The Whole Story Behind Blind Adaptive Equalizers/ Blind Deconvolution

In this chapter the blind deconvolution problem is defined, the dierent applications are given in which the blind deconvolution arises and the dierence between the Polyspectral , Bussgang-type and Probabilistic algorithms is explained.

In this chapter, the Single Input Single Output (SISO) and Single Input Multiple Output (SIMO) system are described. For each system the condition for perfect equalization is given. In addition, the meaning of convolutional noise and Inter- symbol Interference (ISI) often used as a measure of performance in equalizers' applications are explained.

In this chapter the cost function approach for blind deconvolution is explained and the Bayesian estimation techniques are covered. We also show the relationship between the cost function and Bayesian approach and show a derivation from the cost function approach that was recently used successfully in literature.

In this chapter, we prove that for the real valued and two independent quadra- ture carrier cases, no blind adaptive equalizer can be found with perfect equalization performance from the residual Intersymbol Interference (ISI) point of view inde- pendent of the source signal statistics, where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of the equalized output of order three and where the gain between the equalized output and source signal is equal to one. In addition, we give in this chapter the entire mathematical basis that was missing in a recently presented pa- per where a new blind adaptive equalizer was introduced and shown to have an improved equalization performance compared with Godard's algorithm. We also show that the above mentioned blind adaptive equalizer was obtained by using a derivation from the cost function approach.

In this chapter the Bayesian approach for blind deconvolution is explained. We cover the dierent proposed expressions for the conditional expectation known in the literature and explain in detail the advantages and disadvantages of each ob- tained expression.

In the literature, the convolutional noise probability density function (pdf) is modeled as a Gaussian pdf. In this chapter, a new closed-form approximated ex- pression is derived for the conditional expectation based on a new model for the convolutional noise pdf. The new derived expression for the conditional expecta- tion, is based on the Maximum Entropy approach, Edgeworth expansion, Laplace integral method and is valid for the noiseless, real valued and two independent quadrature carrier case. According to simulation results carried out for signal to noise ratio (SNR) SNR = 30 [dB], the new derived expression leads to improved equalization performance for the 16QAM input constellation case.

In this chapter we explain the advantages and disadvantages of the Bayesian and Cost function approach. In doing so we also explain what is the meaning of higher order statistics (HOS) which is widely used in blind equalization, the reason why using HOS (order ≥ 3) in blind equalization and why HOS can not be applied for Gaussian input signals.

Recently, a closed-form approximated expression was proposed for the achievable residual intersymbol interference (ISI) valid for the real valued and two independent quadrature carrier case and for type of blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of order three of the equalized output. Thus the recently proposed expression for the achievable residual ISI can not be applied for input constellations such as the 32QAM or V29 case. In this chapter we propose for the noiseless case, a new closed-form approximated expression for the residual ISI that depends on the step-size parameter, equalizer's tap length, input signal statistics, channel power and that is valid for the general case of input constellations. The new closed-form approximated expression for the residual ISI is applicable for type of blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of order three of the equalized output like in Godard's algorithm. Since the channel power is measurable or can be calculated if the channel coecients are given, there is no need anymore to carry out any simulation with various step-size parameters in order to reach the required residual ISI.

It is very important for a system designer to understand the connection between the equalizer's performance (achievable residual intersymbol interference (ISI) and con- vergence speed) and the various parameters involved in the equalizer's design such as the equalizer's tap length, step-size parameter, channel power and input constel- lation in order to achieve optimal and expected equalization performance. In this chapter, we show the connection between the equalization performance (achievable residual ISI and convergence speed) and the step-size parameter, equalizer's tap length, channel power and input constellation statistics for type of equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of order three of the equalized output.

By choosing a particular equalizer it is useful to know in advance if the chosen equalizer leads to perfect equalization performance. In this chapter, we explain how we can know without carrying out any simulation, if the chosen equalizer leads to perfect equalization performance for the real valued and two independent quadrature carrier case. We derive in this chapter for the real valued and two independent quadrature carrier case, some conditions on the input constellation statistics for which perfect equalization performance is obtained for type of blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps is expressed as a polynomial function of order three. We show also that perfect equalization performance can not be obtained for type of blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps is expressed as a polynomial function of order three, when dealing with the noiseless and 16QAM constellation input case.

Closed-form approximated expressions were recently proposed for the convergence time (or number of iterations required for convergence) and for the Intersymbol Interference (ISI) as a function of time valid during the stages of the iterative deconvolution process. The new derivations are valid for the noiseless, real valued and two independent quadrature carrier case and for type of blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of order three of the equalized output like in Godard's algorithm. The derivations are based on the knowledge of the initial ISI and channel power (which is measurable) and do not need the knowledge of the channel coecients. However, they are not based on strong mathematical foundations and were tested via simulation with Godard's algorithm for two dierent channels only. Thus, it could be argued that they hold only for a particular equalization method and for special cases. In this chapter, we present by simulation the usefulness of the recently derived expressions for more channel types and step-size parameters as well as for another type of equalizer. Thus, the question if the obtained expressions hold also for other types of equalization methods and other channels is answered.

Based on the recently proposed expression for the ISI as a function of time, we derive in this chapter a closed-form approximated expression for the mean square error (MSE) as a function of time (or as a function of iteration number).

In this chapter we present the advantages and disadvantages of the blind serially adaptive equalizer compared with the non-blind adaptive version as well as with the blind but non adaptive approach.

In this section, we bring a simple Matlab program for a serially blind adaptive equalizer.