WANG Qian,ZHANG Chuanding,and XIAN Deyong

Beijing Satellite Navigation Center,Beijing 100094,China

1.Introduction

Satellite navigation systems can provide real-time and accurate positioning,velocity measurements and timing services under all weather conditions and at any location,and nowadays play an important role in human production and life.China has built a regional passive satellite navigation system in the Asian Pacific region.With the need of the national strategy,China aims to complete global navigation satellite system(GNSS)around 2020.

For the global system,the expansion of the offered services’scope results in additional demands on the application environment of satellite navigation.Traditional users receiving terminals acquire a pseudorange value via a single receiving channel,and positioning results are calculated by fusion at this pseudorange level.Although the design structure is simple,it is evident that its precision is inadequate,its reliability is not strong and navigation signal resources are underused in complicate and diverse application environments.

In recent years,vector tracking has been proved as an important way to improve the terminals’processing capacity.The core idea of vector tracking is to make use of information relevance between different channels to integrate independent channel tracking and positioning modules into a single system via Kalman filtering and to output the positioning results directly.Meanwhile,measured values are fed back to modulate the oscillator,thus keeping channel tracking of the satellite signal stable.By analyzing the structure of vector tracking,it can be seen that it is the most thorough processing mode utilizing information fusion.

Inspired by the vector tracking approach,it is possible that the information fusion idea can be adopted at the pseu-dorange level for the signal’s front-end,allowing the receiver to fuse while receiving the signal,thus retaining all the information of the satellite signals without distortion,avoiding informationloss caused by channel isolation,and laying the foundation for diverse application processing at the back-end.The information fusion approach presented in this paper pertains to the front-end signal fusion,which does not outperform the vector tracking algorithm in terms of fusion strength,but has its own uniqueness and application value.

Due to the large fusion strength in vector tracking,the number of state estimation parameters inevitably increases,thus adding more calculation complexity to the Kalman filter.At the same time,the positioning module is integrated into the processing loop,resulting in a single positioning model,which provides general pseudorange positioning.Thus,diverse positioning models such as differential,pseudorange and carrier fusion positioning cannot be realized according to the application requirements.Therefore,today’s baseband signal processing chips of navigation receiver terminals are usually limited to the tracking level of signal processing,rarely extending to the positioning level with the purpose of providing diverse location service functions.

Concerning the research on fusion at the signal and information level,detailed analyses and studies have been carried out by many researchers.In[1–5],an estimation method based on the maximum likelihood estimation of the position was proposed,showing excellent results in reducing the signal’s multi-path and interference effects,and pointing out that the synchronization parameters can be restored at the user location estimation domain.In[6],a design of a joint vector tracking loop using the baseband signal linearization model was proposed,which reduces the interference by synthetically leveraging the principle of interference cancellation and subspace projection,jointly identifying all synchronization parameters of the satellite signal from the baseband signal and demonstrating a strong dynamic tracking ability in ring circuits through experiments.In[7],the tracking ability of the receiving terminal in weak signal and highly dynamic environments via the maximum likelihood estimation was verified through analyzing the estimation precision of important parameters such as time delay,phase,and frequency during the tracking process.

On the basis of the above-mentioned research, avoiding the iterative method of maximum likelihood estimation described by the previous research,this paper sets practical combined baseband processing application model design as the goal,using the least squares as the fusion tool and carrying out global optimization within the scope of several parameters such as code phase,carrier phase and signal amplitude.The estimation error and global optimization effects of this method are investigated through theoretical and experimental analysis.This paper is arranged as follows:in Section 2,starting from the basic navigation signal formula,we introduce the multi-channel joint physical structure and obtain the analytical values of parameters via the least squares method;in Section 3,the measurement error is characterized by a weight matrix and the correctness of this matrix characterization is verified through theory and experiments.In Section 4,the optimization per-formance of multi-channel estimation and the advantages of channel fusion are verified through experiments.Section 5 summarizes the full paper.

2.Signal model

2.1 Multi-channel joint physical structure

The structure of current satellite navigation signals can be divided into three levels:carrier,pseudocode and data message.Among them,carrier and pseudocode parameters are determined in the signal domain while the message is the prerequisite for obtaining the position.In satellite navigation systems,the down link signal is then converted to the base band after frequency conversion by the radio module and is demodulated by the PN code and a carrier.The base band signal can be expressed by

where t is the receiving time;N is the number of the re-ceived satellite signals;ai,τi,?iand fd,iare the amplitude of the signal,the code phase delay,the carrier phase and the carrier Doppler respectively;Si(t-τi)is the modulation code spread of the corresponding signal,and n(t)is the white Gaussian noise.The process of receiver tracking is to obtain the above-mentioned signal parameters from the noisy signal.The scalar loop obtains the parameters of a single branch through an independent and small delay lock loop(DLL)/frequency lock loop(FLL)feedback loop,while the vector loop[8]obtains the parameters of all the branches together with the position via a large vector delay lock loop(VDLL)/vector frequency lock loop(VFLL)feedback loop.In this paper,the multi-parameter joint tracking does not involve location information processing,and none of the channels does not have its corresponding independent discriminator.It obtains various required parameters for tracking in one shot iteration,based on the fusion of multiple DLLs,phase lock loops(PLLs)and FLLs via a least squares solution and filtering.

The vector loop and the proposed multi-parameter joint tracking loop structure are compared as shown in Fig.1.Using the coherent integration time as the processing cycle,the base band signal in the signal channel is Taylor-linearized at first.Then the parameters of each channel are jointly calculated using the least squares method with achieving the most realistic received signal as the optimization objective.The obtained residuals are then fed back to the receiving channel through the Kalman filter.Here,one point to be given special emphasis is that the essence of joint optimization is the information fusion between the channels,the weak signal channel is aided by channels of strong signals,and thus the overall signal tracking capability of the receiver is improved.At the same time,the pseudorange and the carrier observation data obtained by filtering can be used by the receiver to complete the subsequent positioning calculation.

Fig.1 Loop structure comparison

The state vector and measurement vector of the Kalman filter system are

The state transition equation is

where wnis the vector of process noise,IM×Mis the identity matrix,f is the signal frequency.The equation is

where Qkis the Kalman filter gain matrix.

As shown below,there are significant differences between those two structures:the signal discriminator of the vector loop in each channel works independently,but the discriminator designed in this paper is based on the joint optimization of signals of all channels.The vector loop has no dedicated loop filter,tracking and positioning are carried out in one iteration by Kalman filtering,while the updating vector for signal tracking is inversely deduced from the displacement and velocity vectors.In this paper,the signal tracking and positioning solution is divided into two stages, and the synchronization parameters after multichannel joint identification and filtering are fed back to the tracking channel directly.

2.2 Mathematical determination of multi-channel parameters

The mathematical principle of channel fusion is the Newton iterative method.The first step of this method is to Taylor-linearize the non-linear equations near a certain root value and then solve the equations consisting of the multi-channel parameters,and finally to update the estimated value of the root.When the number of effective equations is more than the number of unknowns,the least squares method is adopted to find the unknowns.The final solution minimizes the sum of squares of differences between the estimated signal and the actual signal.

Considering that(1)is anon-linear function,the Newton iterative method cannot be directly used.Ignoring higher order terms,the first order of Taylor expansion can be employed to linearize the non-linear signal.Analyzing(1),it mainly includes four parameters:the carrier Doppler,carrier phase,code phase,and signal amplitude.If the linear Taylor expansions are simultaneously used as estimates of the above-mentioned parameters,the subsequent calculations will be too complex.The carrier Doppler can be solved accurately from the other three parameters.To simplify the algorithm’s complexity,carrier Doppler can be excluded from the discriminator and the Taylor expansions are used for the estimation of the other three parameters.The matrix vector of multi-channels is expressed as

where X is the vector of the received base band signal;is the local replica of the received base band signal vector;N is the noise vector and A,Γ,Φ are the partial derivative matrices of X for the signal amplitude vector a,the code phase delay vector τ and the carrier phase vector ?,respectively.Each element in the latter can be expressed asaccording to the definition of derivative,where i stands for the channel number.

The unknown parameters of(5)are δa,δτ,δ?,which can be uniformly expressed in the form of matrix as θdiscr.In order to facilitate the subsequent derivation and calculation,the simplified matrix representation for(5)is

where G is the derivative matrix consisting of A,Γ,and Φ.To minimize the noise vector N,the least squares for-mula can be directly applied to(6)to solve θdiscr,as shown in

3.Analysis of weight matrix characteristics

3.1 definition and simplified calculation of weight matrix

In Section 2,the measurement error is not considered in the process of finding the residual error of parameters.The measurement error term can be introduced to(6)for numerical analysis.The expression with measurement error added is

In the same way,(9)is obtained by solving(8)via the least squares method

where εpis the measure error vector.

From(9),the weight coefficient matrix is defined as

Matrix H reflects the enlargement degree of measurement errors during parameter estimation,which is the coefficient matrix of measurement noise in the Kalman filtering.Typically,if the number of receiver channels equals N,the size of matrix H is N×3N.The diagonal elements in H reveal the auto-correlation condition of the satellite signal,while non-diagonal elements reveal the cross-correlation condition of the satellite signal.In[9–11],it was showed that the cross-correlation term has a great influence only under the condition that a relatively strong and a weak signal co-exist(above 20 dB)and the signal frequency difference relates to the multiple of dynamic Doppler.In other words,even if strong signals and weak signals exist simultaneously,cross-correlation effects do not always appear.

Next,through the study of the numerical value of the elements in matrix H,the impact of the cross-correlation terms is analyzed.In the experiment,there are 11 receiving channels in the receiver in total,wherein the signal intensity from the 1st to the 7th channel equals 20 dB-Hz,and the signal intensity of the rest of the channels is 40 dBHz.The dynamic acceleration is 5 m/s2and the coherent integration time is 1 ms.Fig.2 reflects the proportion of the diagonal elements’quadratic sum among the entire matrix elements.From Fig.2,we can see that the proportion of non-diagonal elements in the matrix is very small,generally below 10-3.Thus,for facilitating the calculations,matrix H can be diagonalized to diag(H).

Fig.2 Proportion of the diagonal elements

3.2 Characteristics of the weight matrix error

In Section 3.1,the weight matrix definition is derived by means of a mathematical equation.This section deals with how the weight matrix elements reflect the discrimination error through experimental quantitative analysis.The theoretical equation in[12]indicated that,when the discriminator uses a non-coherent power method,the identification error variances of code phase,carrier phase and signal amplitude are expressed as follows:

where d is the correlator spacing,T is the coherent integration time and Ci/N0is the carrier to noise ratio of the i th channel.When the discriminator uses coherent integration,there is no square loss and thus the second term in parentheses in(11)–(13)can be removed.Therefore,an increase of the integration time T yields a more obvious variance reduction.

The experimental conditions are set as follows:eight receiving channels are set in total,wherein the signal intensity from the 1st to the 3rd channel equals 24.5 dB-Hz,and the signal intensity of the rest of the channels is 45 dB-Hz.The curve diagram of the error ratio between the strong and the weak channel is shown in Fig.3.The dynamic acceleration in scene 1 and scene 2 equals 20 m/s2;the dynamic acceleration in scene 3 and scene 4 is 0.2g m/s2.There is no dynamic acceleration in scene 5.The covariance matrices of the amplitude channel,the code phase channel and the carrier frequency channel after Taylor expansion are selected as output elements.The Y axis indicates the ratio of the different covariance elements,i.e.,the code phase error in scene 1 and scene 3,the carrier frequency error in scene 2 and scene 4,and the theoretical calculation in scene 5,between the strong and the weak channels.The calculation equation is the difference between strong and weak signal intensity.

From Fig.3,we can see that the weaker the signal received by the channel,the larger the ratio is,i.e.,the value reflected on the matrix elements is larger.The dynamic stress of the signal is also an important factor on affecting the covariance matrix.When the signal is under low dynamic stress,the experimental results of scene 3 and scene4are close to the theoretical values(112.2);when the signal is under high dynamic stress,the higher stress value causes the experimental results in scene 1 and scene 2 to differ from theoretical value greatly.This characteristic of matrix elements is consistent with the concept of the precision factor derived from the position domain fusion of the general receiver.The validity of error representation of the weight coefficient matrix H is verified by this experiment.Therefore,matrix H can be used to reflect the enlargement degree of the measurement error.This experiment also proves that no matter how strong the signal strength and the dynamic stress are,the amplitude elements in the weight matrix keep constant,which is in line with the theory of(13).

Fig.3 Curve diagram of the error ratio between the strong and weak channels

4.Analysis of the estimation effect

In this paper,multi-channel joint estimation belongs to a signal domain method.The purpose is to overcome the influences of weak signals and the dynamic stress in application environments.In global scope,through the fusion of different channels,the estimated parameters of optimal carrier to noise ratio can be obtained.On this basis,the effect of the key parameter of the phase coherent integration time on performance is analyzed.Finally,the experiment for the position error is conducted by using the weight coefficients.

4.1 Verification experiment based on optimal carrier to noise ratio

Section 2 shows how we obtain the synchronization parameters’residual of all channels in one iteration via the least squares method.It has already been proved in theory that the method can minimize the noise of each channel in global scope,which means that the optimal carrier to noise ratio is obtained for facilitating the subsequent correct message decoding.The following experiment verifies the optimization effect of the least squares method.

The experimental conditions are set as follows: the number of the receiving channels is 8;the coherent integration time is 1ms,the signal dynamic acceleration is 20 m/s2.Fig.4 shows the relationship between the code phase deviation and the output carrier to noise ratio.The output parameters from the least squares method are set as the coordinate origin.As shown from Fig.4,it does not matter whether the carrier noise ratio(CNR)of a channel is high or low,as long as the output parameters deviate from the original setting,the CNR tends to decrease,and the larger the deviation is,the larger the decline is.Because the tracking thresholds of different channels are the same,high CNR channels can certainly tolerate more estimation error than low CNR channels.The amplitude curve of the deviation is symmetric around the coordinate origin.Experiments indicate that such symmetry is independent of the user’s motion state,and the generated dynamic stress during the process of carrier motion can improve the channel tracking threshold.

Fig.4 Relationship between the code phase deviation and the output CNR

4.2 Analysis of the influence of the coherent integration time

Coherent integration time[13]is a key parameter that affects the tracking performance of the loop.The CNR can be improved by extending the coherent integration time,which in turn can improve the stability of the signal tracking loop and reduce the measurement error in the signal tracking process.The extension of the coherent integration time is limited by the bit width of the navigation message and the error of the tracking frequency.Both the dynamic nature of the user equipment and the drift of the crystal oscillator have influence on the performance of coherent integration.The user should select integration time as a trade off between the thermal noise and dynamic conditions.Under the condition of the message data being known,coherent integration time can be extended properly and surpass the bit limitation, which usually is of an order of magnitude of 100 ms at most.

The experimental conditions are set as follows:eight receiving channels are set in total,wherein the signal intensity of all channels equales 25 dB-Hz.The dynamic acce-leration equals2 m/s2.The power spectrum density parameters of the process noise covariance matrix including the crystal oscillator setting on clock drift and clock bias are set by[14].The measurement errors of code phase and carrier Doppler under conditions of different coherent integration time are shown in Table 1,where 10-4chip is equal to 0.147 m.In the loop designed in this paper,increasing of the integration time means the decreasing of the least squares discriminator throughput,and the output parameter of the channel is the mean value of the integration time.As shown from the data in Table 1,extending the coherent integration time can significantly improve the measurement accuracy of the code phase,but the effect on improving the carrier Doppler measurement accuracy is not obvious,and a relatively long integration time(100 ms),on the contrary,increases the error of Doppler measurement.

Table 1 Measurement errors of different coherent integration time

4.3 Analysis of multi-channel fusion influence

In this paper,multi-channel fusion mainly lies in the following two aspects:one is to perform loop-wise mutual aid,which is able to offset the dynamic stress caused by carrier motion,using the characteristics of the three kinds of loops:FLL,PLL,and DLL.Since the adaptability of the FLL to dynamic stress is stronger than that of PLL,FLL can counteract most dynamic stress and carry out PLL traction.The measurement error of the code phase output of the DLL is much larger than that of the carrier phase output of the PLL,and the difference between them reaches 2–3 orders of magnitude.The measured value of Doppler frequency shift output of the PLL can accurately reflect the relative motion speed of the receiver in the radial direction of the satellite signal.Therefore,the impact of high dynamic stress on the DLL can be eliminated with the aid of the PLL.The DLL only needs to adapt to the initial tracking error and the change of the code phase induced by the ionosphere’s time-delay variation.It generally uses a narrow code loop bandwidth for reducing the noise of the code ring and obtaining a higher measurement accuracy.

The auxiliary relationships between different kinds of loops are proved through experiments.The signal intensities of the two scenarios are identical,both being 45dB-Hz;the dynamic values are very different,i.e.,equal to 40 m/s2and 2 m/s2,respectively,with the purpose of examining the response of each loop under different dynamics.Results are shown in Table 2.The measure error of the FLL and the DLL introduced by the scalar method is much larger than that of the multi-channel parameter joint estimation(MPJE)method.The information between the FLL and the DLL is not fused due to the scalar loop.Between the two scenes by MPJE,the ratio of code phase error is 1.71,and the ratio of the carrier Doppler error is 5.98.The FLL is obviously more affected by the dynamics than the DLL.The appearance of such a large deviation is mainly because during the process of multiple loops’joint aiding,the dynamic stress elimination is mostly executed via the FLL.Thus,the carrier Doppler deviation tends to belarger.After the FLL assists the PLL and the PLL assists the DLL,the direct effect on the DLL’s dynamic stress deviation is very small.Thus,the code phase deviation is rel-atively smaller than that of the FLL.It is via this fusion that multiple channels of different loops show their respective advantages.

Table 2 Measure error of different loop under different dynamics

The purpose of the second experiment is to show that the tracking ability of weak signal channels can be improved by using the fusion advantage among channels that strong signals can assist weak signals via the least squares method-based global optimization.Scene settings under conditions of different signal intensities are as follows:scene 1:the level of the received signals from the 1st to the 3rd channel is 25 dB-Hz,and the level of received signals from the 4th to the 8th channel is 45 dB-Hz;scene 2:the level of the received signals from the 1st to the 8th channel is 45 dB-Hz;scene 3:the level of the received signals from the 1st to the 8th channel is 30 dB-Hz;the dynamic values of all three scenes are 20 m/s2.

Experimental results are shown in Table 3.

Table 3 Measure error of code phase under different signal intensities 10-4chip

From Table 3,we can see that the estimation error of the code phase is inversely proportional to the received signal intensity in the channel:the higher the intensity is,the smaller the estimation error is.Furthermore,the estimation error in scene 2 is obviously superior to that in scene 3;although there exist obvious signal intensity differences between the receiving channels in scene 1, the exhibited es-timation error,after integrating global optimization where the weak signal channel is assisted by the strong signal channel,is reduced and approximates that of the strong signal channel.Such an experimental result fully reflects the advantages of channel fusion.For example,the error of Ch8 in scene 1 is only two-thirds of that in scene 3 through information fusion.

4.4 Relation between the weight coefficient and the position error

In Section 3,the weight coefficient matrix of the signal domain is defined.This kind of definition method is in line with the definition of the weight coefficient matrix in the position domain.However,their physical meanings are not the same.Different from the weight coefficient matrix of the signal domain,that of the position domain reflects the geometrical distribution of the visible satellites with respect to users,which has nothing to do with signal intensity.When the element values in the matrix are smaller,the degree of measurement error magnified to positioning error is lower.Position dilution of precision(PDOP)is used to reflect the element of matrix.Fig.5 illustrates the curve of the PDOP factor variation in the position domain under conditions of different dynamic accelerations no matter what is the signal intensity.The sampling frequency of data is 10Hz.The more dramatically the dynamics change,the steeper the corresponding PDOP curve is.

Fig.5 Curve of the PDOP factor under different dynamics

Nevertheless,the weight coefficients of the signal domain are closely related to the signal intensity.The larger the elements in the matrix,the weaker the received signal is.The amplitude and phase parameters can be obtained from signals via the application of the least squares method,while the position information is obtained through applying the least squares method again. The position error is collectively determined by the measurement error and the constellation geometry.(Note that the position error does not include the system errors such as ephemeris error,ionosphere error,and troposphere error.)

The position errors under different signal intensity con-ditions are shown in Fig.6 by MPJE and the scalar method.The sampling frequency of data is 10 Hz.The position error originated from the scalar method is about 16 times as large as that of MPJE especially in the condition of steady state and weak signals through the initial intense vibration.The MPJE makes superior performance in reducing measure error through information fusion compared with the common scalar method.The position error of the strong signal is obviously lower than that of the weak signal when the PDOP factors of two scenes are the same.

Fig.6 Position errors under different signal intensities

Besides,the convergence speed of the weak signal is significantly slower than that of the strong signal in the condition of a same bandwidth.The speed of convergence depends on the bandwidth parameter setting of the filter.The bandwidth parameter in this experiment is determined by the covariance matrix elements of the process noise in Kalman filtering.The setting is the same with that described in Section 4.2 of this paper.

5.Conclusions

Inspired by the idea of a vector tracking algorithm where the signal domain and the position domain are fused,this paper proposes a multi-channel fused parameter estimation method via the least squares algorithm in the signal domain.The method is able to obtain signal amplitude,spreading code and carrier parameters in one iteration and achieves the optimal estimation of CNR across all channels.As analogues to the concept of precision factor and position error in the position domain,the concepts of the weight matrix and estimated error in the signal domain are proposed.Moreover,through experiments,the effectiveness of this parameter estimation method is verified in aspects of signal estimating,channel fusing and position resolving compared with the scalar mode.The research findings of this paperlay a foundation for the design of a practical high-performance base band signal processing prototype in the future.

Thanks for much help from professor Xiaowei Cui and Dr.Jing Liu in Department of Electronic Engineering of Tsinghua University during the process of research.

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