Multicarrier code for the next-generation GPS
- Donglin Wang^{1, 2}Email author,
- Michel Fattouche^{1} and
- Fadhel M Ghannouchi^{1}
https://doi.org/10.1186/1687-1499-2012-185
© Wang et al; licensee Springer. 2012
Received: 6 October 2011
Accepted: 31 May 2012
Published: 31 May 2012
Abstract
This article investigates multicarrier (MC) transmission for next-generation global positioning system (GPS) instead of current spread spectrum signals. A MC code is proposed in this article as an alternative to the coarse/acquisition (C/A) code in GPS. The entire GPS bandwidth for the C/A code is divided into 1 ,024 subcarrier slots. As per our proposed arrangement, each satellite vehicle (SV) takes up only 42 uniformly spaced and non-overlapping subcarrier slots while approximately occupying the same bandwidth as the C/A code in GPS. In this way, the proposed MC code is proved to attain a 4 .73 dB SNR gain compared to the GPS C/A code in terms of Cramer-Rao lower bound for range estimation, which could evidently enhance the GPS receiver's sensitivity. Together with the feature of robustness against multipath effect, the proposed MC code is helpful for urban, tunnel, even indoor and underground positioning. The transmission and reception of the propose MC code is also described, where the range estimation process is explained. Furthermore, the proposed MC code is shown to be robust against narrow band interference. Moreover, the probability of collision between SVs due to Doppler shifts is theoretically analyzed, where the probability of successful positioning is evaluated. Simulation Results show a consistency with our proposed theory.
Keywords
multicarrier code GPS C/A code SNR Gain doppler effect NBI probability of collision1 Introduction
Global navigation satellite system (GNSS) such as the current global positioning system (GPS) or the future European Galileo system can provide a worldwide accurate positioning under an outdoor good environmental conditions [1]. However, GNSS based positioning is not reliable in urban, tunnel, underbridge, indoor or underground environments due to the dense multipath effect as well as non-line-of-sight (NLOS) signal energy attenuation [1]. As it is known, multicarrier (MC) signals, e.g. orthogonal frequency division multiplexing (OFDM) signals [2, 3], are robust against the multipath effect [4–8]. Furthermore, they have a more efficient spectrum usage compared to the current golden codes [9, 10], which thus leads to a SNR gain for range estimation. Therefore, the MC signal is being considered for investigation of the possiblity of its application in next-generation GNSS.
For the best of authors' knowledge, the use of a MC signal as a code of next-generation GNSS is investigated by both [1, 11]. Zanier and Luise [11] discussed about the funamental issues in time-delay estimation of MC signals with applications to next-generation GNSS. This article used a filter bank MC modulation, where the MC modulation has the same power spectral density (PSD) of the current golden code so that they have the same ranging performance. Furthermore, Zanier and Luise [11] derived the fundamental performance of the filter bank MC modulation using Cramer-Rao lower bound (CRLB). Dai et al. [1] tried to propose the OFDM/MC based scheme for the next-generation GNSS. However, authors basically discussed the OFDM communication and the performance of time-of-arrival (TOA) estimation, which are too far from an available GNSS scheme. As we know, it is impossible for all SVs to use the same OFDM signal when the frequency collision between SVs will occur and the data reception will fail without question. Dai et al. [1] only gave a general discussion but did not provide the specific probing signal for each satellite vehicle (SV). Also, it did not compare the OFDM signal with the current GNSS code in terms of ranging and/or positioning accuracy. Furthermore, it did not analyze the scheme's feasibility as an option of the next-generation GNSS.
Differently from the previous literature, this article investigates the possiblity of a unfiltered non-data-bearing MC modulation applied in the next-generation GNSS instead of the current golden code. Specifically, a MC code is proposed as an alternative to the coarse/acquisition (C/A) code for the next-generation GPS. As we all know, GPS transmits three binary codes for target navigation: the pseudo-random noise (PN) based C/A code with 1.023 MHz, the PN-based precise (P) code with 10.23 MHz and the navigation message with 50 bps [12–15]. Among them, the C/A code is for civilian use but performs poor under the dense multipath effect, e.g. positioning in a downtown environment, weak signal detection such as indoor or underground positioning and narrow band interference (NBI).^{a} Consequently, the MC modulation is proposed as a next-generation alternative to the C/A code in order to improve the positioning performance against multipath, weak signal and NBI. Even though the multipath management in the communication area is different from that in the GNSS area, the topic of the multipath management is not addressed in the article.
In our proposed design, the entire GPS bandwidth for the C/A code, i.e. a null-to-null bandwidth 2.048 MHz,^{b} is divided into 1,024 subcarrier slots. Under the assumption^{c} that there are currently 24 SVs in GPS, each SV takes up only 42 uniformly spaced and non-overlapping subcarrier slots as per our proposed arrangement while approximately occupying the whole bandwidth. In this way,^{d} the proposed MC code is proved to attain a 4.73 dB improvement (See Section 3) to that of the C/A code in terms of ranging accuracy. In other words, for a fixed ranging accuracy, the required SNR is 4.73 dB lower than that for the C/A code, indicating a 4.73 dB SNR gain when using the proposed MC codes instead of the current spread spectrum signals. The proposed MC code is also proven to be robust against NBI. Besides, the effect of Doppler shift to positioning process is proved to be negligible.
The remainder of the article is organized as follows. Section 2 describes the proposed MC code for all 24 SVs in GPS. Section 3 proves that the proposed MC code attains a 4.73 dB SNR gain compared to the current C/A code. The transmission and reception of the proposed MC code is described in Section 4, followed by the analysis of NBI effect on the proposed MC code in Section 5. Section 6 thoroughly analyzes the upper bound of Doppler effect on the reception of the proposed probing signals, by defining the probability of collision (POC) and the probability of successful positioning. Simulation results are given in Section 7, followed by conclusion in Section 8.
2 Proposed MC multiple access method for each SV
where A is a constant as shown in Figure 1 and the value of A is determined by the transmitter power.
3 Ranging accuracy and SNR gain
where I_{ D } is given in (1).
MSB in MHz, i.e. $msb\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}\frac{\sqrt{MSB}}{2\pi}\mathsf{\text{MHz}}$, for our proposed MC code with S_{ ID } = 1, 2, ..., 24, and the PN-based C/A code
Probing signal | ID | msb | Signal | ID | msb | Signal | ID | msb | Signal | ID | msb |
---|---|---|---|---|---|---|---|---|---|---|---|
S_{ ID } = 1 | A 1 | 0.5909 | S_{ ID } = 7 | B 3 | 0.5906 | S_{ ID } = 13 | D 1 | 0.5905 | S_{ ID } = 19 | E 3 | 0.5906 |
S_{ ID } = 2 | A 2 | 0.5908 | S_{ ID } = 8 | B 4 | 0.5905 | S_{ ID } = 14 | D 2 | 0.5905 | S_{ ID } = 20 | E 4 | 0.5907 |
S_{ ID } = 3 | A 3 | 0.5908 | S_{ ID } = 9 | C 1 | 0.5905 | S_{ ID } = 15 | D 3 | 0.5905 | S_{ ID } = 21 | F 1 | 0.5907 |
S_{ ID } = 4 | A 4 | 0.5907 | S_{ ID } = 10 | C 2 | 0.5905 | S_{ ID } = 16 | D 4 | 0.5905 | S_{ ID } = 22 | F 2 | 0.5908 |
S_{ ID } = 5 | B 1 | 0.5907 | S_{ ID } = 11 | C 3 | 0.5905 | S_{ ID } = 17 | E 1 | 0.5905 | S_{ ID } = 23 | F 3 | 0.5908 |
S_{ ID } = 6 | B 2 | 0.5906 | S_{ ID } = 12 | C 4 | 0.5905 | S_{ ID } = 18 | E 2 | 0.5906 | S_{ ID } = 24 | F 4 | 0.5909 |
Which indicates that using the proposed MC code in GPS would lead to a 4.73 dB SNR gain compared to the current spread spectrum's gold code.
4 Transmission and reception of the proposed MC code
As known, the data rate of the navigation message in current GPS is 50 bps while the bandwidth used for the proposed MC code is 1.024 MHz. The navigation message is a telemetry message, and the data is transmitted in logical units called frames. For GPS, a frame is 1,500 bits long and takes 30 s to be transmitted. Each frame is divided into five subframes, 300 bits long per subframe. Subframes 1, 2 and 3 contain the high accuracy ephemeris and clock off set data. Subframes 4 and 5 contain the almanac data and some related health and configuration data [12].
The signal is received by Rx RF Front-end and downconverted to a serial baseband signal. The baseband signal is sampled and converted to a parallel data. Passing by DFT, the parallel data is tranformed to frequency domain and passed into parallel/serial converter. The output serial signal, together with the local code generator and local oscillator, is then used for Doppler removal and acquisition [12]. Furthermore, via signal tracking process [12], both the partial delay estimation $\widehat{\tau}$ and NAV are obtained. Given $\widehat{\tau}$ and NAV, the range between transmitter and receiver can be obtained. One note to be pointed out is that the cross-correlation in signal acquistion and tracking is implemented in the frequency domain, which is given in [20].
5 NBI effect
In this section, define the POC as the probability that the NBI hits any part of the mainlobe^{e} of any of the subcarriers. The observation interval (OBI) is selected as 1 ms to evaluate NBI effect, leading to a null-to-null bandwidth of each subcarrier w_{ nn } = 2 kHz. The frequency range out of the available bandwidth 2B = 2.048 MHz, where the probing signal might be hit by a NBI, is w_{ c } = 42w_{ nn } kHz. This is the worst case since the greater the OBI, the narrower the subcarrier band, so the smaller the POC. The NBI effect is represented using POC, where M uncorrelated NBIs are considered, M = 1, 2, ..., 8, and each NBI is referred to be a frequency tone [21].
For example, when M = 1, we have ${p}_{c}=\frac{{w}_{c}}{2B}=4.1\%$ and ${p}_{cc}={\sum}_{k=0}^{3}{C}_{8}^{k}{\left(1-{p}_{c}\right)}^{k}{p}_{c}^{8-k}=5.85\times {10}^{-6}$, so P_{ s } ≈ 100%, indicating that the proposed MC code is robust against NBI.
6 Doppler effect
6.1 SV constellation
6.1.1 Collision analysis between SVs in one orbit
Collision analysis between SVs in each orbit for in-plane receivers, where Collision (S: Sure or N: Never) and CA: central angle
Orbit | Slot ID | CA | DD, kHz | FSBD, kHz | FSAD, kHz | MFR, kHz | Collision |
---|---|---|---|---|---|---|---|
A | A 1-A 2 | 150.11^{o} | -9.5 | 2 | 7.5 | 1 | N |
A 1-A 3 | 103.55^{o} | [7.5, 8.5] | 4 | [11.5, 12.5] | 1 | N | |
A 1-A 4 | 30.13^{o} | [-3.2, -0.7] | 6 | [2.8, 5.3] | 1 | N | |
A 2-A 3 | 106.34^{o} | [-8.6, -7.7] | 2 | [5.7, 6.6] | 1 | N | |
A 2-A 4 | 119.98^{o} | [8.6, 9.1] | 4 | [12.6, 13.1] | 1 | N | |
A 3-A 4 | 133.68^{o} | [-9.4, -9.2] | 2 | [7.2, 7.4] | 1 | N | |
B | B 1-B 2 | 136.64^{o} | [-9.4, -9.3] | 2 | [7.3, 7.4] | 1 | N |
B 1-B 3 | 92.38^{o} | [6.5, 8.0] | 4 | [10.5, 12.0] | 1 | N | |
B 1-B 4 | 31.04^{o} | [-3.3, -0.7] | 6 | [2.7, 5.3] | 1 | N | |
B 2-B 3 | 130.98^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N | |
B 2-B 4 | 105.60^{o} | [7.6, 8.6] | 4 | [11.6, 12.6] | 1 | N | |
B 3-B 4 | 123.42^{o} | [-9.2, -8.8] | 2 | [6.8, 7.2] | 1 | N | |
C | C 1-C 2 | 132.21^{o} | [-9.3, -9.2] | 2 | [7.2, 7.3] | 1 | N |
C 1-C 3 | 98.11^{o} | [7.0, 8.3] | 4 | [11.0, 12.3] | 1 | N | |
C 1-C 4 | 32.13^{o} | [-3.4, -0.8] | 6 | [2.6, 5.2] | 1 | N | |
C 2-C 3 | 129.68^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N | |
C 2-C 4 | 100.08^{o} | [7.2, 8.4] | 4 | [11.2, 12.4] | 1 | N | |
C 3-C 4 | 130.24^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N | |
D | D 1-D 2 | 130.22^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N |
D 1-D 3 | 100.07^{o} | [7.2, 8.4] | 4 | [11.2, 12.4] | 1 | N | |
D 1-D 4 | 32.13^{o} | [-3.4, -0.8] | 6 | [2.6, 5.2] | 1 | N | |
D 2-D 3 | 129.71^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N | |
D 2-D 4 | 98.09^{o} | [7.0, 8.3] | 4 | [11.0, 12.3] | 1 | N | |
D 3-D 4 | 132.20^{o} | [-9.3, -9.2] | 2 | [7.2, 7.3] | 1 | N | |
E | E 1-E 2 | 123.47^{o} | [-9.2, -8.8] | 2 | [6.8, 7.2] | 1 | N |
E 1-E 3 | 105.55^{o} | [7.6, 8.6] | 4 | [11.6, 12.6] | 1 | N | |
E 1-E 4 | 31.09^{o} | [-3.3, -0.7] | 6 | [2.7, 5.3] | 1 | N | |
E 2-E 3 | 130.98^{o} | [-9.3, -9.1] | 2 | [7.1, 7.3] | 1 | N | |
E 2-E 4 | 92.38^{o} | [6.5, 8.0] | 4 | [10.5, 12.0] | 1 | N | |
E 3-E 4 | 136.64^{o} | [-9.4, -9.3] | 2 | [7.3, 7.4] | 1 | N | |
F | F 1-F 2 | 133.68^{o} | [-9.4, -9.2] | 2 | [7.2, 7.4] | 1 | N |
F 1-F 3 | 119.98^{o} | [8.6, 9.1] | 4 | [12.6, 13.1] | 1 | N | |
F 1-F 4 | 30.14^{o} | [-3.2, -0.7] | 6 | [2.8, 5.3] | 1 | N | |
F 2-F 3 | 106.34^{o} | [-8.6, -7.7] | 2 | [5.7, 6.6] | 1 | N | |
F 2-F 4 | 103.54^{o} | [7.5, 8.5] | 4 | [11.5, 12.5] | 1 | N | |
F 3-F 4 | 150.12^{o} | -9.5 | 2 | 7.5 | 1 | N |
6.1.2 Analysis On the visibility of an SV
Visibility analysis for the fixed in-plane GPS receiver, where VI: Visible interval, POV: Probability of Visibility, APOV: Average Probability of Visibility and OX: Orbit X, where X denotes A, B, C, D, E or F
OA | A 2 | A 3 | A 1, A 3 | A 1, A 4 | A 2, A 3 | A 2, A 4 | A 1, A 2, A 4 | A 1, A 3, A 4 |
---|---|---|---|---|---|---|---|---|
VI | 73.32^{ o } | 56.89^{ o } | 30.13^{ o } | 100.06^{ o } | 46.66^{ o } | 30.13^{ o } | 2.89^{ o } | 19.32^{ o } |
POV | 20.40% | 15.80% | 8.40% | 28% | 13% | 8.40% | 0.80% | 5.40% |
OB | B 2 | B 3 | B 1, B 3 | B 1, B 4 | B 2, B 3 | B 2, B 4 | B 1, B 2, B 4 | B 1, B 3, B 4 |
VI | 83.58^{ o } | 70.36^{ o } | 31.04^{ o } | 76.02^{ o } | 22.02^{ o } | 31.04^{ o } | 16.36^{ o } | 29.58^{ o } |
POV | 23.20% | 19.50% | 8.60% | 21.20% | 6.10% | 8.60% | 4.50% | 8.20% |
OC | C 2 | C 3 | C 1, C 3 | C 1, C 4 | C 2, C 3 | C 2, C 4 | C 1, C 2, C 4 | C 1, C 3, C 4 |
VI | 76.76^{ o } | 74.79^{ o } | 32.13^{ o } | 77.32^{ o } | 23.32^{ o } | 32.13^{ o } | 20.79^{ o } | 22.76^{ o } |
POV | 21.30% | 20.80% | 8.90% | 21.50% | 6.50% | 8.90% | 5.80% | 6.30% |
OD | D 2 | D 3 | D 1, D 3 | D 1, D 4 | D 2, D 3 | D 2, D 4 | D 1, D 2,D 4 | D 1,D 3, D 4 |
VI | 74.80^{ o } | 76.78^{ o } | 32.13^{ o } | 77.29^{ o } | 23.29^{ o } | 32.13^{ o } | 22.78^{ o } | 20.80^{ o } |
POV | 20.80% | 21.30% | 8.90% | 21.50% | 6.50% | 8.90% | 6.30% | 5.80% |
OE | E 2 | E 3 | E 1, E 3 | E 1, E 4 | E 2, E 3 | E 2, E 4 | E 1, E 2, E 4 | E 1, E 3, E 4 |
VI | 70.36^{ o } | 83.53^{ o } | 31.09^{ o } | 76.02^{ o } | 22.02^{ o } | 31.09^{ o } | 29.53^{ o } | 16.36^{ o } |
POV | 19.50% | 23.20% | 8.60% | 21.20% | 6.10% | 8.60% | 8.20% | 4.50% |
OF | F 2 | F 3 | F 1, F 3 | F 1, F 4 | F 2, F 3 | F 2, F 4 | F 1, F 2, F 4 | F 1, F 3, F 4 |
VI | 56.88^{ o } | 73.32^{ o } | 30.14^{ o } | 100.06^{ o } | 46.66^{ o } | 30.14^{ o } | 19.32^{ o } | 2.88^{ o } |
POV | 15.80% | 20.40% | 8.40% | 28% | 13% | 8.40% | 5.40% | 0.80% |
APOV | 20.10% | 20.10% | 8.50% | 23.50% | 8.40% | 8.50% | 5.10% | 5.10% |
6.1.3 Approximation of visibility
6.2 Theoretical procedure for calculating POC
Since there is no collision between SVs in one orbit, the in-orbit 'adjacencies'^{f} do not exist and the number of 'adjacencies' is thus reduced. To calculate the POC: denote by M the number of 'adjacencies' of the desired SV "X"; denote by, ${p}_{{M}_{k}}$, k = 1, 2, 3, 4, the probability of having k 'adjacencies' to "X", which is an a priori probability; denote by, ${p}_{{v}_{kj}}$, 1 ≤ j ≤ k, the probability of visibility of j 'adjacencies' to "X" when M = k, which is also an a priori probability; denote by, p_{ k } (collision|j visible), the POC under the condition that there are j visible 'adjacencies' to "X" when M = k, which is a conditional probability.
6.3 A priori probability ${p}_{{v}_{kj}}$
24 SVs are clustered into four categories, each of which contains six SVs
Categories | SVs |
---|---|
I | A 1, B 1, C 1, D 1, E 1, F 1 |
II | A 2, B 2, C 2, D 2, E 2, F 2 |
III | A 3, B 3, C 3, D 3, E 3, F 3 |
IV | A 4, B 4, C 4, D 4, E 4, F 4 |
The desired SV would locate in one of these four Categories. The following parts will consider all cases that the desired SV lies in a specific Catogory. In order to obtain the priori probability ${p}_{{v}_{kj}}$,
- the first step is to get the number of the 'adjacency' of the desired SV. Since the number of 'adjacencies' is evidently changing with the Doppler shift of the desired SV, the number of 'adjacencies' will be obtained with the corresponding Doppler interval of the desired SV.
- For the case that the desired SV has one 'adjacency', referring to Table 3, the visiable range of this 'adjacency' in the percentage can be obtained. This value is ${{p}_{v}}_{{}_{11}}$.
- For the case that the desired SV has two 'adjacencies', referring to Table 3, the visible range of either of two 'adjacencies' can be obtained and this value leads to ${{p}_{v}}_{{}_{21}}$. And the visible range of both 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{22}}$.
- For the case that the desired SV has three 'adjacencies', referring to Table 3, the visible range of any of three 'adjacencies' can be obtained and this value leads to ${{p}_{v}}_{{}_{31}}$. The visible range of any two 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{32}}$. And the visible range of all three 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{33}}$.
- For the case that the desired SV has four 'adjacencies', referring to Table 3, the visible range of any of four 'adjacencies' can be obtained and this value leads to ${{p}_{v}}_{{}_{41}}$. The visible range of any two 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{42}}$. The visible range of any three 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{43}}$. And the visible range of all four 'adjacencies' obtained by reffereing to Table 3 will lead to ${{p}_{v}}_{{}_{44}}$.
The details are presented as follows.
6.3.1 Category I
The changing process of the number of 'adjacencies' of the desired SV ''X'' in Category I (i.e. the SV B 1) with respect to Doppler shift on the SV ''X''
${f}_{{D}_{X}}$, kHz | [-4.75, -3.25] | [-3.25, -1.25] | [-1.25, 0.75] | [0.75, 2.75] | [2.75, 3.25] | [3.25, 4.75] |
---|---|---|---|---|---|---|
M | 4 | 3 | 2 | 1 | 0 | 1 |
'adjacencies' | A 1, A 2, A 3, A 4 | A 2, A 3, A 4 | A 3, A 4 | A 4 | NULL | C 1 |
Referring to the average visible interval, the a priori probabilities can be obtained. Firstly, if ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, only C 1 is the 'adjacency' of B 1. Looking into Table 3, C 1 is visible in the form of the following four combinations: {C 1, C 3}, {C 1, C 4}, {C 1, C 2, C 4}, or {C 1, C 3, C 4}. In total, the visible range for C 1 is 8.5% + 23.5% + 5.1% × 2 = 42.2%. Similarly, when ${f}_{{D}_{X}}\in \left[0.75,2.75\right]\mathsf{\text{kHz}}$, the visible range is also 42.2%. Overall, ${{p}_{v}}_{{}_{11}}=42.2\%$.
When ${f}_{{D}_{X}}\in \left[-1.25,0.75\right]\mathsf{\text{kHz}}$, there are two 'adjacencies' for B 1: A 3 and A 4. Looking into Table 3, exactly one 'adjacency', A 3, is visible in the following three forms: A 3 alone, the combinations of {A 1, A 3} or {A 2, A 3}. In this case, A 3 has a visible range of 37%. Similarly, exactly one 'adjacency', A 4, is visible in the following three forms: the combinations of {A 1, A 4}, {A 2, A 4}, or {A 1, A 2, A 4}. In this case, A 4 has a visibility range of 37.1%. Finally, exactly two 'adjacencies' are visible in the form of the combination of {A 1, A 3, A 4}, which corresponds to a visibility range of 5.1%. Thus, it is concluded that ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$.
When ${f}_{{D}_{X}}\in \left[-3.25,-1.75\right]\mathsf{\text{kHz}}$, there are three 'adjacencies' A 2, A 3 and A 4 of the desired SV B 1. Looking into Table 3, exactly one 'adjacency' A 2 is visible in the form of A 2 alone, which corresponds to a visibility range of 20.1%. Similarly, exactly one 'adjacency' A 3 is visible in the form of A 3 alone or as a combination of {A 1, A 3}, which corresponds to a visibility range of 28.6%. Also, exactly one 'adjacency' A 4 is visible in the form of the combination of {A 1, A 4}, which corresponds to a visibility range of 23.5%. Exactly two 'adjacencies' A 2 and A 3 are visible with a visibility range of 5.1%. Exactly two 'adjacencies' A 2 and A 4 are visible in the form of the two combinations of {A 2, A 4} or {A 1, A 2, A 4}, which correspond to a visibility range of 13.6%. Exactly two 'adjacencies' A 3 and A 4 are visible in the form of the combination of {A 1, A 3, A 4}, which corresponds to a visibility range of 5.1%. Exactly three 'adjacencies' A 2, A 3 and A 4 are always invisible. Thus, it is concluded that ${{p}_{v}}_{{}_{31}}=72.2\%$, ${{p}_{v}}_{{}_{32}}=23.8\%$ and ${{p}_{v}}_{{}_{33}}=0\%$.
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are four 'adjacencies' for B 1: A 1, A 2, A 3 and A 4. Looking into Table 3, exactly one 'adjacency' is visible in the form of A 2 or A 3, which corresponds to the visibility range of 40.2%. Exactly two 'adjacencies' are visible in the form of {A 1, A 3}, {A 1, A 4}, {A 2, A 3} or {A 2, A 4}, which correspond to a visibility range of 48.9%. Exactly three 'adjacencies' are visible in the form of {A 1, A 2, A 4} or {A 1, A 3, A 4}, which correspond to a visibility range of 10.2%. One should note that it is impossible that exactly four 'adjacencies' are visible simultaneously. Thus, it is concluded that ${{p}_{v}}_{{}_{41}}=40.2\%$, ${{p}_{v}}_{{}_{42}}=48.9\%$, ${{p}_{v}}_{{}_{43}}=10.2\%$ and ${{p}_{v}}_{{}_{44}}=0\%$.
6.3.2 Category II
The changing process of the number of 'adjacencies' of the desired SV ''X'' in Category II (i.e. the SV B 2) with respect to Doppler shift on the SV ''X''
${f}_{{D}_{X}}$, kHz | [-4.75, -3.25] | [-3.25, -1.25] | [-1.25, 0.75] | [0.75, 1.25] | [1.25, 3.25] | [3.25, 4.75] |
---|---|---|---|---|---|---|
M | 3 | 2 | 1 | 0 | 1 | 2 |
'adjacencies' | A 2, A 3, A 4 | A 3, A 4 | A 4 | NULL | C 1 | C 1, C 2 |
When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, only C 1 is the 'adjacency' for B 2. Similar to Category I, we have ${{p}_{v}}_{{}_{11}}=42.2\%$. When ${f}_{{D}_{X}}\in \left[-1.25,0.75\right]\mathsf{\text{kHz}}$, there is also only one 'adjacency' to B 2: A 4. Following a similar analysis, it is concluded that is also equal to ${{p}_{v}}_{{}_{11}}=42.2\%$. Consequently, the overall a priori probability for M = 1 is also 42.2%.
When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, C 1 and C 2 are its 'adjacencies'. Referring to Table 3, exactly one 'adjacency' C 1 is visible in the form of the combinations: {C 1, C 3}, {C 1, C 4}, or {C 1, C 3, C 4}, which corresponds to a visibility range of 37.1%. Exactly one 'adjacency' C 2 is visible in the form of C 2 alone, or in the combinations of {C 2, C 3} or {C 2, C 4}, which corresponds to a visibility range of 37%. Exactly two 'adjacencies' are simultaneously visible in the form of the combination of {C 1, C 2, C 4}, which corresponds to a visibility range of 5.1%. Thus, it is concluded that ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$. When ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$, A 3 and A 4 are the 'adjacencies' of B 2. As analyzed in Category I, the same values of ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$ are obtained. Overall speaking, when M = 2, the a priori probabilities are ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$.
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are three 'adjacencies' A 2, A 3 and A 4 of the desired SV B 2. With the same analysis shown in Category I, one can conclude that ${{p}_{v}}_{{}_{31}}=72.2\%$, ${{p}_{v}}_{{}_{32}}=23.8\%$ and ${{p}_{v}}_{{}_{33}}=0\%$. Because B 2 has at most three 'adjacencies', it is evident that ${{p}_{v}}_{{}_{41}}={{p}_{v}}_{{}_{42}}={{p}_{v}}_{{}_{43}}={{p}_{v}}_{{}_{44}}=0\%$.
6.3.3 Category III
The changing process of the number of 'adjacencies' of the desired SV ''X'' in Category III (i.e. the SV B 3) with respect to Doppler shift on the SV ''X''
${f}_{{D}_{X}}$, kHz | [-4.75, -3.25] | [-3.25, -1.25] | [-1.25, -0.75] | [-0.75, 1.25] | [1.25, 3.25] | [3.25, 4.75] |
---|---|---|---|---|---|---|
M | 2 | 1 | 0 | 1 | 2 | 3 |
'adjacencies' | A 3, A 4 | A 4 | NULL | C 1 | C 1, C 2 | C 1, C 2, C 3 |
When ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\cup \left[-3.25,-1.25\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{kHz}}$, only C 1 or A 4 is the 'adjacency' of the desired SV B 3. As shown in Category II, it is obtained that ${{p}_{v}}_{{}_{11}}=42.2\%$.
When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, C 1 and C 2 are the 'adjacencies' of the desired SV B 3. As analyzed in Category II, it is provided that ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$. When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are also two 'adjacencies': A 3 and A 4, and it is shown that also ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$. Overall speaking, when M = 2, the a priori probabilities are ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$.
When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, C 1, C 2 and C 3 are the 'adjacencies' of the desired SV B 3. Referring to Table 3, exactly one 'adjacency' C 1 is visible in the form of the combination {C 1, C 4}, which corresponds to a visibility range of 23.5%. Exactly one 'adjacency' C 2 is visible in the form of C 2 alone or the combination {C 2, C 4}, which corresponds to a visibility range of 28.6%. Exactly one 'adjacency' C 3 is visible in the form of C 3 alone, which corresponds to the visibility range of 20.1%. Exactly two 'adjacencies' C 1 and C 2 are visible in the form of the combinations {C 1, C 2, C 4}, which corresponds to a visibility range of 5.1%. Exactly two 'adjacencies' {C 2, C 3} are visible with a visibility range of 8.4%. Exactly two 'adjacencies' C 1 and C 3 are visible in the form of the combinations of {C 1, C 3} or {C 1, C 3, A 4}, which correspond to a visibility range of 13.6%. Exactly three 'adjacencies' C 1, C 2 and C 3 are always invisible. Thus, it is concluded that ${{p}_{v}}_{{}_{31}}=72.2\%$, ${{p}_{v}}_{{}_{32}}=23.8\%$ and ${{p}_{v}}_{{}_{33}}=0\%$. Also since B 3 has at most three 'adjacencies', it is evident that ${{p}_{v}}_{{}_{41}}={{p}_{v}}_{{}_{42}}={{p}_{v}}_{{}_{43}}={{p}_{v}}_{{}_{44}}=0\%$.
6.3.4 Category IV
The changing process of the number of 'adjacencies' of the desired SV ''X'' in Category IV (i
${f}_{{D}_{X}}$, kHz | [-4.75, -3.25] | [-3.25, -2.75] | [-2.75, -0.75] | [-0.75, 1.25] | [1.25, 3.25] | [3.25, 4.75] |
---|---|---|---|---|---|---|
M | 1 | 0 | 1 | 2 | 3 | 4 |
'adjacencies' | A 4 | NULL | C 1 | C 1, C 2 | C 1, C 2, C 3 | C 1, C 2, C 3, C 4 |
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\left]\cup \right[-2.75,-0.75\right]\mathsf{\text{kHz}}$, only C 1 or A 4 are the 'adjacencies' of B 4. As shown in Category III, it is obtained that ${{p}_{v}}_{{}_{11}}=42.2\%$. When ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\mathsf{\text{kHz}}$, C 1 and C 2 are the 'adjacencies' of B 4. As shown in Category III, ${{p}_{v}}_{{}_{21}}=74.1\%$ and ${{p}_{v}}_{{}_{22}}=5.1\%$. When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, the SVs C 1, C 2 and C 3 are the 'adjacencies' of B 4. As shown in Category III, ${{p}_{v}}_{{}_{31}}=72.2\%$, ${{p}_{v}}_{{}_{32}}=23.8\%$ and ${{p}_{v}}_{{}_{33}}=0\%$. When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, there are four 'adjacencies' C 1, C 2, C 3 and C 4 of the desired SV B 4, which is the same as the four 'adjacencies' A 1, A 2, A 3 and A 4. As shown in Category I, it is obtained that ${{p}_{v}}_{{}_{41}}=40.2\%$, ${{p}_{v}}_{{}_{42}}=48.9\%$, ${{p}_{v}}_{{}_{43}}=10.2\%$ and ${{p}_{v}}_{{}_{44}}=0\%$.
6.4 A priori probability ${p}_{{M}_{k}}$
The desired SV would locate in one of these four Categories. The priori probability ${p}_{{M}_{k}}$ is different when the desired SV lies in a different Category. In a specific Category,
- the first step is to figure out the Doppler interval when the desired SV has one 'adjacency' and to calculate the probability that the desired SV has a Doppler shift within this interval. This percentage leads to the value of ${p}_{{M}_{1}}$.
- The second step is to figure out the Doppler interval when the desired SV has two 'adjacencies' and to calculate the probability that the desired SV has a Doppler shift within this interval. This percentage leads to the value of ${p}_{{M}_{2}}$.
- The third step is to figure out the Doppler interval when the desired SV has three 'adjacencies' and to calculate the probability that the desired SV has a Doppler shift within this interval. This percentage leads to the value of ${p}_{{M}_{3}}$.
- The last step is to figure out the Doppler interval when the desired SV has four 'adjacencies' and to calculate the probability that the desired SV has a Doppler shift within this interval. This percentage leads to the value of ${p}_{{M}_{4}}$.
The details are presented as follows.
6.4.1 Category I
Looking into Table 5, there is one 'adjacency' when ${f}_{{D}_{X}}\in \left[0.75,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}2.75\right]\phantom{\rule{0.3em}{0ex}}\cup \left[3.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}.$ By means of the PDF of the Doppler shift, its integration over the specified interval is obtained as ${p}_{{M}_{1}}=41.3\%$. Similarly, there are two 'adjacencies' when ${f}_{{D}_{X}}\in \left[-1.25,0.75\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{2}}=12.2\%$. There are three 'adjacencies' when ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{3}}=\phantom{\rule{0.3em}{0ex}}14.5\%$. There are four 'adjacencies' when ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{4}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}27.8\%$.
6.4.2 Category II
Looking into Table 6, there is one 'adjacency' when ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \left[-1.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}0.75\right]\phantom{\rule{0.3em}{0ex}}\cup \phantom{\rule{0.3em}{0ex}}\left[1.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}3.25\right]\mathsf{\text{kHz}}$. Similar to Category I, we have ${{p}_{M}}_{{}_{1}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}26.8\%$. There are two 'adjacencies' when ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \left[-3.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-1.25\right]\phantom{\rule{0.3em}{0ex}}\cup \phantom{\rule{0.3em}{0ex}}\left[3.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{2}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}42.3\%$. There are three 'adjacencies' when ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, ${p}_{{M}_{3}}=27.8\%$. Since there are at most three 'adjacencies' for SVs in this Category, ${{p}_{M}}_{{}_{4}}=0\%$.
6.4.3 Category III
Looking into Table 7, there is one 'adjacency' when ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \left[-3.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-1.25\right]\phantom{\rule{0.3em}{0ex}}\cup \phantom{\rule{0.3em}{0ex}}\left[-0.75,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}1.25\right]\mathsf{\text{kHz}}$, we have ${{p}_{M}}_{{}_{1}}=26.8\%$. There are two 'adjacencies' when ${f}_{{D}_{X}}\in \left[-4.75,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-3.25\right]\phantom{\rule{0.3em}{0ex}}\cup \phantom{\rule{0.3em}{0ex}}\left[1.25,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}3.25\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{2}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}42.3\%$. There are three 'adjacencies' when ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, ${p}_{{M}_{3}}=27.8\%$. Additionally, ${{p}_{M}}_{{}_{4}}=0\%$.
6.4.4 Category IV
Looking into Table 8, there is one 'adjacency' when ${f}_{{D}_{X}}\in \left[-4.75,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-3.25\right]\cup \left[-2.75,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}-0.75\right]\mathsf{\text{kHz}}$, ${p}_{{M}_{1}}=41.3\%$. There are two 'adjacencies' when ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{2}}=12.2\%$. There are three 'adjacencies' when ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, ${{p}_{M}}_{{}_{3}}=\phantom{\rule{0.3em}{0ex}}14.5\%$. There are four 'adjacencies' when ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, ${p}_{{M}_{3}}=27.8\%$.
6.5 Conditional probability p_{ k }(collision|j visible)
The conditional probability is related with the null-to-null bandwidth of each subcarrier, w_{ nn }. The smaller the OBI, the bigger the bandwidth, the larger the conditional probability. We consider two cases: (1) the OBI is equal to 1 ms, i.e. the null-to-null bandwidth of each subcarrier is equal to 2 kHz; and (2) the OBI is equal to 20 ms, i.e. the null-to-null bandwidth of each subcarrier is equal to 0.1 kHz.
transmitting signal of the 'adjacencies' and the desired SV. The POC at a specific Doppler bin of the desired SV can then be calculated based on the non-uniform distribution of the Doppler shift. Finally, based on this POC at a specific Doppler bin, the conditional probability p_{ k } (collision|j visible) can be obtained. The details are presented as follows.
6.5.1 Category I
When ${f}_{{D}_{X}}\in \left[-1.25,0.75\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': A 3 and A 4. If A 3 is visible while A 4 is not, looking into Table 3, the Doppler shift on A 3 is in [-4.75, 4,43] kHz and so, relative to B 1, A 3 lies in [-8.75, 0.43] kHz. When ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}\left[-1.25,\phantom{\rule{0.3em}{0ex}}0.43\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{kHz}}$, a collision occurs in the same way as above. In this case, the POC is obtained as {27.3%, 1.28%}. Thus, the integrated POC is {22.9%, 1.08%}. If A 4 is visible while A 3 is not, looking into Table 3, the Doppler shift on A 4 is in [-4.43, 4, 75] kHz and so, relative to B 1, A 4 lies in [-6.73, 2.75] kHz and the POC is approximately {15.8%, 0.78%}. Considering the relative visibility range, the resulting conditional POC for one visible 'adjacency' is p_{2} (collision|1 visible) = {19.4%, 0.92%}. If both A 3 and A 4 are visible, they lie respectively in [0.43, 0.75] and [-6.75, -6.43] kHz relative to B 1. It is evident that B 1 may collide with A 3 while not with A 4. When ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}\left[-1.25,\phantom{\rule{0.3em}{0ex}}0.43\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{kHz}}$, no collision occurs. When ${f}_{{D}_{X}}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}\left[0.43,\phantom{\rule{0.3em}{0ex}}0.75\right]\phantom{\rule{0.3em}{0ex}}\mathsf{\text{kHz}}$, the collision occurs with the following probability: {100%, 26.8%}. Finally, the integrated conditional POC for two visible 'adjacencies' is p_{2} (collision|2 visible) = {29.8%, 4.20%}.
When ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$, there are three 'adjacencies': A 2, A 3 and A 4. If exactly A 2 is visible, it lies in [-7.98, 1.57] kHz, the corresponding POC is {40.6%, 1.90%}. If exactly A 3 is visible, it lies in [-8.43, 0.43] kHz, and the POC is {18.5%, 0.91%}. If exactly A 4 is visible, A 4 lies in [-6.43, -0.02] kHz, and the corresponding POC is {25.1%, 1.24%}. Taking the visibility range into consideration, the conditional POC for one visible 'adjacency' is p_{3} (collisionj 1 visible) = {26.8%, 1.29%}. When A 2 and A 3 are visible while A 4 is not, A 2 lies in [-1.57, -1.25] kHz, which leads to the integrated POC {26.0%, 3.75%}. Any other combinations of two visible 'adjacencies' cannot cause collision. Taking the visibility range into consideration, p_{3} (collision|2 visible) = {5.9%, 0.8%}. Additionally, since the condition of three simultaneously visible 'adjacencies' is false, calculating the POC under this condition is meaningless because its contribution to the collision is 0 when removing the condition.
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are four 'adjacencies': A 1, A 2, A 3 and A 4. When exactly one is visible, A 2 and A 3 lie in [-7.98, -1.57] and [-8.43, -2.02] kHz, respectively. So, we have p_{4} (collision|1 visible) = {26.3%, 1.29%}. The scenarios of two visible 'adjacencies' cannot cause collision except for A 1 and A 4, where A 1 lies in [-9.98, -3.57]kHz, leading to a collision probability of {43.9%, 2.00%}, and A 4 lies in [-6.43, -0.02] kHz, leading to a collision probability of {29.9%, 1.46%}. Since no collision occurs between SVs in one orbit, when A 1 collides with B 1, A 4 doesn't. Also, when A 4 collides with B 1, A 1 doesn't. So the POC for such scenario is {73.8%, 3.46%}. Averaged by the visible range, p_{4} (collision|2 visible) = {35.4%, 1.66%}. For the scenario of three visible 'adjacencies' of A 1, A 2 and A 4, A 1 lies in [-3.57, -3.25]kHz, which provides a collision probability of {20.5%, 2.90%}, while the scenario of A 1, A 3 and A 4 cannot cause collision. So, we have p_{4} (collision|3 visible) = {10.2%, 1.45%}. Additionally, the condition of four simultaneously visible 'adjacencies' is false and its contribution to the collision is 0 when removing the condition.
6.5.2 Category II
When ${f}_{{D}_{X}}\in \left[-1.25,-0.75\right]\mathsf{\text{kHz}}$, there is one 'adjacency' A 4, and the corresponding POC is equal to {23.8%, 1.57%}. When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, there is one 'adjacency' C 1, and the POC is {23.5%, 1.49%}. Considering the occurrence probability, 12.2 and 14.5% for ${f}_{{D}_{X}}\in \left[-1.25,-0.75\right]\mathsf{\text{kHz}}$ and ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, respectively, the average conditional POC is obtained as p_{1} (collision|1 visible) = {23.6%, 1.53%}.
When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': C 1 and C 2. When only C 1 is visible, it lies in [1.25, 10.43] kHz relative to B 2, and the corresponding POC is {15.7%, 0.77%}. When only C 2 is visible, it lies in [3.57, 12.57] kHz relative to B 2, and the corresponding POC is {24.8%, 1.15%}. Taking the visibility range into consideration, the resulting POC for one visible 'adjacency' is {20.3%, 0.96%}.
When both C 1 and C 2 are visible, C 2 lies in [3.25, 3.57] kHz, the corresponding POC is {20.5%, 3.00%}, while C 1 cannot cause any collisions. So, the resulting conditional POC for two visible 'adjacencies' is {20.5%, 3.00%}. On the other hand, when ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': A 3 and A 4, and the POC is {19.4%, 0.92%} for one visible 'adjacency' and {26.0%, 3.75%} for two simultaneously visible 'adjacencies'. Also, the occurrence probability for ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$ and ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$ is 27.8 and 14.5%, respectively. Overall, p_{2} (collision|1 visible) = {20.0%, 0.95%}. and p_{2} (collision|2 visible) = {22.5%, 3.26%}.
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are three 'adjacencies': A 2, A 3 and A 4. When only A 2 is visible, the POC is {43.9%, 2.00%}. When only A 3 is visible, the POC is {18.5%, 0.91%}. When only A 4 is visible, the POC is {24.9%, 1.23%}. Thus, we have p_{3} (collision|1 visible) = {27.6%, 1.32%}. When A 2 and A 3 are visible while A 4 is not, the POC is {20.5%, 2.90%}, while other combinations of two SV's being visible cannot lead to a collision. Thus, p_{3} (collision|2 visible) = {4.5%, 0.62%}. Additionally, the condition of three simultaneously visible 'adjacencies' is false, so its contribution to the POC is 0 when removing this condition.
Since the condition of four 'adjacencies' is also false, its contribution to the POC is also 0 when removing this condition.
6.5.3 Category III
When ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$, there is one 'adjacency' A 4, and the corresponding POC is equal to {23.5%, 1.49%}. When ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\mathsf{\text{kHz}}$, there is one 'adjacency' C 1, and the POC is {23.8%, 1.57%}. Also, the occurrence probability for ${f}_{{D}_{X}}\in \left[-3.25,-1.25\right]\mathsf{\text{kHz}}$ and ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\mathsf{\text{kHz}}$ is 14.5 and 12.2%, respectively. So, the average conditional POC is p_{1} (collision|1 visible) = {23.6%, 1.53%}.
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': A 3 and A 4. When only A 3 is visible, the POC is {24.9%, 1.13%}. When only A 4 is visible, it is {15.7%, 0.77%}. So, the average value is {20.3%, 0.95%}. When both A 3 and A 4 are visible, the POC is {20.5%, 3.00%}. When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': C 1 and C 2. When only C 1 is visible, the POC is {15.7%, 0.77%}. When only C 2 is visible, it is {24.1%, 1.13%}. Thus, taking the visible range into consideration, we have {19.9%, 0.95%}. When both C 1 and C 2 are visible, the POC is {20.5%, 3.15%}. Also, the occurrence probability for ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$ and ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$ is 27.8 and 14.5%, respectively. Overall, the conditional POC is p_{2} (collision|1 visible) = {20.2%, 0.95%} for one visible 'adjacency' and p_{2} (collision|2 visible) = {20.5%, 3.05%} for two visible 'adjacencies'.
When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, there are three 'adjacencies': C 1, C 2 and C 3. When only C 1 is visible, C 1 lies in [2.02, 8.43] kHz relative to B 3, and the POC is {32.9%, 1.63%}. When only C 2 is visible, C 2 lies in [1.57, 10.43] kHz, and the POC is {33.9%, 1.66%}. When only C 3 is visible, C 3 lies in [3.57, 9.98] kHz, and the POC is {48.2%, 2.37%}. Taking the visibility range into consideration, the corresponding conditional POC can be obtained as p_{3} (collision|1 visible) = {37.6%, 1.85%}. The scenarios of two visible 'adjacencies' cannot cause any collisions except for C 2 and C 3, where C 3 lies in [-4.75, -4.43] kHz. This leads to a POC {20.5%, 3.12%}. Taking the visibility range into consideration, we have p_{3} (collision|2 visible) = {4.5%, 0.67%}. Additionally, the condition of three simultaneously visible 'adjacencies' is false, so its contribution to the POC is 0 when removing this condition.
Since the condition of having four 'adjacencies' is false, its contribution to the POC is 0 when removing this condition.
6.5.4 Category IV
When ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$, there is one 'adjacency' A 4, and the corresponding POC is equal to {24.3%, 1.42%}. When ${f}_{{D}_{X}}\in \left[-2.75,-0.75\right]\mathsf{\text{kHz}}$, there is one 'adjacency' C 1, and the POC is {24.1%, 1.64%}. Also, the occurrence probability for ${f}_{{D}_{X}}\in \left[-4.75,-3.25\right]\mathsf{\text{kHz}}$ and ${f}_{{D}_{X}}\in \left[-2.75,-0.75\right]\mathsf{\text{kHz}}$ is 27.8 and 13.5%, respectively. Similarly to the above, the conditional POC is p_{1} (collision|1 visible) = {24.2%, 1.50%}.
When ${f}_{{D}_{X}}\in \left[-0.75,1.25\right]\mathsf{\text{kHz}}$, there are two 'adjacencies': C 1 and C 2. When only C 1 is visible, the POC is {15.8%, 0.78%}. When only C 2 is visible, it is {23.1%, 1.10%}, respectively. Thus, taking the visible range into consideration, we have {19.4%, 0.94%}. When both C 1 and C 2 are visible, the POC is {16.0%, 4.61%}. Similarly, we have p_{2} (collision|1 visible) = {19.4%, 0.94%] and p_{2} (collision|2 visible) = {16.0%, 4.61%}.
When ${f}_{{D}_{X}}\in \left[1.25,3.25\right]\mathsf{\text{kHz}}$, there are three 'adjacencies': C 1, C 2 and C 3. When only C 1 is visible, C 1 lies in [0.02, 6.43] kHz relative to B 4, and the POC is {25.1%, 1.24%}. When only C 2 is visible, C 2 lies in [-0.43, 8.43] kHz, and the POC is {18.5%, 0.91%}. When only C 3 is visible, C 3 lies in [1.57, 7.98] kHz, and the POC is {41.0%, 1.94%}. Taking the visibility range into consideration, the conditional POC is p_{3} (collision|1 visible) = {26.9%, 1.30%}. The scenarios of two visible 'adjacencies' cannot cause any collisions except for C 2 and C 3, where C 3 lies in [1.25, 1.57] kHz. This leads to a POC {26.0%, 4.02%}. Taking the visible range into consideration, p_{3} (collision|2 visible) = {9.1%, 1.42%}. Additionally, the condition of three simultaneously visible 'adjacencies' in this case is false and its contribution to the POC is equal to 0 when removing this condition.
When ${f}_{{D}_{X}}\in \left[3.25,\phantom{\rule{0.3em}{0ex}}4.75\right]\mathsf{\text{kHz}}$, there are four 'adjacencies': C 1, C 2, C 3 and C 4. When exactly one is visible, C 2 and C 3 respectively lie in [2.02, 8.43] and [1.57, 7.98] kHz. So, we have p_{4} (collision|1 visible) = {26.3%, 1.30%}. The scenarios of two visible 'adjacencies' cannot cause any collisions except for C 1 and C 4, where C 1 lies in [-1.98, 4.43] kHz. This leads to a POC {30.0%, 1.46%}, respectively. Taking the visible range into consideration, we have p_{4} (collision|2 visible) = {14.4%, 0.70%}. For the scenario of three visible 'adjacencies' of C 1, C 3 and C 4, C 4 lies in [3.25, 3.57] kHz, which provides a POC {20.5%, 3.12%}, while the scenario of C 1, C 2 and C 4 cannot cause any collisions. So, p_{4} (collision|3 visible) = {10.3%, 1.56%}. Additionally, the condition of four simultaneously visible 'adjacencies' is false and its contribution to the POC is 0 when removing this condition.
6.6 Probability of collision
6.6.1 Observation interval: 1 ms
For 6 SVs in Category I, the a priori probabilities are ${{p}_{c}}_{{}_{1}}=10.2\%$, ${p}_{{c}_{2}}=15.9\%$, ${p}_{{c}_{3}}=20.7\%$, and ${p}_{{c}_{4}}=28.9\%$, respectively. The POC for Category I is therefore obtained as ${p}_{c}^{I}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum}_{k=1}^{4}\left({p}_{{c}_{k}}{p}_{{M}_{k}}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}17.2\%$. Similarly, for 6 SVs in Category II, the a priori probabilities are ${{p}_{c}}_{{}_{1}}=9.96\%$, ${p}_{{c}_{2}}=16.0\%$, ${p}_{{c}_{3}}=24.5\%$, and ${p}_{{c}_{4}}=0\%$, respectively. The POC for Category II is therefore obtained as ${P}_{c}^{I\phantom{\rule{0.3em}{0ex}}I}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum}_{k=1}^{4}\left({p}_{{c}_{k}}{p}_{{M}_{k}}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}15.3\%$. For 6 SVs in Category III, the a priori probabilities are ${{p}_{c}}_{{}_{1}}=9.96\%$, ${p}_{{c}_{2}}=16.0\%$ ${p}_{{c}_{3}}=28.2\%,$ and ${p}_{{c}_{4}}=0\%$ respectively. The POG for Category III is therefore obtained as ${p}_{c}^{I\phantom{\rule{0.3em}{0ex}}I\phantom{\rule{0.3em}{0ex}}I}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum}_{k=1}^{4}\left({p}_{{c}_{k}}{p}_{{M}_{k}}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}17.3\%$. For 6 SVs in Category IV, the a priori probabilities are ${{p}_{c}}_{{}_{1}}=10.2\%$, ${{p}_{c}}_{{}_{2}}=15.2\%$, ${p}_{{c}_{3}}=21.6\%$, and ${p}_{{c}_{4}}=18.7\%$, respectively. The POC for Category IV is therefore obtained as ${p}_{c}^{IV}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum}_{k=1}^{4}\left({p}_{{c}_{k}}{p}_{{M}_{k}}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}14.4\%$.
In other words, the probability for successful positioning is equal to 99.61%.
6.6.2 Observation interval: 20ms
In other words, the probability for successful positioning is approximately equal to 100%.
7 Numerical results
In this section, the theory proposed above is demonstrated by numerical simulations. To verify the SNR gain, Two signals are used for demonstration: a MC code defined by (1) and (2) and the C/A code, where the MC code contains N = 1024 subcarrier slots and one period of the C/A code contains 1, 023 chips. Both signals have an equal energy ξ and an equal null-to-null bandwidth 2B = 2.048 MHz. The signal duration, T_{ s }, is selected as 1.023 ms for both signals.
To test the effect of NBI and Doppler shift to the reception of the proposed MC codes, a computer orbit generation program is employed. In this orbit generation program, the configuration for the numerical results consists of 24 SVs in six circular orbits (eccentricity 0), four SVs in each circular orbit arranging as Figure 6. The six planes are set to have a uniformly 55^{ o } inclination (tilt relative to Earth's equator) and to be separated by 60^{ o } right ascension of the ascending node (angle along the equator from a reference point to the orbit's intersection). And, the orbit altitude is set as 20, 200 km. The GPS receiver in our computer program is located at Calgary, Canada (51^{ o }N and 114^{ o }W), Washington, DC (39^{ o }N and 77^{ o }W), London, UK (51^{ o }N and 0^{ o }W) and Melbourne, Australia (37^{ o }S and 144^{ o }E), respectively.
7.1 SNR gain
7.2 Robustness against NBI
7.3 Probability for successful positioning under Doppler effects
8 Conclusion and future work
The unfiltered MC modulation has been proposed as an alternative for the GPS C/A code in this article. As per our arrangement, each SV takes up only 42 uniformly spaced subcarrier slots while approximately occupying the same bandwidth as the C/A code in GPS. In this way, the proposed MC code was shown to attain a 4.73 dB SNR gain in terms of range estimation compared to the current C/A code. This 4.73 dB SNR gain is significant and possibly helpful for GPS downtown's positioning, tunnel's positioning and even indoor positioning. The transmission and reception of the proposed MC code for the next-generation GNSS are provided in detail, and the range estimation using the proposed MC code is explained. Furthermore, the proposed MC code was shown to be robust against NBI by considering four GPS receivers at different locations, which is also one of the current GPS's disturbing things to overcome. Moreover, the POC and probability of successful positioning under Doppler shifts are thoroughly analyzed in theory and justified by computer simulations. It was shown that the Doppler shift has a negligible effect on the probability of successful positioning when using the proposed MC code as the GPS probing signal. To sum up, from the authors' best knowledge, the proposed MC code could be regarded as an available probing signal for the future next-generation GPS.
However, the dangerous adjacencies, affecting the POC value under Doppler shift, do depend on the strategy adopted to assign the set of subcarriers to the SVs. Our proposed arrangement results in no collision between groups of MC codes that are transmitted by SVs in one orbit. But we have no idea whether it is the best arrangement or not in terms of POC. The future work will focus on the optimization of the subcarrier arrangement for SVs.
Appendix 1
Doppler shift for an in-plane GPS receiver
where v is the speed of SV, i.e. 3.874 km/s; c is the speed of light; f_{L 1}is the carrier frequency of the coarse acquisition (CA) code, i.e. 1575.42 MHz [12]; and θ is the angle between SV's forward velocity and the line of sight from SV to the receiver as shown in Figure 13a. Consider the maximum Doppler shift case, which happens at a point S when SV ascends past the horizon of the receiver. In this case, the radial velocity at R is 0.9 km/s [23] and the resulting maximum Doppler shift is 4.726 kHz. Similarly, the minimum Doppler shift is -4.726 kHz which occurs at a point D when SV descends past the horizon of the receiver. It is also interesting to note that θ changes from 76.5^{ o } when ascending past the horizon at point S to 103.5^{ o } when descending past the horizon at point D.
where w is the angular velocity of SV obtained as $w\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{v}{r+h}$. Figure 8 shows the Doppler shift f_{ D } as a function of γ, where it is noticed that, the closer the position of SV to the zenith of the receiver, the smaller the Doppler shift; and, the farther the position of SV to the zenith of the receiver, the larger the Doppler shift.
Appendix 2
Visible region in the orbital planefor an off -plane GPS receiver
for |β| ≤ 76.1^{ o }. As shown in Figure 5b, the maximum value of α, 153^{ o }, occurs when β = 0^{ o }, i.e. when the receiver lies in the orbital plane of SV; while α = 0^{ o } occurs when |β| ≥ 76.1^{ o }, i.e. when the receiver cannot see SV on the orbit. It can also be shown that the average value of α is 137^{ o } which occurs when β = ± 49^{ o }.
Endnotes
^{a}The C/A code can mitigate NBI by up to 30 dB when the OBI is as usual 1 ms due to the processing gain. Even when the OBI is increased to its maximum 20 ms, the processing gain only comes up to 43 dB [21]. When the receiver is in close proximity to the jammer and the NBI is larger than the processing gain, the positioning performance is poor. ^{b}Here, 2.048 MHz instead of 2.046 MHz is used for calculation convenience. ^{c}If the number of SVs in GPS increases, the number of occupied subcarriers by each SV will get reduced but the designed MC code is similar. ^{d}Doppler shift due to SV motion has a negligible effect on the reception of the MC code, which will be analyzed in detail in Section 6. ^{e}Only the mainlobe of the subcarrier is taken into consideration because the majority of signal energy locates in this area. Interfering on the sidelobe by NBI does not affect the probing signal significantly. ^{f}The 'adjacencies' is defined as the SV which in the frequency domain are close to "X" and can possibly collide with "X". ^{g}This representation denotes p_{1} (collision|1 visible) = 24.3% when the OBI is equal to 1 ms and p_{1} (collision|1 visible) = 1.42% when the OBI is equal to 20 ms. For convenience, we use this representation throughout this article.
Declarations
Acknowledgements
This work is supported by the Alberta Informatics Circle of Research Excellence (iCORE), the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chair (CRC) Program, TRLabs and the Cell-Loc Location Technologies Inc. Canada.
Authors’ Affiliations
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