Alban Goupil

New Algorithms for Blind Equalization: The Constant Norm Algorithm Family

In this paper, a new, efficient class of blind equalization algorithms is proposed for use in high-order, two-dimensional digital communication systems. We have called this family: the Constant Norm Algorithms (CNA). This family is derived in the context of Bussgang techniques. Therefore, the resulting algorithms are very simple. We show that some well-known blind algorithms such as Satos algorithm or the Constant Modulus Algorithm (CMA) are particular cases in our CNA family. In addition, from this class, a new cost function, named Constant sQuare Algorithm (CQA), is derived, which is well designed for QAM. It results in a lower algorithm noise without increasing the complexity. Another advantage of this class lies in the possibility of creating new norms by combining several existing norms in order to benefit from the advantages of each original norm. For example, we present the norm resulting from the combination of the two algorithms, CMA and CQA. Moreover, we highlight that, with regard to the excess mean-square error performance, there is an optimal norm for each constellation, i.e., each modulation, in order to equalize it blindly

Performance analysis of the weighted decision feedback equalizer

In this paper, we analyze the behavior of the weighted decision feedback equalizer (WDFE), mainly from filtering properties aspects. This equalizer offers the advantage of limiting the error propagation phenomenon. It is well known that this problem is the main drawback of decision feedback equalizers (DFEs), and due to this drawback DFEs are not used very often in practice in severe channels (like wireless channels). The WDFE uses a device that computes a reliability value for making the right decision and decreasing the error propagation phenomenon. We illustrate the WDFE convergence through its error function. Moreover regarding the filtering analysis, we propose a Markov model of the error process involved in the WDFE. We also propose a way to reduce the number of states of the model. Our model associated with the reduction method permits to obtain several characteristic parameters such as, error propagation probability (appropriate to qualify the error propagation phenomenon), time recovery and error burst distribution. Since the classical DFE is a particular case of the WDFE (where the reliability is always equal to one); our model can be applied directly to DFE. As a result of the analysis of this process, we show that the error propagation probability of the WDFE is less than that of the classical DFE. Consequently, the length of the burst of errors also decreases with this new WDFE. Our filtering model shows the efficiency of the WDFE.

A geometrical derivation of the excess mean square error for Bussgang algorithms in a noiseless environment

The steady-state excess mean square error (EMSE) is a useful performance criterion to measure how noisy Bussgang algorithms are. Thanks to a simple geometrical interpretation of LMS-like algorithms, the Pythagoras theorem gives us a general equation similar to the fundamental energy conservation of Mai and Sayed (IEEE Trans. Signal process. 48 (1) (2000) 80). Thereafter, a simple, but general, closed form of the EMSE is derived for Bussgang algorithms when they have converged, i.e, when the optimal solution of cost function criterion is obtained. As an example of this closed form, the EMSE computation of the constant modulus algorithm is done and compared with the ones given in the literature.

Variation on variation on Euclid's algorithm

In a paper entitled Variation on Euclids Algorithm for Plynomials, Calvez et al. has shown that the extended Euclids algorithm can be partially obtained by the nonextended one; in fact, it can obtain only two of the three unknowns of the Bezouts theorem. This letter goes further and shows that all polynomials given by the extended Euclids algorithm and all the intermediate values can be obtained directly by the nonextended Euclids algorithm. Consequently, only remainder computations are used. Avoiding multiplications and divisions of polynomials decreases the computational complexity. This variation of Calvez et al. justifies the title of the present letter.

Markovian model of the error probability density and application to the error propagation probability computation of the weighted, decision feedback equalizer

A Markovian model of the error probability density for decision feedback equalizer is proposed and its application to the error propagation probability computation is derived. The model is a generalization of the Lutkemeyer and Noll (see ICC, Porto Carras, Greece, June 1998). It is obtained by the analysis of the Gaussian mixture distribution of the errors which follows a Markov process. The analysis of this process shows that the error propagation probability of the weighted DFE is less than the one of the classical DFE.