# Reciprocity Calibration for Massive MIMO -Part I

## Introduction

In modern communication systems, pilot symbols are transmitted from every antenna of a base station (BS) in the downlink channel and received at the terminal side, then sent back to the BS with channel state information (CSI) to calculate pre-coding coefficients. In Massive MIMO system, such a procedure would significantly degenerate the spectral efficiency due to the amount of feedback information required from the large number of BS antennas. Instead, a more common approach is to compute proper pre-coding coefficients based on uplink CSI based on the reciprocity of the channel when the BS is operating in TDD mode.

In wireless systems, we generally assume that the propagation channel is reciprocal. But the different transceiver radio frequency (RF) chains are usually not. Hence, when utilizing reciprocity to calculate the pre-coding coefficients, we need to estimate the different frequency responses between the uplink and downlink hardware chains. Such a process is called reciprocity calibration [1].

## Channel Reciprocity

Due to the internal electronics of the BS and mobile stations (MS), the measured up/downlink channels are not only determined by the propagation channels, but also influenced by the RF chains of BS/MS. If we let the measured uplink and downlink radio channels between the BS and MS be denoted as

where *p* ∈[0,…,P-1] is the BS antenna index, *q* ∈[0,…,Q-1] is the MS antenna index, *r*^{B} and *r*^{M} represent the BS and MS receiver RF chains, t^{B} and t^{M} represent the BS and MS transmitter RF chains, and H^{U} and H^{D} are the uplink and the downlink propagation channels, respectively.

A relation between the uplink and downlink radio channels can be established as

If the perfect reciprocity of the propagation channel is assumed, which means H^{U}_{p.q} = H^{D}_{q,p} , then we can define calibration coefficient *b _{p,q}* as

The calibration coefficient b_{p,q} allows us to estimate the downlink channel based on the uplink channels. For the non-reciprocity between radio channels, they can be calibrated externally, i.e., by feeding back the downlink channel. However, such approach cannot be applied to a massive MIMO system, since the number of channel estimates to feedback from MSs to the BS scales with *P*.

## Internal Calibration between the BS radios

Instead of relying on external calibration, Massive MIMO system uses an internal reciprocity calibration method, which does not need the feedback from MSs. Considering the calibration coefficient between BS radios defined as

Then

Therefore, calibration between radios *p* and *q* can be achieved if their forward and reverse channels to another BS radio *n* are jointly processed [2]. We set the radio *n* as the reference radio. Then

We define as a relative downlink channel that contains b_{n,q}. Thus, relative downlink channels can be obtained by multiplying the respective uplink channels with their respective calibration coefficients to the reference radio n.

For a distributed Massive MIMO system, we define

Where . is another relative downlink channel, which assures different calibration coefficients are independent from each other [3].

## Reciprocity Calibration for Massive MIMO

To estimate the calibration coefficients b_{p}, we make the *P* antennas transmit a pilot symbol *p* = 1 one-by-one and receive on the other *P – *1 silent antennas. If *y _{p,n}* is the signal received at antenna

*p*and transmitted at antenna

*n*, then

Where *h _{n,p}* is the propagation channel between the BS antennas

*n*and

*p*, and represents an independent zero-mean Gaussian distributed random vector, with a variance of

*N*and circularly symmetric complex variables.

_{0}In the next series of this blog, we are going discuss how to find b_{m} using different methods, including direct-path (argo), Generalized LS and Generalized weighted LS, and the implementation of these methods on Nutaq’s Titan MIMO-6.

## Reference

[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.

[2] C. Shepard, H. Yu, N. Anand, E. Li, T. Marzetta, R. Yang, and L. Zhong, “Argos: Practical many-antenna base stations,” in Proceedings of the 18^{th} Annual International Conference on Mobile Computing and Networking, ser. Mobicom ’12. New York, NY, USA: ACM, 2012, pp. 53–64.

[3] R. Rogalin, O. Bursalioglu, H. Papadopoulos, G. Caire, A. Molisch, A. Michaloliakos, V. Balan, and K. Psounis, “Scalable synchronization and reciprocity calibration for distributed multiuser MIMO,” IEEE Transactions on Wireless Communications, vol. PP, no. 99, pp. 1–17, 2014.