## 4 Equalizer Design Techniques
### Multiple-Input/Multiple-Output Inverse Theorem Methods  
A method for the exact inversion of the RIR--even when it is non-minimum phase--was proposed in [86; 87]. The method is based on a principle called the multiple-input/multiple-output inverse theorem (MINT). With this method, the inverse is constructed from multiple FIR filters, by adding _\"some extra acoustic signal-transmission channels produced by multiple loudspeakers or microphones.\"_ In practice, the MINT states that it is possible to obtain an exact inversion of the room response if the number of loudspeakers is larger than the number of microphones (i.e., measurement points). Thus, the approach is intrinsically multi-channel. Let us consider the case of a system with two loudspeakers and one microphone. Let us indicate with \(G_{1}(z)\) and \(G_{2}(z)\) the transfer functions from the loudspeakers to the microphones, and with \(H_{1}(z)\) and \(H_{2}(z)\) the transfer functions of the equalizers associated to each loudspeaker. Then, for exact inversion of the room response, \(H_{1}(z)\) and \(H_{2}(z)\) must satisfy the following condition:  
\[H_{1}(z)G_{1}(z)+H_{2}(z)G_{2}(z)=1. \tag{2}\]  
As shown in [87], the solution of Equation (2) exists if \(G_{1}(z)\) and \(G_{2}(z)\) are relatively prime (i.e., do not have common zeros), and when the solution exists the orders of \(H_{1}(z)\) and \(H_{2}(z)\) are lower than \(G_{2}(z)\) and \(G_{1}(z)\), respectively. The approach is very powerful because it allows the acausality problem of the equalizer to be overcome. However, the MINT approach also exhibits strong limitations. In [88], the MINT is analyzed under a numerical perspective, studying the condition number of the time domain matrix that is inverted. It is shown that the condition number of the time domain matrix is related to the singular values of the transfer matrix evaluated over frequency. The condition number decreases and the numerical performance is enhanced as the number of loudspeakers is increased. However, the condition number increases _\"at the rate of approximately 1 bit\"_ (i.e., of approximately 6 dB) for each microphone added [88]. Moreover, an analysis of the MINT technique is also presented in [89], discussing the conditions which must be fulfilled for an exact inverse filter matrix to exist. Additionally, [89] demonstrates that the number of loudspeakers must exceed the number of microphones in a manner consistent with the findings of [87]. Moreover, an explicit formula is derived specifying the number of required inverse filter coefficients for the existence of an exact inverse. The paper also investigates the spatial extent of the zones of equalization produced by inverse filtering. It is shown that the equalized zone scales in size in accordance with the acoustic wavelength at the highest frequency of interest.  
The low extent of the equalized zone and the numerical sensitivity to errors in the measured responses appear to be the main limitations of the MINT. An improvement of the method has been proposed in [90], where more control points are considered without increasing the number of inverse filters. Another improvement is discussed in [91], where an iterative method is applied to the MINT considering an optimally-stopped weighted conjugate gradient. To improve the computational efficiency of the MINT, an oversampled subband approach with decimation has been presented in [92].