Statistical Models for Propagation Channels Between Internal BS radios

In the first part of this series blogs, we have discovered that the Massive MIMO system can use internal BS radios to achieve reciprocity calibration instead of relying on external calibrations. The focus of reciprocity calibration is to find out the calibration coefficients bp. A method has been proposed by sounding P antennas with a pilot one by one and receive on the other P 1 silent antennas, as shown below:

If the transmitted pilot is simply 1, then the received signal yp,n at antenna p and transmitted at antenna n is:


Based on this equation, finding the calibration coefficients bp is a statistic estimation problem.

Before we can tackle the problem, we need to know the parameter hn,p, which is the propagation channel between the BS antennas n and p. They are modeled as:

In the equation above, βn,p the channel gain from antenna coupling, which is decided by wavelength and intervals of antenna elements, φn,p  is the channel phase uniformly distributed from 0 to 2π and ωn,p  represents the multipath propagation with no dominant component, which can be modeled as a zero-mean Gaussian distribution.


In Nutaq’s design, we considered reference from other publications and roughly estimate the channel gain from antenna coupling with the equation βn,p = 0.03d-3.7[1].


Direct-Path Method (Argos)

Argos was first proposed by C. Shepard and H. Yu from Rice University. It estimates the calibration coefficients b = [b0,b1,…bp,…bp-1]T only using signals yn,p and yp,n. To simplify the notation system without losing generality, we set the reference antenna index to 0  and b0= 1. Then a least-square method can be applied here by trying to jointly optimize bp  and an,p with the following principle

And the solution to the equation above can be given by



[1] J.Vieira, F. Rusek and F. Tufvesson, “Reciprocity calibration methods for massive MIMO based on antenna coupling,” 2014 IEEE Global Communications Conference, Austin, TX, 2014, pp. 3708-3712.