## 7 Adaptive Single-Point and Multi-Point Equalization
### MISO/MIMO Approaches  
The adaptive RRE techniques proposed in [84, 178, 180] (and many other papers) consider the equalization of a single sound source (i.e., of a single audio reproduction channel), due to the problem of estimating several impulse responses at the same time. If two or more channels are employed, the covariance matrix of a multichannel adaptive algorithm becomes ill-conditioned due to the correlations between the channels for typical reproduction techniques. The ill-conditioning generally causes convergence problems. This was shown, for instance, for stereophonic acoustic echo cancellation [181]. To cope with the non-uniqueness problem, a method to reduce the inter-channel coherence is usually exploited. In this context, many of the techniques used to reduce the channel cross-correlations often introduce significant distortions, which are unacceptable in high-quality sound reproduction systems [181, 182]. Therefore, a suitable technique which is capable of decorrelating the loudspeaker signals and of preserving the audio quality must be considered. The approach in [183] introduces a multichannel solution which also considers the non-uniqueness problem. The room responses are estimated with good accuracy by reducing the inter-channel coherence using a technique that produces only a mild degradation of the sound quality. Specifically, the low-frequency region is decorrelated by exploiting the missing-fundamental phenomenon, while the high frequencies are decorrelated with a second-order time-varying all-pass filter combined with a multiple notch filter [184]. The equalizer is designed in the warped frequency domain to improve the equalization in the low-frequency region and, at the same time, to reduce the computational cost of the design. In [185], the adaptive multichanneland multi-position RRE system briefly introduced in [183] is fully detailed and extended, providing a real-time implementation in commercial Hi-Fi products.  
To improve the convergence speed and the robustness of the adaptive identification algorithm in the presence of low signal-to-noise ratio, the use of a biased adaptive algorithm has recently been proposed in [186] for a MIMO system. In detail, the algorithm is based on the improved proportionate normalized least-mean squares algorithm (IPNLMS) within the conventional filtered-x scheme (IPNLMS-FX), previously introduced for active noise control (ANC) [187], and here extended towards multichannel equalization. However this method requires an a priori estimation of the impulse responses, which is not available in many practical applications. With the same purpose of improving convergence and robustness, a combination of block-based adaptive filters (also employing biased algorithms) was proposed in [188].  
It is worth underlining that if a binaural system is considered, a natural decorrelation among stereo channels is obtained. A stereo representation of an adaptive RRE system can be achieved without channel decorrelation, as reported in [169; 189]. An improvement of this technique is presented in [190], where a subband structure is proposed to reduce the computational complexity of the procedure.