Generalized Least Square Method

In the previous blog, we have discussed the reciprocity calibration for massive MIMO system using direct-path LS methods, or Argos. The calibration coefficients are estimated by the received signal from the single reference antenna.

In this blog, we will show how the generalized LS method is used in the massive MIMO reciprocity calibration. The full set of received signals in the equation

are considered by introducing the a least-square cost function

To find the minimum value of the LS cost function, the gradient of the cost function is set to zero for the calibration coefficient vector b:

which yields the estimated calibration coefficient as

where u1 is the first column of a matrix U,U1 is a matrix made of the N-1 last columns of U. The matrix is defined as

Generalized Weighted LS method

If the antenna coupling gains βn,p is pre-known, as we discussed in the previous blog, then the performance of the estimator for LS method can be improved with this prior information. In this case, the cost function can be rewritten as

The minimum value of the cost function can be found by setting its gradient to zero:

Then

where v1 is the first column of a matrix VV1 is a matrix made of the N-1 last columns of V. The matrix  is defined as

Simulation Results

We have simulated Argos, LS and weighted LS methods in matlab, the results are shown as below

Argos Simulation

MSE of calibration coefficients averaged over the entire array as a function of calibration SNR