Delay Tracking Algorithm for the Pilot Component of GLONASS Perspective Signal with Code Division Multiple Access

The paper studies the issue of the synthesis of the optimal code delay tracking algorithm for the pilot component of GLONASS perspective signal with BOC(1,1) modulation and processing at subcarrier frequencies. The study objective is synthesis and analysis of optimal code delay tracking algorithm of BOC(1,1) modulated signal (pilot component of L1OC GLONASS perspective signal) using the method of additional variable and processing at subcarrier frequencies. Equations describing optimal tracking system with account to multimodality of code delay aposteriori probability density are calculated using the theory of optimal filtering and the method of additional variable. The structural scheme of delay discriminator with processing at subcarrier frequencies is developed. Discrimination and fluctuation characteristics of synthesized delay discriminator are calculated. The results of the proposed tracking algorithm modelling, which demonstrate its performance and accuracy characteristics, are presented.


Introduce the Problem
Since 2012 the Federal Target Program of "GLONASS Maintenance, Development and Use for 2012 -2020 years" is effective in the Russian Federation, based upon which beginning from 2016 new GLONASS signals with code division will be emitted from satellite vehicles GLONASS-K2.In 1 L , 2 L frequency ranges the signals of open access are two-component signals containing pilot and information components, the combining of which into a single navigation signal is carried out according to the method of by-bit time multiplexing (Perov & Harisov, 2010).The information component of this signal has standard binary modulation (BPSK) with a frequency of code ranging symbols of 1,023 MHz, and the pilot component has modulation at subcarrier frequencies BOC(1,1) with a base frequency of b f =1,023 MHz.
It is planned to use signals with similar type of modulation in modernizing navigation satellite system GPS (Interface Specification GPS Space Segment/ User Segment.IS-GPS-200F, 21 September 2011; Interface specification GPS Space Segment/User Segment L1C Interfaces.IS-GPS-800B, 21 September 2011) and in advanced navigation system (European GNSS (GALILEO), 2010;BeiDou, 2011).

Importance of the Problem
BOC-modulation signals tracking has a number of peculiarities associated with the envelope correlation function's several extremums.It leads to the multimodality nature of signal delay aposteriori probability density and there is a problem of possible ambiguity of delay estimation.That is why it is relevant to develop BOC-modulation signal detection and processing algorithms with regard to aposteriori distribution multimodality.

Literature Review
A number of studies are concerned with this topic.Martin N. et al. study the problem of BOC-modulation signal acquisition, the algorithm, in which BOC-modulation signal is considered as a sum of two signals with BPSK modulation at subcarrier frequencies symmetrically located in relation to carrier frequency is described.Heiries V. et al. give the results of this method used to detect signals with BOC(10,5) and BOC(14,2) modulations, which shows their high efficiency.Chen S. et al. (Chen S. et al., 2006) bring the ideology of BOC modulation signal presentation in the form of two signals with BPSK modulation at subcarrier frequencies, it is used to track signal delay and phase.However, proposed processing algorithms are heuristic; they do not result from a strict synthesis.
Fante R. L., David de Castro et al. study signal delay estimation algorithms, wherein besides "early" and "late" reference signals "very early" and "very late" signals are additionally used.Such approach allows extending the discriminator curve of delay discriminator, but does not solve the problems of the multimodality of aposteriori probability density.
Some of modifications of this approach with respect to various types of ВОС-modulations have been described (Musso et al., 2006;Jovancevic et al., 2007;Gyu-In et al., 2007;Jovanovic et al, 2012).
Hodgart et al. (Hodgart et al, 2007;Hodgart et al., 2008) studied BOC modulation signals tracking algorithms with separate signal delay tracking systems related to BPSK ranging code and signal delay related to a digital subcarrier.In such a manner, two delay estimations are united into final estimation according to the corresponding algorithm.It shall be noted that such approach known as "additional variable" was described by Tikhonov V. I. and Harisov V. N. in 1984 and was widely used for signal delay tracking problem solving with extraction delay information from the envelope and signal phase (Perov, A. I., & Harisov, V. N., 2010;Kaplan, E. D., 2006).Its main intended use is disambiguation of estimations driven by the estimated parameter aposteriori probability density multimodality.
As distinct from Hodgart et al. (Hodgart et al., 2007;Hodgart et al., 2008) Tikhonov, V. I. and Harisov V. N. (Tikhionov, V. I., Kharisov, V. N., 2004) used two components tracking (main and additional variables) which is carried out in one tracking loop.It shall be noted that this fact results from the rigorous theory of optimal system filtration synthesis.
Vinogradov et al. (Vinogradov et al., 2006) used the given approach to solve BOC modulation signal delay filtration problems.
In the articles mentioned, the processing of received signals is carried out at carrier frequency using the classic signal phase tracking system.It has been mentioned when receiving BOC modulation signals the processing at subcarrier frequency is possible, that may offer certain advantages.
The objective of the article is the synthesis and analysis of optimal code delay tracking algorithm with BOC(1,1) modulation (L1OC GLONASS signal testing component) using the method of additional variable and with processing at subcarrier frequencies.

Setting the Synthesis Problem
We use optimal filtering theory to synthesize optimal (quasioptimal) tracking system, which implies statistical description of processes and observations.Let us assume there is an additive mixture of navigation signal ( ) and receiver internal noise ( ) n t at the receiver input, the noise is assumed to be white Gaussian with one-way power spectral density 0 N .
L1OC GLONASS signals with code division, which is a two-component signal containing pilot and information components, is considered as a navigation signal (Perov, A. I., & Harisov, V. N., 2010).
The pilot component ( ) It is assumed that sampling of input process by time is carried out in the receiver, thus realization in discrete time , k i t enters signal-processing system (Figure 1) , , where 0 N -receiver internal noise one-way power spectral density.
Figure 1.Indexing time scheme In the paper double time indexing is used, which is necessary for the purpose of correct accounting of input counts , k i y processing in correlators and rarer processing (with a time step T ) in tracking system smoothing filter when synthesizing the tracking system.
It is supposed to form estimations of filtered process k The paper studies the synthesis of the tracking system, which implies retrieval of information only from pilot component of L1OC signal.To carry out this synthesis we write signal L1OC ( ) , where sign ⊗ means an operation of by-bit time multiplexing;  When writing the expression (3) we formally suppose that information component does not depend on the estimated parameter k τ .

, c o s
Considering signal delay the estimated parameter with regard to (3), observations (1) can be written as: When synthesizing delay filtering quasioptimal algorithm we assume that phase and Doppler shift are independently estimated in the receiver, i.e.In this case, the signal delay dynamic model can be written as ( ) where

The Synthesis of Quasioptimal Delay Filtration Algorithm
Where the rate of taking observations is significantly higher than the rate of ˆk τ delay calculations, for the synthesis of optimal filtration algorithm it is necessary to use the theory of optimal filtration with grouping of observations (Perov, A. I., 2012).
Taking into account the periodicity of subcarrier fluctuation ( ) ( ) , the aposteriori density of delay probability density bears multimodal nature.
Let us introduce additional variable d τ , which we will connect with the digital sinusoid of the pilot component, i.e.
Herewith let , for any k .Consequently, the d τ in time is described with equation similar to (5).
Now we shall introduce the expanded vector , for which we will write down the vector equation: where: In the examined task on the time interval, 1,1 , depend on the value of measuring state vector to the moment of time 1 k t − .That is why, if we are interested in the current estimation of state , it's necessary to consider the aposteriori probability density ( ) for which we can write down the equations: ) ) Consequently, now k t with all the available for this moment observations of 0 k Y in process, we will form the estimation of state vector 1 xk− being correspondent to the state vector In accordance with the additional variable method, we firstly consider the aposteriori probability density of the expanded state vector x for one of the periods of periodical argument with use of standard Gaussian approximation in the theory of optimal filtration.The equations describing the change of mathematical expectation [ ] and the matrix of dispersions x, 1 D k − of Gaussian aposteriori probability density are called the equations of optimal (or quasioptimal -when using these or those approximations) filtration.
For the examined setting of the problem, the algorithm of quasioptimal filtration of vector 1 x k − can be presented with the following equations (Tikhionov, V. I., & Kharisov, V. N., 2004): ( ) x, 1 x, 2 ( ) where As ( ) does not depend upon estimated state vector 1 x k − , for further examination suggesting differentiation functions x k − , we will consider the simplified form: Now we shall modify the right part of the formula (11) where -extrapolated estimation of signal phase.
The present expression can be interpreted as the correlation processing of incoming counts Introduce variables: = −Ω, and write down (13) in the following form: With account of ( 14) write down (12) in the following form The presentation already contains the correlation processing at subcarrier frequencies 1 sc ω and 2 sc ω that we will use for further specialization of processing algorithms in the tracking system.Using trigonometric transformations, we will represent (15) in the following form: The formulas ( 16) describe correlators that process the input signal at subcarrier frequencies  16) to (8) it is necessary to conduct the change and represent (15) in the following form: and represent (19) in the following form: The quasioptimal filtration equations (8) sometimes are called "extended Kalman filters".The form of such a presentation is not adequately corresponded to the presentation of classic tracking system, in which there are discriminator and smoothing filter.The discriminator is understood to mean the device, the output process of which contains information about mismatch between true value of signal parameter that is tracked and its estimating value that is formed in tracking system.In general, the discriminator on λ parameter of signal is described with the following formula: ( ) In the considering task the parameters being tracked are the τ signal delay and additional variable d τ .
Taking into account the fact that functions ( ) are defined with formula (11), for which the presentation (21) was obtained, we will write down the formula for discriminator on τ delay (delay discriminator): ( ) When writing down ( 22) we have admitted the assumption about independence of ψ from 1 k τ − .
We can consider the simplified variant of τ delay discriminator, of insert ( ) ( ) 22).In such a case, the τ delay discriminator can be described by simpler formula: When dealing with navigation equipment the calculation of derivative on τ in ( 22) is interchanged with calculation of finite differences: where τ Δ -the mismatch between reference signals of correlators ( 16), i.e. -between the ranging codes ( ) of the present reference signals.
The structural scheme of τ delay discriminator is presented in Figure 2, where ( )  -are the advance and delayed components of correlators ( 16).
The structural scheme of d τ delay discriminator is shown in Figure 3. τ and its stable zeros correspond to the condition ( ) where 2 T π Ω = Ω -the fluctuation period of subcarrier frequency; ( ) The formula (25) depends on the parameter m defining the ambiguousness of estimation.To eliminate this ambiguousness in the additional variable method (Tikhonov, V. I., & Harisov, V. N., 2004) we have taken the algorithm of estimation of the present parameter in the following form: For the resulting valuation of delay ˆres τ there is formula: In the equation of optimal filtration (8) there is derivative ( ) , for which we can write down the following expression: ( ) ( ) ( ) With account of ( 27) the equation ( 8) describing the tracking system takes the following form: The structural scheme of signal delay tracking system is presented in Figure 4, where

Analysis of Discriminator Curves
One of the peculiarities of synthesized delay filtering algorithm is a presence of two time discriminators, one of which ( 22) extracts information from delay of ranging code We shall calculate mathematical expectations of correlation sums ( 16) with account of substituting.
Perform substituting -ratio of signal power to noise one-way power spectral density.
Calculate the average value of envelope's square (17) Subsequently, we can write down: The discriminator curve of ranging code (22) delay discriminator when exchanging of delay differential with finite increment in case of small level of errors 0 The formula for discriminator curve (29) aligns with similar formula for discriminator curve taken for delay discriminator designed for processing signal with modulation BPSK(1) The aperture of discriminator curve (in case of conventionally used value The slope of discriminator curve is equal to  Another important characteristic of quality of tracking systems is being the dispersion of noise of delay discriminator lead to the measuring parameter (Perov, A. I., 2012).We set it as , ef D τ .The present characteristics is univalently linked to the Rao-Cramer's low limit for dispersion when measuring the parameter of signal, i.e. it defines potential accuracy of measurement of correspond parameter (Tikhonov, V. I., & Harisov, V. N., 2004).For the examining ranging code delay discriminator write From the comparison of the present formula with similar formula for dispersion of delay discriminator noise in processing at carrier frequency it follows that for the systems with processing at subcarrier frequency the dispersion is 1.23 times bigger than for system with processing at carrier frequency.
Further, we will consider discriminator on additional variable (24).Its discriminator value can be described with formula In case of small errors 0 ϕ ε ≈ , 0 ω ε ≈ we can write down the approximate expression: ( ) The discriminator (32) on additional variable d τ is alternating with period  From the comparison of ( 33) with ( 30), it follows that if c τ τ Δ = ; the slope of discriminator curve on additional variable is 2 2π times bigger than the slope of discriminator curve on ranging code delay.
The statistical characteristics of discriminators of ranging code delay and additional variable can be used for simplification of dispersion equations (9).The essence of such simplification lies in substitution instead of derivative ( ) . Performing such averaging we will get: Inserting the present formula to the dispersion equations belonging to (9), we will right down the simplified formulas: x The present dispersion equations will be used later, when simulation modelling the synthesized tracking system.
From the analysis of dispersion equations (34), it follows that in steady-state regime In so doing the filtration equation will be

Synthesized Tracking System Modelling
To test the working capacity and characteristics of synthesized tracking system for delay of L1OC GLONASS signal pilot component with processing at subcarrier frequencies we have conducted its modelling by means of digital computer in the environment of MATLAB simulation.The modelling was carried out with use of statistical equivalents method upon the following conditions: ( ) Firstly we will examine the case when delay k τ is described with equation ( 5), the valuation of Doppler frequency shift , D k ω  is known to us, the tracking system is described with transient equation ( 28   We shall note that in the tracking system built on the principle of signals processing at carriers frequency (Perov, A. I., & Harisov, V. N., 2010) the least square value of GLONASS signal pilot component delay tracking error makes ,calc τ σ = 1.72 m for dynamic change of delay, while in case of weak dynamic of delay it makes ,calc τ σ = 0.59 m.Thus, the synthesized system with processing at subcarrier frequencies provides 1.5-1.7 times reduction of the mean square tracking error in set regime.

Conclusion
In this paper, we have synthesized the quasioptimal algorithm of L1OC GLONASS signal pilot component delay filtering with processing at subcarrier frequencies with use of the theory of optimal filtering of information processes and the method of additional variable.
We have taken the equations of optimal filtering and their representation in the form of tracking system including delay discriminator, additional variable discriminator and smoothing filter.
It is shown that synthesized discriminators with processing of pilot component of L1OC signal at subcarrier frequencies have structures that differ from the structures of known delay discriminators with processing at carrier frequency.
There are also structural schemes of synthesized discriminators.We have calculated the discriminator and fluctuation characteristics of delay and additional variable discriminators.In addition, it is shown that discriminator curves have stable equilibrium points.In the steady-state regime in the tracking system we can point out the summarized discriminator, the slope of discriminator curve of which is 2 2π bigger than in delay discriminator only for ranging code, what provides increase of tracking accuracy.
We have also listed the results of simulation modelling of synthesized tracking system.It is specified that in optimal tracking system at signal/noise ratio equal to 40 dbHz the least square value of signal delay tracking error makes ~ 0,35 m (rms) for low-dynamic consumer and ~ 1,0 degrees -for high-dynamic consumer.In the tracking system with processing at subcarrier frequencies in the steady-state regime there is 1,5-1,7 times reduction of the mean square tracking error in comparison with similar tracking error in the system with processing at carrier frequency.
s t has ВОС(1,1) modulation with a base frequency of b f =1,023 MHz and is designed for delay, phase, and Doppler frequency shift measuring.The information component ( ) d s t has BPSK(1) modulation and contains digital information (navigation message).The combination of two components into a single radio signal is carried out on the principle of by-bit time multiplexing.
step in tracking system loop; d T - 0 f -signal carrier frequency; , 1 k τ ξ − -discrete white Gaussian noise with a dispersion of D ξ .The problem of the synthesis of the quasioptimal algorithm of delay filtering k τ in clock time moments k t when processing only the signal pilot component the filtration algorithm (8) we shall use the correlators (14), in which the reference signals are taken with extrapolated valuations

Figure 2 .
Figure 2. The structural scheme of τ delay discriminator

Figure 3 .
Figure 3.The structural scheme of discriminator on d τ additional variable

1z−
means the block of delay per working cycle of tracking system, the block «τ correction» performs the formation of resulting delay estimation ˆres τ in accordance with algorithm (26).

Figure 4 .
Figure 4.The structural scheme of signal delay tracking system have used some approximations, i.e. -when digital sinusoid was changed to natural sinusoid, what is being the definite approximation.The normalized discriminator value depending on standardized specified approximations calculated by means of computer you can see of Figure5.

Figure 5 .
Figure 5. Standardized ranging code delay error

=
. In Figure 6 there is normalized discriminator curve as the function standardized error function d c τ ε τ .

Figure 6 .
Figure 6.The standardized curve on additional variable d τ

Figure
Figure 7. Standardized discriminator curve of summarized Figures 8, 9 there are realizations of τ delay tracking error and additional variable d τ tracking error taken as the result of modelling.The dashed red lines in these Figures mean the limits of confidential least square value of τ delay tracking error and additional variable d τ tracking error calculated by equations of Riccati (34).