Currently, there are two types of interferometers in the world: One is the conventional interferometer using the fixed local signal as a reference signal, and the other is the interferometer using the wavefront clock. In this paper, we propose a new method of the wavefront clock system under development. The new wavefront clock system has a great advantage for the Doppler compensation. The Doppler shift, which is caused by the Earth's rotation, is compensated for during the correlation processing or when signals are received in order to detect fringes (interferometer patterns). Also, the proposed system can be effectively applied to high-frequency (millimeter wavelength), wide-bandwidth, and multibaseline interferometry. The main feature of this method is that the reference signals for the front end and back end of the interferometer system are directly controlled from the observing site according to a calculated a priori delay rate. In this method, fringe stopping and delay tracking can be simultaneously performed on all received frequencies as well as on frequency-converted signals in both the upper sideband (USB) and lower sideband (LSB). Furthermore, this method can be introduced with just a slight modification to the current interferometer systems (including very long baseline interferometry) and can also be applied to a pseudofringe (Doppler shift and delay) simulator to check the correlation processor.
the video analog signal bandwidth for sampling [Hz];
the geometrical delay;
the time difference (in case of VLBI);
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));
the angular frequency of the total local frequency (first local frequency ± second local frequency ±., where plus is USB and minus is LSB).
 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 τg changes with time because of the Earth's rotation, it should be compensated for during signal integration. The angular frequency ωo 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 τg can be expressed as
where τoc is a constant value and τo = τg + τe.
 By substituting equation (2) for equation (1) and neglecting the higher-order term to simplify the situation, we obtain
 To obtain the fringe, the trend of the terms in the left bracket of equation (3) can be transformed into a constant using
Equations (3) and (4) show that if we dynamically shift the delay according to (dτo/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 ωo(dτo/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 (ωo(dτ/dt)t) in the frequency domain still remains after the delay tracking. In order to compensate for the fringe rotation due to ωo(dτo/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.
 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).
 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 θadd 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:
 In existing interferometer systems without the wavefront clock, differentiated frequency, dωo(τg + τe)/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).
 If all interferometer stations set their oscillators and sampling clocks in the same Doppler rest frame, there will be no time-dependent phase or delay drift between their sampled data streams. Generally, delay tracking is performed during correlation processing according to the time label on the attached data stream while digitization is conducted at the station clock rate. If we can control the station clock rate on the basis of the expected changes in the geometrical delay during the observation, delay tracking is not necessary during correlation processing, except for an initial delay offset. When only the sampler's clock at the station is controlled, the fringe rotation still remains because ωo(dτ/dt)t is not affected. If θ (in equation (6)) is changed to satisfy for the actual observation, the fringe rotation in the cross-correlation function (equation (3)) can be canceled:
 if arises from additional phases in the local oscillators and the frequency differences between two stations, ϕx and ϕy are additional phases of the local oscillators at the stations X and Y, and ωx and ωy are the total local angular frequencies, which are defined as the dependent frequencies of each channel in a multifrequency interferometer system. When only the time-related first order is allowed for the equation, it is expressed as
where θ is fixed by offsetting the local frequency by ωo, which is equivalent to the compensated Doppler frequency of the station Y to the station X. Therefore their relation is expressed as
where ωo is the angular frequency of the baseband (same as the total local frequency; see equation (1)), which is equal to ωx. Thus the condition for compensation of the fringe rotation can also be expressed as
Equation (10) represents the condition free from local frequencies. In other words, α is independent of the received frequencies or local frequencies in a general multichannel (and/or frequency) interferometer system. When the local oscillator is driven by the reference frequency of the station, equation (10) is satisfied by controlling the reference frequency rate multiplied by the factor 1 − α[= 1 − (dτo/dt)] at the rate of UTC. The first local frequency multiplied by (1 − α) equals the reference frequency multiplied by (1 − α). If the reference frequency is multiplied by (1 − α), fringe stopping and delay tracking can be carried out simultaneously because they have identical terms as shown in equations (10) and (4). These parameters depend on the reference frequency, not on the received frequency. If the same reference frequency is used to drive the sampler and to label the time, delay tracking is also performed automatically according to changes in the reference frequency rate.
 This is the fundamental idea of our new wavefront clock that can simultaneously carry out delay tracking and fringe stopping just by controlling the reference clock rate; this operation affects all the frequencies at the same time. Fringe stopping is performed on all frequencies as well as in the USB and the LSB, which makes the Doppler compensation in each frequency unnecessary. This is impossible with the conventional interferometer system and the conventional wavefront clock system.
 This new wavefront clock method is advantageous because it can be introduced with a slight modification to the existing interferometers and can be applied to the interferometer experiments with frequencies. The fringe rotation with the wavefront clock is equivalent to changing “(ωy − ωx)” to “(ωy − ωx) plus fringe frequency” at the clock rate based on UTC. It is also calculated by changing “(ωy − ωx) at the clock rate based on UTC” to “(ωy − ωx) at the wavefront clock rate,” which is much simpler. Therefore the phase calibration signals are compatible with the geodetic VLBI. When using this wavefront clock system, it is helpful to set a reference point so as to compensate for the Doppler shift. The reference point can be any arbitrary point (e.g., the array center of the interferometer or the center of the Earth for global-scale interferometers such as VLBI).
 In our wavefront clock system, the reference clock signals for the front end and back end of the interferometer data acquisition system are directly controlled according to a calculated a priori delay rate. As the frequency resolution and signal clarity are important factors in the direct control of the reference clock signal, the wavefront clock system is located between a hydrogen maser and the interferometer system (Figure 2).
 The source of the frequency oscillation is a crystal oscillator [Kiuchi, 1996]. The crystal is used as a voltage-controlled crystal oscillator (VCXO in Figure 3), which is phase locked to the external reference frequency. To obtain high-frequency resolution, the reference signal from the hydrogen maser is multiplied by a phase-locked oscillator (PLO) whose frequency is selected as Freq1. Since this system uses the wavefront clock signal for frequency conversion, sampling, and phase calibration, all of the station's reference signals are phase locked to the wavefront clock. For direct control of the reference signal, the resolution of the frequency is set at (Frequency resolution)/(Freq1) and 7 × 10−13, where Freq1 is 1.4 GHz with the existing milli-Hertz resolution synthesizer. Higher resolution is obtained with Freq1 of the high-frequency PLO and the high-resolution synthesizer. The phase of the synthesizer is refreshed at 40 Hz via VERSAmodule Eurocard (VME) bus extended for instrumentation bus. The VCXO is controlled by the phase-locked loop circuit, and the output frequency depends on the synthesizer frequency that is controlled by the host computer. The parameters given for the wavefront clock system are those for the positions of the reference and target stations, star positions, and start/end time of the operation, including the clock parameters (clock offset and clock rate based on epoch time) and Earth's rotation parameters (ERP). Wavefront clock signals are automatically generated at the clock rate on the basis of UTC. The measured phase stability of the system is shown in Figure 4. The measured phase noise is white phase noise.
Rogers  estimates the fractional loss of coherence due to the instability in the wavefront clock for the T-s integration time:
the loss of coherence;
the angular frequency of local oscillator;
the Allan variance [(standard deviation)2] of white phase noise (T−1) at l s;
the Allan variance [(standard deviation)2] of white frequency noise (T−1/2) at l s;
the constant Allan variance [(standard deviation)2] of flicker frequency noise (T0);
the integration time [s].
 Fringe stopping and delay tracking are executed at the start of the wavefront clock system. The delay (not the delay rate) is the only parameter given for the correlator at the starting time of the observation. The fringe rotator and the imaginary part of the correlator are not used because the wavefront clock correlator is not required to perform bit shift, delay tracking, and 90° phase jump and is free from the coherence losses (loss of three-level approximation of fringe is 4%, fractional bit correction loss is 3.4% [Rogers, 1981; Kiuchi et al., 2000].
2.2. Software for Calculating an A Priori Value
 The Earth's rotation is estimated with the International Earth Rotation and Reference Systems Service (IERS) Earth rotation parameters: (1) wobble matrix W (the polar motion matrix from the conventional international origin pole to the conventional spin axis), (2) diurnal polar motion matrix D (the diurnal polar motion matrix), (3) diurnal rotation matrix S (the diurnal rotation matrix about the instantaneous spin axis), (4) nutation matrix N (the nutation matrix from the conventional true equator and equinox of the date to the mean date), (5) precession matrix P (the precession matrix from the mean equator and the equinox of the date mean of 2000.0.), and (6) aberration matrix A (the aberration correction matrix is derived from a small periodic change in the apparent position of the celestial bodies which is caused by the combination of the movements of the light and observer).
 Given this, assuming that the terrestrial baseline matrix is B, the Earth's rotation can be expressed as
and τg is calculated every second using equation (12) and is approximated with the fourth-order polynomial.
 Verification experiments were performed with two-sideband (2SB) interferometer (VLBI) systems.
3.1. Application of the Wavefront Clock to Millimeter Wave (Over 100 GHz) Interferometry
 We carried out some experiments with the prototype wavefront clock system applicable to the antenna of the Nobeyama Millimeter-wave Array (NMA) with an ultrawide bandwidth correlator (UWBC) [Okumura et al., 2000]. The NMA receiving system is a 2SB system with 90° phase switching cycle at the first local signal. The 90° phase switching cycle (90° phase modulation) on the first local signal is demodulated in the Fourier transform before cross correlation correlator for sideband separation. The wavefront clock system was installed in one element (antenna) of the NMA. The scheme of the NMA observation/correlation system is the same as shown in Figure 1. During these wavefront clock experiments, the function of the additional phase generator of the current NMA system (the so-called fringe rotator and delay tracker) on the wavefront clock element was disabled because the NMA system does not require the proposed wavefront clock system. The correlation process used in the experiments is the same as that in the current NMA system, except for the wavefront clock baseline between the wavefront clock antenna and the reference antenna. In order to deactivate the delay tracking and fringe rotation functions during the correlation process, the wavefront clock baseline needs to be set to zero. By doing this, the position data of the wavefront clock antenna can be interpreted as that of the reference antenna. As this system needs only geometrical delay (τg) for possible compensation for other parameters, an initial geometrical fixed delay for the correlator was given by the wavefront clock controller at the beginning of the experiments. The measurement results are shown in Figure 5.
 In Figure 5, the divided four sections of the right and left charts show the results in 95 GHz (or 86 GHz) band (LSB) (top and bottom left), the results in 107 GHz (or 98 GHz) band (USB) (top and bottom right), the cross-power spectrum (top right and left), and the residual phase (bottom right and left), respectively. The bandwidth of each sideband is 1024 MHz, and integration time is 10 min. From these results, any significant difference cannot be detected between the fringes obtained by the current NMA system and those of the wavefront clock system; however, the detected phase has a slight phase drift. We consider that the major cause of the phase drift is the round trip cable delay compensation system. The cable delay compensation is performed on the fixed frequency, since the system is designed for the fixed reference frequency, not for the variable wavefront clock frequency. It is important to remember that these experiments were carried out with limited coherence length under unfavorable atmospheric conditions because the NMA is a shared facility and experimental research, and developments such as these are allowed only during maintenance season (usually late May), which is not a good season for observation.
 The results show that the wavefront clock system has the following two advantages.
 1. Fringe stopping and delay tracking can be performed at all received frequencies.
 2. Fringe stopping and delay tracking can be performed simultaneously on the upper (USB) and lower (LSB) sidebands.
3.2. Experimental Application of the Wavefront Clock to a Domestic 109-km Baseline VLBI
 We conducted a real-time experiment with the wavefront clock system for the VLBI with a 109-km baseline between Koganei station (Tokyo, Japan) and Kashima station (Ibaraki, Japan) as the reference point. The VLBI system is a 2SB system using image filters for frequency conversion. The wavefront clock system was operated at Koganei station. The Doppler shift of the signal frequency received at Koganei station was compensated for to be equivalent to that received at Kashima station. The only parameter given for the conventional correlator was τg, which was calculated at the wavefront clock start time. Since the wavefront clock system performs fringe stopping at the time of signal reception, the correlator for the wavefront clock VLBI has only a real-part correlation function and does not require any complex correlation function.
 The wavefront clock experiment was conducted with the real-time VLBI system [Kiuchi et al., 2000] for the Keystone project (KSP), with high-speed asynchronous transfer node (ATM) networks (with asynchronous transfer mode adaptation layer 1 corresponding to a constant bit rate protocol) for data transmission. These networks have a transmission capability of up to 2.488 Gbits/s via optical fiber links. Instead of being recorded on magnetic tapes, the measured data is transmitted, via either a 2.488-Gbits/s STM-16/OC-48 (synchronous digital hierarchy/synchronous optical network) or an ATM network, from remote observation stations to the correlation site (Koganei) for real-time processing. Regular geodetic VLBI experiments, which are conducted for 24 hours every other day, reduced the position errors to about 2 mm in the horizontal direction and about 10 mm in the vertical direction [Koyama et al., 1998; Kondo et al., 1998]. The system was designed to allow automated operation throughout the entire process. A block diagram of the real-time wavefront clock VLBI system is shown in Figure 6.
 The signals from the two stations are transmitted to a cross-connect switch (ATM-XC) that merges them into one path. One of the detected fringes in this experiment is shown in Figure 7.
4. Application of the Wavefront Clock to a Fringe Simulator
 The wavefront clock system, which can generate pseudofringes at a desired position, is useful for debugging the correlator system. A block diagram of the simulator is shown in Figure 8.
 The simulator has two frequency conversion systems: system A and system B. RF noise signals from the noise diode are divided and are fed to the two systems via a power divider. A fixed local frequency system is used in system A as in the existing interferometer systems, while the wavefront clock system is used in system B. The wavefront clock system converts white noise to a signal equivalent to that received at the target station. By using these systems, we can obtain two different signals of interferometer data similar to those obtained by the observations at two different stations.
 In this case, the correlated amplitude (high signal-to-noise ratio) is expected to be large because RF noise with no disturbing noise from the noise diode is shared by the two stations. The simulated data contain all the factors (delay tracking, fringe rotation, and clock difference) except for initial delay.
 In order to use this simulator, we generated virtual interferometer data with a desired baseline. The parameters given for the simulator (the wavefront system) were positions (the reference and target stations), star positions, clock parameters (clock offset and clock rate based on epoch time in the case of VLBI), Earth rotation parameters (ERP), start time, and end time. The same parameters are used for the correlator except the clock offset parameter. A simulated signal is automatically generated according to the real-time clock, and τg is added to the clock parameter at the start time of the correlator.
 In the simulation of the interferometer, we set the east-west baseline between station A (Koganei station) and station B (Usuda station) as 111.9 km baseline length. The raw data obtained with the wavefront clock were equivalent to those received at Usuda station, while the data generated with the fixed frequency were equivalent to those received at Koganei station. The two stations used the common reference signal as shown in Figure 8. The ERP values for the a priori calculation software (the wavefront clock system and cross-correlation system) were obtained from the estimation by IERS (www.iers.org). Since our wavefront clock system does not generate initial delay, the a priori value of the initial delay in the correlation process shall be zero. If exact consistency is maintained between the existing interferometer correlation systems and the wavefront clock simulation system, good fringes can be obtained. The results of the gigabit rate correlation process [Kiuchi, 2005] are shown in Figure 9.
 The correlated amplitudes obtained with the wavefront clock method and that obtained with the existing interferometer method are compatible with each other. In addition, pseudofringes were successfully generated, which demonstrates the feasibility of the wavefront clock method. Also, this success will significantly advance the development of the pseudointerferometer signal generator to design and will check the correlation process.
5. Discussions on Sources of Errors
 The problems of the proposed wavefront clock system are given below.
5.1. DC Offset
 DC offset could be a concern for the wavefront clock system and the zero baseline interferometer. DC bias in correlated function is mainly caused by the DC offset of the sampler and the common-mode noise interference in IF, but the impact of these are different among the proposed wavefront clock system, the conventional wavefront clock method (fringe is stopped only at the first local signal), and the zero baseline interferometer.
 Common-mode noise interference in IF is caused by the noise interference in the two stations at the same intermediate frequencies. In the conventional wavefront clock correction (fringe is stopped only at the first local signal), radio interference after the first local signal cannot be rejected since the same IF is applied to every station. In the proposed wavefront clock method, all the conversions (first, second, and so on) are synchronized with the wavefront clock, and different local signal and IF are applied to each station. In the zero baseline interferometer, common-mode noise interference is difficult to reject because a fixed local signal is applied to all stations. This means that the proposed method has a higher noise rejection ratio than that of the conventional wavefront clock method or that of the zero baseline interferometer. In addition to this, the wavefront clock system has another advantage in terms of sampling rate. In the zero baseline interferometer, as the sampling rate is constant, signals inserted into the IF of the two antennas are always in the coordinate phase. On the other hand, in the wavefront clock system, the sampling clock rate changes according to the wavefront clock, and the phase relation of the common-mode noise interference between the two antennas shifts over time. For these reasons, the impact of the common-mode noise interference is smaller in the proposed wavefront clock system compared to the conventional wavefront clock method or the zero baseline interferometer. The impact of DC offset can be compensated for with the 180° switch in the connected interferometers. In VLBI, the DC component is removed after fast Fourier transform processing in the coarse search function (cross) process, but it should be considered as coherence loss. The calculated coherence loss, in the worst case of a 2-mV DC comparator offset for nominal 0 dBm input power, is less than 0.052% [Kiuchi et al., 2000].
 The wavefront clock is correctly set for only one direction in the sky. The wavefront clock system is usually operated with a real-time system (interferometer or real-time VLBI). The most remarkable advantage of the tape-based system is that it is able to provide correlation parameters corresponding to different fringe frequencies by replaying the tapes after observation and drawing contour maps accordingly. On the other hand, the real-time-based system has a real-time fringe monitor to compensate for the disadvantage of not having such a data recording system. Mapping is also available in the real-time-based system by receiving the feedback from the fringe monitor results, but the tape-based system can draw maps much more effectively.
 If the wavefront system is combined with the tape-based observation, the obtained parameters are reproducible as time function and are thus correctable using the VLBI correlation results. By using the wavefront clock, the Doppler effect on the target as well as on its surrounding areas can be canceled and compensated for with a high degree of accuracy.
5.3. VLBI Phase Calibration Signal for Bandwidth Synthesis in the Wavefront Clock Method
 In the geodetic VLBI, the instrument delay of the VLBI station is calibrated by a phase calibration signal (Pcal) which is inserted into the front end to calibrate the phase difference between the channels, while the other type of the existing wavefront clock system (as Figure 1) cannot support the phase calibration signal. In the proposed wavefront clock system, phase calibration signals are always detected at a fixed frequency, as with the existing VLBI method.
 In the wavefront clock system experiments, however, some problems were found in the correlation of the phase calibration signals. Usually, the phase calibration signal consists of 10-kHz and 1-MHz plus 10-kHz sine waves in a video signal (bandwidth 2 MHz). In a VLBI observation with a long baseline, the signal from a star has a Doppler shift due to Earth's rotation, which needs to be corrected by fringe rotation in order to obtain a fringe. In the fringe-stopping process, the phase calibration signal cannot maintain coherence since the phase calibration signal is also rotated. As a result, the cross correlation is not affected by the phase calibration signal. In the wavefront clock, the received signal from stars and the phase calibration signal do not have any Doppler shift because they had already been compensated for at the observation site. Then, in VLBI observations with the wavefront clock system or the zero baseline system that has no distance to cause any Doppler shift, there is no concern for the Doppler frequency between the two stations. However, in return for this, the wavefront clock system has a disadvantage that fringes of the phase calibration signals are detected. In fact, without fringe rotation, as with the wavefront clock system or the zero baseline system, the output of the correlation processor has sine wave components, which are 10 kHz and 1 MHz plus 10 kHz. Figure 10 shows the correlated phase calibration signals in the case of the zero baseline system in conventional processing.
 The transverse axis is the lag bit, and the ordinate axis is the correlated amplitude. The effect of the 10-kHz phase calibration signal appears as DC offset because 10 kHz is a very low frequency compared with the sampling clock. The effect of 1 MHz plus 10 kHz appears as a sine wave (Figure 10, top (labeled 1)), because a cross correlation between sine functions becomes a sine function.
 The effect of the 10-kHz phase calibration signal can be eliminated by software using the acquired phase calibration data produced by the correlation processor, while the effect of 1-MHz plus 10-kHz components can be removed by changing the frequency interval of the phase calibration signal from 1 MHz to 5 MHz (Figure 10, bottom (labeled 2)). This operation is performed just by disconnecting the pulse gate control cable (subminiature C connector). When the frequency interval is changed from 1 MHz to 5 MHz as explained above, the remaining phase calibration signal is 10 kHz only, as another remaining signal of 5 MHz plus 10 kHz is out of the video frequency range. The effect of the phase calibration signal of 10 kHz is removed using a method similar to DC offset.
 The wavefront clock system can provide a simpler correlation process of the acquired data. In this system, the Doppler shift of signals received at different stations can be compensated for with a high degree of accuracy by applying the reference frequency, which changes according to the a priori delay rate. The reference frequency is commonly used in the front end and the back end of the data acquisition terminal of the system. Fringe stopping and delay tracking are simultaneously performed on all the received frequencies and on USB/LSB video frequencies. With these functions of the wavefront clock system, the functions of the correlator can be remarkably simplified, and the function for the correlator is the real-part correlation only.
 The experiment results of the prototype wavefront-clock system are satisfactory so far. With a slight modification of the existing systems, the correlation process can be greatly improved. Although the system still needs further development, we will make every effort to achieve better results.
 We are indebted to Y. Takahashi at the National Institute of Information and Communications Technology (NICT) and M. Momose at Ibaraki University for their helpful technical discussions. We appreciate the help we received from Cosmo-research Inc. in putting together the wavefront clock system.