In an un-echoic chamber, we can generate the reference signals for the calibration method and assume that the reference signal is measured with the corner reflector, which is located at 0-degree (calibration point) of radar. The reason is that the main beam of the antenna is assumed to be in the ideal center. According to the beam pattern of an antenna, the location of the calibration point can be changed. From the received signals at the 0-degree point, we can calculate the calibration parameters and operate the accurate compensation process of each channel. Based on the calibration data, every transmitted signal in MIMO radar is enable to estimate the accurate location of each target in the received signals. It leads to increasing the overall performance of MIMO radar. However, when we receive the reference signal in unexpected conditions, it is hard to analyze the received signals as accurate targets. Thus, it is an important task before radar signal processing. The conventional calibration procedure for MIMO channels is as follows:

1) Measurement for ideal signals: measuring each signal of each channel in an un-echoic chamber.

2) Averaging: applying the averaging technique with the stacked all data.

3) Estimating fractional term: estimating the fractional term of magnitude and phase to compensate for raw data.

4) Extracting final calibration function: Define the final calibration function of each channel with the estimated magnitude and phase results.

Based on the measured signal in an un-echoic chamber, we arrange all data of each channel according to the available angle of MIMO radar. The reflected signal can be analyzed with a virtual array MIMO system. N^th Tx and M^th Rx antennas generate a total $N\times M$ channels and it can be the same as the extended Rx multiple channels. Therefore, we can define the mathematical signal model for the received signals with a range-compressed signal [1] (in FMCW), as shown below:

m = 1, 2, 3, $\cdots$, k-1, k (channel number) where, subscript r means the received signal, a indicates the amplitude of the received signal, W indicates the window function, f₀ is the center frequency of the chirp pulse signal and c₀ is the speed of light. As described above equation, it is necessary to select one channel, which can be used for a reference among the signals received in all channels. Based on the signal information of the selected reference channel, we start to calculate the calibration parameters. The calibration process aims to have the same phase and magnitude of all signals in MIMO channels when the radar receives the reflected signals from the same target. Before the calibration process, even if a signal is received from the same target, the signals of each channel can have a different phase and magnitude depending on unexpected conditions, such as MIMO antenna position error, circuit length error, etc. Thus, these phase and magnitude errors cause the deterioration of the target resolution even though the signal is reflected and received from the same target. In general, the conventional calibration sets the 1^st channel for standard reference with the phase and magnitude terms. The phase and magnitude of each channel are normalized to be shown as the first channel. Equation (2) and (3) indicate the compensation factor, arranged as 1^st channel signal.

In all channels of MIMO, equation (2) represents the compensation matrix matching with the phase of the 1^st channel, and equation (3) represents the compensation matrix matching with the magnitude of the 1^st channel. To increase the accuracy of the above compensation matrices, several chirp pulses are collected from FMCW Rx signals, and the average values thereof are defined as expected calibration factors. where, m is channel number, k is the maximum channel number ($N\times M$), l is the overall number of accumulated signals for averaging. The calibration factors of the phase and magnitude, estimated as the fractional form, can be synthesized and extracted in the form of a complex signal. Therefore, the following equation is finally defined as a general calibration parameter.

By using the extracted calibration parameter, we can receive and calculate the signals of all channels with the expected phase and magnitude when detecting the interested target in MIMO radar.

In order to not only match the target into the expected location but also secure the expected resolution of the target, the calibration process is essential for MIMO channel radar. However, as mentioned before, when the intensity of the unexpected interference increases, it is hard to expect the desired result with the initial calibration parameters, measured in an ideal environment. Therefore, this paper introduces a method of estimating calibration parameters with a simplified process, using a coupling matrix, rather than adding a complex process to derive a more accurate calibration factor.

As mentioned in the previous Section II-A, we try to improve the conventional calibration process based on the initial calibration parameters. It has the advantage of being applicable to existing processes. In other words, it is simple to upgrade the previous calibration method with low cost and low complexity. Based on the calibration parameters estimated in the existing method, a coupling matrix for every chirp can be derived, and we can generate more accurate calibration parameters. Thus, we introduce the overall proposed algorithm, as shown in Figure 2 (b):

1) Measurement for ideal signals: measuring each signal of each channel in an un-echoic chamber.

2) Averaging: applying the averaging technique with the stacked all data.

3) Estimating fractional term: estimating the fractional term of magnitude and phase to compensate for raw data.

4) Extracting final calibration function: Define the final calibration function of each channel with the estimated magnitude and phase results.

5) Applying the coupling matrix with the loop-back process: Extracting the coupling matrix from the conventional calibration model and finding an adequate final model.

As the unexpected signal becomes stronger, the accuracy of calibration parameters can be decreased. Figure 1 illustrates various interference in MIMO radar. In the case of high interference, the standard signal (1st channel) can be contaminated before proceeding with the existing calibration process. It causes unexpected errors in all radar signals. Therefore, to increase the accuracy of calibration results, it is important how modify the calibration parameters according to the various conditions. For example, the radar detection process generates the wrong Direction of Angle (DOA) of each target because the initial signal, calibrated by the inaccurate reference, indicates the wrong direction. Moreover, in Synthetic Aperture Radar (SAR) processing, the interested targets can be blurred and focused on an unwanted area. It leads to reducing the overall quality of the final image. Therefore, in this chapter, we use the covariance matrix and re-calibrated process with a simple and easy method considering the practical situation. The proposed covariance matrix between the received signal and the previously calibrated signal calculates the modified calibration function. And we define its meaning. The proposed matrix is estimated from each mutual signal between each channel. In the previous section, the conventional calibrated signal can be defined as follows:

Based on the above equation, all signals of the MIMO channel are unified as the first channel, considering the phase and magnitude difference. In the existing calibration method, the desired final signal $S_{r, cal}\left(r\right)$ is extracted by multiplying with the equation (5). For explaining the method, equation (5) is defined as the matrix ‘C’. The signal model, defined in (1), shows the standard form, which indicates the received signal model in each channel. The standard form does not consider the interference, using a covariance matrix between channels. Therefore, we can introduce a complex-valued $NM\times NM$ coupling matrix (NM is the total channel number and equal to ‘m’ in Eq. (6)) with angle independent and linear model:

Based on the initial calibration parameters, the covariance matrix model can be re-generated for the received MIMO signals, received every time. To analyze the proposed coupling matrix, we use the originally defined signal, which is calibrated into the standard channel. As shown below, the calibrated signals detect the target (CR, Corner Reflector) with the same phase and amplitude in the initial case:

When the calibrated signal is interfered with by an increase of the unexpected signals, the direction points of each channel also deviate in unexpected directions. To accurately analyze this phenomenon, we first apply the covariance matrix to our proposed process. The covariance matrix generates a square matrix and informs the covariance values between the pairs of each element. Using the calculated matrix, we can find the related tendency between each element which can be seen as the random variance. And also, it indicates the dispersion value and is easy to analyze more accurately. In other words, it helps to understand the interaction parameters between each channel. Therefore, for the first step, the covariance matrix is calculated with the received signal and initial calibrated parameters, as shown in equation (9). It is composed of a m$\times$m square matrix and shows the correlation of each channel that has been compensated by the initial values. In the next step, the initial calibrated signal can be analyzed with vector matching and it leads to concentrating the interest on the expected signals even in the interference situation. The accuracy of calibration can be further improved by applying the vector product between the proposed covariance matrix and the initial calibrated signals of all channels. The vector product amplifies two signals with the same tendency, and simultaneously, weakens two signals, which have a different tendency, compared to the previous value. Based on the initial calibration parameters, the calculation results of the covariance matrix and the received signals are as follows:

Based on the above results, it can be confirmed that the calculated matrix is constructed as the form, which is combined with the compensation values for each channel and the initial calibration signals. Moreover, it can be simplified as shown below:

where, B_n is the amplitude and ${\theta }_n$ are the amplitude and the phase of the received signal, respectively. $S_{r,final\_cal}$ indicates the calibrated results by the proposed technique. In equation (11), the phases that affect each channel are grouped into a common variable, and only the existing initial calibration phase remains. As shown in Figure 3, in this proposed method, the modified calibration point can be changed to the same form as the initial calibration point. The final magnitude must be cubed because the signal is multiplied three times by our proposed calculation. For each channel, the magnitude is amplified to a calibrated signal. And the magnitude deviation of each channel signal is minimized.

That is, when calibrating with the standard channel at a specific angle, it means the reference point can be changed slightly depending on the environment in which signals are transmitted and received. As an aspect of the calibrated phase, it indicates the received signals of all channels can be arranged in the calibration standard point (0-degree). The proposed technique re-generates calibration parameters to improve detection performance, using the covariance matrix of the received signals and the initial calibration parameters, stored in the memory. Thus, our technique can reduce the effect of the current unexpected conditions in the signal processing of radar detection. Through the proposed method, MIMO radars can use existing calibration values without having to calibrate for every transmitted pulse. It is simple and useful for solving environmental conditions because we cannot expect the phase direction of the interference. As mentioned in the previous section, the only saved calibration parameters cannot be adequate to compensate for each signal of all channels, according to various environmental conditions. The results of the proposed technique are better than those of conventional calibration signal processing. In the next section, to verify the performance of the proposed method, we introduce the calibration experiment and show the improved results.