NDT.net Nov 2003, Vol. 8 No.11 |
A number of signal processing techniques exist to improve signal to noise ratios in ultrasonic signals. One such technique is called Complex plane Split Spectrum Processing (CSSP)[1]. This technique is a modification to the Split Spectrum Processing (SSP) technique. CSSP is capable of being able to surpass SSP in terms of improvements in signal to noise ratios, while maintaining linearity in both the amplitude and the energy content of a flaw's signal.
CSSP makes use of an additional mathematical dimension (the complex plane) when solving for the probability that a signal originates from a grain boundary. This paper serves to formalise the CSSP technique published in 2000[1], and shows how shortcomings in SSP can be overcome. This paper also serves to formalise the validation steps required of CSSP such that it may be ported from the laboratory to industrial and medical settings.
As extracted from P. Rubbers and C.J. Pritchard[3]: A real flaw will reflect all wavelengths smaller than the flaw size. However in the case of a grain boundary, not all smaller wavelengths are reflected. This has been shown graphically in Figure 1, where a real flaw is shown as the top solid line, while a grain boundary is shown as the bottom dotted line. Multiple frequencies (in the form of a chirp signal in the figure) can then be transmitted onto these two boundaries. In the case of a real flaw all frequencies are reflected, while in the case of a grain boundary only a subset of the frequencies are reflected. This phenomenon has been termed frequency diversity[2].
Split Spectrum Processing (SSP) uses this frequency diversity to determine if a signal originates from a real flaw or a grain boundary. The implementation of SSP therefore makes use of multiple frequencies to inspect a component. This is best achieved by the use of a broadband signal, which can then be decomposed into a number of frequency bands.
An overview of SSP has been published by P. Rubbers and C.J. Pritchard[3], discussing techniques for the decomposing of a signal into various frequency bands, as well as techniques to re-combine these signals such that the signal to noise (S/N) ratio is improved. It can be shown[3] that each of these techniques improves the S/N ratio at the cost of linearity in the processed signal. Therefore SSP does not allow for the sizing of indications using any conventional means, such as DGS (distance gain size), DAC curves (distance amplitude correction), reference block, scanning method (e.g. 6dB drop off), energy content or even pattern type, since all these techniques require amplitude linearity. The use of SSP in industrial or medical application is therefore limited, and the use of SSP has been relegated to the laboratory.
Fig 1: Frequency diversity of a real flaw versus a grain boundary [3]. | Fig 2: Shortcoming of SSP. |
The basic premise of SSP is that all flaws are linear and parallel to the inspection wave front, as shown in Figure 1. This is rarely the case since flaws usually follow grain boundaries as the flaw propagates through the material. For a general flaw as shown in Figure 2, the high frequency (wavelength l_{2}) will not reflect at the same point as the low frequency (wavelength l_{1}), and may thus be regarded as noise by SSP.
2.1 Hypothosis for CSSP
A number of techniques exist that make use of the complex form of a signal, ranging from Doppler to Holography. In all these techniques, the quadrature (Q) and In-Phase (I) portion of the signal is used during processing. In essence, the In-Phase portion of the wave is the original signal, while the quadrature portion is its causal waveform[4][5], which can be calculated by the use of a Hilbert transform[6]. By combining the original signal (In-Phase portion as the real part) and its Hibert transform (quadrature portion as the imaginary part) we can create a complex signal (analytic signal). The analytic signal can now be used to calculate the instantaneous phase at any point using an arctangent function.
In the case of a pure cosine wave, we would therefore have:
x_{t} = A cos(wt+f)
where:
A | Amplitude of the cosine wave |
w | Angular frequency |
t | Time |
f | Phase of the signal at time=0 |
For this signal the quadrature portion would be (obtained using a Hilbert transform):
y_{t} =A sin(wt+f)
At time t, the phase of this signal is therefore:
F_{t} = tan^{-1}(y_{t}/x_{t}) = (wt+f)
In the case of split spectrum processing this phase information has been neglected. It is the purpose of this paper to demonstrate that by making use of the phase information, all the deficiencies or disadvantages of split spectrum processing can be eliminated.
2.2 Using Phase Information
SSP using Polarity thresholding has been shown schematically in Figure 3 for a two-filter (two frequency) system at various phase differences. At location labelled '1' the signals are in phase, and have thus been detected as an indication using SSP (polarity thresholding). The signal amplitude is however half that of the original signal. At location '2' the signals are 90° out of phase, and the SSP signal is zero, however at phase differences between 90° and 135° the SSP signal incorrectly increases as well. At location '3' the signals are out of phase by 180° and SSP has correctly determined that this part of the signal is noise.
Fig 3: SSP using polarity thresholding on two sine waves. |
Instead of looking at the amplitude graph, the phases of the two signals can be displayed, as well as the difference between the two phases as shown in Figure 4.
Fig 4: Phase of the two sine waves with their phase difference. |
Since a real flaw reflects all frequencies, we can assume that the high and low frequencies shown in Figure 2 must be in phase by less than a pre-determined angle. For a phase difference of 1 radian (57°), we can truncate the above phase diagram as shown Figure 5, where the part of the signal now deemed to be originating from a flaw is in the shaded area:
Fig 5: Phase difference threshold of 1 radian. |
Using this specification, we can now specify that in this shaded area, where the phases are less than 1 radian apart, the original signal represents a real flaw, while the remainder of the signal represents noise, as shown in Figure 6.
Fig 6: simplified CSSP using 1 radian as a threshold. |
Comparing Figure 3 and Figure 6, we can see that the amplitude of the original signal is maintained for flaws, while the incorrect SSP signals between 90° and 135° have been removed. This indicates that the use of phase information increased signal to noise ratio, while maintaining the flaw amplitude in the processed signal.
The equation for the above algorithm can be written as:
Two filter, Simple CSSP using maximum phase spread
A_{i} = X_{i} If |F_{1i} - F_{2i}| < F_{threshold}
A_{i} = 0 If |F_{1i} - F_{2i}| ³ F_{threshold}
where
A | Amplitude of the signal following CSSP recombination (the resultant signal) |
X | Amplitude of the original signal |
i | Subscript for the sample number (related to depth by the sound velocity and the sampling rate) |
F | Time domain phase of a signal |
F_{threshold} | = 57° for the example in Figure 6, but can be specified as a filter parameter |
A_{i} therefore indicates the amplitude of the recombined signal at point i, while F_{1i} indicates the phase of the signal from filter 1 at point i and F_{2i} indicates the phase of the signal from filter 2 at point i.
Returning to Figure 2, we can see that the two frequencies (wavelengths) may be reflected from two different depths. If the difference in depth is equal to a quarter of a wavelength, then the two signals will be 90° out of phase. SSP will then determine that the signals represent noise. CSSP on the other hand will detect the overlapping phase difference at 45°, and the signal will be regarded as a real flaw.
In the above example only two frequency bands were shown. By extension, multiple frequency bands can also be used. Instead of measuring the phase difference between each filtered signal, the spread of the phases need to be quantified to determine the validity of a signal.
This paper proposes one technique, however other possible techniques exist. This proposed technique is based on the maximum spread in the phases. This can be described using a polar diagram as in Figure 7. The maximum spread is therefore the largest angle separating the various phases (i.e. F_{max})
Fig 7: Max phase spread . |
For a system with n filters, a perfectly even distribution of phases would result in F_{max} being equal to 2p/n. F_{max} and therefore the phase threshold F_{threshold} depends on the number of filters n.
Simple CSSP using maximum phase spread
F_{max} = 2p - max (|F_{0i} - F_{1i}| |F_{1i} - F_{2i}|
... |F_{(n-1)i} - F_{ni}| |F_{ni} - F_{0i}|)
A_{i} = X_{i} if F_{max} < F_{threshold}
A_{i} = 0 if F_{max} ³ F_{threshold}
where
A | Amplitude of the signal following CSSP recombination (the resultant signal) |
X | Amplitude of the original signal |
i | Subscript for the sample number (related to depth by the sound velocity and the sampling rate) |
F | Time domain phase of a signal |
F_{threshold} | Specified as a filter parameter by the user depending on the filtering requirements |
n | Subscript for the filter number ordered in increasing phase angle (e.g. between 1 and 6 for a 6 filter system, but arranged in increasing phase for each sample i). |
A_{i} therefore indicates the amplitude of the recombined signal at point i, while F_{ni} indicates the phase of the signal at point i for the filter with the n^{th} largest phase.
While this maximum phase spread may give a clear threshold value to the grain noise, it is possible that certain flaws may be missed when F_{threshold} is decreased to small values (close to 0). This can be overcome by giving the phase spread a weighting function. This is similar to the SSP technique of polarity threshold with scaling: If the phase F_{max} is smaller than F_{zero} but larger than F_{threshold}, a linear interpolation of the amplitude is calculated, resulting in a weighting factor. This weighting factor can be shown as in Figure 8, however the interpolation between F_{zero} and F_{threshold} does not need to be linear, but may be in the form of a quadratic or more complex equation.
Fig 8: Weighting factor from max phase spread. |
CSSP using maximum phase spread
F_{max} = 2p - max (|F_{0i} - F_{1i}| |F_{1i} - F_{2i}|
... |F_{(n-1)i} - F_{ni}| |F_{ni} - F_{0i}|)
weighting factor = 0 ifF_{max} ³ F_{zero}
weighting factor = 1 ifF_{max} £ F_{threshold}
weighting factor = (F_{max} -F_{zero} )/(F_{threshold} - F_{zero}) if F_{threshold} < F_{max} < F_{zero}
A_{i} = weighting factor * X_{i}
2.3 Evaluating CSSP (Maximum Phase Spread)
To compare CSSP with results obtained using various SSP recombination techniques, the RF waveform analysed in [3] is also used here. This signal overlaid with the CSSP result is shown in Figure 9. In this case the CSSP used the maximum phase spread recombination technique, with n=6, F_{threshold} = 1 and F_{zero} = 1.2 (radians).
Fig 9: CSSP with maximum phase spread. |
Fig 10: Normalized phase spread for various locations in A-Scan. |
Figure 10 shows the phase spread at three locations along the A-scan. This phase spread shows seven vectors, one for each band-pass filter, and one for the original signal. In this figure, the clustering of phases at a real flaw can be compared to the random phase distribution caused by grain noise. Note that the magnitude of the largest vector has been normalised in the phasor diagram such as to fill the graph.
Below the phasor diagram is the frequency spectrum of the respective signals, showing the various band-pass filter frequencies. Note that the frequency's vertical scales are linear and scaled to fill the graph.
P. Rubbers and C.J. Pritchard[3] compared a number of SSP techniques using the signal used in Figure 9. The comparison table from [3] has been shown in Table 1, with the addition of the CSSP (maximum phase spread) results. In all the tabulated cases, the signal to noise ratio between the 5mm flat bottom hole, and the previous 15mm of material is compared: The best SSP technique for this specific signal was SSP with polarity thresholding, which has been reprinted in Figure 11 for comparative purposes.
5mm signal | Grain noise | S/N ratio | S/N in dB | |
Original rectified data | 0.057 | 0.05 | 1.14 | 1.1381 |
SSP Minimisation | 0.12627 | 0.042291 | 2.9852 | 9.501 |
SSP Polarity threshold | 0.12627 | 0.037448 | 3.3719 | 10.557 |
SSP Polarity with scaling | 0.12627 | 0.037448 | 3.3719 | 10.557 |
SSP Frequency multiplication | 0.1068 | 0.0339 | 3.1554 | 9.981 |
SSP Square and add | 0.0469 | 0.0427 | 1.0986 | 0.8165 |
SSP Power^{0.25} and add | 0.1809 | 0.1612 | 1.1219 | 0.9986 |
CSSP Maximum phase spread | 0.057 | 0.007 | 8.1429 | 18.2 |
Table 1: Signal to noise ratios of various SSP algorithms, and a CSSP algorithm. |
Fig 11: SSP with polarity thresholding[3]. |
Comparing Figure 9 and Figure 11 we can see that CSSP (maximum phase spread) obtains a better S/N ratio than the best SSP (polarity thresholding) results. In addition to this improvement in S/N ratio, the signal width and energy content is now maintained with CSSP. CSSP can also be used to obtain an envelope signal, by multiplying the weighting factor by the original signal's envelope.
2.4 Flow diagram for CSSP
The flow diagram of CSSP[7] can be presented graphically as in Figure 12. The technique used in this flow diagram to obtain the imaginary part of the analytic signal is a Hilbert transform (FHT).
The Fourier transforms (FFT's) Inverse Fourier transforms (IFFT's) and the frequency domain filters are all based on complex signals, such that the filtered signals are complex and can be used to evaluate the phase angle using an arctangent.
An alternative to the digital method shown in Figure 12, is to perform CSSP using analog techniques: The causal part of the signal can be obtained as per analog Radar (mixer), while delay lines and pass-band filters can implement the signal decomposition. Phase comparisons can then be made using phase comparators, while components also exist to perform the recombination and final multiplication in the analog domain.
Fig 12: Flow diagram for CSSP. |
2.5 Comparison between CSSP and SSP performance
CSSP improves the signal to noise ratios of broadband signals. In addition to this, it has a number of advantages for sizing flaws, and has eliminated the disadvantages of SSP as listed below:
Disadvantage in SSP | Corrected using CSSP |
Non-linear amplitude response: This prevents the use of conventional sizing techniques. | Linear amplitude response. Can use all conventional sizing techniques, such as DAC, DGS, calibration blocks, scanning method (e.g. 6dB drop off) etc. |
Poor detectability of curved or angled flaws. | Improved detectability. |
High frequency output waveform compared to original signal | Output signal has similar frequency components as the original signal. |
Table 2: Disadvantages of SSP corrected using CSSP. |
For CSSP to be accepted as an ultrasonic inspection technique, a number of criteria must be met, and the technique must be validated rigorously according to national standards of governing bodies. These standards may include standards such as BS 4331:1974[8] as the British standard or ASME 5 1998 articles 4 and 5[10] for the USA. More specific guidelines and recommended practice depend on the application, such as EN 1714:1997[9] for the case of welded joints.
This paper has only shown the possible advantages of CSSP in terms of signal to noise ratios (sensitivity). For comprehensive validation of this technique the following trials (on fine grained calibration blocks) are required:
This needs to be followed by validation tests to determine the effect of CSSP on inspection techniques in terms of flaw sizing, flaw characterisation and for component geometry. These validation tests should therefore include the following:
Once these baseline validation tests have been performed, case studies on coarse-grained materials can be put into perspective, and fitness for purpose in industrial and even medical settings can become a reality.
Split Spectrum Processing has been shown in literature to improve signal to noise ratios, but due to its non-linearity, sizing information is lost.
Complex plane Split Spectrum Processing has been shown to also improve signal to noise ratios, while retaining sizing information.
CSSP therefore has the possibility of improving the sensitivity of ultrasonic examinations.
Extensive further work is required in validating CSSP for it to become acceptable as an industrial or medical inspection technique.
The authors would like to thank Eskom Resources and Strategy Research Division for providing the funding that made this research work possible. We would also like to thank Manfred Johannes, Willem Nel, Hugh Neesen and Konrad Hartmann from Eskom Enterprises for their technical support and assistance in this work.
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