On Analog and Digital Representations

Analog and digital frequent our vernacular. The word digital has been slapped on electronic products for the last few decades and conscripted to say, "Buy me! I'm high-tech." Now, everything is digital. Common are digital cameras, digital stereo systems, digital televisions, and digital computers. So what existed before everything became high-tech digital? Analog, of course! Analog and digital remain abstract for many because, frankly, they are abstract mathematical mumbo-jumbo. Contrary to popular belief, digital does not mean better. I intend to demystify the definitions of analog and digital.

Analog and digital are adjectives. The noun that they often modify is signal. A signal is a representation. For example in a hospital, the pulsing line on a heart monitor is a signal that represents a heartbeat. Another example is the light that enters our eyes. Light carries information about color and intensity into our eyes. A light signal is turned into an electrical signal by the retina, and the electrical signals travel along the optic nerves to the brain. The brain then makes sense out of the light-turned-electrical signals and we can identify the objects around us. Light waves conveying color and intensity become uniquely identifiable objects in our minds. Another example of a signal is sound. Just as the retina turns a light signal into an electrical signal, the cilia in the cochlea of the ear turn the mechanical waves that are sound into electrical signals that are then sent to the brain for interpretation. Anything manifest as sound or light is a signal. Signals can be traced out on paper. They can be measured, analyzed, and synthesized. Sometimes they're messy. White noise is violently spiky locally and globally constant. Sometimes they're predictable curvy lines. A note in a song is a bunch of superimposed sinusoidal waves that turn through the same cycles over and over and over again.

The one-word definitions of analog and digital are as follows:

Analog: continuous
Digital: discrete

Continuous means there are no interruptions in the signal. I walk on a curve that represents an analog signal, all points of the signal appear infinitely close to each other. I stand still on one point of the signal. I get curious where the next point is, to see where I'm going. I find what I suspect is the next point. But as I get closer I notice that there is another point between it and the one on which I'm standing. Numbers work the same way. The space between any two numbers in a set of continuous numbers is dense. I will look at the space between 2 and 3. The actual numbers are irrelevant because no matter where I look in the space the view is the same - uniformly dense. I could divide the space into 2, 2.5, 3. Or more finely as 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0. Or even more finely as 2.01, 2.02, 2.03, ... By tacking on significant digits after the decimal point, the resolution of the space gets finer. Continuous implies infinitely fine; there are no breaks in a continuous signal. They can hold water.

The walk along a discrete line is tricky. Between the point on which I stand and the next point there is nothing. An abyss. I balance on one point, and to get to the next point, I must jump. A discrete line is a mathematical oxymoron, but that is beside the point. With discrete space, there are gaps, voids. The space between two numbers in a discrete space is totally empty. The set of integers is an example of a discrete set of numbers. To get from 2 to 3 in the set of integers, I must make the full leap of distance 1, flying over the nothing that is between them. The density implied by continuity has been vacuumed out; leaving nothing.

All real signals are analog. Digital signals are mathematical fanciness. They look good on paper; they dwell in the minds of engineers and in their scrawls. But they are never real. Even in a digital computer, the signals wear digital disguises, but they are analog underneath.

The simplest signals are two-dimensional. One dimension is the amplitude of whatever the signal represents. The other dimension is either time or space. Consider time for the time being. Analog and digital signals inhabit this two-dimensional space - amplitude in one direction, time in the other. The space itself is continuous if the signal is analog and discrete if the signal is digital. The signal as recorded or synthesized in the real world is analog. In a digital computer, data is treated as discrete binary numbers. And the computer's job is to do computations on those discrete binary numbers. The underlying analog nature of the signals is inconsequential; every precaution has been taken in the design of the computer so the data in the computer can be considered discrete.

To get real signals into a computer, it must be first converted from analog to digital. First, the value of the analog signal are sampled at regular intervals of time. I can sample with a pen, paper, and stopwatch. Every t seconds, I write down the value of the signal. Simple. Then, those analog values are quantized to the nearest available digital value. I can quantize too. Let's say I want to quantize to 4 levels represented by the numbers 0, 1, 2, 3. If the value I wrote down is 2.8, I make it 3. Quantization, done. So what was once continuous in time and amplitude, has becomes discrete in time through sampling and discrete in amplitude through quantization. The idea is that if the analog signal is sampled frequently enough, the points in between the sampled points don't really matter. And in the amplitude direction, the number of levels of quantization must be appropriate for representing the signal so the difference between the analog signal and its digital representation is not too big. So digital signals are approximations of analog signals. Analog signals are the real signals that exist in the world.

So why go through all this trouble to convert signals from analog to digital? Computational power. Pure computational power. Digital computers compute very quickly and accurately. And from a simple level of abstraction - two-state devices that encode and do simple math on binary numbers, extremely complex systems have been built.