## 3 Invertibility of the Room Response  
The first research paper on RRE can be attributed to [PERSON] and [PERSON] in 1979 [5]. In their seminal paper, they studied the invertibility of the RTF and implications. Considering the RTF of a synthetic room, they showed that if the reflectivity of the wall is low (below 36%) the RTF is minimum-phase and thus invertible. On the contrary, with larger wall reflection coefficients, as those of typical rooms (in the range 70%-90%), the RIR is non-minimum-phase. However, it is still possible to equalize the minimum-phase part of the room response (i.e., the amplitude response and the minimum-phase part of the phase response) by factoring the RTF \(H(z)\) into a product of a minimum-phase term \(H_{\text{m}}(z)\) and a stable all-pass filter \(A(z)\),  
\[H(z)=H_{\text{m}}(z)\cdot A(z). \tag{1}\]  
The equalization filter is simply computed by taking the inverse \(z\)-transform of the reciprocal of the spectrum of \(H_{\text{m}}(z)\). By listening to the result of the minimum-phase equalization, [PERSON] and [PERSON] reported that \"The room effect had been removed, but a tone, much like a bell chime, sounded in the background\" [5].  
The original approach of [5] is in reality affected by several problems, many discovered by researchers only in later studies. Following the chronological order in which these problems were addressed:  
* When the room response is non-minimum-phase, an exact inverse cannot be implemented with a single sound source, since the inverse is either unstable or acausal.
* The exact equalization of the room response--or of its minimum-phase part--requires very long filters.
* The equalizer is affected by any imperfection in the measurement of the room response [6; 34].
* The room response strongly depends on the location of the loudspeaker and the microphone used for the measurement [6; 31; 32; 33; 34; 48; 49].
* Exact equalization is possible only in one location, and the extent of the equalized zone is just a fraction of the acoustic wavelength [6]. At high frequencies, the equalized zone can be smaller than the inter-aural distance of the ears (around 18 cm).
* The notches of the room response--which are affected by the noise floor--are highly boosted by the equalizer with the generation of an often audible tone-like noise (the bell chime experienced by [PERSON] and [PERSON]) [50; 51; 52].  
* The room response is actually slowly time-varying, affected by humidity and temperature [28; 29] and by movement of people or other obstacles in the enclosure.
* The human ear is sensitive to the excess-phase of the RTF [53].
* The equalizer should preserve the natural roll-off of loudspeakers at low and high frequencies [54; 55]. Amplifying these frequencies could cause an unnatural boost of the loudspeaker response, causing nonlinear effects, energy dissipation, and possible damages.  
In what follows, we will discuss the different solutions that have been devised in order to contrast the above-mentioned problems. In particular, we will review the techniques used to design the equalizer, considering both minimum-phase and mixed-phase equalization strategies, and pre-processing techniques used to contrast the effects of the variations of the room response with position and time. As much as possible, we will try to follow the chronological order in which the techniques were proposed to illustrate the evolution of RRE.