Combining equations 4, 8, and 21, we can derive:

where and are introduced matrices and are diagonal and positive definite.

In order to make the right matrix orthonormal, we make two assumptions:

- The noise is close to white noise in the sense that .
- The signal is orthogonal to the noise in the sense that .

We let denote the right matrix of the last equation in 22, then

where

when .

Since is an orthogonal matrix, then . Since , then , thus . In the same way, since , thus . Then,

when . Thus, we prove that when and are appropriately chosen, and is orthonormal.

2020-02-21