What is a digital system? Is it only binary or decimal, octal as well?

[tggzzz]

A hex digit is only 2 levels but 4 bits in parallel. All things involving hex, octal are also binary in nature. When you use Hex or Octal you're talking about a digital system.

if you want to represent hex digit with single signal it requires 16 levels. Such signal is analog.
Grouping binary signals don't changes signal type it still binary.
So if you're using many binary signals, it doesn't means how much, it still will be digital system.

Your last line contradicts your first line. Think about that.

[radiolistener]

I refer you to the last line of my previous response which - again - you have chosen to omit. Not impressive.

Here it is again: "See the previous responses from myself and many other people".

I don't have goal to make impression on you... 

I just asked you what you're disagree with exactly?
If you don't have clean answer on my question, then there is nothing to discuss...

[radiolistener]

if you want to represent hex digit with single signal it requires 16 levels. Such signal is analog.
Grouping binary signals don't changes signal type it still binary.
So if you're using many binary signals, it doesn't means how much, it still will be digital system.

Your last line contradicts your first line. Think about that.

I don't see any contradiction here. All is pretty clear.
Signal with 16 levels is analog signal.
Several digital signals is digital system.

[TimFox]

Absolutely not.  A system with only 16 levels, and no (legitimate) levels in between them, is digital.
An analog signal moves continuously over its range.
Of course, as young engineers learn when something goes wrong, any voltage on a given wire is an analog signal:  it becomes digital when quantized, but quantizers can be confused by ringing, etc.

[radiogeek381]

Quote from: Radiolistener
Re: What is a digital system? Is it only binary or decimal, octal as well?
….
Just give me answer on the question - where I’m wrong?

As many many people have said, digital is not synonymous with “binary”. There are, in fact, ternary discrete value logics in use today in modern computer design. The primary modern example is in logic validation, automatic test pattern generation, and reset simulations. In these cases the third value is “U” or “undefined/unknown,” which may result from a flip-flop or other storage element that hasn’t been reset. There are a number of formal algebras for ternary logic. Here are two that I’ve seen used:

T and T => T
T and F => F
T and U => U
F and U => F  (conservative F and U => U)

T or T => T
T or F => T
T or U => T.   (conservative T or U => U)
F or U => U

not T => F
not F => T
not U => U

============

There are of course, other ternary logics where “U” does not mean unknown. 

There are programming systems where logical return values (discrete values) take on True, False, and NULL. The latter meaning that no result could be produced.

In all cases “Digital,” among skilled and experienced practitioners of the art, has come to mean “discrete valued” logic. There is nothing magic about binary or boolean logics other than the fact that they are comparatively easy to implement with active devices acting as simple switches or two-level storage elements.

======
Ternary logic is handy for computing devices built from hydraulic components.

At the end of the 1950’s the Soviets built  https://en.wikipedia.org/wiki/Setun an example of a ternary design that was likely motivated by the expense of active devices.

The extension from ternary logic to other logics with multiple symbols is possible, of course.

======
As for qualifications. I did my first digital design for money in 1979. Since then I’ve worked in almost every part of digital design including logic design, modules, system packaging and cooling, signal integrity, electrical design rule checks, validation, performance measurement/evaluation/simulation/modeling, architecture, compilers, modeling and simulation languages, CAD tools, applications programming, computing history, teaching, sales support, project management, and finance. I’ve built widgets in SSI CMOS, TTL, ECL, NMOS, and VLSI CMOS. I’ve taught both logic design and architecture. (For 19 years I worked for a company called “Digital Equipment Corporation.” ) For most of the last 20 years I have been an architect. 

So I know what I’m talking about.

[tggzzz]

Quote from: Radiolistener
Re: What is a digital system? Is it only binary or decimal, octal as well?
….
Just give me answer on the question - where I’m wrong?

As many many people have said, digital is not synonymous with “binary”. There are, in fact, ternary discrete value logics in use today in modern computer design. The primary modern example is in logic validation, automatic test pattern generation, and reset simulations. In these cases the third value is “U” or “undefined/unknown,” which may result from a flip-flop or other storage element that hasn’t been reset. There are a number of formal algebras for ternary logic. ...

As I'm sure you realise, the IEEE Standard Logic Type has 9 values:

  type STD_ULOGIC is (
   ‘U’,   -- uninitialized
   ‘X’,   -- strong 0 or 1 (= unknown)
   ‘0’,   -- strong 0
   ‘1’,   -- strong 1
   ‘Z’,   -- high impedance
   ‘W’,   -- weak 0 or 1 (= unknown)
   ‘L’,   -- weak 0
   ‘H’,   -- weak 1
   ‘-’,   -- don’t care);
and those are necessary when simulating simple gates and buffers.

Where two outputs are connected together, the resolution function is
 TYPE stdlogic_table IS ARRAY(std_ulogic, std_ulogic) OF std_ulogic;
 CONSTANT resolution_table : stdlogic_table :=(
   -- U    X   0   1   Z   W   L   H   -   ------
   -- -------------------------------------------
     (‘U’,‘U’,‘U’,‘U’,‘U’,‘U’,‘U’,‘U’,‘U’), -- U
     (‘U’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’), -- X
     (‘U’,‘X’,‘0’,‘X’,‘0’,‘0’,‘0’,‘0’,‘X’), -- 0
     (‘U’,‘X’,‘X’,‘1’,‘1’,‘1’,‘1’,‘1’,‘X’), -- 1
     (‘U’,‘X’,‘0’,‘1’,‘Z’,‘W’,‘L’,‘H’,‘X’), -- Z
     (‘U’,‘X’,‘0’,‘1’,‘W’,‘W’,‘W’,‘W’,‘X’), -- W
     (‘U’,‘X’,‘0’,‘1’,‘L’,‘W’,‘L’,‘W’,‘X’), -- L
     (‘U’,‘X’,‘0’,‘1’,‘H’,‘W’,‘W’,‘H’,‘X’), -- H
     (‘U’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’,‘X’)  -- -
     );


https://www.vhdl-online.de/courses/system_design/vhdl_language_and_syntax/extended_data_types/standard_logic_type


Radiolistener appears to be under the misapprehension that "Type bit is (‘0’, ‘1’);" is the necessary and sufficient definition of a digital signal.

[TimFox]

Also, "waveform" and "signal" are not synonymous.
Typical digital signals are assembled from binary bits, but a single binary bit is sufficient for only simple results, such as turning on the ceiling light.
A sweet analogy:  cookies are assembled from chocolate chips.
A signal made from four binary bits, with 16 discrete states or values, can represent numbers in hexadecimal, Gray code, and several variations of binary-coded decimal (only 10 states used).
The four bits can be sent serially on a single wire, or in parallel on four wires, for a digital system to process.
A 16-bit signal can give a huge set of still-discrete states:  Unicode is now up to more than 150,000 discrete characters, encoding into 21 bits (heading to 32).
With a signal made from an analog waveform, the value is continuous (in principle), with resolution limited by noise and bandwidth.

[TimFox]

An interesting example of a digital system's readout:
Some early -hp- digital frequency counters such as the 524B, using vacuum tubes, pre-Nixie, used vertical rows of discrete neon bulbs for the most-significant decimal digits, and two moving-needle meters to indicate the two least-significant digits.
Each meter had marks for the 10 states of those two digits:  values of meter current between those marks were not valid.
http://madrona.ca/e/edte/HP524B/viewL.jpg

[Nominal Animal]

the IEEE Standard Logic Type has 9 values:

Interesting!

For those of us who don't think in VHDL, converted to plain text: Connecting two outputs together yields the actual results shown in the following table:

          │ U  X  0  1  Z  W  L  H  -
        ──┼───────────────────────────
        U │ U  U  U  U  U  U  U  U  U    U = Uninitialized
        X │ U  X  X  X  X  X  X  X  X    X = Strong 0 or 1 (unknown)
        0 │ U  X  0  X  0  0  0  0  X    0 = Strong 0
        1 │ U  X  X  1  1  1  1  1  X    1 = Strong 1
        Z │ U  X  0  1  Z  W  L  H  X    Z = High impedance
        W │ U  X  0  1  W  W  W  W  X    W = Weak 0 or 1 (unknown)
        L │ U  X  0  1  L  W  L  W  X    L = Weak 0
        H │ U  X  0  1  H  W  W  H  X    H = Weak 1
        - │ U  X  X  X  X  X  X  X  X    - = Don't care

per tggzzz's VHDL.  It's very logical, too.  (Pun not intended.)

[Nominal Animal]

One useful definition of 'digital system' I like, is 'a system whose inputs, outputs, and internals have a finite number of valid discrete states'.
The corresponding definition for 'analog system' is then 'a system whose inputs, outputs, and internals have continuous ranges of valid states'.
A 'mixed system' is obviously one that includes both.

By 'internals', I am referring to any internal point or feature that is relevant externally, and not just an internal detail like an implementation detail.
By 'discrete', I mean separate, distinct, non-continuous.
By 'valid', I mean that transitions between functionally meaningful states are not considered, and that only states where the system works as intended are considered, and that design errors and unintended states are not considered.
The 'finite' is more a consequence of having discrete states and a limited operational ranges, but it is useful to remember: for any specific digital system, the number of valid discrete states can be determined in a very straightforward manner; it is never infinite.

A 'digital system' does not require that all its components are themselves 'digital systems'.  For example, in many binary logic digital circuits an N-channel MOSFET or NPN BJT can be used to invert the logic signal, even though its internal operation is analog: continuous, not discrete.
Similarly, a purely 'analog system' can contain components that internally are digital: for example, a toggle switch.

What matters is whether at each meaningful point –– so excluding internals of its components! –– the valid states are discrete, or continuous.

As with all human languages, this is not a perfect definition.  Consider an old-style tungsten filament light fixture controlled by a triac dimmer.  It has one discrete state, completely off and unlit, as well as a continuous range of intensities from some lower limit up to maximum.  Thus, it has one discrete state and one continuous range of states.  Instead of having a term to describe this, we just ignore that one discrete state (it being "off" and thus not very interesting, as the same effect can be obtained by removing the entire system), and call it 'analog'.  Plus, because this definition relies on terms used in physics, it can be misunderstood if the terms used in it are not understood.

This leads to the first degree crude approximation of "digital ≃ discrete", "analog ≃ continuous".

As to discrete systems, fundamentally binary ones are the most common, because two is the smallest number of discrete states one can have and still have the ability to change.  Ternary systems exist, but are nowadays very rare.  Most other discrete systems use binary at the lowest level, but combining a number of binary units into larger words, leading to 8-bit bytes and 16-, 32-, and 64-bit words.  Again, those are just the most common ones, others also exist.  Not all discrete systems use binary at their lowest level either: for example, multi-level flash memory cells as used in typical solid state storage nowadays, have a number of different valid states, depending on the amount of charge in that cell, so that each cell specifies the value for more than one bit.

(Physically, any charge is always quantized, because electical charge is discrete, depending on the number of charged particles, with the unit charge having the same magnitude as the charge of an electron or proton have, 1.602176634×10⁻¹⁹ Coulombs.  This is such a small amount of charge that for practical purposes, the amount of charge in a multi-level flash memory cell is considered continuous.  Once again, we must consider the level of complexity at which we are observing things.)

[CatalinaWOW]

I might ask radiolistener: If digital and binary are identically equal, what is the point of having two words?

Radiolistener correctly asserts that the vast majority of implemented digital systems are binary.  And admits that non-Boolean logic systems are possible.  But then incorrectly asserts that the have not been formally defined and implemented.  As Nominal has expanded ternary is fairly well developed, and as radiolistener has expounded on his curriculum vitae, he should be familiar with "fuzzy logic" which was something of a fad a couple of decades ago.

In my personal opinion, developed with a somewhat longer and deeper curriculum vitae, ibinary or Boolean is a subset of digital.  It is by far the most common subset in current practice, and given an engineering approach where small difference can be ignored used as a useful approximation of the whole set.  This rounding approach is widely useful.  But those small differences are where many new ideas are found, and often fertile ground for breakthrough ideas

[radiolistener]

Absolutely not.  A system with only 16 levels, and no (legitimate) levels in between them, is digital.
An analog signal moves continuously over its range.

No. The signal is Analog when it is not Digital. And you cannot categorize signal as Digital if it have more than 2 discrete levels.

The main mistake in your claim is that you're assume that non binary but discrete level signal as Digital, this is incorrect.
Such signal with discrete 16 levels is Discrete, but NOT Digital, because it cannot be represented with binary digit.

And since there is no special domain for N-level discrete signal, where N != 2, such signal falls into the Analog category.

If you develop math and laws for ternary logic, you can involve new domain, for example you can name it "Ternary Domain". You can develop math, logic, laws for that domain, develop algorithms and electronic blocks for such domain, develop and manufacture integrated circuit for such domain. And then you can categorize signal with 3 possible levels to Ternary Domain.

The same for signal with discrete 16 levels, you're needs to develop math, laws and rules, develop electronic blocks for logic operations which working with signal that have 16 levels and then you can categorize such signal with 16 levels to a new invented domain.

But since there is still no Ternary Domain and other type domains in modern electronics, you're have just 2 options: Digital Domain )for signals with 2 discrete levels and Analog Domain for all the rest signals. And this is why you're have only Analog choice for signal with 3 discrete levels. And the same only Analog choice for signal with 16 discrete levels.


Digital Domain is already used for signals with 2 discrete levels and Boolean logic where signal is represented with binary digit. You cannot use digital electronic blocks (such as basic AND, OR, XOR, NOT elements) which are designed to work with 2 discrete levels according to Boolean logic to process signals with 3 or 16 discrete levels.

Such signals are incompatible with Digital Domain.

[radiolistener]

I might ask radiolistener: If digital and binary are identically equal, what is the point of having two words?

They are not identically equals, they have different meaning.

Binary means numbering system with base 2, the same as Decimal means numbering system with base 10 and Hexadecimal is a numbering system with base 16, etc

Digital means digital domain, which covers all math, laws and electronic circuit which are designed to works with basic signals that have only 2 discrete levels and uses Boolean algebra, algorithms and approaches and all others things.

Any thing that works with signals which has more than 2 discrete levels is not covered by Digital Domain and falls to Analog Domain. You cannot apply math and rules or logic elements used in Digital Domain for such signals. Since there is just Digital and Analog domain, all things which cannot be categorized as Digital are falls to Analog category.

For me, this is pretty clear and easy, I don't understand why so clear things leads to an irresistible desire to pull an owl on a globe, that is, to call a non-binary signal as digital... 

It should be pretty obvious that the signal with 16 discrete levels is not compatible with digital circuits and should be treated as analog signal and needs to use ADC / DAC to convert it to digital domain and process within digital domain or use analog circuit to process it in analog domain, there is no other choice....

[radiolistener]

Radiolistener appears to be under the misapprehension that "Type bit is (‘0’, ‘1’);" is the necessary and sufficient definition of a digital signal.

No, I know about special cases, but they are not represent valid state of the signal. I deliberately avoid to talk about it to avoid more confusion for those who don't familiar with digital electronics. They are describe special cases when signal is missing at all or is undefined, or stay in metastable state, which happens in real digital system because it is build with analog circuit and there is no way to build ideal digital system with binary signals.

When I talked that digital signal has binary value, I mean valid signal value, when signal is present, is defined and stable. There is just two values 0 and 1 for valid signal. But real life is a little bit complicated and we needs to deal with all these special cases with missing or undefined signal or metastability issues which happens not because it is digital, but due to analog limitations of our world, where it is impossible do develop ideal digital system and any basic digital logic element is built with analog components and utilize analog nature of our world... 

[radiolistener]

With a signal made from an analog waveform, the value is continuous (in principle), with resolution limited by noise and bandwidth.

Yes, analog signal is continuous and not limited to some fixed discrete values, but it don't means that analog signal which specification has 16 discrete levels is not analog. It is still analog. Discrete/Continuous and Digital/Analog are different properties.

For example digital signal can be Discrete in time or Continuous in time. In both cases the signal is Digital.

The same thing with signal level, analog signal can be Discrete in amplitude or Continuous in amplitude. In both cases the signal is Analog.

So, the Discrete/Continuous property of signal cannot be used to categorize it as Anaglog or Digital.

[Electro Fan]

Perilous as it might be…

Analog is continuously variable.

Digital is binary, 2 states, “on” or “off”, roughly the absence or presence of electricity (as defined by some agreed upon threshold) and as represented by bits (binary digits), 1s and 0s.  Depending on the convention used the 1s and 0s might use 7 bit ASCII, 8 bit EBCDIC, HEX, etc. or various other encoding schemes to represent information in a format meaningful to machines or humans (including data, text, audio, images, and video).  Many conventions can be evolved from the basics (8 bit Bytes, 16 bit Bytes, 32 bit Bytes, 64 bit Bytes, etc) and many A2D and D2A techniques can be employed.  As a result digital can approximate analog and analog can be represented by digital.

[Tomorokoshi]

Let's say the context here is that of a computing system in the most general terms.

Paraphrasing Mr. Blaauw's model of Architecture, Realization, and Implementation, let's be very careful and precise about exactly which aspect of the system is being discussed.*

1. The set of all states described in the Architecture of the system.
2. The methodology by which the set of all states described in the Architecture is Realized in the design.
3. The componentry in the Implementation that allows for the function of the methods from the Realization.

Therefore, when having a discussion like this, it is important that both are talking about the same view into the system. Is one person thinking from the Architecture point of view, while another is thinking from the Implementation point of view, while a third is thinking from the Realization point of view?

Part of the confusion with "digital" and "binary" is understanding where precisely these are considered in the design. I will discuss two computer systems.


The Motorola MC68000 Processor
Any other modern processor could just as well have been chosen. I selected this because:
1. I am familiar with it.
2. I suspect many of you are also familiar with it.
3. It is an elegant design.
4. It is well documented in case someone needs to look something up.
5. None of that analog peripheral nonsense is built-in to confuse things.

In a system the 68000 addresses memory and peripherals through a 23-bit address bus. The upper and lower 8-bit bytes of a 16-bit word are selected using two additional control lines. The 16-bit data bus can return either or both of the 8-bit high and low bytes based on the control lines.

Each of the individual address and data bus signals is specified as either a "High" of nominally 5V or a "Low" of nominally 0V. The input and output drive circuits can reliably transfer using these voltage representations of states. Therefore, while a range is acceptable for the two logic levels, that does not matter in the context of this system.

While the programmer's view of the register set can perform binary operations, the register architecture is grouped into 8-bit bytes, 16-bit words, and 32-bit "longs". Operations are at the byte, word, and long level.

MC68000 Observation 1: To the programmer the digital data is processed in groupings of 8 binary digits known as "bytes".
MC68000 Observation 2: The address and data lines use digital signals in a binary format.
MC68000 Observation 3: The digital signals can be measured as generally discrete values.

All of this is implemented on an integrated circuit.


The IBM 650 Computer
This goes back to the 1950's. Why study the IBM 650? Why study any computer technology? What makes it interesting is the contrast it provides to "modern" systems, not the direct applicability of the system.

Part of the discussion here is that "digital" and "binary" are considered somewhat synonymous. What would it mean for a system to be digital, but not binary?

The IBM 650 is a decimal computer. To the programmer, the architecture is strictly that of a decimal system. For instance, the programmer is not working with base-2 registers that are somehow interpreted to look like base-10 in the fashion of the MC68000 BCD manipulation instructions.

The implementation of a decimal digit is done with 4 binary digits, interconnected to produce a unit that counts from 0 through 9. Is this HEX? No, it isn't: the operations are decimal. 0.1 can be represented exactly.

Homework: Using fixed-point variable representation on a binary computer, work out the value for "0.1".

Just like the MC68000, signal levels in the IBM 650 generally take on two ranges. However, the grouping and controls of the 4 binary digits works out to a decimal digit.

IBM 650 Observation 1: To the programmer the digital data is processed in groups of decimal digits.
IBM 650 Observation 2: The address and data lines use digital signals in the decimal format.
IBM 650 Observation 3: The digital signals can be measured as generally discrete values.

All of this is implemented with vacuum tubes.


Conclusion
Here we have an example with the IBM 650 computer that is digital, but not binary, at the Architecture level. To the programmer, the Implementation and Realization are not important.

Note that we can have discrete circuits multi-level that are not analog. For instance, a monostable multivibrator realized using two triodes could be made to have 10 quasi-stable states. It is not an analog circuit from the point of view of being able to process a continuous time-varying signal such as an audio signal. It would be used as a divide-by-ten circuit.

* Gerrit Blaauw developed the architecture of the IBM 360 computer, and decided on the 8-bit byte.

[radiolistener]

Analog is continuously variable.

To be more precise, analog signal can handle both - continuous and discrete amplitude.
Analog signal can be discrete or continuous in time and can be discrete or continuous in amplitude.
It depends on your requirements for signal but not on Analog/Digital type.

The same digital signal can be continuous and can be discrete in time.
But for amplitude digital signal is always discrete, because uses binary level which is discrete.

Conclusion
Here we have an example with the IBM 650 computer that is digital, but not binary, at the Architecture level. To the programmer, the Implementation and Realization are not important.

No, this is wrong example. It says about decimal representation of digital data on display for human, but that digital data is still binary and requires conversion from binary to decimal in order to show it to the user.

Just open schematic of IBM machine and check, I bet that it uses binary signals, because it is digital 

[Tomorokoshi]

Analog is continuously variable.

To be more precise, analog signal can handle both - continuous and discrete amplitude.
Analog signal can be discrete or continuous in time and can be discrete or continuous in amplitude.
It depends on your requirements for signal but not on Analog/Digital type.

The same digital signal can be continuous and can be discrete in time.
But for amplitude digital signal is always discrete, because uses binary level which is discrete.

Conclusion
Here we have an example with the IBM 650 computer that is digital, but not binary, at the Architecture level. To the programmer, the Implementation and Realization are not important.

No, this is wrong example. It says about decimal representation of digital data on display for human, but that digital data is still binary and requires conversion from binary to decimal in order to show it to the user.

Just open schematic of IBM machine and check, I bet that it uses binary signals, because it is digital 

1. What is the circuit for a binary digit?
2. How many transistors does it use?
3. If it uses more than one transistor, is it still binary?
4. At which level are you discussing the question: Architecture, Implementation, or Realization?

For Blaau's paper on Computer Architecture:
https://www.infra-repository.org/oiar-2013/images/3/3b/Blaauw_Architectuur.pdf

[CatalinaWOW]

With a signal made from an analog waveform, the value is continuous (in principle), with resolution limited by noise and bandwidth.

Yes, analog signal is continuous and not limited to some fixed discrete values, but it don't means that analog signal which specification has 16 discrete levels is not analog. It is still analog. Discrete/Continuous and Digital/Analog are different properties.

For example digital signal can be Discrete in time or Continuous in time. In both cases the signal is Digital.

The same thing with signal level, analog signal can be Discrete in amplitude or Continuous in amplitude. In both cases the signal is Analog.

So, the Discrete/Continuous property of signal cannot be used to categorize it as Anaglog or Digital.

By your definition a digital signal cannot be continuous in time.  There is a discontinuity (the left derivative and the right derivative are not equal) at every state change. 

But I am sure you have a reason that this is not applicable, because you have some alternate definition of continuity.

[CatalinaWOW]

Let's say the context here is that of a computing system in the most general terms.

Paraphrasing Mr. Blaauw's model of Architecture, Realization, and Implementation, let's be very careful and precise about exactly which aspect of the system is being discussed.*

1. The set of all states described in the Architecture of the system.
2. The methodology by which the set of all states described in the Architecture is Realized in the design.
3. The componentry in the Implementation that allows for the function of the methods from the Realization.

Therefore, when having a discussion like this, it is important that both are talking about the same view into the system. Is one person thinking from the Architecture point of view, while another is thinking from the Implementation point of view, while a third is thinking from the Realization point of view?

Part of the confusion with "digital" and "binary" is understanding where precisely these are considered in the design. I will discuss two computer systems.


The Motorola MC68000 Processor
Any other modern processor could just as well have been chosen. I selected this because:
1. I am familiar with it.
2. I suspect many of you are also familiar with it.
3. It is an elegant design.
4. It is well documented in case someone needs to look something up.
5. None of that analog peripheral nonsense is built-in to confuse things.

In a system the 68000 addresses memory and peripherals through a 23-bit address bus. The upper and lower 8-bit bytes of a 16-bit word are selected using two additional control lines. The 16-bit data bus can return either or both of the 8-bit high and low bytes based on the control lines.

Each of the individual address and data bus signals is specified as either a "High" of nominally 5V or a "Low" of nominally 0V. The input and output drive circuits can reliably transfer using these voltage representations of states. Therefore, while a range is acceptable for the two logic levels, that does not matter in the context of this system.

While the programmer's view of the register set can perform binary operations, the register architecture is grouped into 8-bit bytes, 16-bit words, and 32-bit "longs". Operations are at the byte, word, and long level.

MC68000 Observation 1: To the programmer the digital data is processed in groupings of 8 binary digits known as "bytes".
MC68000 Observation 2: The address and data lines use digital signals in a binary format.
MC68000 Observation 3: The digital signals can be measured as generally discrete values.

All of this is implemented on an integrated circuit.


The IBM 650 Computer
This goes back to the 1950's. Why study the IBM 650? Why study any computer technology? What makes it interesting is the contrast it provides to "modern" systems, not the direct applicability of the system.

Part of the discussion here is that "digital" and "binary" are considered somewhat synonymous. What would it mean for a system to be digital, but not binary?

The IBM 650 is a decimal computer. To the programmer, the architecture is strictly that of a decimal system. For instance, the programmer is not working with base-2 registers that are somehow interpreted to look like base-10 in the fashion of the MC68000 BCD manipulation instructions.

The implementation of a decimal digit is done with 4 binary digits, interconnected to produce a unit that counts from 0 through 9. Is this HEX? No, it isn't: the operations are decimal. 0.1 can be represented exactly.

Homework: Using fixed-point variable representation on a binary computer, work out the value for "0.1".

Just like the MC68000, signal levels in the IBM 650 generally take on two ranges. However, the grouping and controls of the 4 binary digits works out to a decimal digit.

IBM 650 Observation 1: To the programmer the digital data is processed in groups of decimal digits.
IBM 650 Observation 2: The address and data lines use digital signals in the decimal format.
IBM 650 Observation 3: The digital signals can be measured as generally discrete values.

All of this is implemented with vacuum tubes.


Conclusion
Here we have an example with the IBM 650 computer that is digital, but not binary, at the Architecture level. To the programmer, the Implementation and Realization are not important.

Note that we can have discrete circuits multi-level that are not analog. For instance, a monostable multivibrator realized using two triodes could be made to have 10 quasi-stable states. It is not an analog circuit from the point of view of being able to process a continuous time-varying signal such as an audio signal. It would be used as a divide-by-ten circuit.

* Gerrit Blaauw developed the architecture of the IBM 360 computer, and decided on the 8-bit byte.

I think you are on the trail of the problem here.  Radiolistener has stated that programmers always think in binary terms.  This may be true of microcontroller programmers, but I can tell you from personal experience that a large class of programmers do not think about the machine representation of signals at all.  This becomes obvious when you get the blank stare look from a programmer who is used to high level languages when asked whether he assumed that the output of the A-D converter was offset binary or twos-complement.  There are many other examples which can be given.

In the world view of radiolistener the classification he defines for analog and digital is complete and sufficient.  It totally works for all use cases he has encountered.  Others here see the world differently. 

Problems like these are common in the world, and characterized well the the Jaines parable of 10 blind men and an elephant.  Each blind man correctly describes the elephant as he has encountered it.  But is has trouble reconciling his own observations with those of the others.

[radiolistener]

By your definition a digital signal cannot be continuous in time.  There is a discontinuity (the left derivative and the right derivative are not equal) at every state change. 

Yes ideal digital circuit is not possible in our analog world. Because ideal digital signal needs to switch from 0 to 1 in zero time interval. This is impossible and this is why digital circuit has bandwidth limitations. That discontinuity happens on time intervals which is outside of working bandwidth. You can ignore it for working bandwidth.

For example, digital AND element has input and output signals which are continuous in time. And these continuous signals are digital.

[radiolistener]

Radiolistener has stated that programmers always think in binary terms.

This is false claim.

I stated that digital circuit always use binary signals. You can't find digital logic element that can accept signal with 10 or 16 discrete levels. It always uses binary. As user you can see decimal values, hexadecimal values, but this is result of conversion from internal binary form to selected numbering system. The digital system always works with digital data stored and transferred in binary form which is represented with 1 and 0.

If circuit uses signal with more than 2 discrete level, this is NOT Digital circuit but Analog or Mixed signal one and pure digital circuit cannot handle such signal...

[tooki]

radiolistener, you really, really need to find some books on information theory and read them. You are basing your beliefs on the domains you know well, and then incorrectly extrapolating that what’s true in your domain is true of all domains.

[radiolistener]

@tooki, if you're think that there is exists digital circuit and computers which working with signals that has 10 or 16 discrete level (or any other more than 2), there is no needs to propose to read books. Just provide example of such circuit.

Can you give me example of such digital logic elements? For example AND element which supports non binary input? Please provide example with datasheet...

If you don't have such example, then there is no sense to talking that it is exists... isn't it?

It seems that some people are sure that digital system can use non binary signals just because their computers can show decimal and hexadecimal digit, this is just misunderstanding. Any digital circuit works with binary signals and binary data, this is obvious. If it shows you decimal value or provide analog output, this is just result of conversion from digital to analog. But original data in digital circuit is stored and processed in binary form.

There is also another misunderstanding from previous topic, where some users tried to convince me that radio link uses digital signal just because it transfers digital data or because their transmitter DAC has discrete voltage levels (more than two) or discrete in time and has some sample rate. This is also wrong. Such radio link is analog, it just carries digital data, but it uses analog signal to carry that data.