## 8 Fixed and Adaptive Wave Domain Equalization
### MIMO and SIMO Approaches  
As an alternative to the wave-theoretic approach discussed so far, the acoustic paths between the loudspeakers and microphones can be interpreted as independent linear time-invariant systems. All resulting transfer functions can be combined together into a multiple-input/multiple-output (MIMO) system. MIMO room equalization approaches differ, amongst others, with respect to the loudspeaker and microphone positions (control points), and the particular technique used to compute the equalization filters. The difference between the wave-theoretic and the MIMO approaches discussed in the sequel is that the computation of equalization filters is not performed in a spatially transformed domain. Although the placement of the loudspeakers and microphones on the border of the listening area is motivated by the HIE, MIMO approaches may depart from this placement. As stated above, a sufficient number of loudspeakers and microphones must be used in order to synthesize and capture the entire sound field up to a given frequency. If the sampling is not dense enough, equalization may only be achieved at or in close vicinity to the microphone positions.  
A non-adaptive MIMO approach which directly emerges from the discretization of the HIE is presented in [213]. The MIMO system is inverted in order to compute equalization filters for global equalization. As an alternative, a local solution is also discussed. A similar approach is followed in [85] for wave field synthesis. Channel shortening has also been investigated in the context of MIMO equalization [214; 215] based on a least-squares solution.  
The computation of equalization filters generally constitutes an inverse problem. Various algorithms have been proposed that improve the numerical and computational efficiency, as well as the numerical conditioning--for instance, a fast iterative MIMO inversion algorithm working in the DFT-domain [216], or a DFT-domain approximation of the MIMO filtered-x algorithm [217]. In [216], a steepest-descent and approximative Gauss-Newton iterative algorithm for the design of a MIMO equalizer is presented. In [218], a method for coping with the low conditioning of the transfer function matrix at some frequencies is proposed. The problem is amended by studying the structure of the MIMO transfer function matrix and replacing its inverse matrix by a pseudo-inverse that allows a range of acceptable solutions. Polynomial-based MIMO formulations of the room equalization problem are discussed in [219; 220] with extensions towards explicitly controlling the number of active loudspeakers used for equalization [221].  
There are also a number of specialized equalization approaches for specific scenarios. For instance, the equalization of multichannel stereophonic systems under the constraint that stereophonic pairs of loudspeakers should have similar transfer functions is discussed in [222; 223; 224]. The approach is split into two stages: (i) equalization of a single path also utilizing the other loudspeakers and (ii) similarity optimization between two channels that are used for stereophonic imaging. The room equalization in cars has been considered in various studies. A non-adaptive MIMO equalization approach utilizing IIR or FIR filters is presented in [225]. The optimization is performed in terms of the overall magnitude response to avoid coloration/tonal issues. A combined room equalization and cross-talk canceling approach for cars is discussed in [226].  
Besides the MIMO approaches reviewed so far, single loudspeaker room equalization approaches have also been investigated which utilize multiple microphones. This constitutes a single-input/multiple-output (SIMO) problem. A non-adaptive polynomial multivariate control approach combined with a constrained mean squared error design and zero clustering is discussed in [149]. A statistic inferential method which considers the statistical variation between the different microphone positions for improved robustness and an enlarged listening area is presented in [227].