GPS Solut (2015) 19:623–638 DOI 10.1007/s10291-014-0420-6
ORIGINAL ARTICLE
Unambiguous combined correlation functions for sine-BOC signal tracking Tao Yan • Jiaolong Wei • Zuping Tang Bo Qu • Zhihui Zhou
•
Received: 25 September 2013 / Accepted: 19 October 2014 / Published online: 1 November 2014 Springer-Verlag Berlin Heidelberg 2014
Abstract Binary offset carrier (BOC) modulated signal has been widely employed in modern global navigation satellite system (GNSS). However, the multiple peaks autocorrelation function of BOC signal leads to tracking ambiguity problem. For high-order BOC signal, the above problem becomes more severe. We propose an unambiguous tracking method based on combined correlation functions for sine-BOC signal. The main feature of this technique is that two new local reference waveforms are used. Based on the cross-correlation functions between the BOC signal and these two local reference waveforms, unambiguous combined correlation functions without any main positive side-peaks are generated. To achieve unambiguous tracking for sine-BOC signal, we present two non-coherent discriminator functions, denoted as V1 and V2. The theoretical expressions of code tracking error in thermal noise are derived. Then, the impact of thermal noise and multipath is analyzed with the help of numerical simulations. Results show that V1 is easier to implement and has relatively better multipath mitigation performance at the expense of some performance degradation in code tracking error. V2 has better code tracking accuracy and provides the most robust code tracking process for highorder sine-BOC signal, but two additional complex correlators are required. Keywords BOC signal Unambiguous tracking Combined correlation function GNSS
T. Yan J. Wei Z. Tang (&) B. Qu Z. Zhou The Department of Electronic and Information Engineering, Huazhong University of Science and Technology, Wuhan, China e-mail:
[email protected]
Introduction With the rapid development and modernization of global navigation satellite system (GNSS), new modulation signals are investigated and introduced. Specifically, the BOC modulated signal was proposed to separate in frequency the military and civilian signals (Betz 1999, 2002). The BOC signal is a binary-phase spreading pseudo-random noise (PRN) code modulated by a sine-phased or cosine-phased square wave subcarrier. The sine-phased BOC signal is generally denoted as sine-BOC(m,n), where m* 1.023 MHz is the subcarrier frequency and n* 1.023 Mchip/s is the PRN code rate. The number of square wave half-periods in each chip is k = 2 m/n, which is referred to as BOC modulation order (Lohan et al. 2006). Due to the square wave subcarrier, the power spectra of the BOC signal are shifted to each side of the center frequency, which provides a high degree of spectral separation with traditional binary-phase shift keying (BPSK) signal. Moreover, thanks to more power allocated to the higher modulating frequencies, the BOC signal has bigger Gabor bandwidth and provides better code tracking performance (Betz 1999, 2002). At present, the BOC signal has been widely used in GNSS. For example, the GPS M code signal exploits the sineBOC(10,5) signal (Barker et al. 2000), and the China’s BeiDou system has selected sine-BOC(14,2) and sineBOC(15,2.5) as the candidate signals at B1 and B3 frequencies (Tang et al. 2010). Nevertheless, when the traditional delay lock loop (DLL) is used in the code tracking process, the multiple peaks auto-correlation function (ACF) of the BOC signal leads to the tracking ambiguity problem, which means that the code tracking loop may lock on the false points, and then, an intolerable tracking error would be introduced (Julien et al. 2004). With the increase in the modulation order, the above problem becomes more severe.
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This is due to the fact that more side-peaks exist in its ACF and the difference among these peaks’ amplitude is smaller. Several methods have been proposed to deal with the tracking ambiguity of the BOC signal. The classic techniques include bump-jump (BJ) method (Fine and Wilson 1999), BPSK-like technique (Martin et al. 2003) and double estimation technique (DET) (Hodgart et al. 2007; Hodgart and Blunt 2007). BJ method needs two additional correlators, very early (VE) and very late (VL). It determines whether the main-peak is locked or not based upon the magnitude comparison of correlator outputs. However, BJ method becomes unreliable in the environment of high noise and multipath, even for the low-order BOC(1,1) signal (Blunt et al. 2007). BPSK-like technique treats each main lobe of BOC signal as an independent BPSK signal after sideband filtering. Although the tracking ambiguity problem does not exist anymore, it sacrifices the higher tracking performance brought by BOC signals. Thus, BPSK-like technique is only applicable to coarse acquisition process. DET uses a subcarrier locked loop (SLL) and a PRN code delay locked loop (DLL) to estimate the subcarrier delay and code delay separately. The tracking ambiguity is corrected by the unambiguous DLL’s delay estimate. Nevertheless, DET is not robust enough for highorder BOC signal (like BOC(14,2)) in the low carrier-tonoise-density ratio (C/N0) case. Moreover, DET is ineffective in terms of multipath rejection. Another way to remove the ambiguity is to design special discriminator functions or local reference waveforms instead of BOC signal, including the autocorrelation side-peak cancellation technique (ASPeCT) (Julien et al. 2011), weighted discriminators method (Kao and Juang 2012), and pseudo-correlation-function-based unambiguous tracking delay lock loop (PUDLL) (Yao et al. 2010). Unfortunately, ASPeCT and weighted discriminators method are restricted to sine-phased BOC(n,n) signal. PUDLL exploits two local step-shape modulated symbols to generate a combined correlation function without sidepeaks. Despite the tracking ambiguity is effectively mitigated, the code tracking performance of PUDLL obviously deteriorates with the increase in BOC modulation order. In addition, some multipath mitigation methods based on strobe pulse or double-delta technique have been proposed for BOC signal. These methods are called ‘‘shaping correlator’’ by Garin (2005). Garin introduced the bipolar reference waveform (BRW) for BOC(n,n) signal tracking. Nunes et al. (2007) proposed the Gating function for BOC signal multipath mitigation. Gating function can decrease the multipath ranging error, but the tracking loop of this method is easy to lose lock due to the effect of thermal noise. Furthermore, it needs the help of a simplified BJ method to avoid false locks.
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All of above research is undoubtedly an important contribution to BOC signal tracking, but it mainly focuses on the low-order BOC signal. We propose an unambiguous tracking method based on combined correlation functions. This method is applicable for general sine-BOC signal. To achieve our method, we design two new local reference waveforms. By using the cross-correlation functions (CCFs) between the BOC signal and these two local reference waveforms, we generate two unambiguous combined correlation functions. Two corresponding non-coherent discriminator functions are proposed, denoted as V1 and V2. V1 is easier to implement and has better anti-multipath performance at the expense of some performance degradation in code tracking error. In contrast with it, V2 shows a more reliable and accurate code tracking process with relatively higher implementation complexity, even for high-order BOC signal. We give first the definition of local reference waveforms and show the combined correlation functions. Then, we provide two non-coherent discriminator functions. On this basis, we further show the unambiguous code tracking implement scheme, analyze the impact of thermal noise and derive the theoretical expressions of code tracking error. The proposed method is compared with existing methods with the help of numerical simulation, including code tracking performance and multipath error performance. Conclusions are summarized in the last section.
Unambiguous combined correlation functions We first review the BOC modulated signal and define the two local reference waveforms. Based on the CCFs between the BOC signal and two local reference waveforms, two combined correlation functions are presented. Proposed local reference waveforms The baseband BOC signal is the product of PRN code waveform c(t) and sine-phase binary subcarrier sc(t), which is denoted as (Rebeyrol et al. 2005) 1 X sBOC ðtÞ ¼ cðtÞ scðtÞ ¼ ci ð1Þki pBOCs ðt i Tc Þ i¼1
ð1Þ where {ci} is the spreading code sequence. k is the BOC modulation order. Tc = 1/Rc is the chip period. pBOCs ðtÞ is the modulated symbol of sine-BOC signal, which is given by sign½sinð2pfs tÞ; 0 t\Tc pBOCs ðtÞ ¼ ð2Þ 0; otherwise where fs is the subcarrier frequency.
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Assume that {ci} has ideal autocorrelation characteristics, and then, the ACF of sine-BOC signal can be expressed as 8
> 1 ; jsjTs > > > Ts < 2k rffiffiffiffiffi RBOC;ref1 ðeÞ ¼ 1 j s Tc j > ð1Þkþ1 1 ; js Tc jTs > > > Ts 2k > : 0; otherwise 8 rffiffiffiffiffi 1 j sj > > 1 ; jsjTs > > > 2k Ts < rffiffiffiffiffi RBOC;ref2 ðeÞ ¼ 1 j s þ Tc j > ð1Þkþ1 1 ; js þ Tc jTs > > > Ts 2k > : 0; otherwise ð7Þ Evidently, both of them have a main-peak located at 0. RBOC;ref1 ðeÞ has a side-perk located at Tc, while RBOC;ref2 ðeÞ has a side-perk located at -Tc. The main-peak and sidepeak have the same amplitude. Figure 2 shows an example of sine-BOC(14,2). As previously mentioned, the ACF of sine-BOC(14,2) has up to 26 side-peaks, but each CCF only includes one side-peak. The number of side-peaks is significantly decreased. Combined correlation functions
ð5Þ ci ð1Þki gref2 ðt iTc Þ
Above, we derive the CCFs of BOC signal and local reference waveforms. Each CCF has still a side-peak, whose
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Fig. 1 The modulated symbols of two local reference waveforms. a k is even, b k is odd
pBOCs t
pBOCs t
Tc
0 g ref 1 t
g ref 1 t
-Ts Ts
-Ts
(a)
Energy normalized correlation function
1 0.8
RB( )
0.6
RBOC,ref1( )
0.4
RBOC,ref2( )
0.2 0
-0.4 -0.6 -0.8 -0.5
0
0.5
Tc-Ts
Tc+Ts
(b)
Tc-Ts
Tc+Ts
Run1 ðeÞ ¼ RBOC;ref1 ðeÞ þ RBOC;ref2 ðeÞ RBOC;ref1 ðeÞ RBOC;ref2 ðeÞ 8 rffiffiffi jsj < 2 1 ; jsj Ts ¼ k Ts : 0; otherwise
ð9Þ
Run1 ðeÞ only exploits the two CCFs. Therefore, its multipath mitigation performance depends on these two CCFs. Only when the multipath delay is close to 0 or Tc, a visible tracking bias error may be introduced due to multipath. This is a significant advantage compared with PUDLL. To further benefit from BOC signal, we multiply the ACF of sine-BOC signal by two CCFs. The results are as follows.
-0.2
-1
Ts
g ref 2 t
g ref 2 t
-1
Tc
0
1
Code delay (chips)
Fig. 2 The ACF of sine-BOC(14,2) and the CCFs between sineBOC(14,2) and two local reference waveforms
amplitude is the same with main-peak. In order to weaken the side-peak, we need to combine the two CCFs. Observing (7), we know that the main-peaks of two CCFs are at the same location, while their side-peaks are at different locations. A natural idea is to multiply the two CCFs, i.e., 8 < 1 j sj 2 RBOC;ref1 ðeÞ RBOC;ref2 ðeÞ ¼ 2k 1 Ts ; jsj\ ¼ Ts : 0; otherwise
RB ðeÞ RBOC;ref1 ðeÞ 8 rffiffiffiffiffi 1 2k 1 jsj j sj > > 1 1 ; jsj Ts > > > k Ts Ts < 2k rffiffiffiffiffi ¼ 1 s 1s > kþ1 1 ; Tc Ts s\Tc > > > 2k Ts k Ts > : 0; otherwise
ð8Þ
RB ðeÞ RBOC;ref2 ðeÞ 8 rffiffiffiffiffi 1 2k 1 jsj j sj > > 1 1 ; jsj Ts > > > k Ts Ts < 2k rffiffiffiffiffi ¼ 1 s 1s > 1k 1þ ; Tc s\ Tc þ T > > > 2k Ts k Ts > : 0; otherwise ð10Þ
However, its main-peak is not as sharp as the ACF of BOC signal, which implies poor code tracking performance. Thus, we do not analyze this case in subsequent discussion. In order to enhance the code tracking accuracy, the main-peak should be as sharp as possible. We study the combination function employed in PUDLL (Yao et al. 2010) and propose a similar unambiguous combined correlation function Run1 ðeÞ:
RB ðeÞ RBOC;ref1 ðeÞ has two negative side-peaks located at both sides of the main-peak, and a positive side-peak close to Tc. RB ðeÞ RBOC;ref2 ðeÞ has the identical main-peak and negative side-peaks. The difference is that its positive side-peak is close to -Tc. Although two negative sidepeaks do not result in false lock, we still want to remove them. Therefore, the second combined correlation function Run2 ðeÞ is defined by
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Run2 ðeÞ has a main-peak and two small positive sidepeaks. After simple derivation, we can obtain that the pffiffiffiffiffi maximum value of side-peaks is 1 4k 2k located at ðTc Ts =2Þ. The amplitude ratio between side-peak and main-peak is 1=ð8kÞ. When k = 4, the amplitude ratio is only 0.03. Compared with the main-peak, two positive side-peaks are so small and far enough away main-peak. Thus, they have little influence on BOC signal tracking. From this point of view, Run2 ðeÞ is unambiguous. Figure 3 shows the two unambiguous combined correlation functions for sine-BOC(14,2). The top is the ideal infinite bandwidth case, and the bottom is the bandwidthlimited case with 36.82 MHz filtering. Results show that the two kinds of combined correlation functions are unambiguous. It is noted that some small side-peaks occur in the second correlation function due to bandwidth limitation, but their amplitudes are small enough to be neglected.
2
Run1
Combined correlation function
ð11Þ
0.4
Run2
0.3
0.2
0.1
0
-1
-0.5
0
0.5
1
Code delay (chips) 0.3 2
Combined correlation function
Run2 ðeÞ ¼ RB ðeÞ RBOC;ref1 ðeÞ þ RB ðeÞ RBOC;ref2 ðeÞ 8 rffiffiffi 2 2k 1 jsj k j sj > > Ts 1 1 ; jsj\ ¼ > > > k k 2k 1 T T s s < rffiffiffiffiffi ¼ 1 jsj 1 j sj > kþ1 1 ; Tc Ts jsj\Tc > > > 2k Ts k Ts > : 0; otherwise
Run1 0.25
Run2
0.2
0.15
0.1
0.05
0 -1
-0.5
0
0.5
1
Code delay (chips)
Unambiguous code tracking method Figure 4 presents the structure of our code tracking loop. Receiver generates the Early (E) and Late (L) versions of two local reference waveforms and BOC signal. The received BOC signal firstly is multiplied by in-phased and quadrature-phase carrier replicas and then correlated with the Early and Late versions of two local reference waveforms. Correlator outputs are obtained after integrating and dumping. It should be noted that only when the second unambiguous combined correlation function is used, the local BOC signal needs to be generated. Therefore, the corresponding part is plotted by dashed line. Without considering multipath and interference, the received BOC IF signal from one satellite can be expressed as (Julien et al. 2011) pffiffiffiffiffiffi rðtÞ ¼ 2C Dðt sÞsBOC ðt sÞcosð2pfIF t þ h0 Þ þ nr ðtÞ ð12Þ where C is the total power of received signal, D(t) is navigation data message, sBOC(t) is defined by (2), fIF is the
Fig. 3 The unambiguous combined correlation functions for sineBOC(14,2). Infinite bandwidth (top) and 36.828 MHz band-limited (bottom)
carrier frequency after down-conversion, h0 is the initial phase of carrier, s is the code delay due to propagation and nr(t) is band-limited white noise and described as (Zhou et al. 2012; Kao and Juang 2012) nr ðtÞ ¼ nc ðtÞcosð2pfIF tÞ ns ðtÞsinð2pfIF tÞ
ð13Þ
where nc(t) and ns(t) are independent Gaussian random processes with the same zero mean and double-sided power spectrum density N0. Since only non-coherent discriminator is considered, navigation data message is ignored. For simplicity, we exploit the baseband complex envelope expression. The baseband signal of r(t) is written as pffiffiffiffiffiffi rbase ðtÞ ¼ 2C sBOC ðt sÞejh0 þ ðnc ðtÞ jns ðtÞÞ ð14Þ After filtering by the front-end filter, the filtered signal is
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Integrate and dump
IE1
Integrate and dump
IL1
Integrate and dump Integrate and dump Integrate and dump Integrate and dump
EBOC
QE1 QL1
E1 L1 Local reference waveform sref1(t) generator
Local BOC generator LBOC
QEBOC
Code NCO
Loop filter
Discriminator
Local reference waveform sref2(t) generator E2
L2 ILBOC Integrate and dump Integrate and dump Integrate and dump Integrate and dump Integrate and dump Integrate and dump
IE2 IL2 QLBOC QE2 QL2
Fig. 4 The proposed code tracking loop
r^base ðtÞ ¼ rbase ðtÞ hðtÞ pffiffiffiffiffiffi ns ðtÞÞ ¼ 2C s^BOC ðt sÞejh0 þ ðn^c ðtÞ j^
ð15Þ
where h(t) is the unit impulse response of filter, H(f) is its transfer function, ‘’ is the convolution operation. The CCFs between s^BOC ðtÞ and local reference waveforms are Z 1 TP R^BOC;ref1 ðeÞ ¼ s^BOCs ðtÞsref1 ðt eÞdt TP 0
ð16Þ ¼ F 1 GBOC;ref1 ðf ÞHðf Þ
R^BOC;ref2 ðeÞ ¼ F 1 GBOC;ref2 ðf ÞHðf Þ where GBOC;ref1 ðf Þ and GBOC;ref2 ðf Þ are the cross power spectrum density between sBOC ðtÞ and two local refer
and ence waveforms. GBOC;ref1 ðf Þ ¼ F RBOC;ref1 ðeÞ
1 GBOC;ref2 ðf Þ ¼ F RBOC;ref2 ðeÞ . F fg and F fg are Fourier transform and inverse Fourier transform, respectively. At each integration period, the discriminator outputs the estimator of code delay. Assuming that the frequency of IF carrier has been correctly estimated, namely f~IF ¼ fIF , then the outputs of complex correlators after integrating and dumping are expressed as
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pffiffiffiffiffiffi d IE1 þjQE1 ¼ 2CR^BOC;ref1 Ds ejDh þnIE1 þjnQE1 2 pffiffiffiffiffiffi d IE2 þjQE2 ¼ 2CR^BOC;ref2 Ds ejDh þnIE2 þjnQE2 2 pffiffiffiffiffiffi d IEBOC þjQEBOC ¼ 2CR^B Ds ejDh þnIEBOC þjnQEBOC 2 pffiffiffiffiffiffi d IL1 þjQL1 ¼ 2CR^BOC;ref1 Dsþ ejDh þnIL1 þjnQL1 2 pffiffiffiffiffiffi d IL2 þjQL2 ¼ 2CR^BOC;ref2 Dsþ ejDh þnIL2 þjnQL2 2 pffiffiffiffiffiffi d ILBOC þjQLBOC ¼ 2CR^B Dsþ ejDh þnILBOC þjnQLBOC 2 ð17Þ where Ds and Dh are the estimation error of code delay and carrier initial phase, and d is the correlator spacing. All the noise terms satisfy Gaussian distribution, and the outputs of I-branches and Q-branches are independent, because they are generated by different noise process nc ðtÞ and ns ðtÞ. For instance, the noise term nIE1 through one integration period is
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nIE1 ¼
1 Tp
Z 0
Tp
629
d n^c ðtÞsrefi t þ dt 2
ð18Þ
where TP is the coherent integration time. Obviously, nIE1 is a Gaussian process with zero mean, and its variance is given by 2 3 !2 Z KTp h i 1 d n^c ðtÞsrefi t þ dt 5 E ðnIE1 Þ2 ¼ E4 Tp ðK1ÞTp 2 Z 1 N0 N0 ^ R1;1 ð0Þ ¼ jHðf Þj2 G1;1 ðf Þdf ¼ Tp 1 Tp ð19Þ G1;1 ðf Þ is the power spectral density of the first local reference waveform, and G1;1 ðf Þ ¼ G2;2 ðf Þ ¼ 2Ts sinc2 ð2pfTs Þ. Considering that Dh 0, the correlator outputs IE1, IL1, IE2, IL2, IEBOC, ILBOC at Ds ¼ 0 satisfy the following joint Gaussian distribution: T
ðIE1 ; IL1 ; IE2 ; IL2 ; IEBOC ; ILBOC Þ N ðl; RÞ
ð20Þ
l and R are defined by
The range of the main-peak in Run1 ðeÞ is from -Ts to Ts. When the correlator spacing is smaller than Ts, without considering the impact of noise terms and filter, substituting (9) and (17) into (22), we obtain 8 8C d d > > 2 Ds; jDsj > > kT T 2 > s s > > > 8C d Ds d d > > \DsTs 1 ; > > > kT T 2 2 s > s > > > 8C d Ds d d 2 > > 4C d d d > > Ts þ Ds ; Ts \DsTs þ > 2 > kT 2 2 2 > > 2 > s > > 4C d d d > > 2 Ts þ þ Ds ; Ts Ds\ Ts > > > kT 2 2 2 : s 0: otherwise ð23Þ From (23), we know that the linear pull-in range of V1 is d=2. V1 ðDsÞ is an odd function. The gain of the dis
8C d 2 criminator V1 is kT Ts . s
pffiffiffiffiffiffi T 2C R^BOC;ref1 ðd=2Þ R^BOC;ref1 ðd=2Þ R^BOC;ref2 ðd=2Þ R^BOC;ref2 ðd=2Þ R^B ðd=2Þ R^B ðd=2Þ 3 2^ R1;1 ð0Þ R^1;1 ðdÞ R^1;2 ð0Þ R^1;2 ðdÞ R^BOC;ref1 ð0Þ R^BOC;ref1 ðd Þ 6 R^ ðdÞ R^1;1 ð0Þ R^1;2 ðd Þ R^1;2 ð0Þ R^BOC;ref1 ðd Þ R^BOC;ref1 ð0Þ 7 7 6 1;1 7 6 ^1;2 ð0Þ ^1;2 ðd Þ ^2;2 ð0Þ ^2;2 ðdÞ ^BOC;ref2 ð0Þ ^BOC;ref2 ðd Þ 7 R R R R R R N0 6 7 6 R¼ TP 6 R^1;2 ð0Þ R^2;2 ðdÞ R^2;2 ð0Þ R^BOC;ref2 ðd Þ R^BOC;ref2 ð0Þ 7 7 6 R^1;2 ðdÞ 7 6 ^ 5 4 R^BOC;ref1 ð0Þ R^BOC;ref1 ðdÞ R^BOC;ref2 ð0Þ R^BOC;ref2 ðdÞ R^B ð0Þ RB ðdÞ R^BOC;ref1 ðdÞ R^BOC;ref1 ð0Þ R^BOC;ref2 ðdÞ R^BOC;ref2 ð0Þ R^B ðd Þ R^B ð0Þ l¼
where R^1;2 is the CCF between sref1 ðtÞ and sref2 ðtÞ in the case of bandwidth limitation. Similarly, the joint distribution of QE1, QL1, QE2, QL2, QEBOC, QLBOC at Ds ¼ 0 is N ð0; RÞ.
For the combined correlation function Run2 ðeÞ, the corresponding non-coherent discriminator function is chosen as 1
Non-coherent discriminator functions
V1 Discriminator output
For these two unambiguous combined correlation functions, we provide two corresponding discriminator functions. When choosing the combined correlation function Run1 ðeÞ, the proposed non-coherent discriminator function is 0 1 ðjIE1 j þ jIE2 j jIE1 IE2 jÞ2 þ A V1 ðDsÞ ¼ @ ðjQE1 j þ jQE2 j jQE1 QE2 jÞ2 0 1 ð22Þ ðjIL1 j þ jIL2 j jIL1 IL2 jÞ2 þ A @ ðjQL1 j þ jQL2 j jQL1 QL2 jÞ2 ¼ 2C R2un1 ðDs d=2Þ R2un1 ðDs þ d=2Þ
ð21Þ
V2
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Code delay (chips) Fig. 5 Discriminator curve for sine-BOC(14,2)
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1.5
Discriminator output
1
Code tracking error (chips)
V1 V2
0.5 0 -0.5 -1 -1.5
V1(Monte Carlo)
-0.5
0
0.5
V2(Monte Carlo) V2(Theory)
0.04
0.02
0 20
-1
V1(Theory)
0.06
25
30
35
40
45
C/N0 (dB-Hz)
1
Code delay (chips) Fig. 7 Code tracking error of the proposed method for sineBOC(14,2)
Fig. 6 Discriminator curve for sine-BOC(10,5)
V2 ðDsÞ ¼
ðIEBOC IE1 þ jIEBOC IE2 jÞþ ðQEBOC QE1 þ jQEBOC QE2 jÞ ðILBOC IL1 þ jILBOC IL2 jÞþ
! !
ð24Þ
ðQLBOC QL1 þ jQLBOC QL2 jÞ
¼ 2C ðRun2 ðDs d=2Þ Run2 ðDs þ d=2ÞÞ On the same assumption, substituting (11) and (17) into (24), we can derive the output of V2 when jDsj k=ð2k 1ÞTs d=2, i.e.,
corresponding correlator spacing are 0.03Tc and 0.1Tc, respectively. We can see that there are no false lock points for V1. V2 includes two positive zero-crossing points except for the origin point due to the two small positive side-peaks near to ±Tc in Run2 ðeÞ. Fortunately, they are weak enough and far away the origin point; thus, they can be neglected. In other words, V1 and V2 are unambiguous. The code tracking error variance is (Kao and Juang 2012)
8 pffiffiffi 4 2C d > > 3k 1 ð2k 1Þ Ds; > > pffiffiffi > Ts k pkffiffiffiTs > > > > 2 2Cd Ds > > p ffiffi ffi > ð 3k 1 Þ ð 4k 2 Þ ; > > Ts k p kTffiffisffi > > > > 2 2Cd Ds > > > p ffiffi ffi ð 3k 1 Þ þ ð 4k 2 Þ ; > > Ts > 1 < k kTs 0 d V2 ðDsÞ ¼ pffiffiffi ð 2k 1 Þ s > B 2 2C d > 2 C > C; > pffiffiffi 1 Ts þ Ds B > A @ > kT 2 > kTs s > > > > 1 > 0 > > d > > p ffiffi ffi ð2k 1Þ s þ > > B > 2 2C d 2 C > C; B > p ffiffi ffi þ Ds 1 þ T þ s > A @ > 2 kT > k T s s :
Noted that (25) is established only when the correlator spacing is smaller than k=ð2k 1ÞTs . V2 ðDsÞ is also an odd pffiffi function. The gain of the discriminator V2 is k4pffiffik2TC s
d 3k 1 ð2k 1Þ Ts . Figures 5 and 6 show the discriminator curves for sineBOC(14,2) and sine-BOC(10,5), respectively. The
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jDsj
d 2
d k d \Ds Ts 2 2k 1 2 d k d Ts Ds\ 2 2k 1 2 k d k d Ts \Ds Ts þ 2k 1 2 2k 1 2
r2 ¼
ð25Þ
k d k d Ts Ds\ Ts þ 2k 1 2 2k 1 2
2BL ð1 0:5BL TP ÞTP r2V KV2
ð26Þ
where BL is the code loop noise bandwidth [Hz], rV is the discriminator output standard deviation, KV is discriminator gain.
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Simulation results and performance analysis In order to evaluate the performance of our method, we analyze its code tracking error in thermal noise and the impact of multipath with the help of theory and simulation. In order to perform a comprehensive assessment, low-order sine-BOC(10,5) and high-order sine-BOC(14,2) are considered. As comparisons, we also present the performance of BJ method, DET, PUDLL and Gating function. Noise performance We simulate the code tracking performance of the proposed method in thermal noise. The non-coherent discriminator functions are used for all the methods. The code loop noise bandwidth BL = 1 Hz, and Tp = 1 ms. For sine-BOC(10,5), the received bandwidth is 30.69 MHz, and correlator spacing d is 0.1chips. For sineBOC(14,2), the received bandwidth is 36.82 MHz, and correlator spacing d is 0.03chips. The ideal brick-wall filter and the first-order loop are used. Some other parameters required by BJ, DET, PUDLL and Gating function are as following. BJ method uses a counter mechanism to achieve the comparison, only when one counter reaches the pre-set threshold; then, the tracker jumps to the new peak. When the tracking loop utilizes a bigger pre-set threshold, the
0.06
Code tracking error (chips)
V1(T) V1(S)
0.05
V2(T) V2(S)
PUDLL
0.04 V2
0.03
V1
BJ
0.02
Gating function
BJ(T) BJ(S) DET(T) DET(S) PUDLL(T) PUDLL(S) Gating function(T) Gating function(S)
0.01 DET
0 20
25
30
35
40
45
C/N0 (dB-Hz) Fig. 8 The theory (T) and simulation (S) results of code tracking error standard deviation for sine-BOC(10,5)
0.03 V1(T)
Code tracking error (chips)
When using discriminator function V1 , the discriminator gain K1 and discriminator output variance r21 are given by (29) and (36), respectively. For discriminator function V2 , it is difficult to derive the accurate expression of discriminator output variance r22 ; thus, we present an approximate result in (46). The corresponding discriminator gain K2 is given by (38). The detail derivation process can be found in the ‘‘Appendix.’’ Taking sine-BOC(14,2) as an example, Fig. 7 shows the curves of code tracking error standard deviation versus C/N0. The received bandwidth is 36.82 MHz, correlator spacing d is 0.03 chips, Tp = 1 ms and BL = 1 Hz. The theoretical and Monte Carlo simulation results are given simultaneously. In the Monte Carlo simulation, assume the estimation error of code delay is zero. Then, the discriminator output standard deviation is obtained statistically. According to (26), the code tracking error is calculated. We can see that Monte Carlo simulation results are in good agreement with theory, even in low C/N0. This is because both of the theoretical analysis and the Monte Carlo simulation are based on the linear model. Therefore, the analysis results in the ‘‘Appendix’’ are efficient enough to evaluate the code tracking performance of our method.
631
V1(S)
0.025
V2(T) V2(S)
0.02 PUDLL
0.015 0.01 0.005 0 20
Gating function V1
V2
BJ
DET
25
BJ(T) BJ(S) DET(T) DET(S) PUDLL(T) PUDLL(S) Gating function(T) Gating function(S)
30
35
40
45
C/N0 (dB-Hz) Fig. 9 The theory (T) and simulation (S) results of code tracking error standard deviation for sine-BOC(14,2)
probability of main-peak tracking is higher. However, Blunt et al. (2007) have pointed out that it is impractical to raise the BJ threshold sufficiently high to deliver reliable tracking. In the simulation, the threshold is set to 10. The tracking performance of DET is mainly determined by the SLL (Hodgart et al. 2007). Thus, in order to have a fair comparison, the early-late spacing of SLL is the same with d. The correlator spacing dcode of DLL can be chosen from Ts to Tc. The smaller dcode indicates better performance. In simulation, the dcode is equal to Ts. The parameter j is used in PUDLL. We set j ¼ 0:3 in simulation, which is proposed by Yao et al. (2010). In order to achieve a more robust tracking, a bigger Gating width is required for Gating function method. In our simulation, the Gating width is set to L = 0.8Ts. In our analysis, the theory and simulation results are shown simultaneously. The theoretical result of DET is
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GPS Solut (2015) 19:623–638 100
35
90
30
Tracking threshold (dB-Hz)
The probability of main-peak tracking (%)
632
80 70 60 50
BOC(10,5),BJ with Threshold=10 BOC(14,2),BJ with Threshold=10 BOC(10,5),DET BOC(14,2),DET
40 30 20 20
25
30
35
40
25 20
V1 V2
15
BJ DET PUDLL Gating function
10 5
45
C/N0 (dB-Hz)
0
sine-BOC(10,5)
sine-BOC(14,2)
Fig. 11 The code tracking threshold Fig. 10 The probabilities of the main-peak tracking versus C/N0
given by (55) in the ‘‘Appendix,’’ which is derived based on the following assumptions: The code delay has been accurately estimated by the DLL, and no false correction occurs. Similarly, the theoretical result of BJ also assumes that no false lock occurs. The theoretical performance of PUDLL and Gating function has been provided by Yao et al. (2010) and Nunes et al. (2007), respectively. The numerical results simulate the actual code tracking process, and every data is obtained statistically through the simulation of duration 100 s. The initial delay estimate error is set to 0. Therefore, the simulation results represent the ‘‘true’’ code tracking performance. Figure 8 shows the code tracking error standard deviation of the above and proposed methods in different C/N0 for sine-BOC(10,5). We can see that the theoretical tracking error of V2 is worse than BJ, DET. However, observing the dashed marked with ‘‘circle’’ and the dashed marked with ‘‘up-triangle,’’ we can find that the true code tracking performance of V2 and PUDLL are the same. In the region of C/N0 \ 25 dB-Hz, V2 can still maintain reliable tracking, while DET begins to lose lock. V1 has relatively worse code tracking performance. However, we should keep in mind that V1 is completely unambiguous. It is more reliable than BJ method with Threshold = 10 when C/N0 \ 29 dB-Hz. Gating function shows the worst code tracking error. Figure 9 shows the code tracking performance for sineBOC(14,2) signal. Seen from the Fig. 9, the theoretical performance of V2 is worse than DET and BJ. In the region of C/N0 [ 27 dB-Hz, the actual performance of DET is also the best. Nevertheless, V2 provides the best performance when C/N0 \ 27 dB-Hz. This is because DET begins to lose lock and frequently perform false corrections. This means that V2 can achieve more robust tracking
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for high-order BOC signal in the lower C/N0 case. The tracking performance of V1 is relatively worse. However, it can also be observed that V1 is far better than Gating function. Note that the theoretical results and simulation results have some deviation, especially in the low C/N0. A main reason is that the loop is losing lock and the discriminator is working in its nonlinear region, but the theoretical analysis is based on the linear model. For V1 and PUDLL, the assumption condition (35) also introduces some deviation (Yao et al. 2010). For BJ method, the true code tracking performance significantly degrades compared with theory results in low C/N0. This is because false locks occur frequently. For DET, it is not possible to estimate the code delay accurately; thus, a loss of discriminator gain is introduced. In most references, the simulations are performed by Monte Carlo simulation. However, it cannot reflect the true code tracking performance. This is the reason why we choose the numerical simulation. The deviation among the theory, Monte Carlo and numerical simulation can also be found in Anantharamu et al. (2009). Figure 10 can further explain the reason. Figure 10 shows the probabilities of locking the main-peak versus C/N0 for BJ method. For DET, Fig. 10 is the probabilities that the corrected operation is performed correctly. Once the difference between DLL’s delay estimate and SLL’s delay estimate exceeds Ts/2, the false corrected operation would be performed, and then, large tracking error is introduced. In order to evaluate the robustness, we can obtain the code tracking threshold from Figs. 8 and 9, which is depicted in Fig. 11. The tracking threshold is the minimum C/N0 value at which a tracking loop is able to maintain a stable lock (Anantharamu et al. 2011). The curve breaking
points in Figs. 8 and 9 are considered to be the tracking threshold (Anantharamu et al. 2009). For sine-BOC(10,5), the tracking threshold of V2 is 22 dB-Hz, which is the lowest. The tracking threshold of DET is 25 dB-Hz. Compared to DET, V2 has 3 dB improvement. Compared with Gating function method, V1 has 6 dB improvement. Similarly, for sine-BOC(14,2), the tracking threshold of V2 is 23 dB-Hz, which is 4 dB lower than the tracking threshold of DET. Compared with Gating function method, V1 has 4 dB improvement. Thus, we can conclude that V2 provides the most robust code tracking process.
633
Multipath error envelope (chips)
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V1
0.06
V2 BJ DET PUDLL Gatimg function
0.04 0.02 0 -0.02 -0.04 -0.06
Impact of multipath Multipath causes the distortion of the correlation functions, which results in the offset of zero-crossing point in discriminator curve. Thereby, a tracking bias error is introduced. To analyze the multipath mitigation performance, similar with Julien et al. (2011), we exploit a general model. There is only one multipath with some amplitude attenuation relative to the direct signal. The phase difference between multipath and direct signal is 0 or 180. In simulation, the multipath to direct ratio (MDR) is -6 dB. Namely, the amplitude of multipath is a half of direct signal’s amplitude. When considering the effect of multipath, discriminator function V1 is rewritten as 12 R^BOC;ref1 ed a1 R^BOC;ref1 ed s1 þ C B 2 2 B C B C B C d d B R^BOC;ref2 e a1 R^BOC;ref2 e s1 C B C 2 2 B C B C D1 ðeÞ¼B C d d B R^ C a1 R^BOC;ref1 e s1 B BOC;ref1 e C 2 2 B C B C B C @ d d A R^BOC;ref2 e a1 R^BOC;ref2 e s1 2 2 0 12 R^BOC;ref1 eþd a1 R^BOC;ref1 eþd s1 þ C B 2 2 B C B C B C d d B R^BOC;ref2 eþ a1 R^BOC;ref2 eþ s1 C B C 2 2 B C B C B d d C B R^ C a1 R^BOC;ref1 eþ s1 B BOC;ref1 eþ C 2 2 B C B C B C @ d d A R^BOC;ref2 eþ a1 R^BOC;ref2 eþ s1 2 2 0
where a1 is the MDR, s1 is the multipath delay relative to direct signal. The multipath error envelope is obtained by solving equation D1 ðeÞ ¼ 0 (Kao and Juang 2012). For discriminator function and V2, the multipath error envelope can be solved in the same way.
0
0.5
1
1.5
Multipath delay (chips) Fig. 12 Multipath error envelope for sine-BOC(10,5)
Figures 12 and 13 show the multipath error envelope and average multipath error curves for sine-BOC(10,5), respectively. The front-end bandwidth and the early-late spacing are the same as before. As it can be observed, discriminator function V1 can completely mitigate the medium-delay and far-delay anti-phase multipath. In addition, V1 can almost completely suppress the mediumdelay multipath (whose delay is from 0.3 chips to 0.7 chips for BOC(15,10)). The multipath performance of V2 is close to DET and better than BJ. This can be found clearly in Fig. 13. Figure 14 shows the multipath error envelope for sineBOC(14,2). When the multipath delay is from 0.1 chips to 0.9 chips, V1 can efficiently mitigate multipath, which is the same level with Gating function, and far better than BJ, DET and PUDLL. Figure 15 provides the average multipath error for sine-BOC(14,2). Only when multipath delay is from 0.02 chips to 0.11 chips, V1 show slightly worse. Once multipath delay is more than 0.1chips, V1 shows great advantages compared with BJ, DET and PUDLL. The far-delay multipath has more effect on V1, but the far-multipath has lower occurrence probability. Also, Fig. 15 shows that the anti-multipath performance of V2 is better than DET and BJ method for sineBOC(14,2). We also note that Gating function has better multipath mitigation performance than V1 for short-delay and fardelay multipath. However, for high-order sine-BOC(14,2) signal, the advantage of Gating function is not great. More importantly, the previous analysis shows that the code tracking performance of Gating function is poor. The tracking loop of Gating function is unstable and easy to lose lock in the environment of noise. From this sense, the multipath performance of V1 is still attractive.
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V1 0.035
V1
V2 BJ DET PUDLL Gatimg function
0.03 0.025 0.02 0.015 0.01 0.005 0
0
0.5
1
1.5
V1
Multipath error envelope (chips)
V2 BJ DET PUDLL Gatimg function
0.005 0 -0.005 -0.01 -0.015 0
0.2
0.4
0.6
BJ DET PUDLL Gatimg function
0.008
0.006
0.004
0.002
0
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 15 Average multipath error for sine-BOC(14,2)
Fig. 13 Average multipath error for sine-BOC(10,5)
0.01
V2
0.01
Multipath delay (chips)
Multipath delay (chips)
0.015
Average multipath error (chips)
Average multipath error (chips)
0.04
0.8
1
1.2
Multipath delay (chips)
Fig. 14 Multipath error envelope for sine-BOC(14,2)
high-order sine-BOC(14,2) signal, when the multipath delay varies from 0.1 chips to 0.9 chips, V1 has the same level multipath mitigation performance with Gating function method. In terms of code tracking performance, V1 shows worse than DET, BJ and PUDLL, but V1 is more reliable than BJ method with Threshold = 10. In addition, the tracking accuracy of V1 is significantly better than Gating function. On the whole, the multipath performance of V1 is attractive. Discriminator function V2 provides a good code tracking performance. In the high C/N0 case, the tracking accuracy of V2 is comparable with DET. In the lower C/N0, V2 is the most reliable. Even though tracking the high-order BOC(14,2) signal, the tracking threshold of V2 has 4 dB improvement. In terms of multipath mitigation, V2 shows a slightly better average multipath performance than DET, BJ and PUDLL.
Conclusions
Appendix: Derivation of code tracking error variance
An unambiguous tracking method based on combined correlation functions has been proposed for sine-BOC signals. The proposed technique exploits two new local reference waveforms. To achieve unambiguous tracking, we provide two non-coherent discriminator functions, denoted as V1 and V2. The performance of code tracking and anti-multipath has been analyzed and compared with existing methods. Based on simulation results, we summarize the following conclusions: Discriminator function V1 can almost completely suppress the medium-delay multipath, and its average multipath performance is better than DET, BJ and PUDLL. For
The non-coherent discriminator functions are used in our method, so we can set Dh 0. In this derivation, we take some simplified hypothesis. First, we assume that filter is ideal, i.e.,
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H ð f Þ ¼ H ðf Þ ¼ jH ð f Þj In the case of infinite bandwidth, it is easy to prove that R1;2 ð0Þ ¼ 0; R1;2 ðdÞ ¼ R1;2 ðdÞ ¼ 0 RBOC;ref1 ðeÞ ¼ RBOC;ref2 ðeÞ
when jej Ts
Thus, they are also approximately established in bandwidth limitation case, i.e.,
GPS Solut (2015) 19:623–638
Z
1
635
Hðf ÞGBOC;ref1 ðf Þejpfd df
1
Z
1
Z1 1 Z1 1 Z1 1
Hðf ÞGBOC;ref1 ðf Þejpfd df Hðf ÞGBOC;ref2 ðf Þejpfd df
ð27Þ
Let fX ð xÞ be the probability density function of X, according to (31), we derive out pffiffiffiffiffiffi N0 ^ ^ fX ð xÞ¼P 2CRBOC;ref1 ðd=2Þ; R1;1 ð0Þ;x ;x[0 ð32Þ TP
Hðf ÞGBOC;ref2 ðf Þejpfd df Hðf ÞGBOC;ref1 ð f Þcosðpfd Þdf
1
Based on the above assumptions, we next provide a simple derivation of the approximate code tracking error variation. Discriminator function V1 Ignoring the noise terms, V1 can be expanded to the following equation, d d 2 2 ^ ^ V1 8C RBOC;ref1 Ds RBOC;ref1 Ds þ 2 2 ð28Þ Following (16), the CCF is converted to frequency domain, then the discriminator gain at Ds ¼ 0 is dV1 K1 ¼ dDsDs¼0 Z 1 0 1 Hðf ÞGBOC;ref1 ð f Þ cosðpfd Þdf C B 1 C ¼ 64pC B A @Z 1 f Hðf ÞGBOC;ref1 ð f Þ sinðpfd Þdf 1
ð29Þ Noted that (29) is the discriminator gain in the case of bandwidth limitation. Let X ¼ ðjIE1 j þ jIE2 j jIE1 IE2 jÞ 0 Y ¼ ðjIL1 j þ jIL2 j jIL1 IL2 jÞ 0 Z ¼ ðjQE1 j þ jQE2 j jQE1 QE2 jÞ 0 W ¼ ðjQL1 j þ jQL2 j jQL1 QL2 jÞ 0
According to (20) and (21), IE1 and IE2 can be seen as the independent and identically distributed Gaussian random variables pffiffiffiffiffiffi N0 N 2C R^BOC;ref1 ðd=2Þ; R^1;1 ð0Þ TP
where 2 ð2xlÞ x l 1 P l; r2 ; x ¼ pffiffiffiffiffiffi e 2r2 Q 2 r 2pr 2 x x þl ð Þ 1 2 2 2þl 2r p ffiffiffiffiffi ffi e Q þ r 2pr
Similarly, the probability density functions of Y; Z and W are pffiffiffiffiffiffi N0 ^ ~ ^ fY ðyÞ ¼ P 2C RBOC;ref1 ðd=2Þ; R1;1 ð0Þ; y ;y[0 TP N0 fZ ðzÞ ¼ P 0; R^1;1 ð0Þ; z ; z [ 0 TP N0 ^ fW ðwÞ ¼ P 0; R1;1 ð0Þ; w ; w [ 0 TP ð34Þ Based on a similar assumption with the derivation of PUDLL (Yao et al. 2010), the correlation coefficient between X 2 and Y 2 is approximately equal to . qR2E R2L R^21;1 ðdÞ R^21;1 ð0Þ ð35Þ Thus, the variance of discriminator V1 is ¼ E V12
2 1 qR2E R2L E X 4 þ E Z 4 E2 X 2 E2 Z 2
r2V1
ð36Þ
ð30Þ
then V1 ¼ ðX 2 Y 2 Þ þ ðZ 2 W 2 Þ.E½V1 ¼ 0. Now, we discuss the distribution of X. For different values of IE1 and IE2, X is calculated by 8 0\IE1 \IE2 2IE1 ; > > > > 2IE ; IE2 \IE1 \0 or > 1 > < 0; IE1 \0 IE2 ð31Þ X¼ 2IE2 ; 0\IE2 \IE1 or > > > > IE2 ; IE1 \IE2 \0 or > > : 0; IE2 \0 IE1
ð33Þ
where E Xi ¼
Z
1
xi fX ðxÞdx;
0
E Zi ¼
Z
1
zi fZ ðzÞdz
ð37Þ
0
Substituting (29) and (37) into (26), we can obtain the theoretical code tracking error variance of discriminator function V1. Discriminator function V2 Similar to discriminator function V1, the gain of discriminator function V2 at Ds ¼ 0 is
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00 Z
1 Hðf Þf G ðf Þsin ð pfd Þdf BOC;ref1 BB C BB 1 C K2 ¼ 16pCBB C Z 1 @@ A Hðf ÞGB ðf Þcosðpfd Þdf 1 0Z 1 11 Hðf ÞG ð f Þcos ð pfd Þdf BOC;ref1 B CC B 1 CC þB CC Z 1 @ AA Hðf ÞfGB ðf ÞsinðpfdÞdf 1
we have h i h i
E ðXIE XIL Þ2 ¼ 2 E ðIEBOC IE1 Þ2 ðE½IEBOC IE1 Þ2 ð43Þ Equation (43) can be calculated by the characteristic function of joint Gaussian distribution, namely
1
ð38Þ where GB ðf Þ ¼ F fRB ðeg is the power spectral density of the BOC signal. Equation (38) provides the discriminator gain under bandwidth limitation, which is different with the one in the case of infinite bandwidth. Let
ð44Þ pffiffiffiffiffiffi where l1 ¼ 2C R^BOC;ref1 d2 ; R^BOC;ref1 ð0Þ , r2 ¼ N0 R^1;1 ð0Þ. R^1;1 ð0Þ
XIE ¼ IEBOC IE1 þ jIEBOC IE2 j XIL ¼ ILBOC IL1 þ jILBOC IL2 j XQE ¼ QEBOC QE1 þ jQEBOC QE2 j
h i E ðIEBOC IE1 Þ2 ¼ 1 þ 2q2 r4 þ l21 þ l22 þ 4l1 l2 q r2 þ l21 l22 2 ðE½IEBOC IE1 Þ2 ¼ l1 l2 þ qr2 h i E ðXIE XIL Þ2 ¼ 2 1 þ q2 r4 þ l21 þ l22 þ 2l1 l2 q r2
pffiffiffiffiffiffi l2 ¼ 2C R^B d2 ,
q¼
Tp
In a similar way, the second part of r2V2 is ð39Þ
XQL ¼ QLBOC QL1 þ jQLBOC QL2 j
h i E ðXQE XQL Þ2 ¼ 2 1 þ q2 r4
ð45Þ
Using (40), (44) and (45), the variance of discriminator V2 output can be approximately expressed as
then V2 ¼ ðXIE XIL Þ þ ðXQE XQL Þ. Obviously, E ½ V2 ¼ 0
r22 ¼ 2 2 þ 2q2 r4 þ l21 þ l22 þ 2l1 l2 q r2
E½ðXIE XIL Þ ðXQE XQL Þ ¼ 0 h i h i r2V2 ¼ E V22 ¼ E ðXIE XIL Þ2 þ E ðXQE XQL Þ2 ð40Þ
ð46Þ
Substituting (38) and (46) into (26), we obtain the theoretical code tracking error variance of discriminator function V2.
The first part of r2V2 is expanded to h i h i h i E ðXIE XIL Þ2 ¼ E ðIEBOC IE1 Þ2 þ E ðIEBOC IE2 Þ2 þ 2E½IEBOC IE1 jIEBOC IE2 j h i h i þ E ðILBOC IL1 Þ2 þ E ðILBOC IL2 Þ2 þ 2E½ILBOC IL1 jILBOC IL2 j ! E½IEBOC IE1 ILBOC IL1 þ E½IEBOC IE1 jILBOC IL2 j 2 þE½ILBOC IL1 jIEBOC IE2 j þ E½jIEBOC IE2 jjILBOC IL2 j
Considering that the correlator outputs of I-branch follow the joint Gaussian distribution, we take the following approximation,then
ð41Þ
The code tracking error variance of DET The two dimensional correlation function of DET in the case of infinite bandwidth is (Hodgart et al. 2007)
h i h i h i h i E ðIEBOC IE1 Þ2 ¼ E ðIEBOC IE2 Þ2 ¼ E ðILBOC IL1 Þ2 ¼ E ðILBOC IL2 Þ2 E½IEBOC IE1 jILBOC IL2 j ¼ E½ILBOC IL1 jIEBOC IE2 j E½IEBOC IE1 jIEBOC IE2 j ¼ E½ILBOC IL1 jILBOC IL2 j rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i h i h i E½jIEBOC IE2 jjILBOC IL2 j E½jIEBOC IE2 jE½jILBOC IL2 j E ðIEBOC IE2 Þ2 E ðILBOC IL2 Þ2 ¼ E ðIEBOC IE2 Þ2 E½IEBOC IE1 ILBOC IL1 E½IEBOC IE1 E½ILBOC IL1 ¼ ðE½IEBOC IE1 Þ2
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ð42Þ
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vðDs; Ds Þ trcðDs ÞKðDsÞ
637
ð47Þ
where Ds is the estimation error of code delay in DLL, Ds
r2DET ¼
BL ð1 0:5BL TP ÞTs2
is the estimation error of subcarrier delay in SLL, KðeÞ ¼ 1 Tjecj is the ACF of BPSK signal, and trcðÞ is a continuous triangular cosine of periodicity 2Ts. Considering the effect of bandwidth limitation, Eq. (47) is rewritten as ^ðDsÞ r cðDs ÞK vðDs; Ds Þ t^
ð48Þ
^ðeÞ ¼ F fGK ð f ÞH ð f Þg, and GK ð f Þ ¼ F fKðeÞg. where K t^ r cðeÞ ¼ p82 cos pe Ts .
16K^ð0Þ p2
27 K^2 ð0Þ p2 C=N0
1 þ Tp C=N 0
ð55Þ
(55) is established based on the assumption that the estimation error of code delay in DLL is close to zero.
References
1
The code tracking performance of DET is mainly determined by the SLL. The E and L correlator outputs of in-phased and quadrature-phase branches in SLL are pffiffiffiffiffiffi ^ðDsÞcosðDhÞ þ nIE IE 2C t^rcðDs d=2ÞK pffiffiffiffiffiffi ^ðDsÞcosðDhÞ þ nIL IL 2C t^rcðDs þ d=2ÞK ð49Þ pffiffiffiffiffiffi ^ðDsÞsinðDhÞ þ nQE QE 2C t^ rcðDs d=2ÞK pffiffiffiffiffiffi ^ðDsÞsinðDhÞ þ nQL QL 2C t^ rcðDs þ d=2ÞK where these noise terms nIE , nIL , nQE and nQL satisfy Gaussian process with zero mean and variance N0 ^ð0Þ.E½nIE nIL ¼ E½nQE nQL ¼ N0 t^ ^ r c ð 0Þ K Tp t^ Tp r cð0ÞKðd Þ. The discriminator output of SLL is V ðDs Þ ¼ IE2 þ QE2 IL2 þ QL2
ð50Þ
Assume that the code delay has been accurately estimated by the DLL, i.e. Ds 0 and Dh 0, then ^2 ð0Þ t^rc2 ðDs d=2Þ t^rc2 ðDs þ d=2Þ V ðDs Þ ¼ 2CK pffiffiffiffiffiffi ^ð0Þðt^rcðDs d=2ÞnIE trc ^ ðDs þ d=2ÞnIL Þ þ 2 2CK 2 þ nIE n2IL þ n2QE n2QL ð51Þ The discriminator gain at Ds ¼ 0 is ^ 2 ð 0Þ dV 28 C K pd ¼ sin K¼ 3 dDs Ds ¼0 p Ts Ts
ð52Þ
Due to Ds 0, the noise part in (52) is pffiffiffiffiffiffi ^ð0Þt^rcðd=2ÞðnIE nIL Þ þ n2 n2 þ n2 n 2 2C K IE IL QE n2QL ð53Þ The variance of discriminator function V is given by ! ^ ð 0Þ 28 CN0 2 pdTs ^2 16K 1 2 rV ¼ 4 sin þ K ð 0Þ p2 Tp C=N0 p Tp Ts ð54Þ Substituting (52) and (54) into (26), we have
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Tao Yan is a PhD student at the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, China. He received his BS degree from Huazhong University of Science and Technology in 2010. He is interested in design method of GNSS signals and signal processing algorithm for GNSS receivers.
Jiaolong Wei received BE, MS and PhD degrees from Huazhong University of Science and Technology, Wuhan, China, in 1986, 1990 and 2004, respectively. He is currently a professor, PhD supervisor and vicedean of the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include wireless communications and networks, intelligence computation and satellite navigation.
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Zuping Tang received BE, MS and PhD degrees from Huazhong University of Science and Technology, Wuhan, China, in 2002, 2005 and 2009 respectively. He is currently a lecturer of the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include GNSS signal design theory, signal quality evaluation and GNSS receiver technique. Bo Qu is a PhD student in Communication and Information System at Huazhong University of Science and Technology, where he received his BS and MSc degrees in from the Department of Electronic and Information Engineering in 2007 and 2009, respectively. His scientific research work focuses—among other topics— on GNSS signal vulnerability and receiver performance with special focus on multipath or inference effects and mitigation. Zhihui Zhou is a PhD candidate at the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, China. He received his BS degree from Huazhong University of Science and Technology in 2011. He is interested in nextgeneration GNSS signals and signal processing algorithm of GNSS receivers.
