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

APPENDIX B

PROOF OF THEOREMS FROM CHAPTER 3

This appendix presents proofs for key steps in the derivation of the structured model. It provides some insight into the mapping between tensor and matrix elements, the derivation of the wideband coupling coefficients, and the structured model synthesis equation.

B.1 PROOF OF (3.53)

We begin by hypothesizing that the Kronecker structure will allow us to line up the proper (m, n, d)th elements of the (i, j, k)th eigenvectors with the (m, n, d)th element of H, and we write the multiplication out explicitly to make sure that we get to the summation formula:

Using the definition for the Kronecker product, we multiply out each element.

B.2 PROOF OF (3.54)

We start off by expanding the definition of the wideband coupling coefficient ω_ijk,

Because both terms in the square brackets are scalars, we can therefore switch their order.

The elements of H are random variables, but the elements of the eigenvectors are not, therefore

B.3 PROOF OF ...