# Augmenting Environmental Interaction in Audio Feedback Systems+
## 3 System Design and Validation
### Acoustic Resonance with Timbre  
The length of the prevailing path in a feedback loop principally determines the acoustic resonance of the audio feedback, and this path is predominantly influenced by the distance between a microphone and a loudspeaker. We chose the resonant frequency as the simplest characteristic representation of acoustic resonance, and approximated it by the position of maximum energy in the spectrum. The distance between a microphone and a loudspeaker is simulated by appending a silence to the beginning of the impulse response, in order to delay the direct sound and the following reflections. Figure 8 compares the distances and the approximations of resonant frequencies.  
Timbre, a target control feature, is the quality of sound that distinguishes different sounds of the same frequency and amplitude. Our system purposes to control timbre of the feedback sounds in terms of brightness, overall distribution of energy over frequency, by the cutoff frequency parameter of the low-pass filter. Spectral centroid is used as a measure of brightness, and Figure 9 shows the controlled different timbres through the comparison of spectra and spectral centroids: since the cutoff frequency is the attenuating position, higher cutoff frequency generates brighter sounds and lower cutoff frequency generates darker sounds.  
Figure 8: Resonant frequency approximations by the position of maximum energy in the spectra of the simulated feedback signals. Distance is simulated as the length of a silence appended to the beginning of the impulse response.  
Figure 7: (**a**) Low threshold and (**b**) average peak amplitude curves of the feedback signals according to volume level of ambient noise. Ambient noise is simulated using different volumes of white noises.  
The approximation of resonant frequency is mapped to the cutoff frequency parameter of the low-pass filter. An intriguing point is that the change in timbre (brightness) mostly influences the position of maximum energy in a spectrum, which is used as the input information. This enables evaluation of this mapping by observing changes in the approximations of resonant frequencies. However, the resonant frequency does not always behave in proportion to cutoff frequency change: timbre (brightness) may show abrupt changes by only a slight change in cutoff frequency (the _nonlinearity_ explained in [2]). Thus, we designed the cutoff frequency to adaptively change depending on variation of the frequency approximation.  
Similarly to the previous tempo control designs (Section 3.1), two opposite mappings were also investigated in accordance with the changing direction. For example, if the estimated frequency is bigger than a threshold, the cutoff frequency parameter gradually increases (proportional mapping) or decreases (reflected mapping) using a ramp function; on the other hand, when the estimated frequency is lower than a threshold, the parameter decreases (proportional mapping) or increases (reflected mapping). The effect of changing the distance is thereby intensified by the proportional mapping and reduced by the reflected mapping: Figure 10 evaluates the two mappings.  
Figure 10: Approximations of spectral centroids and resonant frequencies from the simulated feedback signals when the cutoff frequency of the low-pass filter is (**a**,**d**) constant and (**b**,**e**) controlled with the proportional mapping and (**c**,**f**) reflected mappings. Vertical dot lines in the upper figures represent the positions where the approximated resonant frequencies deviate from the thresholds. These thresholds are depicted as the horizontal dot lines in the lower figures.  
Figure 9: Plots of the magnitude spectrum of the normalized feedback signals when the cutoff frequency of the low-pass filter is (**a**) 300 Hz and (**b**) 6000 Hz. Spectral centroids are 3327 Hz and 3965 Hz respectively. The distance is set to 4 meters.