[music-dsp] Bandlimiting, Aliasing and Reconstructed Signals

Charles Turner vze26m98 at optonline.net
Thu Dec 23 08:56:30 EST 2010

```Happy Holidays Everyone!

I wanted to ask a question provoked by reading the SuperCollider Users' mailing list, which had me thinking I didn't understand the underlying concepts that were being discussed.

I went back and looked at stuff like Hamming's _Digital Filters_ and Oppenheim's _Signals and Systems_, although I'm greatly challenged by the math most of the time. Also, the discussions of sampling theory are just that: the authors are happy to shift back and forth from A->D and D->A to clarify a general theory, but as I'm interested in SYNTHESIS, taking away the finer points of signal "reconstruction" isn't always easy.

Here's my limit case: let's assume some typical laptop with CD-quality sound generation capability with a sample rate of 44.1khz and sample size of 16 bits. I create a sinusoidal waveform on the computer with a period of 4,410hz. I choose to create this waveform by feeding 4,410 divisions of the unit circle into a sine function. In other words, I calculate a unique value for each sample of the period at the sample rate of the laptop's D->A converter.

As this waveform is sent out the DAC, I assume it's subjected to a zero order hold of approximately 0.023 milliseconds. The DAC may also do it's own filtering of the signal before going out to a set of speakers.

My questions are:

1) Is the synthesized signal aliased? If so, how can we anti-alias it?

2) Is the signal band-limited? If not, do we want it to be, and how do we do it?

I'd also ask the same question about a similarly synthesized square wave. That may seems a bit simple, but there was the assertion on the SC-list that "smoothing" (I think that was the word) helped a loudspeaker figure out where it needed to be at a given point in time. I understand generally the point the poster was making, but isn't this a slippery slope? Not all speakers are designed with the same frequency response, so unless we tailor waveform synthesis to the specific characteristics of a loudspeaker, aren't we in danger of smoothing either too much or too little?

Also, a square wave is a square wave: it has sharp transitions. What timbral or spectral components of a square wave are intrinsic to its waveform, and what is introduced by a particular DAC and speaker combination? Or in other words, is the acoustic result of a synthesized square wave its resultant output, or is it something that "sounds good"?