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60 50 40 30 20 10 20 15 10 5 0 –5 –10 –15 0 20 40 60 80 100 120 Scan position (arbitrary unit) Amplitude (mV) Amplitude (mV) 0 20 40 60 80 100 120 Scan position (arbitrary unit) (a) (b) F IGURE 3. Overall schematic diagram of adaptive noise cancellation. Reference signal d i Output y i Error signal ε i Adaptive filter H ( z ) y i + Input signal u i
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(9) where p n is the noise power. When the filter is optimized, the error is a minimum p n and the filter output y i = s i is completely noise free. Figure 4a shows a magnetic flux leakage signal obtained during tests of gas transmission pipelines. The measurements are corrupted by a periodic noise caused by helical variations introduced by the manufacturing process. The result of applying adaptive filtering (Fig. 3) is shown in Fig. 4b. 5 Signal Restoration Signal restoration procedures are used when the distortion processes that introduce specific artifacts in the signal are known and can be expressed in the form of a mathematical function. Two classes of signal restoration procedures are discussed below: (1) low frequency trends and (2) the effect of the transducer footprint. These distortions can be eliminated by using restoration procedures such as detrending and deconvolution, which are described next. Detrending Noise and trends are common forms of distortion that are often present in eddy current and magnetic flux leakage signals. Trends are low frequency changes in the signal levels caused by several factors including instrument drift and gradual variations in probe orientation. In the case of eddy current nondestructive testing, low frequency trends are introduced in the signal because of gradual variations in probe liftoff. The raw eddy current signal in Fig. 2a shows typical distortion introduced by slowly varying trends. A commonly used technique for eliminating such artifacts is based on zero phase high pass filters, which can be implemented by using the discrete cosine transform. Discrete Cosine Transform The discrete cosine transform is a special case of the discrete fourier transform 1 where the basis functions consist of cosines instead of complex exponentials. The discrete cosine transform of an N -dimensional discrete time signal x is given by Eq. 10: E E s y p i i i n ε 2 2 [ ] = ( ) + 192 Electromagnetic Testing F IGURE 4. Results obtained from application of adaptive noise cancellation algorithm: (a) raw magnetic flux leakage data; (b) output after noise cancellation. 20 (0.8) 40 (1.6) 60 (2.4) 80 (3.0) 100 (3.6) Specimen width, mm (in.) 2 4 6 8 10 12 Scan position (arbitrary unit) (a) (b) 20 (0.8) 40 (1.6) 60 (2.4) 80 (3.0) 100 (3.6) 2 4 6 8 10 12 Scan position (arbitrary unit) Specimen width, mm (in.)
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(10) where B ( k ) denotes the N transformed values and where the coefficients for c ( k ) are given by: (11) and: (12) for 1 k N – 1. To get rid of the low frequency trends, the low frequency coefficients in the transformed signal B ( k ) are set to zero and the signal is reconstructed. This technique results in removing the low frequency trends and the mean without affecting the phase information, crucial in eddy current signals. Figure 5 shows the results of discrete cosine transform based
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