Estructura en relación con la actividad

For any one channel of detection, individual points in the measurement convey neither displacement nor velocity. To determine these values one must calculate either the frequency of oscillations (speed) or the total number of oscillations (displacement). The short time Fourier Transform (STFT) is most commonly used to determine the velocity of a flyer over a period of time from the interferogram.14 If the flyer direction needs to be ascertained then the complex STFT must be used instead. When the highest simultaneous time and frequency resolution is required then the frequency of peaks in the raw interferogram is best used to determine the flyers speed because it is not limited by the Fourier uncertainty limit. Last, if one wishes to directly determine the displacement of a flyer from an interferogram then phase analysis between quadrature channels can be used. Codes used to produce velocity histories using each of these methods can be found in appendix A.

In all cases, normalization and base line subtraction from the raw signals is used to eliminate zero frequencies from the Fourier transform following Equation 3.8.10

( ) = ( ) ( ) (3.8)

Where ( ) is the corrected interferrogram, is the raw signal, ( ) is a linear correction to the baseline and N designates each channel. In practice we begin this fit from the first detected oscillation through the end of data collection. Otherwise the initial baseline has too much weight in the fitting routine. is a channel dependent normalization that corrects for

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detector sensitivities and system losses along the different detector paths. The normalization constants for our system were determined from a long region of constant interference intensity.

Short Time Fourier Transform

If one is interested in determining the speed of a flyer plate the most robust method is to use a short time Fourier transform (STFT). This produces a power spectrum ( ) for each time point within the window duration t. Because the Fourier transform is discrete this also results in a level of uncertainty Δf that is proportional to the window size by ≥ .15 Following from the work by Dolan we use a Hamming window because our signal to noise is typically better than 10:1.14

A sample spectrogram is shown in Figure 3.3(a) with 0.08 ns time steps and a 10 ns Hamming window function used for the Fourier transform. This image is quite data intensive because at each time point a full Fourier transform has been produced. One should note that because a Fourier transform is used there will be peaks in intensity at every integer multiple of the flyer’s speed. These are harmonics and can be easily discerned because the true flyer speed will not have further harmonic traces beneath it at ½ the apparent flyer velocity. Fitting the maximum at each time step using a parabola and taking the maximum one produces the line out shown in Figure 3.3(b). Because the line out only tracks the maximum velocity at each time point these require much less computational power to render, and it is possible to plot more than one on top of another.

One can change the STFT window size for finer determination of the flyer’s speed, but this will come at the cost of time resolution. Shown in Figure 3.4 are lineouts using 2, 5 and 10 ns window durations. Focusing on impact it is clear that the velocity is much smoother for the larger window durations. Most notable is the reduction in oscillations after impact which aclear in the 2 ns line out. To determine the speed of an object accurately, a very long window should be used, but if one wants to measure short duration motions, such as the sustained shock, from a line out then a short window must be used for the highest precision. This means that data must often be processed multiple times in order to analyze different features.

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Complex STFT

Complex STFT (cSTFT) analysis of interferograms is carried out in a similar manner as for the standard STFT, except that the signal used must contain both real and complex components. To create the complex interferogram we use the 120° phase offset in our dual detection system to produce a pair of quadrature, or 90° out of phase signals. Through Euler’s formula these are then create one vector of complex interferometric data on which the cSTFT is run.

Dolan’s publications and Scandia report adequately cover the mathematical edifice for producing quadrature signals from triature signals, so only the solution is presented here in equations 3.9 and 3.10.10

( ) = sin(120°) ∗ ( ) (3.9) ( ) = cos(120°) ∗ ( ) − ( ) (3.10)

Where ( ) is one quadrature signal and ( ) is the other quadrature signal. This transformation is exhibited in Figure 3.5. In Fig. 3.5 (a) the corrected signals are 120° out of phase and in Fig. 3.5 (b) they are now 90° out of phase. This change is most easily seen by looking at the peak position of one signal as the other crosses the zero axis. In quadrature detection one signal should peak when the other is at zero.

The quadrature signals are then used to produce a single complex interferogram through Euler’s formula.

= cos( ) − ( ) = ( ) − ( ) (3.11)

From this the flyer direction can be ascertained via the sign of the phase velocity, or using the cSTFT this will be apparent through positive or negative frequency components. Looking closely at Fig. 3.5(b) one will notice that the frequency of oscillations slow down and then speed up again. From only STFT analysis one does not know if this is an object that slows down and then goes backward, or if it begins moving forward again. The complex interferogram in three dimensions is shown in Fig. 3.6. From this one can see that the phasor spins counter clockwise

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as the flyer is moving forward, stops, and then begins spinning clockwise. This indicates that the flyer has changed direction.8

In Figure 3.7 the same data as Figure 3.3 is analyzed again using the cSTFT method. One will notice that the spectrogram now contains both positive and negative velocity components. Fitting the peak of the Fourier transform at each time step again produces a line out (Fig. 3.7(b)). In this case the direction of the flyer is determined inherently by the program and no user assumptions are necessary. For standard flyer experiments this type of analysis is unnecessary because the user can easily deduce which direction an object is moving. But when a flyer spalls in flight and multiple pieces impact one after another this gives crucial information by showing multiple crossovers from positive to negative velocities as each piece hits spaced in time.

The downside to this analysis compared to the STFT method is that better signal to noise is necessary to work up data. In poorer quality data the program will often jump from forward to negative motions sporadically because of subtle changes in the relative phase of the two signals due to noise.

Phase Analysis

Another way to analyze the data directly is through what is called phase analysis. This technique has been thoroughly discussed by Dolan et. al.10 Starting with the quadrature signals

( ) and ( ) one can determine the complex phasors phase through equation 3.12.

( ) = _{( )}( ) (3.12)

This in turn is related to the physical position of the flyer through equation 3.13

( ) = ( ) + [ ( ) ( )] (3.13)

where ti denotes the initial time of motion and λ0 is the laser wavelength. Through differentiation this leads the velocity of the flyer in time. We follow the suggestion of Dolan and

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use a Savitzky Golay filter to calculate the derivative.16 This in essence applies a smoothing filter to the raw data and derivative while performing the differentiation.

A flyer history using this technique is shown in Figure 3.8. While this method does inherently determine the flyer plate direction of motion, it is very sensitive to signal to noise and the relative phase of the two signals. Based on Dolan’s work it is possible that this would perform better with a third channel of detection, but in our system the cSTFT is superior in every case tested.

Peak Finding

For the most accurate simultaneous measurement of flyer speed and duration of motion we use a peak finding algorithm. This program determines the frequency of interferogram oscillations through the peak-to-valley time and has higher simultaneous resolution compared to the STFT analysis because it avoids performing a Fourier transform. Half an oscillation occurs every time that the flyer moves 0.388 μm and these periods determine the time resolution of the velocity history. So it is important to keep in mind that this process does not have a consistent time resolution throughout, and velocity resolution will depend on the signal to noise of the raw interferogram.

The program, which can be found in appendix A, functions by first running a function to determine the rough position of each peak and valley. It then uses a polynomial fit about each peak to determine their exact position. These peak times are converted into a frequency and subsequently a flyer speed. A sample flyer history is shown in Figure 3.9. As shown, this analysis method can determine the flyer speed from launch through impact just like the other methods. One advantage it has over the STFT method is that it handles lower speeds better. Shown in Figure 3.10 is a comparison of the same data set analyzed by STFT using a 2 ns window and peak finding. The STFT works well where there is a high frequency signal that contains at least one full oscillation per window size. But at the low velocities (<0.5 km s-1), where there is only a partial oscillation per analysis window, the STFT suffers and had additional noise. Because the peak finding routine has variable time resolution based on the period of the fringes, this does not have any problem here.

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The one downside to the peak finding routine is that it requires at least 5:1 signal to noise to operate well. But when this is achieved it is the most robust algorithm for processing data.

3.4 Concluding Remarks 

In this chapter the fundamentals of PDV analysis have been discussed. The reader should now understand why window corrections are necessary when measuring and impact through a transparent material and the advantages of 3x3 mixing for phase shifted signals. Furthermore, a series of analysis techniques have been discussed. The take home message is that when there is good signal to noise (at least 5:1) then peak finding should be used to process the data. But if the flyer velocity is the only measurement needed from the PDV data then a STFT routine will work adequately.

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