CONTENIDO DE LA PROPUESTA

A matched filter will detect the signal to which it is matched with the optimum signal-to-noise ratio when the noise can be considered white [146], and can be applied to the received signal using cross-correlation. Practically, this is usually performed on discrete data using a FFT [156] due to the considerable speed benefits. In this case, the signal is not a one-dimensional signal, but a two- dimensional parabola in a B-scan, defined by equation 7.1. For a given B-scan, sis fixed,c1 and

c2are assumed to be only able to take the values of the bulk compression or shear waves, and by

varyingx, a parabola filter can be described for a choseny. Using cross-correlation, this parabola

matched filter can simply and quickly be applied to the B-scan, and this will detect parabolas generated by point-like scatterers. However, the form of the parabola will vary with the depth of

(a) 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Offset (cm)

Time Relative to Zero Offset (

µ s) Depth 2.0cm Depth 8.0cm (b) −4 −2 0 2 4 −2.5 0 2.5 5 7.5 Offset (cm)

Time Relative to Zero Offset (

µ

s) Depth 2.0cm Depth 8.0cm

Figure 7.2: Parabola shapes for defects at depths,y, of 2cm and 8cm, are shown for thec1=c2=

3226m/s case (a) which is symmetric, and thec16=c2case (b) withc1=3226m/s andc2=5932m/s

which is asymmetric (the speeds are experimentally measured shear and compression wave speeds respectively, for a mild steel sample). The separation between the emitter and receiver,s, is 8cm.

The zero offset time for each parabola has been subtracted from the parabola to form∆t(x, y)as

defined by equation 7.2. As the offset,x, is varied,∆t(x,8cm)−∆t(x,2cm)is always positive for

(a) as long as−s/2≤x≤s/2, but varies between positive and negative values for (b).

the scatterer,y, as can be seen in figure 7.2. Therefore, the parabola filter can only be considered

matched for the particular depth it was created for. To match perfectly for a large depth range of closely spaced depths, would require application of a large number of matched filters to the B-scan, which would be prohibitively computationally expensive. If a large increment is used for the change in depth between parabola filters, it is possible that the filter will hinder rather than help analysis due to the potentially poor correlation between the signal in the B-scan and any of the matched filters applied. This work describes how to cover the depth range of interest, whilst minimising the number of required matched filters, and keeping the difference between the matched filter and the actual parabola within bounds that retains the effectiveness of the matched filter.

Approximations to the shape of the parabola have been made by other workers [17], but this is not a requirement in this case, as the full description of the parabola can be used. The implement- ation of the parabola matched filter described here relies on the ultrasound signal having a finite and reasonably constant bandwidth, such that the pulse amplitude remains at the same polarity (either positive or negative) for a known minimum time, T, that is consistent along the length

of a given parabola (but not necessarily consistent between separate parabolas). As an example, consider a signal which along the time dimension consists of a single cycle of a sine wave, of period

2T, as in figure 7.3a, which has also had a customised raised cosine window applied. The pulse

has negative amplitude for timeT and then positive amplitude for timeT. For a perfect match,

the parabola matched filter would have the exact same shape as the parabola in the B-scan, but it is sufficient in many cases if the filter parabola and the actual parabola are merely the same sign along the length of the parabola. If the matched filter is an infinitesimally thin line, as long as it does not deviate from the parabola formed by the sine wave by more than T /2 in either the

positive or negative time direction, positive parts of the sine wave are always contributing to the point that the matched filter will form, not the negative parts (or vice versa). This means the filter can stray from the exact shape, in either direction, by a maximum of half the time for which the amplitude does not change sign (from positive to negative or vice versa). The result is that a single parabola acting as a matched filter can act upon a range of parabolas at different depths in the B-scan whilst maintaining effectiveness (figure 7.3b), as the matched filter is sufficiently close to the same shape as the actual parabola that it only includes the part of the parabola that is

(a) 0 1 2 3 4 −0.8 −0.6 −0.4 −0.2 0 0.2 Time (µs) Amplitude (arbitrary) (b) 0 1 2 3 4 26 27 28 29 30 31 Offset (cm)

Time Relative to Zero Offset (

Figure 7.3: Consider an example simulated signal of centre frequency 500kHz (a); the signal is negative for 1μs (the region between the first and second dashed lines) and then positive for

1μs (the region between the second and third dashed lines), and therefore T = 1µs. A single

prototype parabola can be used to filter a range of depths if the signal has a period of this length. Assuming that the received parabolas observed will have a temporal width of 1μs, for the symmetric

c1 = c2 =3226m/s case with separation 8cm, bounds are shown (dashed and dot-dashed lines)

at ±T /2 from the parabolas for scatterers atx= 0cm with depths of 1.6cm and 2.6cm (b). The

prototype parabola (exactly matched for a depth of 2cm) is cross-correlated with the B-scan in both temporal and spatial dimensions, and it can be seen that it would sit within either set of bounds if appropriately shifted in the temporal domain (solid lines). The matched filter would respond to both the positive and negative regions of the signal.

all one polarity (either positive or negative), not a mix of the two. A relatively small number of parabola filters can then cover the entire depth range of interest.

Returning to the example signal of a single cycle of a sine wave, it is still the case that the closer the matched filter to the parabola of interest, the stronger the filter response will be, as if the two have the exact same shape, then all the highest amplitude parts of the sine wave (close to 1.6μs or 2.4μs in figure 7.3a) will contribute to the peak of the signal after matched filtering,

making that peak larger. With cross-correlation, the response seen to a perfect match will be the auto-correlation of the matched filter, but in this case the matched filter is a infinitesimally thin line, whereas the actual parabola must have a finite width in the temporal direction, and this means the auto-correlation of the matched filter is not particularly relevant. In practice, rather than actually being infinitesimally thin, the filter is one sampling period wide, and if the actual parabola were also this wide (if the pulse was temporally very sharp), then the matched filter and the parabola shape would have to match exactly, and the response would nearly be a Dirac delta function in the two-dimensional B-scan. This is a worst case scenario for the technique, asT →0

and the number of required parabola filters tends to infinity. The technique works much better if the signal is closer to a rectangular function, as then the response is the same as long as the filter shape does not move beyond the bounds of the parabola formed from the rectangular function for small deviations from the correct depth.

The filters can be applied for eachc1andc2combination of relevance, and the results combined.