^{1}

^{*}

^{1}

^{*}

Recently, two expressions (for the noiseless and noisy case) were proposed for the residual inter-symbol interference (ISI) obtained by blind adaptive equalizers, where the error of the equalized output signal may be expressed as a polynomial function of order 3. However, those expressions are not applicable for biased input signals. In this paper, a closed-form approximated expression is proposed for the residual ISI applicable for the noisy and biased input case. This new proposed expression is valid for blind adaptive equalizers, where the error of the equalized output signal may be expressed as a polynomial function of order 3. The new proposed expression depends on the equalizer’s tap length, input signal statistics, channel power, SNR, step-size parameter and on the input signal’s bias. Simulation results indicate a high correlation between the simulated results and those obtained from our new proposed expression.

Blind equalization is used in various applications such as: signal processing, digital communication, speech and image processing. Generally, a communication system may be presented by a signal transmitted via a com- munication channel added with white noise as illustrated in

degradation in performance caused by the ISI, a blind adaptive equalizer, may be implemented in those systems [

Up to now, the performance of a chosen equalizer (the achievable residual ISI) for biased input signals could be obtained only via simulation. According to [

In this paper, we propose for the noisy and biased input signal case a closed-form approximated expression for the residual ISI that depends on the equalizer’s tap length, input signal statistics, channel power, SNR, step- size parameter and on the input signal’s bias. Since the channel power is measurable, there is no need anymore to carry out any simulation with various step-size parameters in order to reach the required residual ISI.

The paper is organized as follows: after having described the system under consideration in Section 2, the closed-form approximated expression for the achievable residual ISI is introduced in Section 3. In Section 4, simulation results are presented and the conclusion is given in Section 5.

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

1. The input sequence

2. The biased input sequence mean is

3.

4. The unknown channel

5.

6. The noise

7. The variance of

The transmitted sequence

where “

where D is a constant delay and

where

where

where

Substituting (4) into (5) yields:

where

where

Next, let us define:

where

since

according to assumption 6 from this section,

Therefore, we may have with the help of (11):

where

where

equalizer’s tap length. The operator

In this section, a closed-form approximated expression is derived for the residual ISI valid for biased input signals.

Theorem: Consider the following assumptions:

1. The source signal

2. The convolutional noise

3. The convolutional noise

4. The gain between the source and equalized output signal is equal to one.

5. The convolutional noise

6.

7. The signal

8. The added noise

9. The channel

10.

11. The signal

12. The equalizer’s output noise

13.

The residual ISI expressed in dB units may be defined as:

where

or

and

where

where

where

variance of

Comments:

1. It should be noted that assumptions 2 - 5, are precisely similar to those made by [

2. It should be mentioned out that our expression for the residual ISI (14) looks quite similar to the residual ISI expression given in [

Proof:

By using (5), (8) and (12),

where

Substituting (19) into (18) and by using (4) we obtain:

By substituting (8) into (20) and using the relation of

Please note that

where

From (23) we obtain:

which may be written as:

By using assumptions 6, 7, 8 and 11 (from this section) we obtain:

Let us define:

Our next step is developing the following expression:

which is a part of the expression for B (27). Since the channel’s impulse response decays in time, (28) may be written as:

By substituting (29) into (27) we obtain:

Next, we turn to calculate

We recall the expression for

Therefore, for the latter stages of the de-convolution process

From (31), we obtain:

Next, the expectation operator is applied on (32):

By substituting

From assumption 12 (in this section) we obtain:

It should be pointed out that (35) looks similar to the equalizer's output noise variance at [

here (35),

Next, we turn to calculate the expression for the residual ISI applicable for the biased case. For that purpose,

we calculate first the expression for

Thus, by using (8), we obtain:

which may be written as:

From (38) we obtain:

By applying the expectation operator on both sides of (39) and using assumption 11 (from this section), we obain:

Thus, the expression for

By using assumption 3 (from this section), (41) may be written as:

Next, we turn to calculate

Substituting (43) into (42) we obtain:

Substituting (44) into (40) leads to:

For the ideal case,

multiplications of different index elements. Then (45) may be written as:

By using (43), we may write (46) as:

From (7), (47) may be written for

From assumptions 2 and 13 (from this section), we may use the relation:

This completes our proof.

In this section, our new proposed expression for the residual ISI (14) was tested via simulation, where we used Godard’s algorithm [

where,

Two different input sources were considered: 1) A biased 16QAM, a modulation using ± {1, 3} levels for in- phase and quadrature components in addition to a given bias. The bias is the same for the real and imaginary axes. 2) A uniformly distributed input signal within [−0.5, 1.5] for the x and y axis where the two axis are independent. The following five different channels were used:

Channel 1: The channel parameters were taken according to [

Channel 2: The channel parameters were determined according to [

Channel 3: The channel parameters were determined according to [

Channel 4: The channel parameters are:

Channel 5: The channel parameters were determined according to [

Figures 2-9 are the simulated performance of (50) for the biased 16QAM input case, namely the ISI as a function of iteration number for various step-size parameters, channel characteristics, various SNR values and for three different biases, compared with the calculated residual ISI expression (14) proposed in this paper. According to Figures 2-9, the residual ISI obtained by (14) is very close to the simulated results.

Figures 10-18 are the simulated performance of (50) for the biased 16QAM input case, namely the ISI as a function of iteration number for various equalizer's tap length, channel characteristics, various SNR values and for two different biases, compared with the calculated residual ISI expression (14). Figures 10-18 show a high correlation between the simulated results and those calculated with (14).

Figures 21-23 illustrate the simulated performance of (50) for the biased 16QAM input case, namely the ISI as a function of iteration number for various biases, two different SNR values and three channel cases, compared with the calculated residual ISI expression (14). Figures 21-23 show a high correlation between the simulated results and those calculated with (14).

Figures 24-26 illustrate the simulated performance of (50) for the uniformly source input, namely the ISI as a function of iteration number for three different values for the SNR, compared with the calculated residual ISI expression (14). Figures 24-26 show a high correlation between the simulated results and those calculated with (14).

In this paper, we proposed an approximated closed-form expression for the residual ISI obtained by blind adaptive equalizer, where the error of the equalized output signal may be expressed as a polynomial function of order 3. This new expression is valid for the noisy and biased input case and depends on the step-size parameter, equalizer’s tap length, SNR, channel power and input signal statistics. This new proposed expression may be considered as a general closed-form expression for the residual ISI, where the previous proposed expressions from the literature are only special cases of it. Simulation results have shown a high correlation between the simulated results for the residual ISI and those that were calculated from our new proposed expression.