Two noob questions

[71 dB]

Let's say we have sampling frequency fs = 1 kHz => Nyquist limit fn = fs/2 = 500 Hz.

Now we take samples of 100 Hz sinusoidal analog signal. We get the sample points describing the analog signal, but for mathematical reasons we would have gotten se exact same sample point data if we had sampled an unfiltered 900 Hz (or 1.9 kHz, 2.9 kHz, 3.9 kHz,...) analog signal. That's what the spectrum "replicas" mean. The digital world takes a part of the analog world limited by the Nyquist limit and "pretents" it is all there is and copies it over and over like two mirrors facing each other copies the space between them over and over. The real world exists only once between the mirrors and the copies are an illusion. Now, when we transform the digital sample point data back to an analog signal using proper reconstruction filtering, we get 100 Hz sinusoidal back. If the original signal was 100 Hz analog signal, our analog => digital => analog chain did well and all is good, but if it came from 900 Hz signal, things obviously went south! Anti-alias filters ensure thing don't go south in the analog to digital part and anti-imaging filter ensure things don't go south in the digital to analog part (because high frequency noise gets generated depending on what kind of DAC we use).

 

[danadam]

This filtered signal is what's going to the sampler, and what comes out of the sampler is what I call the "sampled signal". Isn't this a thing?

I think your "sampled signal" might be what dspguide.com calls "impulse train" (figure 3-5 is actually in the next chapter):

Our overall goal is to understand what happens to the information when a signal is converted from a continuous to a discrete form. The problem is, these are very different things; one is a continuous waveform while the other is an array of numbers. This "apples-to-oranges" comparison makes the analysis very difficult. The solution is to introduce a theoretical concept called the impulse train.

Figure 3-5a shows an example analog signal. Figure (c) shows the signal sampled by using an impulse train. The impulse train is a continuous signal consisting of a series of narrow spikes (impulses) that match the original signal at the sampling instants. Each impulse is infinitesimally narrow, a concept that will be discussed in Chapter 13. Between these sampling times the value of the waveform is zero. Keep in mind that the impulse train is a theoretical concept, not a waveform that can exist in an electronic circuit. Since both the original analog signal and the impulse train are continuous waveforms, we can make an "apples-apples" comparison between the two.

Now we need to examine the relationship between the impulse train and the discrete signal (an array of numbers). This one is easy; in terms of information content, they are identical. If one is known, it is trivial to calculate the other. Think of these as different ends of a bridge crossing between the analog and digital worlds.

 

[VNandor]

I think your "sampled signal" might be what dspguide.com calls "impulse train" (figure 3-5 is actually in the next chapter):

That's definitely it. I recognize the irony in not using the correct term for it right after saying people should use the correct terms to avoid confusion.

Keep in mind that the impulse train is a theoretical concept, not a waveform that can exist in an electronic circuit. Since both the original analog signal and the impulse train are continuous waveforms, we can make an "apples-apples" comparison between the two.

Something that can actually exist is pulse amplitude modulated signals (which apparently isn't used for AD conversion) which is what I found to be a deceptively similar concept to the impulse train. I just assumed that must be what sampler would be doing, and there would be a separate circuitry that holds the peaks for the quantizer.

 

[gregorio]

As the troll seems to have gone away, @old tech might find it useful if I clarifying/expand on my first post (https://www.head-fi.org/threads/two-noob-questions.966489/#post-17347463).

The process of digitisation and reconstruction is very complex, although “complex” is a relative term and there are certainly far more mathematically complex processes routinely done by computers/devices these days. However, it’s complex enough that even professionals tend to have an understanding limited to their specific field. A DSP software developer for example would likely have a very good understanding of the DSP math involved but not necessarily such a good understanding of digital signals, clocking or the other factors involved. Even an actual mathematician would be limited in their understanding unless they had access to all the exact math implemented by a particular set of devices and a reasonable understanding of psychoacoustics and physics. There are very few people who have a comprehensive understanding of the whole process (inclusive of all the mathematical ramifications and of the necessary ancillary components). Dan Lavry appears to be one who does, Paul Frindle another and a few others I’ve come across in the last 30 years but there really aren’t many, and in my field (sound/music engineering), there doesn’t seem to be any at all, because that level of knowledge/understanding isn’t necessary to our jobs. Apart from those few, the rest of us have to make do with “mental model”, at least in some respects. Mental models can of course just be bad (entirely wrong/incorrect) but even good mental models almost inevitably break down (are incorrect/inaccurate) at some level or in some respect. None of the explanations given to you in this thread that I can see are either entirely wrong nor entirely correct (because they’re all mental models) but hopefully some, or all together will have aided your understanding (your mental model is hopefully better than it was). Although that’s not a given because some of the explanations appear to contradict others, so the end result could be more confusion than you started with!

In addition, the situation is further confused by the fact that when we get down to the level of detail discussed, the terminology isn’t clearly defined. For example, I’ve used the term “digital signal”, VNandor used the term “sampled signal”, 71dB used the term “digital sample point data” and danadam has introduced the “impulse train”. All of these terms effectively mean the same thing if you’re just after a better basic “mental model” but at the level of detail that’s been discussed, they all also mean quite different things (even “digital signal” could potentially mean 3 different things off the top of my head). From what I can tell, these differences in the terms used account for all the apparent contradictions/disagreements in this thread so far (with the exception of the trolling of course).

G