I have been looking for options to add noise cancelling to my existing normal headphone, however there does not seem to be any product that will do this. Why isn't it possible to make a device that can listen to sound and output the right sounds to add noise cancelation to other headphones?
Active noise cancelling works (and i'm simplifying) by taking sound from the headphone exterior and playing that, out of phase with itself. This causes cancellation. As tetsujin alluded to, the scope we are working with is really small, and really fast. If there's a 40hz sound, then it needs to get 40hz, 180° out of phase, to your ears. This needs to happen at the same time the actual outside sound gets there. You can't have an adapter somewhere, because even a millisecond would reduce cancellation enough that it doesn't work.
When it's built into the headphones, it can just delay the sound for some minute fraction of a second, and that would do it. If the Mic is somewhere else, it would become infinitely more complex. To the point that no one makes anything like that.
EDIT: Be sure to check AJs comment below, as it clarifies the issue a bit more.
In short, no, It's not possible.
The distances & times are measure in microns & nanoseconds. Any retro-fit kit would need to be made for each specific headphone model, which is simply not financially viable, even if physically possible.
Active noise cancellation requires recording the outside signal with a microphone and injecting an out of phase version of what would reach your ears into the headphone signal. This works best and cheapest with a fixed known mechanical combination of a specific microphone and specific headphone.
Of course, noise cancellation can also be adaptive: hands-free phone speaker systems work in that manner in order to do echo cancellation. The suppression is worse and artifacts are introduced, also the computational requirements are much higher.
The quality tradeoffs will be acceptable if the goal is speech intelligibility but are not really suitable for high-quality music consumption.
It would be very easy to do. The Bose QC20s have the NC hardware in the audio cables, and not the earbuds. There are similar designs from Audio Technica. Acura also includes NC functions in the stereo systems of their cars.