Problem 3: Non unique solutions

For certain situations the solution of the adaptive optimization is not well defined:

• For partials near dc Phase affects amplitude.

• For nearby partials ( $\Delta_\Omega < \frac{2\pi}{N}$, $N$=block size) phase and frequency can affect amplitude.

Solution:

• Minimize a regularized objective function:

  
  

  $$E = \hbox{Min} \longrightarrow E + \lambda R(p) = \hbox{Min}$$
  
  

• The regularization term $R(p)$ is a function of the partial parameters and is choosen to favor the desired solutions.

Example:

• To ensure zero phase for partials close to dc we use

  
  

  $$R(p)= \sum((A_i+a_in)^2 )$$
  
  

  for all partials $i$ and times $n$ with instantanous frequency $\Omega(n) <\frac{2\pi}{N}$

• Phase deviation from zero has to be compensated by increase in amplitude parameters to achieve the same solution
  $\Rightarrow$ the regularization term enforces zero phase.