We often teach or learn about filters in continuous time, but then need to implement them in discrete time (e.g., in code) on data acquired at discrete sample times. This notebook shows one way to design and implement a simple first-order low-pass filter in discrete time. The example is written in Python and uses Matplotlib.

Let $x_k\in\mathbb{R}$ be the signal that we wish to filter, where $k=0,1,2\ldots$ is the time index. Let $T>0$ be the sample period for the discrete-time filter. Let $\omega_c>0$ [rad/s] be the low-pass filter's $-3$ dB cutoff frequency. Then the filtered signal $y_k$ is (approximately) given by $$y_k \approx \left(\frac{2-T\omega_c}{2+T\omega_c}\right)y_{k-1} + \left(\frac{T\omega_c}{2+T\omega_c}\right)\left(x_k+x_{k-1}\right),$$ where $x_{k-1}$ and $y_{k-1}$ are the original and filtered signals, respectively, at one time step prior.

• first-order-lpf.ipynb (Jupyter notebook)

Here is an sample of the output.

This is relatively standard material. Supplementary web links are given in the Jupyter notebook.

You may wish to cite this work in your publications.

Joshua A. Marshall, How to Implement a First-Order Low-Pass Filter in Discrete Time, 2021, URL: https://github.com/botprof/first-order-low-pass-filter.

You might also use the BibTeX entry below.

@misc{Marshall2021,
  author = {Marshall, Joshua A.},
  title = {How to Implement a First-Order Low-Pass Filter in Discrete Time},
  year = {2021},
  howpublished = {\url{https://github.com/botprof/first-order-low-pass-filter}}
}

Source code examples in this notebook are subject to an MIT License.