I. Introduction

A blind channel equalizer is a scheme inspired in an adaptive algorithm trying to detect the transmitted data sequence which has been distorted through its travel within the communications channel. The algorithms of the blind equalizer have been widely used in fields such as signal processing, image reconnaissance, design of control systems, and objects location in the natural space, among others.

The implementation of adaptive equalizers allowing the compensation of the channel interference caused by the Inter-Symbol Interference [ISI] and noise dates to the 1960s decade. The pioneer works by Widrow-Holf (1960), Lucky (1966), Saltzberg (1968), and Lugannani (1969) are a good example of researchers who contributed to the algorithms conception, being improved nowadays. This has improved the performance in the signal reception presenting distortions (Aquino, 2012).These algorithms are divided in two large groups: the ones using a training signal —so-called supervised—; and blind or self-taught algorithms called unsupervised. The first ones, mentioned in the works presented by Jojoa (2003), Aquino (2012), Erdogmus & Principe (2002), and Rocha (2005) use a training sequence available in the receiver. Consequently, they can calculate the error signal as the difference between the training signal and the signal outside the equalizer; allowing the periodic adjustment of the equalizer coefficients.

The main training adaptive algorithms are the Least Mean Square [LMS], Least Square [LS], and Recursive Least Square [RLS]. The LMS algorithm is important due to its low computational cost, although it offers a relatively slow convergence speed (Madeira, 2005). Regardless of the fact that this supervised equalization method presents good results relative to detection and convergence speed, it is not very employed because the use of training sequences reduces the channel capacity; hence, the idea of working with self-taught algorithms arose. These last ones have been proposed by Madeira (2013); Rolim (2005); Shalvi & Weinstein (1990); Neves, Attux, Suyama, Miranda, & Romano (2006).These algorithms do not use training sequences, only the statistical knowledge of the transmitted signal (Benveniste, Goursat, & Ruget,1980).

Some of the self-taught algorithms are the Constant-Modulus Algorithm [CMA] (Romano, Attux, Cavalcante, & Suyama, 2016), the Lucky direct choice algorithm (1966), Sato’s algorithm (1975), and the Multi-Mode Algorithm [MMA]. The most employed nowadays is the CMA; it is computationally more complex and it is considered as a self-taught version of the LMS algorithm due to their similarities. Furthermore, it handles an adaptation weight that defines the convergence speed and the robustness degree: the higher the adaptation weight, the higher the convergence speed and lower robustness (Madeira, 2013).

This research article implements a blind channel equalization scheme using the ARγ algorithm by employing higher order statistics from the Bussgang algorithm and the root mean error evaluation in the symbol detection. It is divided in three sections: the first one shows the blind channel equalization methods with the Bussgang algorithms, the following section shows the implementation of a blind equalizer by using the ARγ algorithm with the higher-level statistics of the Bussgang algorithms; and the last section presents the implementation results and the conclusions.

II. Materials and Method

Within the digital communication systems, it is necessary to recover the transmitted signal, which has been affected by the harmful effects by the communications channel. For this reason, it is necessary to include in the receiver an adaptive equalizer capable to mitigate these effects. The general model of a discrete time communication system is based on a sequence of symbols transmitted through a communications channel; this last introduces interferences, generating in the receiver a distorted vision of the transmitted data sequence. Figure 1 shows the general block diagram of a discrete time communication system using non-supervised equalizing.

Figure 1
Unsupervised equalization basic scheme

The communication system using an equalizer is formed by an input data sequence named c(n), which is not gaussian, independent, and identically distributed; the channel including the transmission/modulation and reception/demodulation systems is represented though H(z),v(n) represents the AWGN noise (additive, white, and gaussian), u(n) is a signal with noise at the exit of the channel, e(n) is the error signal, and y(n) is the signal at the exit of the equalizer. For this type of models, it is appropriate to represent the distortions caused by the channel using a linear model described by the response to the finite unitary impulse. The transfer function is presented in (1).

(1)

where:

N is the length of the unitary channel response; and

h(k,n) are the coefficients in the n instant.

For these types of models, it is important to consider the following aspects when implementing the channel equalizing. First, the Inter-Symbol Interference [ISI], which is a phenomenon occurring due to the channel dispersive effect; it is the result of the convolution between the transmitted signal and its response to the unitary impulse. In most of the wireless communications applications, this dispersive effect is observed in the signal as propagation delays caused by the reflection of the signal through several trajectories. This generates a detection of time desynchronized signals. The second aspect to consider is the intrinsic noise present in every communications channel, which affects the equalizer performance when the signal detection is performed. The last one is the channel model due to limitations detecting a signal transmitted via a non-linear channel (or no-minimal phase channel); these channels can be time-variant such as the fading channels.

When the implementation of the adaptive equalizer is performed, it is supposed that the supervised techniques are limited to the use of systemic resources, where working with training sequences to calculate the error signal and updating the equalizer coefficients is not optimal. This is mainly caused because of the bandwidth consumption and the reduction in the convergence speed. Consequently, we used the blind equalization method —or unsupervised— by using statistics of the transmitted signal to calculate the error signal and correct the channel dispersive effects. This equalization technique is characterized by using methods based on Second Order Statistics [SOS], Higher Order Statistics [HOS], and methods based in information theory.

A. Blind Channel Equalization Methods

Methods Based on Second Order Statistics

The methods based on second order statistics use the features of the so-called seasonality cycle in the u(n)signal to calculate the error signal in the communications channel. This kind of method does not present phase ambiguity, which allows to obtain better performance and detection of the transmitted symbols. Its main disadvantage relies on the exact estimation of the channel order and the data estimation without noise; therefore, it does not present an adequate performance in real situations. Some techniques using SOS include the linear prediction ones, based on subspaces, and least-squares (Romano et al., 2016).

Methods Based on Statistics with Order Higher than Two

The function of this method is to show that the random processes of the y(n) and c(n) signals (Figure 1) have the same features in their any-order moments, i.e., E{|y|k }=E{|c|k}for all k>0, where the equalizer can adjust its coefficients seeking that the moments of the y(n) signal are the same than the ones in the c(n) signal.

This kind of methods also uses the fourth order moments, known as curtoses. These ones allow a correct detection of the signal if E{|y|4 }=E{|c|4}. A method using these superior order statistics is the constant module criteria where, in certain circumstances, is very similar to the Shalvi-Weinstein criteria. This method employs the moments of order two and four of the transmitted signal to correctly detect the signal in the exit (Madeira, 2005).

Methods Based on the Information Theory

The methods based on the information theory use the Probability Density Function (PDF) seeking that the signal equalization is optimal, always that the PDF of the y(n)signal is the same than the PDF of c(n). This allows to minimize the error entropy between these two random sequences (Erdogmus & Principe, 2002; Lathi, 1998).

One kind of method using the information theory to estimate its parameters is the Maximum Likelihood [ML] one, it also uses the PDF of two signals, say y(n) and c(n).Nevertheless, a good estimation of the ML function demands higher computational resources, making more complex the simulation processes.

B. Bussgang Channel Blind Equalization Algorithms

The Bussgang algorithms use SOS over the transmitted signal and they also use an updated equation in the coefficients, given by (2).

(2)

where:

w(n) is the parameters vector in the instant n,

x(n) is the equalizer input vector,

μ is the step size, and

ψ[y(n) ]corresponds to the memory-less non-linear estimator of the transmitted signal c(n).

This estimator (see Table 1) uses superior order statistics of x(n)to obtain an approximation to the transmitted symbol. Within the Bussgang algorithms using SOS, we can mention the Direct Decision Algorithm [DDA], the Sato algorithm, and the constant module algorithm.

Table 1
Estimators and their corresponding cost functions

C. ARγ Algorithm Based on the Stochastic Gradient

The regressive Accelerator algorithm version γ was born from the continuous time accelerator algorithm proposed in 1998 by Pait (Jojoa, 2003). When this algorithm is discretized, it generates three new versions known as complete accelerator algorithm [AAC], progressive accelerator algorithm [APCM], and regressive accelerator algorithm [ARCM], where the efforts were pointed out to reduce the computational complexity of ARCM. From here, the ArY arose. It has features such as having three adjustment parameters (Alpha, Gamma, and M1) and it is able to achieve a good convergence speed, entailing to the reduction of the final measurement error (Solarte, 2012). Equations (3), (4), (5), (6), and (7) represent theARγ algorithm.

(3)

(4)

(5)

(6)

(7)

where:

x(n) represents the input signal vector;

w(n) is the coefficients vector of the adaptive filter;

d(n) the scalar, which is the desired signal in the instant n;

e(n) the scalar, which is the error signal in the instant n;

g(n) the auxiliary scalar in the instant n;

q(n) the auxiliary vector;

w0 the optimum coefficients vector; and

α, γ and m1 are fixed adjustment parameters.

III. Results

The implementation of the blind equalizer using the ARγ algorithm was developed using statistics with order higher than 2 to perform the estimation of the transmitted data. We considered the linear estimators of the Bussgang algorithms to adapt the ARγ to the blind equalization mode. This allowed us to execute the calculation of the Mean Squared Error [MSE]. Using this MSE value, we gathered the corresponding values to the variables g(n), q(n) and w(n) of the ARγ. The setup parameters of this algorithm are calculated by setting up a parameter and modifying the other two in a range between 0.1 and 6; by increasing the values in steps of 0.1, a minimal squared error is obtained. The transmitted symbols constellation is 4-QAM, the channel coefficients are h= [.4 1 -.7 .6 .3 -.4 .1], the length of the data is L=2000, and the signal to noise relation is 25 dB.

A. Estimator of the Direct Decision Algorithm Adopted to the ARγ One

Equations (8), (9), (10), (11) and (12) represent the ARγ algorithm with decision statistics.

(8)

(9)

(10)

(11)

(12)

The blind channel equalization using the ARγ in the direct decision configuration was implemented by setting up a value for Gamma between 0.01 and 1 and variating the two adjustment parameters —i.e., Alpha and M1— in an interval between 0.01 and 6 with steps of 0.1. With this, we gathered a minimal squared error of 0.022331, with an Alpha value of 0.41 and M1 in 5.8. The results entail that for those Alpha, Gamma, and M1 values, a better estimation of the transmitted data and a considerable reduction of the squared error are obtained (see Table 2, Figure 2 and Figure 3).

Table 2
Variation in Arγ parameters with decision statistics

Figure 2
Equalization with decision statistics

Figure 3
MSE with variations in gama and alfa

B. Estimator of the Sato Algorithm Adapted to the ARγ One

Equations (13), (14), (15), (16), (17) and (18) represent the ARγ algorithm with Sato statistics.

(13)

(14)

(15)

(16)

(17)

(18)

The second experiment was performed using the ARγ with statistics of the Sato algorithm by variating the Alpha, Gamma, and M1 setup parameters. We observed that for a Gamma value of 0.03, Alpha in 0.16, and M1 in 5.9, we obtained a minimal squared error of 0.0020. This result shows that with the use of the Sato statistics —which include the order 2 moments of the transmitted signal for the calculation of the error signal defining the new equalizer coefficients—, the data estimation is more optimum with a minimum squared error (see Table 3, Figure 4 and Figure 5).

Table 3
Variation in Arγ parameters with decision statistics

Figure 4
Figure 4. Equalization with Sato statistics

Figure 5
Figure 5. MSE with variations in alpha and M1

C. Estimator of the Constant Module Algorithm Adapted to the ARγ One

Equations (19), (20), (21), (22), (23) and (24) represent the ARγ algorithm with statistics of the Constant Module Algorithm [CMA] one.

(19)

(20)

(21)

(22)

(23)

(24)

The last experiment was performed by implementing a channel equalizer with the ARγ algorithm using the statistics of the CMA, which include the moments of order 4 and 2 of the transmitted signal. The results show that with the following values: Gamma = 0.03; Alpha = 5.71, and M1 = 5.8, a minimal squared error of 0.0014 is obtained. This last value represents a better performance relative to the Sato and decision algorithms. With these results, we obtained an equalization with values much closer to the transmitted ones with a minimum squared error (see Table 4, Figure 6 and Figure 7).

Table 4
Variation in Arγ parameters with CMA statistics

Figure 6
Equalization with CMA statistics

Figure 7
MSE with variations in alpha and M1

IV. Conclusions

The blind channel equalization has been employed as a channel estimation technique to solve problems related with detection of the transmitted signal in telephonic and wireless channels, where it is not practical to employ training sequences to perform a correct estimation of the data. Therefore, this method is known as an estimation method of the output signal working with superior order statistics of the input signal and in absence of training sequences, which optimizes the channel capacity.

The results of the blind equalizer implementation by using the ARγ algorithm and the superior order statistics of the Bussgang algorithms show the blind channel equalization through a minimal squared error and with larger estimation of the transmitted data. This, by considering that the variation of the algorithm setup parameters provides minimal values of the mean squared error. It is important to consider that for Gamma values larger than 1, the algorithm becomes unstable and the detection process presents ambiguities.

The analysis in the different proposed scenarios reflect that the simulated adaptive algorithm depends on the setup parameters and the use of the second and superior order statistics to perform a close estimation of the transmitted data.

Finally, from the obtained results in the simulations, we can conclude that the statistics of the constant module algorithm present better convergence speeds in the calculation of the mean squared error: we obtained an MSE of 0.0014 with Alpha = 5.71 and M1 = 5.8. Consequently, a better detection of the transmitted symbols is obtained compared to the Sato and decision algorithms, which allows a better reliability in the detection of the transmitted symbols.

References

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Notes

Author notes

Additional information

Cómo citar: Hurtado, J., & Jojoa, P. (2018). Effects of blind channel equalization using the regressive accelerator algorithm version γ, Sistemas & Telemática, 16(46), 9-20. doi:10.18046/syt.v16i46.3009