# Modeling ADC in Active Noise Cancellation

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?

I am also working on ANC, with my understanding may be I could provide some clarity for you.

I could not see your point exactly on ADC modelling, but as far as I know this ADC transfer function will be covered in secondary path modelling.

"However, there is also the transfer function from the noise source to the adaptive filter."

Do you mean modelling of the reference input microphone, as far as I can see only transfer function involved between noise source to adaptive filter is reference input microphone function.

The way you are using secondary path description is also a bit confusing to me.

But here is what I know from this paper page 945

Here P(z) is the unknown primary acoustic path between reference input sensor and error sensor.

Below text directly quoted from the reference provided

Summing junction represents the acoustical superposition in the space from cancelling loudspeaker to error microphone, where the primary noise is combined with the output of the adaptive filter.

Therefore, it is necessary to compensate for the secondary-path transfer function from y(n) to e(n) which includes the digital-to-analog (D/A) converter, reconstruction filter, power amplifier, loudspeaker, acoustic path from loudspeaker to error microphone, error microphone, pre-amplifier, anti-aliasing filter, and analog-to digital (A/D) converter.

The compensation as you might have already known is done by offline or online modelling techniques.

If you want to include your primary path modelling as well i.e reference microphone TF, you can do that or if you are considering feedback effect, then again in the feedback compensation this reference microphone transfer function is included.

Please correct me if I am wrong.
Thanks for bringing this up!

• Sorry I was too lazy to make a diagram. My second paragraph is referring to the path from x(n) to W(z). I'm not concerned about the microphone. The issue is the anti-aliasing filter in the ADC. W(z) will attempt to invert this TF. As far as I can tell, there is no mention of this as an issue, but it causes major problems in my simulations. Can this be modeled in secondary path as you say?? I'll need to try. The quote you provided discusses only the anti-aliasing filter of error ADC. – Todd Jan 5 '16 at 13:48
• What are components involved in the path from x(n) to W(z), Microphone, and microphone wire. Are you considering A/D for first mic also in this path. Is that what you are trying to model ? – charansai Jan 5 '16 at 15:57
• I am also curious how you are trying to model the ADC that you are referring to? Are you using system identification individually ? – charansai Jan 5 '16 at 15:59
• My application isn't acoustics and it isn't quite for noise cancellation either. There is really only the ADC from x to w to worry about. My model of the ADC is just a linear phase LPF with the transition band near the Nyquist rate. I also tried moving the transition band just above the Nyquist rate (and then decimate to lower sample rate) so the ADC is effectively all-pass but allowing some small amount of aliasing at the upper end. In either case, W(z) requires many more taps and the cancellation impaired. – Todd Jan 5 '16 at 17:21
• Ohh, there I cant help! But in actual ANC as far as I know, the x to w ADC model along with reference microphone model will be included in feedback path modelling of ANC. – charansai Jan 5 '16 at 17:43

You are correct that the system will attempt to invert the ADC filter. In acoustics this is not usually a problem because there is not much energy at those frequencies. If your application is not a standard acoustic system, there may be an opportunity to put a copy of the ADC filter in the plant path (this is normally not possible because the error summation only exists in the acoustic domain). Alternatively you can put a gentle low pass filter on the error signal so the adaptive filter will not respond to near-Nyquist frequencies. This will slow down the adaptation speed slightly.