Channel 2 according to [22] :

$\begin{array}{c}h\left[n\right]=(-0.0144,0.0006,0.03427,-0.3090,-0.03842,\\ \text{}0.8376,0.04163,0.4247,0.02976,0.08122,0.04764,\\ \text{}0.0411,0.0162,0.0063).\end{array}$

Channel 3 according to [32] :

$h\left[n\right]=(\begin{array}{ll}\mathrm{0,}\hfill & \text{for}n<0\hfill \\ -\mathrm{0.4,}\hfill & \text{for}n=0\hfill \\ 0.84\stackrel{\dot{}}{0}{.4}^{n-1}\mathrm{,}\hfill & \text{for}n>0\hfill \end{array}$

Figures 7-10 show the simulation results for the ISI, MSE, MConE and Accumulated Error (8) respectively for various tap length values ( $N=11,13,15$ ), SNR = 20[dB] and ${\mu}_{G}=3\times {10}^{-5}$ . According to Figure 8 it is very difficult to see for which equalizer’s tap length ( $N$ ), better equalization performance is obtained from the MSE point of view. From Figure 7 the equalization performance from the ISI point of view is very close for the case of $N=13$ and $N=15$ . However, according to Figure 9, the difference in the equalization performance between the various equalizer’s tap length is seen very clearly. The difference in the equalization performance for the various equalizer’s tap length is also seen in Figure 10 in the short range while in the long term the difference in the equalization performance resembles the difference in equalization performance as is seen in Figure 7.

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 ( ${\mu}_{G}=4\times {10}^{-5}$ for channel

Figure 7. ISI as a function of iteration number for various equalizer’s tap length. The averaged results were obtained from 50 Monte Carlo trials.

Figure 8. MSE as a function of iteration number for various equalizer’s tap length. The averaged results were obtained from 50 Monte Carlo trials.

Figure 9. MConE as a function of the window length for various equalizer’s tap length. The averaged results were obtained from 50 Monte Carlo trials.

Figure 10. Error Accumulation as a function of iteration number for various equalizer’s tap length. The averaged results were obtained from 50 Monte Carlo trials.

Figure 11. ISI as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 12. MSE as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 13. MConE as a function of the window length for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 14. Error Accumulation as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

2 and ${\mu}_{G}=5\times {10}^{-5}$ for channel 3) where the equalizer’s tap length was set to 13 ( $N=13$ ) and SNR = 20 [dB]. According to Figure 12 it is very difficult to see for which channel better equalization performance is obtained from the MSE point of view. From Figure 11 the equalization performance from the ISI point of view is very close for the two channel cases (channel 2 and channel 3). However, according to Figure 13 the difference between the equalization performance for the two channels is seen very clearly. In addition the difference in the equalization performance for the two channels is also seen in Figure 14 in the short range while in the long term the difference in the equalization performance resembles the difference in the equalization performance as is seen in Figure 11.

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 ( ${\mu}_{G}=4.4\times {10}^{-5}$ , $N=15$ for channel 2 and ${\mu}_{G}=7\times {10}^{-5}$ , $N=11$ for channel 3) where SNR = 15 [dB]. According to Figure 16 it is very difficult to see for which channel better equalization performance is obtained from the MSE point of view. From Figure 15 the equalization performance from the ISI point of view is very close for the two channel cases (channel 2 and channel 3). However, according to Figure 17 the difference between the equalization performance for the two channels is seen very clearly. In addition the difference in the equalization performance for the two channels is also seen in Figure 18 in the short range while in the long term the difference in the equalization performance resembles the difference in the equalization performance as is seen in Figure 15.

Figure 15. ISI as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 16. MSE as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 17. MConE as a function of the window length for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 18. Error Accumulation as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 19. ISI as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

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 ( ${\mu}_{G}=4.5\times {10}^{-5}$ , $N=9$ for channel 1 and ${\mu}_{G}=4\times {10}^{-5}$ , $N=19$ for channel 2) where SNR = 20 [dB]. According to Figure 20 it is very difficult to see for which channel better equalization performance is obtained from the MSE point of view. From Figure 19 the equalization performance from the ISI point of view is very close for the two channel cases (channel 1 and channel 2). However, according to Figure 21 the difference between the equalization performance for the two channels is seen very clearly. In addition the difference in the

Figure 20. MSE as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 21. MConE as a function of the window length for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 22. Error Accumulation as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 23. ISI as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

equalization performance for the two channels is also seen in Figure 22 in the short range while in the long term the difference in the equalization performance resembles the difference in the equalization performance as is seen in Figure 19.

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 ( ${\mu}_{G}=1\times {10}^{-5}$ , $N=13$ for channel 1 and ${\mu}_{G}=3\times {10}^{-5}$ , $N=7$ for channel 2) where SNR = 20 [dB]. According to Figure 24 it is very difficult to see for which channel better equalization performance is obtained from the MSE point of view. From Figure 23 the equalization performance from the ISI point of view is very

Figure 24. MSE as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 25. MConE as a function of the window length for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 26. Error Accumulation as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 27. ISI as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

close for the two channel cases (channel 1 and channel 3). However, according to Figure 25 and Figure 26 the difference between the equalization performance for the two channels is seen very clearly.

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 ( ${\mu}_{G}=1\times {10}^{-5}$ , $N=13$ for channel 2 and ${\mu}_{G}=3\times {10}^{-5}$ , $N=7$ for channel 3) where SNR = 20 [dB]. According to Figure 28 it is very difficult to see for which channel better equalization performance is obtained from the MSE point of view. From Figure 27 the equalization performance from the ISI point of view is very close for the two channel cases (channel 2 and channel 3). However, according

Figure 28. MSE as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 29. MConE as a function of the window length for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

Figure 30. Error Accumulation as a function of iteration number for two channel cases. The averaged results were obtained from 100 Monte Carlo trials.

to Figure 29 and Figure 30 the difference between the equalization performance for the two channels is seen very clearly.

5. Conclusion

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.

Acknowledgments

We thank the Editor and the referee for their comments.

Abbreviations

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

Conflicts of Interest

The authors declare no conflicts of interest.

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