting factors. We will also show that for highest precision applications the formula for the ionospheric carrier phase advance should be adopted for BOC signals and we discuss the possibility to track single side-lobes of the BOC signal spectrum. FUTURE NAVIGATION SIGNALS A future RTK receiver will face a signal scenario outlined in [10], [4] and [9]. The signals differ in many parameters like carrier frequency, modulation scheme and data rate such that at a first glance for each signal a detailed analysis should be performed. However, signal tracking and the multipath performance depend to a large extent only on the modulation scheme. Other parameters are of less importance, resp. their influence can be easily included. Thus we subdivide all signals according to their modulation scheme into binary phase shift keying (BPSK) and BOC signals. The first type of signals is described by the code rate (or chipping rate) f c and is denoted as BPSK( f c ). BOC signals are characterized additionally by the subcarrier frequency f s and are denoted as BOC( f s, f c ). Both constants are multiples of 1.03 MHz.

2 Tab. 1 Considered Galileo and GPS signals Signal Frequency [10.3 MHz] Min. Rec. Power [dbw] Typical C/N 0 [dbhz] Modulation Code Chip Length [m] Data Rate [bit/s] L BPSK(1) E-L1-E1, OS BOC(,) E5a BPSK(10) E-L1-E1, PRS BOC(14,) , DF From the GPS and Galileo signals we select four to demonstrate the methods developed in the following sections. The signals are listed in Tab. 1. Each of the signal is representative for all those future navigation signals having the same modulation scheme (with respect to the code and phase measurement accuracy and multipath performance). The well known L1 C/A code signal uses a BPSK(1) modulation scheme which will also be used by the future L civil signal. A BPSK(10) modulation scheme is used by the currently emitted P code on L1 and L, by the future L5 signal and by the Galileo E5a and E5b signal. A BOC(,) signal might be modulated on the Galileo E1-L1-E carrier and could be part of the open service (OS), a BOC(14,) might be modulated on the same carrier and could be part of the public regulated service (PRS). It should be emphasized that the first three signals of Tab. 1 are representative for all freely accessible signals of the modernized GPS and Galileo. Thus they are of special importance for a future civil RTK system. Although the BOC(14,) will be most likely part of the PRS we include it in our analysis too, because it shows the most complex autocorrelation function (ACF) and the most significant difference to the well known BPSK signals. For all signals we choose a minimum received power and a data rate listed in Tab. 1. Also data-free (DF) channels are considered. For Galileo signals it should be noted that these values are only preliminary. The relation of the typical signal-to-noise ratio C/N 0 to the minimum received power can be determined by looking at a L1 C/A-code receiver, which shows typical C/N 0 values of 45 dbhz. Thus the difference between C/N 0 and the minimum received power is -05 dbws if a minimum received power of 160 dbw is assumed for the current L1 C/A code signal. GPS/GALILEO RECEIVER SIMULATION To gain insight into the process of tracking the new navigation signals and to assess their performance we implement a Matlab/Simulink simulation of a GNSS receiver. The simulation consists of two parts which run at a different sample rate. The simulation of the receiver frontend runs at a high sample rate of 1.6 GHz sufficient to simulate all effects at the RF level. The simulation of the tracking channel runs at a sample rate being about MHz and mimics a SWC receiver. The division of the tracking process into parts is based on writing the emitted satellite signal s(t) as st () = ck ptf ( c k). (1) k = The time is denoted as t, the pseudo random noise (PRN) code sequence is c k and p(φ) is the single chip function. For example if a BOC signal is considered p(φ) takes the form p( φ) = rect( φ)sign(sin( πφ f / f )) cos( πφ f / f ). () s c RF c The carrier frequency is f RF, the code rate is f c and the subcarrier frequency f s. The function rect represents a rectangular pulse of unit amplitude and duration. Bandpass filtering of the signal within the satellite to keep the power spectral density within the allocated frequency band is not included in (1). However, it will be included in the receiver frontend (cf. Fig. 1).

3 The signal s(t) is a convolution of the code sequence c k with the waveform of a single chip p(φ). Therefore it is sufficient to consider the effect of ionospheric delays, filtering and even down conversion on p(φ) only. This simplifies the calculations since p(φ) is different from 0 only if 0<φ<1. At any stage the full navigation signal can be reconstructed via convolution of the modified single chip function with the code sequence. Frontend Simulation The frontend simulation includes bandpass filtering of the signal, which might occur in the satellite or in the receiver, and down conversion. As discussed above, it is sufficient to feed only the single chip function into the receiver frontend. The frontend converts the real-valued signal from f RF to the IF. Bandpass filtering occurs within the satellite payload and within the receiver frontend. Provided the filters in the satellite payload do not distort the signal too much and that they are wider than the receiver frontend filters, the effect of the payload filters would only correspond to a renormalization of the signal-to-noise level. However, it should be noted that the bandpass filter in the receiver frontend can be modified to include also the effect of the satellite payload. A implementation of a combined GPS/Galileo frontend for L1 is shown in Fig. 1. An IF of 0 MHz is used. The incoming signal is bandpass filtered by a finite impulse response filter of order 104 and a bandwidth of 3 MHz and is multiplied by the complex nominal carrier with the frequency f RF. The output of the multiplier is lowpass filtered and the resulting complex signal p IF is stored for further processing. The output p IF is resampled with the sample rate of the tracking channel simulation. Note that all parameters of the frontend (e.g. the bandwidth of the filters) can easily be changed and adopted for tracking of other navigation signals. Fig. 1 Matlab/Simulink simulation of the GNSS receiver frontend for L1, resp. E-L1-E1. Both filters in the frontend have a constant group delay within the passband region and thus no signal deformation is expected. It should however be noted that the behaviour of analog filters used in current RTK receivers differs quite significantly from this ideal behaviour [7]. The net effect of both filters in Fig. 1 on the navigation signal (within the passband region) is a constant delay, called receiver hardware delay. Tracking Channel Simulation The block diagram of the tracking channel is shown in Fig.. In addition to Matlab/Simulink, this simulation uses the programming language C to implement time critical parts (e.g. signal generator, code generator). It runs in a batch processing mode to improve the performance. At each processing step a number of n f samples is processed within the channel. The received satellite signal at the in-phase (I) and quadra-phase (Q) channels at IF level is generated using the IF signal generator james. It uses the output p IF of the frontend simulation to give a realistic signal. The simulation uses floating-point numbers to represent the signal samples. The effect of analog to digital conversion is not included into the simulation. Gaussian white noise is added to the IF signal. From () one sees that the power of the emitted signal is 1. Reduction of the signal power along the propagation path is not considered, only filtering within the frontend removes parts of the signal power. The variance σ N ² of the Gaussian noise is chosen to achieve the required signal-to-noise ratio C/N 0 at the output of the IF signal generator, C/ N0 sf SA N σ =. (3)

4 Fig. Tracking channel simulation of the GNSS receiver The signal power at IF level is denoted as s, f SA is the sample rate. Signal plus noise are multiplied in the module Doppler Removal with a sin and cos wave (i.e. a complex sinusoidal wave) which are generated within that module. The input f, which is connected to the PLL output, steers the internal frequency generator. If the PLL locks perfectly onto the signal the I-channel output of the module Doppler Removal contains the whole navigation signal and the Q- output contains only noise. The code correlation module contains three complex correlators consisting of the code generators and so-called dotproduct units, which multiply the incoming signal with the reference (early, punctual and late) codes and output the sum. In each step a number of n f samples is processed and thus the predetection integration time T D is given by the product of n f with the inverse of the sample rate f SA. The code correlation module can track BPSK as well as BOC signals. The latter are correlated with the full reference signal including also the subcarrier square wave This tracking mode is called ACF tracking in contrast to side-lobe tacking described below. For BOC signals it is optimal if the narrow center peak of the code autocorrelation function is tracked to achieve a high signal-to-noise ratio [1]. This can be well controlled in this simulation by properly initializing the receiver code pseudorange. The DLL is realized with an early-power minus late-power or an early minus late discriminator and with a first order loop filter [3]. The coefficients of the filter are chosen according to [16]. Normally a tracking loop bandwidth B DLL of 1 Hz is maintained. The PLL uses a non-coherent arctan, a non-coherent arctan or a coherent discriminator and also a first order filter. The tracking loop bandwidth B PLL is usually 18 Hz. Aiding of the DLL by the PLL is not implemented for this investigation. The tracking channel outputs the estimated carrier phase and code pseudorange with a rate of 1/ T D. With this output a statistical analysis can be performed. For example the pseudorange code measurement accuracy is obtained by evaluating the standard deviation value. CODE TRACKING It addition to simulating the tracking channel we calculate the code accuracy for a non-coherent early-power minus latepower code discriminator by an analytical formula. Since such a calculation is tedious (cf. [5]) we present only the results. To do this, we have to define the correlation function of the incoming signal chip function at IF level with the reference chip function at IF level p REF,

5 Fig. 3 Correlation function R B (φ) for the signals listed in Tab. 1 plotted as a function of chip offset φ. R B ( φ) = p IF ( φ + φ) p REF ( φ ) dφ. (4) φ = For example, for BOC signals p REF takes the form p ( φ) = rect( φ )sign(sin( πφ f / f )). (5) REF s c The correlation functions for all considered signals are shown in Fig. 3. The BPSK signals show one peak, the BOC signals have f s /f c -1 maxima. One nicely sees the effect of bandlimiting the signal which is more expressed for the signals having a high code rate (e.g. BPSK(10)) or a high subcarrier frequency (e.g. BOC(14,)). Furthermore we need the autocorrelation function of the reference code R ( φ) = p ( φ + φ) p ( φ ) dφ (6) C REF REF φ = and the S-function (also named code discriminator in the following) of the early-power minus late-power code discriminator ( B B ) S( φ) = γ R ( φ d/ ) R ( φ + d / ). (7)

6 Fig. 4 Early-power minus late-power S-function S(φ) for the signals listed in Tab. 1 plotted as a function of code offset φ. For all graphs an identical value of 0.05 is chosen for the discriminator spacing d and γ equals -1. The low value of d is chosen to track the central peak of the BOC(14,) correlation function. The value of the constant γ must be chosen such that S(φ) = φ for small values of φ and we would like to point out that for calculating code pseudorange accuracy values we assume that the code pseudorange itself (i.e. the distance between receiver and satellite) vanishes. This implies that S(0) = 0. Fig. 4 shows the S-function for all signals. An identical discriminator spacing is chosen to allow a visual comparison of the graphs. With this definition we evaluate the pseudorange code measurement accuracy σ NELP 1 for an early-power minus latepower discriminator as σ ( n R ( d ) ( R (0) R ( d) ) σ ( R (0) R ( d) )) 16BDLLTDc σ N nf NELP f B C C N C C fc γ = +. (8) The value is given in m and the constant c denotes the speed of light. It is useful to compare the obtained values against the Cramer-Rao lower bound which is given by (cf. [14]) 1 NELP means Non-coherent Early Late Power following [].

7 σ / ( ) f S( f) df IF. BIF c BDLL LB = π fc C/ N 0 B (9) The one-sided bandwidth of the signal at baseband is denoted as B IF and the normalized power spectral density as S(f). Betz et al. claim in [] that in the limit of a vanishing discriminator spacing the early-power minus late-power discriminator (8) should give the same results as (9). However, we were not able to verify this statement; indeed we find an optimal correlator spacing which is different from 0 (see below). Betz et al. claim that the effect of bandlimiting the noise is very important for a narrow discriminator spacing. We investigated this but have to conclude that this effect can be seen only if a sample rate (much) higher than B IF is employed. Indeed Betz et al. work with a continuous time (i.e. an infinite sample rate). We can show that in the limit n f ˆ and f SA ˆ such that T D remains constant our analytical approach, but not exactly (8), gives the same results as (9) if the noise at IF level is assumed to be bandlimited. However, in our simulation, which runs with a sample rate being typically MHz, white noise is generated at the IF level, thus we cannot see the effect of bandlimiting the noise in the simulation. The sample rate of the simulation can be increased and the noise at IF level can be bandlimited, but both operations are computationally very demanding and have not been investigated. To illustrate (8) and (9) we evaluate them for the signals listed in Tab. 1. For the discriminator spacing d, the predetection integration time T D and for the two-sided bandwidth B W of the frontend bandpass filter we use the values listed in Tab.. B DLL is chosen to be 1 Hz and the sample rate f SA of the simulation equals B W. For the one-sided bandwidth of the signal at baseband level B IF in (9) we use B W /. In addition to that we perform a Monte-Carlo (MC) simulation to verify (8). We chose code and phase Doppler shifts according to a relative velocity of 800 m/s between receiver and satellite. The code sequence is chosen to be a C/A code (PRN 5) for simplicity. Since interference effects are not considered here, the actual code sequence shall not be of importance. We run the Matlab/Simulink simulation of Fig. 10 times, each time for 10 s and analyze the difference of the estimated pseudorange minus the true pseudorange. The difference is biased by the receiver hardware delay but the bias has no influence on the estimated standard deviation values. The accuracy of the MC standard deviation values is given by the standard deviation of the standard deviation, obtained via comparing the results from the 10 independent runs. The results are shown in Tab. 3. We see that (8) agrees well with the simulation results, but we also see that the Cramer- Rao lower bound is not reached except for the BOC(14,) signal. To investigate this we try to find the optimal correlator spacing d, i.e. the one with the minimum tracking error. We evaluate (8) as a function of d and show the result in Fig. 5. For all 4 curves we find that they are bounded from below by (9) and all show an increase for very small values of d. The two BPSK signals on L1 and E5a show one minima whereas the two BOC signals on E1-L1-E have a more complicated structure. The peaks of the two BOC graphs correspond to points where the early-power minus latepower S-function has a vanishing derivative. This occurs if the early and late tracking points are located at the positive or negative peaks of the code correlation function R B or at points when R B crosses the x-axes. It is interesting to note that the values for the BOC(14,) signal are very near the Cramer-Rao lower bound. The curve for L1 is proportional to the square-root of d for d > Tab. Receiver parameters Signal d B W T D L MHz 0 ms E-L1-E1, OS 0. 3 MHz 5 ms E-L1-E1, PRS 1/14 3 MHz 1 ms E5a 1 4 MHz 0 ms Tab. 3 Code tracking pseudorange standard deviation [m] Signal Cramer-Rao (9) SWC Formula (8) MC Simulation L / E-L1-E1, OS / E-L1-E1, PRS / E5a /

8 Fig. 5 Pseudorange code measurement accuracy plotted as a function of discriminator spacing calculated from (8). Other signal parameters are taken from Tab. 1 and Tab.. From Fig. 5 the optimal correlator spacing with respect to thermal noise can be extracted and lies below 0.1 for the BPSK(1) and for the two BOC signals. These low values are explained by the high signal bandwidth of the signal at IF level. Note however, that code tracking might not be stable for such low values and a highly dynamic signal coming from a highly variable motion of receiver and/or satellite. This point has to be investigated in the future. For the BPSK(10) signal the optimal correlator spacing is about 0.4, but this curve is very flat. PHASE TRACKING Carrier phase tracking of BOC signals is similar to carrier phase tracking of BPSK signals since the punctual code correlator used for that purpose (cf. Fig. 1) practically removes the complete code structure of the signal (this is fully achieved in the idealized case of an infinite bandwidth signal and if the DLL tracking error vanishes). Thus we expect that the standard formula for the phase measurement variance σ Ph using a Costas phase discriminator (cf. [1]) σ Ph c BPLL 1 = 1+ 4π C/ N0 C/ N0T frf is a good approximation as it is for BPSK signals. D We show in Fig. 6 the E5a pseudorange phase measurement accuracy as a function of the signal-to-noise ratio calculated via (10) and from a MC simulation. The MC simulation runs for 100 s (see section on code tracking). We use an arctan PLL with a tracking loop bandwidth of 18 Hz and a predetection integration time of 1 ms. The simulation results and (10) are in good agreement especially for C/N 0 >40 dbhz. The carrier phase could not be tracked for C/N 0 values below 3 dbhz and cycle slips occur. Fig. 7 shows the same graph (upper blue line and red circles plus error bars) for the E-L1-E1 BOC(14,) signal and three different PLL discriminators. Again (10) resp. (13) is a reasonable well approximation. Note however, that here the simulation time is only 10 s, causing larger error bars compared to Fig. 6. (10)

9 Fig. 6 E5a pseudorange phase measurement accuracy plotted as a function of signal-to-noise ratio. The blue line refers to (10), the red circles plus error bars to the MC simulation. Fig. 7 E-L1-E1, PRS pseudorange phase measurement accuracy plotted as a function of signal-to-noise ratio. The blue lines refer to (10) and (13), the red circles plus error bars to a MC simulation of an arctan discriminator (11), the green circle plus error bars to a MC simulation of an arctan discriminator and the black circles plus error bars to a MC simulation of a coherent PLL (1). Tracking of Data-Free Channels The modernized GPS and Galileo will broadcast navigation signals, where only the code is modulated on the carrier frequency but no data. The main benefit of such data-free channels is that there is no need for the phase discriminator to eliminate the unknown data bit. If I p denotes the punctual I-channel value and Q p the punctual Q-channel value the output of an arctan phase discriminator S ph is given by

10 Sph = atan Qp Ip (11) and the division removes the unknown data bit [3]. As a negative consequence another tracking point occurs at 0.5 cycles, besides the point at 0 cycles. This ambiguity can be reduced by replacing the arctan function in (11) with an arctan function which gives the arc tangent of Q p /I p, taking into account which quadrant the point (Q p, I p ) is in. Thus the discriminator output can be twice as large as in the case of (11) without causing cycle slips or a loss-of-lock. Equivalently one can say, that the tracking error threshold of the PLL is increased by 6 db [4]. The performance at low C/N 0 values can be further improved if a coherent phase discriminator described by S Q n R ( ) ph = p f B (0) (1) is used [1]. In that way the so-called squaring loss is avoided and the variance of the phase measurements is approximately given by (cf. [1]) σ Ph c = 4π frf BPLL C/ N 0. (13) Fig. 7 and Fig. 8 show the improved tracking quality of the arctan discriminator and of the coherent discriminator. We track the identical IF signal with three independent tracking channels. The first channel is equipped with an arctan phase discriminator (11) and an early-power minus late-power code discriminator, the second one with an arctan phase discriminator and an early-power minus late-power code discriminator and the third one uses a coherent phase discriminator (1) and a coherent early minus late code discriminator. In Fig. 8 two cycle slips (at 3 s and 4. s) occur in the first (arctan) tracking channel which do not show up in the other two tracking channels. In Fig. 7 the pseudorange phase measurement accuracy as a function of signal-to-noise ratio is shown. The accuracy is determined from a MC simulation running 10 times, each time for 1 s. Only points without cycle slips are shown. One sees that the arctan and the coherent PLL can track the signal at a lower signal power. Furthermore (10) and (13) are a reasonable good description of the measurement accuracy, but longer simulation runs are necessary to get a better accuracy especially at low C/N 0 values. Fig. 8 E-L1-E1, PRS estimated phase pseudorange plotted as a function of time for a signal-to-noise ratio of 7 dbhz. The blue line refers to an arctan phase discriminator (11), the green to an arctan discriminator and the red line to a coherent phase discriminator (1).

11 Ionospheric Influence On BOC Signals The influence of a homogeneous ionosphere on GPS navigation signals has been well examined [11]. In a mm accurate approximation, neglecting scintillation effects and terms of the order f -n, n>, the ionosphere increases the measured code pseudorange and decreases the phase pseudorange by the following values BPSK 40.3TEC BPSK 40.3TEC ICODE = and I CODE =. (14) frf frf The constant TEC is the total electron content along the signal propagation path. Both formulas refer to the carrier frequency f RF. Since BOC signals, especially the BOC(14,), have the maximum power spectral density at the left and right edge of the allocated frequency band, one might ask if (14) is still valid. Furthermore one could ask if the code correlation function (4) is deformed by the ionosphere, such that code tracking is not possible for high TEC values. Both problems have been investigated in [13] and it is shown that the code correlation function does not change its shape significantly even for high TEC values. As a result code tracking is not affected by the ionospheric influence. Furthermore, the standard formula (14) for the ionospheric code delay gives the delay with mm accuracy. The conclusions are different for the ionospheric phase advance since the phase measurement error due to thermal noise is usually of the order of 1 mm. Therefore it is suggested to replace the standard ionospheric advance formula (14) by I BOC PHASE 40.3TEC 1 1 = + f f f f + f RF RF s RF s (15) to account for the separation of the two main peaks of the BOC signal spectrum. The process of carrier phase tracking itself (cycle slips, signal-to-noise ratio) is not negatively influenced by the ionospheric advance. SIDE-LOBE TRACKING In addition to conventional code tracking (ACF tracking) of BOC signals described above, tracking of just one peak of the BOC spectrum (i.e. side-lobe tracking) has been suggested in various places in the literature, especially in the context of signal acquisition. This method will be described in the following, but no simulation, as in the case of ACF tracking, has been performed. This will be performed within the near future. For BOC signals like the BOC(14,) with extreme spectrum splitting, side-lobe tracking might be a useful means to monitor the ACF tracking for false locks in the code correlation function R B. The procedure described in [1] and [8] involves filtering the side-bands and then acquiring them. However, it can be shown that this side-band filtering is not necessary. Not having to implement a special side-band filter has many advantages: 1) lower implementation complexity and ) no additional distortions are introduced, which is especially important when switching between sideband and ACF mode. Side-band tracking can be realized by correlating the BOC signal with a reference signal, which is only modulated by the code (i.e. without the square-wave). This can of course not happen on the carrier frequency f RF, i.e. on L1 in the case of the BOC(14,) as the code and the code with the square-wave are perpendicular. The reference signal must have a different frequency. An IF f IF different from zero must be chosen and the PLL of Fig. must be initialized with a nonvanishing frequency offset even in the case of no Doppler shift. Intuitively one would assume the correct frequency offset somewhere near the maximum of the spectral side-lobes, although it is not obvious exactly for what reference frequency a correlation will happen. Therefore the frequency offset must be introduced as an unknown in the analysis, which makes the whole derivation very lengthy. Suffice it to say that this analysis can be done and the result is that a correlation does, in fact, occur at the offset frequencies, i.e. at f IF ± nf s, n being an integer. Note that these frequencies do not correspond to the maximum of the spectral side-lobes. The maximum of the spectral side-lobes at IF level are located at approximately 1 fmax fif ± n/+ n π 6 fs π. (16)

12 Normalized correlation function Delay in [chips] Fig. 9 Correlation function between the code alone and a BOC modulated code. The blue and red curve correspond to the real part of a BOC(,) and a BOC(14,). The magenta and the green curve are the corresponding imaginary parts. In Fig. 9 the theoretical correlation functions at the carrier offset are shown for BOC(,) and BOC(14,) signals. Note that the imaginary part is not zero for this kind of cross-correlation. Further note that the amplitude of the correlation function in Fig. 9 is larger than 0.5, meaning that more than half the available power in the signal is received. There are points in the correlation functions where the derivative is zero. This could potentially become a problem when constructing the S-function. Therefore care must be taken when choosing the correlator spacing, as a bad choice might result in a S-function with zero slope at the tracking point (see also Fig. 5). Another concern is that even with a proper choice of d, the tracking loop might get stuck on one if those plateaus. This, however, is not a stable state and noise and other fluctuations would rapidly kick the tracking loop off the zero slope. MULTIPATH PERFORMANCE The presence of multipath signals generally results in ranging errors (code multipath) and carrier phase errors (carrier multipath). The actual ranging or carrier phase errors error depend on various signal and receiver parameters: Signal type (e.g. rectangular chip shape, raised cosine, binary offset carrier) Signal bandwidth Code rate Relative power levels of multipath signals (signal attenuation α due to reflection) Actual number of multipath signals Geometric path delay of multipath signal Correlator spacing d (early minus late) Type of discriminator (e.g. early minus late, early-power minus late power, dot product) Carrier frequency f RF (carrier multipath) Within the framework of this paper, the main focus will be drawn to the code multipath performance of the signals described above. The influence of multipath will be illustrated by means of multipath error envelopes. In these diagrams, the resulting ranging errors are plotted as a function of the geometric path delay (i.e. the geometric path length difference between the direct and the delayed signal component). The computation of multipath error envelopes is based on the assumptions, that the direct signal component is always available (no shadowing effects), that only one multipath signal is present and that the multipath signal undergoes an attenuation of 3 db (α=0.5) [17].

13 In addition to the multipath environment, the multipath performance strongly depends on the characteristics of the incoming signal, namely signal type, signal bandwidth and code rate f c. Signal type and bandwidth determine the shape of the signal s correlation function R B. The correlation functions for the signals discussed above are illustrated in Fig. 3. and can be used to set up the corresponding code discriminators, i.e. the S-functions (7). Although different types of discriminators can be implemented, the following analyses base on the assumption that an early minus late discriminator is used. This type of discriminator is obtained by subtracting a late copy of the correlation function from a corresponding early copy. The correlator spacing between early and late code is set to d=0.05 for all signals, thus allowing an intuitive comparison of each signal s multipath performance. The code discriminators which are used to determine the code multipath performance are illustrated in Fig. 4. Pseudorange determination is generally performed by tracking the zero-crossing of the code discriminator. The presence of multipath signals results in a shift of the tracking point along the x-axis. The resulting offset R can be deemed to be the ranging error caused by the multipath signal. This effect is illustrated in Fig. 10 for the BOC(,) signal. There, a multipath signal with a relative path delay (PD) of 0.5 chips (dotted blue line) superimposes with the direct signal component (solid blue line). As a result, the resulting discriminator (dashed red line) which is used to track the incoming signal is obviously distorted. Additionally, a significant shift R of the zero-crossing (tracking point) along the x-axis can be observed. The resulting multipath error R in chips can be converted to m by simply multiplying with the code chip length c/f c. The multipath error envelopes for each type of signal can be computed by constantly increasing the geometric path delay of the multipath signal (beginning with a relative path delay of 0 m) and determining the corresponding zerocrossing offset R. The error envelopes have been computed twice, firstly for a multipath carrier phase shift of 0 with respect to the direct signal component and secondly for a shift of 180. These computations were carried out for the four signals listed in Tab. 1. The results are summarized in Fig. 10. While maximum multipath errors occur for the BPSK(1) signal, the BOC(14,) shows the most promising multipath performance with ranging errors less than m. The BPSK(10) and the BOC(,) signal exhibit similar maximum ranging errors. As already mentioned, multipath performance is also characterized by the sensitivity in terms of relative path delay. With respect to this criterion, the BPSK(10) signal shows the best performance being sensitive only for multipath signals with a relative path delay of less than 30 m. The BPSK(1) signal, however, is sensitive for multipath signals with a relative path delay of up to approximately 300 m. Fig. 10 Deformation of an early-minus-late discriminator by the influence of multipath (left ) and multipath error envelopes for the BPSK(1), BPSK(10), BOC(,) and BOC(14,) signal. The correlator spacing between early code and late code is d=0.05 for all signals. The correlator spacing has been set to d=0.5 for visualisation purposes

14 The illustration of the multipath error envelopes in Fig. 10 is based on a very narrow correlator spacing. Since the multipath error envelope for each signal was computed by using identical values for the correlator spacing (d=0.05), the multipath performances can be easily compared. On the other hand, the use of narrow correlator spacing together with short code chip lengths (as it is the case for the BPSK(10) signal) might not be feasible because these two parameters define the tracking threshold of the DLL. This threshold is directly proportional to the correlator spacing d and the code chip length c/f c. Thus, further examination is required to determine optimum correlator spacing for each signal. CONCLUSIONS A set of techniques (MC simulation, analytical formulas and algorithms) has been developed to assess the performance of BOC and BPSK navigation signals. The techniques have been applied to assess the performance of a future GNSS receiver tracking a selection of signals. The pseudorange code measurement accuracy due to thermal noise has been analyzed by an analytical formula and by a MC simulation. Both methods are in an excellent agreement. The accuracy has been calculated as a function of the discriminator spacing and the optimal spacing with respect to thermal noise can be extracted. Note that other design parameters (e.g. DLL tracking stability) might give other values and in the receiver design process a compromise must be found. The pseudorange phase measurement accuracy due to thermal noise is evaluated for a BPSK(10) and for a BOC(14,) signal via a MC simulation. The results are in a reasonable agreement with the well known formula for a Costas phase discriminator (10) resp. for a coherent phase discriminator (13). Both do not depend on the modulation scheme. The performance gain of an arctan phase discriminator and of a coherent phase discriminator to track data-free channels at low signal-to-noise ratio values was demonstrated. For BOC signals an adopted formula for the ionospheric advance is given and the possibility to track only one side-lobe of the BOC signal spectrum (i.e. side-lobe tracking) has been outlined. The code multipath performance has been derived by algorithms analyzing the code correlation function (4) which was generated via the simulation of the GNSS receiver frontend. This provides an easy way to generate the bandlimited correlation functions. The multipath envelopes for the 4 different signals have been compared under the assumption of an identical discriminator spacing. We conclude that tracking of BOC signals is possible with techniques developed originally for BPSK signals. The performance of the BOC signals (especially code multipath and code accuracy) is quite different as compared to the well known GPS signals. Consequently the new signals must be evaluated in more detail as well as the possibility to implement new tracking techniques. In addition to that also other parameters (e.g. acquisition, navigation data demodulation, interference, tracking stability, carrier multipath, ) must be considered to assess the overall performance of a future RTK system. ACKNOWLEDGMENTS The investigations and developments of a future GNSS RTK receiver are supported within the scope of the research project FKZ: 50NA0003 in contract with DLR. REFERENCES [1] Barker B., J. Betz, J. Clark, J. Correia, J. Gillis, S. Lazar, K. Rebhorn and J. Straton (000): Overview of the M- Code Signal, Proc. ION 000, National Technical Meeting, January. [] Betz J. and K. Kolodziejski (000): Extended Theory of Early-Late Code Tracking for a Bandlimited GPS Receiver, Navigation, Vol. 47, No. 3, pp

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