## 1 Introduction  
The convergence performance of an adaptive filtering is decisively affected by the choice of certain characteristic parameters of the filter. This is one of the main motivations behind the high number of adaptive algorithms and structures that are continually introduced in the literature to provide improvements of convergence and steady-state performance in certain scenarios. The design of an adaptive filter may be even very risky when no a priori information is provided on the model of the signal to be processed. One solution to this problem is represented by the adaptive combination of adaptive filters [1; 2; 3; 4; 5; 6], which exploits the diversity of parallel adaptive branches. Combined structures of filtering have been applied to a wide variety of online adaptive filtering applications [7; 8; 9; 10; 11; 12; 13; 14; 15] due to their capabilities of detecting and selecting instant by instant the best-performing filter. To this end, the analysis of the mean squared performance aims at showing the effectiveness of adaptive combined schemes and how to exploit them the best [6; 16; 17; 18; 19; 20]. In this paper, we focus on the combination of adaptive filters in the multichannel case. In particular, the combined structure that we take into account is composed of a convex combination of two adaptive multiple-input single-output (MISO) filters [8]. MISO filters are widely used in several applications, such as adaptive beamforming [21; 22], OFDM systems [23], channel equalization [24], and audio and speech applications [25; 26]. These have been chosen as a baseline scheme of the multichannel case,but the combined MISO structure, and the related analysis thereof, can also be easily extended to multiple-input multiple-output (MIMO) filters, as in [11], since an MIMO filter can be seen as a parallel of MISO filters. In order to adapt MISO filters, we adopt a multichannel affine projection algorithm (APA) [27; 28], which, being characterized by a projection order, allows us to generalize the analysis with respect to both gradient-based and Hessian-based adaptive algorithms. Due to its robustness, the APA has been widely implemented with variable parameters to further improve performance [29; 30; 31; 32; 33]. A first novel insight in this paper concerns the combination of two multichannel APA filters differentiated according to the projection order. The combination of APAs using different projection orders has been addressed in [8; 34; 35; 36; 37; 38], where, however, the combination takes into account only the current projection. Conversely, the proposed approach, which is also generalized with respect to the other filter parameters, considers a mixing parameter for each projection order. Moreover, in the choice of differentiating the projection order, and in particular setting a unitary order for one of the two MISO filters, the diversity between gradient-based and Hessian-based algorithms is exploited, thus yielding a performance improvement [1; 16; 39]. Some attempts in that sense were also provided in [40], but without a theoretical performance analysis. In this paper, instead, a performance analysis is conducted in-depth, thus representing itself another novel contribution. Indeed, in order to show the capabilities of the proposed adaptive scheme, we derive a steady-state analysis of the combined algorithm, thus deriving a theoretical value of the excess mean squared error (EMSE) based on the energy conservation arguments [41]. In the literature, many efforts have been made to analyze the performance of single-channel APA filters [42; 43] and of combined least mean squares (LMS)-like algorithms [1; 6]. The use of the proposed scheme leads to further non-straightforward assumptions that increase the complexity of deriving a theoretical value for the EMSE. Experimental results prove, on the one hand, the validity of the steady-state EMSE analysis presented and, on the other hand, the effectiveness of the proposed combined filtering architecture.