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Are signals modeled either digitally or analogously or can signals modeled as both?

signalmodeler
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2 Answers2

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Any signal can be modeled, but not necessarily uniquely. Any finite duration (i.e. real-world) signal is not bandlimited unless we assume it is - meaning, it has infinite valid representations.

This applies to both analog and digital. Specific cases:

  1. Signal is analog: the analog representation is true by definition, but we still won't have a mathematical description (formula) for it unless we assume a bandlimit.
  2. Signal is digital: the digital representation is true by definition, but we still won't have a formula unless we assume a bandlimit.
  3. We have a formula: if its Fourier Transform is bandlimited, we can create a unique digital and analog representation. If not, then we require an infinitely long analog representation.
  4. We don't have a formula: can't model uniquely either in analog or digital (i.e. can't convert), as in cases 1 and 2.

Note, when I say "representation", I mean the raw source rather than its measurements, as measurements are finite and cannot be analog/continuous. I also use "analog" and "continuous" interchangeably, and "digital" and "discrete". Lastly, a bandlimited signal is necessarily periodic (repeats infinitely and lasts forever).

If we don't care about uniqueness, then we can represent / make formula for anything in digital or analog. As for how we can assume a bandlimit, see these posts.

OverLordGoldDragon
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  • Thanks, just a small thing, is "bandlimit" unique or very unique to signal processing? I just never came across it in English before. – signalmodeler Aug 10 '21 at 05:49
  • @signalmodeler Bandlimit just means "finite range of frequencies", i.e. $F(\omega)=0$ outside of some range $[\omega_0, \omega_1]$. Due to how Fourier Transform works, the uniqueness statement is universal about (nearly) all functions rather than being just a signals thing. – OverLordGoldDragon Aug 10 '21 at 05:52
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A common definition of a signal is "a function that conveys information about a phenomenon".

By "modelled digitally" you probably mean a discrete-time discrete-valued representation. By "modelled analogously" you probably mean a continuous-time continuous-valued representation.

If you mean that you would like to have a closed form, then no, this is not possible for all arbitrary signals. For example it is not possible for random noise.

If alternatively by modelling digitally/analogously you mean, if it's possible to built digital or analog circuits to generate arbitrary signals, the answer is a clear no. A simple counterexample would be a signal that extends into the past, for which you would need a real acausal system, which is not possible. Another simple counterexample are signals with infinite energy.

As you can see, this question is far too broad and needs clarification to get any helpful results.

juliusd
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