## 1. Two Different Wavefront Clock Methods

[2] In existing interferometers, including very long baseline interferometry (VLBI) [*Rogers*, 1970, 1981; *Rogers et al.*, 1984; *Whitney et al.*, 2004], the cross-correlation function between two signals (X and Y) is expressed as

where

*B*the video analog signal bandwidth for sampling [Hz];

*τ*_{g}the geometrical delay;

*τ*_{e}the time difference (in case of VLBI);

*θ*an additional phase due to a local oscillator; (

*θ*has a positive sign in upper sideband (USB) and a negative sign in lower sideband (LSB));*ω*_{o}the angular frequency of the total local frequency (first local frequency ± second local frequency ±., where plus is USB and minus is LSB).

[3] Integration of equation (1), which is related to time, is necessary to detect fringes of weak radio sources with a sufficient signal-to-noise ratio. Since the geometrical delay *τ*_{g} changes with time because of the Earth's rotation, it should be compensated for during signal integration. The angular frequency *ω*_{o} is the total local frequency (first local frequency, second local frequency, and so on), and it changes according to channels in the case of a multifrequency interferometer system. Here *τ*_{g} can be expressed as

where *τ*_{oc} is a constant value and *τ*_{o} = *τ*_{g} + *τ*_{e}.

[4] By substituting equation (2) for equation (1) and neglecting the higher-order term to simplify the situation, we obtain

[5] To obtain the fringe, the trend of the terms in the left bracket of equation (3) can be transformed into a constant using

[6] Equations (3) and (4) show that if we dynamically shift the delay according to (*dτ*_{o}/*dt*)*t* in the cross-correlation function, the effect of the geometrical delay will not be reflected in the right-hand side of equation (3), except for the term *ω*_{o}(*dτ*_{o}/*dt*)*t*. The compensation of the delay shift is called delay tracking and is performed by shifting the acquired digital data during correlation processing. The purpose of the delay tracking is to correct the data in the time domain, not in the frequency domain. Therefore fringe rotation (*ω*_{o}(*dτ*/*dt*)*t*) in the frequency domain still remains after the delay tracking. In order to compensate for the fringe rotation due to *ω*_{o}(*dτ*_{o}/*dt*)*t* during the integration process, the counter phase (frequency) rotation is necessary. This frequency offset method is called “fringe stopping.” Since the fringe rotation rate depends on the received frequency, as shown in equation (3), fringe stopping must be performed for each frequency channel, though this process may complicate the correlation processing.

[7] Delay tracking and fringe stopping are performed separately in the conventional wavefront clock system, as opposed to the new system which is able to perform them simultaneously, as will be discussed later. In the existing wavefront clock system, delay tracking is performed by changing the rate of the clock for data sampling in accordance with an expected delay, and fringe stopping is carried out for each frequency channel by either shifting the local frequency or by adding an extra phase to the local oscillator. Existing interferometer systems use this conventional wavefront clock system (see Figure 1).

[8] In equation (3), the cosine function representing delay tracking and fringe stopping can be canceled by *θ*. If *θ* is considered as a fixed value in the conventional interferometer system, Doppler compensation is allowed only at the correlation site. On the other hand, if a local signal phase *θ*_{add} is added, it is considered to contribute to the fixed local signals at the observation site. The additional local signal phase is obtained as follows:

[9] In existing interferometer systems without the wavefront clock, differentiated frequency, *dω*_{o}(*τ*_{g} + *τ*_{e})/*dt*, is added to the data acquired by the station Y during correlation processing to cancel the Doppler shift. In the existing wavefront clock system, it is added to the local frequency during data acquisition to cancel the Doppler shift. The proposed new wavefront clock system can perform delay tracking and fringe stopping simultaneously by controlling the rate of the reference frequency at the station (an element in the interferometry), which is used for both the local oscillator and the sampler (Figure 2).