# Is digital necessarily discrete in both amplitude and time?

Is digital necessarily discrete in both amplitude and time?

Or rather it is necessarily discrete only in time (but not necessarily in amplitude)?

• Every definition of digital signals is that it's both. If you find a different one, clearly say that you mean that. But Uroc327's answer is spot on: Usually it means discrete in both and you'd do good in adhering to that canonical definition. Mar 9, 2022 at 14:51
• Does this answer your question? What is the difference between continuous, discrete, analog and digital signal? Mar 9, 2022 at 14:52
• I object. The accepted answer explicitly defines digital as time- AND value-discrete. So, yes, it does answer your question. Mar 9, 2022 at 15:33
• I avoid asking questions that others have answered already, then not reading the answers, then saying the answer don't answer my question ;) Mar 9, 2022 at 15:48
• The answer I linked to answers your question: it literally has "digital" in bold and above that what it means, "discrete t, discrete y(t)". I'm not blaming you for not finding it at all! It's just that your question is in fact answered by the answer. Though I must admit it was easy to find for me, I searched for definition of digital and read the first post that seemed to refer to a definition. Mar 9, 2022 at 16:23

## 3 Answers

Digital by definition means signals expressed using “digits” and those digits are typically “0” and “1”. This means a fixed point representation and need not be discrete-time to be digital (but most commonly is).

Therefore the one test is, is it expressed using “fixed point” representations; are the amplitude values quantized? If it is, it is digital. You can then go on to define if it is discrete time or continuous time.

Using that description, a discrete time system need not be “digital” if we haven’t quantized the amplitudes for each sample. (Such as a continuous time sample/hold).

As @MarcusMueller points out here, in Discrete Time Systems by Oppenheim & Schaefer, the authors define "digital systems" as being both discrete in time and discrete in frequency. In my own use, I would specifically distinguish the two interfaces of a D/A converter as being discrete in time and discrete in magnitude on the digital side, and being discrete in magnitude and continuous in time on the analog side (if we consider prior to reconstruction filtering the typical stair-case output of a DAC). With these thoughts in mind I would argue the typical convention for "Digital" with respect to signal processing is that it be both Discrete in Time and Discrete in Magnitude, and as @AlexTP defines, countably finite in magnitude (able to be described from a finite number of digits).

• @MarcusMueller (Continuing here)...I am actually content with O&S definition as being complete: Digital is simply discrete in time and discrete in magnitude. I don't see the reason to condition it to also be a finite number of magnitudes (and similarly for finite time). Yes that is a condition to be physically realizable, but we often use descriptions from infinite sets- I don't think that is a reasonable requirement to exclude a fictitious infinite digital system, no different than our use of infinite spans for frequency and time. Mar 19, 2022 at 15:07
• The usage in "Oppenheim & Schaefer" is less a definition than a statement of what digital signals are discussed. It doesn't imply that continuous-time digital signals don't exist, just that you won't learn about them in that book (where the title immediately tells you that the scope is discrete time). Jan 11, 2023 at 19:57
• @BenVoigt Yes that’s a great comment. I agree with you—- as far as what establishes a definition I am not sure, perhaps we can ask what a reasonable engineer would think of when you say “digital system” and if you would need to clarify if time is discrete or not. I suspect in most cases if we don’t clarify that then discrete would be assumed and that it would generally be safe for us to do so. In the cases where time is not discrete, it should be clarified for the same reason. Jan 11, 2023 at 20:34

It's a matter of definition. Usually one defines digital to be discrete in both, discrete time to be (possibly) amplitude continuous and quantized to be (possibly) time continuous.

• What about the output from a grey-coded absolute encoder? That's continuous in time, and I'd certainly call it digital. Mar 10, 2022 at 0:11
• Did you mean Gray-coded? Mar 10, 2022 at 20:13

I would add to the notion of discreteness that the discrete symbols encoding the data or signal should also be finite, or taking a limited number of values in some set called symbol dictionary or alphabet, made for instance of numbers/digits or letters.