Programa de Auditoría

In the search for higher sensitivity and resolution, using pulse compression is proposed as processing method [50]. Pulse compression is a technique that is often used in radar systems since it is designed to enhance the detection sensitivity, resolution and reduce noise. The cross correlation g(τ) of the reference signals(t) and the perceived signal h(t) can be determined by Equation 2.23 or 2.24. The first is in the time domain while the latter is in the frequency domain.

g(τ) =

Z ∞

−∞

s(t)h(τ+t)dt (2.23)

g(τ)∝ F−1_{[S(ω)∗H(ω)] (2.24)}

As an explanatory example, the Cross Correlation (CC) of a sine signal and a cosine signal is shown in Figure 2.9. With increasing time delayτ, the red cosine signal is shifted to the right resulting in different values for the correlation. The resulting correlation value is shown in Figure 2.9b. Since these signals are periodical in nature, the cross-correlation is also periodical. The period of the signals is set to 2 s, the time shift between the sine and cosine signal is thus 0.5 s. This results in a peak in the cross correlation at a delay time of 0.5 and 2.5 s. The length of the signals is limited, resulting in a value of zero if the time delayτ is larger than the duration of the reference signals(t). Non-periodical signals are more suitable for CC since these signals have one cross correlation peak instead of a periodical peak value as can be seen in Figure 2.10.

(a) Sine signal with a length of 6 s and a cosine signal with a length of 2 s.

(b) The Cross Correlation of the signals in Fig- ure 2.9a.

Figure 2.9: Graphical explanation of Cross Correlation (CC).

The time delay and value of the cross correlation main peak are the characteristics that can be deter- mined for each pixel resulting in a peak correlation image and a peak delay image respectively. The time delay of the main peak is determined using Equation 2.25.

τCCpeak =τ|{g(τ)=max(g(τ))} (2.25)

The CC peak value is dependent on the magnitude of both the perceived signalh(t) and the reference signal s(t). The amplitude of the reference signal can be set to one. However, the amplitude of the measured signal is directly dependent on the heat distribution on the surface. Uneven heating therefore greatly influences the CC peak value. In 2011, Tabatabaei et al. suggested the cross correlation phase as a method to normalize the CC peak value [51]. The cross correlation phase can be obtained by dividing the cross correlation of the signal and the reference by the cross correlation of the signal and the quadrature of the reference as is shown in Equation 2.26. The quadrature of the reference signal is determined by performing a Hilbert transform which is the term between square brackets in the denominator of Equation 2.26. ϕCC = F−1_{(S(ω)∗H(ω))} F−1_{([−isgn(ω)S(ω)]∗H(ω)) τ=0} (2.26) Both the time shift, the magnitude of the CC peak and the CC phase can be a measurement of the defect depth [50][51]. In recent years, several experiments have been conducted on CC as analysis method for thermography, all showing the improved Signal to Noise Ratio (SNR) in comparison with LT [52][53][54]. Applications have expanded to concrete inspection [55] and human breast cancer in- spection [56].

All non periodical signals are in theory suitable for pulse compression using CC. However there are several signals which result in a good Peak Sidelobe Level (PSL). The Peak Sidelobe Level (PSL) can be determined using Equation 2.27.

PSL = 20 log(Side lobe peak

Main lobe peak) dB (2.27)

Several signals have been applied in thermography:

ˆ Linear chirp signal (Figure 2.11a)

ˆ Digital chirp signal (Figure 2.11b)

ˆ Quadratic chirp signal (Figure 2.12a)

ˆ 7-bit Barker signal (Figure 2.12b)

The auto correlation function of a chirp signal is shown in Figure 2.10. It can clearly be seen that the side lobes around the correlation peak are relatively large. Reduction of the side lobe amplitude results in improved detection performance and depth resolution [57][58]. Three options for reducing the side lobe energy are proposed [20][59]. The first method is replacing the linear chirp with a digitized chirp signal (DFMTWI) as Mulaveesala proposed in 2016 [60]. In DFMTWI the input signal is converted into the digital form that can be seen in Figure 2.11b. The advantage of using a digitized form of the chirp signal is the higher amount of energy injected at the fundamental frequency leading to an increased amplitude of the reflected signal [60]. The second method is replacing the the linear chirp signal with a quadratic chirp while the third method is to use a 7 bit Barker coded signal. Both signals are shown in Figure 2.12. Barker coded signals are widely used in radar and ultrasonic systems, in 2011 the use of this binary phase code has been implemented for NDT purposes by Tabatabaei and Mandelis [20]. Seven different Barker codes are known. A Barker code has a maximum cross correlation side lobe peak of one. In total 11 different Barker codes are known. The 7-bit Barker code is the optimal length for optical thermography [20].

The result of the quadratic chirp and the 7-bit Barker code signals on the correlation function is shown in Figure 2.10. The reduced PSL in theory results in a higher SNR leading to a high contrast between defective and non-defective regions [59][61].

Many efforts have been put into the use of a Barker sequence codes in NDT over the past 10 years [20][52]. In 2018, Fei Wang et al. used a truncated-correlation of a pulsed chirp signal with a laser heat source for qualitative three-dimensional visualization of surface cracks by changing the reference signal delay time [62]. BCTWI has been applied to CFRP and GFRP materials, verifying the detectability of defects with this method and the superiority of the method in comparison with LT [20][52][53][63][64]. However no effort has been done to link the thermography results to defect characters and further research is required on this subject.

(a)Linear chirp signal (b)Digital chirp signal

Figure 2.11: Chirp signals (frequency from 0.5 Hz to 2.5 Hz in 10 sec).

(a)Quadratic up chirp signal (b)7 bit Barker signal