Bandpass Hilbert Transformer

Project goal: design and implement a finite impulse response (FIR) band-pass Hilbert transformer that works in near real-time on 44100 Hz audio signals.

Context: This was originally a project for the third-year course Zajemanje in obdelava podatkov (Data Acquisition and Processing) at the Faculty of Mathematics and Physics and the University of Ljubljana, Slovenia, which I later put online for a public audience.

Demo

Here is the filter's bandpass and phase-shifting effect on a sequence of sinusoidal audio signals sampled at 44100 Hz—to hear the audio signals you can (carefully) unmute the video. The dotted blue line shows the inputted audio waveform and the red line shows the filtered output.

realtime-demo.mp4

Here are the filter specifications:

Frequency band	Attenuation	Phase shift
Below 500 Hz	Less than −40 dB	-
1000 Hz to 2000 Hz	0 dB with up to 1 dB ripple	π/2 radians
Above 2500 Hz	Less than −40 dB	-

As shown in the video, the filter passes and phase-shifts signals in the 1000 Hz to 2000 Hz passband and attenuates signals in the stopbands below 500 Hz and above 2500 Hz. The filter specs are clearly quite lenient, but that misses the point somewhat. The project is a pedagogical exercise in implementing a filter "by hand" rather than an attempt at designing high-performance, production-ready software, so the filter specifications are less important than the process of manually computing the filter kernel, applying a window function, implementing the overlap-add method for online convolution, etc.

Project structure

• The report directory contains the LaTeX source files for the project report
• The media directory holds figures and a video demonstrating real-time filtering.
• The src directory contains the project source code. Here is a breakdown of what each file does:
  • constants.py stores project-wide filter specifications
  • kernels.py contains functions for offline computation of test signals, window functions, and filter kernels and frequency responses
  • plotting.py generates the figures used in the LaTeX report
  • realtime.py is used to test the filter in real time using an audio signal from the computer microphone
  • hilbert.mplstyle is a basic style sheet for Matplotlib plots

Dependencies

The project is programmed in Python 3. Both the offline component (computing filter kernels and generating figures) and real-time component require:

• NumPy: for implementation of the fast Fourier transform and common mathematical functions
• Matplotlib: for plotting

The real-time component additionally requires:

• PortAudio: for capturing audio from the computer's microphone
• PyAudio: provides Python bindings for the C code in PortAudio
• Tkinter: for the real-time plotting interface

To locally compile the LaTeX report you'll need:

• A LaTeX installation (e.g. TeXLive)
• The pdflatex and latexmk programs, which should be included with most LaTeX installations
• The biber backend for BibLaTeX (to manage the bibliography)

Warning: the audio capture involved in the real-time filtering component may cause headaches depending on your operating system and hardware. I got everything working on macOS, which uses the Core Audio API, but ran into troubles with clipping and latency on a Linux machine using PulseAudio. Your mileage may vary.

Building

To locally build the figures and LaTeX report:

1. First change into the src directory and run python plotting.py, which will generate figures and save them to the media directory.
2. Once figures are generated, change into the report directory and run latexmk report.tex. You'll probably want to configure latexmk to use pdflatex as the LaTeX engine for compilation, since the report uses PDF graphics.

License

All original writing, figures, and video, including the contents of the report and media directories, are licensed under the Creative Commons Attribution-NonCommercial 4.0 International License. The project source code, i.e. the contents of the src directory, is licensed under the MIT License.