## 1 Introduction  
In this paper, the blind adaptive deconvolution problem is addressed, which arises in many applications, such as seismology, underwater acoustic, image restoration and digital communication [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. In digital communication applications, the problem is often called blind adaptive equalization. Non-blind adaptive equalizers, unlike blind adaptive equalizers, require training symbols to generate the error that is fed into the adaptive mechanism which updates the equalizer's taps. Therefore, blind adaptive equalizers have some important advantages compared with the non-blind version: (1) Simplified protocols in point-to-point communications, avoiding the retransmission of training symbols after abrupt changes of the channel. (2) Higher bandwidth efficiency in broadcast networks. (3) Reduced interoperability problems derived from the use of different training symbols. It is well known that inter-symbol interference (ISI) is a limiting factor in many communication environments, where it causes an irreducible degradation of the bit error rate thus imposing an upper limit on the data symbol rate [36]. In order to overcome the ISI problem, an equalizer is implemented in those systems [34; 35; 36; 37; 38; 39; 40; 41]. In this work, the T-spaced blind adaptive equalizer is considered for the single-input-single-output (SISO) case where the sampling rate is equal to the symbol rate (thus referredto as T-spaced where T denotes the baud, or symbol, duration). Please note that a fractionally-spaced equalizer (FSE) where the sampling rate is higher than the symbol rate can be modeled with a single-input-multiple-output (SIMO) system. In addition, a SIMO system can be modeled with a parallel combination of T-spaced blind adaptive equalizers. Thus, improving the equalization performance of a T-spaced blind adaptive equalizer for the SISO case may also lead to equalization performance improvement for the SIMO case. As already mentioned above, blind adaptive equalizers do not use any training symbols to generate the error that is fed into the adaptive mechanism which updates the equalizer's taps. Instead of those training symbols, an estimate of the desired response is obtained via the use of a nonlinear transformation to sequences involved in the adaptation process. Now, very often, blind adaptive equalization algorithms are classified according to the location of their nonlinearity in the algorithm chain [42]. According to Reference [42], there are three different types: (1) polyspectral algorithms, (2) Bussgang-type algorithms, (3) probabilistic algorithms. In the first method, the nonlinearity is located at the output of the channel, right before the equalizer. Thus, the nonlinearity actually estimates the channel. This estimation is fed into the adaptive mechanism which updates the equalizer's taps. In the second type, the nonlinearity is situated at the output of the equalizer. Here, the nonlinearity can just be the estimation of the source signal via the the use of the conditional expectation (the expectation of the source input given the equalized or deconvolutional output), or the nonlinearity can be a predefined cost function that holds some information of the ISI. Thus, minimizing this predefined cost function with respect to the equalizer's taps may lower the residual ISI and help in the symbol recovery process. Since Bussgang-type algorithms often have shorter convergence times than polyspectral methods, which need larger amounts of data for an equivalent estimation variance, they are more popular [42]. In the third class of algorithms, directly locating the nonlinearity is more problematic compared to the first two groups since the nonlinearity is combined with the data detection process. While these algorithms can extract considerable information from relatively little data, this is often accomplished at a huge computational cost [42].