2.5 Simulation Results
2.5.1 Test signal Corresponding to Local Mode
Table 2.1: Mean and Variance of the estimated mode for the improved Prony, the modified TLS-ESPRIT and the Proposed Method for outlier (=10×peak value) placed towards the end of the signal at different SNR (Test signal- local mode)
Improved Prony
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2 (Variance) µ(Mean) σ2(Variance)
40 0.0229 9.206×10−8 1.006 1.984×10−9
30 0.0229 9.209×10−7 1.006 1.984×10−8
20 0.0231 9.226×10−6 1.006 1.983×10−7
Modified TLS-ESPRIT
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2 (Variance) µ(Mean) σ2(Variance)
40 -0.0869 3.292×10−8 1.0011 4.342×10−9
30 -0.0869 3.267×10−7 1.0011 3.464×10−8
20 -0.0870 3.301×10−6 1.0011 3.396×10−7
Proposed Method (Robust Modified Prony)
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2 (Variance) µ(Mean) σ2(Variance)
40 -0.1000 5.571×10−8 1.0000 1.482×10−9
30 -0.1000 5.622×10−7 1.0000 1.471×10−8
20 -0.1000 5.774×10−6 1.0000 1.456×10−7
Performance comparison among all the methods for this test signal for the estimated attenuation factor and frequency with the outlier injected at the end of the sample is shown in Table 2.1. It is ob- served that the estimated mean value of attenuation factor degrades highly for improved Prony while the proposed method provides much more close results ( i.e, estimated mean value of attenuation factor is equal to−0.1) than that of modified TLS-ESPRIT (which has a mean value of−0.0869 for SNR= 40 dB and SNR=30 dB and−0.087 for SNR= 20 dB). For estimated frequency all the methods gives good results with the proposed method estimating much more accurate results as depicted by its mean and variance values as provided in Table 2.1.
2.5 Simulation Results
Table 2.2: Mean and Variance of the estimated mode for the improved Prony, the modified TLS-ESPRIT and the Proposed Method for outlier (=10×peak value) placed towards the beginning of the signal at different SNR (Test signal- local mode)
Improved Prony
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 0.0093 6.484×10−8 1.0024 1.651×10−9
30 0.0093 6.364×10−7 1.0024 1.657×10−8
20 0.0095 6.354×10−6 1.0024 1.576×10−7
Modified TLS-ESPRIT
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 -0.0676 1.831×10−8 1.0028 3.144×10−9
30 -0.0676 1.787×10−7 1.0028 2.450×10−8
20 -0.0676 1.752×10−6 1.0028 2.454×10−7
Proposed Method (Robust Modified Prony)
Estimated Attenuation factor (True value= -0.1) Estimated Frequency (True value= 1 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 -0.1000 1.533×10−7 1.0000 3.896×10−9
30 -0.1000 1.513×10−6 1.0000 3.932×10−8
20 -0.1000 1.551×10−5 1.0000 3.857×10−7
Performance check of all the methods for this test signal corresponding to the estimated mean value of attenuation factor and frequency for outlier injected towards the beginning of the sample is shown in Table 2.2. Improved Prony gives incorrect attenuation factor while the proposed method gives correct mean, i.e, the estimated mean value of attenuation factor is equal to−0.1 as compared to the modified TLS-ESPRIT which is equal to−0.0676. For estimated frequency, it is observed that all the methods gives good results even with high noise level with the proposed method giving more accurate mean, i.e, estimated mean value of frequency is equal to 1 Hz.
Table 2.3: Estimated Attenuation factor and Frequency for the improved Prony, the modified TLS-ESPRIT, ERA/Prony and the Proposed Method for test signal without noise and without outlier (Test signal- local mode)
Methods Attenuation factor (True value= -0.1) Frequency (True value= 1 Hz)
Improved Prony -0.1000 1.000
Modified TLS-ESPRIT -0.1000 1.000
ERA/Prony -0.1000 1.000
Proposed Method -0.1000 1.000
Table 2.4: Estimated Attenuation factor and Frequency for the improved Prony, the modified TLS-ESPRIT, ERA/Prony and the Proposed Method for test signal without noise but with outlier (=10× peak value) (Test signal- local mode)
Methods Attenuation factor (True value= -0.1) Frequency (True value= 1 Hz)
Improved Prony 0.0229 1.0006
Modified TLS-ESPRIT -0.0869 1.0011
ERA/Prony -0.1223 1.0009
Proposed Method -0.1000 1.0000
Performance comparison for all the methods for the test signal without noise and without outlier is shown in Table 2.3. It is observed that all the four methods gives accurate attenuation factor and frequency estimates. Performance analysis is also done for test signal without noise, but with outlier (whose value is 10 times the peak signal value) as shown in Table 2.4. Improved Prony method provides more erroneous estimates in presence of large outliers as seen in the estimated attenuation factor value from Table 2.4, while the proposed method is better as compared to the modified TLS- ESPRIT and ERA/Prony method in estimating the attenuation factor and frequency.
2.5 Simulation Results
Table 2.5: Mean and Variance of the estimated mode for the improved Prony, the modified TLS-ESPRIT, ERA/Prony and the Proposed Method for outlier (=10×peak value) placed towards the beginning of the signal at different SNR (Test signal- inter-area mode)
Improved Prony
Estimated Attenuation factor (True value= -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR (dB)µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 0.0365 1.477×10−7 0.3973 2.747×10−9
30 0.0365 1.482×10−6 0.3973 2.733×10−8
20 0.0368 1.482×10−5 0.3973 2.813×10−7
Modified TLS-ESPRIT
Estimated Attenuation factor (True value= -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR (dB)µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 -0.0811 4.105×10−8 0.3981 5.475×10−9
30 -0.0811 3.992×10−7 0.3981 4.793×10−8
20 -0.0811 3.953×10−6 0.3981 4.712×10−7
ERA/Prony
Estimated Attenuation factor (True value= -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR(dB) µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 -0.0929 2.249×10−8 0.3995 1.630×10−8
30 -0.0929 2.189×10−7 0.3995 1.668×10−7
20 -0.0929 2.259×10−6 0.3995 1.672×10−6
Proposed Method (Robust Modified Prony)
Estimated Attenuation factor (True value= -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR (dB)µ(Mean) σ2(Variance) µ(Mean) σ2(Variance)
40 -0.07 5.148×10−8 0.4000 1.336×10−9
30 -0.07 5.093×10−7 0.4000 1.351×10−8
20 -0.07 5.055×10−6 0.4000 1.339×10−7
Table 2.6: Mean and Variance of the estimated mode for the improved Prony, the modified TLS-ESPRIT and the Proposed Method for outlier (=10×peak value) placed towards the end of the signal at different SNR (Test signal- inter-area mode)
Improved Prony
Estimated Attenuation factor (True value = -0.07) Estimated Frequency (True value = 0.4 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2 (Variance)
40 0.0736 2.922×10−7 0.4032 4.243×10−9
30 0.0737 2.999×10−6 0.4032 4.203×10−8
20 0.0744 3.024×10−5 0.4032 4.315×10−7
Modified TLS-ESPRIT
Estimated Attenuation factor (True value = -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2 (Variance)
40 -0.1230 1.072×10−7 0.4036 1.048×10−8
30 -0.1230 1.070×10−6 0.4036 1.001×10−7
20 -0.1230 1.067×10−5 0.4036 9.796×10−7
Proposed Method (Robust Modified Prony)
Estimated Attenuation factor (True value = -0.07) Estimated Frequency (True value= 0.4 Hz)
SNR (dB) µ(Mean) σ2(Variance) µ(Mean) σ2 (Variance)
40 -0.07 1.157×10−7 0.4000 2.727×10−9
30 -0.07 1.154×10−6 0.4000 2.729×10−8
20 -0.07 1.179×10−5 0.4000 2.733×10−7