Accelerated Collective Detection Technique for Weak GNSS Signal ...
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Accelerated Collective Detection Technique for Weak GNSS Signal Environment Lakshay Narula, Keshava P. Singh
Mark G. Petovello
Department of Electronics Engineering IIT-(BHU) Varanasi, India [email protected]
PLAN Group, Department of Geomatics Engineering University of Calgary Calgary, Canada [email protected]
Abstract—Collective Detection is an Assisted-GNSS (A-GNSS) technique for direct positioning, where the information from all satellites in view is combined to enable rapid acquisition and a direct navigation solution. This technique is shown to perform effectively in weak signal environments like indoor navigation, reducing the required signal strength by 10 to 20 dB-Hz. When the signal from satellites cannot be acquired individually, Collective Detection constructively adds information from each satellite together, thus improving sensitivity and directly leading to a position solution. In a sense, the vector-based approach used generally in tracking, is extended to the acquisition stage. However, the existing Collective Detection techniques are computationally intensive and thus have limited practical applications. Also, the transmit-time assistance provided to the receiver is assumed to be of sub-millisecond accuracy, which is not a feasible assumption. This paper looks at these limitations of Collective Detection and aims to mitigate them under the assumption that at least one satellite can be acquired individually. It has been shown that the proposed Accelerated Collective Detection is faster and more efficient than the traditional scheme, and performs equally well in terms of accuracy.
Also the combination of receiver clock instability and user dynamics have to be sufficiently low in order to maintain the coherence of the incoming signal over the integration interval. More recently, it has been proposed that combining information from multiple satellites is also a feasible solution, instead of longer integration from individual satellites. Axelrad et al. [3] have shown that to simultaneously (but separately) acquire four satellites individually with a probability of detection (Pd) of 0.9, a power level of at least 41.6 dB-Hz is required, and for acquisition of single satellite individually it is close to 40 dB-Hz. However, for combined acquisition of four satellites, this requirement goes down to 36.2 dB-Hz which is a significant improvement, as shown in Fig. 1.
Keywords—collective detection; coarse-time estimation; weak signal navigation.
I.
INTRODUCTION
Satellite navigation has become ubiquitous in today’s world, and position and location requirements are not limited to only military and surveying purposes. This demand has now extended to indoor locations and weak signal environments like deep urban canyons. The GNSS signal strength in these scenarios is nominally 10 – 25 dB weaker [1] than the lower bound of -130 dBm observed in open-sky conditions. Hence, a lot of effort is being directed towards development of techniques for acquisition of weak signals in GNSS. To this end, increasing the coherent integration time has proved to be an effective solution [2]. It uses the fact that the noise in which the GNSS signal is buried is zero-mean Gaussian, and when integrated over a longer period of time, accumulates slower than the signal of interest, and thus the required signal-to-noise ratio (SNR) is improved. However, the integration time is limited by the data-bit transitions in the GNSS signal’s navigation message (if present), and requires advanced techniques like data-bit wipe-off for mitigation [2].
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Fig. 1. Theoretical probability of acquisition for a single satellite and combined acquisition. All satellites are assumed to be at the same input C/N0. Figure taken from [3].
The combined acquisition may be done by coherent combination of signals from multiple satellites, as proposed by DiEsposti et al. [4] but this involves large computational effort because carrier phase domain has to be searched. Alternatively, non-coherent integration of individual Cross-Ambiguity functions (CAF, also known as correlograms) may be employed, as proposed by Axelrad et al. [3] and Zhe et al. [5].
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The present work aims to improve the scheme of noncoherent combination of multiple satellites, known as Collective Detection. Some of the shortcomings of the algorithms presented in current literature are identified and subsequent attempts are made to mitigate them. In general, however, it is shown that computational complexity of the algorithm and assumption of very accurate time-assistance are the two key issues with the traditional Collective Detection scheme. To this end, the main contributions of this work are: development of an algorithm that accelerates Collective Detection, and estimation of the coarse-time error as a fifth unknown in the navigation solution.
The pseudorange can be converted to an equivalent codephase using
The rest of the paper is organized as follows. Section II provides a background about the current literature and their shortcomings, and also presents the motivation for this work. Section III describes the proposed modified Collective Detection algorithm and the assumptions and considerations about the same. Section IV elaborates on the kind of tests simulated to compare the performance of the algorithms and section V discusses the results obtained. Section VI concludes the overall contributions of this work.
1. For every point in the position/clock-bias domain search space, compute the differences relative to the center of the grid: ('N, 'E, 'U, 'b).
II.
BACKGROUND
Collective Detection is an Assisted-GNSS (A-GNSS) technique that requires a-priori information about the approximate position of the receiver and the current GNSS time [6]. This aiding may be provided by a cell-phone tower or by other communication channels like WiFi. In contrast to most A-GNSS approaches, Collective Detection combines correlograms of individual satellites, which are otherwise too weak to be acquired by conventional techniques, to yield a position solution [3]. This is done by summing the signals from all satellites for specific position and bias values. In particular, a search space is set up around the estimated position and bias of the receiver with realistic uncertainties ('Nmax, 'Emax, 'Dmax and 'bmax) for each dimension. There are three search dimensions for position and a fourth dimension of the common clock-bias, which needs to be estimated because of the drifting clock of the receiver. The assumption here is that the transmittime is very accurately known such that the satellite’s position can be accurately computed. Thereafter, projections are done from the code-phase/Doppler domain (of each satellite) to position/clock-bias domain for each candidate North-EastDown-Clock value in the search space. The data from all satellites is then summed in the position/clock-bias domain. More specifically, for each satellite, if U is the pseudorange at the center of the search space, then the range-offset (difference between U and the new pseudorange) at a location separated by ('N, 'E, 'U, 'b) from the center of the search space is calculated as οߩ ൌ െ ሺܽݖሻ Ǥ ሺ݈݁ሻ οܰ െ ሺܽݖሻ Ǥ ሺ݈݁ሻ οܧ ሺ݈݁ሻ ο ܦ οܾ … (1) where, az is the azimuth of the satellite and el is the elevation of the satellite. As a result, the pseudorange at the new position, U’, is given by ᇱ
ߩ ൌ ߩ οߩ
… (2) … (2)
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߬Ƹ ൌ ݉݀ሺߩᇱ ǡ ܶ ሻ
… (3) where, ࣎ො is the computed code-phase and ࢀࢉ is the period of the ranging code (i.e., 1 ms for GPS L1 C/A code). The correlator value corresponding to the code-phase computed in equation (3) can thus be effectively projected from the code-phase (or correlator) domain to the position/clock-bias domain. With this in mind, the Collective Detection algorithm can be described as follows:
2. For each satellite, compute its position using the ephemeris data and then use this to compute the pseudorange that a receiver placed at that point in the search grid would observe using equations (1) and (2). Use the computed pseudorange to compute the received code-phase. 3. For all satellites, take the correlator values at the computed code-phases (interpolating as necessary) and sum them noncoherently to obtain the collective correlogram. The summation is effectively the projection into the position/clock-bias domain. 4. Select the position/clock-bias value that has the largest power as the final solution. A single iteration of the Collective Detection algorithm is shown in Fig. 2.
Fig. 2. Receiver position, satellite position and clock-bias determine delay for SV.
The details of the Collective Detection approach are presented in [3], [5], [7], [8] and [11], and were only summarized above. This scheme has been shown to provide positioning information accurate up to 50 meters with signal strength of 20 dB-Hz using 1 ms GPS snap-shot with good satellite geometry [3], which is around 20 dB lower than the nominal GPS signal strength. For acquisition of such signals, a coherent integration time of more than 20 ms might be required to acquire them individually. However, the traditional Collective Detection scheme described above faces two key limitations: (a) With appropriate assumptions about the a-priori information available to the A-GPS, the number of points to be evaluated in the traditional Collective Detection scheme is
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huge and requires large computational time. This is mainly due to the uncertainty in the clock-bias, which is assumed to lie between ±0.5 ms. This translates to a search range of about 300 km in clock-bias domain alone and limits the adoption of Collective Detection to many applications due to its computational complexity. (b) The traditional Collective Detection scheme assumes the knowledge of accurate transmit-time of the signals being received, which is not practical. This assumption has been made previously because including the fifth variable of coarsetime would increase the complexity many-fold due to the increase in the search space. The computational complexity of traditional Collective Detection can be reduced to a certain extent by refining the final solution in stages, with decreasing step size [3]. More specifically, initially the search space is centered on the a priori rough position of the receiver, and a coarse search is done, with step size as large as 300 m in the position domain. The search is then re-centered around the peak correlation value and a medium or fine search is done to narrow down to the final solution. The refinement scheme is used in this paper, and the stages are summarized in TABLE I. The values of initial uncertainties are chosen based on the typical assistance data provided by Global System for Mobiles (GSM) [6]. The largest step size in any dimension cannot be larger than the equivalent length of one ranging code (300 m for GPS L1 C/A) [9], and clock-bias dimension must be searched with a maximum step size of half the equivalent length of one ranging code [8]. TABLE I DESCRIPTION OF SEARCH SPACE STAGES USED Item North/East(m) N/E Step Size(m) Down(m) Down Step Size(m) Clock-bias(m) Bias Step Size(m) Number of points
Step 1
Step 2
Step 3
Step 4
±3,000 150 ±600 200 ±150,000 150 23,545,767
±1,000 50 ±300 100 ±15,000 150 2,365,167
±500 25 ±150 50 ±7,500 75 2,365,167
±250 12.5 ±75 25 ±3,000 30 2,365,167
been made to estimate the coarse-time error in the navigation solution in his thesis. A scheme to accelerate the traditional Collective Detection has also been proposed in [7], which has been called as Hybrid Detection. Therein, for each satellite that can be individually acquired, it has been proposed that one dimension in the Collective Detection search space may be eliminated. Singular Value Decomposition has been used to theoretically prove the same. However, several issues that may arise due to Hybrid Detection have not been analyzed. In particular, only preliminary results relating to this scheme have been shown, and no quantitative analysis is provided. Coarse-time error has not been considered in case of Hybrid Detection. The present work aims at alleviating the computational load associated with Collective Detection by cutting down on the common clock-bias search space, which, with reference to TABLE I, is by far the major factor that increases the complexity of the algorithm. The requirement for the proposed algorithm is that at least one satellite must be strong enough to be acquired individually by the acquisition block. This is a practical assumption because even in indoor scenario a few low elevation satellites may be directly visible to the receiver through windows. Considering the large common clock-bias dimension and its corresponding computational complexity, some of the previous works have either assumed perfect knowledge of the clock-bias (Axelrad et al. [3]; DiEsposti, [4]), or assumed that the clockbias will not be greater than a few chips, e.g. ±50 chips (Bradley et al., [8]). However, with the proposed acceleration in place, the present work assumes an uncertainty over the complete clock-bias domain. This is important because clockbias information cannot be easily provided through external aiding and so it is often not practical to make any assumptions on this dimension. Also, once the computational load is reduced, the coarsetime error may also be estimated as a fifth unknown. Hence, the fine-time assistance assumption is also dealt with in this paper and results of the simulation are presented. III.
Even though this refinement process is faster than performing a fine search on the complete initial search space in one stage, it is clear that the traditional Collective Detection method is still computationally challenging. Projections of almost 30 million points need to be done to arrive at a navigation solution. As an extension, there is a need for further acceleration in the Collective Detection algorithm. Furthermore, J. W. Cheong [7] has analyzed the effect of coarse-time error on the positioning estimate of Collective Detection and has also conducted a case study on the typical errors in the time-assistance. It has been reported that on an average an error of around 200 ms can be expected in the transmit-time information provided to the receiver. This may lead to additional errors in the pseudorange measurements, which will be different for different satellites. These errors may be as high as 150 m as the velocity of the satellites, relative to the user, may be close to 800 m/s [6]. However, no attempt has
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ACCELERATED COLLECTIVE DETECTION
This section presents the contributions, related assumptions and other considerations of the paper relating to the improvements made in the existing traditional Collective Detection schemes. A. Reduction in clock-bias search domain This sub-section only considers the case where fine-time assistance is provided such that coarse-time estimates are not needed. This is done to better explain the acceleration algorithm proposed in this paper that, although ultimately aimed at the coarse-time estimation scenario, is nevertheless applicable to the fine-time assistance case as well. The scenario where coarse-time needs to be estimated is addressed in sub-section B below. Returning again to TABLE I, it is clear that the clock-bias dimension is much larger than the other three dimensions: 2,001 points for clock-bias versus 41 points for the North and East directions and 7 points for the vertical direction. Hence,
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the motivation is to estimate the clock-bias information without having to consider the entire search space. To this end, in this work, an algorithm has been developed such that if at least one satellite has a line-of-sight (LOS) component (or even a non-line-of-sight (NLOS) component with sufficiently small delay) that is strong enough to be acquired individually by an acquisition algorithm, then the clock-bias search space can be reduced considerably. This is done by computing an approximate clock-bias from the strong signals and using this to dramatically reduce the range of clock-bias values to be searched. With this in mind and assuming a single strong satellite is acquired, the clock-bias can be estimated as follows: 1. For every point in the position domain search space (note that the clock-bias domain is ignored at this stage), compute the position difference relative to the center of the grid: ('N, 'E, 'U).
2 seconds from the actual transmit-time [6]. So errors as large as 1.6 km can be expected in the satellite range due to inaccurate transmit-time information. This error arises because without extracting the transmit time of the signal from the navigation message (as would ideally be done), the transmit time needs to the estimated from the receiver time. By extension, errors in the receiver time translate nearly one-toone to errors in transmit times. Since the transmit time is needed to compute the satellite position, the end result is a range error arising from an incorrect computed range to the satellite. As an example, Fig. 3 shows that a position error of 1 km is obtained in the horizontal plane if the coarse-time error is 500 ms. Note that the plot only shows the north and east components; the height and clock-bias are being held to their final estimated values.
2. For each satellite, compute its position using the ephemeris data and then use this to compute the geometric range (i.e., b = 0) a receiver placed at that point in the search grid would observe. Use the computed geometric range to compute the geometric code-phase. 3. Using the strong satellite’s correlogram, extract the actual/measured code phase. This is possible because the strong signal should have only one large correlator output that is easy to identify. 4. Once the geometric and measured code-phases are available, the two can be differenced to yield an estimate of the clock bias. The new search space in the clock-bias domain is centered at the value obtained from step 4 above, and a nominal range of few meters, rather than the 300 km range in case of the traditional algorithm, may be used for searching. This algorithm is termed as Accelerated Collective Detection. Once the clock-bias value is calculated for a particular point in the search space, projections into the position/clockbias domain proceed as in the original algorithm. The only difference is that the range of clock-bias values to be searched is dramatically reduced. In this paper, once the clock-bias value is calculated, projections are done for all values of clock-bias within ±50 m of this value, at steps of 12.5 m. If more than one satellite with strong signals is present, the procedure explained above is repeated for each one strong satellite. This is necessary because, except for the true position, the clock-bias estimates from each strong satellite would be different. In this case, averaging clock-bias estimates across all strong satellites would not necessarily yield a more accurate estimate. B. Estimating coarse-time error The reduction in clock-bias search domain now allows for estimation of the coarse-time error as a fifth unknown without making the computational effort extremely challenging, unlike the traditional method. The relative velocity between a GPS satellite and receiver may be as high as 800 m/s and the timeassistance provided to A-GNSS may have a maximum offset of
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Fig. 3. Collective correlogram with coarse-time error of 500 ms. Note that the centre of the search space is the true receiver location. The error is close to 1 km.
To estimate coarse-time error, a 5-dimensional search space is considered. The offset from the center of the search space is now given by ('N, 'E, 'U, 'b, 't), where 't is the coarsetime offset from the center of search space. To include the coarse-time dimension, the complete Accelerated Collective Detection algorithm is performed for each value of 't. To do this, it should be noted that for different 't values the computed satellites positions are different. However, only for the correct value of the coarse-time, all the satellites collectively agree to the position/clock-bias solution. In this work, simulations have been done both with and without solving for coarse-time error and its impact on accuracy is noted. It is shown that even if an initial error of within ±2 seconds is introduced in the initial time-assistance, the algorithm can estimate the coarse-time error and provide a navigation solution that is as good as the one with fine timeassistance. C. Confidence Metric This paper also introduces a ‘confidence metric’ for Collective Detection that represents the confidence that the algorithm has in the navigation solution it outputs. A higher
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confidence does not necessarily prove a higher accuracy; an algorithm may be very confident about an incorrect solution it proposes. A better interpretation of the confidence metric would be to see it as the inertia for providing the same solution again if some noise is introduced in the position/clock-bias correlograms. So, a better confidence metric implies that if some random noise were to be added to the correlogram, it would still output the same result. As a consequence, an incorrect solution with a low confidence is better than an incorrect solution with a high confidence. Confidence metric is also an indicator of how prominent the peak is in the collective correlogram. The confidence metric is computed as “the distance from the maxima in the search grid where all the correlators are less than X percent of the maximum correlator.” So, if all correlators within two steps of the maxima are less than X (= 85, say) percent of the maximum peak, and the step size is 150 m, then the confidence metric is (2 * 150) = 300 m. Similarly, if all correlators in the collective correlogram are greater than 85% of the highest correlator value, the confidence metric would be infinite. It must be noted that a lower value of confidence metric (in m) is better than a higher value.
acquire only 1 to 3 satellites, then the Collective Detection algorithm can be accelerated using the proposed algorithm. In this paper, the FFT-based parallel code-phase search has been used for acquisition, with 1 ms coherent integration time and 5 non-coherent integrations. For this acquisition block, only the signals stronger than 40 dB-Hz can be acquired. As shown in Fig. 4, no distinct peak can be observed in the CAF for a 35 dB-Hz signal, whereas a distinct peak is observed for CAF with 45 dB-Hz signal. Hence, for this algorithm, 45 dBHz may be classified as a strong signal whereas 35 dB-Hz may be characterized as weak. Although a signal with 35 dB-Hz power may be acquired easily using advanced techniques, this was not done for this paper as the focus was to compare the two Collective Detection algorithms. If more sensitive acquisition techniques are employed, both the algorithms should perform better at lower signal strengths, and also, accelerated Collective Detection can be done in more GNSS hostile environments. Regardless, the computational efficiencies observed with the proposed Accelerated Collective Detection algorithms should still be realized.
A better confidence metric signifies the following: x
If some of the individual correlograms were to get corrupted by some errors, the algorithm would still be stable and accuracy will be maintained. On the other hand, a worse confidence metric indicates that random errors might change the position solution significantly.
x
The highest correlator peaks, corresponding to the correct position solution, will be concentrated in a smaller region and hence it is easier to calculate a position solution. If a lot of correlators have similar values, the precision of the algorithm is diluted as the correct solution becomes hard to distinguish.
D. Dependence on acquisition sensitivity Another important factor to be considered while using Collective Detection is the acquisition strategy being used to acquire the satellites. The weakest signal that can be acquired is heavily dependent on the acquisition scheme being employed. Some of the commercial High-Sensitivity GPS (HSGPS) receivers can acquire signals as weak as -148 dBm (22 dB-Hz) during a cold start (ublox-M8, [10]), whereas if the traditional FFT-based acquisition is used with 1 ms integration, only the signals above -130 dBm (40 dB-Hz) can be acquired (as shown in Fig. 1). Hence, a signal power that is considered to be weak for one acquisition strategy may actually be strong signal for a different acquisition scheme. If more than four satellites can be acquired individually by the acquisition block, then using Collective Detection is not necessary as the normal estimation techniques are well proven to give a good positioning accuracy. If none of the satellites are strong enough to be acquired individually, then the clock-bias search space cannot be reduced and the traditional Collective Detection algorithm needs to be employed to arrive at a navigation solution. However, if the acquisition block can
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Fig. 4. Cross-ambiguity functions for (a) 35 dB-Hz and (b) 45 dB-Hz signals, showing that 45 dB-Hz can be considered a strong signal and 35 dB-Hz can be considered a weak signal.
E. Other factors and considerations This section mentions other factors that affect the performance of Collective Detection, but have not been analyzed in this paper. x
The effect of satellite geometry on Collective Detection has not been considered in this work. The focus of the paper is to compare and contrast the traditional Collective Detection scheme with the proposed Accelerated Collective Detection technique. The effect of satellite geometry will be identical in both cases (for the same number of unknowns) and, as
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with all satellite-based positioning methods can have a significant effect of position quality. x
For a similar reason, no ionospheric or tropospheric error model was employed to reduce errors in measurement. Although it is important to take care of this source of error in an actual receiver, this paper assumes a similar effect of these errors in both traditional and Accelerated Collective Detection cases. IV.
SIMULATOR DATA SETUP
A Spirent GSS7700 GPS simulator was used to generate GPS data for the comparison of the Accelerated Collective Detection algorithm proposed and the traditional Collective Detection algorithm. The scenario details are summarized below and in TABLE II: x Sampling rate – 12.5 Msps x Centre frequency of antenna – 1575.42 MHz x 16-bit integer complex samples x Initial receiver position – (51.0N, -114.0E) x Receiver velocity – 5 m/s in North and East directions TABLE II SIMULATED SCENARIO DETAILS PRN 5, 11, 18 12, 17, 18, 22, 26, 28 9, 15
LOS power (dB-Hz) 45 35 35
NLOS power (dB-Hz) 40
Fig. 5 shows the acquisition results for the data set used. As mentioned above, PRN 5, 11 and 18 have strong signals and can be acquired using standard methods. Satellites with strength 35 dB-Hz are not acquired and are almost as noisy as the satellites not visible to the receiver. V.
RESULTS AND DISCUSSION
The results presented in this section compare the traditional Collective Detection procedure with the proposed Accelerated Collective Detection algorithm. For the case of traditional Collective Detection, only the 4-variable case has been considered, as it is computationally challenging to solve for the fifth unknown. For Accelerated Collective Detection, cases of 1, 2 and 3 strong signals have been considered for the 4variable case. When considering 1 or 2 strong signals, the other strong signals are completely ignored by the algorithm. Also, coarse-time error has been estimated for cases of 1 and 3 strong signals. A. Accelerated Collective Detection refinement process As mentioned in section II, an iterative refinement procedure is employed to evaluate the navigation solution. This is presented in Fig. 6 for the two strong-satellites case, where initially the a priori position is off from the actual position by about 1 km in both North and East directions. Eventually, the algorithm narrows down to the correct position solution.
NLOS delay (m) 100
TABLE II suggests that a mild indoor scenario has been simulated where some low elevation satellites (PRN 5, 11 and 18) may be directly visible to the receiver through windows and other satellites have weak signals and considerable multipath.
Fig. 6. Example of the refinement procedure for Accelerated Collective Detection for the case with two strong satellites. Height and clock-bias values are held fixed at their final estimates. The two ridges intersect at the center, which is the correct position solution. Note that the sub-figures (a), (b), (c) and (d) correspond to Steps 1, 2, 3 and 4 in TABLE I. Fig. 5. Acquisition results for the data set used. Note that only PRN 5, 11 and 18 can be acquired, which is consistent with the data set scenario.
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It is obvious that in Fig. 6, there are two “ridges” that intersect at the correct N-E position solution. To explain these, it is noted that for all points in the graph, the height and clockbias estimates are held to their final estimated values (i.e., the
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. figure is effective a 2D slice through a 4D space). The final clock-bias estimate, by definition, introduces a bias to the true range and that bias will manifest as a line/ridge in the horizontal plane. With this in mind, the ridges in Fig. 6 correspond to those of the two strong satellites used. The reason there are no values away from these lines is precisely because the strong satellites are used to determine (and limit) the range of clock-bias values to search. This is not to say that these points were not searched at all, but rather that when they are searched, the computed clock-bias is different than the final estimated value and is thus not shown. B. Comparison of computational complexity The major contribution of this work is the proposed acceleration in the Collective Detection algorithm. Fig. 7 shows a comparison of the number of points in the search space at which projections need to be made. These numbers are calculated by summing the number of search points over all 4 steps of the refinement process. Clearly, there is a dramatic reduction in the computational challenge of the Collective Detection procedure. It should be noted that for Accelerated Collective Detection, the number of points also depends on the number of strong satellites being used for clock-bias estimation, as for multiple strong satellites, the complete Accelerated Collective Detection procedure is repeated. Although a higher number of strong satellites lead to higher computational effort in Accelerated Collective Detection, it helps to achieve better accuracy and confidence metric, as will be shown later. Nevertheless, this computational effort is still very low as compared to the traditional case, as shown in Fig. 7.
Fig. 7. A comparison of computational complexity of traditional and accelerated Collective Detection methods. (X-variable-Y stands for Collective Detection with X variables and Y strong satellites).
It is intuitive that computation load required for the 5variable algorithm should be higher than the 4-variable Accelerated Collective Detection algorithm, and is evident in Fig. 7. C. Accuracy of navigation solution In this section, it is shown that using the Accelerated Collective Detection scheme, the accuracy of the navigation solution remains almost as good as the traditional Collective Detection method. Fig. 8 shows the cumulative error histogram for both traditional and Accelerated Collective Detection algorithms, with fine time-assistance assumed for both cases.
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Fig. 8. Comparison of positioning accuracy with fine time-assistance.
The initial a priori estimate of position was set by a random offset within ±1 km in the N-E domain, and the resulting horizontal positioning errors were noted. For the Accelerated Collective Detection algorithms with four unknowns, 1000 runs were conducted for better statistical analysis. However, only 50 runs were possible for the traditional method, owing to its computation time (see Fig. 7). TABLE III shows the 50 percentile and 95 percentile errors for all techniques discussed herein, including the five-state case. It is interesting to note that the Accelerated Collective Detection algorithm gives higher accuracy than the traditional method when more than one strong satellite is present. This may be due to the following: x
As shown in TABLE I, the smallest clock-bias step size for traditional Collective Detection method is taken to be 30 m. However, for the Accelerated Collective Detection, this can be calculated to a higher degree of accuracy using the strong satellite(s).
x
As will be presented in the next sub-section, the confidence metric is better for the Accelerated Collective Detection with more than one strong satellite as the projections are done only for the useful values of clock-bias. This reduces the noise in the collective correlogram and makes the peak more prominent and robust to errors, and hence easier to distinguish. TABLE III COMPARISON OF 50/95 PERCENTILE ERRORS Algorithm
Traditional, 3-strong signals 4-variables, 1-strong signal 4-variables, 2-strong signals 4-variables, 3-strong signals 5-variables, 1-strong signal 5-variables, 3-strong signals
50 Percentile error (m)
95 Percentile error (m)
41.01 45.05 28.96 20.65 37.04 24.79
60.04 100.69 76.42 33.73 107.94 40.35
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. For the single strong satellite case, only one ridge is present on the N-E search space, and hence no intersection of ridges takes place. This results in a poorer confidence metric. As a result, the peak is difficult to distinguish and can be easily corrupted by errors.
scenarios. Here, however, this has not been taken into account, as the focus of the work is only to compare the two algorithms on a level field. For the results presented in this paper, the position was calculated using the top 2% correlator peaks in the collective correlogram, and taking the equally-weighted average of the positions of all such correlators. D. Comparison of confidence metric This section shows that another advantage of the Accelerated Collective Detection scheme is that it results in a much improved confidence metric if more than one strong satellite is visible to the receiver. This is because in this method, once the clock-bias value has been estimated from the strong satellite, projections are done only for the useful clockbias values. This reduces the addition of noisy projections into the collective correlogram. Also, when more than 1 strong satellite is present, the ridges intersect at the correct position solution and add up. As a consequence, the peak is more prominent and is not easily corruptible to errors/noise in the individual correlograms.
Fig. 9. Horizontal positioning accuracy after estimation of coarse-time error.
Fig. 9 shows the cumulative error histogram for the 5variable Accelerated Collective Detection algorithm, over 100 runs. Here, a random time-assistance error was introduced within ±1 second of the actual value. As is clear from Fig. 9 and TABLE III, this does not affect the accuracy of the accelerated algorithm in a dramatic way, at least for the data set under consideration. The algorithm narrows-down to the correct transmit-time and produces results with accuracy as good as the fine time-assistance case. It is noted that the errors reported for traditional Collective Detection in this section are larger than those presented in Axelrad et al. [3]. The following reasons are thought to be the cause of this discrepancy: x Due to the large computational complexity of Collective Detection, the algorithm is generally made to narrowdown to the final solution in steps, with decreasing stepsize as the uncertainty decreases. The initial coarse iterations have larger step size. Axelrad et al. have only presented the errors in the last fine iteration, which has an uncertainty of ±900 m. However, in actual situations the uncertainty is larger, so this paper reports the error of the overall algorithm, beginning with an uncertainty of ±3 km. x The data set being considered has considerable multipath components, which in some cases is even stronger than the LOS signal (PRN 9 and 15). This may lead to false peaks in the collective correlogram and results in errors in the navigation solution. x Axelrad et al. have presented the accuracy statistics of different techniques that may be used for calculation of final position from the collective correlograms, like, ‘peak-power method’, ‘signal-limited method’ and ‘noise-limited method’, and it was shown that the ‘peakpower method’ performs poorly for weak signal
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Fig. 10. Comparison of confidence metric for (a) Traditional Collective Detection, (b) Accelerated Collective Detection with 3 strong satellites and (c) Accelerated Collective Detection with 1 strong satellite. The likelihood is normalized and the X and Y axis have the same values of errors within ±500m.
Fig. 10 shows sample correlograms for both traditional Collective Detection and Accelerated Collective Detection for one and three strong satellites, at the same step in the narrowdown process with fine assistance provided. It is evident that the number of correlators with a high value is lower in the correlogram with better confidence metric (Fig. 10 b). This makes it easier to estimate a position solution using maximum likelihood technique, than the case where a large number of closely spaced correlators have similar high values (Fig. 10 c). The correlogram with one strong satellite (Fig. 10 c) has poor confidence in its solution, but when the same procedure is
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. repeated for strong satellites at different locations in the sky, the correlators add at the correct position solution and improves the confidence metric, as seen in the correlogram with three strong satellites (Fig. 10 b). Although the ridges from all satellites intersect in the traditional Collective Detection, the confidence metric is not very impressive (Fig. 10 a). This is because in the traditional method, for each possible value of clock-bias, projections are done at all NEU points. The projections for incorrect value of clock-bias are not useful, and thus add noise to the collective correlogram. This results in a poorer confidence metric.
error as a fifth-unknown in the navigation solution, without making this task extremely computationally challenging. Results show that in practical scenarios with coarse-time assistance, the positioning accuracy is not compromised by because the coarse-time error can be estimated. A confidence metric has also been proposed for the Collective Detection algorithm, and it has been shown that in case of more than 1 strong satellite signal being available, an improved confidence metric is observed for the Accelerated Collective Detection algorithm as compared to the traditional method, which results in a more prominent peak in the correlogram and subsequent improvement in the accuracy of the navigation solution. ACKNOWLEDGMENT The authors of this paper would like to thank Vijay Kumar Bellad and Eric Tiantong Ren, students at the PLAN Group, University of Calgary, for their invaluable help in the collection of the data set required for this work. The authors also thank Lakshmi Mukundan, Harsh Sharma and Akash Jain, students at the Department of Electronics Engineering, IIT(BHU), Varanasi, who helped in this research with the best of their abilities. REFERENCES
Fig. 11. Comparison of confidence metric for traditional and accelerated algorithms. (X-variable-Y stands for Collective Detection with X variables and Y strong satellites).
Fig. 11 shows a comparison of confidence metric for the different algorithms discussed in this paper. It is worth recalling that a lower confidence metric implies that the navigation solution is more precise (and not necessarily more accurate). It is noted that the confidence metric is worst for the Accelerated Collective Detection scheme with 1 strong satellite signal. This is expected since no intersection of ridges takes place in this method. However, with two or three strong signals, the confidence metric is better for Accelerated Collective Detection and this also contributes to better positioning accuracy, as seen in TABLE III. VI.
CONCLUSIONS AND FUTURE WORK
Example results on a simulator generated mild indoor scenario data set suggest that Accelerated Collective Detection is a promising upgrade over the traditional Collective Detection algorithm, under the assumption that at least one satellite can be individually acquired. If fine time-assistance is available and one satellite is acquired by the acquisition algorithm, the accelerated algorithm is almost 75 times faster than the traditional method and does not compromise on the accuracy of the solution, with a mean error of approximately 45 m for both cases. In case of 2 or 3 strong satellites available to the receiver, it is interesting to note that the accuracy of the position solution for Accelerated Collective Detection is higher than that for the traditional method, and at the same time the accelerated method is much faster than the traditional method. As the Accelerated Collective Detection is much faster than the traditional method, it allows the estimation of coarse-time
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[1]
Dedes, G., Dempster, A.G., "Indoor GPS positioning - challenges and opportunities", VTC-2005-Fall, Sept., 2005, pp. 412-415. [2] Borio, D., O’Driscoll, C., and Lachapelle, G., "Coherent, Non-coherent, and Differentially Coherent Combining Techniques for Acquisition of New Composite GNSS Signals," IEEE Transactions on Aerospace and Electronic Systems, Vol. 45, No. 3, July 2009, pp. 1227–1240. [3] Axelrad P., Bradley B. K., Donna J., Mitchell M., Mohiuddin S. "Collective Detection and Direct Positioning Using Multiple GNSS Satellites", Navigation: Journal of The Institute of Navigation, Vol. 58, No. 4, Winter 2011, pp. 305-321. [4] DiEsposti, R., "GPS PRN Code Signal Processing and Receiver Design for Simultaneous All-in-View Coherent Signal Acquisition and Navigation Solution Determination", Proceedings of the 2007 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2007, pp. 91–103. [5] Zhe He, V. Renaudin, M. G. Petovello and G. Lachapelle, "Use of High Sensitivity GNSS Receiver Doppler Measurements for Indoor Pedestrian Dead Reckoning", Sensors 2013, 13, pp. 4303-4326. [6] van Diggelen, F., A-GPS: Assisted GPS, GNSS, and SBAS, Norwood, MA, Artech House, 2009. [7] Cheong J. W., "Signal Processing and Collective Detection for Locata Positioning System", Ph.D. Thesis, School of Surveying & Spatial Information Systems, UNSW, March, 2012. [8] Bradley, B. K., Axelrad, P., Donna, J., and Mohiuddin, S., "Performance Analysis of Collective Detection of Weak GPS Signals," Proceedings of the 23rd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2010), Portland, OR, September 21– 24, 2010, pp. 3041–3053. [9] Borre, K., Akos, D., Bertelsen, N., Rinder, P., and Jensen, S., A Software-Defined GPS and Galileo Receiver, A Single-Frequency Approach, ISBN:978-0-8176-4390-4, Birkhauser, Basel, Switzerland, 2007. [10] "Product Catalog 15, Wireless & Positioning Solutions 2014", 2014, p. 39. [11] Langley, R. B., “Innovation: Collective Detection”, GPS World, January 2010.
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Accelerated Collective Detection Technique for Weak GNSS Signal Environment Lakshay Narula, Keshava P. Singh
Mark G. Petovello
Department of Electronics Engineering IIT-(BHU) Varanasi, India [email protected]
PLAN Group, Department of Geomatics Engineering University of Calgary Calgary, Canada [email protected]
Abstract—Collective Detection is an Assisted-GNSS (A-GNSS) technique for direct positioning, where the information from all satellites in view is combined to enable rapid acquisition and a direct navigation solution. This technique is shown to perform effectively in weak signal environments like indoor navigation, reducing the required signal strength by 10 to 20 dB-Hz. When the signal from satellites cannot be acquired individually, Collective Detection constructively adds information from each satellite together, thus improving sensitivity and directly leading to a position solution. In a sense, the vector-based approach used generally in tracking, is extended to the acquisition stage. However, the existing Collective Detection techniques are computationally intensive and thus have limited practical applications. Also, the transmit-time assistance provided to the receiver is assumed to be of sub-millisecond accuracy, which is not a feasible assumption. This paper looks at these limitations of Collective Detection and aims to mitigate them under the assumption that at least one satellite can be acquired individually. It has been shown that the proposed Accelerated Collective Detection is faster and more efficient than the traditional scheme, and performs equally well in terms of accuracy.
Also the combination of receiver clock instability and user dynamics have to be sufficiently low in order to maintain the coherence of the incoming signal over the integration interval. More recently, it has been proposed that combining information from multiple satellites is also a feasible solution, instead of longer integration from individual satellites. Axelrad et al. [3] have shown that to simultaneously (but separately) acquire four satellites individually with a probability of detection (Pd) of 0.9, a power level of at least 41.6 dB-Hz is required, and for acquisition of single satellite individually it is close to 40 dB-Hz. However, for combined acquisition of four satellites, this requirement goes down to 36.2 dB-Hz which is a significant improvement, as shown in Fig. 1.
Keywords—collective detection; coarse-time estimation; weak signal navigation.
I.
INTRODUCTION
Satellite navigation has become ubiquitous in today’s world, and position and location requirements are not limited to only military and surveying purposes. This demand has now extended to indoor locations and weak signal environments like deep urban canyons. The GNSS signal strength in these scenarios is nominally 10 – 25 dB weaker [1] than the lower bound of -130 dBm observed in open-sky conditions. Hence, a lot of effort is being directed towards development of techniques for acquisition of weak signals in GNSS. To this end, increasing the coherent integration time has proved to be an effective solution [2]. It uses the fact that the noise in which the GNSS signal is buried is zero-mean Gaussian, and when integrated over a longer period of time, accumulates slower than the signal of interest, and thus the required signal-to-noise ratio (SNR) is improved. However, the integration time is limited by the data-bit transitions in the GNSS signal’s navigation message (if present), and requires advanced techniques like data-bit wipe-off for mitigation [2].
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Fig. 1. Theoretical probability of acquisition for a single satellite and combined acquisition. All satellites are assumed to be at the same input C/N0. Figure taken from [3].
The combined acquisition may be done by coherent combination of signals from multiple satellites, as proposed by DiEsposti et al. [4] but this involves large computational effort because carrier phase domain has to be searched. Alternatively, non-coherent integration of individual Cross-Ambiguity functions (CAF, also known as correlograms) may be employed, as proposed by Axelrad et al. [3] and Zhe et al. [5].
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The present work aims to improve the scheme of noncoherent combination of multiple satellites, known as Collective Detection. Some of the shortcomings of the algorithms presented in current literature are identified and subsequent attempts are made to mitigate them. In general, however, it is shown that computational complexity of the algorithm and assumption of very accurate time-assistance are the two key issues with the traditional Collective Detection scheme. To this end, the main contributions of this work are: development of an algorithm that accelerates Collective Detection, and estimation of the coarse-time error as a fifth unknown in the navigation solution.
The pseudorange can be converted to an equivalent codephase using
The rest of the paper is organized as follows. Section II provides a background about the current literature and their shortcomings, and also presents the motivation for this work. Section III describes the proposed modified Collective Detection algorithm and the assumptions and considerations about the same. Section IV elaborates on the kind of tests simulated to compare the performance of the algorithms and section V discusses the results obtained. Section VI concludes the overall contributions of this work.
1. For every point in the position/clock-bias domain search space, compute the differences relative to the center of the grid: ('N, 'E, 'U, 'b).
II.
BACKGROUND
Collective Detection is an Assisted-GNSS (A-GNSS) technique that requires a-priori information about the approximate position of the receiver and the current GNSS time [6]. This aiding may be provided by a cell-phone tower or by other communication channels like WiFi. In contrast to most A-GNSS approaches, Collective Detection combines correlograms of individual satellites, which are otherwise too weak to be acquired by conventional techniques, to yield a position solution [3]. This is done by summing the signals from all satellites for specific position and bias values. In particular, a search space is set up around the estimated position and bias of the receiver with realistic uncertainties ('Nmax, 'Emax, 'Dmax and 'bmax) for each dimension. There are three search dimensions for position and a fourth dimension of the common clock-bias, which needs to be estimated because of the drifting clock of the receiver. The assumption here is that the transmittime is very accurately known such that the satellite’s position can be accurately computed. Thereafter, projections are done from the code-phase/Doppler domain (of each satellite) to position/clock-bias domain for each candidate North-EastDown-Clock value in the search space. The data from all satellites is then summed in the position/clock-bias domain. More specifically, for each satellite, if U is the pseudorange at the center of the search space, then the range-offset (difference between U and the new pseudorange) at a location separated by ('N, 'E, 'U, 'b) from the center of the search space is calculated as οߩ ൌ െ ሺܽݖሻ Ǥ ሺ݈݁ሻ οܰ െ ሺܽݖሻ Ǥ ሺ݈݁ሻ οܧ ሺ݈݁ሻ ο ܦ οܾ … (1) where, az is the azimuth of the satellite and el is the elevation of the satellite. As a result, the pseudorange at the new position, U’, is given by ᇱ
ߩ ൌ ߩ οߩ
… (2) … (2)
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߬Ƹ ൌ ݉݀ሺߩᇱ ǡ ܶ ሻ
… (3) where, ࣎ො is the computed code-phase and ࢀࢉ is the period of the ranging code (i.e., 1 ms for GPS L1 C/A code). The correlator value corresponding to the code-phase computed in equation (3) can thus be effectively projected from the code-phase (or correlator) domain to the position/clock-bias domain. With this in mind, the Collective Detection algorithm can be described as follows:
2. For each satellite, compute its position using the ephemeris data and then use this to compute the pseudorange that a receiver placed at that point in the search grid would observe using equations (1) and (2). Use the computed pseudorange to compute the received code-phase. 3. For all satellites, take the correlator values at the computed code-phases (interpolating as necessary) and sum them noncoherently to obtain the collective correlogram. The summation is effectively the projection into the position/clock-bias domain. 4. Select the position/clock-bias value that has the largest power as the final solution. A single iteration of the Collective Detection algorithm is shown in Fig. 2.
Fig. 2. Receiver position, satellite position and clock-bias determine delay for SV.
The details of the Collective Detection approach are presented in [3], [5], [7], [8] and [11], and were only summarized above. This scheme has been shown to provide positioning information accurate up to 50 meters with signal strength of 20 dB-Hz using 1 ms GPS snap-shot with good satellite geometry [3], which is around 20 dB lower than the nominal GPS signal strength. For acquisition of such signals, a coherent integration time of more than 20 ms might be required to acquire them individually. However, the traditional Collective Detection scheme described above faces two key limitations: (a) With appropriate assumptions about the a-priori information available to the A-GPS, the number of points to be evaluated in the traditional Collective Detection scheme is
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huge and requires large computational time. This is mainly due to the uncertainty in the clock-bias, which is assumed to lie between ±0.5 ms. This translates to a search range of about 300 km in clock-bias domain alone and limits the adoption of Collective Detection to many applications due to its computational complexity. (b) The traditional Collective Detection scheme assumes the knowledge of accurate transmit-time of the signals being received, which is not practical. This assumption has been made previously because including the fifth variable of coarsetime would increase the complexity many-fold due to the increase in the search space. The computational complexity of traditional Collective Detection can be reduced to a certain extent by refining the final solution in stages, with decreasing step size [3]. More specifically, initially the search space is centered on the a priori rough position of the receiver, and a coarse search is done, with step size as large as 300 m in the position domain. The search is then re-centered around the peak correlation value and a medium or fine search is done to narrow down to the final solution. The refinement scheme is used in this paper, and the stages are summarized in TABLE I. The values of initial uncertainties are chosen based on the typical assistance data provided by Global System for Mobiles (GSM) [6]. The largest step size in any dimension cannot be larger than the equivalent length of one ranging code (300 m for GPS L1 C/A) [9], and clock-bias dimension must be searched with a maximum step size of half the equivalent length of one ranging code [8]. TABLE I DESCRIPTION OF SEARCH SPACE STAGES USED Item North/East(m) N/E Step Size(m) Down(m) Down Step Size(m) Clock-bias(m) Bias Step Size(m) Number of points
Step 1
Step 2
Step 3
Step 4
±3,000 150 ±600 200 ±150,000 150 23,545,767
±1,000 50 ±300 100 ±15,000 150 2,365,167
±500 25 ±150 50 ±7,500 75 2,365,167
±250 12.5 ±75 25 ±3,000 30 2,365,167
been made to estimate the coarse-time error in the navigation solution in his thesis. A scheme to accelerate the traditional Collective Detection has also been proposed in [7], which has been called as Hybrid Detection. Therein, for each satellite that can be individually acquired, it has been proposed that one dimension in the Collective Detection search space may be eliminated. Singular Value Decomposition has been used to theoretically prove the same. However, several issues that may arise due to Hybrid Detection have not been analyzed. In particular, only preliminary results relating to this scheme have been shown, and no quantitative analysis is provided. Coarse-time error has not been considered in case of Hybrid Detection. The present work aims at alleviating the computational load associated with Collective Detection by cutting down on the common clock-bias search space, which, with reference to TABLE I, is by far the major factor that increases the complexity of the algorithm. The requirement for the proposed algorithm is that at least one satellite must be strong enough to be acquired individually by the acquisition block. This is a practical assumption because even in indoor scenario a few low elevation satellites may be directly visible to the receiver through windows. Considering the large common clock-bias dimension and its corresponding computational complexity, some of the previous works have either assumed perfect knowledge of the clock-bias (Axelrad et al. [3]; DiEsposti, [4]), or assumed that the clockbias will not be greater than a few chips, e.g. ±50 chips (Bradley et al., [8]). However, with the proposed acceleration in place, the present work assumes an uncertainty over the complete clock-bias domain. This is important because clockbias information cannot be easily provided through external aiding and so it is often not practical to make any assumptions on this dimension. Also, once the computational load is reduced, the coarsetime error may also be estimated as a fifth unknown. Hence, the fine-time assistance assumption is also dealt with in this paper and results of the simulation are presented. III.
Even though this refinement process is faster than performing a fine search on the complete initial search space in one stage, it is clear that the traditional Collective Detection method is still computationally challenging. Projections of almost 30 million points need to be done to arrive at a navigation solution. As an extension, there is a need for further acceleration in the Collective Detection algorithm. Furthermore, J. W. Cheong [7] has analyzed the effect of coarse-time error on the positioning estimate of Collective Detection and has also conducted a case study on the typical errors in the time-assistance. It has been reported that on an average an error of around 200 ms can be expected in the transmit-time information provided to the receiver. This may lead to additional errors in the pseudorange measurements, which will be different for different satellites. These errors may be as high as 150 m as the velocity of the satellites, relative to the user, may be close to 800 m/s [6]. However, no attempt has
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ACCELERATED COLLECTIVE DETECTION
This section presents the contributions, related assumptions and other considerations of the paper relating to the improvements made in the existing traditional Collective Detection schemes. A. Reduction in clock-bias search domain This sub-section only considers the case where fine-time assistance is provided such that coarse-time estimates are not needed. This is done to better explain the acceleration algorithm proposed in this paper that, although ultimately aimed at the coarse-time estimation scenario, is nevertheless applicable to the fine-time assistance case as well. The scenario where coarse-time needs to be estimated is addressed in sub-section B below. Returning again to TABLE I, it is clear that the clock-bias dimension is much larger than the other three dimensions: 2,001 points for clock-bias versus 41 points for the North and East directions and 7 points for the vertical direction. Hence,
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the motivation is to estimate the clock-bias information without having to consider the entire search space. To this end, in this work, an algorithm has been developed such that if at least one satellite has a line-of-sight (LOS) component (or even a non-line-of-sight (NLOS) component with sufficiently small delay) that is strong enough to be acquired individually by an acquisition algorithm, then the clock-bias search space can be reduced considerably. This is done by computing an approximate clock-bias from the strong signals and using this to dramatically reduce the range of clock-bias values to be searched. With this in mind and assuming a single strong satellite is acquired, the clock-bias can be estimated as follows: 1. For every point in the position domain search space (note that the clock-bias domain is ignored at this stage), compute the position difference relative to the center of the grid: ('N, 'E, 'U).
2 seconds from the actual transmit-time [6]. So errors as large as 1.6 km can be expected in the satellite range due to inaccurate transmit-time information. This error arises because without extracting the transmit time of the signal from the navigation message (as would ideally be done), the transmit time needs to the estimated from the receiver time. By extension, errors in the receiver time translate nearly one-toone to errors in transmit times. Since the transmit time is needed to compute the satellite position, the end result is a range error arising from an incorrect computed range to the satellite. As an example, Fig. 3 shows that a position error of 1 km is obtained in the horizontal plane if the coarse-time error is 500 ms. Note that the plot only shows the north and east components; the height and clock-bias are being held to their final estimated values.
2. For each satellite, compute its position using the ephemeris data and then use this to compute the geometric range (i.e., b = 0) a receiver placed at that point in the search grid would observe. Use the computed geometric range to compute the geometric code-phase. 3. Using the strong satellite’s correlogram, extract the actual/measured code phase. This is possible because the strong signal should have only one large correlator output that is easy to identify. 4. Once the geometric and measured code-phases are available, the two can be differenced to yield an estimate of the clock bias. The new search space in the clock-bias domain is centered at the value obtained from step 4 above, and a nominal range of few meters, rather than the 300 km range in case of the traditional algorithm, may be used for searching. This algorithm is termed as Accelerated Collective Detection. Once the clock-bias value is calculated for a particular point in the search space, projections into the position/clockbias domain proceed as in the original algorithm. The only difference is that the range of clock-bias values to be searched is dramatically reduced. In this paper, once the clock-bias value is calculated, projections are done for all values of clock-bias within ±50 m of this value, at steps of 12.5 m. If more than one satellite with strong signals is present, the procedure explained above is repeated for each one strong satellite. This is necessary because, except for the true position, the clock-bias estimates from each strong satellite would be different. In this case, averaging clock-bias estimates across all strong satellites would not necessarily yield a more accurate estimate. B. Estimating coarse-time error The reduction in clock-bias search domain now allows for estimation of the coarse-time error as a fifth unknown without making the computational effort extremely challenging, unlike the traditional method. The relative velocity between a GPS satellite and receiver may be as high as 800 m/s and the timeassistance provided to A-GNSS may have a maximum offset of
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Fig. 3. Collective correlogram with coarse-time error of 500 ms. Note that the centre of the search space is the true receiver location. The error is close to 1 km.
To estimate coarse-time error, a 5-dimensional search space is considered. The offset from the center of the search space is now given by ('N, 'E, 'U, 'b, 't), where 't is the coarsetime offset from the center of search space. To include the coarse-time dimension, the complete Accelerated Collective Detection algorithm is performed for each value of 't. To do this, it should be noted that for different 't values the computed satellites positions are different. However, only for the correct value of the coarse-time, all the satellites collectively agree to the position/clock-bias solution. In this work, simulations have been done both with and without solving for coarse-time error and its impact on accuracy is noted. It is shown that even if an initial error of within ±2 seconds is introduced in the initial time-assistance, the algorithm can estimate the coarse-time error and provide a navigation solution that is as good as the one with fine timeassistance. C. Confidence Metric This paper also introduces a ‘confidence metric’ for Collective Detection that represents the confidence that the algorithm has in the navigation solution it outputs. A higher
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confidence does not necessarily prove a higher accuracy; an algorithm may be very confident about an incorrect solution it proposes. A better interpretation of the confidence metric would be to see it as the inertia for providing the same solution again if some noise is introduced in the position/clock-bias correlograms. So, a better confidence metric implies that if some random noise were to be added to the correlogram, it would still output the same result. As a consequence, an incorrect solution with a low confidence is better than an incorrect solution with a high confidence. Confidence metric is also an indicator of how prominent the peak is in the collective correlogram. The confidence metric is computed as “the distance from the maxima in the search grid where all the correlators are less than X percent of the maximum correlator.” So, if all correlators within two steps of the maxima are less than X (= 85, say) percent of the maximum peak, and the step size is 150 m, then the confidence metric is (2 * 150) = 300 m. Similarly, if all correlators in the collective correlogram are greater than 85% of the highest correlator value, the confidence metric would be infinite. It must be noted that a lower value of confidence metric (in m) is better than a higher value.
acquire only 1 to 3 satellites, then the Collective Detection algorithm can be accelerated using the proposed algorithm. In this paper, the FFT-based parallel code-phase search has been used for acquisition, with 1 ms coherent integration time and 5 non-coherent integrations. For this acquisition block, only the signals stronger than 40 dB-Hz can be acquired. As shown in Fig. 4, no distinct peak can be observed in the CAF for a 35 dB-Hz signal, whereas a distinct peak is observed for CAF with 45 dB-Hz signal. Hence, for this algorithm, 45 dBHz may be classified as a strong signal whereas 35 dB-Hz may be characterized as weak. Although a signal with 35 dB-Hz power may be acquired easily using advanced techniques, this was not done for this paper as the focus was to compare the two Collective Detection algorithms. If more sensitive acquisition techniques are employed, both the algorithms should perform better at lower signal strengths, and also, accelerated Collective Detection can be done in more GNSS hostile environments. Regardless, the computational efficiencies observed with the proposed Accelerated Collective Detection algorithms should still be realized.
A better confidence metric signifies the following: x
If some of the individual correlograms were to get corrupted by some errors, the algorithm would still be stable and accuracy will be maintained. On the other hand, a worse confidence metric indicates that random errors might change the position solution significantly.
x
The highest correlator peaks, corresponding to the correct position solution, will be concentrated in a smaller region and hence it is easier to calculate a position solution. If a lot of correlators have similar values, the precision of the algorithm is diluted as the correct solution becomes hard to distinguish.
D. Dependence on acquisition sensitivity Another important factor to be considered while using Collective Detection is the acquisition strategy being used to acquire the satellites. The weakest signal that can be acquired is heavily dependent on the acquisition scheme being employed. Some of the commercial High-Sensitivity GPS (HSGPS) receivers can acquire signals as weak as -148 dBm (22 dB-Hz) during a cold start (ublox-M8, [10]), whereas if the traditional FFT-based acquisition is used with 1 ms integration, only the signals above -130 dBm (40 dB-Hz) can be acquired (as shown in Fig. 1). Hence, a signal power that is considered to be weak for one acquisition strategy may actually be strong signal for a different acquisition scheme. If more than four satellites can be acquired individually by the acquisition block, then using Collective Detection is not necessary as the normal estimation techniques are well proven to give a good positioning accuracy. If none of the satellites are strong enough to be acquired individually, then the clock-bias search space cannot be reduced and the traditional Collective Detection algorithm needs to be employed to arrive at a navigation solution. However, if the acquisition block can
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Fig. 4. Cross-ambiguity functions for (a) 35 dB-Hz and (b) 45 dB-Hz signals, showing that 45 dB-Hz can be considered a strong signal and 35 dB-Hz can be considered a weak signal.
E. Other factors and considerations This section mentions other factors that affect the performance of Collective Detection, but have not been analyzed in this paper. x
The effect of satellite geometry on Collective Detection has not been considered in this work. The focus of the paper is to compare and contrast the traditional Collective Detection scheme with the proposed Accelerated Collective Detection technique. The effect of satellite geometry will be identical in both cases (for the same number of unknowns) and, as
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with all satellite-based positioning methods can have a significant effect of position quality. x
For a similar reason, no ionospheric or tropospheric error model was employed to reduce errors in measurement. Although it is important to take care of this source of error in an actual receiver, this paper assumes a similar effect of these errors in both traditional and Accelerated Collective Detection cases. IV.
SIMULATOR DATA SETUP
A Spirent GSS7700 GPS simulator was used to generate GPS data for the comparison of the Accelerated Collective Detection algorithm proposed and the traditional Collective Detection algorithm. The scenario details are summarized below and in TABLE II: x Sampling rate – 12.5 Msps x Centre frequency of antenna – 1575.42 MHz x 16-bit integer complex samples x Initial receiver position – (51.0N, -114.0E) x Receiver velocity – 5 m/s in North and East directions TABLE II SIMULATED SCENARIO DETAILS PRN 5, 11, 18 12, 17, 18, 22, 26, 28 9, 15
LOS power (dB-Hz) 45 35 35
NLOS power (dB-Hz) 40
Fig. 5 shows the acquisition results for the data set used. As mentioned above, PRN 5, 11 and 18 have strong signals and can be acquired using standard methods. Satellites with strength 35 dB-Hz are not acquired and are almost as noisy as the satellites not visible to the receiver. V.
RESULTS AND DISCUSSION
The results presented in this section compare the traditional Collective Detection procedure with the proposed Accelerated Collective Detection algorithm. For the case of traditional Collective Detection, only the 4-variable case has been considered, as it is computationally challenging to solve for the fifth unknown. For Accelerated Collective Detection, cases of 1, 2 and 3 strong signals have been considered for the 4variable case. When considering 1 or 2 strong signals, the other strong signals are completely ignored by the algorithm. Also, coarse-time error has been estimated for cases of 1 and 3 strong signals. A. Accelerated Collective Detection refinement process As mentioned in section II, an iterative refinement procedure is employed to evaluate the navigation solution. This is presented in Fig. 6 for the two strong-satellites case, where initially the a priori position is off from the actual position by about 1 km in both North and East directions. Eventually, the algorithm narrows down to the correct position solution.
NLOS delay (m) 100
TABLE II suggests that a mild indoor scenario has been simulated where some low elevation satellites (PRN 5, 11 and 18) may be directly visible to the receiver through windows and other satellites have weak signals and considerable multipath.
Fig. 6. Example of the refinement procedure for Accelerated Collective Detection for the case with two strong satellites. Height and clock-bias values are held fixed at their final estimates. The two ridges intersect at the center, which is the correct position solution. Note that the sub-figures (a), (b), (c) and (d) correspond to Steps 1, 2, 3 and 4 in TABLE I. Fig. 5. Acquisition results for the data set used. Note that only PRN 5, 11 and 18 can be acquired, which is consistent with the data set scenario.
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It is obvious that in Fig. 6, there are two “ridges” that intersect at the correct N-E position solution. To explain these, it is noted that for all points in the graph, the height and clockbias estimates are held to their final estimated values (i.e., the
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. figure is effective a 2D slice through a 4D space). The final clock-bias estimate, by definition, introduces a bias to the true range and that bias will manifest as a line/ridge in the horizontal plane. With this in mind, the ridges in Fig. 6 correspond to those of the two strong satellites used. The reason there are no values away from these lines is precisely because the strong satellites are used to determine (and limit) the range of clock-bias values to search. This is not to say that these points were not searched at all, but rather that when they are searched, the computed clock-bias is different than the final estimated value and is thus not shown. B. Comparison of computational complexity The major contribution of this work is the proposed acceleration in the Collective Detection algorithm. Fig. 7 shows a comparison of the number of points in the search space at which projections need to be made. These numbers are calculated by summing the number of search points over all 4 steps of the refinement process. Clearly, there is a dramatic reduction in the computational challenge of the Collective Detection procedure. It should be noted that for Accelerated Collective Detection, the number of points also depends on the number of strong satellites being used for clock-bias estimation, as for multiple strong satellites, the complete Accelerated Collective Detection procedure is repeated. Although a higher number of strong satellites lead to higher computational effort in Accelerated Collective Detection, it helps to achieve better accuracy and confidence metric, as will be shown later. Nevertheless, this computational effort is still very low as compared to the traditional case, as shown in Fig. 7.
Fig. 7. A comparison of computational complexity of traditional and accelerated Collective Detection methods. (X-variable-Y stands for Collective Detection with X variables and Y strong satellites).
It is intuitive that computation load required for the 5variable algorithm should be higher than the 4-variable Accelerated Collective Detection algorithm, and is evident in Fig. 7. C. Accuracy of navigation solution In this section, it is shown that using the Accelerated Collective Detection scheme, the accuracy of the navigation solution remains almost as good as the traditional Collective Detection method. Fig. 8 shows the cumulative error histogram for both traditional and Accelerated Collective Detection algorithms, with fine time-assistance assumed for both cases.
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Fig. 8. Comparison of positioning accuracy with fine time-assistance.
The initial a priori estimate of position was set by a random offset within ±1 km in the N-E domain, and the resulting horizontal positioning errors were noted. For the Accelerated Collective Detection algorithms with four unknowns, 1000 runs were conducted for better statistical analysis. However, only 50 runs were possible for the traditional method, owing to its computation time (see Fig. 7). TABLE III shows the 50 percentile and 95 percentile errors for all techniques discussed herein, including the five-state case. It is interesting to note that the Accelerated Collective Detection algorithm gives higher accuracy than the traditional method when more than one strong satellite is present. This may be due to the following: x
As shown in TABLE I, the smallest clock-bias step size for traditional Collective Detection method is taken to be 30 m. However, for the Accelerated Collective Detection, this can be calculated to a higher degree of accuracy using the strong satellite(s).
x
As will be presented in the next sub-section, the confidence metric is better for the Accelerated Collective Detection with more than one strong satellite as the projections are done only for the useful values of clock-bias. This reduces the noise in the collective correlogram and makes the peak more prominent and robust to errors, and hence easier to distinguish. TABLE III COMPARISON OF 50/95 PERCENTILE ERRORS Algorithm
Traditional, 3-strong signals 4-variables, 1-strong signal 4-variables, 2-strong signals 4-variables, 3-strong signals 5-variables, 1-strong signal 5-variables, 3-strong signals
50 Percentile error (m)
95 Percentile error (m)
41.01 45.05 28.96 20.65 37.04 24.79
60.04 100.69 76.42 33.73 107.94 40.35
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. For the single strong satellite case, only one ridge is present on the N-E search space, and hence no intersection of ridges takes place. This results in a poorer confidence metric. As a result, the peak is difficult to distinguish and can be easily corrupted by errors.
scenarios. Here, however, this has not been taken into account, as the focus of the work is only to compare the two algorithms on a level field. For the results presented in this paper, the position was calculated using the top 2% correlator peaks in the collective correlogram, and taking the equally-weighted average of the positions of all such correlators. D. Comparison of confidence metric This section shows that another advantage of the Accelerated Collective Detection scheme is that it results in a much improved confidence metric if more than one strong satellite is visible to the receiver. This is because in this method, once the clock-bias value has been estimated from the strong satellite, projections are done only for the useful clockbias values. This reduces the addition of noisy projections into the collective correlogram. Also, when more than 1 strong satellite is present, the ridges intersect at the correct position solution and add up. As a consequence, the peak is more prominent and is not easily corruptible to errors/noise in the individual correlograms.
Fig. 9. Horizontal positioning accuracy after estimation of coarse-time error.
Fig. 9 shows the cumulative error histogram for the 5variable Accelerated Collective Detection algorithm, over 100 runs. Here, a random time-assistance error was introduced within ±1 second of the actual value. As is clear from Fig. 9 and TABLE III, this does not affect the accuracy of the accelerated algorithm in a dramatic way, at least for the data set under consideration. The algorithm narrows-down to the correct transmit-time and produces results with accuracy as good as the fine time-assistance case. It is noted that the errors reported for traditional Collective Detection in this section are larger than those presented in Axelrad et al. [3]. The following reasons are thought to be the cause of this discrepancy: x Due to the large computational complexity of Collective Detection, the algorithm is generally made to narrowdown to the final solution in steps, with decreasing stepsize as the uncertainty decreases. The initial coarse iterations have larger step size. Axelrad et al. have only presented the errors in the last fine iteration, which has an uncertainty of ±900 m. However, in actual situations the uncertainty is larger, so this paper reports the error of the overall algorithm, beginning with an uncertainty of ±3 km. x The data set being considered has considerable multipath components, which in some cases is even stronger than the LOS signal (PRN 9 and 15). This may lead to false peaks in the collective correlogram and results in errors in the navigation solution. x Axelrad et al. have presented the accuracy statistics of different techniques that may be used for calculation of final position from the collective correlograms, like, ‘peak-power method’, ‘signal-limited method’ and ‘noise-limited method’, and it was shown that the ‘peakpower method’ performs poorly for weak signal
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Fig. 10. Comparison of confidence metric for (a) Traditional Collective Detection, (b) Accelerated Collective Detection with 3 strong satellites and (c) Accelerated Collective Detection with 1 strong satellite. The likelihood is normalized and the X and Y axis have the same values of errors within ±500m.
Fig. 10 shows sample correlograms for both traditional Collective Detection and Accelerated Collective Detection for one and three strong satellites, at the same step in the narrowdown process with fine assistance provided. It is evident that the number of correlators with a high value is lower in the correlogram with better confidence metric (Fig. 10 b). This makes it easier to estimate a position solution using maximum likelihood technique, than the case where a large number of closely spaced correlators have similar high values (Fig. 10 c). The correlogram with one strong satellite (Fig. 10 c) has poor confidence in its solution, but when the same procedure is
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NOTE: Some figures on this page do not display on older PDF viewers; if you are experiencing problems, try upgrading your viewer. repeated for strong satellites at different locations in the sky, the correlators add at the correct position solution and improves the confidence metric, as seen in the correlogram with three strong satellites (Fig. 10 b). Although the ridges from all satellites intersect in the traditional Collective Detection, the confidence metric is not very impressive (Fig. 10 a). This is because in the traditional method, for each possible value of clock-bias, projections are done at all NEU points. The projections for incorrect value of clock-bias are not useful, and thus add noise to the collective correlogram. This results in a poorer confidence metric.
error as a fifth-unknown in the navigation solution, without making this task extremely computationally challenging. Results show that in practical scenarios with coarse-time assistance, the positioning accuracy is not compromised by because the coarse-time error can be estimated. A confidence metric has also been proposed for the Collective Detection algorithm, and it has been shown that in case of more than 1 strong satellite signal being available, an improved confidence metric is observed for the Accelerated Collective Detection algorithm as compared to the traditional method, which results in a more prominent peak in the correlogram and subsequent improvement in the accuracy of the navigation solution. ACKNOWLEDGMENT The authors of this paper would like to thank Vijay Kumar Bellad and Eric Tiantong Ren, students at the PLAN Group, University of Calgary, for their invaluable help in the collection of the data set required for this work. The authors also thank Lakshmi Mukundan, Harsh Sharma and Akash Jain, students at the Department of Electronics Engineering, IIT(BHU), Varanasi, who helped in this research with the best of their abilities. REFERENCES
Fig. 11. Comparison of confidence metric for traditional and accelerated algorithms. (X-variable-Y stands for Collective Detection with X variables and Y strong satellites).
Fig. 11 shows a comparison of confidence metric for the different algorithms discussed in this paper. It is worth recalling that a lower confidence metric implies that the navigation solution is more precise (and not necessarily more accurate). It is noted that the confidence metric is worst for the Accelerated Collective Detection scheme with 1 strong satellite signal. This is expected since no intersection of ridges takes place in this method. However, with two or three strong signals, the confidence metric is better for Accelerated Collective Detection and this also contributes to better positioning accuracy, as seen in TABLE III. VI.
CONCLUSIONS AND FUTURE WORK
Example results on a simulator generated mild indoor scenario data set suggest that Accelerated Collective Detection is a promising upgrade over the traditional Collective Detection algorithm, under the assumption that at least one satellite can be individually acquired. If fine time-assistance is available and one satellite is acquired by the acquisition algorithm, the accelerated algorithm is almost 75 times faster than the traditional method and does not compromise on the accuracy of the solution, with a mean error of approximately 45 m for both cases. In case of 2 or 3 strong satellites available to the receiver, it is interesting to note that the accuracy of the position solution for Accelerated Collective Detection is higher than that for the traditional method, and at the same time the accelerated method is much faster than the traditional method. As the Accelerated Collective Detection is much faster than the traditional method, it allows the estimation of coarse-time
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