Because equation 7 is a singular value decomposition (SVD) of the signal matrix
, the left matrix in equation 7 is a unitary matrix:
|
(21) |
Combining equations 4, 8, and 21, we can derive:
|
(22) |
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
|
(23) |
where
|
(24) |
when
.
|
(25) |
Since
is an orthogonal matrix, then
. Since
, then
, thus
. In the same way, since
, thus
.
Then,
|
(26) |
|
(27) |
|
(28) |
when
.
Thus, we prove that
when and are appropriately chosen, and
is orthonormal.
2020-02-21