I would like to understand how to create an active noise canceling system to reduce the noise coming from a street road bump. The advantage is that the area the noise is coming from is very small.
Noise cancelling systems work by cancelling out the noise - coming from potentially all of the environment - within a limited spatial location, usually the size of a human ear. The sound within such a limited location can be seen strictly as the temporal oscillations of the local air pressure, and not as a wave of it.
In contrast, it is not possible to create a sound-cancelling system that will take the noise that originates inside a limited spatial location and cancels it out over all of space. There are two fundamental reasons for this: wave interference and, ultimately, conservation of energy.
When you allow the sound to propagate into the surrounding space, you can no longer treat it as locally homogeneous oscillations: you have a source of waves which you're trying to cancel out by adding a phased-out source. You then have two wave sources, and these will interfere according to the usual rules of wave phenomena: you will have regions where their effects cancel out, and regions where they add up with each other to make a stronger (in fact, by a factor of 4) effect:
You can of course dream up fancier systems than just a single ANC source. However, they will all fall foul of this problem. The fundamental reason for this is, of course, conservation of energy. Your noise source is emitting energy into the environment. You are further using your ANC sources to put out even more energy into it. This energy has to end up somewhere - it can't just vanish. Wave interference simply causes a 're-routing' of energy from places where you want it to be quiet into places that necessarily will become louder. For more con this, see What happens to the energy when waves perfectly cancel each other?.