3.1 Phase-difference measurements between a roundtrip signal without carrier recovery and return signal in a satellite
A block diagram of the system depicting a simple roundtrip method without carrier recovery is shown in Fig. 2. The delay model generated by this system is shown in Fig. 3.
It is assumed that all of the frequencies transmitted from a satellite are produced by the same frequency standard, and the frequency is assumed to be
, with phase
. Each frequency signal that is transmitted from the antenna of the satellite has its own initial phase
(i = 1, 2, 3: indicating frequency). The phase of the signal transmitted from the satellite is given by
where
is the delay from the frequency standard to the frequency synthesizer for
is the delay from the synthesizer to the antenna of the satellite;
correlates with the initial phase of the frequency standard of the satellite. This standard is dependent on how the transmitting signal is produced, and it is difficult to accurately judge the phase. For this reason,
is taken as an unknown constant, and
is the phase variation of a carrier signal generated by the frequency standard. The variation is almost nullified in the difference between the roundtrip and return signals, although the phase variation within the period of the roundtrip time persists. This phase variation is discussed later in the article.
The transmitted signals from the antenna of the satellite reach a landing unit after a propagation time
Taking into account the delay
from the antenna of lander 1 to the mixer, the phase of the signal at the mixer is derived by
In the mixer, the frequency signal
with initial phase
is mixed with the received signal, and the frequency changes to
Mixing the frequency signal
with the received signal at lander 1 is the simplest method for reserving the initial phase
in the received signal at the satellite. The phase variation of the frequency standard of lander 1 is included in
The phase of the transmitted signal from the antenna of lander 1 is obtained by
where
is the delay from the frequency standard to the mixer of lander 1, and
is the delay from the mixer to the antenna in lander 1.
The transmitted signals from lander 1 reach the satellite after a propagation time of
The
is slightly different from the
because the satellite and lander 1 move during the propagation time
However, the difference is very small and can be easily corrected by the rough predicted orbit of the satellite, such that
is presumed to be equal to
here. The phase of the received signal at the antenna of the satellite is expressed as
The signal received at the satellite is called the roundtrip signal. Taking into account the delay
from the antenna to the mixer of the satellite, the phase of the roundtrip signal can be expressed as,
The signal which returns directly from the antenna in the satellite is named the return signal and reaches the mixer after a delay
The phase of the return signal in the satellite at the input of the mixer is expressed as
where
is the phase variation of the carrier signal and is caused by the instability of the frequency standard in the satellite. This is not the same as the
because
is the phase variation that is delayed by the roundtrip time.
When the return signal is mixed with the local frequency standard
a phase can be obtained as
where the phase variation caused by the instability of the frequency standard in the satellite is included in
(the phase variation of
is very small compared with
), and
is the delay from the frequency standard to the mixer in the satellite.
After mixing with the local frequency standard, the return signal is compared with the roundtrip signal in the satellite. The phase difference obtained as the result of phase comparison is
where
The frequency standard is usually designed such that
is small enough. The values
and
are minor and can be ignored as they are of no significance. The term of the
is also minimal and is neglected hereafter. For a long period, the phase difference of
may drift to a certain extent, and the phase value of
cannot be omitted. Further, we provide a method for the estimation of this value. By using the estimated phase value, the differential phase value can be modified to remove the phase caused by
Substituting
to
can be rewritten as
The phase of
is proportional to the lower frequency
instead of the RF (radio frequency).
Finally, the measured phase difference can be obtained as
where,
.
Assuming that the
is small and can be measured separately and corrected prior to the launch, Eq. (8) can be expressed as,
is different for the variable frequency
, but remains constant as long as the temperature of this system and the voltage of power supply do not vary. The entire delay would be of a few nanoseconds.
For lander 2, Eq. (9) can also be derived; however, the delays for lander 2, which are
and
, are different from
and
and, therefore,
and
cannot be estimated separately from Eq. (9). In order to determine
, calibration on the ground is required.
There is a method, in principle, to determine the
separately, and this is called a phase calibrator. A pulse is inserted at the front end and produces many frequency components after a band-pass filter. The phases are measured by the tone signal that is produced by the pulse at near-signal frequencies, and the delay produced in the instruments can also be calculated. This instrument is, of course, kept at a constant temperature. However, in reality, it is rather difficult to determine
separately with high accuracy, although the instrumental total delay variation can be monitored to a certain extent. This could help predict the
separately.
3.2 Calibration of the instrumental delay on the ground by using a common frequency standard
We consider a method to calculate
in Eq. (8). If the phase with known
on the ground is measured before launch,
can be obtained. The standard frequency signal is often used for both units of the lander and the satellite. The delay of each cable supplying the frequency standard is, of course, well calibrated. Then,
can be considered a known constant,
the delay difference of the two cables, and
the frequency standard of the satellite and the lander. The
is zero because of the same frequency standard. Figure 4 depicts the concept of the calibration on the ground. The phases are measured from R = R1 to R2, which are known. Thereafter, a black line can be obtained from R1 to R2. The observed phase at R is
where, Ni is the number denoting cycle ambiguity. If this system were an instrument for measuring range, all values other than the first should be zero. The phase at R = 0 in the figure would be
. Then, the measured phase
without 2π ambiguity should be
Since
is known, the
in Eq. (10b) can be obtained at three frequencies, and each of
(i = 1, 2, 3) is obtained.
For example, if R is 50∼100 m, fi = 2.2 GHz, f0 = 40 MHz, and a phase measurement error is of a few degrees, the
can be estimated with an error of a few degrees from Eq. (10b). The measurements for i = 1 , 2, 3 are carried out at the same time. The phases at three frequencies are corrected by the observed
, and the three phases at these frequencies give the correct range.
On the other hand, each lander transmits the signal given by Eq. (3). The phase difference of the transmitted signals from the two landers is given by
The first term provides the range difference; the fourth term is obtained from Eq. (8) after determining the range; the second and third terms represent the phases caused by the delay difference in the landers. These terms should be well calibrated. The fourth term is rather small compared with the third one. In particular, the change of the second term with temperature is the main source of error. The characteristics with temperature should be obtained prior to this calculation.
Equation (10c) indicates that an original signal is generated at a satellite, and the signal arrives at the Earth through a lander. In order to correct the resultant delay from the satellite to the lander, precise ranging is conducted by the NBV (narrow-band VLBI). From this perspective, a frequency standard in a lander only plays the role of a simple frequency converter. The common signal transmitted from the satellite goes through a lander and arrives at the Earth. The delay (or phase) from the satellite to the lander is estimated separately and is corrected for the total delay of the path from the satellite to the lander.
A measurement test for
can be conducted after the calibration of the instrumental delay on the ground by using independent frequency standards. Although the measured phase
varies in this test measurement because
fluctuates with time, the correct distance R can be estimated after
has been corrected by using observed phases at three frequencies.
3.3 The effect of a phase variation of the frequency standard in the satellite
The signal fi is generated from the frequency standard of the satellite. The phase noise whuency standard of the satellite. The phase noise when
sat overlaps to the fi signal gives
where
is the Allan standard deviation. The phase noise is almost cancelled by the correlation of the roundtrip and return signals, but it remains for τ, which is almost equivalent to the roundtrip duration. If we consider the distance between the satellite and the lander to be 2000 km, and
, then the phase noise is estimated from Eq. (11), to be about 0.1° by taking the roundtrip time as τ.
Since
can be determined with an accuracy of about 10°, the accuracy of iVLBI depends on the stability of
Particular sources of error are phase changes of RF frequency devices and cables with changes in temperature. Several steps are undertaken in order to minimize these errors. These devices should be installed in a temperature-controlled box. Their characteristics at a specific temperature should be measured carefully on the ground prior to launch. After launch, the temperature should be monitored again. Further calibration can be done for large variations in temperature.
3.4 Phase difference between the signals received in the orbiter from the two landers
The phase difference between the signals received from both landers is obtained from Eq. (8) as
where
is the phase difference between the received signals from the two landers. The second term of Eq. (12) is the delay difference of the cables from the frequency standard to the mixer. This term can be taken as zero as long as the cables are of the same length. Therefore, the cables are to be maintained at same length to the extent possible during manufacturing. The phase difference can be calibrated on the ground, and a marked difference in the phase difference can be corrected. Then, Eq. (12) can be rewritten as
The effect of the delay and phase instability of the frequency standard in the satellite is cancelled in Eq. (12b). The first term was previously obtained by calibration on the ground. Equation (12b) suggests that the delay changes and phase instability of the frequency standard in the satellite do not directly affect the phase difference of the signals from the two landers.
3.5 A simple roundtrip system with carrier recovery
Equations (1) through to (12b) can be applied to the system with a carrier recovery in a lander. This system takes advantage of low power-distribution loss and low transmitting power, but it consists of rather complicated circuits. An example of a roundtrip system with carrier recovery is shown in Fig. 5.
This circuit produces a pure signal that has the same frequency and phase as the signal received by the antenna. There are two groups of output signals. As shown in Fig. 5, one group of signals is mixed with the standard frequency on the lander. This is called a mixed signal. The other group of signals is the transferred pure signal. The turnaround ratio is given by k/l, and this is called a transferred signal. From the transferred signal, the Doppler shift can be obtained on the orbiter with high accuracy. Thereafter, the standard frequency difference of the orbiter and the lander can be obtained from the mixed and transferred signals. In the following, we describe how the phase value caused by the frequency difference can be calculated.
As the standard frequency difference in each observation period can be estimated, the phase caused by the frequency difference can be calculated as
Where, T
k
is the time interval between two continuous observation periods.