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Signal Processing Methods

Three characteristics of the received signals from the FE models will be analysed: the amplitude of the first backwall signal (BW1), the amplitude of the pit signal (PS) and the extracted wall thickness (WT). BW1 and pit signal phase changes have not been considered.

6.3.1 Normalised Amplitude Definitions

Figure 6.11 a) Schematic of a spot measurement setup b) illustration of the windowing used in the method to determine the backwall and pit amplitudes

131 An overview of the method to determine the backwall and pit amplitudes is illustrated in Figure 6.11b, whereby the time trace is segmented into two windows covering the 1st

backwall echo (BW1) and the pit signal. The maximum amplitudes within either the BW1 or pit signal window is taken as the ABW1 amplitude or the APS pit signal amplitude respectively.

Both of the amplitude values are calculated (in dB) relative to ABW1,ND, i.e. the 1st backwall

echo without a defect present. The Hilbert envelope of the raw time traces, senv[n], are

used to calculate the amplitudes. The following outlines the methods which were used to define the BW and pit signal windows as well as how the BW1 amplitude or the pit signal amplitudes, ABW1 and APS, are calculated. One drawback of using the windowing method is

that it creates β€˜deadzones’, see appendix 9.6 for more details.

6.3.1.1 Determination of the First Backwall Amplitude

First, the peak location of the BW1 without a defect is determined using the known nominal thickness, b:

π‘›π΅π‘Š1,𝑁𝐷,π‘π‘’π‘Žπ‘˜= π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯

𝑛 (𝑠𝑒𝑛𝑣,𝑁𝐷[(𝑏 𝑐⁄ ) 𝑓𝑔 𝑠< 𝑛 < (3𝑏 𝑐⁄ ) 𝑓𝑔 𝑠 ])

(6.1)

where senv,ND[n] is the enveloped signal with no defect present, cg is the group velocity, fs is

the sampling frequency and (𝑏 𝑐⁄ ) 𝑓𝑔 𝑠 is a threshold which occurs approximately at the

midpoint between the input signal and the BW1 echo. The start of the BW1 window can then be defined:

π‘›π΅π‘Š π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘ = π‘›π΅π‘Š1,𝑁𝐷,π‘π‘’π‘Žπ‘˜βˆ’ 𝑁𝑖𝑛𝑝𝑒𝑑 π‘‘π‘œπ‘›π‘’π‘π‘’π‘Ÿπ‘ π‘‘/2 (6.2)

where Ntoneburst is the sample width of the input toneburst and is equal to 𝑁𝑐𝑦𝑓𝑠 𝑓 𝑐𝑒𝑛

⁄ , Ncy are

the number of cycles and fcen is the centre frequency. Similarly, the end of the BW1 window

is determined:

π‘›π΅π‘Š π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑 = π‘›π΅π‘Š1,𝑁𝐷,π‘π‘’π‘Žπ‘˜+ 𝑁𝑖𝑛𝑝𝑒𝑑 π‘‘π‘œπ‘›π‘’π‘π‘’π‘Ÿπ‘ π‘‘/2 (6.3)

The limits of the BW1 window are then used to find the maximum value of senv,ND[n] and

senv[n] which occur within the BW1 window. The relative value of the BW amplitude is given

132 π΄π΅π‘Š1= 20 π‘™π‘œπ‘”10

π‘šπ‘Žπ‘₯(𝑠𝑒𝑛𝑣[π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘ < 𝑛 < π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑])

π‘šπ‘Žπ‘₯(𝑠𝑒𝑛𝑣,𝑁𝐷[π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘< 𝑛 < π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑])

(6.4)

6.3.1.2 Determination of the Pit Signal Amplitude

The window used to define the pit signal starts where the input signal ends and ends when the BW1 signal without a defect begins. First, the peak location of the input toneburst using the known nominal thickness, b, is determined.

𝑛𝑖𝑛,π‘π‘’π‘Žπ‘˜= π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯

𝑛 (𝑠𝑒𝑛𝑣,𝑁𝐷[𝑛 < (𝑏 𝑐⁄ ) 𝑓𝑔 𝑠 ])

(6.5)

Next the start of the pit signal window is located using the known width of the pulse:

𝑛𝑃𝑆 π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘= 𝑛𝑖𝑛,π‘π‘’π‘Žπ‘˜+ 𝑁𝑖𝑛𝑝𝑒𝑑 π‘‘π‘œπ‘›π‘’π‘π‘’π‘Ÿπ‘ π‘‘/2 (6.6)

The end of the pit signal window is equal to the start of the BW1 window:

𝑛𝑃𝑆 π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑= π‘›π΅π‘Š π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘= π‘›π΅π‘Š1,𝑁𝐷,π‘π‘’π‘Žπ‘˜βˆ’ 𝑁𝑖𝑛𝑝𝑒𝑑 π‘‘π‘œπ‘›π‘’π‘π‘’π‘Ÿπ‘ π‘‘/2 (6.7)

Then using the limits of the pit signal window the maximum value of senv[n] which occurs

within the pit signal window is determined. The relative value of the pit signal amplitude is given in the dB scale, relative to the BW1 echo with no defect.

𝐴𝑃𝑆 = 20 π‘™π‘œπ‘”10(

π‘šπ‘Žπ‘₯(𝑠𝑒𝑛𝑣[𝑛𝑃𝑆 π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘< 𝑛 < 𝑛𝑃𝑆 π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑])

π‘šπ‘Žπ‘₯(𝑠𝑒𝑛𝑣,𝑁𝐷[π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ π‘ π‘‘π‘Žπ‘Ÿπ‘‘< 𝑛 < π‘›π΅π‘Š1 π‘€π‘–π‘›π‘‘π‘œπ‘€ 𝑒𝑛𝑑]))

(6.8)

6.3.2 Time of Flight Method to Estimate Wall Thickness

In this study cross correlation is used to extract a value of wall thickness. An overview of the method is shown in Figure 6.12. First a threshold, nWT threshold, where the input toneburst

ends is defined, see Figure 6.12a.

π‘›π‘Šπ‘‡ π‘‘β„Žπ‘Ÿπ‘’π‘ β„Žπ‘œπ‘™π‘‘ = 𝑛𝑖𝑛,π‘π‘’π‘Žπ‘˜+ 𝑁𝑖𝑛𝑝𝑒𝑑 π‘‘π‘œπ‘›π‘’π‘π‘’π‘Ÿπ‘ π‘‘/2 (6.9)

nin,peak is determined using equation 6.5. Two new signals are created, one of which

contains the signal before the threshold (input toneburst signal, sin[n], Figure 6.12b), and

another which contains the signal after the threshold (BW1 signal and a potential pit signal, sBW1[n], Figure 6.12c). The removed part of the signal is replaced with zeroes:

133 𝑠𝑖𝑛[𝑛] = 𝑠[𝑛 > π‘›π‘Šπ‘‡ π‘‘β„Žπ‘Ÿπ‘’π‘ β„Žπ‘œπ‘™π‘‘] = 0 (6.10)

π‘ π΅π‘Š1[𝑛] = 𝑠[𝑛 < π‘›π‘Šπ‘‡ π‘‘β„Žπ‘Ÿπ‘’π‘ β„Žπ‘œπ‘™π‘‘] = 0 (6.11)

The two time traces are cross correlated with the input toneburst, stone[n]6, which is used

to generate the FE toneburst7:

𝐢𝑖𝑛[π‘˜] = βˆ‘ 𝑠𝑖𝑛[𝑛]π‘ π‘‘π‘œπ‘›π‘’[𝑛 + π‘˜] 2π‘βˆ’1 𝑛= 0 (6.12) πΆπ΅π‘Š1[π‘˜] = βˆ‘ π‘ π΅π‘Š1[𝑛]π‘ π‘‘π‘œπ‘›π‘’[𝑛 + π‘˜] 2π‘βˆ’1 𝑛= 0 (6.13)

The arrival times of the input toneburst, tin, and the BW1 echo, tBW1, are found using the

argmax() function: π‘˜in,max= π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯ π‘˜ (𝐢𝑖𝑛[π‘˜]) (6.14) 𝑑𝑖𝑛 = π‘˜π‘–π‘›,π‘šπ‘Žπ‘₯ 𝑓𝐢 (6.15) π‘˜π΅π‘Š1,π‘šπ‘Žπ‘₯= π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯ π‘˜ (πΆπ΅π‘Š1[π‘˜]) (6.16) π‘‘π΅π‘Š1= π‘˜π΅π‘ŠπΌ,π‘šπ‘Žπ‘₯ 𝑓𝐢 (6.17)

where fC is the sampling frequency of the cross correlations (Cin and CBW1). If the sampling

frequency of stone[n] is the same as s[n] then fc = fs. The method of calculating the wall

thickness will change depending on the transducer setup being modelled. If the same set of

6 To simplify the cross correlation s

tone[n] is resampled to have the same sampling frequency as s[n] 7 The input signal is effectively auto-correlated and therefore t

in = 0; however, there are certain setups where this is not a given (such as the waveguide setup) therefore the method has been described in full.

134 nodes generates and receives the signal (i.e. pulse-echo) then the wall thickness, b, can simply be calculated using:

𝑏 = 𝑐𝑔.

π‘‘π΅π‘Š1βˆ’ 𝑑𝑖𝑛

2

(6.18) When the transducer setup under investigation has a different set of Tx and Rx nodes (i.e. pitch catch) then the distance travelled by the pulse must take account of the separation, d, between the Tx and Rx node sets [143]:

𝑏 =1

2(βˆšπ‘π‘”(π‘‘π‘–π‘›βˆ’ π‘‘π΅π‘Š1)(𝑐𝑔(π‘‘π‘–π‘›βˆ’ π‘‘π΅π‘Š1) βˆ’ 2𝑑))

(6.19)

Furthermore the input signal, sin, will not necessarily be represented in the signal received

at the Rx nodes. In the case of the waveguide setup the surface wave between the Tx and Rx node sets will produce tin [143].

Figure 6.12. Illustration of method used to extract wall thickness, using a time trace a) which is divided into two: b) the input signal in the time trace is cross correlated with the input toneburst2 to find a first time value and c) the first backwall signal is cross correlated with the input toneburstto find the second time value.

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