Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this community
What I'm looking for is the mathematical proof of why we can calculate derivatives as a simple convolution of a mask, and how do we get that mask? I know it has something to do with how the derivative is approximated
Can anyone direct me to some book or anything?
There's no mathematical proof per se; the fact is that finite differences are one method of approximating derivatives. Remember the definition of a derivative:
$$ f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} $$
The above equation states that to calculate the derivative of a function $f$ at position $x$, one takes the difference between the function evaluated at $x$ and the function evaluated at some small distance $\Delta x$ away. The ratio of the difference in the function value between the two points and the distance traveled along the $x$-axis is defined to be the derivative, as the distance between the two points becomes infinitesimally small.
A finite-difference derivative method, then, is just a way of approximating the above relationship using some finite spacing instead of the infinitely-small spacing implied by the limit. This type of approach makes sense for real-world applications where you might not analytically know the function $f(x)$ (and therefore can't evaluate its behavior across arbitrarily-small intervals.
With that said, there are a number of ways to approximate the derivative of a signal in this way; one example is the causal first-order difference:
$$ x'(t) \approx \frac{x(t) - x(t-T)}{T} $$
This is simple to implement, but performs best if $T$ is small relative to the inverse of the signal's bandwidth. A slightly more complicated method can provide better results in some cases (the second-order difference):
$$ x'(t) \approx \frac{x(t) - 2x(t-T) +x(t-2T)}{T^2} $$
I've used the "backward second-order difference" form above to show how they could be applied using causal filters, but note that they will have a resulting delay commensurate with the order of the approximation.
Assuming you want the derivative of a continuous band-limited (below Fs/2) signal represented by some sample points, then the sampling/reconstruction theorem can be used to demonstrate that an infinite convolution can be used to reconstruct the derivative of the signal as well as the signal. More realistically, a windowed convolution, using the same window as you would use for Sinc reconstruction or filtering, can be used for derivative-of-Sinc reconstruction. This finite length convolution kernel would then produce the derivative of the reconstructed signal at the sample points (and a poly-phase version could also estimate the derivative between sample points).
A mask can be created by computing the derivative of the Sinc reconstruction function, sampling that, and windowing it.
