Simplified Kalman filtering carrier tracking method compatible with digital phase-locked loop

Abstract

The invention discloses a simplified Ka lman filtering carrier tracking method compatible with a digital phase-locked loop, and the method comprises the steps: carrying out the approximation of the innovation of Ka lman filtering, building the corresponding relation between the Ka lman filtering gain and the coefficient of a digital phase-locked loop filter, and changing the coefficient of the filter to simulate the gain convergence process of the Ka lman filtering. And the digital phase-locked loop achieves the performance similar to that of Ka lman filtering carrier tracking. According to the method, the performance similar to that of a Ka lman filtering tracking method can be achieved on the basis that an existing digital phase-locked loop implementation framework is not changed.

Description

Technical Field

[0001] The present invention relates to the field of satellite navigation technology, and more particularly to a simplified Kalman filtering carrier tracking method compatible with a digital phase-locked loop. Background Art

[0002] Stable tracking of the carrier phase of the received signal is the basis for GNSS (Global Navigation Satellite System) receivers to perform positioning calculations. Satellite navigation signal carrier tracking mainly includes two methods: digital phase-locked loop and Kalman filtering.

[0003] The digital phase-locked loop is based on the analog phase-locked loop theory and consists of a digital numerically controlled oscillator, a phase detector (including correlation accumulation and carrier phase) and a loop filter. Taking the second-order digital phase-locked loop as an example, its mathematical model is as follows: Figure 1 shown.

[0004] Figure 1 Where a1 and a2 are the coefficients of the loop filter, t c is the loop update period. Kalman filtering is the optimal method for signal parameter estimation. When performing carrier tracking, the carrier tracking method based on Kalman filtering first divides the continuous signal into segments and then updates the signal at a certain period t. c When only the first-order dynamics is considered, the signal r of each segment is k (t) can be expressed as follows:

[0005]

[0006] Where θ 0,k 、ω k Respectively represent [kt c ,(k+1)t c ]Carrier initial phase and Doppler frequency within the time period.

[0007] It is easy to know that the carrier initial phase and Doppler frequency in adjacent time periods satisfy the following rules:

[0008]

[0009] Where w ω,k Represents Doppler frequency jitter, which has a mean of 0 and a variance of Gaussian distribution.

[0010] The Kalman filter carrier tracking method does not directly estimate the carrier initial phase of the received signal, but estimates the carrier initial phase error between the received signal and the local signal. When the signal is generated locally, only the Doppler frequency is adjusted, and the first-order dynamic form is also used. Then, according to formula (2), the carrier initial phase error ε between the received signal and the local signal at different time periods can be obtained: k The changing rules are as follows:

[0011]

[0012] Where, and Respectively represent [kt c ,(k+1)t c ] period of time, the carrier initial phase and Doppler frequency of the local signal.

[0013] The carrier initial phase error and Doppler frequency are used as the state vector x of the Kalman filter. k =[ε k ω k ] T , the following state transfer equation can be established:

[0014] x k+1 =Φx k -u k +w k (4)

[0015] Where, is the external input of Kalman filter, Φ represents the state transfer matrix, w k =[0w ω,k ] T Represents the process noise with covariance matrix Q. The specific expressions of Φ and Q are:

[0016]

[0017] Local Signals k (t) and the received signal r k (t) related value y k It can be used to estimate the phase difference between the two, and its expression is:

[0018]

[0019] Where, Indicates the Doppler frequency difference between the received signal and the local signal, n y,k Represents the noise component in the correlation value.

[0020] Without considering the π ambiguity, the phase error φ k There is the following relationship between and the state vector:

[0021]

[0022] In the formula, Re(y) and Im(y) represent the real part and imaginary part of the complex number y, respectively, and v φ,k represents the observation noise,

[0023] The mean is 0 and the variance is Gaussian distribution.

[0024] According to formula (7), the following observation equation can be established:

[0025] z k = Ηx k +v φ,k (8)

[0026] Where, Denotes the observation vector, H=[1t c / 2] represents the observation matrix.

[0027] Based on formulas (4) and (8), the update expression of the Kalman filter tracking method can be obtained:

[0028]

[0029] Where, and are the state vectors x k+1 and x k The optimal estimate of Δ k+1 represents [(k+1)t c ,(k+2)t c ] new information during the period, G k =[g ε,k g ω,k ] T Represents the gain matrix of Kalman filter. Assume that the covariance matrix of Kalman filter state vector is P k , then the gain matrix G k The iterative process is as follows:

[0030]

[0031]

[0032] Expanding the matrix form shown in formula (9), we can obtain:

[0033]

[0034] The Kalman filter carrier tracking method does not limit the local signal Doppler frequency Here we assume that In order to make ε in formula (3) k+1 The solution is equal to 0, that is:

[0035]

[0036] Formulas (12) and (13) are the mathematical models of the Kalman filter carrier tracking method. Its implementation block diagram (the red dotted arrow in the figure reflects the loop data flow) is as follows: Figure 2 shown.

[0037] contrast Figure 1 and Figure 2 It can be seen that the Kalman filter carrier tracking method is not compatible with the implementation of the digital phase-locked loop. In order to apply the Kalman filter carrier tracking method on the existing digital phase-locked loop framework, it is necessary to propose a simplified Kalman filter carrier tracking method that is compatible with the digital phase-locked loop. Summary of the Invention

[0038] The object of the present invention is to provide a simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop, so as to overcome the defects of the prior art.

[0039] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0040] A simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop (DPLL) is disclosed. The method approximates Kalman filter innovations, establishes a corresponding relationship between Kalman filter gain and digital phase-locked loop filter coefficients, and simulates the gain convergence process of the Kalman filter by changing the filter coefficients, thereby enabling the digital phase-locked loop to achieve performance similar to that of Kalman filter carrier tracking.

[0041] Furthermore, the method specifically comprises the following steps:

[0042] S1. Set the process noise variance of Kalman filter carrier tracking and the observation noise variance

[0043] S2. Set parameter t according to the carrier tracking update period. c According to the uncertainty of the carrier initial phase and Doppler frequency in the initial tracking stage, the covariance matrix is ​​set to P k The initial value of k is 1;

[0044] S3. Iteratively calculate the gain matrix G when k = 1 to 1000 according to the following formula k =[g ε,k g ω,k ] T :

[0045]

[0046] Where, P k and P k-1 They are [kt c ,(k+1)t c ] period and [(k-1)t c ,kt c ] period Kalman filter state vector covariance matrix, H = [1 t c / 2] represents the observation matrix, Φ is the state transfer matrix, the matrix superscript -1 represents the matrix inversion, the superscript T represents the matrix transposition, and Q is the process noise matrix. Their expressions are:

[0047]

[0048] S4, according to the gain matrix G k =[g ε,k g ω,k ] T , calculate the loop filter coefficient g' ω,k and g' ε,k :

[0049] g′ ω,k =g ω,k / t c

[0050] g′ εk =g εk / t c

[0051] S5, calculate the loop filter coefficient g' ω,k 、g' ε,k Stored as an array in the receiver's memory, where k = 1 to 1000;

[0052] S6. Before loop tracking, read the loop filter coefficient g' from the memory ω,k 、g' ε,k ;

[0053] S7, use g' in the kth update cycle ω,k 、g' ε,k as loop filter coefficients, and use g' ω,1000 、g' ε,1000 As the final coefficient of the digital phase-locked loop.

[0054] Furthermore, the method is based on a mathematical model of a Kalman filter carrier tracking method, wherein:

[0055] According to the formula Get the new information Δ k+1 and the phase detection result φ k+1 The following relationship exists:

[0056]

[0057] When the signal Doppler frequency changes slowly and the loop tracks stably, there is Δ k+1 ≈φ k+1 , the phase detection result is approximately treated as the new information for filtering, and the Doppler frequency of the local signal is approximately:

[0058]

[0059] The mathematical model of the Kalman filter carrier tracking method is simplified according to the data of the Doppler frequency approximation of the local signal.

[0060] Compared with the prior art, the present invention has the advantage that the present invention can achieve performance similar to that of the Kalman filter tracking method without changing the existing digital phase-locked loop implementation framework. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0062] Figure 1 It is the mathematical model diagram of the digital phase-locked loop.

[0063] Figure 2 It is a mathematical model diagram of the Kalman filter carrier tracking method.

[0064] Figure 3 It is a mathematical model diagram of the simplified Kalman filter tracking method of the present invention. DETAILED DESCRIPTION

[0065] The preferred embodiments of the present invention are described in detail below with reference to the accompanying drawings so that the advantages and features of the present invention can be more easily understood by those skilled in the art, thereby making a clearer and more precise definition of the protection scope of the present invention.

[0066] This embodiment discloses a simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop. The method approximates the Kalman filter innovations, establishes a correspondence between the Kalman filter gain and the digital phase-locked loop filter coefficients, and changes the filter coefficients to simulate the gain convergence process of the Kalman filter, so that the digital phase-locked loop achieves performance similar to that of Kalman filter carrier tracking.

[0067] According to formula (12), we can get the new information Δ k+1 and the phase detection result φ k+1 The following relationship exists:

[0068]

[0069] When the signal Doppler frequency changes slowly, Under the condition of stable tracking of the loop, it is approximately considered that Then according to formula (13) Substituting the above approximation into formula (14), we get Δ k+1 ≈φ k+1 , the phase detection result is approximately treated as the new information for filtering, and the Doppler frequency of the local signal is approximately:

[0070]

[0071] According to formula (15), we can Figure 2 The mathematical model of the Kalman filter carrier tracking method shown is simplified to Figure 3 form.

[0072] Among them, g' ω,k =g ω,k / t c , g' ε,k =g ε,k / t c Obviously, Figure 3 The simplified Kalman filter tracking method shown is similar to Figure 1 The digital phase-locked loop structure shown is consistent, and a2=g' ω,k , a1=g' ε,k This makes it possible to implement Kalman filter carrier tracking based on the existing digital phase-locked loop framework.

[0073] According to the above principles, the simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop in this embodiment specifically includes the following steps:

[0074] Step S1: Set the process noise variance of Kalman filter carrier tracking and the observation noise variance

[0075] Step S2: Set the parameter t according to the carrier tracking update period. c According to the uncertainty of the carrier initial phase and Doppler frequency in the initial tracking stage, the covariance matrix is ​​set to P k The initial value of is k=1.

[0076] Step S3, iteratively calculate the gain matrix G when k=1~1000 according to the following formula k , and calculate g' ω,k and g' ε,k :

[0077]

[0078] Where, P k and P k-1 They are [kt c ,(k+1)t c ] period and [(k-1)t c ,kt c ] period Kalman filter state vector covariance matrix, H = [1 t c / 2] represents the observation matrix, Φ is the state transfer matrix, the matrix superscript -1 represents the matrix inversion, the superscript T represents the matrix transposition, and Q is the process noise matrix. Their expressions are:

[0079]

[0080] Step S4: According to the gain matrix G k =[g ε,k g ω,k ] T , calculate the loop filter coefficient g' ω,k and g' ε,k :

[0081] g′ ω,k =g ω,k / t c

[0082] g′ εk =g εk / t c

[0083] Step S5: The calculated loop filter coefficient g' ω,k 、g' ε,k Stored in the receiver's memory as an array, where k = 1 to 1000.

[0084] Step S6: Before loop tracking, read the loop filter coefficient g' from the memory ω,k 、g' ε,k.

[0085] Step S7: Use g' in the kth update cycle ω,k 、g' ε,k as loop filter coefficients, and use g' ω,1000 、g' ε,1000 As the final coefficient of the digital phase-locked loop.

[0086] By implementing the present invention, it is possible to achieve performance similar to that of the Kalman filter tracking method without changing the existing digital phase-locked loop implementation framework.

[0087] Although the embodiments of the present invention are described in conjunction with the accompanying drawings, the patent owner may make various changes or modifications within the scope of the appended claims. As long as they do not exceed the scope of protection described in the claims of the present invention, they should be within the scope of protection of the present invention.

Claims

1. A simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop, characterized in that: The method approximates the innovation of the Kalman filter, establishes a corresponding relationship between the Kalman filter gain and the filter coefficient of the digital phase-locked loop, and changes the filter coefficient to simulate the gain convergence process of the Kalman filter, so that the digital phase-locked loop achieves performance similar to that of the Kalman filter carrier tracking.

2. The simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop according to claim 1, wherein: The method specifically comprises the following steps: S1. Set the process noise variance of Kalman filter carrier tracking and the observation noise variance S2. Set parameter t according to the carrier tracking update period. c According to the uncertainty of the carrier initial phase and Doppler frequency in the initial tracking stage, the covariance matrix is ​​set to P k The initial value of k is 1; S3. Iteratively calculate the gain matrix G when k = 1 to 1000 according to the following formula k =[g ε,k g ω,k ] T : Where, P k and P k-1 They are [kt c ,(k+1)t c ] period and [(k-1)t c ,kt c ] period Kalman filter state vector covariance matrix, H = [1 t c / 2] represents the observation matrix, Φ is the state transfer matrix, the matrix superscript -1 represents the matrix inversion, the superscript T represents the matrix transposition, and Q is the process noise matrix. Their expressions are: S4, according to the gain matrix G k =[g ε,k g ω,k ] T , calculate the loop filter coefficient g' ω,k and g' ε,k : g′ ω,k =g ω,k / t c g′ ε,k =g ε,k / t c S5, calculate the loop filter coefficient g' ω,k 、g' ε,k Stored as an array in the receiver's memory, where k = 1 to 1000; S6. Before loop tracking, read the loop filter coefficient g' from the memory ω,k 、g' ε,k ; S7, use g' in the kth update cycle ω,k 、g' ε,k as loop filter coefficients, and use g' ω,1000 、g' ε,1000 As the final coefficient of the digital phase-locked loop.

3. The simplified Kalman filter carrier tracking method compatible with a digital phase-locked loop according to claim 1, wherein: The method is based on the mathematical model of the Kalman filter carrier tracking method, in which: According to the formula Get the new information Δ k+1 and the phase detection result φ k+1 The following relationship exists: When the signal Doppler frequency changes slowly and the loop tracks stably, there is Δ k+1 ≈φ k+1 , the phase detection result is approximately treated as the new information for filtering, and the Doppler frequency of the local signal is approximately: The mathematical model of the Kalman filter carrier tracking method is simplified according to the data of the Doppler frequency approximation of the local signal.

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