Digital Signal Processing Reference
In-Depth Information
Using the individual columns
j
of W
(s),
a
and replacing the summation with
the expectation operation, the above equation can be written as:
By substituting:
based on the assumption that the source signals are statistically independent
in the sparse domain the above equation can be written as:
and
Then the estimated
a
matrix is:
There is no closed form solution to minimize both L2 and L1 norm of (4)
simultaneously. However, we can solve this system iteratively by applying
the Linear Equality Constraints (LEC) optimization technique [10] by noting
that (9) can be used as the set of linear constraints. The LEC corresponds to:
The LEC in essence corresponds to finding under the linearity
constraint such that the (L1 norm) is minimized. The LEC
optimization problem can be solved by applying the line search together with
the projection gradient method. One of the ways to find the lines or direction
of lines is by applying Armijo rules of line search. We applied this technique.
