The calibration is generally required for array proecssing and direction finding. The calibration contains three parts:

1. Antenna coupling calibration

2. Near-range calibration

3. Rx channel gain/phase calibration

The first one is due to the coupling between antenna elements and the power leakage from Tx to Rx. The latter is removed by subtracting the average of the very close-range bins to the radar. The former is presented on all received signals since it determines the output induced current of each antenna by the incident wave. The second one appears only when the objects are close. Actually, there is a relationship between the antenna aperture size and the far-field distance of a radar:

$d_{far-field} = 2 \left(\frac{D^2}{\lambda}\right)^2 \quad\quad (27)$

where D is the largest dimension of the anteann. Rx channel gain/phase mismatch is resulted in by imbalance in the gain/phase of different receiver paths. This is $$\Gamma$$ in eq11. There are different methods to calibrate the received signal from all virtual receivers. Generally, they are classified into two major groups of train-based and non-train-based methods. The first requires a reference object in a specific distance to the radar, while for the second one, there is no need for the test object. The train-based method cannot be updated over time once the calibration coefficients are derived, while the non-train-based method can be updated. We used the idea in 1 for non-train-based method. 2 has introduced a method for coupling matrix estimation, which does not depend on the shape of the matrix, and it is claimed that this method can be used for any type of antenna array configuration. However, their method is not mathematically correct. The primer reason for inability in coupling matrix estimation is that usually the number of unknown variables are more than the number of known equations. Thus, the problem either has infinite solutions or it does have any depending on the equations. We will mention this later. In fact, the calibration in 2 compensates the Rx channel gain/phase and the antenna gains (above item 3). Thus, we have to further be careful about the antenna coupling and near-range errors. Although the angular estimations in 2 or 1 are high-resolution, we used Capon beamforming as a comparison. Basically, these works used MUSIC method while the Capon beamformer exploits a completely different concept.

## Joint receiver calibration and angle of arrival estimations 2¶

The method relies on knowing the number of objects that are present in eq11. For our radar, this is a vector of samples received form all virtual antennas at a single range. Assume that there are K objects at a specific range. We want to obtain their angle of arrivals i.e. $$\theta_k,\phi_k$$ for $$k=1,\cdots,K$$ and $$K<=L-1$$. In 2, the authors claimed that they developed a system model and an algorithm in which they obtain the angle of arrivals in the presence of the antenna coupling and the Rx channel calibrations. Particularly, they alleged that the algorithm could find the coupling matrix of any array configurations. To do so, they assumed that the received vector x(n) has a covariance of R such that its eigenvalue decomposition gives a unitary matrix of D. If the eigenvalues are in descending order, we can choose the signal subspace the first K columns of D. So, the remaining columns are spanning the noise space, $$e(t_f,t_s)$$. If the noise space of D is a matrix of U, then the following condition should be satisfied:

$<U,x(t_f )> =0 \quad\quad (28)$

which means that the noise space is orthogonal to the signal space. Equivalently, it can be shown that the following must be true:

$J = ||U^H C \Gamma A ||^2 = \sum_{k=1}^{K} ||U^H C \Gamma \boldsymbol{a}(\theta_k,\phi_k )||^2 = 0 \quad\quad (29)$

Numerically, J should be as close as possible to zero. So, it is possible to use descent methods to reduce the value of J in the iteration. However, this equation has three unknowns, i.e. $$C,Γ, a(θ_k,ϕ_k )$$. The proposed algorithm is as follows:

1. Set the initial values of C, $$\Gamma$$

2. Set find K maximum of $$p = ||U^H C \Gamma \boldsymbol{a}(\theta_k,\phi_k )||^{-2}$$ corresponding to K pair of $$(\theta_k,\phi_k)$$.

3. Find an optimum Γ by solving the following problem when we know the K steering vectors from the previous step:

$\begin{split}g = \min_{\Gamma} \sum_{k=1}^{K} ||U^H C \Gamma \boldsymbol{a}(\theta_k,\phi_k )||^2 , \\ s.t. \quad \Gamma_{11} = 1 \quad\quad \quad (30)\end{split}$
4. Finding optimum C by knowing both Γ and the steering vector:

$\begin{split}h = \min_{C} \sum_{k=1}^{K} ||U^H C \Gamma \boldsymbol{a}(\theta_k,\phi_k )||^2 , \\ s.t. \quad C_{11}=1 \quad\quad \quad (31)\end{split}$

If we do not have any constraint on the shape of C, then the optimum solution would be:

$c = -\hat{M}^{*} m, \quad \quad (32)$

where $$\hat{M}^{*}$$ is pseudo-inverse of $$\hat{M}$$, and $$\hat{M}$$ is a matrix containing all columns of M except the first column and M is:

$M = (A^T (\Gamma^{i+1})^T) \otimes U^H$

where $$\otimes$$ is the Kronecker product. In fact, $$\hat{M}$$ does not have a pseudo-inverse since it is is not a full column rank matrix . Hence, this method cannot be used to estimate the coupling matrix. However, the groundwork 1 has a solution to the problem for individual cases of the array structure like a uniform rectangular array.

Note that $$\Gamma$$ can model the gain/phase shifts due to the near-range effect; however, the steering vector is for far-field. So, generally, the joint calibration and estimation method can be applied for distant objects, for example, farther than 2 m.

## Direction Finding in the Presence of Mutual Coupling 1¶

Many factors contribute to changing the response of the sensor array over time: gradual changes in the behavior of the sensor itself and of the electronic circuitry between the sensor and the output of the digitizer (due to thermal effects, ageing of components, etc.), changes due to the environment around the sensor array (e.g., the impact of metal objects near an antenna array on its beam pattern).

We develop an eigenstructure-based method for simultaneously estimating the DOA’s and the unknown coupling, gain, and phase parameters. They made the following assumptions which are valid for our system:

1. The signals and the noise processes are stationary and ergodic over the observation period.

2. The columns of $$B = C \Gamma A$$ are linearly independent.

3. The signals are not perfectly correlated.

4. The noise is uncorrelated with the signals, and its covariance matrix is full rank and is known except for a constant multiplication.

The general form of received signal covriance matrix by including the mutual coupling matrix (MCM), C, can be expressed similar to eq18 as the following:

$R_x = C \Gamma A(\boldsymbol{\theta}) R_s A^H(\boldsymbol{\theta}) \Gamma^H C^H + R_n \quad \quad (34)$

The signal covariance $$R_s$$ is the middle-term expectation in eq18 where we have different received powers for different Dopplers. In fact, $$R_x$$ is fully characterized by $$2MN - N^2 + 1$$ since it is a semi-positive definite matrix with N (eigenvalues, eigenvector) pairs for N targets with N different Dopplers. Specifically, there are N+1 real eigenvalues *, and $$2NM - N^2 -N$$ parameters that define N complex eigenvectors satisfying the orthonormality constraints. On the other hand, the number of unknowns in $$C, \Gamma, A(\theta)$$ and the noise is $$rN+N^2+1+2(M-1)+P$$ where r=1 for azimuth only estimation and it is 2 for both elevation and azimuth estimation, N^2 is the unknowns in R_s, 1 is the noise power and 2(M-1) is the number of unknowns in $$\Gamma$$ (knowing $$\Gamma_{11} =1$$ and they are complex) and P is the number of unknowns in MCM. Though, the following should be true:

$\begin{split}2 M N - N^2 +1 & \geq rN + N^2 + 1 + 2(M-1) + P \\ M & \geq \frac{2N^2+rN+P-2}{2(N-1)} \quad\quad \quad\quad (35)\end{split}$

So, this means that the unknowns, especially the channel mismatches and the MCM, can be found if and only if eq35 is satisfied.

The orthogonality of the signal and noise spaces suggest that there should be a set of angles, coupling and mismatch coefficients such that:

$J_c = \sum^{N}_{n=1} || \hat{U}^H C \Gamma \boldsymbol{a}(\theta_n) ||^2 = 0\quad\quad (36)$

where $$\hat{U}$$ is the estimated U –-noise space bases. If N and $$\theta$$ are known, solving the problem for a general form of C is very difficult to satisfy the condition for the invertibility of the resulting matrixes in the solution. However, 1 suggests that if we are considering C for a ULA or UCA, one can take the matrix out of the product for forming a vector out of all unknown in C and right multiplying it with a matrix made of the elements in $$\Gamma a(\theta_n)$$. The approach is very interesting, but it cannot be used for a general array geometry. But Ahmet in 3 has a general lemma by which any matrix of C multiplied to a vector x can be written by a new matrix Q(x)c where c is a vector composed of all unknowns in C. Due to array symmetry and the network symmetry in a passive network, we can reduce the number of mutual coupling coefficients from $$M^2$$ to less than $$M^2/2$$. If the distinct elements in C are $$\boldsymbol{c} = {c_1, c_2, \cdots, c_L}$$, then $$C \boldsymbol{a} (\theta)$$ can be expressed as:

$\begin{split}C \boldsymbol{a} (\theta) & = T \boldsymbol{c} \\ [T]_l & = E_l \boldsymbol{a} (\theta) \quad\quad \quad\quad (37)\end{split}$

where $$[T]_l$$ is the l’th column of transformation matrix. To show this, consider that:

$C = \sum^{L}_{l=1} E_l \boldsymbol{c} \quad\quad (38)$

Therefore, it is straightforward to derive eq37.

The proposed algorithm in 1 consists of the 3 steps:

1. Initialization: finding $$\hat{R}_x$$, setting $$C, \Gamma$$.

2. Estimating N target DOAs by finding the peaks in the following map:

$P(\theta) = \frac{1}{||\hat{U}^H C \Gamma \boldsymbol{a}(\theta)||^{-2}}$
3. Finding $$\Gamma$$ by minimizing $$J_c$$ and a constraint that $$\Gamma_{11} = 1$$

4. Finding C by applying the matrix-vector conversion based on the array type (only ULA and UCA in this paper) and minimizing $$J_c$$ for $$c_{11}=1$$ or any linear constraint of $$w^T c=a$$ (c is the vectorized form of unknown coupling coefficients).

Converges as it minimizes the objective function, $$J_c$$, at each iteration since the objective function is convex.

## Channel Gain/phase compensation and frequency shift calibration¶

The presence of channel mismatches makes steering vectors to be lienarly dependent, therefore, it increases the error in angle detection. This is intuitively understandable by looking at the steering vectors as the bases of the angle space. In idea, each target direction can be expressed as a linear combination of all steering vectors; however, breaking orthogonality between them changes the space and the target representation. It can be shown that for small random perturbations of gain and phase of channels, $$\Gamma$$, the angle detection error is not high, and it is negligible. In the presence of high phase/gain imbalances across channels, on the other hand, the direction finding fails to detect an accurate target location, and the background noise is very high.

In FMCW radars, any delay difference between the generator to the receiver mixer among channels makes a frequency shift such that it is different for each channel. Thus, a target at the range R from a reference channel is shifted to R=Delta R_i for the i’th channel. Although the shift depends on the FMCW chirp slope, K, it is constant for all channels:

$\Delta f_i = \frac{K}{K_{calib}} \Delta f_{i, calib} \quad\quad (39)$

where $$\Delta f_i$$ is the new frequency shift when the radar is calibrated for the chirp slope of $$K_{calib}$$ .

The train-based calibration requires an object as a reference to be placed at zero azimuth and elevation angles. The distance of the target should be greater than what is computed in eq27, in order to avoid near-range effect. Then, following the below steps gives the MIMO calibration for gain/phase mismatches and the frequency shifts:

1. Putting a corner reflector at a far range on the broadside

2. Activating transmitters in sequence such that in each chirp interval one Tx is active, and receiving from all receivers

3. Finding the exact range profile peak corresponding to the target assuming that there is no other reflection §

4. Applying fine tuned frequency matching to detect sub-frequency resolution frequency shifts with respect to the first channel

5. Analyzing the stability of frequency shifts for each transceiver over time. They should be stable enough to trust the measurement.

*

one if for the noise variance.

matrixes should be at least full-column rank to have a solution.

one must obtain the range peak by using frequency match filtering to get the exact range profile peak location.

§

one has to make sure that there is no other reflections from the particular range that the corner reflector is.