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

[1]. 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 [2] [3]. 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 [4].

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

References

[1] M. Ahmed Ouameur, “Massive MIMO – Part 6: Estimation and capacity limits due to transceiver impairments,” http://www.nutaq.com/blog/massive-mimo-%E2%80%93-part-6-estimation-and-capacity-limits-due-transceiver-impairments, 2014[2] M. Ahmed Ouameur, “Massive MIMO – Part 3: Capacity, coherence time and hardware capability,” http://www.nutaq.com/blog/massive-mimo-%E2%80%93-part-3-capacity-coherence-time-and-hardware-capability, 2014[3] 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[4] 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[5] 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[6] R. Muller, L. Cottatellucci and M. Vehkapera, “Blind Pilot Decontamination,” SUBMITTED TO IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, 2014[7] 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[8] 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[9]  Beal, M.J., Variational Algorithms for Approximate Bayesian Inference, PhD. Thesis, Gatsby Computational Neuroscience Unit, University College London, 2003.[10] http://www.nutaq.com/products/titanmimo-4