Digital and analog. We often think of them, mistakenly, as equivalent things in different domains. But that’s a misunderstanding, and we often use the terms incorrectly.

For instance, run a mic through a preamp and get a signal, we know that’s an analog signal. And if we run it through an ADC and store it in a computer, it’s digital. But if we encode that signal as up and down voltage fluctuations at a high rate of speed, and run it over a wire, such as the AES-EBU standard, arguments break out over whether the signal is digital or analog.

The arguments stem from the terms being misunderstood, with better terms available.

Analog

The definition of analog is, “Something that bears an analogy to something else; something that is comparable.” The electrical signal from a mic is an analog of the compression and rarefaction in the air from the movement of air through your vocal cords. We can amplify it and it pushes a speaker that moves air and then our eardrums.

Digital

Literally, digits are your fingers, for counting. Something represented by numbers is digital. But numbers are an abstractions. Long ago, most of the world standardized on symbols for a 10-based (for our ten fingers) system of number representation. We can say, “twenty-six, and write down “26”. Or “0x001A”. In binary computer memory, we can store the equivalent bits as approximate voltages, or save to a hard drive in a group of magnetic states. And transfer them over a serial connection as fluctuations of voltage. In a way, these fluctuations of voltage and magnetic states are analogs of the digital values we want to represent.

So, fundamentally analog signals are analogs, and digital signals are analogs. No wonder some argue that sending a digital signal over a wire is an analog signal. It is truly both. Perhaps we might not be using the best terms to describe what comes out of a mic and what comes out of a computer.

Continuous

Continuous implies no separation of values. A mic signal simply flows and fluctuates over time. It’s not a series of stable states, it’s a continuum.

Discrete

Digital audio data are discrete—individual samples updated at a fixed rate. But we can also have sampled analog as well, and it is also discrete. We can store each sample in a purely analog form—without intermediate conversion to a digit—in a bucket brigade device (BBD), essentially holding each analog voltage in a capacitor.

“Sampled” implies discrete, but in particular a series of discrete values over time (we’ll only discuss uniform sampling, where the timing is a fixed rate).

Digital is always discrete or sampled. A number is a symbol, and while we can have a series of symbols that change often, they can never be continuous. The analog lowpass filter in a DAC is what ultimately converts the discrete samples to a continuous signal.

Analog can be continuous or discrete (inside analog delay stomp boxes, for instance), though we almost always use analog as a synonym for continuous. But making the mistake of thinking they are the same thing is where more complex discussions go wrong.

So, back to the argument of whether a signal stored on a hard drive, or SSD, or in RAM are truly digital, yes they are. And they are analog, because numbers are ideas and we can’t store ideas, only analogs of ideas. It gets more confusing when transferring a digital signal over a wire, because we can’t compartmentalize electrical flow, it’s always continuous. (Strictly speaking, electron flow itself is discrete, but we’re not operating at that level—it would be horrendously intolerant of noise anyway to deal with discrete flow—which is why I say “electrical” and not “electrons”.) It’s a discrete digital signal, transported over a continuous flow of analog voltages.

But I wouldn’t call that an analog signal, it’s still a digital signal. If you amplify it and play it back, that’s when it will hit home that it’s not an analog of the music signal, it’s an analog of the states of the digital signal.

We’ve seen that samples can represent a signal with greater peak amplitude than the peak sample values.

If those sample values were higher, the output would exceed “1.0”, or 0 dB full-scale—I’ll refer to these as “overs”. This brings two questions: Can DACs reproduce such a signal? And, what is the worst case peak between samples?

There is nothing inherent in analog design that ties the maximum digital input value to the maximum voltage a DAC can put out. But exactly how much higher the output can be depends on the design, and it’s certainly possible that there isn’t enough headroom to handle overs typical of the vast library of music available.

How bad can it be?

There is no absolute limit to how high an inter-sample peak can be. Yikes.

Intuitively, we can understand that high frequencies are more of a problem with such peak than low, since a low frequency sine wave would have many samples per cycle, while high frequencies have few, leaving curves less well defined. As such, we can take the worst-case high frequency signal, which is alternating positive and negative extreme values, and introduce a discontinuity that causes and overshoot of the waveform. That’s illustrated in this article, True Peak Detection, which includes a proof that there is no limit to the potential overshoot.

All is not lost

First, real music doesn’t include sustained high frequencies at maximum amplitude. The unbounded test signal described was at the Nyquist frequency, a signal we’re technically not defined to handle, and relies on increased duration to increase the peak, at maximum sample values. Even dropping the frequency slightly would ruin the phase relation and produce maximum sample values far closer to the analog signal it represents.

For music, inter-sample overs are most likely on transient peaks, where it’s extremely difficult to tell whether the signal clipped. It doesn’t mean we shouldn’t strive to reproduce them without clipping, or avoid them in the music production process. And there is another way to avoid them even if your equipment can’t handle these overs gracefully—I’ll get to that in a moment.

Avoiding overs in production

The sure place to avoid inter-sample overs is at the mastering stage (or he mix stage if there is no formal mastering). A true peak meter, which might oversample at 8x and check for overs, can let you know of any potential problem. I suspect about +3 dB is limit of what most music with overs might hit—in absence of true peak detection, you could hedge your bets by backing your mix down a couple of dB.

Handling overs in DACs

While you’re unlikely to hear inter-sample overs in the first place, it’s possible that some equipment or software handles overs poorly—in the worst case, an oversampling converter that doesn’t handle magnitudes greater than 1.0 well numerically, for instance. However, this is unlikely with modern equipment, particularly using modern DAC chips.

Some DACs boast the ability to contend with overs in their design. Benchmark Audio claims to handle +3.5 dB, for instance. But few manufacturers list specs on their headroom.

And finally…digital gain control

You can use digital gain control to reduce the sample magnitude before the DAC. You probably do it every day already. Do you keep your iPhone turned up all the way, or backed down a bit? Do you have a digital control knob, or remote volume control on your DAC for home listening? Computer output turned down a little from maximum output? Reducing just 2-3 dB from maximum output adds that much headroom to the DAC’s analog output for handling peaks. If you run a DAC with no digital gain control directly into a pre-amp, and control the analog level, you don’t have this protection, but more and more we control gain digitally, ahead of the conversion.

In closing

It’s unlikely inter-sample overs have ruined any listening experiences for you—you probably haven’t even noticed them. And if you listen using digital volume at least a few dB below its maximum, it’s unlikely any music you’re listening to will cause inter-sample overs. Ultimately, inter-sample overs are important to understand, but not a major threat.

Let’s say we want to test the output of a DAC while it passes a pure sine wave. The “pure sine wave” requirement is a bit more difficult than it might seem. In the general case, it’s not possible to create a sine wave that is free of quantization error. You could scale the wave so that the amplitude peaks are on exact values, but other samples of the wave would fall between quantization levels and require numerical rounding. This is not an overwhelming problem, typically, because all of the error is confined to the ± 1 lsb area. For a signal at or near full scale and 24-bit, the total error is well below the noise level of any analog audio circuit playing it back. But the signal to noise ratio necessarily gets worse as the signal level is reduced, so a sine at the lowest possible levels has terrible S/N.

If we were to do similar tests with square waves (more precisely, bandlimited square waves, to satisfy the sample theorem), we could do much better—if we select frequencies with strict limitations. Specifically, by synchronizing the frequency to use periods that are multiples of the sampling period, and with the phase aligned so that the square wave transitions occur exactly half-way between samples. In order words, data patterns like 1,1,-1,-1,1,1,-1,-1… Put more generally, switching between a value of A for n samples, and -A for n samples, where A is any exact bit value, and n is an integer number of samples.

For example, toggling between a positive value for four samples and it’s corresponding negative value for four samples yields a perfect bandlimited square wave of frequency one-eighth of the sample rate and no DC offset—6 kHz for a 48 kHz sample rate. The bandlimited square wave would have three total harmonics below half the sample rate (a fourth would sit right on the Nyquist frequency).

Sine waves

That should seem obvious, for square waves, but what about sines? First, let’s consider a square wave as described, but for higher frequencies, which would have fewer harmonics—if we can get to the point of a single harmonic below Nyquist, we’d have a pure sine wave! The highest frequency obtainable with the square wave method we’re using would be alternating samples, A,-A,A,-A… But that would be a square wave at the Nyquist rate, equivalent to a sine wave at the Nyquist rate, and we’d expect a proper converter to wipe that out. Next up is A,A,-A,-A…, half Nyquist. That would indeed be a single harmonic—the fundamental at one-quarter the sample rate, 12 kHz for 48 kHz SR. And completely free of quantization. Here’s a representation of how the sine wave relates to the samples:

Note that once converted to analog, the resulting sine wave has inter-sample peaks greater than we’d expect from the digital values. Inter-sample peaks are always a fact of life with digital audio, I’ll publish an article on them soon. But in this case, we could change the phase and align the peaks on exact sample boundaries, resulting in peaks that are exactly represented by the sample values, while retaining the error-free quality:

Now, we’re identified one perfect sign wave, at one-quarter the sample rate. Any others? Not from square waves, because we can’t go higher, and if we go lower we introduce more harmonics. But let’s think about the sine and why we have a perfect one. The answer is that one-quarter works because it features two things we need: it produces a sine wave, and it only uses values that are an exact bit value—either an exact level of our choosing and it’s negative, or zero.

Ultimately, we have only a few choices that can produce a sine from only 0, A, and -A. In addition to one-fourth the sample rate, one-third and one-sixth satisfy the requirements:

What about other integer factors? We have to skip 2 (Nyquist), we’ve seen 3, 4, and 6; 5 doesn’t work, because although it’s symmetrical, it produces two levels per side—both can’t be exact sample values. If you try anything over 6 (3 samples per half-cycle) you’ll find it provides too many sample points per half cycle to maintain a single value (±) along with zero.

So that’s it—1/3, 1/4, and 1/6. For a 48 kHz sample rate, we can have perfect 16, 12, or 8 kHz sines. The A can be any value, from a single lsb to the maximum word size. But remember, positive for DACs doesn’t go quite to “1.0”, so balance the maximum negative level accordingly, and also take into account inter-sample peaks. However, it seems these special cases of sines are mainly useful only for very small amplitudes, so those two caveats can be ignored in that case. For full-scale sines and 24-bit converters, the imperfections from calculation and quantization are small, and in particular much smaller than the minimum analog circuitry noise, so we would normally calculate our sine tones with traditional sine computation (we might use a sine function, or an FFT especially for multiple harmonic tones).

Calculating values

To calculate the exact values needed for a sine wave of amplitude ± 1.0, use \(sin(\frac{2πx}{div})\), where x is the integer sample number (0, 1, 2…), and div is the divisor of the sample rate (3, 4, or 6 for “perfect” sine waves). You can multiply the sample values by a constant to scale them to the desired signal size. The idea, of course, is to round them to exact samples values, symmetrically around zero. You’ll get this automatically with rounding towards zero. And don’t dither these special cases—they have no noise, so you’ll only introduce noise with dither.

This article was the result of thought and discussion here.