4.4.1 Conguration
The simulations were performed using Matlab2019a. The observed lter was a high-pass FIR lter with4taps:[0.2, −0.5, 0.7, 0.1]. The applied sampling frequency was10kHz.
To investigate the complex-valued RSC4BI-algorithm that was presented in Sec. 4.2, the following probe signals were generated with0.1s record length:
• complex-valued Gaussian white noise withN2(0.1,1) andN2(0.01,1),
• IQ signal modulated by 16-QAM with and without AWGN, and
• IQ signal modulated by 256-QAM with and without AWGN.
In the case of the noisy signals, the previously described complex-valued Gaussian white noise was added to it N2(0.1,1)
. The output signal of the observed lter was generated through convolution (31) to obtain a reference signal for the investigations. The remaining part of the conguration and the assumptions during the simulations were the same as in Sec. 3.5.1.
4.4.2 Runtime of the procedures
The simulations were performed on a desktop computer containing an AMD Ryzen 7 2700X processor with 32 GB RAM. Each method was evaluated10,000times, and the runtime of every calculation was separately measured. The observation window included N = 20 samples while the lter length was changed between the values K = (10,20,30) because the calculation complexity of R[n] is dependent only onK. The averages of the measured runtimes were presented in Fig. 48. The bar chart is completed by additional crosses to illustrate the medians and the5−95% percentiles as well.
As Section 4.3.1 stated, the complex-valued RSC4BI-algorithm has lower computational requirements than the other methods. If the number of the lter taps increases, then the runtime increases slower than in the case of classic matrix inversion or solving the system of linear equations. However, it should be noted that the solution using a system of linear equations was the fastest whenK was low as it is an optimised built-in function of Matlab but it grew faster with increasing values ofKthan the RSC4BI method.
K = 10 K = 20 K = 30 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Runtime (s)
RSC4BI
System of linear equations Matrix inversion
Figure 48: Runtime of complex-valued RSC4BI on computer (applied clock frequency:3.7GHz)
4.4.3 Root Mean Square Deviation of complex-valued RSC4BI-algorithm
This second simulation was based on Monte Carlo method to investigate the performance of the algorithm by applying dierent probe signals. The WF contained K = 20 taps while the observation window enclosed N = 20 samples. As a result, the RMSD of the reference and the modelled output signals was calculated.100,000 simulations were performed, and their results were averaged. The delay elements and the observation window were empty at the beginning; therefore, there was a decaying transient. Thus, the rst 15ms of the signal was cut o to obtain the truncated average of RMSD over the time (marked by red line). Furthermore, the outliers of the 100,000 simulations being not inside the percentile range[0.001,99.999] were removed because the error of the calculation may rarely result high outliers, which distorted the RMSD.
These results are presented in Fig. 4954, where the RMSD and its average are displayed in dB in the case of the various probe signals. The deviation was the lowest if the input signal was only noise (Fig. 5354). However, it has to be noted that the signal level of noise was lower than the IQ signals, which leaded to lower RMSD if only noise was present.
Studying the pairs of Fig. 4950 and Fig. 5152, the eect of the additive noise can be investigated.
Opposite to Sec. 3.5.3, the QAM signals had xed possible values according to their constellation tables, or they were around the constellation points when noise was present; therefore, the AWGN did not have signicant eect to these signals because they were not close to zero (cf. Sec. 3.2.5). Furthermore, comparing Fig. 42 and Fig. 53 or Fig. 43 and Fig. 54, the simulations proved that using the same type of probe signals, both the real- and the complex-valued algorithm gave the same RMSD.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)
-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -101.47 dB
Figure 49: RMSD of RSC4BI-algorithm for 16-QAM signal over time
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s) -200
-180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -94.97 dB
Figure 50: RMSD of RSC4BI-algorithm for noisy 16-QAM signal over time
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)
-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -96.98 dB
Figure 51: RMSD of RSC4BI-algorithm for 256-QAM signal over time
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s) -200
-180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -98.38 dB
Figure 52: RMSD of RSC4BI-algorithm for noisy 256-QAM signal over time
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)
-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -116.12 dB
Figure 53: RMSD of RSC4BI-algorithm for complex Gaussian white noise over time (µ= 0.1andσ2= 1)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s) -200
-180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -136.55 dB
Figure 54: RMSD of RSC4BI-algorithm for complex Gaussian white noise over time (µ= 0.01andσ2= 1)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)
-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0
RMSD (dB)
Truncated cumulated average: -55.38 dB
Figure 55: RMSD of outliers for complex Gaussian white noise over time (µ= 0.1andσ2= 1)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s) -350
-300 -250 -200 -150 -100 -50 0
RMSD (dB)
0.068 0.069 0.07 0.071
-300 -250 -200 -150 -100 -50
Figure 56: RMSD of a single outlier for complex Gaussian white noise over time (µ= 0.1andσ2= 1)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)
-350 -300 -250 -200 -150 -100 -50 0
RMSD (dB)
RSC4BI Matrix inversion System of linear equation
Figure 57: Comparison of the dierent methods depending on lter length if complex signals applied (K= 20)
4.4.4 Root Mean Square Deviation of the outliers
As it is mentioned in Sec. 4.4.3, some simulations had to be removed as outliers. Their RMSD is presented in Fig. 55, which gure is the complement of Fig. 53. It can be seen that although the RMSD of the outliers had a still acceptable order of magnitude, but its variance was signicantly larger leading to the above mentioned distortion.
One of the outliers is presented in Fig. 56 as well. It can be observed that the RMSD was suciently low in general, but there were some peaks that were caused by the temporary calculation errors. Studying the enlarged part of the gure, it can be seen that despite this error, the algorithm was still convergent, and after the peak, the RMSD decreased to its previous range.
4.4.5 Root Mean Square Deviation comparison of the dierent methods
The output of the RSC4BI-algorithm was compared not only to the convolved reference signal but to the two conventional methods presented in Sec. 3.5.1 as well. These results are shown in Fig. 57 for K =N = 20. In this case, the precision of the calculations depended on the algorithm, and as Fig. 57 shows, the RMSD of the system of linear equations was lower compared to the inversion, and RSC4BI gave the highest RMSD.
IncreasingK, the solution of the system of the linear equations became unterdeterminated (Sec. 3.5.4);
therefore, the WF may not nd the optimal solution. The performed simulations demonstrated that those methods which are based on matrix inversion or on RSC4BI can not generate the proper
0 0.005 0.01 0.015 0.02 0.025 0.03 Time (s)
-400 -200 0 200 400 600 800 1000 1200
RMSD (dB)
K = 10 K = 20 K = 30 K = 40
Figure 58: RMSD of RSC4BI if the imaginary part of the main diagonal not removed
lter coecients in general. At the same time, the system of the linear equations method calculated the corresponding results because it is an optimised algorithm without using matrix inversion (cf. Sec. 3.5.1).
4.4.6 Round-o error of the main diagonal in the inverse of the auto-correlation matix As it was shown in Sec. 4.2.5, the imaginary part of the main diagonal in R−1 had to be removed after updating the inverse of a matrix by RSC4BI to avoid the round-o error and its accumulation. In this simulation, the behavior of the algorithm was investigated if these imaginary parts were not set to zero. The results of these simulations are presented in Fig. 58 for various values ofK whileN = 40and the input signal was a Gaussian white noise withN2(0.1,1).
As it can be seen, when K was much lower than N, then the RMSD was also low, and it did not diverge, but the results could be distorted (seeK= 20in Fig. 58) if no correction was applied in the main diagonal ofR−1. By adding more taps to the lter (K= [30,40]), the algorithm became unstable, despite that its stability and correctness were demonstrated previously (Sec. 4.4.3), when the main diagonal was R−1 is compensated (cf. theK=N cases in Fig. 53 and in Fig. 58 theK= 40curve).