#### 3.1. Hardware Calibration Scheme

From the analysis in

Section 2.1, we know that the correlator should be calibrated in order to implement the cross-correlation measurement accurately. We developed a calibration system, as illustrated in

Figure 3. The noise source and the attenuator are used to provide a power adjustable correlated noise signal for the system. It should be noted that the correlated noise calibration standards (CNCS) developed in [

31] could also be adopted if the statistical property of the noise signal needs to be adjustable. The magic T serves as a power divider for the noise signal, and the delay trimmers are used to tune the time delay between the RF paths. The vector modulator (VM) module [

32] is employed to control the phase of the local oscillator signals for the receivers. We assume that the outputs of the signal generator and the VMs are

${\varphi}_{LO}$ is the phase difference between the LOs of the two receivers.

T is the period VM1 needs to sweep the phase of

${V}_{VM1}(t)$ between 0–360°.

N is the total number of phase states in the 0–360° phase sweep, and

n is the

nth phase state,

$n=1,2,\cdots N$. From the expression of

${V}_{VM1}(t)$, it experiences a 0–360° phase sweep during the period of time

T, and there are

N phase states in the phase sweep. The phase of

${V}_{VM2}(t)$ is fixed, but a 180° phase-switching scheme [

14,

19,

21,

24,

33] can also be employed in VM2, which would enable the removal of the offset drift [

33] and the cross-talk presented in the imaging system [

19].

The two receivers and the complex correlator in

Figure 3 can be regarded as a single baseline interferometer; for an interferometer baseline which uses a complex correlator with a rectangular passband of radian bandwidth

B, the outputs are [

19,

34]

where

${R}_{o}$ is the normalization parameter,

${\tau}_{g}$ and

${\tau}_{i}$ are the time delay presented at RF and IF between the two receivers, respectively,

${\omega}_{LO}$ and

${\omega}_{IF}$ are the LO and IF radian frequency, respectively.

In our calibration system,

${\tau}_{i}$ is small due to the high level of symmetry in the IF path, and

${\tau}_{g}$ is tunable, while

${\varphi}_{LO}$ can be adjusted by using the VMs. Following this, the outputs of the correlator in the calibration system can be formulated as

It can be seen that, due to the phase sweep of LO1, the outputs of the complex correlator depend on the phase difference between VM1 and VM2. By sweeping

${\varphi}_{LO}$ from 0–360°,

${V}_{real}({\varphi}_{LO})$ and

${V}_{imag}({\varphi}_{LO})$ can be plotted on a polar diagram. This curve is named as a correlation circle and is shown in

Figure 4. The outputs of the complex correlator describe an ellipse. The center of the ellipse is the DC offset presented in the in-phase and quadrature correlating subunit of the complex correlator. The major axis (

$2{r}_{real}=2{G}_{real}\mathrm{sinc}\left[B({\tau}_{g}-{\tau}_{i})\right]$) and the minor axis (

$2{r}_{imag}=2{G}_{imag}\mathrm{sinc}\left[B({\tau}_{g}-{\tau}_{i})\right]$) of the ellipse are proportional to the gain of the low frequency amplification circuit in the in-phase and quadrature correlating subunit, respectively. Meanwhile, the axial ratio (

$AR=\frac{{r}_{real}}{{r}_{imag}}$) reflects the amplitude imbalance between the in-phase and quadrature correlating subunit, and the closer the axial ratio is to 1, the better the amplitude symmetry is.

Based on the above descriptions, by sweeping the phase difference between the LOs of the receivers while measuring the outputs of the complex correlator, the outputs amplitudes and DC offsets of the correlator can be determined. Subsequently, the correlator is able to be calibrated by tuning the digital potentiometers in the low frequency amplification circuit. The flow chart of the calibration procedure is illustrated in

Figure 5.

It can be noted that after a 0–360° phase sweep is finished for

${\varphi}_{LO}$, the corresponding measured

${V}_{real\_m}({\varphi}_{LO})$ and

${V}_{imag\_m}({\varphi}_{LO})$ can be used to calculate the actual outputs characteristics (

${r}_{real\_m},{r}_{imag\_m},{C}_{real\_m},{C}_{imag\_m},A{R}_{m}$) of the complex correlator, which are formulated as

Following this, whether one increases or decreases the resistance values of the digital potentiometers and the corresponding adjusting step size depends on the differences between the desired outputs characteristics and the actual ones. The calibration process is finished when the output characteristics of the complex correlator are acceptable. It should be noted that the best achievable outputs characteristics could be obtained when the tolerances for the desired outputs characteristics are extremely small. Finally, the Data-Word (which is the data that is used to set the resistance value of the digital potentiometer) are stored.

#### 3.2. Quadrature Errors and Residual DC Offsets Calibration Algorithm

The calibration scheme presented above is a hardware-based calibration method, and the best achievable outputs characteristics of the complex correlator rely on the resolution of the digital potentiometers. Due to finite resolution, the amplitude imbalance and the residual DC offsets may still exist, but they are very small. However, the quadrature phase error caused by the non-ideal behavior of the RF circuit cannot be cancelled by the hardware-based calibration scheme.

For a non-ideal complex correlator, we assume that the amplitude of the quadrature correlating subunit is

$g$ times that of the in-phase correlating subunit and that the quadrature phase error is

$\epsilon $. The quadrature errors (

$g$ and

$\epsilon $) can be determined and further calibrated by applying an exactly 90° phase change to the LO of one of the receivers [

24]. This is an efficient way to calibrate the quadrature errors, but the DC offsets of the correlator are neglected and this approach requires a relatively high phase shift accuracy for the LO. To improve the reliability and accuracy for the quadrature errors calibration, a calibration algorithm based on a LO phase shift scheme is developed, capable of obtaining the averaged quadrature errors and less demanding for the phase shift accuracy of the LO. Furthermore, the residual DC offsets following the hardware calibration are also considered in this algorithm. The procedure for this calibration algorithm is as follows. Firstly, we assume that the phase states of LO1 and LO2 in

Figure 3 are

${\varphi}_{LO}$ and 0°, respectively;

${\tau}_{g}={\tau}_{i}=0$, and

$\rho =a{e}^{i\theta}$ is the actual cross-correlation of the correlated signals injected into the receivers. Therefore, the output of the non-ideal complex correlator is

Following this, the phase is shifted by approximately 90° for LO1. It should be noted that if the VM is adopted and calibrated to control the phase state of the LOs, the phase shift error can be less than 1°. The detailed calibration process for the VM can be seen in our previous work [

32].

Subsequently, the output of the non-ideal complex correlator would be

${r}_{q}({\varphi}_{LO})$ is formulated under the assumption that the phase state of LO1 is changed from ${\varphi}_{LO}$ to ${\varphi}_{LO}+{90}^{\xb0}$.

From [

24], the quadrature errors could be determined as

Theoretically, $g$ and $\epsilon $ are the inherent property of the complex correlator that are decided by the hardware itself. Due to ${\varphi}_{LO}+{90}^{\xb0}$ phase state cannot be exactly obtained for LO1, and there is a phase error between the desired ${\varphi}_{LO}+{90}^{\xb0}$ phase state and the actual one. It should be noted that this phase error is a function of the desired phase state, since the phase shift accuracy of the VM varies with the desired phase state. For instance, the phase errors may be 0.8° and 0.5° for the phase states at 60° and 80°, respectively. Therefore, the phase shift error of the VM is included in $\epsilon ({\varphi}_{LO})$ and $g({\varphi}_{LO})$.

In practice, by assuming that an exactly 90° phase change is obtained for LO1 and that the DC offsets (

${C}_{real}$,

${C}_{imag}$) are equal to zero,

$\epsilon ({\varphi}_{LO})$ and

$g({\varphi}_{LO})$ can be calculated by using Equation (21) and (22). Indeed, due to the non-exact 90° phase change and the nonzero DC offsets, there would be some deviations between the estimated quadrature errors (

$g({\varphi}_{LO})$ and

$\epsilon ({\varphi}_{LO})$) and the actual ones (

$g$ and

$\epsilon $). However, these deviations usually being small,

$g({\varphi}_{LO})$ and

$\epsilon ({\varphi}_{LO})$ could be used to estimate

$g$ and

$\epsilon $, respectively. Since the estimated quadrature errors are dependent on

${\varphi}_{LO}$,

$g({\varphi}_{LO})$ and

$\epsilon ({\varphi}_{LO})$ averaged over

${\varphi}_{LO}$ (

$\overline{g}$ and

$\overline{\epsilon}$) would be more suitable for determining

$g$ and

$\epsilon $.

$\overline{g}$ and

$\overline{\epsilon}$ are formulated as

As discussed in

Section 3.1, the DC offsets of the complex correlator could be estimated via the outputs of the correlator after a 0–360° phase sweep is adopted for

${\varphi}_{LO}$. Subsequently, the function of

$g({\varphi}_{LO})$ and

$\epsilon ({\varphi}_{LO})$ can be obtained, after which

$\overline{g}$ and

$\overline{\epsilon}$ are determined.

After adopting the calibration algorithm, the actual cross-correlation obtained is

Furthermore, in order to investigate the performance of the quadrature errors calibration algorithm, the output phase error [

15] of the complex correlator is introduced. The output phase error is formulated as

where

$\varphi $ is the phase of the cross-correlation between two correlated signals injected into the complex correlator.

Given that the outputs (

${V}_{real}$ and

${V}_{imag}$) of the complex correlator are able to reconstruct the cross-correlation phase, the output phase error can subsequently be used to estimate the phase reconstruction performance of the complex correlator. For an actual complex correlator, the phase reconstruction accuracy is affected by DC offsets and quadrature errors. Therefore, if these aspects are known, the phase reconstruction accuracy could be improved. Meanwhile, the output phase error can also be calibrated and decreased. Considering the DC offsets and the quadrature errors, the calibrated output phase error can be written as

Since the DC offsets and the quadrature amplitude error could be reduced significantly by using the proposed hardware calibration, the output phase error is usually dominated by quadrature phase errors.