. We start by discussing potential tips and techniques for reducing matrix-based manipulation/computation burdens. Recall from Part 3 of our Massive MIMO series that when considering real hardware capabilities and realistic channel coherence time, one might expect eigen-beamforming to outperform zero-forcing in terms of achieved capacity  . Therefore, to achieve a high capacity when using low-cost hardware to acquire channel state information, detect the received information symbols, and construct beamforming weights within a short coherence time, one needs to implement computationally demanding techniques like minimum mean square error (MMSE) which involve matrix inversion .
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 , 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 .
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”  and (ii) “Approximate matrix inversion for high throughput data detection in large scale MIMO uplink” . These works rely on efficient series expansion for matrix inversion. To my knowledge this technique was first introduced by Nicolas Le Josse . Using the Cayley-Hamilton theorem , a matrix inversion has been approximated with a finite sum of a weighted matrix polynomial .
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  and . 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  a Newman series approximation is adopted. A special case when the series is limited to two terms is presented in .
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 .
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) :
 M. Ahmed Ouameur, “Massive MIMO – Part 6: Estimation and capacity limits due to transceiver impairments,” https://nutaq.com/blog/massive-mimo-%E2%80%93-part-6-estimation-and-capacity-limits-due-transceiver-impairments, 2014 M. Ahmed Ouameur, “Massive MIMO – Part 3: Capacity, coherence time and hardware capability,” https://nutaq.com/blog/massive-mimo-%E2%80%93-part-3-capacity-coherence-time-and-hardware-capability, 2014 C. Shepard, N. Anand, and L. Zhong, “Practical Performance of MU-MIMO Precoding in Many-Antenna Base Stations,” http://www.ruf.rice.edu/~mobile/publications/shepard2013cellnet.pdf N. Shariati, E. Björnson, M. Bengtsson and Mérouane Debbah, “Low-Complexity Polynomial Channel Estimation in Large-Scale MIMO with Arbitrary Statistics,” Submitted to Journal of Selected Topics in Signal Processing – Special Issue on Signal Processing for Large-Scale MIMO Communications, January 2014 H. Quoc Ngo, E.G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012 R. Muller, L. Cottatellucci and M. Vehkapera, “Blind Pilot Decontamination,” SUBMITTED TO IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, 2014 M. Wu, Bei Yin ; A. Vosoughi, C. Studer, “Approximate matrix inversion for high-throughput data detection in the large-scale MIMO uplink,” IEEE International Symposium on Circuits and Systems (ISCAS), 2013 Le Josse L., Laot Christophe and Amis K., “Efficient Series Expansion for Matrix Inversion with Application to MMSE Equalization,” IEEE Communications Letters, Volume No 12 , Issue No 1, January 2008 Beal, M.J., Variational Algorithms for Approximate Bayesian Inference, PhD. Thesis, Gatsby Computational Neuroscience Unit, University College London, 2003. https://nutaq.com/products/titanmimo-4