How do audio readers interpolate?

Question

$124$ Hz sampled at $256$ Hz has "beats", but $128$ doesn't - a phenomenon of this question:

Whether these are "legitimate" beats is up to interpretation, but suppose the goal is the underlying continuous-time function and we are bandlimited, and beats are supposed to reflect amplitude - then it's not legitimate.

Question is, what interpolation do audio readers use, if any? For above example, a .wav generated by scipy with 256 samples yields no beats in either case, but if we use 2048 samples with frequencies scaled proportionally (and beats graphically verified), there are beats.

Above with 8 samples is just a brief bleep, despite the sampling rate specifying a 1 sec duration, so one might think they're converted directly to voltages, but... electricity is fast enough that I don't have to hear 2048 samples at all.

Extra detail

What I'm really trying to figure out is when to favor calling the left figure "beats". I'm not the only one who thinks they're beats; librosa.stft:

The culprit is imperfect analyticity. What I'm unsure of, is whether the spectrogram should show a flat line; again, I think it depends on interpretation, and I'm trying to interpret it for audio.

(As a side note, if imperfect analyticity yields "more accurate" amplitude in some sense, I wonder if it does the same for instantaneous frequency.)

Audio code

import numpy as np
from scipy.io.wavfile import write
N = 2048
t = np.linspace(0, 1, N, 0)
f0, f1 = N//2 - 4, N//2
x0 = np.cos(2np.pi  f0 * t)
x1 = np.cos(2np.pi  f1 * t)
A = np.iinfo(np.int16).max
x0, x1 = x0 * A, x1 * A
sr = int(1/np.diff(t)[0])
write('x0.wav', sr, x0.astype(np.int16))
write('x1.wav', sr, x1.astype(np.int16))

Answer 1

What kind of interpolation is used depends on the particular implementation of each device.

Once I tried to answer this question for my computer's sound card. What I did was to produce a wav file filled with zeros, except for a single one in the middle. I connected my sound card's output to an oscilloscope and configured the trigger.

What I saw was a truncated sinc wave, seven sampling intervals long (IIRC, it might have been a bit more or less). The conclusion was that the impulse response of my sound card's interpolator was a sinc pulse with a few taps.

My guess would be other devices use different impulse responses. They might use a longer sinc, or splines, or even linear interpolation in the case of very cheap devices. If you have a scope (there are cheap USB ones these days), you can repeat this experiment and find out what your specific device is doing.

Answer 2

there are beats.

There are generally no beats. This is purely an artifact on how you draw it: which is simply by connecting neighboring samples with straight lines. This implies that you assume that the value of $x(nT+\tau)$ can be obtained by linear interpolation between the two neighboring samples $x(nT)$ and $x((n+1)T)$. This assumption is wrong and it's especially wrong at higher frequencies.

Values between samples can be accurately calculated using Whitaker-Shannon interpolation. See: https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula

How do audio readers interpolate?

Audio readers don't interpolate at all. There is no need to, since all information is fully contained in the samples.

When you play the audio back, the DAC will apply an anti aliasing low pass filter which provide interpolation, so it will look smooth again in the analog domain.

In rare cases you can get audible beats for sine waves with frequencies very close to Nyquist. This is simply caused by the actual anti-aliasing not being "good enough". The ideal interpolation filter is an ideal low pass filter, which can't be implemented in practice.

Let's look at a trivial example $x_0[n]=\cos(n\frac{\pi}{2})$ and $x_1[n]= \cos(n\frac{\pi}{2} +\frac{\pi}{4})$. Obviously these have the same amplitude and the same frequency but if we draw them "naively" they look very different.

If your frequency would be slightly higher or lower than $pi/2$ you would actually see some "beating" but this is just a drawing artifact and nothing else.