On Wed, Jun 5, 2013 at 12:17 AM, Dennis Tillman <email@example.com> wrote:
My understanding is that that isn't exactly true. An analog signal isThat is incorrect. The exact same problem exists in the
continuous-domain Fourier transform, and basically in any other thing
that assumes an unbounded continuous domain. This issue is called the
"boundary problem". It has nothing to do with there being discrete
points. It's about the fact that at some boundary (here, the start or
the end) on one side you have enough information to reason about
what's happening, and on the other side of that point you don't.
Notice I said "information". According to the Shannon-Nyquist Sampling
Theorem (which is true), the discrete-time signal contains exactly the
same information as the continuous-time signal, up to Fs/2. Since we
don't care about what's above Fs/2, that's *exactly* what we want.
Discrete and continuous Fourier transforms yield exactly the same
results, but to compare you'd have to do a lot of pencil-and-paper
computation, so I'm not going to be able to show you something
The Sampling Theorem has very big weaknesses. For one thing, it
assumes an infinitely-long oversampling filter during regeneration. Or
more generally, when going from digital back to analog, it says it's
*possible* to reconstruct the signal, but doesn't say *how*.
Technology has shown it's absolutely impossible. In digital audio, you
could encounter this waveform (assume 1-bit sampling depth):
What does it look like? Well, according to the Sampling Theorem, it's
a sinewave which has had about 50 cycles during that span of time. To
arrive at this conclusion, you have to analyze a long chunk of digital
signal to even have the information needed to do it. That's because
future samples in a digital signal may change your interpretation of
the current sample you're interested in, and it's a major weakness of
the Sampling Theorem. In fact, the closer the frequency you're trying
to convey is to Fs/2, the more data you need to interpret it correctly
and reconstruct it in analog - to have full sampling capability up to
Fs/2 you have to make an infinitely-long oversampling filter, which is
This non-determinism of analog waveforms produced according to the
Sampling Theorem is just one weakness. Another (perhaps linked, I
don't know..) is how badly it degrades.
In analog, and this is especially applicable in audio, you have the
following main forms of degradation:
1. noise. Noise is usually distributed in some uniform way and it just
means the signal is a bit less clear. Noise is extremely easy to
ignore, our minds are well trained for ignoring constant partials in
signals. If you don't trust me, try this experiment that will use an
optical signal. You're staring at a monitor right now. Is it dirty? Is
the backlight equally bright everywhere? Our eyes normally don't track
objects on the screen which helps hide this. Open up a new Notepad (it
has a white background) and maximize it. Then open a second one, and
make the window tiny, one inch across. Now move that new window while
the first one is in the background creating a white field of light.
While moving the smaller window, *track it with your eyes*. The object
needs to be this big, the mouse cursor itself wouldn't work, it's too
small. If you succeed at tracking the smaller window with your eyes
while it's moving, you should see all the dirt you've been staring at
for months without even knowing it's there.
2. non-linearity. This creates *harmonic* distortion which means that
even the error is in some way related to the input signal. In music,
it means the erroneous part of the signal is *musically related* to
the original signal, therefore musically pleasing. Don't trust me
distortion is musical? Ask Jimmy Hendrix.
In digital, you have however the following main forms of degradation:
1. quantization noise. It depends on the *level* of the signal and on
how many times it crosses a specific boundary. This is also called
"bit crushing". The error signal is a high-frequency "zipper" tone
that's difficult to ignore. If you want an example, find a wood-grain
table and slide your finger nail across it. You'll hear a zip. This
happens in exactly the same way as quantization: a specific signal (in
this case, the position of your finger in space) repeatedly crosses
sampling boundaries. If you use something that's more finely-grained,
like a very fine plastic-top office desk, the zipper noise becomes
less audible because there are more, smaller, finer "steps" in the
grain. If you use glass there's nothing because it's so fine. Now
here's the kicker: the best DACs in the world are unable to create
steps so fine that we are unable to hear them. A lot of analog
filtering has to happen for quantization noise to be removed
2. Aliasing. This one's absolutely terrible. The error signal's
frequency content is in no way musically related to the desired
signal. In fact, as your melody progresses, this one plays *against
you*. It's like having a drunken monkey sit beside you on the piano,
and it bashes keys exactly when you press keys.
I currently don't know of anything that could alleviate those issues.
There's no other way of digitally representing a signal sampled from
the real world that I know of which doesn't have those problems. I
have heard of mip-mapped sampling, but I haven't tried it yet and I
don't know whether it helps, in theory there are some things it could
be better at.
Just my 2 euro-cents.
In certain cases the discontinuity can be minimized, for example if the