O(whatever) refers to the asymptotic behavior as n gets arbitrarily large. If n is bounded, big O notation doesn’t really mean anything because everything is O(1).

I understand big O however Fourier bio O runtime is bound by the length of the signal, not by maximum value of the signal.

The photonic processor in question can only process some maximum M number of bits worth of signal in a given time T though. To have a throughput of better than M/T you need more time or more/better processors (i.e., for large enough inputs the runtime isn't actually O(1) ).

What I was pointing out is that digital implementations have this exact same property -- if the input is bounded in the sense that there are a maximum number of bits being fed into your digital computer then the maximum runtime is a constant equal to the maximum runtime for any of those finite inputs -- i.e., the runtime is O(1) for the inputs you're actually able to provide (the exact same inputs you could provide to the photonic processor to achieve a constant-time Fourier transform). To scale up beyond that threshold you still need more time or more/better processors.

Are we talking about analog computers?

Yes.