Hearing Aid Acoustic Feedback Solution #5 – Notch Filtering
Because acoustic feedback is not evident at all high frequencies, it would appear appropriate to reduce gain at the specific frequency or frequency range at which it occurs, without affecting significantly adjacent frequencies. A technique to perform this is the use of a notch filter. A notch filter (more appropriately called a stopband, band-stop, or band-rejection filter) passes all frequencies except those in a stop band centered on a center frequency. The amplitude response of a notch filter is flat at all frequencies except for the stop band on either side of the center frequency (Figure 10). The standard reference points for the roll-offs on each side of the stop band are the points where the amplitude has decreased by 3 dB, to 70.7% of its original amplitude.
Figure 10. Notch Filter centered at 2800 Hz (Agnew, 1996)
In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a band-stop filter with a narrow stopband (high Q factor).
The Q (Quality) Factor
The Q factor is a closely related item of a filter. High Q notch filters eliminate a single frequency or narrow band of frequencies. Q is the Quality Factor of either a notch filter or of a bandpass filter. It is defined as the center frequency of a filter divided by the bandwidth. The bandwidth is the frequency of the upper 3-dB roll-off point minus the frequency of the lower 3-dB roll-off point. The -3 dB points and -20 dB points are determined by the size of the stop band in relation to the center frequency, in other words, the Q of the filter.
Generally, the higher the Q-factor, the more exact the notch. A notch filter with a low Q-factor may effectively notch out a range of frequencies, whereas a high Q notch filter will only delete the frequency of interest.
Three values of Q for a 10 kHz notch filter are shown in Figure 11 below. Please note that this frequency range is being used for illustrative purposes only and obviously does not reflect the frequency range of hearing aids.
Figure 11. Different values of Q for a notch filter.
While not shown here, the phase response of a notch filter shows the greatest rate of change at the center frequency. The rate of change becomes more rapid as the Q of the filter increases.
A common misconception of notch filters is that many people think that the higher the Q, the deeper the notch. This is not true. The depth of the notch depends on the matching of components. The Q affects only the location of the -3 dB points, or in other words, the stop band bandwidth.
So, what is the ultimate limit? Can any Q be possible? Unfortunately, no. At very high Q values the response of the circuit will begin to have overshoot and undershoot that will destroy the integrity of the notch. The frequency that was supposed to be rejected may actually be amplified.
Relative to hearing aids, the actual shape of the transition region between the unity gain area and the notch, and width of the notch, are determined by the hearing aid designer.
The application of a notch filter to hearing aids is illustrated in Figure 12, which shows the original frequency response and then the frequency response with a notch filter applied {{2}}[[2]] Agnew, J. Acoustic feedback and other audible artifacts in hearing aids, Trends in Amplification, 1996, Vol. 1 (2)[[2]].
Figure 12. The application of a notch filter to a hearing aid showing the effects on the frequency response both without and with the notch filter.
Does Notch Filtering Help Control Acoustic Feedback?
Agnew (1993) {{1}}[[1]] Agnew, J. Application of a notch filter to reduce acoustic feedback. Hearing Journal, 1993, vol. 46 (3):37-40,42-43[[1]] described using a variable notch filter with power BTE hearing aids. The center frequency of the notch was adjustable using a potentiometer, from 1500 to 5000 Hz. Additional gain of 8 to 10 dB prior to feedback was provided for the lower frequencies, but smaller increases in gain before feedback were measured between 2000 and 4000 Hz.
Digital control of analog hearing aids offered opportunities for use of notch filters having different bandwidths. The hearing aids could be programmed to reduce gain in the frequency bands (and also had a tendency to reduce peaks in the frequency response) where feedback occurred.
Notch Filter Problems
In actual practice, notch filters suffer from at least two shortcomings that must be overcome. 1) Because oscillation is likely to occur at more than a single frequency, if the notch is intended to reduce the primary frequency, feedback may occur at a secondary frequency and again produce an oscillation problem. 2) Acoustic feedback is not a static phenomenon, as was mentioned previously. The in-situ environment, including both the wearer factors (talking, chewing, etc.) and interactions with the environment (moving in and out of certain environments, distance to walls, etc.) create factors that alter the frequency or frequencies causing feedback. If one attempts to cover all these with a wider notch filter, it still may not cover all the frequencies causing oscillation, and even worse, is most likely to reduce the desired frequency amplification necessary for speech communication.
Because of these potential limitations, attempts have been made to utilize more than a single notch filter. Unfortunately, results have been consistent with the problems related to # 2 in the previous paragraph.
Bonus To This Discussion About Filters
Although this series of blogs relates to use of the notch filter to manage feedback, this discussion can also identify the use of the Q factor for bandpass filters as well (consider this an added benefit because it also has implications for hearing aids, although not for feedback purposes).
Three values of Q for a 10 kHz bandpass filter are shown in Figure 13. Again, please note that this frequency range is being used for illustrative purposes only and does not reflect the frequency range of hearing aids.
Figure 13. Different Q values for a bandpass filter.
• For a filter with a Q of 0.1, the -3 dB points are at about 1 kHz and 100 kHz and a center frequency of 10 kHz.
• For a filter with a Q of 1, the shape of the curve is different, looking like a rounded 90 degree angle.
• For a filter with a Q of 10, the response of the bandpass filter dramatically changes.
Next blog on acoustic feedback will continue to focus on feedback approaches that have been used with hearing aid.
