## 3 Theoretical Analysis
### Algorithm Flow
Some of the detection results are presented in Table 1. The dominant factors corresponding to each frequency component in the detection results obtained by the proposed method were calculated according to Equation (19) and are provided in Table 1.

\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Before**} & \multicolumn{4}{c}{**Prony Algorithm**} & \multicolumn{4}{c}{**The Proposed Method**} \\ \cline{2-10}  & **Amplitude** & **Frequency** & **Phase/** & **Attenuation** & **Amplitude** & **Frequency** & **Phase/** & **Attenuation** & **Dominant** \\ \multicolumn{10}{c}{**SNR**} \\ \hline \multirow{5}{*}{5 dB} & 368.3924 & 779.0099 & 34.1670 & \(-\)538.5595 & 50.2379 & 49.9946 & 71.7320 & \(-\)0.1279 & 392.7905 \\  & 149.8316 & 793.3605 & 179.6763 & \(-\)135.9869 & 24.8951 & 149.9963 & 107.1586 & \(-\)0.3297 & 75.5083 \\  & 59.1708 & 30.6002 & 102.8747 & \(-\)141.3244 & 8.7083 & 137.8659 & 141.9504 & \(-\)0.1860 & 47.3565 \\  & 50.6543 & 50.0278 & 70.6035 & \(-\)0.1404 & 6.1268 & 349.9532 & 160.7258 & \(-\)1.2100 & 5.0635 \\  & 36.9181 & 149.7132 & 123.6631 & \(-\)4.0898 & 2.5826 & 319.0668 & 44.8012 & \(-\)66.7384 & 0.0522 \\  & 31.0460 & 287.0820 & 143.5194 & \(-\)122.5444 & 1.7748 & 343.3577 & 170.2391 & \(-\)15.2113 & 0.1167 \\  & 23.1498 & 355.5101 & 92.8171 & \(-\)67.1172 & 1.5152 & 198.5234 & 109.4213 & \(-\)54.4115 & 0.0278 \\  & 22.8510 & 188.2917 & 171.7149 & \(-\)53.1641 & 1.4871 & 258.8410 & 6.7160 & \(-\)72.0078 & 0.0207 \\ \hline \multirow{5}{*}{10 dB} & 292.7002 & 184.8522 & 153.7994 & \(-\)740.6918 & 50.4065 & 49.9963 & 72.3137 & \(-\)0.1008 & 500.0645 \\  & 83.3545 & 734.6794 & 103.8009 & \(-\)411.8509 & 25.1007 & 150.0043 & 107.9841 & \(-\)0.2800 & 89.6454 \\  & 52.8242 & 49.9528 & 74.4297 & -0.3731 & 8.6343 & 137.9901 & 143.9876 & \(-\)1.7352 & 5.0336 \\  & 41.6969 & 182.4927 & 29.8205 & \(-\)595.6980 & 6.2607 & 350.0049 & 160.4478 & \(-\)0.6625 & 9.7520 \\  & 40.7923 & 150.1790 & 99.2792 & \(-\)4.8672 & 6.3925 & 512.2301 & 14.6780 & \(-\)898.9631 & 0.0071 \\  & 11.2334 & 350.1351 & 163.7297 & \(-\)12.0463 & 4.0890 & 372.4281 & 52.4083 & \(-\)227.411 & 0.0180 \\  & 9.2335 & 135.9086 & 122.6909 & \(-\)43.4386 & 2.9818 & 478.3799 & 164.9383 & \(-\)414.3335 & 0.0072 \\  & 6.4724 & 240.6855 & 1.8089 & \(-\)49.8908 & 1.3560 & 929.1278 & 46.8287 & \(-\)195.8049 & 0.0069 \\ \hline \multirow{5}{*}{15 dB} & 85.2741 & 579.5170 & 54.9866 & \(-\)620.8631 & 50.3869 & 49.9994 & 72.0592 & \(-\)0.0903 & 557.9945 \\  & 50.9132 & 50.0017 & 71.9242 & \(-\)0.1827 & 25.0419 & 149.9997 & 108.0167 & \(-\)0.2814 & 88.9904 \\  & 37.7624 & 121.4600 & 40.8577 & \(-\)171.2520 & 8.5718 & 138.0054 & 143.8190 & \(-\)1.5998 & 5.3580 \\  & 27.2627 & 150.0270 & 106.7519 & \(-\)0.9241 & 6.3054 & 350.0015 & 160.4092 & \(-\)0.4510 & 13.9809 \\  & 17.0381 & 138.8344 & 58.7591 & \(-\)105.0894 & 1.4600 & 113.8934 & 180.0000 & \(-\)586.8984 & 0.0107 \\  & 11.6107 & 383.2363 & 94.7416 & \(-\)118.2871 & 0.6265 & 103.0310 & 38.6674 & \(-\)716.9492 & 0.0009 \\  & 7.3542 & 349.9924 & 162.0018 & \(-\)1.5335 & 0.6158 & 803.2629 & 61.8563 & \(-\)808.0462 & 0.0008 \\  & 3.9676 & 349.7109 & 146.4665 & \(-\)63.7995 & 0.4230 & 457.5418 & 90.0185 & \(-\)303.4712 & 0.0014 \\ \hline \multirow{5}{*}{16 dB} & 49.8536 & 50.0016 & 71.8437 & -0.0719 & 50.3943 & 49.9998 & 72.0175 & \(-\)0.0937 & 537.8260 \\  & 24.6096 & 149.9883 & 107.9125 & \(-\)5.0973 & 24.9830 & 150.0001 & 107.9953 & \(-\)0.3016 & 82.8349 \\  & 14.1595 & 545.9081 & 107.3173 & \(-\)451.70 Simulation signal \(y(t)\) is known to comprise four frequency components at 50 Hz, 150 Hz, 138 Hz and 350 Hz, respectively. The corresponding amplitudes are 50.4 V, 25 V, 8.5 V and 6.2 V and the corresponding phases are 72\({}^{\circ}\), 108\({}^{\circ}\), 144\({}^{\circ}\) and 160.2\({}^{\circ}\). In Table 1, for the harmonic detection results of the 5 dB noisy signal, the Prony algorithm merely detects the two frequency components at 50.0278 Hz and 149.7132 Hz, both close to the simulation signal.