Non-Periodic Sparse Time-Frequency Method
3.2. Numerical Examples
STFR and periodic STFR, non-periodic STFR uses the dictionary explicitly in the algorithm. We use the method introduced in [34] to solve (3.1.10) numerically.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1 0 1 2 3 4 5 6 7 8
Figure 3.1. Signal with a Linear Trend: The horizontal axis is the time variable and the vertical one is the signal itself.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.2. Extracted Trend: The linear trend is in blue and the extracted trend is in red. Except the right boundary, the error is small in the extracted trend.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.3. Extraction of the first IMF: The extracted IMF is in red. As can be seen, except from the boundaries the extraction is faithful.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.4. Extraction of the second IMF: The extracted second IMF is in red. It is almost indistinguishable from the high-frequency original IMF.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6 7 8
Figure 3.5. Signal with a Quadratic Trend: The horizontal axis is the time variable and the vertical one is the signal itself.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.6. Extracted Trend: The quadratic trend is in blue and the extracted trend is in red. There is almost no error in the extraction.
Example 5. To further test the method, we ran the algorithm on a signal that con- tained some intrawave modulation in one of the IMFs (intrawave signals are addressed in greater detail in the next chapter). The signal used (see Figure 3.9) is described
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.7. Extraction of the first IMF: The extracted IMF is in red. As can be seen, except from the boundaries the extraction is faithful.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.8. Extraction of the second IMF: The extracted IMF is in red. Like the previous example, the extraction is accurate, except near the right boundary.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Figure 3.9. Signal with a Hump-like Trend: The horizontal axis is the time variable and the vertical one is the signal itself.
in mathematical terms as f(t) = 1
1.2 + cos (2πt) + 1
1.5 + sin (2πt)cos (32πt+ 0.2 cos (64πt)).
The trend is like a hump. The IMFs were again extracted with high accuracy except near the boundaries (see Figures 3.10, 3.11).
Example 6. To check the stability of the algorithm with noise perturbation, we ran a series of tests that added white noise (represented by χ(t)) to an IMF. In this example, the signal used (see Figure 3.12) was:
f(t) = cos (60πt+ 10 sin (2πt)) +χ(t).
The extracted IMF was compared to the original one (see Figure 3.13). Even in the presence of heavy noise, the extraction is still acceptable, showing the stability of the non-periodic STFR method in the presence of noise.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.10. Extracted Trend: The trend is in blue and the ex- tracted trend is in red. There is almost no error in the extraction, except near the boundaries.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5
Found IMF Vs real IMFs
Figure 3.11. Extraction of the IMF: The extracted IMF is in red.
Since the signal has intrawave modulation, the extraction has slight phase lags seen near the peaks and troughs. Still, the extraction is faithful.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−6
−4
−2 0 2 4 6
Figure 3.12. Signal with Noise Perturbation: The horizontal axis is the time variable and the vertical one is the signal itself.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1.5
−1
−0.5 0 0.5 1 1.5
Found IMF Vs real IMFs
Figure 3.13. Extraction of the IMF: The extracted IMF is in red.
Even in the presence of noise perturbation, the generality of the extrac- tion is acceptable.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−5 0 5 10
Figure 3.14. Signal with a Quadratic Trend Polluted with Noise Perturbation: The horizontal axis is the time variable and the vertical one is the signal itself.
Example 7. To further test the method’s stability, we tested it on the signal from Example 4 plus white noise (see Figure 3.14)
f(t) = 6t2+ cos10πt+ 10πt2+ cos
60πt 0≤t≤0.5 80πt−10π 0.5≤t≤1
+χ(t).
The trend is extracted fairly well, (see Figure 3.15). The IMFs are also acceptable (see Figures 3.16, 3.17). In fact, there is no observable phase error in extraction. Although there are jumps near the peaks and troughs of the IMFs, the trends of the IMFs are extracted properly, except for some boundary error. This further demonstrates the stability of the non-periodic STFR method in the presence of noise.
When compared with other STFR methods, the only shortcoming of the Non- Periodic STFR method is the speed of the algorithm (see Table 3.1). In particular, non-periodic STFR can be seen as a strong trend detector compared to even the EMD\EEMD [58].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.15. Extracted Trend: The linear trend is in blue and the extracted trend is in red. Due to the presence of noise, the extracted trend deviates from the original trend slightly.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.16. Extraction of the first IMF: The extracted IMF is in red. The only part of the extraction that is not completely acceptable is the right boundary of the extraction.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1 0 1 2 3 4 5 6
Found IMF Vs real IMFs
Figure 3.17. Extraction of the second IMF: The extracted IMF is in red. Here, the noise perturbation has more effect on the IMF extraction. However, the generality of the extraction is still acceptable.
Periodic STFR TV STFR Non-Periodic STFR Accuracy in envelope
Extraction High Medium-High Medium
Accuracy in IF
Extraction High Medium-High Medium
Accuracy in IMF
Extraction High High High
Speed High Medium Medium
Non-Periodic Data No Yes Yes
Noise Stability High Low High
Boundary Error for
Non-Periodic Data High Medium Low
First Guess
Initialization No Yes No
Table 3.1. Comparison of the STFR Methods
CHAPTER 4