A closed-form approximated expression for the residual ISI obtained by blind adaptive equalizers applicable for the non-square QAM constellation input and noisy case

Recently, closed-form approximated expressions were obtained for the residual Inter-Symbol Interference (ISI) obtained by blind adaptive equalizers valid for the real or two independent quadrature carrier case such as the 16 Quadrature Amplitude Modulation (QAM) input. In this paper we propose for the complex and dependent quadrature carrier case (such as the 32QAM source), a closed-form approximated expression for the achievable residual ISI which depends on the step-size parameter, equalizer´s tap length, input signal statistics, channel power and SNR. This approximated expression is applicable for blind adaptive equalizers where the error is fed into the adaptive mechanism, which updates the equalizer´s taps and can be expressed as a polynomial function up to order five of the equalized output. Godard´s algorithm for example, applies a third order polynomial function to the adaptation mechanism of the equalizer thus belongs to the above-mentioned type of equalizers. Since the channel power is measurable, or can be calculated if the channel coefficients are given, there is no need to perform any simulation with various step-size parameters and different values of SNR to reach the required residual ISI for the dependent quadrature carrier input case.