Continuous Case
Let $ z \left( t \right) = \left( h \ast x \right) \left( t \right) $. What is the derivative of $ z \left( t \right) $ with respect to $ x \left( t \right) $?
Discrete Case
Given $2$ vectors $ x \in \mathbb{R}^{n} $ and $ h \in \mathbb{R}^{m} $, their convolution given by
$$ z = h \ast x $$
What would be the gradient of $ z $ with respect to $ x $? And what would be the gradient w ith respect to $ x $ of the following quadratic cost function?
$$ \frac{1}{2} {\left\| h \ast x - y \right\|}_{2}^{2} $$