AW: DCO's, Anti-Aliasing, and Filters

[Haible Juergen]

	> ANY and ALL digital signals have aliasing...There is no way around
it.

Not quite. If you have sampled a band limited analogue signal
at a sample rate larger than twice its bandwidth, you have
no aliasing. 

	>This is because between the samples, or dots that represent the
	>waveform, ANYTHING can happen, and we would never know about it
from the
	>digital recording. 

We can know, if we have a *little* extra information: that the original
signal was properly band limited, and where the band is located
in relation to the sampling frequency. (The second one is just
a theoretical problem, as we can asume the audio was in the
base band, and not, say, between 1Mhz and 1.005 Mhz.)

	>We say that this is a straight line, but more often, it is not. 

Actually, it is *never*. If it were, it would not be band limited.
Theory says that if the original signal was band limited, you
can interpolate between the points with a sin(x) / x function
(not straight lines), and get the original signal without error.
(Speaking of time discretisation, not the error you get from
limited level, i.e. number of bits, resolution.)

	> The
	>natural curve of the original analog signal is gone.

It's perfectly restored when you use the interpolation described
above.
In the frequency domain, that means a brickwall
low pass filter. Sure, there is no such filter in the real world,
but there is no limit approximating it, either. (other than a 
certain time delay thru the filter)

 The trick is that 3 pieces of information work together:
(1) The discrete points
(2) the "agreement" that the signal was properly band limited
(3) the agreement that the base band is used
These 3 things *together* will restore the full information.

When you receive the digital signal (i.e. only the "dots"), you
will asume that the sender has "agreed" to point (2) and (3),
because it would not have been reasonable to sample at
that specific rate otherwise. But you cannot tell *if* sampling
was done "reasonable", i.e. in conformance with Nyquist's
theorem. So you just do the obvious: Apply your sin(x) / x
interpolation (low pass filter). If sampling was ok, you can
restore the original signal (given your filter is a good enough
approximation). If it was not, you have aliasing.

	>So, what happens is that we have a recorded signal that has all the
	>elements to represent a harmonic below the information. Like not
having
	>enough dots when playing connect the dots.. If you dont have enough
dots
	>to accurately represent your image, you could think that a picture
of a
	>cow was a teacup.. This is in essence what aliasing is.

Nice description. While it does not apply to the simple sample
and playback operation, it can surely happen when you start
to process this once sampled signal, transpose it, resample it,
and what else they are doing in those hideous contraptions.  

JH.