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Design and Implementation of a Blind Adaptive Equalizer Using Frequency Domain Improved Square Contour Algorithm

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Abstract

Improved Square Contour Algorithm (ISCA) is a blind adaptive equalization technique which employs estimated outputs from the decision device along with the dispersion constant in its cost function, thereby, forcing the equalizer outputs to lie on multiple zero error square moduli instead of a single contour in Square Contour Algorithm (SCA). This leads to greater ISI suppression and an improved convergence rate. In this paper, frequency domain implementation of block based ISCA is developed. Simulation results show that the FDISCA exhibits similar performance as time domain ISCA but with a significantly lower computational complexity. However, by employing normalization in the frequency bins, a noteworthy improvement in the convergence rate for FD-ISCA is observed. Complete hardware architecture of FD-ISCA along with its simulation results is also discussed.

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Authors and Affiliations

  1. National University of Science & Technology, Islamabad, Pakistan

    Sana Abid Khan & Shahzad Amin Sheikh

Authors
  1. Sana Abid Khan
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  2. Shahzad Amin Sheikh
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Correspondence to Sana Abid Khan.

Appendix A

Appendix A

As mentioned before, for blind equalizer adjustment, a scheme to minimize a special type of non-MSE cost function involving the statistical characteristics of the transmitted data and the output of the equalizer needs to be employed. The generalized expression for the cost function for ISCA is given by [5]:

$$ {J}_{ISCA,n} = \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2p$}\right.\kern0.5em E\left\{{\left[{\left(\left|{z}_{R,n}+{z}_{I,n}\right|+\left|{z}_{R,n} - {z}_{I,n}\right|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right]}^p\right\} $$
(1)

where p is an integer and its value depending on performance and complexity of implementation is usually set as p = 2 [11]. The error signal is found using the Stochastic Gradient Decent (SGD) Algorithm as:

$$ {e}_n=\raisebox{1ex}{$\partial {J}_n$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right. $$
(2)

Substituting p = 2 in (1) and computing its gradient:

$$ \begin{array}{l}\raisebox{1ex}{$\partial {J}_n$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.=\kern0.5em \raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.\kern0.5em \left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\kern0.5em E\left\{{\left[{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right]}^2\right\}\right]\\ {} = \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\ *\ 2*\left\{{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right\}\ \raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.E\left\{{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right\}\\ {} = \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\ *\ 2*\left\{{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right\}\ *\\ {}\kern1em \left\{\raisebox{1ex}{$\partial E$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.\left[{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2\right] - \raisebox{1ex}{$\partial E$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.\left[{\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right]\right\}\end{array} $$

Since the value of Λ n (2R ISCA )2 does not depend on the tap weights wn,\( \raisebox{1ex}{$\partial $}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.\left[{\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right]=0 \). So,

$$ \begin{array}{l}\raisebox{1ex}{$\partial {J}_n$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.=\kern0.5em \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\ *\ 4*\left\{{\left(|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right)}^2 - {\varLambda}_n{\left(2{R}_{ISCA}\right)}^2\right\}\ *\\ {}\kern4em \left(\left|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}\right|\right)\ *\left\{\raisebox{1ex}{$\partial E$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.\left[|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n} - {z}_{I,n}|\ \right]\right\}\end{array} $$

We know from the properties of signum function that \( \frac{d}{dx}\left|{x}_r+{x}_i\right|=c\operatorname{sgn}\left({x}_r+{x}_i\right)=\operatorname{sgn}\left({x}_r\right)+j*\operatorname{sgn}\left({x}_i\right) \). Thus,

$$ \begin{array}{l}e(n)\kern0.5em =\kern0.5em \raisebox{1ex}{$\partial {J}_n$}\!\left/ \!\raisebox{-1ex}{$\partial {w}_n$}\right.=\kern0.5em \left\{{\left(\left|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n}-{z}_{I,n}\right|\right)}^2-{\varLambda}_n*{R}_{ISCA}^2\right\}*\left(\left|{z}_{R,n}+{z}_{I,n}|+|{z}_{R,n}-{z}_{I,n}\right|\right)*\\ {}\kern7.25em \left\{\left(\operatorname{sgn}\left({z}_{R,n}+{z}_{I,n}\right)+\operatorname{sgn}\left({z}_{R,n}-{z}_{I,n}\right)+j\left(\operatorname{sgn}\left({z}_{R,n}+{z}_{I,n}\right)+\operatorname{sgn}\right({z}_{R,n}-{z}_{I,n}\right)\right\}\end{array} $$
(3)

is the error function used to compute the tap weight vectors.

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Khan, S.A., Sheikh, S.A. Design and Implementation of a Blind Adaptive Equalizer Using Frequency Domain Improved Square Contour Algorithm. J Sign Process Syst 85, 235–247 (2016). https://doi.org/10.1007/s11265-015-1072-7

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