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.






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