[music-dsp] Bandlimiting, Aliasing and Reconstructed Signals

robert bristow-johnson rbj at audioimagination.com
Thu Dec 23 11:11:29 EST 2010

On Dec 23, 2010, at 8:56 AM, Charles Turner wrote:

> 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.

that is effectively the same as sampling a continuous-time sinusoidal  
signal.  since your sampling rate is 10 times faster than the sine  
wave's frequency, which well exceeds 2, then you've sampled it  
sufficiently fast and there should be no aliasing

> 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.

not if the DAC is a "one-bit" sigma-delta DAC (which are the cheap way  
of making audio or "multimedia" DACs).

> My questions are:
> 1) Is the synthesized signal aliased?

no.  not that particular signal.  now if you sampled a square wave,  
that would be a different story.

> If so, how can we anti-alias it?
> 2) Is the signal band-limited?

since it's a simple sine wave, it is bandlimited to the frequency of  
the sine wave (which is 1/10 Fs or 1/5 Nyquist).

> 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's different.  if you simply sample a hypothetical square wave,  
you'll have some nasty sounding aliases.  some of the square wave's  
harmonics will fold over around the Nyquist frequency and will be  
harmonic no longer.

> 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.

but bandlimited square waves are different than a continuous-time  
square wave.

> 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"?

do you know about, or how to compute Fourier series for simple  
waveforms?  that, plus understanding the sampling and reconstruction  
theorem will help.


r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."

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