Hi Dennis,

On Wed, Jun 5, 2013 at 12:17 AM, Dennis Tillman <dennis@ridesoft.com> wrote:

My understanding is that that isn't exactly true. An analog signal is

continuous. A physical sampler has to start and to end. The start and end

points are discontinuous. A Fourier transform can be performed on the analog

signal, but the Discrete Fourier Transform of the sampled signal will have

problems with the first and the last samples because they are discontinuous.

That 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

conclusive.

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):

1111111111111111111111111111101111111111111111111111111111111

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

impossible.

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.

D.

In certain cases the discontinuity can be minimized, for example if the

waveform being sampled is periodic and if by coincidence or design exactly

one or N (where N is an integer) periods is sampled then both the Fourier

transform and the Discrete Fourier Transform will yield the same results.

Beyond this point I suggest you discuss this in greater depth with someone

who does sampling for a living. I am satisfied that we have put to rest the

myth about "dip" in the 7D20. My focus is back to impossible things only a

7000 scope can do. See the latest photos I will publish in a few minutes.

Dennis

-----Original Message-----

From: cheater00. Sent: Tuesday, June 04, 2013 11:58 AM

<snip>

in a physical sampler (ideally) conversion from continuous to discrete time

and back gives you a flat frequency response up to Fs/2.

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