DCO's, Anti-Aliasing, and Filters

[Brian Towles]

I been reading this thread for a while, but now I'm moved to words ... ;)

Rob wrote:

> ANY and ALL digital signals have aliasing...There is no way around it.
> 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. This is really where aliasing occurs. Its about the
> space between the dots.
>
> We say that this is a straight line, but more often, it is not. The
> natural curve of the original analog signal is gone.
>
> The best way to describe aliasing is to think of what aliasing means in
> dictionary terms:something is acting or taking on the characteristic of
> something else.
>
> In our terms, it means that there is insufficient numerical frequency to
> accurately represent all the harmonic characteristics of our original
> information.

This is _not_ strictly true ... however, you are stepping into the realm of
fourier transforms and mathematical representations of signals.  Most
DIYer's probably have little practical need for what's mathematically going
on here, but it's not bad to know :)  When you mention the straight-line
interpolation of the output signal, you're sighting a problem with the
reconstruction of a signal.  As long as the original sample met the Nyquist
condition ( fmax <= 2*fsample ) and if you ignore the effects of quantizing
the signal level, the digital representation stores _all_ of the signal's
information.  So, a discrete set of points can accurately represent an
analog signal.  However, the problems arise when the signal must be
recovered.  Theory tells us that an ideal low-pass filter can perform this
operation, but realizing this filter is impossible.  So, people make
approximations with high-order filters ....

> 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.
> In the picture example, the dots are represented in x and y, whereas in
> our example, they are represented by y vs. time.
>
> So, from what I have seen, all anti-aliasing filters do is limit
> bandwidth. The anti-aliasing filter is a bandpass filter with a steep
> rolloff after the upper and lower limits of our hearing (10hz to 25khz)
> that makes the aliasing inaudible, or at least lessened..
> But, if you did not record with sufficient sampling rate essentially you
> are just going to make muddy semi-sinusoidal noise. Try it sometime..
> I hope this helps anyone who was lost in the shuffle of the discussion.

These bandpass filters you mention are just a practical implementation of
the ideal filters I mentioned earlier.  Also, the linear approximation you
noted relates to using a first-order filter on the DAC output.
Higher-order filters give better and better approximations. So, now I
finally come to the two possible definitions of aliasing:

1. When you sample a signal whose frequency content is greater than half of
your sampling frequency, aliasing occurs ( this is Nyquist stuff ).  No
matter what ADC and DAC techniques you use, the signal is inaccurate and
cannot be perfectly reconstructed - I'd call this aliasing.

2. If you perfectly sample a signal, but use an approximate filter after
the DAC, some extra frequency information will be passed through the filter
- I wouldn't call this aliasing.

There exist more problems when you begin to consider the quantization
effects of ADC and DAC, but we'll let that go ...

Hope that helps and if I'm wrong, shoot me down! ;)

Brian