An electric guitarist usually has a pedalboard that they use to make different guitar tones. Let's approach this like an electrical engineer might.
Time is a straight arrow: call it t.
The guitarist plays, and their guitar pickup records a signal. At any time t, measure the value of this signal's waveform: call it x.
That signal can go into a pedal that maps each value to create an output signal. Call it f(x, t).
The pedal is a linear system when it satisfies f(x1 + x2, t) = f(x1, t) + f(x2, t). In English, you can blend two guitar signals going into a single linear pedal, and it will sound the same as if you used a separate pedal for each guitar.
The pedal is a time-invariant system when it satisfies f(x, t1) = f(x, t2). In English, a time-invariant pedal does not warble over time.
The magic happens when a system is both linear and time-invariant. These systems are so special, we call them "LTI" for short. Feed an LTI system a single, really loud, POP! Something like a clap, or a gunshot. Record the sound of this gunshot, and call it "impulse_response.wav". This file characterizes EVERYTHING about the system. If you know how the system responds to an impulse, you will know how the system responds to any other sound. All you have to do is convolve the sound with the impulse response, i.e. use the contents of impulse_response.wav as a set of weights and pass the sound through it.
The echoes you hear when you speak in a large room is a real-world example of an LTI system. People will go to a cathedral, an open field, or a large tunnel, set up a microphone, and fire a blank pistol. They can go home, convolve the sound of their guitar or voice or whatever with the impulse response, and it will sound exactly like they're in that space! There are whole communities of people making and sharing impulse responses out there, and I just think it's the coolest thing ever.
For any guitarists reading this, I'll bring this back to the pedalboard. Another cool thing about LTI systems is that when you chain multiple of them together, the order does not matter. Reverb, delay, EQ, and wah are all LTI effects. If you've ever wondered why pedal ordering sometimes matters and sometimes doesn't, this is why.
> Another cool thing about LTI systems is that when you chain multiple of them together, the order does not matter. Reverb, delay, EQ, and wah are all LTI.
Delay and reverb may be LTI in your definition, but they very much do matter what order they are in, every time.
Delay before reverb is going to give a more staccato sound, whereas the reverb before delay is probably just going to wash the whole thing out if you’re not careful on your settings.
I suspect you are right when it comes to analog circuits, but they are definitely LTI in a typical digital implementation. I checked this by adding reverb+delay and delay+reverb to two copies of a track in REAPER. I inverted the phase of one, played them together, and they cancelled out.
This method of convolution of the sound is nowadays utilised in "profiling" amps like the ones made by Kemper. There are these particular amps, effects, speakers, and/or their specific combos, which are considered sacred/sweet/holy which are highly sought after. But most of them cost a lot. There's also the problem faced by touring bands where they have to lug around huge setups. And then there's the issue of maintenance and repairs. These profiling amps help musicians capture the sound of their favorite setups and replicate it in a compact setup. Hugely popular, if not for the price. There are still purists who swear by the sound of 'analog' setups, but for most of the purposes the sound is identical. Fascinating stuff.
> The pedal is a time-invariant system when it satisfies f(x, t1) = f(x, t2). In English, a time-invariant pedal does not warble over time.
> Reverb, delay, EQ, and wah are all LTI effects.
The math doesn't work, assuming scalar f, x, t1, t2 as implied by their description. For a delay the x passed at t1 has no correlation to what f calculates to. It needs vectors F(X, t) and X I think.
You found it! I rewrote my comment like six times because I think the rigorous notation where the signal is a function of t is a little messy for a HN comment. I tried picking a notation that was less intimidating and wasn't too obviously inconsistent.
It's worth mentioning that in a typical pedalboard you're probably going to see zero linear effects. The closest ones are probably an EQ and pure digital Reverbs and Delays. Almost all other effects have tons of saturation.
I thought I remembered a single platform where people upload their IRs, but I can't find it. I remember that one being too experimental for my taste. The best-sounding IRs are usually released in sets by a single individual, and mass-compiled in forums like these [0].
Time is a straight arrow: call it t.
The guitarist plays, and their guitar pickup records a signal. At any time t, measure the value of this signal's waveform: call it x.
That signal can go into a pedal that maps each value to create an output signal. Call it f(x, t).
The pedal is a linear system when it satisfies f(x1 + x2, t) = f(x1, t) + f(x2, t). In English, you can blend two guitar signals going into a single linear pedal, and it will sound the same as if you used a separate pedal for each guitar.
The pedal is a time-invariant system when it satisfies f(x, t1) = f(x, t2). In English, a time-invariant pedal does not warble over time.
The magic happens when a system is both linear and time-invariant. These systems are so special, we call them "LTI" for short. Feed an LTI system a single, really loud, POP! Something like a clap, or a gunshot. Record the sound of this gunshot, and call it "impulse_response.wav". This file characterizes EVERYTHING about the system. If you know how the system responds to an impulse, you will know how the system responds to any other sound. All you have to do is convolve the sound with the impulse response, i.e. use the contents of impulse_response.wav as a set of weights and pass the sound through it.
The echoes you hear when you speak in a large room is a real-world example of an LTI system. People will go to a cathedral, an open field, or a large tunnel, set up a microphone, and fire a blank pistol. They can go home, convolve the sound of their guitar or voice or whatever with the impulse response, and it will sound exactly like they're in that space! There are whole communities of people making and sharing impulse responses out there, and I just think it's the coolest thing ever.
For any guitarists reading this, I'll bring this back to the pedalboard. Another cool thing about LTI systems is that when you chain multiple of them together, the order does not matter. Reverb, delay, EQ, and wah are all LTI effects. If you've ever wondered why pedal ordering sometimes matters and sometimes doesn't, this is why.