1. Abstract
    An experimental verification of general relativity conception was made using an atomic maser in a rocket attaining an altitude of 1,800 metres.
The signal of the maser was monitored on the ground, so that the effect of gravitational potential on the frequency of the maser was measured.
The resulting data was processed through a careful prediction and elimination of the Doppler shift and other error resources, so that the
gravitational blue shift is directly observed. The experiment is described including a consummate discussion of navigation algorithm applied in the
processing procedure. The authors believe that this is a direct high accuracy test of the general relativistic phenomena using an airborne clock.

XXXXtroduction
    A rocket is constructed and launched carrying an atomic frequency standard as the payload. The frequency of the signal from the atomic
frequency standard is examined on the ground.
    Another atomic frequency standard is used as a comparison to monitor the change in frequency of the received signal from the payload.
A group of variables that will influence the change are eliminated so that the resulting data, representing the relativistic shifts, are recovered and

recorded.
    The objective of the experiment is to test general relativity concept by measuring directly the effect of gravitational potential on the

frequency of a proper clock, in this case the atomic frequency standard.
    In this experiment, a gravitational effect amounting to 5.6e-13 was measured.



    The predicted proportion change in frequency is expressed in equation [a].Where β is the velocity/c and r is the displacement of the rocket
relative to theground base. ais the centrifugal acceleration of the ground station while ε represents thepropagation vector of the rocket-to-ground
signal. Our knowledge of the relative velocityand displacement of the rocket is obtained from the flight data recorder installed in thepayload. As
we chose the ground base as the navigation frame, the movement of the groundbase in the geographic frame was straightly eliminated in the
navigation system.
    In equation [1], the first term is the gravitational blue shift, the second term expressesthe Doppler shift. The last term describes the effect of the
rotation of the earth during thepropagation of the signal. In the elimination of the second and third terms, out knowledge ofthe rocket's velocity
and position are obtained from the FDR(Flight Data Recorder) while theknowledge of the velocity and position of the ground base is gained from
the Earth Modeland GLarLng of the launching site.
    The specific procedure is exhibited in the materials and methods section.

3. Materials and Methods

    The data was first generated and collected after the hardware processing of ground base, which procedure is labeled as data
acquisition. The recorded data was then stored in the computer for further processing, labeled as Data processing.
Data acquisition
  a) Algebra description
    The frequency signal transmitted from the payload is fixed at f0 + △f.
    △f was set to be 50mHz (Figure [d]).
    The signal received on the ground base is labeled as f1. It is predicted to be



    after the elimination of errors.
    The ground station processed f1 with a standard signal of frequency f0 , which was generated by the atomic frequency standard.
Heterodyne-beat method was applied thus a signal with frequency f1 - f0  was sampled by a high speed analog to digital converter as f2 and
processed by the digital signal processor. A standard signal of frequency△f is generated by atomic frequency standard and processed in the
computer, a series of effects including doppler shift are taken into account.


  

    The final signal, labeled △fR, is expressed by equation [2].
    The frequency of f2 was estimated by 4-parameter estimation algorithm and recorded in the computer. The method of 4-parameter
estimation will be discussed later.
    
    Therefore, the resulting data was obtained. The predicted behaviour of this final signal is shown in figure [6].

  b) Technical details
    A transmitter and a superheterodyne receiver were built specially for the experiment.
    The signal from the on-board AFS(Atomic Frequency Standard) was directly amplified and transmitted. The output power of the amplifier
was +38dBm and the frequency was 63.8978MHz.
    The structure of the receiver is shown below.

  

    The signal from the antenna at the ground base was filtered and directly amplified by a LNA(Low Noise Amplifier) and mixed with FS1 by
mixer 1 to obtain an IF(Intermediate Frequency) with a frequency of 5MHz. The IF signal was processed by an AGC(Automatic Gain Control) circuit
so that the amplitude was stabilized. This processed IF signal was mixed with FS2 by mixer 2 to obtain a 50mHz signal and was converted into
digital signal by a high speed ADC. The exact frequency of the signal sampled by the ADC was estimated through 4-parameter estimation
algorithm. FS1 and FS2 were generated by the AFS at the ground base.

    1) The realization of heterodyne-beat method
    The kernel of heterodyne-beat method is the shift of spectrum. As a result, a mixer is used to obtain the difference in frequency of two
signals. In this application, two integrated mixer circuits AD831 were used.
    The following images indicate a test result for the mixer circuit. The left image indicates the signal of RF input of the mixer and another
one shows the local oscillation input of the mixer. They were generated by two DDS(Direct Digital Synthesis) circuits. The clock standard of the DDS
circuits was connected to an AFS. The frequencies of them were 7MHz and 7.00001MHz.


  

    The following image shows the output of the mixer which equals to the difference between two frequency signals connected to the local

oscillation input and the RF input. Here, the difference was 10Hz.

  


    2) The design of AGC circuit

    An AGC circuit was used to stabilize the amplitude of IF signal.
    The input signal was demodulated and filtered into a voltage signal represents the strength of the signal. This voltage signal was used to

control the gain of a VGA(Variable Gain Amplifier).

  

    In this case, the wave-detector was AD8307 and the VGA was AD603.
    A test for the AGC module is shown below. Channel 1 was connected to the output of the AGC module, channel 2 was connected to the
input. The input of the AGC module was connected to a function generator.

      
    
    As the waveform shown on the oscilloscope, although the amplitude of the input signal to the AGC was changed, the amplitude of output
signal of it remained the same.
    A test result for the transmitter and the receiver is shown below.
    The transmitter was placed 800 metres away from the receiver. The image on the left shows the output signal from the transmitter and
another image shows the signal sampled from the output of LNA of the receiver.

  

    The image below shows the signal sampled from the output of the AGC module, where frequency = IF = 5MHz.

  
2) Method of estimation of frequency
   a) Definition of 4 parameters
Assume the sampled signal S(t) is given by
    
    Where A0' is the ideal amplitude of the signal, ω0' is the ideal frequency of the signal, C0' is the ideal DC offset of the signal and θ0' is
the ideal phase of the signal.
    The signal can be expressed by the equation
    

    Where
    
    Suppose the magnitude of the signal sampled during time tk(k = 0, 1, 2, ...) is y(k), is given by
  
    b) Method of 3-parameter estimation
    Suppose the sampled voltage value of the signal at time tk is y(k), k = 1, 2, 3 ...N-1. The amplitude of sine, amplitude of cosine and DC
offset is defined as A,B and C. The RSS(Residual Sum of Squares) between the estimation value and actual value is given by
  
    Where N is the length of samples, set
  
    The solution for X is given by the least square solution below:
.  
c) 4-parameter estimation algorithm
    The idea of successive approximation is applied in this algorithm. First, a rough frequency is given, 3-parameter estimation algorithm is
applied to the sampled signal. The cosine amplitude, sine amplitude, DC offset and estimated RSS are obtained. The operation is repeated with
different frequencies so a serial of estimated RSS are obtained. One of those set of obtained result with minimum estimated RSS is the value of
actual frequency. The detailed steps are shown below.
    1) Determine the frequency of the signal roughly though DFT(Discrete Fourier Transform), label this frequency as f0.
    2) Set the domain of iteration to be ωdl and ωdu, where ωdl is the lower boundary, given by ωdl = f0 - fclk / N. ωdu is the upper
boundary, given by ωdu = f0 + fclk / N. fclk is the frequency of sampling clock and N is the length of DFT.
    3) Set ω0 = ωdu - ωdl. 2M+1 points (M∈N*) are samples between ωdl and ωdu with equal intervals. 3-parameter estimation algorithm
is used here to compute the RSS of this group of samples.
    4) Find and record the minimum value of RSS of samples in step 3 This minimum value is corresponding to the actual frequency.
    Repeat operations 2 to 4 until the precision of the estimation reaches the required level.
    A picture of the receiver is shown below.



2) Data processing

  a) Downlink signal

    The navigation system provided data of dynamics with a sample rate of 1,200 samples per second. The data was given in terms of
angular velocity and acceleration in the on-board coordinate. The six groups of parameters are ωx, ωy, ωz, ax, ay and a--z, respectively where the
X-axis if the mean axis of the rocket.
    In the determination of change in angle, △θ and change in velocity, △V, the cubic spline function is adopted in curve fitting before
integration. This method of Simpson's rule provides six groups of data: △θx, △θy, △θz, △Vx, △Vy and △Vz. As a result, the behaviour of the rocket
between samples are predicted and considered.

    In the determination of attitude angle, the method of Quaternion is applied.
    The quaternion numbers at time tm+1 are given in equation [9], where △θx, △θy and △θz are the output of change in angle and vector Φ

is the rotation vector, which is given by equation [9]
    
    

    In equation [9] and [10], the angular velocity of the rocket is assumed to fit cubic function. However,  the actual angular velocity does not

fit a cubic function.
    Equations [9] and [10] do not achieve minimum shift of algorithm. After the parachute deployment, the rocket was suspended in the
descending stage. Thus the rocket is likely to experience coning motion, which means that the rocket vibrate about the equilibrium position at
small angles. The coning motion is the worst working environment for the SINS(Strapdown Inertial Navigation System) as it will cause serve shift of
the Math Platform.

For optimization algorithms, the following improvements are made.
    O-XYZ represents the reference frame R, which is the on-board frame when the rocket is in equilibrium.
    
    Let b(tm-1) and b(tm) to be the instantaneous on-board frame at time tm-1 and tm.
    According to Euler Theorem, O-XYZ can be regarded as a rotating transformation of b(tm) or b(tm-1) with rotating vector Q(t) given in

equation [11].
    
  

    The shift on-board frame can be regarded as a rotation transformation of the ideal on-board frame, which is the frame when the rocket

is in equilibrium. The samples of △θ is grouped in three again.
    In each group, the samples are labeled as .
    
  

    Equation [12] is an improved  form of equation [3]. By selecting proper constant k1 and k2, the effect of the coning motion is minimised.
Here, the ideal values for k1 and k2 are 0.45 and 0.675.
    Therefore, the attitude of the rocket is found through the optimized quaternion algorithm. The Eular angles are found by equation [13].

    

    The quaternions are supposed to be standardized. However, resulting form calculation errors and other factors, the quaternion numbers
gradually loses standability. The standardization of quaternion numbers is applied at the end of each period of attitude refreshment. The formula
for standardization is given by equation [8].

    

    Where  is the standardization value and  is the value after attitude refreshment.
    So far, the discussion of the rocket's dynamic is in the on-board coordinate. However, the final results have to be expressed in the
navigation frame, which sets the ground base as the origin.
    
  

Equation [15] gives the coordinate transformation matrix (attitude matrix) from on-board frame to navigation frame in terms of quaternion.
    The initial extraction quaternion numbers are thus given by the initial attitude matrix obtained in initial azimuth alignment.

  

    The velocity of the rocket at time tm in the navigation frame, Vm, is given by equation [16], containing a series of error compensations.
Vm-1 is the velocity in the same frame at time
tm-1. Cm-1 is the coordinate transformation matrix at time tm-1. Vm-1 is the compensation velocity caused by while△Vg/corm is the

compensation velocity caused by the deleterious acceleration. △Vsfm is the compensation velocity caused by ecific force.
    
  

    Where△Vm us the change in velocity during period [tm-1, tm].
    △Vrotm is the compensation velocity caused by rotation effect.
    △Vsculm is the compensation velocity caused by sculling motion.

    Due to air current, gustiness and other factors, the rocket experiences vibrations during the flight. Those factors cause a highly dynamic

working environment for the payload. Therefore, the velocity has to be compensated so that the sculling effect and the rotation effect are
eliminated. Otherwise, the calculation of velocity will involve serve errors. When it comes to position determination, there two error resources
contribute to scroll errors. Here△Vrotm and△Vsculm represent the compensation velocities due to the rotation effect and the sculling effect
respectively. the rotation effect happens when the direction of linear velocity rotates in a three-dimensional coordinate. The sculling effect is
caused by the angular vibration and linear vibration are in phase and of same frequency on the rocket. This is quite similar to the sculling motion:
on one hand, the syrup vibrates periodically about the lateral axis of the boat. On the other hand, the boat forges ahead along the direct-axis in
an intermittent behaviour.

    The original expression for △Vrotm and △Vsculm are:
    
  


    The optimized formula for △Vrotm and △Vsculm are

  


    The optimized algorithm for sculling effect rotation effect, in velocity determination as well as the correction for conning motion in
attitude determination make sure that the motion of the layload is precisely calculated in spite of the unstable motion of the rocket. Hence the
cancelling of doppler effect and second-order general relativity conception shift are more reliable. The specific precision level is related in the
discussion section.

The final expression for Vm is given by equation [22]:

  

    Considering that all the compensation dosages have been taken into account the calculation of Vm and that the data is discrete with
equal time internals, the data of displacement, is obtained through numerical integration.
    As the dynamic data is determined, the following equations are substituted into
    equation [2].
    
  

    Where c is the velocity of light, Wen is the angular velocity of the earth in the navigation frame and  is the radius vector of the earth at
the launching spot.
    The frequency of this signal is plotted against time in Figure [5].
    The navigation algorithm aims to calculate the velocity of the rocket in the navigation frame. However, the accelerometer does not tell
deleterious acceleration and relative acceleration of the rocket. Therefore, the compensation velocity has to be estimated from the measure value.



    Where g is the gravitational acceleration of the launch site.
     In equation [24], the second term represents the centripetal force of the navigation frame, which rotates about the earth. The third term
is the coloris acceleration due to the interference of the  and . The coloris acceleration is when the rocket experiences a relative velocity to the
navigation frame, while the navigation frame rotates itself.
    Substituting the data obtained from the FDR module, which are ωx, ωy, ωz, ax, ay and az during the flight into the equation [24], data of
velocity is obtained.
    A compensated with the data of frequency monitored from the ground base, the dynamic data is applied in the following Doppler-
cancelling system.
    The doppler shift is given by equation [25]



    Theta is given by



    In the former process, the dynamic data of each sample is recorded with its corresponding frequency monitored.
    Thus doppler shift effect of  the downlink signal eliminated. Now this signal is sipposed to be given by the following equation



    This is the pure relativistic shift of the downlink signal.

    b) Comparison group

    The data of the transmitter on the payload, was real time recorded, which is linearly related to the time standard on the payload. As the
payload experiences a relative velocity to the navigation frame, the clock effect is considered. The following cancellation applies to the special
Relativity Conception. For the time base, the original time interval between two pulses tm-1 and tm is △    t.



    The data of the time base experienced former processing with the dynamic data, thus the actual transmitted signal is calculated from the
corrected time base in the computer. This signal is labeled as fair. Fair is processed to predicted relativistic shift.

实验误差分析

    During the experiment, a series of error resources were introduced, which fall into two categories: the errors generated in the data acquisition process and Data processing progress.
    
Error of data acquisition

       1) Time standard
    
    a)Difference of the time standards

    During the experiment, two sets of time standards were applied: one on the payload and the other in the ground base. The one on the payload generates the downlink signal and the comparison group. The one in the ground base is used in the first comparison with the downlink signal to generate the signal f1 - f0.

    So the error caused by the difference between the two time standards were only introduced in the first order comparison.
    During the pre-launch testing, the frequency of two time standards were adjusted. One of the zero beat was shown in figure .

    b ) During the light time of downlink signal

          i) Change in frequency of signal

    When the EM wave travels in the troposphere(which covers the space within an altitude of 12 kilometres while the apogee of the rocket is within 2 kilometres), the velocity of light, c, is not a constant but a vector. It varies with graphic parameters.



    Where n is the refractive index of the troposphere is given by equation [54]



    Where T is the absolute temperature, p is the absolute pressure, ew is the absolute humidity(in terms of hPa)
Equation [53] is substituted into each sample, after this correction,  , is 5%.

ii) Path variation

    The propagation vector in equation [1],  is assumed to be a straight line. However, due to the nonuniform variation of air particles, refraction take place. So the real propagation path of the downlink signal is a curve rater than a straight line. The actual path is given below.



    Where rm is the displacement of the rocket from the centre of the earth, which is given
by  . Where  is the transformation matrix that map the navigation frame to the earth frame.
    


    Where L and  are the longitude and latitude of the launch site.


    
    After correction, the error in path determination can be ignored.

Error of frequency estimation

    The received signal was discretized and the frequency of it was calculated though algorithms by digital circuits. During this process, errors are introduced. In this section, the error introduced by the frequency estimation algorithm will be discussed.
    Assume the actual value of four parameters of the sample signal is ω0, A0, B0  and C0, the sampled value of the signal at time tk is y(k), the RSS of them is given by
    
△ω = ω - ω0
△A = A - A0
△B = B - B0
△C = C - C0
[60]

    The difference in frequency between the estimation value and actual value is r(k) at time tk. The value of r(k) is given by


    Substitute the equation [64] to the equation [63], equation [65] is obtained. This equation gives the error of this algorithm.
    


    In this experiment, the sample rate was 120 million samples/s. The frequency was calculated per 10 second. The SNR of the signal was 50dB+. As a result, the accuracy of this algorithm reached a level of 1×10-12 Hz.


6. Conclusion


    An experimental verification of the gravitational blueshift has been successfully achieved as a test of the general relativity conception.






---------------------------------------------------------------------------------------------


哦也发完收工

93大哥都玩到这份上了。。。我真感到自卑了。。。唯一欣慰的是能看懂帖子。。。

93啊，kc多扶扶低端火箭啊，别搞得像中国一样，高端科技和低端普及成了鲜明的对比

膜拜。这是神一级别的任务……除了专业名词外还凑合看得懂

专业的模型和数据处理

1. Abstract
    An experimental verification of general relativity conception was made using an atomic maser in a rocket attaining an altitude of 1,800 metres.
The signal of the maser was monitored on the ground, so that the effect of gravitational potential on the frequency of the maser was measured.
The resulting data was processed through a careful prediction and elimination of the Doppler shift and other error resources, so that the
gravitational blue shift is directly observed. The experiment is described including a consummate discussion of navigation algorithm applied in the
processing procedure. The authors believe that this is a direct high accuracy test of the general relativistic phenomena using an airborne clock.

XXXXtroduction
    A rocket is constructed and launched carrying an atomic frequency standard as the payload. The frequency of the signal from the atomic
frequency standard is examined on the ground.
    Another atomic frequency standard is used as a comparison to monitor the change in frequency of the received signal from the payload.
A group of variables that will influence the change are eliminated so that the resulting data, representing the relativistic shifts, are recovered and

recorded.
    The objective of the experiment is to test general relativity concept by measuring directly the effect of gravitational potential on the

frequency of a proper clock, in this case the atomic frequency standard.
    In this experiment, a gravitational effect amounting to 5.6e-13 was measured.



    The predicted proportion change in frequency is expressed in equation [a].Where β is the velocity/c and r is the displacement of the rocket
relative to theground base. ais the centrifugal acceleration of the ground station while ε represents thepropagation vector of the rocket-to-ground
signal. Our knowledge of the relative velocityand displacement of the rocket is obtained from the flight data recorder installed in thepayload. As
we chose the ground base as the navigation frame, the movement of the groundbase in the geographic frame was straightly eliminated in the
navigation system.
    In equation [1], the first term is the gravitational blue shift, the second term expressesthe Doppler shift. The last term describes the effect of the
rotation of the earth during thepropagation of the signal. In the elimination of the second and third terms, out knowledge ofthe rocket's velocity
and position are obtained from the FDR(Flight Data Recorder) while theknowledge of the velocity and position of the ground base is gained from
the Earth Modeland GLarLng of the launching site.
    The specific procedure is exhibited in the materials and methods section.
    In equation [1], the first term is the gravitational blue shift, the second term expresses the Doppler shift. The last term describes the
effect of the rotation of the earth during the propagation of the signal. In the elimination of the second and third terms, out knowledge of the
rocket's velocity and position are obtained from the FDR(Flight Data Recorder) while the knowledge of the velocity and position of the ground
base is gained from the Earth Model and GLarLng of the launching site.
    The specific procedure is exhibited in the materials and methods section.

3. Materials and Methods

    The data was first generated and collected after the hardware processing of ground base, which procedure is labeled as data
acquisition. The recorded data was then stored in the computer for further processing, labeled as Data processing.
Data acquisition
  a) Algebra description
    The frequency signal transmitted from the payload is fixed at f0 + △f.
    △f was set to be 50mHz (Figure [d]).
    The signal received on the ground base is labeled as f1. It is predicted to be



    after the elimination of errors.
    The ground station processed f1 with a standard signal of frequency f0 , which was generated by the atomic frequency standard.
Heterodyne-beat method was applied thus a signal with frequency f1 - f0  was sampled by a high speed analog to digital converter as f2 and
processed by the digital signal processor. A standard signal of frequency△f is generated by atomic frequency standard and processed in the
computer, a series of effects including doppler shift are taken into account.


  

    The final signal, labeled △fR, is expressed by equation [2].
    The frequency of f2 was estimated by 4-parameter estimation algorithm and recorded in the computer. The method of 4-parameter
estimation will be discussed later.
    
    Therefore, the resulting data was obtained. The predicted behaviour of this final signal is shown in figure [6].

  b) Technical details
    A transmitter and a superheterodyne receiver were built specially for the experiment.
    The signal from the on-board AFS(Atomic Frequency Standard) was directly amplified and transmitted. The output power of the amplifier
was +38dBm and the frequency was 63.8978MHz.
    The structure of the receiver is shown below.

  

    The signal from the antenna at the ground base was filtered and directly amplified by a LNA(Low Noise Amplifier) and mixed with FS1 by
mixer 1 to obtain an IF(Intermediate Frequency) with a frequency of 5MHz. The IF signal was processed by an AGC(Automatic Gain Control) circuit
so that the amplitude was stabilized. This processed IF signal was mixed with FS2 by mixer 2 to obtain a 50mHz signal and was converted into
digital signal by a high speed ADC. The exact frequency of the signal sampled by the ADC was estimated through 4-parameter estimation
algorithm. FS1 and FS2 were generated by the AFS at the ground base.

    1) The realization of heterodyne-beat method
    The kernel of heterodyne-beat method is the shift of spectrum. As a result, a mixer is used to obtain the difference in frequency of two
signals. In this application, two integrated mixer circuits AD831 were used.
    The following images indicate a test result for the mixer circuit. The left image indicates the signal of RF input of the mixer and another
one shows the local oscillation input of the mixer. They were generated by two DDS(Direct Digital Synthesis) circuits. The clock standard of the DDS
circuits was connected to an AFS. The frequencies of them were 7MHz and 7.00001MHz.


  

    The following image shows the output of the mixer which equals to the difference between two frequency signals connected to the local

oscillation input and the RF input. Here, the difference was 10Hz.

  


    2) The design of AGC circuit

    An AGC circuit was used to stabilize the amplitude of IF signal.
    The input signal was demodulated and filtered into a voltage signal represents the strength of the signal. This voltage signal was used to

control the gain of a VGA(Variable Gain Amplifier).

  

    In this case, the wave-detector was AD8307 and the VGA was AD603.
    A test for the AGC module is shown below. Channel 1 was connected to the output of the AGC module, channel 2 was connected to the
input. The input of the AGC module was connected to a function generator.

      
    
    As the waveform shown on the oscilloscope, although the amplitude of the input signal to the AGC was changed, the amplitude of output
signal of it remained the same.
    A test result for the transmitter and the receiver is shown below.
    The transmitter was placed 800 metres away from the receiver. The image on the left shows the output signal from the transmitter and
another image shows the signal sampled from the output of LNA of the receiver.

  

    The image below shows the signal sampled from the output of the AGC module, where frequency = IF = 5MHz.

  
2) Method of estimation of frequency
   a) Definition of 4 parameters
Assume the sampled signal S(t) is given by
    
    Where A0' is the ideal amplitude of the signal, ω0' is the ideal frequency of the signal, C0' is the ideal DC offset of the signal and θ0' is
the ideal phase of the signal.
    The signal can be expressed by the equation
    

    Where
    
    Suppose the magnitude of the signal sampled during time tk(k = 0, 1, 2, ...) is y(k), is given by
  
    b) Method of 3-parameter estimation
    Suppose the sampled voltage value of the signal at time tk is y(k), k = 1, 2, 3 ...N-1. The amplitude of sine, amplitude of cosine and DC
offset is defined as A,B and C. The RSS(Residual Sum of Squares) between the estimation value and actual value is given by
  
    Where N is the length of samples, set
  
    The solution for X is given by the least square solution below:
.  
c) 4-parameter estimation algorithm
    The idea of successive approximation is applied in this algorithm. First, a rough frequency is given, 3-parameter estimation algorithm is
applied to the sampled signal. The cosine amplitude, sine amplitude, DC offset and estimated RSS are obtained. The operation is repeated with
different frequencies so a serial of estimated RSS are obtained. One of those set of obtained result with minimum estimated RSS is the value of
actual frequency. The detailed steps are shown below.
    1) Determine the frequency of the signal roughly though DFT(Discrete Fourier Transform), label this frequency as f0.
    2) Set the domain of iteration to be ωdl and ωdu, where ωdl is the lower boundary, given by ωdl = f0 - fclk / N. ωdu is the upper
boundary, given by ωdu = f0 + fclk / N. fclk is the frequency of sampling clock and N is the length of DFT.
    3) Set ω0 = ωdu - ωdl. 2M+1 points (M∈N*) are samples between ωdl and ωdu with equal intervals. 3-parameter estimation algorithm
is used here to compute the RSS of this group of samples.
    4) Find and record the minimum value of RSS of samples in step 3 This minimum value is corresponding to the actual frequency.
    Repeat operations 2 to 4 until the precision of the estimation reaches the required level.
    A picture of the receiver is shown below.



2) Data processing

  a) Downlink signal

    The navigation system provided data of dynamics with a sample rate of 1,200 samples per second. The data was given in terms of
angular velocity and acceleration in the on-board coordinate. The six groups of parameters are ωx, ωy, ωz, ax, ay and a--z, respectively where the
X-axis if the mean axis of the rocket.
    In the determination of change in angle, △θ and change in velocity, △V, the cubic spline function is adopted in curve fitting before
integration. This method of Simpson's rule provides six groups of data: △θx, △θy, △θz, △Vx, △Vy and △Vz. As a result, the behaviour of the rocket
between samples are predicted and considered.

    In the determination of attitude angle, the method of Quaternion is applied.
    The quaternion numbers at time tm+1 are given in equation [9], where △θx, △θy and △θz are the output of change in angle and vector Φ

is the rotation vector, which is given by equation [9]
    
    

    In equation [9] and [10], the angular velocity of the rocket is assumed to fit cubic function. However,  the actual angular velocity does not

fit a cubic function.
    Equations [9] and [10] do not achieve minimum shift of algorithm. After the parachute deployment, the rocket was suspended in the
descending stage. Thus the rocket is likely to experience coning motion, which means that the rocket vibrate about the equilibrium position at
small angles. The coning motion is the worst working environment for the SINS(Strapdown Inertial Navigation System) as it will cause serve shift of
the Math Platform.

For optimization algorithms, the following improvements are made.
    O-XYZ represents the reference frame R, which is the on-board frame when the rocket is in equilibrium.
    
    Let b(tm-1) and b(tm) to be the instantaneous on-board frame at time tm-1 and tm.
    According to Euler Theorem, O-XYZ can be regarded as a rotating transformation of b(tm) or b(tm-1) with rotating vector Q(t) given in

equation [11].
    
  

    The shift on-board frame can be regarded as a rotation transformation of the ideal on-board frame, which is the frame when the rocket

is in equilibrium. The samples of △θ is grouped in three again.
    In each group, the samples are labeled as .
    
  

    Equation [12] is an improved  form of equation [3]. By selecting proper constant k1 and k2, the effect of the coning motion is minimised.
Here, the ideal values for k1 and k2 are 0.45 and 0.675.
    Therefore, the attitude of the rocket is found through the optimized quaternion algorithm. The Eular angles are found by equation [13].

    

    The quaternions are supposed to be standardized. However, resulting form calculation errors and other factors, the quaternion numbers
gradually loses standability. The standardization of quaternion numbers is applied at the end of each period of attitude refreshment. The formula
for standardization is given by equation [8].

    

    Where  is the standardization value and  is the value after attitude refreshment.
    So far, the discussion of the rocket's dynamic is in the on-board coordinate. However, the final results have to be expressed in the
navigation frame, which sets the ground base as the origin.
    
  

Equation [15] gives the coordinate transformation matrix (attitude matrix) from on-board frame to navigation frame in terms of quaternion.
    The initial extraction quaternion numbers are thus given by the initial attitude matrix obtained in initial azimuth alignment.

  

    The velocity of the rocket at time tm in the navigation frame, Vm, is given by equation [16], containing a series of error compensations.
Vm-1 is the velocity in the same frame at time
tm-1. Cm-1 is the coordinate transformation matrix at time tm-1. Vm-1 is the compensation velocity caused by while△Vg/corm is the

compensation velocity caused by the deleterious acceleration. △Vsfm is the compensation velocity caused by ecific force.
    
  

    Where△Vm us the change in velocity during period [tm-1, tm].
    △Vrotm is the compensation velocity caused by rotation effect.
    △Vsculm is the compensation velocity caused by sculling motion.

    Due to air current, gustiness and other factors, the rocket experiences vibrations during the flight. Those factors cause a highly dynamic

working environment for the payload. Therefore, the velocity has to be compensated so that the sculling effect and the rotation effect are
eliminated. Otherwise, the calculation of velocity will involve serve errors. When it comes to position determination, there two error resources
contribute to scroll errors. Here△Vrotm and△Vsculm represent the compensation velocities due to the rotation effect and the sculling effect
respectively. the rotation effect happens when the direction of linear velocity rotates in a three-dimensional coordinate. The sculling effect is
caused by the angular vibration and linear vibration are in phase and of same frequency on the rocket. This is quite similar to the sculling motion:
on one hand, the syrup vibrates periodically about the lateral axis of the boat. On the other hand, the boat forges ahead along the direct-axis in
an intermittent behaviour.

    The original expression for △Vrotm and △Vsculm are:
    
  


    The optimized formula for △Vrotm and △Vsculm are

  


    The optimized algorithm for sculling effect rotation effect, in velocity determination as well as the correction for conning motion in
attitude determination make sure that the motion of the layload is precisely calculated in spite of the unstable motion of the rocket. Hence the
cancelling of doppler effect and second-order general relativity conception shift are more reliable. The specific precision level is related in the
discussion section.

The final expression for Vm is given by equation [22]:

  

    Considering that all the compensation dosages have been taken into account the calculation of Vm and that the data is discrete with
equal time internals, the data of displacement, is obtained through numerical integration.
    As the dynamic data is determined, the following equations are substituted into
    equation [2].
    
  

    Where c is the velocity of light, Wen is the angular velocity of the earth in the navigation frame and  is the radius vector of the earth at
the launching spot.
    The frequency of this signal is plotted against time in Figure [5].
    The navigation algorithm aims to calculate the velocity of the rocket in the navigation frame. However, the accelerometer does not tell
deleterious acceleration and relative acceleration of the rocket. Therefore, the compensation velocity has to be estimated from the measure value.



    Where g is the gravitational acceleration of the launch site.
     In equation [24], the second term represents the centripetal force of the navigation frame, which rotates about the earth. The third term
is the coloris acceleration due to the interference of the  and . The coloris acceleration is when the rocket experiences a relative velocity to the
navigation frame, while the navigation frame rotates itself.
    Substituting the data obtained from the FDR module, which are ωx, ωy, ωz, ax, ay and az during the flight into the equation [24], data of
velocity is obtained.
    A compensated with the data of frequency monitored from the ground base, the dynamic data is applied in the following Doppler-
cancelling system.
    The doppler shift is given by equation [25]



    Theta is given by



    In the former process, the dynamic data of each sample is recorded with its corresponding frequency monitored.
    Thus doppler shift effect of  the downlink signal eliminated. Now this signal is sipposed to be given by the following equation



    This is the pure relativistic shift of the downlink signal.

    b) Comparison group

    The data of the transmitter on the payload, was real time recorded, which is linearly related to the time standard on the payload. As the
payload experiences a relative velocity to the navigation frame, the clock effect is considered. The following cancellation applies to the special
Relativity Conception. For the time base, the original time interval between two pulses tm-1 and tm is △    t.



    The data of the time base experienced former processing with the dynamic data, thus the actual transmitted signal is calculated from the
corrected time base in the computer. This signal is labeled as fair. Fair is processed to predicted relativistic shift.






//=============================GOOGLE==============翻译


1。摘要
    实验验证了广义相对论的概念的一个使用原子激射器在达到海拔1800米的火箭。
脉泽的信号在地面上进行了监测，使测量脉泽频率的引力势的影响。
通过认真的预测和消除多普勒频移和其他错误的资源，所以由此产生的数据处理
直接观测到引力蓝移。实验说明，包括精湛的导航算法讨论中的应用
处理过程。作者认为，这是直接使用机载时钟广义相对论现象的高精确度测试。

XXXXtroduction
    建造和发射火箭是携带有效载荷原子频率标准。从原子的信号频率
频率标准，是研究在地面上。
    另一个原子频率标准是作为一个比较，监测有效载荷从接收到的信号频率的变化。
一组变量的变化将影响被淘汰，因此产生的数据，代表相对论的转变，是恢复和

记录。
    这项实验的目的是测试广义相对论的概念，通过直接测量引力势的影响

合适的时钟，在这种情况下，原子频率标准的频率。
    在这个实验中，引力的影响金额为5.6e - 13进行了测量。



    预测在频率比例的变化是在方程表示[A]。β是速度​​/ C和R是火箭的位移
相对theground基地。 AIS地面站的离心加速度，而ε代表thepropagation火箭地面载体
信号。我们对火箭的相对velocityand位移的知识是从飞行数据记录仪安装在thepayload。至于
我们选择了地面导航框架，在地理框架groundbase运动是直线在淘汰
导航系统。
    在方程[1]，第一项是引力的蓝移，第二届expressesthe多普勒频移。最后一项描述的效果
地球自转的信号thepropagation期间。在消除第二个和第三个条件，出知识ofthe火箭的速度
和立场，得到的FDR（飞行数据记录器）而获得了从地面基地的速度和位置theknowledge
地球Modeland GLarLng发射场。
    具体程序是展出的材料和方法“一节。
    在方程[1]，第一项是引力蓝移，第二项表示的多普勒频移。最后的术语描述
地球自转的影响，在信号的传播。消除在第二个和第三个条件，走出了知识
火箭的速度和位置是从FDR（飞行数据记录器），而获得知识的速度和地面的位置
基地是从地球模型和发射场GLarLng获得。
    具体程序是展出的材料和方法“一节。

3。材料和方法

    首先，数据生成和地面基地的硬件处理程序标记为数据收集后
收购。记录的数据，然后存储在计算机作进一步处理，数据处理标记。
数据采集
  A）代数描述
    从有效载荷传输信号频率是固定在F0 +△F.
    △f被设置为50MHZ（图[D]）。
    地面基座上接收到的信号标记为F1。据预测，到



    后消除的错误。
    地面站进行处理，这是由原子频率标准生成一个标准信号频率f0 F1。
外差拍方法应用于信号频率f1 - F0，F2作为一个高速模拟 - 数字转换器采样和
处理数字信号处理器。一个标准信号的频率△f是所产生的原子频率标准，并在处理
考虑到计算机，一系列包括多普勒频移的影响。


    

    最终信号，标有△FR，表示方程[2]。
    f2的频率估计4参数估计算法，并在计算机记录。 4参数的方法
估计将在后面讨论。
    
    因此，产生的数据。在这最后的信号预测的行为是在图所示[6]。

  B）的技术细节
    一个发射器和一个超外差接收机内置专门的实验。
    从主板上的AFS（原子频率标准）的信号直接放大和传输。放大器的输出功率
为38 dBm的频率为63.8978MHz。
    接收器的结构如下所示。

    

    从天线在地面基地信号过滤和直接扩增由一个LNA（低噪声放大器）和混合FS1
搅拌机1获得中频（IF）频率为5MHz。中频信号处理AGC（自动增益控制）电路
使幅度稳定。这处理中频信号混合搅拌机2与FS2获得一个50MHz的信号，并转换成
高速ADC的数字信号。通过4参数估计的ADC采样信号的确切频率估计
算法。 FS1和FS2产生AFS在地面基地。

    1）实现差拍法
    差拍方法的核心是频谱的转变。因此，混频器是用来获取在两个不同的频率
信号。在此应用中，两个集成混频器电路AD831。
    下面的图像显示了混频器电路的测试结果。左边的图像显示的混频器的RF输入和其他信号
一个显示混频器的本振输入。他们所产生的两个DDS（直接数字合成）电路。 DDS的时钟标准
电路连接到一个AFS。他们的频率分别为7MHz和7.00001MHz。


    

    下面的图像显示混频器的输出等于两个频率信号连接到本地之间的差异

振荡输入和RF输入。在这里，不同的是为10Hz。

  


    2）AGC电路的设计

    的AGC电路被用来稳定中频信号的振幅。
    输入信号转换成电压信号的解调和过滤信号强度。这个电压信号被用来

控制增益一个VGA（可变增益放大器）。

    

    在这种情况下，波检测AD8307 VGA的AD603。
    如下所示的AGC模块的测试。通道1连接到输出的AGC模块，通道2被连接到
输入。的AGC模块的输入连接到函数发生器。

      
    
    由于波形显示在示波器上，虽然被改变输入信号的幅度的AGC，输出幅度
它的信号保持不变。
    下面是一个发射器和接收器的测试结果。
    从接收器，发射器放置在800米外。左边的图像显示从发射机的输出信号，并
另一幅图像显示从LNA的接收器的输出采样的信号。

  

    下面的图像显示信号采样的AGC模块的输出，频率= IF = 5MHz的。

  
2）的频率估计方法
   a）定义的4个参数
假设采样信号s（t）由下式给出
    
    其中A0是理想的信号振幅，ω0“是理想的信号频率，C0”是理想的直流偏移的信号和θ0“
信号的理想阶段。
    信号可以表示方程
    

    在哪里
    
    假设在时间TK（k = 0，1，2，...）为Y（K），采样的信号的幅度是由
  
    b）方法3参数估计
    假设时间TK信号采样的电压值是Y（K），K = 1，2，3，...，N - 1。振幅的正弦，余弦和DC的幅度
偏移量是定义为A，B和C.估计值与实际值之间的RSS（残差平方和）由下式给出
  
    其中N为样本的长度，设置
  
    X的解决方案是由最小二乘下面的解决方案：
。
C）4参数估计算法
    逐次逼近的想法是在这个算法中应用。首先，一个粗略的频率是3参数估计算法
适用于采样信号。余弦幅度，正弦幅度，直流偏移和估计RSS获得。反复操作
所以估计RSS序列的不同频率。这些所得与最低估计RSS结果之一是价值
实际频率。具体步骤如下所示。
    1）确定信号的频率大约尽管DFT（离散傅立叶变换），标签频率为F0。
    2）设置域迭代ωdl和ωdu，其中ωdl下边界，ωdl= F0 - FCLK / Nωdu上层
边界，给予ωdu= F0 + FCLK / N FCLK是采样时钟频率，N为DFT的长度。
    3）设置ω0=ωdu - ωdl。 2M 1点（M∈N *）是相等的间隔之间ωdl和ωdu样本。 3，参数估计算法
这里使用的是计算这组样品的RSS。
    4）查找并记录在步骤3中的RSS样品的最低值，最低值是对应的实际频率。
    2至4重复操作，直到估计的精度达到所要求的水平。
    接收图片所示。



2）数据处理

  A）下行信号

    导航系统提供了每秒1200样本的采样率的动态数据。在数据
角速度和加速度在主板上的协调。参数的六组ωx，ωy，ωz，AX，AY和A - Z，分别
X轴，如果火箭的平均轴。
    在角度变化的决心，△θ和改变速度，△V，三次样条函数曲线拟合前通过
整合。辛普森法则的方法，提供了6组数据：△θx，△θy，△θz，△VX，△VY和△的Vz。因此，火箭的行为
样本之间的预测和考虑。

    在测定的姿态角，四元数的方法是应用。
    时间tm +1四元数方程[9]，其中△θx，△θy和△θz改变角度和向量的输出Φ

旋转向量，这是由公式[9]
    
    

    在方程[9]和[10]，假设火箭角速度，以适应立方米功能。然而，实际的角速度不

适合立方米功能。
    方程[9]和[10]没有达到最小频移算法。降落伞后，火箭在暂停
降阶段。因此，火箭很可能经验圆锥的议案，这意味着火箭振动的平衡位置
小角度。圆锥的议案罪（捷联惯导系统）最糟糕的工作环境，因为它会导致服务转移
数学平台。

优化算法，进行了以下改进。
    O - XYZ代表参考架R，当火箭处于平衡状态，这是主板上的框架。
    
    让B（TM - 1）和B（TM）是瞬时板TM - 1和TM时间框架。
    根据欧拉定理，可视为一个旋转变换旋转向量q（T），B（TM）或b（TM - 1 O - XYZ）

方程[11]。
    
    

    当火箭板上的转变可视为一个理想的旋转变换板上的帧，这是帧帧

处于平衡状态。样品的△θ是再次分为三个。
    在各组中，样品标示为。
    
    

    方程[12]是一个改进形式方程[3]。通过选择适当的常数K1和K2的圆锥运动的影响降到最低。
在这里，K1和K2的理想值是0.45和0.675。
    因此，火箭的态度是通过优化的四元数算法。欧拉角方程[13]。

    

    四元是应该加以规范。然而，造成形式计算错误等因素影响，四元数号码
逐渐失去standability。四元数的标准化应用在各个时期态度小食的结束。计算公式
标准化是由方程[8]。

      

    哪里是标准化的价值和态度茶点后的价值。
    到目前为止，火箭的动态的讨论是在主板上的协调。然而，最终的结果都表示要在
导航框架，它设置为原点的地面基地。
    
    

公式[15]给出了从板上帧帧在四元数的导航坐标变换矩阵（的态度矩阵）。
    从而初步提取四元数由最初的态度在初始方位对准中获得的矩阵。

    

    时间在导航框架，VM TM火箭的速度，是由方程[16]，包含了一系列的错误补偿。
VM - 1是在同一时间帧的速度
TM - 1。 CM - 1在时间TM - 1的坐标变换矩阵。 VM - 1△VG /球茎是造成补偿速度

补偿的速度造成的有害加速度。 △Vsfm ecific力所造成的赔偿速度。
    
    

    △Vm的我们在期内的变化速度[TM - 1，TM]。
    △Vrotm补偿速度造成旋转的效果。
    △Vsculm sculling议案引起的补偿速度。

    由于气流，阵风和其他因素的影响，火箭的经验，在飞行过程中的震动。这些因素导致一个高度动态

有效载荷的工作环境。因此，速度已得到应有的补偿，使sculling效果和旋转效果
淘汰。否则，计算速度将涉及服务的错误。当涉及到位置的确定，有两种错误资源
滚动错误作出贡献。这里的△Vrotm和△Vsculm代表由于旋转效果和补偿速度sculling效果
。旋转效应发生时的线速度方向旋转一个三维坐标。 sculling效果
角振动和直线振动造成火箭在相位和频率相同。这是颇为相似sculling议案：
一方面，糖浆振动定期船的横向轴。另一方面，锐意进取的船沿直轴
间歇性的行为。

    △Vrotm和△Vsculm的原始表达式是：
    
  


    △Vrotm和△Vsculm的优化公式

  


    sculling效果旋转效果，速度决心以及精读的议案，修正优化算法
态度，决心确保layload的议案火箭不稳定的议案，但恰恰是在计算。因此，
取消多普勒效应和二阶广义相对论的概念转变，更可靠。在具体的精度水平相关
讨论部分。

为VM的最后一个表达式是由公式[22]：

  

    鉴于已考虑到所有的补偿剂量的VM的计算和数据是离散的
相等的时间内部，流离失所的数据，通过数值积分获得。
    由于动态的数据是确定的，下面的公式代入
    方程[2]。
    
  

    其中，C为光速，温家宝是在导航框架角速度和地球是地球半径向量
发射现场。
    这个信号的频率与时间绘制图[5]。
    导航算法的目的是计算速度的火箭在导航框架。然而，加速度不告诉
有害加速度和相对加速度的火箭。因此，补偿的速度，从测量值估计。



    其中，g是重力加速度发射场。
     在方程[24]，第二项代表的导航框架，其中约地球旋转的向心力。第三个任期
coloris加速度的干扰。当火箭经历了相对速度coloris加速
导航框架，而旋转导航框架本身。
    代从罗斯福模块获得的数据，这些数据在入方程[24]航班ωx，ωy，ωz，AX，AY，AZ，数据
速度获得。
    一个从地面基地监测的频率数据补偿，动态的数据是应用在以下多普勒
取消系统。
    多普勒频移方程[25]



    西塔是由



    在前者的过程中，每个样品的动态数据记录其相应的监测频率。
    因此，下行信号的多普勒转移效应淘汰。现在由下式给出这个信号是sipposed



    这是纯粹的下行信号的相对论​​的转变。

    b）比较组

    上的有效载荷发射的数据，实时记录，这是线性相关的有效载荷的时间标准。由于
有效载荷的经验，一个导航框架的相对速度，时钟效果。以下取消适用于特殊
相对论的构想。时基，两个脉冲之间的TM - 1和TM的原始的时间间隔△T.



    经历时基数据与动态数据的前处理，从而计算实际传输的信号是从
纠正在计算机的时间基准。这个信号被标记为公平。博览会是处理预测相对论转变。

引用第8楼jrcsh于2011-11-05 21:59发表的  :
1. Abstract
    An experimental verification of general relativity conception was made using an atomic maser in a rocket attaining an altitude of 1,800 metres.
The signal of the maser was monitored on the ground, so that the effect of gravitational potential on the frequency of the maser was measured.
The resulting data was processed through a careful prediction and elimination of the Doppler shift and other error resources, so that the
gravitational blue shift is directly observed. The experiment is described including a consummate discussion of navigation algorithm applied in the
.......

google太威武了。。。我的AD831变成搅拌机了Orz

引用第4楼qharryq于2011-11-05 19:16发表的  :
93大哥都玩到这份上了。。。我真感到自卑了。。。唯一欣慰的是能看懂帖子。。。

我是看不懂 你看得懂？

干得漂亮，赞一个

温家宝都成发射现场了- -。。。。。。

少了一张figure d，补上。。。 = -

太厉害了。

实验误差进阶分析..

其实我代表个人觉得英文不好又不想学英语的还是不要混神马科技爱好了

初始对准的误差，和后期的噪声以及漂移相比，不在一个数量级上。
对于200RMB的倾角传感器来说，角度误差很容易做到小于0.05度。
而ADIS16365这种接近5k的家伙，一小时的随机角度游走，差不多到20deg/rt(hr)
以前是陀螺角速度推算经纬度，当然会有很多问题。现在有GPS完全无压力。
不要搞了这么久，最大的误差项没有消灭，其他的折腾了半天

提醒一下，
其实惯性导航的算法现在非常成熟了，各方研究焦点还是集中在器件以及与GPS的紧组合上。
KCSA目前计划山寨光纤陀螺，就是基于这个考虑。不过光纤陀螺问题很多，还得慢慢折腾才行

引用第17楼warmonkey于2011-11-16 19:25发表的  :
初始对准的误差，和后期的噪声以及漂移相比，不在一个数量级上。
对于200RMB的倾角传感器来说，角度误差很容易做到小于0.05度。
而ADIS16365这种接近5k的家伙，一小时的随机角度游走，差不多到20deg/rt(hr)
以前是陀螺角速度推算经纬度，当然会有很多问题。现在有GPS完全无压力。
不要搞了这么久，最大的误差项没有消灭，其他的折腾了半天
.......

整个15楼都在分析导航误差

上升阶段导航过程小于10秒，数据用来消除狭义相对论，下降阶段为GPS/INS组合解算位置，0楼有提到
ADIS16365提供1200sa/s的动态数据，3子样解算后速率120sa/s，计算出火箭相对于地面基站的速度矢量，消除多普勒效应，1楼有提到
解算出预测多普勒频偏及其他误差以后降采样到0.1Hz，其他误差包括Path variation、Propagation in speed of signal等 见3楼。
地面用4参数估计法解算频率，10s更新一次，0.1Hz，见2楼

最后计算结果见2楼，实验数据吻合广义相对论


你们继续搞“最高科技含量”的火箭[s:274]

路过~~~............