I'm interested in numerically differentiating a noisy signal.
I saw that Stephen Boyd (Convex Optimization, page 312) uses the so-called total variaton method, based on the smoothing function
$\phi_{tv} = \sum\limits_{i=1}^{n-1}\left|\hat{x}_{i+1}-\hat{x}_i \right| = \left\|D \hat{x} \right\|_1$
The question is: is it more accurate to numerically differentiate the signal de-noised by using this method, or are there more advanced techniques to differentiate noisy data?
Thanks!