This is more of a general question and a chance for me to explore this topic a bit deeper. Do you know of any 'real world' situation where it might be necessary to detect the presence of an unknown tone, and then apply a notch filter to remove the tone? In these situations, are any of these notches applied 'automatically' (i.e. the need for the notch is detected and then the filter is applied)?
Miscellaneous notch filter questions:
1) Why are notch filters used instead of simply either a LPF of HPF?
2) Are notch filters typically implemented in practice through IIR or FIR?
Thank you for taking time to read my question,
Notch filters are effectively IIR filters. They are of a class called 'recursive filters', because the notch filtere uses a part of the output together with the input. For a readable account please see http://www.dspguide.com/ch19.htm.
The notch filters are used to stop a very narrow frequency band, say, for example the 50 or 60 Hz line frequency signal present in many data series. The goal is to remove just the line frequency but as little of the neighbouring frequencies that may carry signal of interest. Typical short LPF or HPF do not achieve these narrow band rejection, without being infinitely long. This is what recursion is able achieve with a few coefficients.
A notch is designed to reject a single frequency. It helps if one think of a notch in the frequency domain as an inverted impulse in freq domain. Just as an impulse in time domain contains all frequencies (hence the term 'impulse response' of a device), a notch in the frequency domain results in an infinite series in the time domain. Therein lies the skill in notch filter design, to make the notch filter as sharp as possible in the frequency domain, balanced with keeping the time equivalent as short as possible, i.e. shorter than infinite.
No, notch filters needn't be IIR, although in many situations IIR will result in less computation and less delay, at the expense of phase distortion.
One use for an automatic notch filter is in communications. You can have two near carriers that beat against one another creating a pure tone of the difference (heterodyne). The filter can notch this out. A simple method that is used is to use an LMS adaptive filter. You can Google this to find more.
This type of filter is popular for amateur radio.
You use a notch when you don't want to lose too much of the background signal, only that part of the spectrum which is being wiped out by the interfering tone.
You seem to be implying in your example that the interfering tone is very stable, and that you first detect it, then design a notch filter, and then apply that notch. In practice what you will want to do to remove a narrowband interfering tone is to lock onto it and subtract it (sometimes called frequency agile filtering).
Locking onto a sinusoid is a non-LTI process, typically performed by a PLL and not a LTI filter. Using LTI you would have to open the BW of the notch wide enough to capture the interfering sinusoid as it moves around, and thus remove too much of the desired signal.
Tones can arise spontaneously due to positive feedback, such as the acoustic feedback in PA systems and hearing aids. Adaptive notch filtering is one standard technique for suppressing such tones - the system detects one or more tones, determines their frequencies, and creates notch filters centred at these frequencies. IIR notch filters are generally preferred, as FIR notch filters introduce significant delay and are more computationally expensive.