In this blog post series on signal processing for massive MIMO applications, we address the research issues outlined in Part 7 of our previous series on massive MIMO
Eigen value decomposition (EVD) based blind channel estimation techniques will be considered from two perspectives in the upcoming technical notes: (i) as a potential approach to improve spectral efficiency since it requires no or a minimal number of pilot symbols [5], and (ii) implementation complexity. The interest in subspace projection based techniques stems from the fact that channel estimation based on the presence of pilot symbols suffers from pilot-contamination effects in a multi-cell multi-user MIMO systems with very large antenna arrays, which turns out to be an artifact of linear techniques such least square [6].
A survey of recent literature on linear channel estimation and detection comes up with two notable works: (i) “Low-complexity polynomial channel estimation in large scale MIMO with arbitrary statistics” [4] and (ii) “Approximate matrix inversion for high throughput data detection in large scale MIMO uplink” [7]. These works rely on efficient series expansion for matrix inversion. To my knowledge this technique was first introduced by Nicolas Le Josse [8]. Using the Cayley-Hamilton theorem [8], a matrix inversion has been approximated with a finite sum of a weighted matrix polynomial [8].
For the sake of illustration, consider a covariance matrix of the form 
. The structure of this matrix is exploited to propose low complexity techniques in [4] and [7]. The following series expansion is used:
Where the scaling factor φ satisfies |1-φλi |<1 for all eigen values λi of the covariance matrix. Equation (1) lends itself well for recursive matrix-matrix multiplication. In [7] a Newman series approximation is adopted. A special case when the series is limited to two terms is presented in [7].
The implementation approach using polynomial expansion can resort to pipelined systolic array for supporting high throughputs. This is easily portable in FPGA-based processing boards like the TitanMIMO testbed [10].
Another approach, which to our knowledge has not been addressed in the literature yet, can be exploited in the case of distributed arrays. Such a case is supported by the TitanMIMO system where different TitanMIMO clusters can collaborate in such a way that the inversion of the large matrix can be perceived as an inverse of a partitioned matrix as follows (in the case of two TitanMIMO clusters) [9]:
Where
