I'm tinkering in Matlab with a problem that's very similar to active noise cancellation. In the literature, the secondary path is described as the transfer function from the output of the adaptive filter to the error input sensor. The algorithm needs to model this path to obtain good results.
However, there is also the transfer function from the noise source to the adaptive filter. The ADC will always have some transition band below the Nyquist rate. When I simulate the ADC in Matlab, the adaptive filter tries to invert the ADC response as well as model the system I'm trying to cancel. In fact, if the system is purely a delay, this is exactly the inverse system identification problem.
In some applications, the ADC can be ignored because the signal may already be digital, but I don't think that is ever the case with ANC. Why is this not mentioned as being a problem? Is it because, in practice, the impulse response of the ADC is short compared to the system response?