CHARACTERISTICS AND PERFORMANCE OF AWT

To better illustrate limitations with the CWT and the advantages with the proposed AWT, an analytical signal is considered. It is a combination of two closely-spaced frequency-modulated sinusoidal functions and two closely-spaced delta functions, which represents the most demanding signal on both time and frequency resolution. The example signal 𝑥(𝑡) can be mathematically expressed into:

𝑥(𝑡) = 𝑐𝑜𝑠[2𝜋2000𝑡 + 100 𝑠𝑖𝑛(2𝜋10𝑡)]

+ 𝑐𝑜𝑠[2𝜋2600𝑡 + 100 𝑠𝑖𝑛(2𝜋10𝑡)] ⁡ + ⁡⁡𝐾[𝛿(𝑡 − 0.145) + 𝛿(𝑡 − 0.15)] (10) in which 𝐾 = 10 is a gain factor, representing the relative significance on the delta and sinusoidal functions. The signal is sampled at 10 kHz in data analysis or ∆𝑡 = 10⁻⁴ s. Its theoretical scalogram in time-frequency domain is presented in Figure 2.9.

Figure 2.9. Theoretical scalogram of the combined sinusoidal and delta signal

The signal was analyzed with CWT (𝐹_𝑐 = 2 Hz). The obtained scalogram is shown in Figure 2.10. It can be seen form Fig. 10 that the lowest and worst frequency resolution is achieved at time instants 0, 0.1 and 0.2 s, while the highest and best frequency resolution is at 0.05 and 0.15 s. The algorithm as described in Figure 2.5 is thus applied to improve the frequency and time resolution in the range as shown in Figure 2.10.

Figure 2.10. CWT representation of the signal

At time 0 s, the two frequency components are actually mixed together with a frequency range of 2590 Hz to 4170 Hz, resulting in a frequency bandwidth or ridgeline thickness of 1580 Hz, centered at 3380 Hz as shown in Figure 2.11(a). In this case, the scaling factor is 𝑎₁ = ^{𝐹𝑐𝐹𝑠}

𝐹₁ = 2 ×¹⁰⁰⁰⁰

3380 = 5.92 and the ridgeline thickness of a single frequency component is ^{𝐹𝑠∆𝜓̂}

𝑎1 =^{𝐹1∆𝜓̂}

𝐹𝑐 = ³³⁸⁰

2√2 = 1195 Hz. Since the actual ridgeline thickness exceeds the thickness of single component, the scaling factor must be updated as 𝑎₁^′ = ^𝐹𝑠

(1580−^{𝐹𝑠∆𝜓̂}

𝑎1 )√2

= 18.37. The center frequency can thus be updated as 𝐹_𝑐 = ^{𝐹1𝑎1′}

𝐹𝑠 = Time (s)

Frequency (kHz)

0 0.05 0.1 0.15 0.2

1 2 3 4 5

3380×18.37

10000 = 6.2 Hz, which is larger than 0. Therefore, 𝐹(0) = 6.2 Hz. The final step is to determine the window length, 𝑇₀ = ^2𝐹(0)

𝐹𝑙 ∆_𝜓= ^2×6.2

1000√2= 0.0088 s, where 𝐹_𝑙 = 1000 Hz.

(a) 0 s (b) 0.05 s

Figure 2.11. Two instantaneous spectra without the effect of delta functions

At time 0.05 s, the instantaneous spectrum as presented in Figure 2.11(b) indicates a clear separation of two frequency components with a ridgeline thickness of 350 Hz and 550 Hz at peak frequencies of 1,000 and 1,600 Hz, respectively. The calculated scaling factors at the two peak frequencies are 20 and 12.5, and their corresponding ridgeline thicknesses are 354 Hz and 566 Hz, respectively. Since both identified ridgeline thicknesses are smaller than the calculated, the two frequency components are well separable in frequency domain. In this case, the time resolution can be improved with a new scaling factor at the second peak: 𝑎₂^′ = ^{𝐹𝑠(𝐹2+𝐹1)}

2√2𝐹2(𝐹2−𝐹1)= 10000(1600+1000)

2√2×1600(1600−1000)= 9.58, which is less than the original scaling factors at the two peaks. The center frequency at this time is then calculated as 𝐹(0.5) = 𝐹_𝑐 = ^{𝐹2𝑎2′}

𝐹𝑠 = ^1600×9.58

10000 = 1.53 Hz, and the window length is updated to 𝑇_0.5 =^2𝐹(0.5)

𝐹𝑙 ∆_𝜓= ^2×1.53

1000√2 = 0.0022 s.

In addition to the above two extreme cases for the cosine functions, two special conditions must be discussed: energy leakage and delta functions. It can be seen from Figure 2.11 that energy leakage due to signal discretization and integration truncation is present at a frequency range of 500 Hz - 1000 Hz. The low-frequency peaks can be directly filtered out when their amplitudes are below 10%. Otherwise, they can be separated from other signal peaks by verifying that the frequency bandwidth of each peak at 5% remains significantly less than the theoretical thickness calculated from 𝐹_𝑠/𝑎√2. It can be seen from Figure 2.9 that the two delta functions occur at 0.145 s and 0.15 s. Figure 2.12 shows the instantaneous spectrum at 0.15 s when energy is distributed over a broadband of frequency with local concentrations at the frequencies of cosine functions. In this case, the signal threshold must be determined to be above the stationary component by identifying the baseline of the frequency distribution as indicated in Figure 2.12.

Figure 2.12. The instantaneous spectrum at 0.15 s with the effect of delta functions

Once the instantaneous spectra at all times have been analyzed by following the procedure given in Figure 2.5, the time-varying center frequency of the AWT can be determined and presented in Figure 2.13(a) as a function of time that is defined by the beginning of each window. The time-varying window length is proportional to the center

frequency as indicated in Figure 2.5. The variation of 𝑞_𝑘 over time is presented in Figure 2.13(b), which ensures that the energy distribution in the time and frequency plane is consistent with that of the CWT. It can be observed from Figure 2.13(b) that the normalization factor significantly fluctuates around the two delta functions due to their locally added energy. Using the obtained center frequency and window length, the AWT of the signal in Equation (10) results in the scalogram as shown in Figure 2.13(c). The AWT reveals a smooth transition at time instances of the discretized signal and exhibits higher resolution than the CWT as shown in Figure 2.10 in the high frequency range where improved frequency resolutions are used. Figure 2.13(c) is a high fidelity representation of Figure 2.9 in the time-frequency domain of interest.

(a) Time-varying center frequency (b) Time-varying normalization factor

(c) Scalogram of the signal by AWT

Figure 2.13. Characteristic AWT parameters and results

To demonstrate the advantages of AWT over STCWT, the signal represented by Equation (10) was divided into 9 segments, each assigned an optimized center frequency of the Morlet mother wavelet. The STCWT scalogram of the signal is presented in Figure 2.14 by stitching together those of the non-overlapping segments. It can be clearly observed from Figure 2.14 that the overall scalogram is discontinuous and not smooth, leaving behind the stitching traces or sudden frequency jumps at transition times between the center frequencies due to end effects. Although direct, simply stitching the STCWTs would thus be unsatisfactory.

Figure 2.14. The STCWT scalogram by segmental stitching

To quantify the dispersion of ridgelines in scalogram, each instantaneous spectrum (wavelet coefficient) as shown in Figures 2.10 and 2.13(c) at any time, excluding the two delta functions for simplicity, was normalized by its total area and then treated as a probability density function so that the standard deviation of the spectral distribution over frequency can be evaluated. For comparison, two theoretic frequency curves as shown in Figure 2.9 excluding the delta functions were considered as two zero-thickness ridgelines in a spectrum of two pulses. The standard deviation of the theoretic spectrum can be

determined to be 300 Hz. Figure 2.15 shows the standard deviation (Std. Dev.) of spectral distributions as a function of time with the solid (blue) and dotted (red) lines representing the CWT and AWT, respectively. Figure 2.15 also includes a constant deviation of the two theoretic frequency lines in dashed line (green).

Figure 2.15. Standard deviations of CWT and AWT over time and its comparison with theoretic value

It can be seen from Figure 2.15 that the deviation of AWT is less fluctuating and smoother than that of CWT. This is because the frequency resolution has been improved by the AWT in most part of the time-frequency domain. At 0.05 s and 0.15 s, the deviation of CWT, 305 Hz, is slightly smaller than that of AWT, 309 Hz since the frequency resolution in the AWT was locally sacrificed for improved time resolution as discussed in the previous section, both exceeding the theoretic value, 300 Hz. At 0, 0.1, and 0.2 s, where CWT has the lowest frequency resolution, the deviation of CWT spectra reaches a maximum of 406 Hz in comparison with 315 Hz for the proposed AWT or approximately 22% reduction from the deviation of CWT. The variation range of the AWT, 309 Hz to 322 Hz, is only 4.3% of its lowest deviation while the range of the CWT, 305 Hz to 406 Hz, is 33.1% of its lowest deviation. The reduction in variation will significantly improve

the detectability of delamination as demonstrated in the following application example.

Note that the end effect was not taken into account during the comparison.