1
$\begingroup$

In Chapter 1 of Oppenheim & Schafer's Digital Signal Processing, the authors state that not all discrete-time signals are the result of sampling continuous-time signals, and thus they are making the choice not to "force results from analog systems theory into a discrete framework" for the textbook.

Can anyone think of examples of naturally occurring digital signals that are not the result of sampling?

$\endgroup$
3
  • 1
    $\begingroup$ Look down at your hands. There is a reason they are called "digits". Use any base you'd like, but those are indeed digital signals you can use everyday if you so chose. $\endgroup$
    – Envidia
    Jun 19, 2023 at 22:20
  • $\begingroup$ I guess you are asking about naturally occurring discrete-time signals, rather than naturally occurring signals with discrete values like integers or such. Is that right? $\endgroup$ Jun 20, 2023 at 6:58
  • $\begingroup$ Did you mean sampled time signals, i.e. signals that exist as a sequence of discrete quantities? Or did you mean actual digital signals, i.e. signals that exist as quantities that are encoded digitally? If you meant sampled time, you should probably edit your question for clarity. $\endgroup$
    – TimWescott
    Jun 20, 2023 at 14:17

3 Answers 3

3
$\begingroup$

I believe that Oppenheim and Schafer were thinking of the creation (synthesis) of waveforms in making that statement: Simply, we need not sample an actual analog signal to create a digital signal. However the OP's question is interesting in limiting this further to "naturally occurring".

Without arguing too much about what "naturally occurring" actually means, consider these situations that could apply to being discrete in time (discrete-time signals), and finally to being both discrete in time and discrete in magnitude (digital signals):

Creating an animation by writing an image on each page of a book and flipping through the result, presents each image as a discrete sample.

Dripping water from a faucet creates sounds as discrete events.

Lightning is an example of a single impulse, and the image from a single lightning flash is a single sample.

Clapping, and finger snapping.

The above represent events at discrete events in time which are therefore "discrete-time signals". "Digital signals" have the property of being both discrete in time and discrete (quantized) in magnitude. Mortgage payments are an example of values that are discrete in time (paid on discrete days) and quantized in magnitude (to the closest penny).

Another example is taking a cross country trip and recording our distance to the closest mile on each day.

Please see this related post.

$\endgroup$
2
  • $\begingroup$ Thanks so much for your thoughtful answer. I added "naturally occurring" because of course you can make a digital signal in the abstract by letting x(n) equal something digital. I was curious specifically what the authors meant, and I think your answer makes sense. On my end, I found it conceptually confusing as any course I've taken always emphasized the derivation of digital signals from analog signals. At some point, it's maybe more of a philosophy question than engineering if quantizing amplitude means a signal is truly discrete in value, but I see the validity of the argument. $\endgroup$
    – Camellia99
    Jun 20, 2023 at 16:51
  • $\begingroup$ Yes I agree with you. I think the distinction was unnecessary to teaching the material and find it better personally to keep a very strong relationship with the analog world (for our own intuition). $\endgroup$ Jun 21, 2023 at 1:58
3
$\begingroup$

Look at yourself in the mirror! Gazillions of cells that constitute your body — though not all of them — have DNA strands, which can be viewed as (quaternary) digital signals of the type $$\{1,\dots,n\} \to \{ \texttt{A}, \texttt{C}, \texttt{G}, \texttt{T} \}$$ where the values are the $4$ nucleobases: adenine ($\texttt{A}$), cytosine ($\texttt{C}$), guanine ($\texttt{G}$), thymine ($\texttt{T}$).


$\endgroup$
1
  • 1
    $\begingroup$ Nice example! .. $\endgroup$ Jun 29, 2023 at 18:25
1
$\begingroup$

Digital = discrete in values, discrete over domain (e.g. time, position; i.e. input space)

  • Particle energy level over time: takes on discrete values; values change by absorbing or emitting particles, which are discrete in time. (Whatever the process, discrete in time is forced by conservation of energy: discrete in value and continuous in time means infinitely rapid changes in energy over time - i.e. infinite power).
  • Anything discrete-valued with finite domain-derivative: number of passengers on a bus, number of migrating birds, number of fish in a lake. The number of fish in a lake cannot change continuously, as that's adding or removing infinite fish over any finite time interval. Number of beavers in a continent vs position of center in a continent - since there's finite continents and each is non-zero sized.
  • Most conditional signals: how many smartphones I can buy vs how much money I have. This signal is important to Apple. Apple's price adjustment with each release makes it a 2D digital signal, and this signal is important to political commentators who complain of consumerism. The number of episodes they mention Apple in in a given week makes it a 3D digital signal, and this signal is important to whoever else extends this chain meant to illustrate a point. "Smartphones unnatural" - ant war logistics.

One can define a "more digital" signal that's not only discretely realized over a continuous domain, but whose domain itself is discrete - i.e. incapable of realizing most values. There, prior examples can be made qualifying by a redefinition of the domain - e.g. number of fish in a lake per year. Then, "the most digitalest" signal is one where a continuous definition, even if we wanted, is invalid:

  • Mean number of children fathered vs number of mates. Rabbit population control.
  • Any cumulative value vs discrete-valued realization: height of stack of pennies vs number of pennies, charge vs number of electrons. Penny height is up to manufacturer, but not once it's already set. Electron charge is up to the simulation, and it changing means bigger problems than "what's natural and digital".
$\endgroup$
1
  • $\begingroup$ Meant in general mathematical sense, clarified $\endgroup$ Jun 29, 2023 at 12:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.