# Derivative filter in Python

In a research paper (section 2.1, 3rd paragraph) about EEG, the authors note that the spectral amplitude profile of a signal is inversely proportional to frequency. To correct for this trend, they propose to apply a derivative filter to the signal.

My question is: Is there a Python function which implements such a derivative filter ? Is the savgol_filter function from the Scipy module suited to this task ? If not, how could I design such a filter in Python ?

• Can you link to that paper? derivative only means that it's the derivative of some other filter, and as far as I can tell, this tells me nothing about what the frequency response of the filter would look like – Marcus Müller Sep 28 '17 at 9:05
• @MarcusMüller : I added the link in the post. The paper does not say which filter is used. – Pouteri Sep 28 '17 at 9:07
• @MarcusMüller Derivation has a magnitude frequency response $|\omega|$ and a phase frequency response $\frac{\pi}{2}$ for positive frequencies and $-\frac{\pi}{2}$ for negative frequencies. – Olli Niemitalo Sep 28 '17 at 12:57
• @OlliNiemitalo a filter that implements derivation, certainly, but a filter that is a derivative of another filter is something else, and because I've confused these two terms myself, I asked for original reference :) – Marcus Müller Sep 28 '17 at 14:21

## 2 Answers

Let $f(x)$ be a signal band-limited to frequencies $(-\pi,\, \pi)$. Given $f(x)$ as input, the same $f(x)$ is given as output by a system that has as its impulse response the sinc function:

$$\operatorname{sinc}(x) = \left\{\begin{array}{ll}1&\text{if }x = 0,\\ \frac{\sin(\pi x)}{\pi x}&\text{otherwise.}\end{array}\right.$$

Taking the derivative $f'(x)$ of signal $f(x)$ is a linear time-invariant operation. By associativity, $f(x)$ is differentiated by a system that has as its impulse response the derivative of $\operatorname{sinc}(x):$

$$\operatorname{sinc}'(x) = \left\{\begin{array}{ll}0&\text{if }x = 0,\\ \frac{\cos(\pi x)}{x} - \frac{\sin(\pi x)}{\pi x^2}&\text{otherwise.}\end{array}\right.$$

Both of $\operatorname{sinc}(x)$ and $\operatorname{sinc}'(x)$ are band-limited to $(-\pi,\, \pi)$ and can thus be sampled at integer $x$ without aliasing. Sampling $\operatorname{sinc}'(x)$ at integer $x$ gives the ideal discrete-time impulse response for you to use in filtering samples of $f(x)$ at integer $x$ to obtain samples of $f'(x)$. Figure 1. $\operatorname{sinc}(x)$ (red) and its derivative $\operatorname{sinc}'(x)$

The infinitely long ideal impulse response can be multiplied by a window function to obtain a realizable impulse response. SciPy can calculate several types of window functions and do finite-impulse-response (FIR) filtering with an arbitrary impulse response by scipy.signal.convolve. You need to delay the impulse response to make it that of a causal system.

• Nice theory. But don't the very long tails make this difficult in practice ? If I'm not mistaken, the sum |sinc'(k)| at positive integers is 1 + 1/2 + 1/3 ... diverges. – denis Nov 10 '17 at 15:39
• @denis It diverges, but it only means that for some "pathological" bounded inputs like ..., -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, ... the output grows without bound as the window length is increased. It's the same kind of situation as with sinc interpolation, see question: Signal values we will 'miss' between sampling instances during sampling of band limited signals. – Olli Niemitalo Nov 10 '17 at 23:18

Here are some links and notes on various derivative filters, probably TL;DR .

For uniformly spaced data, use np.gradient ; this calculates central differences $(x_{i+1} - x_{i-1}) \, / \, h$, and fairs the edges nicely. For better filters, combine np.gradient s at different spacings, e.g.
$^4/_3 \ (x_{i+1} - x_{i-1}) \ - \ ^1/_3 \ (x_{i+2} - x_{i-2})$; see Finite_difference_coefficient.

Although it's common to use np.diff without thinking, one-sided differences $(x_{i+1} - x_i) \, / \, h$ amplify high-frequency noise — try [1 -1 1 -1 ...] . Use np.gradient instead.

You could use Savitzky-Golay. But you usually want to add constraints at various frequencies, e.g. derivfilter( [1 -1 1 -1 ...] ) = 0: playing around vs. off-the-shelf.

For very noisy data, there seem to be two main approaches:

1. filter first, e.g. with an order-2 Butterworth low-pass filter, before anything else
2. fit splines, which have smooth derivatives; see spline-fitted-1d-data on SO.