|NDT.net Apr 2006 Vol. 11 No.4|
Using Phase Slope for Arrival Time DeterminationBerndt H. (email@example.com)
scEye Research & Development, Oakland, CA, USA
Johnson G.C. (firstname.lastname@example.org), Schniewind A.P.(email@example.com)
University of California, Berkeley, CA, USA
14th International Symposium on Nondestructive Testing of Wood
AbstractPhase slope is routinely used as a means of sound velocity determination in other branches of ultrasonic nondestructive testing. The advantages are the very high time resolution obtainable and the elimination of amplitude effects. In Wood NDT, phase information is still underutilized, possibly due to the complex changes ultrasonic signals undergo on traversal of wood specimens. This paper will demonstrate how phase unwrapping of Short Time Fourier Transforms (STFT) of ultrasonic signals can be used to precisely determine arrival times, independent of attenuation and interference effects. The algorithm used will be explained and its validity demonstrated through various experiments. The combination of STFT and phase unwrapping has the potential of adding other useful measurables to the toolbox of Wood NDT practitioners, and this paper will touch on some of those.
IntroductionIn many applications of nondestructive materials evaluation, precise determination of the time location of a signal is essential . In most cases, one will need to know the location relative to some other signal. In ultrasonic nondestructive testing of wood, this time difference is usually found by some analysis of the time-domain signals directly. Threshold crossing, also known as “first break” determination, is often the method of choice. Energy peak location has also been used. Each of these methods is prone to errors due to changes in the shape of the pulses as they traverse a wood sample. Other methods for determination of relative time locations make use of frequency domain information as well. Deconvolution by spectral division and complex cepstral analysis are two examples. The topic of this paper is the use of phase information for finding time shifts. The procedure is well established for applications with other materials . We will show in the following that phase slope information can successfully be utilized in wood nondestructive testing as well.
Experimental ProcedureThe experimental data for this study were obtained from waterlogged Southern pine samples, ultrasonically scanned in a water tank. The measurements were carried out with 5 MHz, focused broadband transducers connected to a pulser-receiver. The signals were captured with a 12-bit, 20 MSPS oscilloscope board plugged into a micro computer. The material, apparatus, and techniques have been described in detail in previous publications [1, 2, 3].
Figure 1 shows several views of the water tank, the scanning apparatus, and the transducer positioning guides.
Measurements reported here are based on a series of scans of a sample cut for transmission in the longitudinal direction. The original sample dimensions were 100 mm by 70 mm by 49.6 mm. The latter dimension is the transmission path length. We recorded the signals transmitted through a 10 mm by 10 mm area in the center of the sample, scanned in 0.2 mm steps – 0.1 mm for the thickest and thinnest sample stages. Reference signals were recorded with identical transducer positions but without the wood sample in the sound path. This procedure was repeated four times, after reducing the thickness of the sample each time, by cutting away a slice from the face nearest the receiving transducer.
Signal ProcessingFor signal processing purposes, one can regard a wood sample as a “black box” which converts the input signal into an output signal. In the vast majority of cases, it is safe to assume that wood acts as a linear, time-invariant (LTI) system  with respect to ultrasonic pulses. The changes imparted on the signal on traversing the wood sample can then be expressed by a transfer function, h(t), which when convolved with the input, x(t), yields the output signal, y(t):
For the purposes of this discussion, consider the overall system of the wood sample under investigation as being composed of simpler subsystems. If the original pulse is split into several copies of the original upon transmission, there is a subsystems for each pulse image, and they all transmit in parallel. Each of these parallel subsystems can be considered as a cascade of a delay filter, d(t), and an attenuation filter, a(t). The attenuation filter is chosen to be time-neutral, so that if it were applied to the input signal by itself, no time shift would occur. If the sound speed is independent of frequency, the delay filter is simply a shifted unit impulse, d(t) = d(t - t0) , so that for each received pulse image, the transfer function looks like
Applying the Fourier transform to the output signal, , and using the convolution and time shift properties, this becomes
where the right hand side of the second equality uses the polar representation of complex numbers. One sees that the time shift in the signal transforms to a phase shift in the Fourier domain which is linear in frequency. This allows us to determine the time shift between two pulses by comparing the phase curves of their respective Fourier transforms. Given two received pulses, s1 with time delay t1 , and s2 with time delay t2 , and their respective phase curves , the difference in their time shifts can be found from
Therefore, if the sound speed is independent of frequency, the difference in pulse arrival times can be found by determining the slope of the (linear) difference between their phase curves. A difficulty, both practical and theoretical, arises from the fact that the phase is known only to within Modulo 2!. There is no “correct” way to recover the continuous phase, every method requiring some assumptions. There is, however a wealth of successful methods available in the literature, and most signal processing toolboxes provide some built-in algorithm for it. The calculations in this study use a simple method from the documentation of a Mathematica application .
The theory presented above applies only to complete signals containing a single image of an ultrasonic pulse. If several copies of the outgoing pulse are mixed, the phase information will also be mixed and the interpretation of phase slope becomes difficult. If the multiple pulse images are well separated, simple windowing of the signal will allow the above theory to be applied . If the separation of pulse images is poor, interpretation of the results has to proceed with caution. In particular, the phase slope must be constant, or nearly so, for a range of window positions in order for us to have confidence in the pulse separation.
ResultsAs a first test of the proposed method for determining relative arrival times of ultrasonic pulses, we conducted an experiment to verify the linearity of our measurement apparatus and to determine the sound speed in water for our transmission line. We collected the transmitted signals for ten different transducer separations, with ten replications in each position. Using one signal from the set taken at the smallest separation, and hence the shorted transmission time, we determined the phase curve differences to all the other signals, and obtained the slopes of these difference curves via linear regression. We then plotted the relative transducer separations against the time shifts and again performed a linear regression to find the slope of this curve, which is the sound speed in our measurement tank.
Figure 2 shows the results of this experiment. The slope of the best fit line is 1.475 mm/_s (1475 m/s), with a standard error of 0.001 and an R2 of 0.9999. This result inspires confidence in the measurement apparatus and the signal processing method, and provides us with a water sound speed reference which will be needed for further analyses. Short time Fourier transforms (STFT) are a simple form of time-frequency analysis [4, 6]. A signal is analyzed by multiplying it with an appropriate window sequence, thereby focusing on a section of the whole signal. Windowing introduces artifacts in the analysis due to edge effects. The simple “box car” rectangular window for example – a sequence of ones – introduces high frequency artifacts in the amplitudes of the Fourier transform. Other commonly used windows, like Hamming and Hanning, were designed with the goal to reduce these side nodes. Numerical experimentation showed that for the purposes of phase slope determinations, the choice of window class had no significant effect, while the window width and taper were important factors. In the analyses reported here, we used various “super-Gaussian” windows [Weaver], defined as
For the time shift analysis presented later, we generated window sequences with the parameters A = 2 and N = 3, sampled at intervals of 0.2 t. A window constructed in this way will, in the correct position, select the whole water-transmitted pulse with only minor distortions. Figure 3 shows a scaled version of the window sequence and its effect on the sample.
To make sure the windowed signal still has the same phase properties as the raw signal, we took one of the water-transmitted signals, swept the window through the whole length of it, and determined the slopes of the unwrapped phase curves for each window position.
Figure 3 shows the results of this procedure. The phase slope, indicated by dots plotted at the sample point corresponding to the center of the respective window, increases as expected with advancing window position. There is, however, a plateau of constant or nearly constant phase slope for a wide range of overlap between the window (long and short dashed line) and the ultrasonic pulse (solid line). A fairly small portion of the pulse’s energy (gray filled area) already contains enough of the phase information to give the phase curve of the windowed signal a slope equal to the phase slope of the whole signal (dashed horizontal line).
For the thinnest sample stage (4.1 mm) used here, the transmitted signal shows three clearly separated images of the outgoing pulse, as can be seen in Figure 4. The three pulse images are due to the longitudinal p-wave, a shear wave, and the second arrival (first reflection) of the pwave, respectively, as discussed previously [1, 3]. The time shifts for the first and third pulse images will be used in the discussion later on.
Using the same approach as for the water-transmitted signal, we swept the window through the length of the signal and determined the phase difference slopes for each window location. The resulting phase slopes are plotted at the respective window center location in Figure 4. It can be seen that each of the three pulse images is associated with a run of nearly equal phase difference slopes. For signals at this thickness stage, the energy content of the windowed signal, when plotted against the window position, shows distinct local peaks for the three pulse images discernible to the eye. We can safely use the phase slope determined for windows positioned at these local maxima for time shift determinations.
Applying the same analysis to the second-thinnest (14.1 mm) sample stage reveals a more complex situation (Figure 5). The first p-wave arrival appears to be made up of two separate, slightly shifted copies of the pulse. This splitting is indicated by two discernible plateaus in the phase difference slopes. The amplitude spectrum of a windowed version of this arrival also shows the indentations typical for interference between two closely spaced pulses. As the sample thickness increases, this smearing-out of the first arrival continues. We may assume that it is due to forward-scattering, mode conversion, or other mechanisms that cause the signal energy to travel by an increasing number of different sound paths.
Since the close analysis and interpretation of these scattering phenomena are beyond the scope of this study, we concentrated our investigation on the earliest arrival. We chose the window position for the phase difference slope determinations as the “first break” of the signal, using a threshold well above the noise level. Used this way, phase slope becomes a means of compensating for the problems inherent in the “first break” arrival time determination. To demonstrate the effect of combining these two approaches, we compared the time shift determined from the “first break” method alone to the phase slope-corrected time shift.
Figure 6 shows the superpositions of an image of the reference pulse, shifted by the amount determined with the respective method, and the wood-transmitted pulse for the case of greatest difference between the two methods. This signal was collected at the original sample thickness of 49.6 mm.
It appears that in this example, the “first break” arrival time is a bit underestimated. The shape of the reference pulse does not match up with the waveform at that point in time, while the time shift determined using the phase slope correction lines up very nicely with significant signal features. This interpretation is, however, strictly subjective. More in-depth studies would be required to show conclusively that using phase slope corrections leads to real improvements in arrival time determination. For this study, we would like to demonstrate linearity and selfconsistency for the phase difference slope method of time shift determination.
Figure 7 shows the time shifts obtained from phase difference slopes plotted against the sample thickness. Linear regression of these data yielded the solid line superimposed on the sample points. The slope of that line is Dt/h , where h is the thickness of the sample and Dt = t wood - twater. We obtain the sound speed in the wood sample by the following argument: To simplify the notation, we define the slowness as the inverse of the sound speed, i.e., s =1/dv. The signal needs h · swater to traverse the thickness of the sample. In water, the signal needs water h · s wood to traverse the same distance. Hence, Dt = h(swood - swater), and we can calculate the sound speed in wood related to a particular phase difference slope as the inverse of
swood = swater + (Dt/h)
The value of the slope determined by linear regression is -0.499342, with a standard error of 0.0002 and an R2 of 0.9999. The sound speed in wood calculated from this value of the slope is 5.579 mm/_s (5579 m/s). The very good linearity of the time shift with sample thickness indicates that we are truly measuring a phase velocity. The linearity and the high reproducibility of the phase slope difference method for finding the time shift – very little scatter for sets of 100 independent measurements – indicate that this approach for arrival time determination works rather well.
Having established that the method works in principle, consider the arrival of the first echo in the thin sample shown in Figure 4 as a test of self-consistency. For the first echo of the fast mode, the corresponding slope would be echo wood water Dt/h = 3swood - swater , since the pulse has to traverse the sample thickness three times. Using this relation, we calculated the expected slope for the first echo arrivals of the p-wave and plotted it in Figure 6 (dashed line) together with the time shifts obtained from the phase delay method applied to the thinnest sample (see Figure 4). The match is quite good, with predicted arrival being a little earlier than measured arrivals.
ConclusionsWe demonstrated the validity of using phase slope for time shift determinations on a set of signals transmitted through water. We showed that applying windows to the sample maintains the phase information for reasonably well separated pulses. For the type of wood sample and ultrasonic signals used here, the phase slope method can be used successfully for high-resolution time shift measurements on thin wood samples – of the order of 5 to 10 wavelengths thickness. At greater path lengths, ultrasonic samples transmitted through wood become smeared out in time, with the phase slopes of windowed pulses indicating the presence of multiple pulse images closely spaced. Other STFT analyses confirm the presence of multiple copies of the outgoing pulse. Various time-frequency methods, like wavelet transforms and STFT, can be applied in the analysis of these pulse multiples. Phase slope determinations can add a useful tool to that set of methods. Separating these pulse multiples created by forward scattering may provide useful new parameters for ultrasonic property evaluation of wood.
We showed that when combined with “first break”, threshold crossing arrival time determinations, phase slope can be used to correct for errors introduced by attenuation and interference effects. The high time resolution of the phase slope method is a significant asset in this application. We used this approach to obtain a precise estimate of the phase velocity of pwaves in the longitudinal direction of waterlogged Southern pine.