Curiously, it doesn't seem to have anything to do with C39. Unless the input is being strobed at the resonant frequency (in which case it's like a bandpass'd charge pump of sorts), which I guess could be.
If you want to be very literal about it, you can instantiate each op-amp as an integrator, and just do the math (add/subtract inputs, accumulate outputs) with a few DSP blocks.
DSP scares a lot of people, but it is literally exactly that simple!
There are some gotchas, like needing a sample rate high enough that it doesn't go unstable (i.e., a few times the "ping" frequency), and enough bits that the integrator doesn't end up stuck on an LSB. That shouldn't be a problem, most DSP blocks have more than enough bits, I'd think 12-16 would be more than enough here even for an unoptimized implementation (and I want to say 18 bits is a more common size, at least in some product lines??).
Tim