# Mathematical proof of relationship between derivative and convolution

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?

• Hint: In the frequency domain, both taking the derivative and convolution are multiplications with frequency-dependent factor. – Niki Estner Sep 26 '13 at 11:24

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.