A new closed approximated formed expression for the achievable residual ISI obtained by adaptive blind equalizers for the noisy case

Recently, a closed approximated formed expression was derived by the same author for the achievable residual intersymbol interference (ISI) for the noiseless case that depends on the step-size parameter, equalizer's tap length, input signal statistics and channel power. This approximated expression 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 the equalized output. It should be pointed out that Godard's algorithm for example, belongs to the mentioned type of blind equalizers. As mentioned before, this expression was derived for the noiseless case, thus can not be helpful for the wireless case where the expected signal to noise ratio (SNR) may have values down to 10 dB. In this paper, a closed approximated formed expression is derived for the achievable residual ISI that depends on the step-size parameter, equalizer's tap length, input signal statistics, channel power and on the SNR. Since the channel power is measurable or can be calculated if the channel coefficients are given, there is no need anymore to carry out any simulation with various step-size parameters and different values for the SNR in order to reach the required residual ISI.