From Merriam Webster:
Analog
- of, relating to, or being a mechanism or device in which information is represented by continuously variable physical quantities
- something that is similar or comparable to something else either in general or in some specific detail : something that is analogous to something else
Digital
- composed of data in the form of especially binary digits
- of, relating to, or using calculation by numerical methods or by discrete units
In terms of storing, transmitting and playing audio, each term is ambiguous, yet their different meanings are similar, which leads to confusion.
Analog
The key phrase with meaning 1 is “continuously variable”. A turntable needle tracking a record groove, a tape deck head responding to fluctuating magnetic fields on tape, are both continuously variable. Reflective and non-reflective spots on a CD are not continuously variable – it either reflects a laser beamed at it, or it doesn’t. A square wave transmitted along a wire is not continuously variable – it is either at its max voltage, or its min voltage, nothing in between.
However, if we look more closely at the last two examples, we realize that they really are continuous. A reflective spot on a CD doesn’t reflect back 100% of the light; it’s not perfectly smooth, some of the light is scattered and lost. Conversely, a non-reflective spot does reflect back some tiny amount of the light, even though it absorbs or scatters most of it. And while all the spots of the same type (reflective or not) are similar, they are not exactly the same; each is unique. A square wave does not switch from high to low, or low to high, instantaneously. That would require infinite rise time, which is impossible. And as it approaches the new voltage, it will overshoot or undershoot just a bit before it stabilizes. So the voltage actually does vary continuously from the high to low value, even if it spends 99.99% of its time at a high or low value. In this sense neither of these are as discrete as they first seem; they are almost but not quite discrete, but actually continuous.
In fact, the universe at the super-macro atomic scale at which we perceive and manipulate it, is continuous. It only becomes discrete at the subatomic/quantum level.
The other sense of “analog” is that it is an “analog” of, or actually resembles the thing it represents (closely related to the word “analogy”). A magnetic tape that encodes music, has a strong field where the music is loud and a weaker field where it is quiet. A turntable needle tracking a record groove physically moves over a bigger amplitude when the music is loud, smaller when it is quiet. The shape of the groove itself resembles the waveform of the music being played.
Music as we experience it, and as it passes through air as vibrations and pressure changes, is continuously varying. Analog storage of music fulfills both definitions of the term: it is continuously varying, and it physically resembles the music (in some way, directly or indirectly).
Digital
The first definition refers to binary digits. However, this does not fully capture the sense of what it really means. The rational numbers are continuously varying, in the sense that they are infinitely dense: between any two of them, no matter how close they are, lie infinitely many more. Mathematically, the rational numbers are not a true “continuum”, as they have holes – by holes I mean numbers that we know must exist since they are the solution to simple algebra problems, yet are not rational. For example, the square root of 2.
Yet pragmatically speaking, this is a distinction without a difference. It is impossible to detect the difference between rational and real numbers through observation or measurement of the physical world. For every irrational number R, for any small value ε, we can pick a rational number Q so that | R – Q | < ε. We can pick ε smaller than any means of physical observation or measurement. Indeed, ε can be smaller than relativistic uncertainty principles permit. So even in theory, not just in practice, it is impossible to discern the difference in the physical universe. The difference between rational and real numbers does exist, but it is a mathematical distinction, not a physical one.
So for purposes of analog vs. digital, the notion of “infinitely dense” is a sufficient interpretation of what “continuous” means. Numbers can be continuous. Of course, numbers can also be non-continuous or discrete: like the counting numbers.
Even binary digits can be continuous. Every rational number can be expressed in binary digits, though some of them require infinitely many binary digits. For example, 1/7 in binary is 0.001001001… but that is still a well defined and valid number. When people use the term “binary digit” they often mean stored in a computer. But binary is simply a numbering system. It can be, but doesn’t have to be, stored in a computer.
However, in a computer the manifestation (storage, transmission, computation) of numbers is necessarily finite. Thus these numbers cannot be “infinitely dense”, which means they cannot be continuous. They are discrete numbers – even floating points, because they have finite resolution.
Because numbers can be either continuous or discrete, they are a poor concept on which to base the definition of “digital”. So much for Webster’s definition 1; that colloquial usage leads to confusion.
A better concept is definition 2: that of being “discrete units”. Storing or encoding data as a set of discrete states. We often use computers and binary, but the number of states is immaterial: it can be 2 (binary), 3 (trinary) or whatever.
In short, a big part of the confusion around the term “digital” is this: Just because it uses numbers, doesn’t mean it must be discrete. And just because it’s discrete, doesn’t mean it must use numbers.
Digital audio is discrete, and it uses numbers. The first is essential, the second is an incidental convenience.
Once again: truly discrete phenomena do not exist in our universe at the super-macro atomic scale. Discrete means a set of states with nothing in between the states. This is easy to understand from an abstract logical perspective. But shifting between physical states cannot be instantaneous, because that would require an infinite rate of change, which requires infinite energy/power.
Manifestation vs Meaning
Put differently: what it is vs what it means
Put differently: the signal, versus the information
A signal is a phenomena in the physical universe that encodes information. The signal can be a radio wave, a telephone transmission, handwriting on paper, scratches on a clay tablet, etc. Signals contain or encode information. The sender translates or encodes the information into the signal. The receiver decodes the information from the signal.
All signals are continuous, by the simple fact that they exist in our universe which does not have discrete phenomena at the macro-atomic scale. But this does not imply that all signals are “analogues” of the information they encode. Some are, some are not. That depends on the encoding.
The same signal could have different meaning, depending on the encoding. If the sender and receiver do not agree on the encoding, they may believe the message has been successfully transmitted, when it has not. The receiver might decode the signal in a different way than the sender encoded, and thus receive a different message.
This illustrates the fact that the signal, and the message or information it contains, are two different things. One is physical, the other logical. Signals cannot be discrete, but messages/information can be either discrete or continuous.
We can encode discrete messages into continuous signals. For example: consider the discrete binary message 11111100110 (which happens to be 2022, the current year). We can encode this into a series of voltage pulses each of fixed duration, encoding each 1 as 1.0 V and each 0 as -1.0 V. The receiver can easily extract the 1s and 0s from the signal.
However, somebody who doesn’t know the encoding scheme may receive this signal and not even know whether it contains a message, let alone what that message is.
Advantage: Digital
So what is the big deal behind digital audio? Why back in the 1980s was it called “perfect sound forever”? It’s neither perfect nor forever, so where did that phrase come from?
When encoding information into a signal, and decoding it from a signal, discrete states have certain advantages over continuous varying. Consider the above example of positive and negative voltage pulses. The receiver doesn’t care exactly what the voltage is. He can interpret any positive voltage as a 1, and any negative voltage as a 0. If the signal gets distorted, like a bunch of ripple added to it, or the peaks vary between 0.7 and 1.3 instead of exactly 1.0, it won’t change the message. The receiver will still receive it perfectly intact. Of course, under extreme conditions the signal could be so distorted that the message is lost, but that takes a lot of distortion. Encoding information as discrete states is robust, and in most normal cases delivers perfectly error-free messages.
Now consider an analog encoding of information, like a turntable needle tracking a record groove or a tape deck head detecting the magnetic field on tape passing by. Here, the encoding is continuously variable. Every tiniest wiggle or variation has meaning, it is part of the message. Indeed, high quality audiophile equipment is designed to respond to even those smallest subtle signal changes. If the signal gets distorted, even slightly, the distortion becomes part of the message. This encoding is not robust; it’s much more difficult to tell the difference between the encoded message, and any distortion that signal may have suffered.
Summary
For precision, let’s use “discrete” instead of “digital” and “continuous” instead of “analog”.
Digital audio is information. The original music is a continuous phenomena, and is encoded into information as discrete states. Those discrete states are encoded into continuous form for physical storage and transmission, and can be decoded back into discrete states. We use this discrete encoding because it is more robust, relatively immune from imperfections and distortions in transmission and storage, which makes possible perfect transmission that is not possible with information encoded in continuous forms.
In short, digital audio is “digital”, but the means by which we store and transmit it is “analog”. We encode the digital audio information into analog form or signals, and decode or extract the digital information from the analog signals.
The theory and physics of discrete vs. continuous information and signals has been known since Claude Shannon and others developed information theory back in the early 20th century. The Shannon-Whittaker interpolation formula which is the basis for analog to digital to analog conversion, was known at least since the 1930s. So why didn’t digital audio exist until the 1980s? The reason is computing power – or lack thereof. The range and resolution of human hearing is high enough that it requires a lot of digital data to attain sonic transparency. We knew how to do it, but it took decades for computing technology to get fast enough to process the volume of digital data required.