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Up to now, the Mean Square Error (MSE) criteria, the residual Inter-Symbol Interference (ISI) and the Bit-Error-Rate (BER) were used to analyze the equalization performance of a blind adaptive equalizer in its convergence state. In this paper, we propose an additional tool (additional to the ISI, MSE and BER) for analyzing the equalization performance in the convergence region based on the Maximum Time Interval Error (MTIE) criterion that is used for the specification of clock stability requirements in telecommunications standards. This new tool preserves the short term statistical information unlike the already known tools (BER, ISI, MSE) that lack this information. Simulation results will show that the equalization performance of a blind adaptive equalizer obtained in the convergence region for two different channels is seen to be approximately the same from the residual ISI and MSE point of view while this is not the case with our new proposed tool. Thus, our new proposed tool might be considered as a more sensitive tool compared to the ISI and MSE method.

In data communication, signals transmitted between remote locations often encounter a signal-altering physical channel (in wired communications or in wireless communications). These physical channels may cause signal distortion, including echoes and frequency-selective filtering of the transmitted signal [

An effective way to overcome the ISI is using adaptive equalization technology. An adaptive equalizer is an inverse filter which reduces the effects of ISI by deconvolving the transmitted data sequence from the time varying channel response [

When a training session is impossible or very costly, blind equalizers are a convenient solution. Blind equalization algorithms are essentially adaptive filtering algorithms designed such that they do not require the external supply of a desired response to generate the error signal in the output of the adaptive equalization filter [

Today, in order to analyze the equalization performance, namely, to see how much the equalizer overcomes the ISI, the ISI, the MSE or the BER are simulated. The ISI, BER and MSE provide long term statistical information in the steady state region (convergence state). Thus, for instance, there may be cases where two different simulation results obtained in the convergence region with different channels (but using the same algorithm for reducing the ISI), may lead approximately to the same residual ISI but may have different short term statistical information. Namely, in the short term, there may be seen different amounts of errors for the two different channels. Thus, one channel is preferable over the other. Therefore, the following question may arise: is it possible to get also short term statistical information of the blind adaptive equalization performance in the convergence region?

A major topic of discussion in standard bodies dealing with network synchronization [

The purpose of this work is to provide an additional tool (additional to the ISI, MSE and BER) for diagnosing equalization performance in the steady state region based on the MTIE method used in the telecommunication area. Simulation results will show that our new proposed tool provides us short term as well as long term statistical information and is able to show differences in the equalization performance comparison obtained in the convergence state even when it is quite difficult to see it with the MSE and ISI method.

The paper is organized as follows: after having described the system under consideration in Section II, Section III describes our new proposed tool for analyzing the equalization performance in the convergence region based on the MTIE. In Section IV simulation results are given using our new proposed tool compared with the existing methods (MSE, ISI) and Section V is our conclusion.

In this section we consider the system described in

1) The input sequence

2) The unknown channel

3) The equalizer

4) The noise

For simplicity, we use in this paper only the 16QAM constellation input (

where

where

of the equalizer (

where

In this section we introduce our new proposed tool for the blind equalization performance analysis based on a network clock synchronization measurement method, namely, the MTIE measurement method.

For a given clock, the time error function

Thus, the

An example of a MTIE measurement of a clock is shown in

Next, we adopt the concept of the MTIE measurement method from the telecommunication area to the world of equalization performance. Based on (2), we denote

Since

where

An example for a MConE measurement belonging to an equalization process in the convergence state is shown in

The resemblance between the example in

In this section we present several simulation results using the MConE tool for obtaining the blind adaptive equalization performance using Godard’s algorithm [

As already mentioned earlier in this paper, our new proposed tool for diagnosing equalization performance in the steady state region might be considered as a more sensitive tool compared to the ISI and MSE method. But, it should be kept in mind that not every difference seen in the equalization performance comparison with our new proposed tool automatically leads to errors in the recovered symbols. Thus, to see this, we denote in the following Error Accumulation as

where

Channel 1 according to [

Channel 2 according to [

Channel 3 according to [

Figures 7-10 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for various tap length values (

Figures 11-14 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for two different channels (channel 2 (CH2) and channel 3 (CH3)) and for two different step sizes (

2 and

Figures 15-18 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for two different channels (channel 2 (CH2) and channel 3 (CH3)) and for two different step sizes and equalizer’s tap length (

Figures 19-22 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for two different channels (channel 1 (CH1) and channel 2 (CH2)) and for two different step sizes and equalizer’s tap length (

equalization performance for the two channels is also seen in

Figures 23-26 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for two different channels (channel 1 (CH1) and channel 3 (CH3)) and for two different step sizes and equalizer’s tap length (

close for the two channel cases (channel 1 and channel 3). However, according to

Figures 27-30 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for two different channels (channel 2 (CH2) and channel 3 (CH3)) and for two different step sizes and equalizer’s tap length (

to

In this paper, we proposed a new tool for analyzing the equalization performance in the convergence state which can be considered as an additional tool to the literature known methods (ISI, MSE, BER). The new proposed tool is based on the MTIE criterion that is used for the specification of clock stability requirements in telecommunications standards. This new tool preserves the short term statistical information unlike the BER, ISI and MSE method. Thus, our new proposed tool can supply us short term as well as long term statistical information. Simulation results have shown that with our new proposed tool, difference in the equalization performance comparison was clearly seen in the convergence state while this was not the case with the MSE and ISI method. Thus, our new proposed tool for analyzing the equalization performance in the convergence state might be considered as a more sensitive tool compared to the ISI and MSE method.

We thank the Editor and the referee for their comments.

Suissa, G. and Pinchas, M. (2017) A New Equalization Performance Analyzing Method for Blind Adaptive Equalizers Inspired by Maximum Time Interval Error. Journal of Signal and Information Processing, 8, 42-64. https://doi.org/10.4236/jsip.2017.82004

MSE- Mean Square Error

ISI- Intersymbol Interference

BER- Bit Error Rate

MTIE- Maximum Time Interval Error

FIR- Finite Impulse Response

QAM- Quadrature Amplitude Modulation

TE- Time Error

TIE- Time Interval Error

SNR- Signal to Noise Ratio

ConE- Convolution Error

MConE- Maximum Convolution Error

E A- Error Accumulation

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