The humbucker sound is usually warmer (i.e. cuts more high frequencies), without interference noise (hum), louder and compressed than a single coil. Is it possible to get the same sound with a single coil if it goes through a noise gate (to clear the hum), a compressor and a boost (and maybe EQ)?

My question is not only if there are a bunch of pedals and plugins that would achieve that, but also if in theory that would work (e.g. using DSP to achieve those effects with any needed parameters).

That is not an "opinion" question. I don't want to discuss if you (or an expert listener) can distinguish the two. It is an engineering question: Can the two output signals be made indistinguishable in spectrum and dynamics?

I've worked on this kind of thing a lot to try to have the most flexible sound without switching guitars all the time. Here's what I found:

First, you can't make them indistinguishable. You can just make them reasonably close for live use.

Second, it's much easier and more effective to play a humbucker guitar and use an EQ pedal or effect to scoop out the low mids and boost the highs to make it sound like a single coil, instead of the other way around.

Third, you could get a humbucker with a coil tap feature so you actually switch from humbucker to single coil on the same guitar. The biggest challenge with these is the noticeable change in output level from the guitar when you switch.

Finally, you could use P-90 style pickups. They are technically single coil but they are so overwound that when you max out the guitar volume they sound thicker like a humbucker, and if you lower the volume a bit, you get a normal single coil sound.

First off, I recently heard of a pedal that does something like this; it's called the Keyztone Exchanger. The reviewer was pretty enthusiastic.

A few caveats, off the top of my head:

• a humbucker cancels all kinds of external signals that induce a voltage in the pickup; the power mains is the most notorious, but not the only kind of source of noise. Once the noise is in the signal, you cannot just get rid of it using a noise gate - as long as you play, and the noise gate is open, you'll hear the hum and hiss underneath the notes. If you're only interested in the hum from the power grid, you could selectively filter for that frequency and a few of its harmonics, but that might damage your signal as well.


• you'd probably need a very specific EQ. The humbucker acts as a comb filter, canceling a specific set of frequencies that depend on the distance between the rows of polepieces.

"Single coil" and "Humbucker" aren't standards for frequency spectra. There are enough variables within each technology—everything from the type of magnet to the number of coil windings to the material and thickness of the coil wire—to make a huge amount of variation possible. And that takes none of the other very important variables of tone into consideration: the density of the guitar's wood, the type and thickness of the strings, the position of the pickup along the strings, the electronics inside the guitar, the closeness of the pickup to the strings, the playing technique, and probably lots of other stuff I can't think of off the top of my head.

Depending on the alternating current service in the country where the guitar is plugged in, the hum comes in at 50 Hz or 60 Hz. The lowest fundamental frequency of a guitar is 82.4 Hz (E2), so the brunt of the hum can be easily and safely removed (with a parametric EQ notched at the appropriate frequency) without fear of affecting the tone. (The waveform of alternating current in electric power circuits is almost always a sine wave, so its impact at other frequencies is theoretically nothing and, in practice, dependent on the local conditions.) As you know, that alone wouldn't make a "typical" single coil pickup sound like a "typical" humbucker. (A noise gate is nothing but a volume-triggered on/off switch. In theory, it does nothing whatsoever to change the sound.)

Assuming you could exactly establish the frequency response of a humbucking pickup, and each of those frequencies was sufficiently present in the signal from a single coil pickup, you could probably create an EQ curve which makes the waveforms look alike in a Fourier analysis. But that's just for one note played at one loudness.

As you alluded to, the frequency response changes with the amplitude. So you'd need a compressor to mimic the exact behavior of the humbucking pickup or a sophisticated dynamic EQ.

But even then, it wouldn't be identical. Even if you have the same player, guitar, pick, pickup, strings, amplitude, and note, you wouldn't ever get the exact same thing twice. There are way too many variables at play.

It is an engineering question: Can the two output signals be made indistinguishable in spectrum and dynamics?

No.

Why?

Well, the single coil always picks up the hum of the electrical grid. Both when you don't play, and when you do. The humbucker always cancels this noise, both when you don't play, and when you do. Consequently, the gate is insufficient to make the two signals indistinguishable. You'd need to filter out the grid's hum even when you are playing.

Also, the single coil picks up every environmental field, not just the 50Hz noise of the grid. Again, the humbucker cancels all these environmental fields out. So, depending on your environment, you have some noise across all the frequencies in the single coil's spectrum that are just not there in the humbucker's spectrum. Consequently, you cannot just filter out all the noise frequencies, you must remove the noise from every band. And some of this noise is not predictable, so you can't filter it out without muting the entire guitar.

Obviously, you can put in some signal processing that fools most humans with the input of a single coil. And maybe you are such a DSP-wizard that you can even achieve a more pleasant sound with the single coil + DSP combo. But it won't be technically indistinguishable from the humbucker's sound.