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 [email protected]
"Imagination is more important than knowledge."
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