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Recently, closed-form approximated expressions were obtained for the residual Inter Symbol Interference (ISI) obtained by blind adaptive equalizers for the biased as well as for the non-biased input case in a noisy environment. But, up to now it is unclear under what condition improved equalization performance is obtained in the residual ISI point of view with the non-biased case compared with the biased version. In this paper, we present for the real and two independent quadrature carrier case a closed-form approximated expression for the difference in the residual ISI obtained by blind adaptive equalizers with biased input signals compared with the non-biased case. Based on this expression, we show under what condition improved equalization performance is obtained from the residual ISI point of view for the non-biased case compared with the biased version.

We consider a blind deconvolution problem in which we observe the output of an unknown, possibly nonmini- mum phase, linear system from which we want to recover its input using an adjustable linear filter (equalizer) [

In this paper, we derive for the real and two independent quadrature carrier case a closed-form approximated expression for the difference in the residual ISI obtained by blind adaptive equalizers with biased input signals compared with the non-biased case. This expression depends on the step-size parameter, equalizer’s tap length, input signal statistics, channel power and signal to noise ratio (SNR). In addition, this expression is valid for blind adaptive equalizers where the error 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 and where the gain between the input and equalized output signal is equal to one as is in the case of Godard’s algorithm [

The paper is organized as follows. After having described the system under consideration in Section II, the condition for which improved equalization performance is obtained from the residual ISI point of view for the non-biased case compared with the biased version is introduced in Section III. In Section IV simulation results are presented and the conclusion is given in Section V.

The system under consideration is the same system as shown in [

1) The input sequence

2) The mean of the input sequence

3)

4) The unknown channel

5)

6) The noise

7) The variance of

The transmitted sequence

where “

where p[n] is the convolutional noise that arises due to the use of non-ideal equalizer’s coefficients (blind equalization) instead of the ideal set and

where

where

According to [

where

and

where

where

Please note that (5) can be also applied for the non-biased case by substituting

according to (5) that improved equalization performance may be obtained from the residual ISI point of view for biased input signals compared to the non-biased version. But, on the other hand, the expression for

In this section, we first derive a closed-form approximated expression for the difference in the residual ISI obtained by blind adaptive equalizers with biased input signals compared to the non-biased case. Then, based on this new derived expression, we derive the condition for which improved equalization performance is obtained from the residual ISI point of view for the non-biased input case compared to the biased version. In the following we denote

Theorem:

where

Proof:

Based on (14) we have:

With the help of (11) and (18) we may write:

From (19) we may conclude that

From (20) we have:

Next by substituting

where

and b is given in (16). Based on (22) we may write:

Next we turn to find the relationship between

where

From (25) and (24) we may have:

which can be also written as:

where

The solution for

According to [

Simulation results carried out in [

where

Next, by using (5) we may have:

Now we substitute (32) into (34) and obtain:

where

This completes our proof.

then improved equalization performance is obtained from the residual ISI point of view for the non-biased input

case compared to the biased version. In the following we will show the relationship between

Thus, according to (38), we may conclude that if

tained from the residual ISI point of view for the biased input case compared to the non-biased version. Please note that (37) depends on the step-size parameter, equalizer’s tap length, input signal statistics, channel power, signal to noise ratio and on the properties of the chosen blind equalizer via

In this section,

where,

A biased 16QAM, a modulation using ±{1, 3} levels for in-phase and quadrature components in addition to a given bias was considered. The bias for the real and imaginary axes were the same. In our simulation we used

the channel given in [

Figures 2-5 are the simulated performance of (39) for the biased 16QAM input case, namely the ISI as a function of iteration number for various SNR values and two different biases, compared with the non-biased case. Figures 6-9 are the zoomed versions of Figures 2-5 respectively. According to

difference in the residual ISI between the biased and non-biased case is

According to

According to

According to

Based on Figures 6-9, the simulated results for

In this paper, we derived for the real and two independent quadrature carrier case a closed-form approximated expression for the difference in the residual ISI obtained by blind adaptive equalizers with biased input signals compared to the non-biased case. This expression depends on the step-size parameter, equalizer’s tap length, input signal statistics, channel power, SNR and chosen equalizer via

We thank the editor and the referee for their comments.