High dynamic bidirectional time comparison device and control method thereof

Abstract

This invention discloses a high-dynamic bidirectional time comparison device and its control method. The device includes an automatic gain control amplifier circuit, a quadrature I / Q demodulator, a mixer, a variable gain amplifier, an analog-to-digital converter, and a processor. The automatic gain control amplifier circuit is connected to the quadrature I / Q demodulator, the quadrature I / Q demodulator is connected to the mixer, the mixer is connected to the variable gain amplifier, the variable gain amplifier is connected to the analog-to-digital converter, and the analog-to-digital converter is connected to the processor. The high-dynamic bidirectional time comparison device and its control method of this invention employ a receiving loop with high dynamic tracking capability to track the received signal, thereby reducing the impact on time synchronization accuracy in high-dynamic scenarios.

Description

Technical Field

[0001] This invention relates to the fields of remote time synchronization, satellite navigation, signal processing, and embedded system development, and particularly to a high-dynamic bidirectional time comparison device and its control method. Background Technology

[0002] A two-way time comparison system is a device designed based on a two-way time comparison signal modulator / demodulator for remotely measuring the clock difference between atomic clocks at two different locations. Its block diagram is shown below. Figure 1 As shown.

[0003] The time comparison modem consists of a main control computer, a signal modulation module, a signal demodulation module, and a time interval counter module. Taking station A as an example, during remote comparison system operation, modem A receives a 1PPS second pulse signal from atomic clock A, and the signal modulation module generates a 70MHz time comparison baseband signal containing relevant data. This baseband signal is then converted into a high-frequency radio frequency (RF) transmission signal by an up-conversion module. This signal is transmitted by an antenna, relayed by a communication satellite, and received by the antenna at station B. The RF received signal at station B is then down-converted to a 70MHz time comparison baseband signal. This baseband signal is demodulated by the modem's signal demodulation module, and the time interval counter module measures the time. The main control computer then performs relevant calculations and data comparisons to ultimately determine the time difference between stations A and B.

[0004] The calculation method for the time difference Δt between two stations is as follows:

[0005]

[0006] The parameters in this formula are as follows: Figure 1 As shown, TA represents the time difference between the 1PPS second pulse signal measured by the atomic clock at station A and the received 1PPS second pulse signal at station B, and TB represents the time difference between the 1PPS second pulse signal measured by the atomic clock at station B and the received 1PPS second pulse signal at station A. These differences can be obtained through information exchange and related calculations between stations A and B. and These are the delay times of the transmitting and receiving modules at stations A and B, respectively, which can be measured in advance and are fixed values. These represent the propagation delay caused by the signal transmitted from antenna A to the satellite, and the delay caused by the signal transmitted from the satellite to antenna A, respectively. and These represent the propagation delay caused by the signal transmitted from the B-site antenna to the satellite and the propagation delay caused by the signal transmitted from the satellite to the B-site antenna, respectively. Normally, the uplink and downlink propagation delays between the antenna and the satellite at the same station are the same. In the above formula The value is zero, so it can be ignored.

[0007] To meet the needs of data exchange and comparison between stations in a remote time comparison system, as well as the measurement of pseudorange and other related parameters, a system was designed as follows: Figure 2 The time comparison signal is shown.

[0008] like Figure 2 The time comparison signal shown is generated by modulating three layers of signals using BPSK, with a baseband signal frequency of 70MHz. The first layer is a data frame signal, primarily used for data exchange and information transmission between stations. The second layer is a spreading code signal, mainly generated by pseudo-random code generators with different initial phases. This is used for inter-station comparison in code division multiple access (CDMA) mode, and also for measuring pseudorange between stations. The final layer is a carrier signal, primarily used to modulate and generate a 70MHz baseband signal for convenient signal transmission.

[0009] In a two-way time synchronization system, when receiving signals, the demodulation module of the modem needs to capture and continuously track the received signals to strip away the carrier and spreading code, obtain frame data information, and measure the carrier phase and pseudorange. Therefore, the effectiveness of its capture and tracking directly affects the accuracy of the system's measurement of relevant parameters. The better the signal capture and tracking, the better the quality of the separated signal, resulting in more accurate measurement of relevant parameter values ​​and ultimately a more precise time difference calculation.

[0010] With the increasing demand for time synchronization, the application scenarios of two-way time comparison systems are no longer limited to the initial static fixed platforms. Many dynamic platforms, such as shipborne, vehicle-mounted, and missile-borne platforms, have put forward the requirement for remote high-precision time synchronization. However, in highly dynamic scenarios, the demodulation modules of existing time comparison devices have low noise and poor time synchronization accuracy, which cannot meet the operational requirements. Summary of the Invention

[0011] Compared to static scenarios, in dynamic scenarios, due to factors such as the Doppler effect, the frequency of the signal received by the demodulation module changes significantly with its own motion state. This change in the received signal frequency affects the loop's tracking performance, thus reducing the carrier-to-noise ratio of the demodulated signal and ultimately lowering the accuracy of time synchronization between the two stations. Therefore, it is necessary to design a receiving loop with high dynamic tracking capability to track the received signal and reduce the impact on time synchronization accuracy in highly dynamic situations. Since the Doppler effect on the spreading code signal can be eliminated with carrier assistance, the demodulation module's ability to acquire and track the received signal is primarily determined by its ability to acquire and track the signal carrier.

[0012] To address this issue, the present invention provides a high-dynamic bidirectional time comparison device for demodulating the time comparison baseband signal received by the system, thereby solving the problem that the low noise level of the demodulated signal in a high-dynamic scenario leads to insufficient time synchronization accuracy.

[0013] This invention provides the following technical solution:

[0014] In one aspect, this specification provides a high-dynamic bidirectional time comparison device, including an automatic gain control amplifier circuit, a quadrature I / Q demodulator, a mixer, a variable gain amplifier, an analog-to-digital converter, and a processor. The automatic gain control amplifier circuit is connected to the quadrature I / Q demodulator, the quadrature I / Q demodulator is connected to the mixer, the mixer is connected to the variable gain amplifier, the variable gain amplifier is connected to the analog-to-digital converter, and the analog-to-digital converter is connected to the processor, wherein:

[0015] The automatic gain control amplifier circuit is used to determine a first gain based on the strength of the time-comparison baseband signal, amplify the time-comparison baseband signal according to the first gain to obtain an amplified signal, and send the amplified signal to the quadrature I / Q demodulator.

[0016] The quadrature I / Q demodulator is used to demodulate the amplified signal to obtain an I-channel signal and a Q-channel signal, wherein the I-channel signal and the Q-channel signal are orthogonal to each other.

[0017] The mixer is used to perform frequency mixing processing on the I-channel signal and the Q-channel signal to obtain an intermediate frequency signal;

[0018] The variable gain amplifier is used to determine a second gain according to the control command of the analog-to-digital converter, and amplify the intermediate frequency signal according to the second gain to obtain an amplified intermediate frequency signal;

[0019] The analog-to-digital converter is used to send control commands to the variable gain amplifier to convert the amplified intermediate frequency signal into a digital signal;

[0020] The processor is used to process the digital signal, and to continuously track and strip the digital signal using a carrier loop to obtain Doppler frequency shift measurement values, carrier phase measurement values, and frame data bits.

[0021] Optionally, the processor includes a carrier stripping module, a pseudo-code stripping module, a noise reduction module, a pseudo-code generator control module, a carrier generator control module, a pseudo-code generator, and a carrier generator, wherein:

[0022] The carrier stripping module is used to receive the digital signal, which is an I-channel signal and a Q-channel signal. A local carrier is generated using a carrier generator, and the local carrier is subjected to a 90° phase transformation to obtain a sine carrier signal and a cosine carrier signal. The sine carrier signal and the cosine carrier signal are multiplied by the I-channel signal and the Q-channel signal, respectively, to obtain an i-channel signal and a q-channel signal. The i-channel signal and the q-channel signal are then sent to the pseudocode stripping module.

[0023] The pseudo-random code stripping module is used to receive the i-channel signal and the q-channel signal, generate three pseudo-random code signals with different phases, perform calculations on each pseudo-random code signal with the i-channel signal and the q-channel signal respectively to obtain six signals, and send the six signals to the noise reduction module.

[0024] The noise reduction module is used to receive the six signals, process the six signals through coherent integration over a preset time to obtain a noise-filtered signal, and send the noise-filtered signal to the pseudocode generator control module and the carrier generator control module.

[0025] The pseudo-code generator control module is used to receive the noise-filtered signal, obtain frame data bits and pseudo-code generator control signal based on the noise-filtered signal, and send the pseudo-code generator control signal to the pseudo-code generator. The pseudo-code generator control signal is used to control the pseudo-code generator to generate a local pseudo-random code that is in phase with the received pseudo-random code in real time.

[0026] The carrier generator control module is used to receive the noise-filtered signal, obtain the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal based on the noise-filtered signal, and send the carrier generator control signal to the carrier generator. The carrier generator control signal is used to control the carrier generator to generate a local carrier with the same frequency and phase as the received carrier in real time.

[0027] Optionally, the carrier generator control module includes a carrier phase detector, a second-order frequency-locked auxiliary third-order phase-locked loop filter, and a Kalman filter module, wherein:

[0028] The carrier phase detector is used to receive the noise-filtered signal and obtain the phase difference and frequency difference between the local carrier and the received carrier based on the noise-filtered signal.

[0029] The second-order frequency-locked auxiliary third-order phase-locked loop filter is used to receive the phase difference and frequency difference between the local carrier and the received carrier, lock the phase difference and frequency difference between the local carrier and the received carrier within the first 20 seconds of receiving them, obtain the acceleration integral accumulation result and the velocity integral accumulation result, and send the acceleration integral accumulation result and the velocity integral accumulation result to the Kalman filter module.

[0030] The Kalman filter module is used to receive the acceleration integral accumulation result and the velocity integral accumulation result, perform Kalman filtering based on the acceleration integral accumulation result and the velocity integral accumulation result to obtain the control quantity of the carrier generator, convert the control quantity of the carrier generator into a carrier generator control signal, and send the carrier generator control signal to the carrier generator.

[0031] Optionally, the second-order frequency-locked auxiliary third-order phase-locked loop filter performs loop filtering according to the following formula:

[0032]

[0033] Where k represents the k-th iteration, S1 is the cumulative result of the acceleration integral, S2 is the cumulative result of the velocity integral, and T s Φ is the sampling time of the filter. e f is the phase difference between the local carrier and the received carrier. e f is the frequency difference between the local carrier and the received carrier. d B is the frequency adjustment amount of the carrier generator. Lf B is the bandwidth of the second-order frequency-locked loop. Lp Let a2, a3, b3, ω be the bandwidth of the third-order phase-locked loop. nf ω np These are the filter parameters.

[0034] Optionally, the Kalman filter module performs Kalman filtering according to the following formula: Prediction phase:

[0035] The predicted value of the state vector estimation is

[0036]

[0037] The mean square error prediction value of the state vector estimation is

[0038] P k,k-1 =AP k-1 A T +Q

[0039] Update and correction phase:

[0040] The filter gain value is

[0041] Kk =P k,k-1 H T HP k,k-1 H T +R k ) -1

[0042] The state vector estimate is

[0043]

[0044] The root mean square error of the state vector estimate is

[0045] P k =(IK k H)P k,k-1

[0046]

[0047]

[0048] in, This is the estimated state vector value at step k. P is the estimated state vector value at step k-1. k Let P be the mean square error of the state vector estimate at step k. k-1 The mean square error of the state vector estimate at step k-1. P is the predicted value of the state vector estimation at step k-1. k，k-1 Let ω be the mean squared error prediction of the state vector estimate at step k-1, A be the state transition matrix, Q be the variance of the estimated noise, and ω be 2π·70·10. 6 T s q is the sampling time of the filter. b The carrier phase noise introduced by the processor's crystal oscillator, q d The carrier frequency noise introduced by the processor's crystal oscillator, q a K represents the acceleration of the system platform relative to the satellite. k Let H be the current Kalman filter gain, H be the state observation matrix, and R be... k Z represents the observation error at step k. k C / N0(k) is the observed value of the filter, and C / N0(k) is the measured value of the carrier-to-noise ratio of the received signal at time k. P P within a time interval of length K nw The mean of (k), T coh Let P be the integration time, M be the computation window length, and P be the time to integration. nw (k) represents the ratio of narrowband power to wideband power of the signal at time k, f dx(k) is the control quantity of the carrier generator, x1(k) is the phase difference between the local carrier and the received carrier, x2(k) is the carrier Doppler frequency, x3(k) is the rate of change of the carrier Doppler frequency, I is the identity matrix, and C is the speed of light in vacuum.

[0049] In a second aspect, the present invention provides a method for controlling a high-dynamic bidirectional time comparison device, comprising:

[0050] The automatic gain control amplifier circuit determines a first gain based on the strength of the time-comparison baseband signal, amplifies the time-comparison baseband signal according to the first gain, obtains an amplified signal, and sends the amplified signal to the quadrature I / Q demodulator.

[0051] The quadrature I / Q demodulator demodulates the amplified signal to obtain an I-channel signal and a Q-channel signal, wherein the I-channel signal and the Q-channel signal are orthogonal to each other.

[0052] The mixer performs frequency mixing processing on the I-channel signal and the Q-channel signal to obtain an intermediate frequency signal;

[0053] The variable gain amplifier determines a second gain according to the control command of the analog-to-digital converter, and amplifies the intermediate frequency signal according to the second gain to obtain an amplified intermediate frequency signal;

[0054] The analog-to-digital converter sends control commands to the variable gain amplifier to convert the amplified intermediate frequency signal into a digital signal;

[0055] The processor processes the digital signal and uses a carrier loop to continuously track and strip the digital signal to obtain Doppler frequency shift measurements, carrier phase measurements, and frame data bits.

[0056] Optionally, the processor processes the digital signal, employing a carrier loop to continuously track and strip the digital signal to obtain Doppler frequency shift measurements, carrier phase measurements, and frame data bits, specifically including:

[0057] The carrier stripping module receives the digital signal, which is an I-channel signal and a Q-channel signal. It generates a local carrier using a carrier generator, performs a 90° phase transformation on the local carrier to obtain a sine carrier signal and a cosine carrier signal, and multiplies the sine carrier signal and the cosine carrier signal with the I-channel signal and the Q-channel signal respectively to obtain an i-channel signal and a q-channel signal. The i-channel signal and the q-channel signal are then sent to the pseudocode stripping module.

[0058] The pseudo-random code stripping module receives the i-channel signal and the q-channel signal, generates three pseudo-random code signals with different phases, performs calculations on each pseudo-random code signal with the i-channel signal and the q-channel signal respectively, obtains six signals, and sends the six signals to the noise reduction module.

[0059] The noise reduction module receives the six signals, processes the six signals through coherent integration over a preset time to obtain a noise-filtered signal, and sends the noise-filtered signal to the pseudocode generator control module and the carrier generator control module.

[0060] The pseudocode generator control module receives the noise-filtered signal, obtains frame data bits and pseudocode generator control signal based on the noise-filtered signal, and sends the pseudocode generator control signal to the pseudocode generator. The pseudocode generator control signal is used to control the pseudocode generator to generate a local pseudo-random code that is in phase with the received pseudo-random code in real time.

[0061] The carrier generator control module receives the noise-filtered signal, and obtains the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal based on the noise-filtered signal. The carrier generator control signal is sent to the carrier generator to control the carrier generator to generate a local carrier with the same frequency and phase as the received carrier in real time.

[0062] Optionally, the carrier generator control module receives the noise-filtered signal, obtains the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal based on the noise-filtered signal, and sends the carrier generator control signal to the carrier generator. The carrier generator control signal is used to control the carrier generator to generate a local carrier in real time that has the same frequency and phase as the received carrier, specifically including:

[0063] The carrier phase detector receives the noise-filtered signal and obtains the phase difference and frequency difference between the local carrier and the received carrier based on the noise-filtered signal.

[0064] The second-order frequency-locked auxiliary third-order phase-locked loop filter receives the phase difference and frequency difference between the local carrier and the received carrier, locks on the phase difference and frequency difference between the local carrier and the received carrier within the first 20 seconds of receiving them, obtains the acceleration integral accumulation result and the velocity integral accumulation result, and sends the acceleration integral accumulation result and the velocity integral accumulation result to the Kalman filter module.

[0065] The Kalman filter module receives the acceleration integral accumulation result and the velocity integral accumulation result, performs Kalman filtering based on the acceleration integral accumulation result and the velocity integral accumulation result to obtain the control quantity of the carrier generator, converts the control quantity of the carrier generator into a carrier generator control signal, and sends the carrier generator control signal to the carrier generator.

[0066] Optionally, the second-order frequency-locked auxiliary third-order phase-locked loop filter performs loop filtering according to the following formula:

[0067]

[0068] Where k represents the k-th iteration, S1 is the cumulative result of the acceleration integral, S2 is the cumulative result of the velocity integral, and T s Φ is the sampling time of the filter. e f is the phase difference between the local carrier and the received carrier. e f is the frequency difference between the local carrier and the received carrier. d B is the frequency adjustment amount of the carrier generator. Lf B is the bandwidth of the second-order frequency-locked loop. Lp Let a2, a3, b3, ω be the bandwidth of the third-order phase-locked loop. nf ω np These are the filter parameters.

[0069] Optionally, the Kalman filter module performs Kalman filtering according to the following formula:

[0070] Prediction phase:

[0071] The predicted value of the state vector estimation is

[0072]

[0073] The mean square error prediction value of the state vector estimation is

[0074] P k,k-1 =AP k-1 A T +Q

[0075] Update and correction phase:

[0076] The filter gain value is

[0077] K k =P k,k-1 H T HP k,k-1 H T +R k ) -1

[0078] The state vector estimate is

[0079]

[0080] The root mean square error of the state vector estimate is

[0081] P k =(IK k H)P k,k-1

[0082]

[0083] in, This is the estimated state vector value at step k. P is the estimated state vector value at step k-1. k Let P be the mean square error of the state vector estimate at step k. k-1 The mean square error of the state vector estimate at step k-1. P is the predicted value of the state vector estimation at step k-1. k，k-1 Let ω be the mean squared error prediction of the state vector estimate at step k-1, A be the state transition matrix, Q be the variance of the estimated noise, and ω be 2π·70·10. 6 T s q is the sampling time of the filter. b The carrier phase noise introduced by the processor's crystal oscillator, q d The carrier frequency noise introduced by the processor's crystal oscillator, q a K represents the acceleration of the system platform relative to the satellite. k Let H be the current Kalman filter gain, H be the state observation matrix, and R be... k Z represents the observation error at step k. k C / N0(k) is the observed value of the filter, and C / N0(k) is the measured value of the carrier-to-noise ratio of the received signal at time k. P P within a time interval of length K nw The mean of (k), T coh Let P be the integration time, M be the computation window length, and P be the time to integration. nw (k) represents the ratio of narrowband power to wideband power of the signal at time k, f d x(k) is the control quantity of the carrier generator, x1(k) is the phase difference between the local carrier and the received carrier, x2(k) is the carrier Doppler frequency, x3(k) is the rate of change of the carrier Doppler frequency, I is the identity matrix, and C is the speed of light in vacuum.

[0084] The high-dynamic bidirectional time comparison device provided in this embodiment of the invention uses a receiving loop with high dynamic tracking capability to track the received signal, thereby reducing the impact on time synchronization accuracy in high-dynamic situations and solving the problem that the low noise carrier of the demodulated signal from the demodulation module in high-dynamic situations leads to the inability of time synchronization accuracy to meet the working requirements. Attached Figure Description

[0085] Figure 1 This is a block diagram of the components of a bidirectional time comparison system in the prior art;

[0086] Figure 2 This is a schematic diagram of the time comparison signal in a bidirectional comparison system in the prior art;

[0087] Figure 3 This is a schematic diagram of the high dynamic bidirectional time comparison device in an embodiment of the present invention;

[0088] Figure 4 This is a schematic diagram of the processor structure in an embodiment of the present invention;

[0089] Figure 5 This is a schematic diagram of the structure of a second-order frequency-locked auxiliary third-order phase-locked loop filter in an embodiment of the present invention;

[0090] Figure 6 This is a schematic diagram illustrating the principle of Kalman filtering in an embodiment of the present invention.

[0091] Other features and advantages of the invention will be set forth in the description which follows, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention may be realized and obtained by means of the structures particularly pointed out in the written description, claims, and drawings. Detailed Implementation

[0092] To make the objectives, technical solutions, and advantages of this specification clearer, the technical solutions of this specification will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this specification, and not all of them. Based on the embodiments in this specification, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this specification.

[0093] As described herein, the term “comprising” and its various variations can be understood as open-ended terms that mean “including but not limited to”, and the term “one embodiment” can be understood as “at least one embodiment”.

[0094] The inventors discovered that the demodulation module of existing time comparison devices has low noise and poor time synchronization accuracy, which cannot meet the working requirements. In view of this, in this embodiment of the invention, a receiving loop with high dynamic tracking capability is used to track the reception, thereby reducing the impact on time synchronization accuracy in high dynamic situations.

[0095] Figure 3 The schematic diagram illustrates the structure of a high-dynamic bidirectional time comparison device according to one embodiment of this application. The time comparison baseband signal received by the system is first amplified by an AGC (Automatic Gain Amplifier), the gain of which varies with the signal strength. The amplified signal is then demodulated by an orthogonal I / Q demodulator into two mutually orthogonal I and Q signals. The I and Q signals are then mixed by a mixer to down-convert the frequency to 0.99MHz. The intermediate frequency signal is amplified by a VGA (Variable Gain Amplifier) ​​and then acquired as a digital signal by an ADC (Analog-to-Digital Converter), where the amplification gain of the VGA is controlled by the ADC. The acquired digital signal is then transmitted by the ADC to a processor for processing. The processor is a Xilinx ZYNQ chip, which integrates a programmable logic device (PL side) and an ARM-based MCU (PS side), and the two communicate via an AXI bus. To achieve optimal operating speed, the PL (Programmable Logic Device) in this system is primarily used for logic control, signal generation, and parallel integer operations, while the PS (Power Switch) assists with floating-point operations and overall control. This approach leverages the advantages of programmable logic devices (PLDs) in terms of speed and stability in parallel data processing while avoiding their disadvantages of high resource consumption and slow speed in floating-point operations. Furthermore, the integrated bus within the chip improves the efficiency of data interaction between the PL and PS, resulting in a faster demodulation algorithm compared to an FPGA+DSP processor combination connected via an external bus. This also significantly reduces the difficulty of program development. Finally, the demodulated data is transmitted to the host computer via a serial port.

[0096] In practice, the dynamic signal acquisition, tracking, and demodulation algorithm based on Kalman filtering is mainly implemented in the processor, and its structure is as follows: Figure 4 As shown.

[0097] Depend on Figure 4 It can be seen that the I and Q signals after quadrature demodulation are sampled by the ADC and converted into digital signals before entering the processor (ZYNQ).

[0098] In one implementation, the processor first generates a DDS at the PL (Programmable Logic Device) terminal as a carrier generator (NCO) to produce a local carrier. This local carrier is then phase-shifted by 90° and split into two orthogonal carrier signals: a sine carrier and a cosine carrier. These two signals are multiplied by the sampled I and Q signals, respectively, and mixed to obtain the i and q signals. This process shifts the center frequencies of the i and q signals to zero, thus completing the stripping of the intermediate frequency carrier signal (including the Doppler shift caused by platform dynamics).

[0099] In one implementation, the pseudo-random code generator generates three slightly different phase pseudo-random code signals: E (early), P (prompt), and L (late), respectively, using a shift register. These three signals are then correlated three times with the i and q signals to generate i... E i P i L and q E q P q L There are six signals in total. This process is equivalent to despreading the i and q signals, removing the pseudo-random code that serves as the spreading code. For i... P and q P For these two signals, after removing the pseudo-random code, only the information of the frame data bits will be retained.

[0100] In one implementation, i E i P i L and q E q P q L Since these six signals contain only bit information, noise can be removed by coherent integration over a certain period of time. The noise-filtered signals are I... E I P I L and Q E Q P Q L .

[0101] In one implementation, I E With Q E I L With Q L The amplitude calculations are performed separately to obtain signals E and L. The calculation method is as follows: and Then, coherent integration is performed on the E and L signals for a certain period of time to filter out noise, and the filtered signal is used as the input of the code phase discriminator for phase detection. The phase detection principle is to calculate the position of the main peak of the pseudo-random code autocorrelation function, i.e., the difference between the current local pseudo-random code phase and the received pseudo-random code phase. The calculation method is the code phase difference. The code phase difference output by the code phase discriminator serves as the input to the code loop filter, used to calculate the control input for the pseudo-random code generator. This control generates a pseudo-random code generator control signal, which in turn controls the generator to produce local pseudo-random codes in real-time that are in phase with the received pseudo-random codes. This forms a closed loop for continuous tracking and stripping of the pseudo-random code signal. Simultaneously, the pseudo-random code loop can also output measured values ​​of code phase and pseudorange based on the control input calculated by the loop filter.

[0102] In one implementation, I P and Q P Two signals are input to the phase detector, which calculates the phase difference Φ between the local carrier and the received carrier. e and frequency difference f e Both the input and output signals are calculated from the loop filter to determine the control input for the carrier generator. These signals are used to ensure that the frequency and phase of the local carrier generated by the carrier generator are consistent with the received carrier. This forms a closed-loop control, guaranteeing continuous tracking and stripping of the received carrier by the carrier loop, and outputting measurements such as Doppler shift and carrier phase, while simultaneously demodulating the frame data bits.

[0103] For received signals with high dynamic range and superimposed Doppler shift, the carrier loop is mainly needed for signal stripping. The performance of the carrier loop determines the dynamic performance of the demodulator module. The acquisition and tracking performance of the carrier loop for the received signal is primarily determined by its loop filter.

[0104] In one implementation, to achieve rapid signal acquisition and stable, accurate tracking, the carrier loop uses two types of loop filters during the acquisition and tracking phases. The acquisition and tracking phases of the carrier loop are primarily distinguished by the loop's runtime; generally, the first 20 seconds after the loop receives the signal constitute the acquisition phase, after which it transitions to the tracking phase. Since the acquisition phase requires the loop to quickly lock onto the received signal, but not necessarily high locking accuracy, a high-bandwidth second-order frequency-locked loop filter with an auxiliary third-order phase-locked loop is used for signal acquisition. Its structure is as follows: Figure 5 As shown.

[0105] Specifically, when performing calculations on discrete digital signals in the processor, the integral of acceleration and velocity is calculated using an iterative accumulation method. The overall calculation formula for the loop filter is as follows:

[0106]

[0107] In the above formula, k represents the k-th iteration, S1 is the result of the acceleration integral accumulation, and S2 is the result of the velocity integral accumulation. s The sampling time of the filter is given. The input is the phase difference Φ of the phase detector output. e and frequency difference f e The filter output is the frequency adjustment amount f from the carrier generator. d The remaining constants, such as a, b, and ω, are filter parameters, and the loop bandwidth is determined by them, as shown in the following formula:

[0108] The bandwidth of the second-order frequency-locked loop is:

[0109]

[0110] The bandwidth of the third-order phase-locked loop is:

[0111]

[0112] To capture signals with large frequency fluctuations in high-dynamic scenarios, the bandwidth of the capture loop filter should be set as high as possible, depending on the actual situation, to improve its ability to capture high-dynamic signals. However, higher bandwidth means more noise signals enter the loop, resulting in poorer filtering performance and reducing the loop's signal tracking accuracy.

[0113] In one implementation, after the acquisition phase is completed, a Kalman filter is used as the loop filter in the tracking phase. The integral results S1 and S2 calculated in the acquisition phase are used as the initial values ​​for the Kalman filter calculation. The principle of the Kalman filter is as follows: Figure 6 .

[0114] Specifically, in the first round, given an initial value for a state vector and its mean square error, the predicted value of the next state vector estimate and its predicted mean square error are calculated. Then, the previous predicted values ​​are updated and corrected using the current observations, thereby calculating the estimated value of the next state vector and its mean square error. The obtained state vector estimate is used to calculate the control input f of the carrier generator. d The calculation of the mean square error of the state vector estimation, along with the mean square error of the state vector estimation, is used as an initial value for the calculation of the next round of predictions, thus forming a continuous iterative cycle. The calculation formula is as follows:

[0115] Prediction phase:

[0116] The predicted value of the state vector estimation is

[0117]

[0118] The mean square error prediction value of the state vector estimation is

[0119] P k,k-1 =AP k-1 A T +Q (5)

[0120] Update and correction phase:

[0121] The filter gain value is

[0122] K k =P k,k-1 H T HP k,k-1 H T +R k ) -1 (6)

[0123] The state vector estimate is

[0124]

[0125] The root mean square error of the state vector estimate is

[0126] P k =(IK k H)P k,k-1 (8)

[0127] In the above formula, and P k，k-1 These are the state vector estimates used at step k-1, respectively. The predicted value at step k is calculated using the mean square error P. I is the identity matrix, and A is the state transition matrix, a third-order square matrix used to calculate the predicted value of the state vector estimate. In this system, its value is...

[0128]

[0129] Where T s The sampling time of the filter. Q is the variance of the estimated noise. In this system, the values ​​of Q are as follows:

[0130]

[0131] Where ω equals 2π·70·10 6 T s q is the sampling time of the filter. b The carrier phase noise introduced by the processor's crystal oscillator is determined by the baseband signal frequency (70MHz) and the type of crystal oscillator. dThe carrier frequency noise introduced by the processor's crystal oscillator is determined by the baseband signal frequency (70MHz) and the type of crystal oscillator, q a This represents the acceleration of the system platform relative to the satellite's motion.

[0132] Specifically, in the gain equation of the Kalman filter, K k K represents the current Kalman filter gain, primarily used for subsequent calculations of the current state vector estimate. k The value is determined by the current mean square error prediction value P. k，k-1 and the state observation matrix H and observation error R k Calculations show that K k The value of K is changed in real time based on the changes in the values ​​of both. In a carrier loop with Kalman filtering, the Kalman filter gain K... k The value of is the loop bandwidth. For receiving high-dynamic signals, the tracking loop needs a sufficiently large bandwidth, covering the frequency changes caused by the Doppler effect. However, increasing the loop bandwidth reduces the tracking accuracy, thus reducing the accuracy of the device's time synchronization. Traditional tracking loops have a fixed bandwidth, such as a second-order frequency-locked loop assisted by a third-order phase-locked loop for signal acquisition. Although this bandwidth can be set to a higher value to achieve rapid acquisition and pulling of high-dynamic signals, the loop needs to accurately track the signal after pulling to improve time comparison accuracy. High-precision time synchronization requires minimizing the bandwidth of the receiving tracking loop during demodulation; a smaller bandwidth loop will experience loss of lock when the signal frequency changes abruptly in a high-dynamic environment. Therefore, to maximize loop tracking accuracy while ensuring normal signal tracking, a signal tracking algorithm that can change the loop bandwidth in real time according to the dynamic characteristics of the received signal needs to be designed. The loop bandwidth K after adding Kalman filtering is... k The mean square error prediction value P can be estimated based on the current calculated state vector of the loop. k,k-1 Adjustments are made in advance so that the bandwidth can be adjusted to maintain the loop's locked state when the signal is dynamic, and the bandwidth can be reduced to improve accuracy when the signal is stable, so that the loop can achieve good signal tracking performance.

[0133] Specifically, the state observation matrix H in this system is:

[0134]

[0135] Observation noise R k The value is related to the measured value of the carrier-to-noise ratio of the received signal at that time. The calculation method is as follows: The carrier-to-noise ratio is calculated as follows.

[0136]

[0137] Among them, P nw (k) represents the ratio of narrowband power to wideband power of the signal at time k, T coh For the integration time, μ p P within a time interval of length K mw The mean of (k), M is the calculation window length, M is usually 20, K is 50, and C is the speed of light in vacuum.

[0138] Observation noise R k The calculation method is shown below.

[0139]

[0140] Z in the state vector estimation equation k Z is the observed value of the filter, which in this system is Z. k The filter input at time k is the phase difference Φ between the local carrier and the received signal carrier. e .

[0141] The Kalman filter used in this system is a third-order filter, therefore its state vector is also a third-order X. k = [x1(k), x2(k), x3(k)], where x1, x2, and x3 represent the phase difference, carrier Doppler frequency, and rate of change of the carrier Doppler frequency, respectively. Therefore, the filter output, which is the control quantity f of the local carrier generator (NCO), is... d for

[0142]

[0143] The Kalman filter will use the current control quantity f d (k) is converted into a corresponding control signal and sent to the local carrier generator (NCO). The NCO receives the signal and makes corresponding adjustments, thus completing one closed-loop control cycle. During loop operation, this process is continuously iterated based on the state of the received signal to achieve continuous tracking of the received signal carrier.

[0144] The above embodiments construct a high-dynamic bidirectional time comparison device, which uses a receiving loop with high dynamic tracking capability to track the received signal, thereby reducing the impact on time synchronization accuracy in high-dynamic situations and solving the problem that the time synchronization accuracy cannot meet the working requirements due to the low noise carrier of the demodulated signal in the demodulation module in high-dynamic situations.

[0145] Example 2

[0146] Based on the same technical concept, this invention also provides a method for controlling a high dynamic bidirectional time comparison device. Since the principle of the above method in solving the problem is similar to that of the high dynamic bidirectional time comparison device, the implementation of the above method can refer to the implementation of the device, and the repeated parts will not be described again.

[0147] This invention provides a method for controlling a high-dynamic bidirectional time comparison device, the method being executed by a processor and including the following steps:

[0148] Step 1: The automatic gain control amplifier circuit determines a first gain based on the strength of the time-comparison baseband signal, amplifies the time-comparison baseband signal according to the first gain, obtains an amplified signal, and sends the amplified signal to the quadrature I / Q demodulator.

[0149] Step 2: The quadrature I / Q demodulator demodulates the amplified signal to obtain an I-channel signal and a Q-channel signal, which are orthogonal to each other.

[0150] Step 3: The mixer performs frequency mixing on the I-channel signal and the Q-channel signal to obtain the intermediate frequency signal.

[0151] Step 4: The variable gain amplifier determines the second gain according to the control command of the analog-to-digital converter, and amplifies the intermediate frequency signal according to the second gain to obtain the amplified intermediate frequency signal.

[0152] Step 5: The analog-to-digital converter sends a control command to the variable gain amplifier to convert the amplified intermediate frequency signal into a digital signal.

[0153] Step 6: The processor processes the digital signal and uses a carrier loop to continuously track and strip the digital signal to obtain Doppler frequency shift measurement value, carrier phase measurement value, and frame data bits.

[0154] In summary, in this embodiment of the invention, the high-dynamic bidirectional time comparison device uses a receiving loop with high dynamic tracking capability to track the received signal, thereby reducing the impact on time synchronization accuracy in high-dynamic situations and solving the problem that the low noise carrier of the demodulated signal from the demodulation module in high-dynamic situations leads to insufficient time synchronization accuracy to meet the working requirements.

[0155] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the invention.

[0156] Those skilled in the art will understand that, in addition to implementing the apparatus and its various devices, modules, and units provided by the present invention in the form of pure computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the apparatus and its various devices, modules, and units provided by the present invention function as logic gates, switches, application-specific integrated circuits, programmable logic controllers, etc. Therefore, the apparatus and its various devices, modules, and units provided by the present invention can be regarded as structures within hardware components or as software modules for implementing the method.

[0157] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A high-dynamic bidirectional time comparison device, characterized in that, The system includes an automatic gain control amplifier circuit, a quadrature I / Q demodulator, a mixer, a variable gain amplifier, an analog-to-digital converter (ADC), and a processor. The automatic gain control amplifier circuit is connected to the quadrature I / Q demodulator, the quadrature I / Q demodulator is connected to the mixer, the mixer is connected to the variable gain amplifier, the variable gain amplifier is connected to the ADC, and the ADC is connected to the processor. The automatic gain control amplifier circuit is used to determine a first gain based on the strength of the time-comparison baseband signal, amplify the time-comparison baseband signal according to the first gain to obtain an amplified signal, and send the amplified signal to the quadrature I / Q demodulator. The quadrature I / Q demodulator is used to demodulate the amplified signal to obtain an I-channel signal and a Q-channel signal, wherein the I-channel signal and the Q-channel signal are orthogonal to each other. The mixer is used to perform frequency mixing processing on the I-channel signal and the Q-channel signal to obtain an intermediate frequency signal; The variable gain amplifier is used to determine a second gain according to the control command of the analog-to-digital converter, and amplify the intermediate frequency signal according to the second gain to obtain an amplified intermediate frequency signal; The analog-to-digital converter is used to send control commands to the variable gain amplifier to convert the amplified intermediate frequency signal into a digital signal; The processor is used to process the digital signal, and to continuously track and strip the digital signal using a carrier loop to obtain Doppler frequency shift measurement values, carrier phase measurement values, and frame data bits.

2. The apparatus according to claim 1, characterized in that, The processor includes a carrier stripping module, a pseudo-code stripping module, a noise reduction module, a pseudo-code generator control module, a carrier generator control module, a pseudo-code generator, and a carrier generator, wherein: The carrier stripping module is used to receive the digital signal, which is an I-channel signal and a Q-channel signal. A local carrier is generated using a carrier generator, and the local carrier is subjected to a 90° phase transformation to obtain a sine carrier signal and a cosine carrier signal. The sine carrier signal and the cosine carrier signal are multiplied by the I-channel signal and the Q-channel signal, respectively, to obtain an i-channel signal and a q-channel signal. The i-channel signal and the q-channel signal are then sent to the pseudocode stripping module. The pseudo-random code stripping module is used to receive the i-channel signal and the q-channel signal, generate three pseudo-random code signals with different phases, perform calculations on each pseudo-random code signal with the i-channel signal and the q-channel signal respectively to obtain six signals, and send the six signals to the noise reduction module. The noise reduction module is used to receive the six signals, process the six signals through coherent integration over a preset time to obtain a noise-filtered signal, and send the noise-filtered signal to the pseudocode generator control module and the carrier generator control module. The pseudo-code generator control module is used to receive the noise-filtered signal, obtain frame data bits and pseudo-code generator control signal based on the noise-filtered signal, and send the pseudo-code generator control signal to the pseudo-code generator. The pseudo-code generator control signal is used to control the pseudo-code generator to generate a local pseudo-random code that is in phase with the received pseudo-random code in real time. The carrier generator control module is used to receive the noise-filtered signal, obtain the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal based on the noise-filtered signal, and send the carrier generator control signal to the carrier generator. The carrier generator control signal is used to control the carrier generator to generate a local carrier with the same frequency and phase as the received carrier in real time.

3. The apparatus according to claim 2, characterized in that, The carrier generator control module includes a carrier phase detector, a second-order frequency-locked auxiliary third-order phase-locked loop filter, and a Kalman filter module, wherein: The carrier phase detector is used to receive the noise-filtered signal and obtain the phase difference and frequency difference between the local carrier and the received carrier based on the noise-filtered signal. The second-order frequency-locked auxiliary third-order phase-locked loop filter is used to receive the phase difference and frequency difference between the local carrier and the received carrier, lock the phase difference and frequency difference between the local carrier and the received carrier within the first 20 seconds of receiving them, obtain the acceleration integral accumulation result and the velocity integral accumulation result, and send the acceleration integral accumulation result and the velocity integral accumulation result to the Kalman filter module. The Kalman filter module is used to receive the acceleration integral accumulation result and the velocity integral accumulation result, perform Kalman filtering based on the acceleration integral accumulation result and the velocity integral accumulation result to obtain the control quantity of the carrier generator, convert the control quantity of the carrier generator into a carrier generator control signal, and send the carrier generator control signal to the carrier generator.

4. The apparatus according to claim 3, characterized in that, The second-order frequency-locked auxiliary third-order phase-locked loop filter performs loop filtering according to the following formula: Where k represents the k-th iteration, S1 is the cumulative result of the acceleration integral, S2 is the cumulative result of the velocity integral, and T s Φ is the sampling time of the filter. e f is the phase difference between the local carrier and the received carrier. e f is the frequency difference between the local carrier and the received carrier. d B is the frequency adjustment amount of the carrier generator. Lf B is the bandwidth of the second-order frequency-locked loop. Lp Let a2, a3, b3, ω be the bandwidth of the third-order phase-locked loop. nf ω np These are the filter parameters.

5. The apparatus according to claim 3, characterized in that, The Kalman filter module performs Kalman filtering according to the following formula: Prediction phase: The predicted value of the state vector estimation is The mean square error prediction value of the state vector estimation is P k,k-1 =AP k-1 From T +Q Update and correction phase: The filter gain value is K k =P k,k-1 H T (HP k,k-1 H T +R k ) -1 The state vector estimate is The root mean square error of the state vector estimate is P k =(I-K k H)P k,k-1 in, This is the estimated state vector value at step k. P is the estimated state vector value at step k-1. k Let P be the mean square error of the state vector estimate at step k. k-1 The mean square error of the state vector estimate at step k-1. P is the predicted value of the state vector estimation at step k-1. k，k-1 Let ω be the mean squared error prediction of the state vector estimate at step k-1, A be the state transition matrix, Q be the variance of the estimated noise, and ω be 2π·70·10. 6 T s q is the sampling time of the filter. b The carrier phase noise introduced by the processor's crystal oscillator, q d The carrier frequency noise introduced by the processor's crystal oscillator, q a K represents the acceleration of the system platform relative to the satellite. k Let H be the current Kalman filter gain, H be the state observation matrix, and R be... k Z represents the observation error at step k. k C / N0(k) is the observed value of the filter, and C / N0(k) is the measured value of the carrier-to-noise ratio of the received signal at time k. p P within a time interval of length K nw The mean of (k), T coh Let P be the integration time, M be the computation window length, and P be the time to integration. nw (k) represents the ratio of narrowband power to wideband power of the signal at time k, f d x(k) is the control quantity of the carrier generator, x1(k) is the phase difference between the local carrier and the received carrier, x2(k) is the carrier Doppler frequency, x3(k) is the rate of change of the carrier Doppler frequency, I is the identity matrix, and C is the speed of light in vacuum.

6. A method for controlling the high dynamic bidirectional time comparison device according to any one of claims 1-5, characterized in that, include: The automatic gain control amplifier circuit determines a first gain based on the strength of the time-comparison baseband signal, amplifies the time-comparison baseband signal according to the first gain, obtains an amplified signal, and sends the amplified signal to the quadrature I / Q demodulator. The quadrature I / Q demodulator demodulates the amplified signal to obtain an I-channel signal and a Q-channel signal, wherein the I-channel signal and the Q-channel signal are orthogonal to each other. The mixer performs frequency mixing processing on the I-channel signal and the Q-channel signal to obtain an intermediate frequency signal; The variable gain amplifier determines a second gain according to the control command of the analog-to-digital converter, and amplifies the intermediate frequency signal according to the second gain to obtain an amplified intermediate frequency signal; The analog-to-digital converter sends control commands to the variable gain amplifier to convert the amplified intermediate frequency signal into a digital signal; The processor processes the digital signal and uses a carrier loop to continuously track and strip the digital signal to obtain Doppler frequency shift measurements, carrier phase measurements, and frame data bits.

7. The method according to claim 6, characterized in that, The processor processes the digital signal, continuously tracking and stripping it using a carrier loop to obtain Doppler frequency shift measurements, carrier phase measurements, and frame data bits, specifically including: The carrier stripping module receives the digital signal, which is an I-channel signal and a Q-channel signal. It uses a carrier generator to generate a local carrier and performs a 90° phase transformation on the local carrier to obtain a sine carrier signal and a cosine carrier signal. The sine carrier signal and the cosine carrier signal are multiplied by the I-channel signal and the Q-channel signal, respectively, to obtain an i-channel signal and a q-channel signal. The i-channel signal and the q-channel signal are then sent to the pseudocode stripping module. The pseudo-random code stripping module receives the i-channel signal and the q-channel signal, generates three pseudo-random code signals with different phases, performs calculations on each pseudo-random code signal with the i-channel signal and the q-channel signal respectively, obtains six signals, and sends the six signals to the noise reduction module. The noise reduction module receives the six signals, processes the six signals through coherent integration over a preset time to obtain a noise-filtered signal, and sends the noise-filtered signal to the pseudocode generator control module and the carrier generator control module. The pseudocode generator control module receives the noise-filtered signal, obtains frame data bits and pseudocode generator control signal based on the noise-filtered signal, and sends the pseudocode generator control signal to the pseudocode generator. The pseudocode generator control signal is used to control the pseudocode generator to generate a local pseudo-random code that is in phase with the received pseudo-random code in real time. The carrier generator control module receives the noise-filtered signal, and obtains the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal based on the noise-filtered signal. The carrier generator control signal is sent to the carrier generator to control the carrier generator to generate a local carrier with the same frequency and phase as the received carrier in real time.

8. The method according to claim 7, characterized in that, The carrier generator control module receives the noise-filtered signal, and based on the noise-filtered signal, obtains the Doppler frequency shift measurement value, the carrier phase measurement value, and the carrier generator control signal. The carrier generator control signal is then sent to the carrier generator. This carrier generator control signal controls the carrier generator to generate a local carrier in real time that has the same frequency and phase as the received carrier. Specifically, it includes: The carrier phase detector receives the noise-filtered signal and obtains the phase difference and frequency difference between the local carrier and the received carrier based on the noise-filtered signal. The second-order frequency-locked auxiliary third-order phase-locked loop filter receives the phase difference and frequency difference between the local carrier and the received carrier, locks on the phase difference and frequency difference between the local carrier and the received carrier within the first 20 seconds of receiving them, obtains the acceleration integral accumulation result and the velocity integral accumulation result, and sends the acceleration integral accumulation result and the velocity integral accumulation result to the Kalman filter module. The Kalman filter module receives the acceleration integral accumulation result and the velocity integral accumulation result, performs Kalman filtering based on the acceleration integral accumulation result and the velocity integral accumulation result to obtain the control quantity of the carrier generator, converts the control quantity of the carrier generator into a carrier generator control signal, and sends the carrier generator control signal to the carrier generator.

9. The method according to claim 8, characterized in that, The second-order frequency-locked auxiliary third-order phase-locked loop filter performs loop filtering according to the following formula: Where k represents the k-th iteration, S1 is the cumulative result of the acceleration integral, S2 is the cumulative result of the velocity integral, and T s Φ is the sampling time of the filter. e f is the phase difference between the local carrier and the received carrier. e f is the frequency difference between the local carrier and the received carrier. d B is the frequency adjustment amount of the carrier generator. Lf B is the bandwidth of the second-order frequency-locked loop. Lp Let a2, a3, b3, ω be the bandwidth of the third-order phase-locked loop. nf ω np These are the filter parameters.

10. The apparatus according to claim 3, characterized in that, The Kalman filter module is performed according to the following formula. Kalman filtering: Prediction phase: The predicted value of the state vector estimation is The mean square error prediction value of the state vector estimation is P k,k-1 =AP k-1 From T +Q Update and correction phase: The filter gain value is K k =P k,k-1 H T (HP k,k-1 H T +R k ) -1 The state vector estimate is The root mean square error of the state vector estimate is P k =(I-K k H)P k,k-1 in, This is the estimated state vector value at step k. P is the estimated state vector value at step k-1. k Let P be the mean square error of the state vector estimate at step k. k-1 The mean square error of the state vector estimate at step k-1. P is the predicted value of the state vector estimation at step k-1. k，k-1 Let ω be the mean squared error prediction of the state vector estimate at step k-1, A be the state transition matrix, Q be the variance of the estimated noise, and ω be 2π·70·10. 6 T s q is the sampling time of the filter. b The carrier phase noise introduced by the processor's crystal oscillator, q d The carrier frequency noise introduced by the processor's crystal oscillator, q a K represents the acceleration of the system platform relative to the satellite. k Let H be the current Kalman filter gain, H be the state observation matrix, and R be... k Z represents the observation error at step k. k C / N0(k) is the observed value of the filter, and C / N0(k) is the measured value of the carrier-to-noise ratio of the received signal at time k. P P within a time interval of length K nw The mean of (k), T coh Let P be the integration time, M be the computation window length, and P be the time to integration. nw (k) represents the ratio of narrowband power to wideband power of the signal at time k, f d x(k) is the control quantity of the carrier generator, x1(k) is the phase difference between the local carrier and the received carrier, x2(k) is the carrier Doppler frequency, x3(k) is the rate of change of the carrier Doppler frequency, I is the identity matrix, and C is the speed of light in vacuum.

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