Multiple-Input Multiple-Output Channel Models: Theory and Practice

APPENDIX A

AN INTRODUCTION TO TENSOR ALGEBRA

This appendix presents a brief introduction to tensor algebra and the higher-order singular value decomposition (HOSVD). Tensor algebra is used in Chapter 3 to extend the idea of correlation to receive-transmit-delay space and to develop a novel wideband MIMO channel model.

Multiple-antenna channels are often expressed using multidimensional quantities such as vectors, matrices, and tensors. Here, we include a brief overview of tensor algebra and introduce notations used throughout the text. This section focuses specifically on notations included in this text only and as a result is very limited in scope. It is not necessary to have a good understanding of tensor algebra to understand the subjects covered in this text. It is, however, important to understand the notation and operations covered in this section to do so. Tensor algebra is a very mature topic and is the subject of many textbooks and theses, including Refs. 136 and 137.

The chapter begins by discussing the HOSVD, which is a multidimensional extension to the matrix singular value decomposition (SVD). In presenting the HOSVD, we summarize some important tensor operations used in the rest of the text. The HOSVD also serves as the inspiration for the structured model.

A.1 NOTATION AND TENSOR ORDER

We denote scalars using lowercase letters with no indices attached; for example, {a, b, c,…}. We denote vectors using bold lowercase letters {a, b, c,…}. We denote matrices using bold ...