Evaluation of Data/Pilot Tracking Algorithms for GPS L2C Signals Using Software Receiver Kannan Muthuraman, Surendran Konavattam Shanmugam and Gérard Lachapelle Position Location Navigation (PLAN) Group, Department of Geomatics Engineering, University of Calgary, Alberta, Canada. BIOGRAPHY Kannan Muthuraman is a PhD student in the Department of Geomatics Engineering, University of Calgary. He received his B.E. in Electronics and Communication Engineering from the College of Engineering, Guindy, Anna University, Chennai, India in 2006. His research interests include GNSS software receiver design for modernized signals. (
[email protected]) Surendran K Shanmugam is a PhD candidate in the same department. He received his MSc in Electrical and Computer Engineering from University of Calgary in 2004 and bachelor’s degree in Electronics and Communication Engineering from College of Engineering, Anna University, Chennai, India 2001. His research interests include GNSS receiver design and ground based wireless location. (
[email protected]) Dr. Gérard Lachapelle is a professor and CRC/iCORE Chair in Wireless Location in the above department, where he heads the PLAN group. He has been involved with GPS development and application since 1980. For more information visit http://plan.geomatics.ucalgary.ca. ABSTRACT The current GPS constellation is being modernized to overcome the limitations of legacy GPS signals. L2C is the civilian signal added to L2 band as a part of modernization efforts. The major change in L2C signal structure is the inclusion of the pilot channel along with the data channel. In the context of tracking, the pilot channel carries the same information about frequency, phase and code errors. This information could be used along with that of the data channel for better tracking performance. This paper investigates the performance of Data/Pilot combined carrier-frequency tracking. A detailed analysis of existing methods to combine the frequency discriminators on data and pilot channel is done. Hybrid discriminators that make ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
use of the data and pilot channel’s coherent integration output directly are proposed. Consequently, the performance of different possible combinations of the discriminators to form estimates of frequency error is analysed. They are evaluated under various / levels. The advantages of using different discriminator combinations are demonstrated based on / . INTRODUCTION Civilian GPS usage is ever growing with new applications emerging at an increasingly higher rate. As the applications rise in number so does the demand in terms of performance and the need for more accurate measurements even under harsh and weak signal environments. The performance of legacy GPS signals is severely degraded in such environments. Due to the presence of data bits on the C/A code, coherent integration time cannot be extended beyond the data-bit period of 20 ms. Furthermore, the phase or frequency discriminators used in the tracking loops should be insensitive to the data bit transitions to ensure the stability of the loops. For carrier phase tracking, Costas loops should be used at the expense of increasing the tracking threshold by 6 dB and the introduction of the πphase ambiguity (Kaplan 2006). Novel methods to get better tracking performance under such harsh environments using L1-C/A code belong to an active field of research. As a part of GPS modernization, new signals are added at the L2 & L5 frequency. L2C is the civilian signal transmitted at 1227.5 MHz and L5 at 1176.45 MHz. L2C is of our primary concern in this paper because of the availability of three IIR-M satellites (PRN 12, 17 & 31) currently transmitting the signal. The L2C signal carries a chip-by-chip time multiplexed moderate length Code (CM) with a code-length of 10230 chips and a longer length Code (CL) with 767, 250 chips. Both CM and CL are generated using a short-cycled 27 bit Linear Feedback Shift Register (LFSR). A unique property of CM/CL code is that the number of ones and zeros are equal and 1 / 11
synchronization between CM and CL code leaves behind only one of 75 possible combinations between them. The navigation data is modulated only on the CM code at 50 bps (20 ms) and hence the CL code serves as a pilot channel (IS-GPS-200-D). This breaks the bond between the data bit and coherent integration time, thereby allowing us to getter better estimates even with lower / and also to use a pure PLL to track the pilot channel (CL) ( Van Dierendonck 1995). The CM and CL codes are clocked at 511.5 KHz, thereby the time multiplexed code is at the same rate as the C/A code, namely 1.023 MHz. Thus the code-period for CM is 20 ms and that for CL is 1.5 s. Once we are in lock with the CM code, since the CM code is perfectly aligned with the data bit, we are in bit-synched mode i.e., there is no need for separate bit synchronization algorithm. Acquisition of the L2C signal is achieved initially by acquiring the code-phase and approximate Doppler of the CM code and subsequently, using this information to search over one among the 75 possibilities for CL code acquisition. This works well under moderate / conditions. Direct CL code acquisition is mostly preferred at lower / (Fontana et al 2001). To track a time multiplexed code in the receiver, one possible implementation is zero-padding (Tran & Hegarty 2003), as shown in Table 1. Table 1 - Local Code Generation for L2C CM CM CM Code CL CL CL Code CM CL CM CL Tx Code CM 0 CM 0 CM–Zero 0 CL 0 CL Zero-CL Hence the changes that has to be brought in an existing L1 C/A code and carrier tracking loops for L2C tracking is that (i) the local NCO should be modified to track the Doppler associated with L2 frequency, and (ii) the inclusion of zero-padded local code generators. This could be imagined as two independent tracking loops for every PRN (data and pilot channel for every satellite). Normally tracking loops are initialized with carrier frequency- and code tracking. Phase tracking is initiated after achieving frequency lock (Kaplan 2006). Carrier frequency tracking is preferred at the initial stages over phase tracking as this helps in reducing the time to achieve frequency acquisition, which is rather slow in phase-locked loops. For example, in a second order PLL, the time taken for frequency acquisition is proportional to the square of ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
the initial frequency error. But when we use a frequency discriminator, the pull-in time reduces to the logarithm of the initial frequency error (Gardner 2005). It should be emphasized that, though independent tracking loops for data and pilot channel might work well, it is advantageous to combine the data and pilot channel information. The pilot channel conveys the same information – code phase, carrier phase offset and carrier frequency offset – as a data channel. Further the noise corrupting the pilot channel (CL) coherent integration outputs is statistically independent of the data channel coherent integration outputs (CM) (Van Dierendonck et al. 1992). Hence it is advantageous to combine the information from the data and pilot channels. Thus we have two independent measurements for code phase, carrier phase offset and the carrier Doppler. Hence it is more attractive to look at combined data-pilot channel tracking, which uses the information from both data and pilot channel to form the estimates of the tracking parameters. Most work in data-pilot combined tracking has been done for the L5 signal where the data modulation is present on the in-phase channel while the pilot channel is transmitted in quadrature, against time-multiplexed data/pilot channel transmitted in the in-phase channel for L2C. Spilker & Van Dierendonck (1999) proposed a way to combine the data and pilot channel information in the context of L5 code tracking by combining the data and pilot channel coherent integration outputs in a non-coherent manner before feeding them to the discriminators. Hegarty (1999) proposed to run independent discriminators on the data and pilot channels and then combine their outputs using weights that are inversely proportional to their output variances, again in the context of L5. This was later extended to L2C as well by Tran & Hegarty (2002). Alternatively, there also exists a possibility where one could use the data and pilot channel coherent integration outputs directly to derive the frequency error estimates. This paper mainly deals with the possible combinations of the coherent integration outputs that could be used to derive the frequency error. Consequently, different possible combinations of the discriminators are implemented and analysed in comparison with carrier-frequency tracking performance obtained using either data or pilot channel alone. GENERAL FLL THEORY Consider the input signal
of the form 2 / 11
exp
(1)
represents the where represents the data bit, intermediate frequency in rad/sec after down-conversion, the amplitude, the time delay and the phase of the input signal. Let the internal reference signal be of the form ̂ exp
(2)
where, ̂ , & are the estimated code phase delay, carrier frequency and phase, respectively. The error in estimate is given as Δ
, Δ
, Δ
̂
where, Δ , Δ & Δ corresponds to the errors in carrier frequency, phase and code delay estimates respectively. In the absence of data bit transitions in the given coherent integration time , and perfect code synchronization ̂ Δ 0 , the correlated signal integrated over the time period is given by
Δω sin 2 Δω 2
exp
Δω
Δϕ
2
(3)
where is the coherent integration output at time . Since this output depends on the amplitude of the signal, it is necessary to normalize it before feeding it to the discriminator. On normalization with respect to the amplitude | |, one gets exp
Δ
2
(5)
If the two coherent integration outputs belong to the same data bit interval, i.e. , where then the cross product discriminator output is 2
2Δ
Δ (6)
Δ Δ where, Δ is the gain of the frequency discriminator that varies as a function of the frequency error. For the frequency discriminator to provide reliable estimates about the frequency error, the gain has to be positive and should be within the main lobe of the function (Ling 1996). Further, Figure 2 clearly gives the relation between the coherent integration time and the range of frequency error over which the cross product discriminator’s output is valid. For example, if a coherent integration time of 1 ms is chosen, its reliability is guaranteed within the bandwidth of 500 . This limits the coherent integration time that could be used to track the signal during the initial stages. For using a coherent integration time of 20 ms, which is possible in a pilot channel, the Doppler bin should be identified within 25 at the acquisition stage. There is also a possibility of false lock in carrier frequency tracking when the gain tends to zero, namely Δ 0. Thus in order to reduce the burden at the acquisition stage, a minimum possible coherent integration time depending on the / of the signal is used. Further, in applications demanding dynamic stress tolerance, shorter coherent
(4)
Δ
The cross product discriminator, which makes use of the current and the previous normalized coherent integration outputs to derive the frequency error, is shown in Figure 1. The discriminator output is given as (* denoting the complex conjugate)
Normalize I+jQ
x
.
Figure 1- Cross Product Frequency Discriminator ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
Figure 2 - Normalized Cross Product Frequency Discriminator Gain Versus Frequency Error 3 / 11
integration time is more advantageous (Kaplan 2006). The other possible discriminator which provides unity gain over the same frequency error range (Kaplan 2006) as in a cross product discriminator is the ATAN2, as shown in Figure 3. If there is a change in the data bit, , then the cross product discriminator contains the data bit term as
2
2 Δ
Δ
(7)
Hence to remove the effect of data bit, the dot-product should be used (Kaplan 2006 & Van Dierendonck 1995). The dot-product discriminator output is given as
exp Δ
exp Δ
(8)
Here it should be noted that the dot-product type of discriminator brings in an additional reduction in the frequency range as compared to the cross product discriminator. This is because, for cos Δ to produce the exact effect of the data bit transition, the term should be positive. For this condition to be satisfied, Δ
1 4
(9)
The effect on the gain curve ( Δ discriminator is shown in Figure 4.
of the frequency
It could be deduced from the above equations that the
Figure 3 - Normalized ATAN2 Frequency Discriminator Gain Versus Frequency Error ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
reference frequency or the internal carrier signal frequency is assumed to be the same over the two integration intervals, current and previous. From the theory, the updates should be performed only once in every two , but in reality most receivers update every . DATA-PILOT COMBINED TRACKING The digitized band-limited received signal is carrier-wiped off and then correlated with the CM code in the data channel and the CL code in the pilot channel. Thus the incoming noise is spread by two different codes, namely CM and CL (unfiltered local codes). In fact all CM and CL codes belong to the subset of one longer pseudo-random code from a 27 bit LFSR with fixed taps, which is differentiated based on the different initial and final states of the LFSR (IS-GPS-200-D). Thus the noise crosscorrelation at the coherent integration outputs of the data and pilot channel depends only on the CM and CL crosscorrelation properties (Van Dierendonck et al. 1992). For of 1 ms, with the data collected from the IIR-M satellites, the normalized noise cross-correlation of the data and pilot channel’s coherent integration outputs is plotted in Figure 5. Normalization is done with respect to the average of the peaks of the autocorrelation of coherent integration outputs. It is observed that the noise at the coherent integration outputs of the two channels is statistically independent (cross correlation less than 20 log 0.009 ~-40 dB). Thus the frequency discriminators on the data and pilot channel making use of this tends to make estimates of the same frequency error but corrupted by independent noise. It is advantageous to combine the frequency discriminator outputs to get fine
Figure 4 - Normalized Dot Product Frequency Discriminator Gain Versus Frequency Error 4 / 11
estimates of the frequency error. In Tran and Hegarty (2002) the discriminator outputs are combined using weights that are inversely proportional to their variances. While applying the weights to effect the combination, it should be noted that the summation of all weights should be unity to preserve the exact frequency error. If & represents the variance of the data and pilot discriminators respectively, then the weighting factors and of the channel , , discriminators are calculated as (The weights are given subscripts in accordance to the channel from which the current and previous coherent integration outputs are used)
,
,
(10)
where - denotes the output of the discriminator combination, here the suffix “Pure” is used to refer the combination result of “Pure Discriminators” , i.e., those which makes use of their respective channel coherent integration outputs to extract their version of the frequency error. (The suffix used to denote the type of the discriminator - DotProduct/ATAN2/Cross Product - is dropped hereafter for ease of notations) 1 ,
,
1
,
,
(11)
1
(12)
(13)
(14)
This simplifies to ,
,
The major factor that has to be considered while combining the discriminator outputs is that the cross product and ATAN2 discriminator’s output are reliable over twice the frequency error range as compared to the dot product discriminator. Thus the initial frequency error should be resolved within
at the acquisition stage when
combining the data and pilot channels. Further, when using the cross product discriminator on the pilot channel and dot product on the data channel, the gain of the frequency discriminator remains the same over its operating range of frequency errors. But when the ATAN2 discriminator is used on the pilot channel, it provides unity gain. This does ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
Figure 5 – Normalized noise cross correlation between CM and CL coherent integration outputs not pose a serious problem as long as we are in the linear region of the main lobe of the curve or at lower / where the ATAN2 discriminator’s output is too noisy (Julien 2005). ON-THE-FLY VARIANCE ESTIMATOR As discussed in the previous section, the discriminator outputs have to be weighted based on their output variance. This varies with the type of discriminators used and the / of the received signal. The / of the GPS signal varies as a function of many parameters such as altitude and elevation of the satellite, user environment etc. Hence it is necessary to estimate the variance at the outputs of the discriminators on-the-fly. Here the variance estimator as proposed in Moir (2001) is used. It should be ensured that the mean of the signal should be removed before being fed to the variance estimator. This is implemented using a recursive approach with the forgetting factor , as 1 –
(15)
where, is the input signal whose mean has to be estimated and is the estimated-mean at an instant . For our application, the value of 0.95 was found to fit well in a trial-and-error test. Initially the variance estimation is tested using known AWGN signal at the input. Then it was implemented at the output of the FLL Discriminators (Dot-Product – Data Channel & Atan2 - Pilot Channel), before the loop filter and tested using good / signals collected under LOS conditions. Figure 6a (PRN 31 – High / ) & Figure 6b 5 / 11
Figure 6a - Variance Estimator Outputs for PRN 31 (PRN 12 – Moderate / ) shows the standard deviation as calculated by the variance estimator for 2 ms. The disadvantage of using ATAN2 at low / is more visible for PRN 12 (Figure 6b). This is mainly attributed to the self-normalization process of ATAN2 as discussed in Julien (2005). Thus, in the analysis of the data-pilot combined tracking, along with the results that use ATAN2 on the pilot channel, the effect of the cross-product discriminator alternative is also provided. HYBRID FLL DISCRIMINATOR This section explains how the frequency error could also be obtained from the current data channel coherent integration output and the previous pilot channel coherent integration output and also the other possible combination. The integration time in the data and pilot channel should be the same for meaningful frequency error estimate using hybrid discriminators. The normalized coherent integration output of the data channel, similar to (4), is given by
exp
Δ
Δ
2
(16)
Since the frequency error and the phase error remains the same in the pilot channel too, the normalized coherent integration output for the pilot channel is given by the following equation, which is similar to (16), except for absence of data bit: exp
Δ
2
Δ
ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
(17)
Figure 6b - Variance Estimator Outputs for PRN 12 The hybrid discriminator output given as
is
,
,
(18)
where
(19)
and Δ Δ
,
(20)
Thus the hybrid discriminator behaves like a discriminator using either the data or the pilot channel information. Similarly the other combination will yield identical & results. Let this be denoted by . It should be , emphasized that the coherent integration outputs at the input of the hybrid discriminators & , are corrupted by independent noise. Hence it , is expected that the weighted combination of the two hybrid discriminators should have similar performance as that of the combination of the pure discriminators. There is also a possibility of augmenting these proposed hybrid discriminators to the existing combination of the pure discriminators. COMBINATION OF DISCRIMINATORS
PURE
AND
HYBRID
Figure 7 shows the combination of the four discriminators. The weights of the corresponding discriminators are denoted using the notation . For example , 6 / 11
uses current data channel coherent integration output and previous pilot channel’s coherent integration output. ,
From the inputs entering the discriminator, the noise at the outputs of & should be , , independent as we are using different coherent integration outputs to derive the frequency error. The same applies for the outputs of the Pure-Discriminators & . But the noise at the outputs of the Pure and Hybrid discriminator combination shows a gradual decorrelation with / . This is shown in Figure 8. Optimum weights could be calculated using the covariance matrix of the four discriminator outputs due to the presence of noise cross correlation at higher / . But in a GPS receiver, the FLL is normally on track only for a few seconds and the extra computational burden is neither feasible nor attractive. Hence, a simple mean of the four discriminators is sufficient.
Figure 8 – Noise cross-correlation between combination of pure and hybrid discriminators for 4 ms RESULTS TEST METHODOLOGY
If additional computational complexity can be tolerated, the variance estimation on the fly could be used. In such cases the weighting should be done with the similar conditions as stated above, namely ,
,
,
,
1
To validate and evaluate the performance of the proposed combination, different data sets were collected using a GPS hardware simulator, namely the Spirent 7700, as well as live line-of-sight (LOS) data for validation purpose.
(21)
The live and simulator signals are passed through the RFfront end of a NovAtel Euro-3M card which brings down the signal to an IF frequency of 70.1 MHz. The signal is digitized at a sampling rate of 20 MHz with a 2-bit (I&Q) quantization using a FPGA and data acquisition card. The data collected in this form is then re-mapped to {-3, -1, +1,
and 1
&
,
1
,
,
,
Normalize
x
.
.
, .
x x
.
.
.
,
∑ ,
Hybrid Discriminators ,
Normalize
x
.
Figure 7: Combination of Four Discriminators (Pure + Hybrid) ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
7 / 11
Figure 9: CM Code Acquisition for PRN 31 Figure 10: CM/CL Code Tracking for PRN 31 +3} using a software decoder. The approximate Doppler and code-phase offset are obtained using the acquisition modules in MATLAB™ platform. This information along with the decoded data is then processed using a modified version of the PLAN Group’s GSNRx™ (GNSS Software Navigation Receiver) software in Visual C++ platform. L2C – ACQUISITION The CM Code is acquired first using the zero-padded local code and this information is later used to determine one of the 75 possible chunks of CL code. Figure 9 shows the CM acquisition results for the data collected using live data from the IIR-M Satellite PRN 31 with a good / as on 28Feb07. L2C – CM/CL TRACKING The tracking loops are initially validated under high / conditions, collected from the IIR-M Satellites PRN 12, 17 & 31. In the software receiver, correlation begins at the local code epoch, so that the loss due to data bit boundary is minimized. Figure 10 shows in-phase and quadrature coherent integration outputs for the same data set with a coherent integration of 5 ms. The in-phase output of CL is expected to stay positive, but as shown in Figure 10, there is a similar behaviour in the CM code as well. This implies that there is no data modulation on the CM code. The same is observed in different data sets collected for the same PRN’s on 28Feb, 1Mar & 7Mar07. Our observations regarding the absence of data bit modulation in the data ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
channel is also confirmed with the USCG Navigation Center, Navigation Information Service. The signals from the IIR-M satellites are mainly used to validate the software receiver. The evaluation of the discriminator combinations is performed using signals collected using the hardware simulator. Using the simulator allows the received signal power levels to be fixed, thus enabling performance analysis with respect to / . PERFORMANCE COMPARISON OF DIFFERENT DISCRIMINATOR COMBINATIONS The GPS hardware simulator is configured to generate signals for all the satellites in view with a constant predetermined received signal power on L2, while that for the L1 is fixed at -160 dBW. This allows the receiver to track the L1 signals for all data sets, which are used to ensure correct frequency lock on L2. The / of L2 is dropped in steps of 3 dB per data set collected. The output signals from the simulator are also monitored using a L2C capable NovAtel OEM-V3 receiver. This is used to get an estimate of / on L1 & L2. Hence the results are presented as an average across eight PRN’s per / . Initially the software receiver is modified to run in FLL+DLL mode continuously for the entire data set available (~60 s of data set for each / ). This allows one to get fair estimates of the required parameter. Coherent integration time is chosen to be the minimum required to maintain the frequency lock. The parameter used to analyse and compare the performance of the different discriminator combinations is the variance measured at the output of the combination before being fed to the loop filter. The specifications include a FLL Noise bandwidth of 4 Hz and a delay locked loop employing an 8 / 11
envelope detector with a 1-chip (E-L) spacing running independently on the data and pilot channels with a noise bandwidth of 1.5 Hz. Based on the type of discriminators on the pilot channel (ATAN2 or Cross Product), the results obtained are classified into two different cases. The results include the performance of “data-channel-only” (CM Only) and “pilotchannel-only” tracking (CL Only) to emphasize the need for data/pilot combined tracking. Finally the reduction in the noise variance achieved using the four discriminator combinations is compared with that obtained from the combination of the respective pure discriminators alone. CASE 1: USING ATAN2 DISCRIMINATOR ON THE PILOT CHANNEL Different discriminator combination output variances when the ATAN2 is used in the pilot channel are shown in Figure 11. The essence of data-pilot combined tracking is clear from the figure. The advantage could be quantified as a minimum of 2 dB reduction in noise variance as compared to the tracking data channel alone. When the pilot channel is only used to derive the frequency error with the same coherent integration time, uncertainty in the estimate increases with / as compared to data channel only tracking. Though the pilot channel allows one to integrate longer than 20 ms to reduce the noise, in that case the gain curve for the frequency discriminator, as shown in Figure 3, gets narrower demanding fine frequency resolution at the acquisition stage.
Figure 11 - Case 1 - Comparison of Output Variances of Different Discriminator Combination with ATAN2 on Pilot Channel ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
All the discriminator combinations provide similar performance at higher / . The reduction in noise variance by using a combination of pure discriminators is minimal for lower / as compared to other combinations. This is attributed to the presence of the ATAN2 on the pilot channel, whose measurement gets noisier at lower / . The combination of hybrid discriminators provides much better reduction in noise variance at lower / since it makes use of the dot product type discriminators to extract the frequency error where the normalization is done externally. The advantage of using the combination of four discriminators to extract the frequency error against that of using the hybrid discriminator combination is visible only for / less than 36 dB-Hz. This is mainly because of the existing noise cross-correlation between the outputs of the four discriminators at higher / . But as / drops, noise decorrelation results in a further reduction in noise variance. CASE 2: CROSS PRODUCT DISCRIMINATOR ON PILOT CHANNEL Utilizing pilot channel information alone to track carrier frequency provides similar performance as compared to utilizing data channel information alone. The hybrid discriminator combination performs similar to that of the pure discriminator combination across all / . As in the previous case, the combination of four discriminators performs almost the same as the other two combinations at higher / . For / less than 36 dB-Hz, there is a gradual improvement in reduction of noise as compared to
Figure 12 - Case 2 - Comparison of Output Variances of Different Discriminator Combination with Cross Product on Pilot Channel 9 / 11
using either the combination of hybrid or pure discriminators only. This is similar to the previous case. The advantage of using combination of four discriminators in comparison to the other combinations is discussed in detail in the next section. PERFORMANCE COMPARISON DISCRIMINATOR COMBINATION
OF
FOUR-
Figure 13 shows the effective reduction in noise variance by using the combination of four discriminators against that of using the respective combination of pure discriminators only. Both cases are addressed. For the first case, namely using ATAN2 on the pilot channel, the inclusion of the hybrid discriminator provides us with two additional and less noisy measurements and the noise cross-correlation at the output of four discriminators drops with / . Thus one gets significant reduction in the noise variance. For the second case, all the discriminator outputs carry almost the same amount of uncertainity. Hence the reduction in noise variance comes only from the reduction in the noise cross-correlation at the discriminator outputs with / . Hence this effect is pronounced only at lower / . GENERALIZATION To generalize the above results across / for a constant coherent integration time under ideal conditions, a simulation of different discriminator combinations was performed. Data-modulated (50 bps & random phase) carrier was time multiplexed with the pilot carrier wave at 1.023 MHz with a sampling rate of 20 MHz (Perfect Code wipe-off was assumed). Since the discriminator output variance at steady state is required, the local carrier is assumed to be perfectly frequency synchronized with the incoming signal (with random phase error). The noise corrupting the channels is assumed to be independent AWGN. Then the correlated samples are accumulated for coherent integration time ( ) of 5 ms. The discriminator outputs are weighed using the variance estimating loops. For every / the standard deviation is estimated from 2000 discriminator output samples. It is clear from Figure 14 that the performance degradation of the ATAN2 is significant for / lower than 33 dBHz. The performance of the dot product on the data channel and the two different hybrid discriminators is similar across / . The cross product discriminator performs slightly better than the dot product discriminator. At higher / , all the combinations exhibit identical performance, ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
Figure 13 - Comparison of Reduction in Noise Variances Using Four Discriminator Combinations against Respective Combination of Pure Discriminators still it should be noted that they provide reduction in the noise variance as compared to using a single discriminator. There is a clear difference in performance between the combination of pure discriminators that use ATAN2 on the pilot channel and the combination of hybrid discriminators as / drops. The combination of pure discriminators that use the cross product discriminator on the pilot channel performs similar to the combination of hybrid discriminators. The combination of four discriminators with ATAN2 on the pilot channel shows better performance than using either the combination of pure discriminators or hybrid discriminators at lower / . And as expected the best performance at lower / is obtained
Figure 14 - Comparison of Different Discriminator / Discriminator Combination Performance for
10 / 11
by using a combination of four discriminators that use a cross product discriminator in the pilot channel. This validates the previous observations that it is the gradual noise decorrelation as the / drops that provide a better combination. Thus the simulation verifies the results obtained using the hardware data. CONCLUSIONS Utilizing information from the data and pilot channels for carrier frequency tracking has been shown to be advantageous at all / . The combination of the proposed hybrid discriminator has been shown to exhibit similar performance as the combination of pure discriminators (with a cross product on the pilot channel). A reasonable advantage comes in using all the four discriminators at lower / . In such cases, depending on the application, the weight estimation could be done either using a covariance matrix or the variance estimation loops or a simple mean of the discriminator outputs. Simple mean of discriminators is sufficient for all the combinations at higher / , as the individual discriminator output variances do not change significantly. Rather at moderate and lower / , weight estimation based on the discriminator output variances will help to exclude noisy measurements. This is particularly useful when there is a loss of lock in either the data or the pilot channel due to false acquisition. If the implementation is done in software, covariance matrix based weight estimation could be adopted. Possible extensions of the hybrid discriminators to carrier phase and code tracking is currently being investigated. ACKNOWLEDGEMENTS The Informatics Circle of Research of Excellence is acknowledged for its financial support. REFERENCES Fontana, R.D., W. Cheung, P.M. Novak and T.A. Stansell Jr. (2001), The New L2 Civil Signal, ION GPS 2001, 1114 September 2001, Salt Lake City, UT. Gardner, F.M. (2005), Phase Lock Techniques, Third Edition, Jhon Wiley & Sons, Inc., USA.
Julien, O. (2005), Carrier Phase Tracking of Future Data/Pilot Signals, ION GNSS 2005, Long Beach, CA, September 13-16, 2005. Kaplan, E. (2006), Understanding GPS: Principles and Applications, Artech House, Chapter 5, pp 153-241. Ling, F. (1996), Convergence and Optimum MSE of Digital Frequency Locked Loop for Wireless Communications, IEEE 46th Vehicular Technology Conference, 1996. 'Mobile Technology for the Human Race'., 28 April-1 May 1996, Volume 2, pp.1215 – 1219 Moir,T.J. (2001), Automatic Variance Control and Variance Estimation Loops, Journal of Circuits, Systems and Signal Processing, Jan 2001, pp. 1-10, Volume 20 No 1 Spilker Jr, J.J. and A.J. Van Dierendonck (1999), Proposed New Civil GPS Signal at 1176.45 MHz, IONGPS 1999, 14-17 September 1999, Nashville, TN, pp 1717-1725 Tran, M., and C.Hegarty (2003), Performance Evaluations of the New GPS L5 and L2 Civil (L2C) Signals, ION NTM 2003, 22-24 January 2003, Anaheim, CA, pp 521-535 Tran, M., and C.Hegarty (2002), Receiver Algorithms for the New Civil GPS Signals, ION NTM 2002, 28-30 January 2002, San Diego, CA, pp 778-789 Van Dierendonck, A.J. (1995), GPS Receivers, B.Parkinson and J.J.Spilker, Jr., eds., Global Positioning System: Theory and Applications, Volume 1, Chapter 8. American Institute of Aeronautics and Astronautics, Inc., Washington D.C., USA Van Dierendonck, A.J, P.Fenton and T.Ford (1992), Theory and Performance of Narrow Correlator Spacing in a GPS Receiver, Navigation: Journal of the Institute of Navigation, Vol 39, No.3, Fall, pp.265-283. IS-GPS-200-D, Interface Specification IS-GPS-200, Revision D, Interface Revision Notice (IRN)-200D-001, 7 March 2006, Navstar GPS Space Segment / Navigation User Interface.
Hegarty, C.J. (1999), Evaluation of the Proposed Signal Structure for the New Civil GPS Signal at 1176.45 MHz, MITRE Center for Advanced Aviation System Development, Working Note June 1999, Chapter 3. ION GNSS 2007, Fort Worth, TX, Sep 25-28 2007
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