Eddy Deegan wrote:In the case of tape, I'm assuming that the limited frequency reproduction capabilities of the physical medium result in that 'exactly the same' output. I'm not sure I fully understand what 'band-limit' means in this context though.
In reality, everything is 'band-limited', meaning it all has a usable working frequency range (or bandwidth) and cannot work with signals outside that range. This is true of microphones, tape recorders, mixing consoles, loudspeakers, ears and so on...
However, for all of these things, the transition from the 'pass-band' -- the region where it works as intended -- to the 'stop-band' where it doesn't work at all, is usually relatively gentle (6 or 12dB/Octave, typically) -- at least initially -- and so may extend over several octaves.
In contrast, the transition for a basic 44.1k digital system has to be the infamous 'brick-wall' filter where it goes from working perfectly at 20kHz, to being completely deaf at 22.05kHz (in theory, at least).
So one of the compromises that has to be accepted when using digital is that very hard limit on the frequency range that can be accommodated -- hence, band-limited.
Of course, if you choose to run the digital system at 96kHz its band limits will be far beyond the natural limits of (almost) any microphone / mixer / loudspeaker / ear anyway, so we won't notice...
My assumption (I'm making a few!) is that the 'resolution' of a tape would be the smallest coherent unit of time that can be represented meaningfully on the grains of oxide as they pass over the read head
The size and proximity (density per unit length, in other words) of the magnetic domains determines the signal/noise ratio and equates broadly to the wordlength of a digital system. The width of the gap in the replay head defines the upper frequency bandwidth limit (known as the 'extinction frequency, actually).
If... the output of a sawtooth oscillator [is] processed by a sweeping analogue filter then I would imagine that a lot of information is lost between samples at 44.1kHz although that information is probably of no real consequence.
No, nothing is lost at all... and this is a fundamental aspect of sampling theory that eludes many people -- and understandably so as it is a difficult concept only really provable with complex maths. Although it is easy to demonstrate in practice.
For something to happen between
samples, that something must inherently involve frequencies above half the sample rate -- but we've already accepted in our contract to
go digital that those frequencies aren't allowed in to the system in the first place by virtue of the band-limiting policy that is part and parcel of working with any sampled system.
If those very high frequencies were allowed in (usually because the input filtering was omitted), the reconstruction process gets confused and they get translated to much lower frequencies -- a process we call aliasing. It's that tinkly distorted effect you hear if you and record a voice message into one of those electronic greeting cards and then play it back!
However, the waveform recreated by a DAC would not be exactly the same as the input was even if you couldn't hear the difference. I'd applied the same logic to tape although perhaps the physical constraints of tape make that distinction moot.
Perhaps the more obvious example to consider is a pure square wave which, as we all know, contains odd harmonics that go on forever (and certainly way above 20kHz). So if you have an analogue synth that generates a perfect square wave (none can, of course, but many will get very close), and you look at that output signal on a 'scope it looks lovely and square wave-ish. Then you record it into a digital system, play it back and look at it on the 'scope again, and it comes out looking all wriggling and much less square-wave-ish.
And all the digital-haters shout out, there you are... it's missed stuff and it's not accurate!
But actually, all they're seeing is the effect of the analogue brick-wall filters at the input and output of the digital system -- the filters that restricts the bandwidth to slightly less than half the sample rate. If you remove the higher frequency harmonics from a square wave you get a wriggly square wave! Basic physics 101.
Of course, if you use a digital systems that samples at a much higher rate, many more of the odd harmonics will be included and the replayed square wave will look
much more like the square wave generated by the analogue synth. But it won't sound
any different because we can't hear any of the harmonics above 20kHz anyway.
But you can easily prove that the digital system handles everything up to its prescribed frequency limit with sine waves. Feed in a 20kHz sine wave, and you'll get a 20kHz sine wave out. (if you're trying to do this with 'scope's on computer screens be very wary of their own display sampling limitations, though!
I'm sure I'm missing something here, but I am interested in understanding more. Would you mind elaborating on this a bit Hugh?
The very best video demonstration of all this that I've found is by Monty on Xiph.org. it should be compulsory viewing! https://ftp.osuosl.org/pub/xiph/video/Digital_Show_and_Tell-360p.webm
...And Monty actually demonstrates the square wave thing about 17 minutes into that video! :D
And there's my own effort at a written explanation here: https://www.soundonsound.com/techniques/digital-problems-practical-solutions