Digital vs. Analog Circuits: Not a Binary Choice!

CS 441 Lecture, Dr. Lawlor

Counting on your fingers uses "digits" in the computational sense; digital storage uses discrete values, like the fingers which are either up or down.  A 25%-raised pinky finger does not represent one-quarter, it represents zero!  This means your fingers can bounce around a bit, and you can still tell which number you're on.  Lots of systems are digital:
The other major way to represent values is in analog.  Analog allows continuous variation, which initially sounds a lot better than the jumpy lumpy digital approach.  For example, you could represent the total weight of sand in a pile by raising your pinky by 10% per pound.  So 8.7 pounds of sand would just by an 87% raised pinky.  8.6548 pounds of sand would be an 86.548% raised pinky.   Lots of systems are also analog:
Note that in theory, one pinky can represent weight with any desired degree of precision, but in practice, there's no way to hold your pinky that steady, or to read off the pinky-height that accurately.  Sadly, it's not much easier to build a precise electronic circuit than it is to build a more-precise pinky.

In other words, the problem with analog systems is that they are precision-limited.  To store a more precise weight, your storage device must be made more precise.  Precision stinks.  The real world is messy, and that messiness screws up electrical circuits like it screws up everything else (ever hear of clogged pipes, fuel injectors, or arteries?).  Messiness includes noise and the gross term "nonlinearity", which  just means input-vs-output is not a straight line--so the system's output isn't the same as its input.

Yes, it's always possible to make your system more precise.  The only problem is cost.  For example, here's a review of some excellent, shielded, quality AC power cables for audio equipment.   These cables supposedly pick up less noise than ordinary 50-cent AC plug.  But the price tag starts at $2500--for a 3-foot length!

Note in many cases the desired output is indeed highly nonlinear--the output isn't simply proportional to the input.  If you're designing an analog audio amplification circuit, nonlinearity is considered "bad", because it means the output isn't a precise duplicate of the input, and the circuit has lost signal quality (think of the rattling base thumping down the street!).  But most computations are nonlinear, so an analog circuit to do those computations should also be nonlinear.  Such analog computers, without any digital logic, have actually been built!  In some sense, the best possible simulation of a mechanical system is the system itself, which can be considered a mechanical analog computer simulating... a mechanical analog computer.  The downside of such a "simulation" is noise, repeatability, and design and fabrication cost.

Luckily, digital systems can be made extraordinarily complex without encountering noise problems.  Digital systems scale better because to gain precision in a digital system, you don't have to make your digits better, you just add more digits.  This quantity-instead-of-quality approach seems to be the dominant way we build hardware today.  But be aware that analog computers might make a comeback in a power-limited world--considering that a single transistor can add, multiply, and divide, digital logic might not be able to compete!  Also, there are potential computability differences between digital and analog computers.

How many levels?

OK.  So digital computation divides the analog world into discrete levels, which gives you noise immunity, which lets you build more capable hardware for less money.  The question still remains: how many of these discrete levels should we choose to use?
Two levels is the cheapest, crappiest system you can choose that will still get something done.  Hence, clearly, it will invariably be the most popular!

For a visual example of this, here's a little TTL inverter-via-NAND circuit:
trivial TTL 7400 circuit

Here's the chip's input-vs-output voltage curve, measured using the "Input" and "Output" wires shown above.
oscilloscope trace, TTL 7400 chip

The Y axis is voltage, with zero thorough five volts shown.  The X axis is time, as the input voltage sweeps along.  Two curves are shown: the straight line is the input voltage, smoothly increasing from slightly negative to close to 5 volts.  The "output" curve is high, over 4v for input voltages below 1v; then drops to near 0v output for input voltages above 1.3v.  High voltage is a "1"; low voltage is a "0".  So this circuit "inverts" a signal, flipping zero to one and vice versa.  Here's the digital input vs output for this chip:

Here's the trace of another chip of the same type.  Note the curve isn't exactly the same!
oscilloscope trace, 7400 chip

These two chips don't behave in exactly the same way, at least seen from the analog perspective of "how many volts is it sending out?".  But seen from the digital perspective, 0 or 1 output for 0 or 1 input, they're identical!