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I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$.

The bound has to be evaluated taking the second derivatives of a function $l(x; A,u)$ wrt the unknown parameters $A$ and $u$, where

$$ l(x; A, u) = - \frac{1}{\sigma^2} \left[ \left( x-As(u)\right)^H \left( x-As(u)\right) \right] $$

Extending the product inside the function, I get: $$ l(x; A, u) = - \frac{1}{\sigma^2} \left[ x^Hx - x^H As - xs^HA^H - s^HA^HAs \right] $$

I should take the second derivative wrt $A$. I've seen in other questions that, for minimization problems, minimization can be performed taking the derivative wrt the complex conjugate and setting it to zero. (see Derivative of conjugate transpose of matrix )

The first derivative wrt $A$ should be: $$ \frac{\partial\, l(x; A, u)}{\partial A} = - \frac{1}{\sigma^2} \left[ - x^H s - s^HA^Hs \right] $$

Is the second derivative wrt to $A$ equal to $0$ or am I doing something wrong?

Any help would be greatly appreciated.

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