Basic Digital Signal Processing: Filters

Digital filters are ubiquitous in musical signal-processing. The design and implementation of filters is currently an very active area of research. Digital filter theory forms the foundation for understanding much of contemporary DSP work.

Links

In addition to the filter objects and instruments available in Max/MSP and RTcmix, you can find many more at:

maxobjects.com

a fairly comprehensive database for third-party Max/MSP objects (commercial, shareware and free). This is also a good place to search for objects that may do a particular task that isn't done by the distributed Cycling '74 objects.

The following are links to pages covering aspects of digital filter theory:

Luke's filter web page -- good basic tutorial by CMC's own Luke DuBois!
Digital Filters: An Introduction -- from "DSP Tutor", a more mathematical description of filter theory
Yehar's digital sound processing tutorial for the braindead! -- great name, pretty good info
Introduction to Digital Filters -- very mathematical (but very good) treatment, from Julius O. Smith

Applications and Examples

Fun filter examples from class:

week6a-examples.sit -- StuffIt archive with the class max/msp and rtcmix patches

individual patches as text files (for Windows users)
- lores1.txt
- cycleres1.txt
- butter.txt
- BUTTER.sco
  
  NOTE: the "BUTTER.sco" file contains the rtcmix script used in the "butter.txt" patch. You will need to download both of them.

We began class by discussing the basic types of filters:

lo-pass (low-pass): only let frequencies below a certain cutoff frequency get through, with a certain roll-off amount of attentuation for frequencies above the cutoff. Often these lo-pass filters have a resonant 'bump' right at the cutoff point.
hi-pass (high-pass): inverse of lo-pass. duh.
band-pass: only let frequencies in a certain pass-band get through the filter, specified by a center frequency and width. The width is usually given in Hz, but occasionally it is specified as the "Q" of the filter. I'll leave it as an exercise for you, the intrepid reader of this document to figure out what the heck a "Q" is.
band-reject: inverse of the band-pass filter.

Next we looked at how filters are implemented digitally, using filter equations. Happy discussion of Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) equations followed, including how past inputs ("feed forward", in FIR filters) and past outputs ("feedback", IIR filters) are used in filter design.

Two things are useful to try when looking at the action of a filter equation. The first is to look at how the filter operates on the lowest frequency possible and the highest frequency possible for a given sampling rate. The lowest frequency possible is essentially an unchanging number (called DC -- Direct Current)... the frequency is so doggoned low that it doesn't even oscillate! Assuming an amplitude range of our samples from -1.0 to 1.0, this would be represented as the following number stream:

1.0 1.0 1.0 1.0 1.0 1.0 1.0 . . .

The highest frequency possible for a given sampling rate sits right at the Nyquist frequency (1/2 the sampling rate). Assuming again the -1.0 to 1.0 range, and assuming that we got really lucky and our samples lined up exactly right, it would yield the following number stream:

1.0 -1.0 1.0 -1.0 1.0 -1.0 1.0 . . .

Cranking these numbers through simple filter equations (such as those discussed in Luke's tutorial above) show how these linear equations can modify a time-domain number stream. Amazing stuff!

The second thing to try is to follow what happens to a unit impulse (also called the Dirac delta function (band-limited in this case) if you want to impress your mathematical friends) when sent into the filter equation. This could be represented by the following number stream:

0.0 0.0 1.0 0.0 0.0 0.0 . . .

(note the "1.0" impulse.) Seeing what happens to this stream after being 'processed' through a simple filter equation will demonstrate why FIR filters are called 'finite' and IIR filters are 'infinite'. Here's a question for you -- are the comb filters we discussed last week FIR or IIR filters?

For most of the work we do, we won't need to know about filter equations and various digital filter design approaches (Butterworth filters, elliptical filters, biquad filters, etc.) directly. Instead we will probably be more concerned with acoustically-tangible qualities like the center frequency, cutoff frequency, bandwidth, etc. But it's sort of fun to fool around with these equations. Some people find it to be such astounding fun that they make a living at it.