Seamless looping of a signal without pops

Question

I'm trying to seamlessly loop an audio signal. When I loop the current audio signal I have, I hear audible clicks or pops right as it loops back to the beginning. I want to be able to loop the audio signal without having any pops.

I have tried changing the signal near the end by writing an interpolation which follows this formula $$ B_n=\frac{(A_n d_1) + (Cd_2)}{d_1+d_2} $$ Where $B$ is the output, $A$ is the audio signal itself, and $C$ is the first signal value within the audio signal. $d_1$ equals to how many signal values until the first sample. $d_2$ equals how many signal values since the specified start. And I choose when the specified start is based on how many milliseconds I wish for the interpolation to be. In this case I gave 15ms a try. So, the specified start will be 15ms before the end of the signal.

And in simple terms; it's a crossfader that is linearly transitioning from the signal source $A$ onto the single value $C$.

The end result is this.

The transition points are highlighted via the cursor. Also, do not mind that both signals are not exactly the same. For the upper signal, I cut a portion of the start so that it would start at a non-0 value.

Since the crossfade is fairly quick for human ears (15ms), I did not manage to hear any disintegration of audio. But is this good practice? I am new to signal processing and cannot tell whether or not this will work for all types of signals I might have. Is there any other way this could be done in a more "proper" manner?

Answer 1

But is this good practice?

Yes, that's a perfectly reasonable approach.

Will this will work for all types of signals I might have ?

There are a LOT of signals out there and there is always an outlier. In your case, I think the most "vulnerable" would be low frequency sine waves. A 40 Hz sine wave has a period of 25 ms and looping one at an inconvenient point without audible artifact is tricky. So this all boils down to: how likely are these signal for your application, what level audibility is acceptable and what are the constraints around resources, looping points & length, etc.

Is there any other way this could be done in a more "proper" manner?

1. If you have some leeway with the exact loop length and loop points, try to put the loop points at the nearest zero crossing. Ideally both zero crossing go in the same direction.
2. You can do an energy cross fade instead of a linear cross fade. This will minimize the volume dip in the middle of the cross fade, but at 15ms it's not going to make much of an audible difference.

Answer 2

Fully agreeing with Hilmar, to add to

Is there any other way this could be done in a more "proper" manner?

Well, you can try to make sure that the spectrum of your signal stays the same, without the discontinuity. The way I'd approach that mathematically:

1. Take a low-pass filter. In this case, preferably a FIR, for you'll need the impulse response to be done after a finite time. Let it have impulse response length $L$
2. Take the last $L$ samples of your signal, and copy them, prepending them to the beginning of your signal.
3. Filter the whole thing with your FIR
4. Crop away the first $L$ samples again. The result should have the same length as the original signal.

This ensures that there's no discontinuity at the end of your signal. The idea is very much stolen directly from cyclic prefix OFDM, where it is used to make it look like the channel (== an FIR filter) is applied in a cyclic matter to the signal. Here, you want your channel to suppress the high-frequency content that leads to "pops".

Notice that the effect is quite similar to your static value, just that it doesn't linearly interpolate to the start sample, but linearly combines a weighting of the last $L-n$ samples with the first $n$ samples (which is the definition of a digital FIR filter).

Answer 3

Let me edit your statement to totally change its meaning, but which question I think you should ask.

Is there ... a ... "proper" manner?

If you had asked this, and had meant "is there any universally accepted yardstick by which one processes audio?" Well, no. You do what sounds good (if it's for artistic purposes), or what's most intelligible most of the time (for communications purposes), or what tires people out the least (for human factors purposes), etc.

Is there any other way this could be done in a more "proper" manner?

If there's no defined universal "proper", then there's no "more proper".

If you can stand to lose a bit of information, then instead of fading in the very first sample of your audio, it may sound better to fade in the first section of your audio. I'm going to mess with your notation, but let's say you have an audio clip $A_n$ starting at sample 0 and continuing to sample $N$, and that you're going to transition over $m$ samples. Then at last $m$ samples you go:

$$B_n = \frac{N - n}{m} A_n + \frac{n - m + N}{m} A_{n - m + N}$$

This should smoothly taper the end of the clip to 0 at the same time it's taking the beginning of the clip at 0 and smoothly ramping it up to full volume.

It may or may not sound better. If you're trying to do something like play one sustained trumpet note forever, then probably not because the note's beginning and end are highly correlated, and the components are almost guaranteed to not be in phase at the transition. But if you're trying to have a speech clip repeat over and over again, or the same musical passage, then probably, because the components won't be highly correlated, so they can be mashed together without constructive or destructive interference.