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FIG. 12. AF function of the echo of PVC tube with an increasing synthetic white noise. In Secs. II and II, we explained formation of character- istic spectral patterns from interferences between primary

Figure 12 AF function of the echo of PVC tube with an increasing synthetic white noise. In Secs. II and II, we explained formation of character- istic spectral patterns from interferences between primary

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TABLE I. Parameters used to compute the theoretical form functions of three cylindrical shells: a PVC tube, a steel pipe, and a thin walled aluminum flas! (aluminum bottle). Note that all acoustical parameters can be found in Ref. 21. lator can provide a good alternative solution to analytic computation, with excellent agreement between theoretical and numerical models. and b (outer and inner diameters, respectively), the density, and the acoustic properties (c, and c,) of the cylinder mate- rial. As a result the form function f,, is a function of the frequency, the dimensions of the cylinder, and its acoustic properties. To be more precise, the form function has the dependencies f..(ka,k,,kp,a,b), where k=27f/Cyaterr Kz =2af/c), and kp=27f/c,. So for a given cylinder where di- mensions and material are fixed, f,, is reduced to a function of the frequency alone. It is this dependency we will study in Secs. IN-V.
TABLE I. Parameters used to compute the theoretical form functions of three cylindrical shells: a PVC tube, a steel pipe, and a thin walled aluminum flas! (aluminum bottle). Note that all acoustical parameters can be found in Ref. 21. lator can provide a good alternative solution to analytic computation, with excellent agreement between theoretical and numerical models. and b (outer and inner diameters, respectively), the density, and the acoustic properties (c, and c,) of the cylinder mate- rial. As a result the form function f,, is a function of the frequency, the dimensions of the cylinder, and its acoustic properties. To be more precise, the form function has the dependencies f..(ka,k,,kp,a,b), where k=27f/Cyaterr Kz =2af/c), and kp=27f/c,. So for a given cylinder where di- mensions and material are fixed, f,, is reduced to a function of the frequency alone. It is this dependency we will study in Secs. IN-V.
FIG. 1. (Color online) Theoretical form functions |f,,| for the PVC tube and the steel pipe.
FIG. 1. (Color online) Theoretical form functions |f,,| for the PVC tube and the steel pipe.
FIG. 2. (Color online) Dispersion plot of the phase and group velocities associated to the first Lamb waves Ay and Sy in a steel plate with 13 mm thickness.
FIG. 2. (Color online) Dispersion plot of the phase and group velocities associated to the first Lamb waves Ay and Sy in a steel plate with 13 mm thickness.
FIG. 4. (Color online) FDTD simulation showing the normalized sound pressure (in dB) for the echo of a PVC tube insonified with a linear chirp. Figure 4 provides a snapshot of the backpropagation of the simulated acoustic wave 1.1 ms after its interaction with the PVC tube, which is situated to the right of the image. The transmit signal is a linear chirp traversing 80-130 kHz over
FIG. 4. (Color online) FDTD simulation showing the normalized sound pressure (in dB) for the echo of a PVC tube insonified with a linear chirp. Figure 4 provides a snapshot of the backpropagation of the simulated acoustic wave 1.1 ms after its interaction with the PVC tube, which is situated to the right of the image. The transmit signal is a linear chirp traversing 80-130 kHz over
FIG. 3. Echo formation for shell cylinder following the GTD. Display of the geometrical path taken by the specular echo and the secondary echo.
FIG. 3. Echo formation for shell cylinder following the GTD. Display of the geometrical path taken by the specular echo and the secondary echo.
FIG. 5. (Color online) Matching between the analytical and numerical power spectra of the pressure field computed at the center of the PVC cylindrical shell.
FIG. 5. (Color online) Matching between the analytical and numerical power spectra of the pressure field computed at the center of the PVC cylindrical shell.
FIG. 6. Display of the normalized sound pressure expressed in dB. Visual- ization of the interaction between the incoming acoustic wave and the PVC shelled cylinder and the echo formation inside the target. In the lower an- notated diagram the position of the cylinder is indicated by the bold circle.
FIG. 6. Display of the normalized sound pressure expressed in dB. Visual- ization of the interaction between the incoming acoustic wave and the PVC shelled cylinder and the echo formation inside the target. In the lower an- notated diagram the position of the cylinder is indicated by the bold circle.
FIG. 7. Reflection by a concave spherical mirror, construction of the source image A’ from the source A in the axis CS. The values for CA, and CA, obtained using the acoustic response were given previously and were —50 and +45 mm, respectively. Using the optical equation Eq. (6) the source
FIG. 7. Reflection by a concave spherical mirror, construction of the source image A’ from the source A in the axis CS. The values for CA, and CA, obtained using the acoustic response were given previously and were —50 and +45 mm, respectively. Using the optical equation Eq. (6) the source
chirp components are well separated and the frequency dis- tribution is clearly bimodal, designated as type C.
chirp components are well separated and the frequency dis- tribution is clearly bimodal, designated as type C.
FIG. 8. (Color online) Display of two of the biomimetic pulses DC1 and DC6: (a) time signal of the DC1 pulse; (b) power spectrum of the DC1 pulse; ( time signal of the DC6 pulse; and (d) power spectrum of the DC6 pulse. Comparing the PVC tube returns for both transmit sig- nals, similar gross echo structures are seen. A first main echo We are especially interested in the detailed spectral echo structures elicited using wideband pulses for discrimination purposes. The targets of interest here are the open-ended cyl- inders used for building the analytic and numerical models in Secs. II and III. As previously noted, these objects have simi- lar target strengths (e.g., 1.6 dB difference at 100 kHz) and
FIG. 8. (Color online) Display of two of the biomimetic pulses DC1 and DC6: (a) time signal of the DC1 pulse; (b) power spectrum of the DC1 pulse; ( time signal of the DC6 pulse; and (d) power spectrum of the DC6 pulse. Comparing the PVC tube returns for both transmit sig- nals, similar gross echo structures are seen. A first main echo We are especially interested in the detailed spectral echo structures elicited using wideband pulses for discrimination purposes. The targets of interest here are the open-ended cyl- inders used for building the analytic and numerical models in Secs. II and III. As previously noted, these objects have simi- lar target strengths (e.g., 1.6 dB difference at 100 kHz) and
FIG. 9. (Color online) Spectrograms of PVC tube and steel pipe target echoes computed with a 100 ys sliding Gaussian window for pulses DC1 and DC6.
FIG. 9. (Color online) Spectrograms of PVC tube and steel pipe target echoes computed with a 100 ys sliding Gaussian window for pulses DC1 and DC6.
FIG. 10. (Color online) Matching between the theoretical and empirical DBS echoes power spectra for (a) steel pipe and (b) PVC tube. The high empirical response below 80 kHz due mainly to reverberation from the tank floor.
FIG. 10. (Color online) Matching between the theoretical and empirical DBS echoes power spectra for (a) steel pipe and (b) PVC tube. The high empirical response below 80 kHz due mainly to reverberation from the tank floor.
TABLE III. Tolerances permitted in physical parameters of the PVC tube in order to keep the positions of resonance peaks and notches within +1 kHz band. be achieved in parameter estimation given certain prior as- sumptions, such as possible target shape, size, and material properties. Under the assumption of a cylindrical target, we can begin to extract size information from the wideband spectral response by relating the notch spacing Af in the form function |f,,| to the time spacing Ar between the two main echoes through the relation AfAt=1.°° In the case of the PVC tube we find a value of Af=4.5 kHz. The corre- sponding time spacing Afpyc is 222 ys, equating to a cylin- der diameter (cf. Eq. (3)) of 166 mm. These results are in line with expectations from the timing information given in Sec. IVB 1.
TABLE III. Tolerances permitted in physical parameters of the PVC tube in order to keep the positions of resonance peaks and notches within +1 kHz band. be achieved in parameter estimation given certain prior as- sumptions, such as possible target shape, size, and material properties. Under the assumption of a cylindrical target, we can begin to extract size information from the wideband spectral response by relating the notch spacing Af in the form function |f,,| to the time spacing Ar between the two main echoes through the relation AfAt=1.°° In the case of the PVC tube we find a value of Af=4.5 kHz. The corre- sponding time spacing Afpyc is 222 ys, equating to a cylin- der diameter (cf. Eq. (3)) of 166 mm. These results are in line with expectations from the timing information given in Sec. IVB 1.
FIG. 11. (Color online) Matching between the theoretical and empirical DBS echoes power spectra for the aluminum bottle with (a) DC1 pulse and (b) DC6 pulse. The empirical spectrum has been computed using a single echo of the bottle.
FIG. 11. (Color online) Matching between the theoretical and empirical DBS echoes power spectra for the aluminum bottle with (a) DC1 pulse and (b) DC6 pulse. The empirical spectrum has been computed using a single echo of the bottle.
TABLE IV. (Top) Confusion matrix relative to the DC1 pulse echoes and (bottom) confusion matrix relative to the DC6 pulse echoes. These matrices have been computed with 40 measurements per target. Let F be the spectrum of the broadband echo. Let (Witicfi.y] be the location in frequency of the notches of F. To F we associate AF such that corresponds to the most significant frequencies of the DBS. The 2 kHz sliding window just ensures that the notches do not overlap in the computation of the AF function. From the location of the notches, the AF function is computed using Eq. (7) for each echo spectrum. In order to classify the tar- gets one reference AF’. is needed for each target. For the cone, tilted cone, rocket head, brick (length), and brick (width) target a single broadside echo has been recorded as the reference. The reference functions AF,.. for the PVC tube and steel pipe are computed from synthetic echoes. Thanks to the models proposed in Sec. II C, the timing between the specular echo and the secondary echoes can be estimated with excellent precision. Synthetic echoes for these targets are built simply using the timing and the relative phase shift between the primary and the secondary echoes. Thus, the PVC tube and steel pipe AF,,, functions come solely from the notch locations given by analytical models.
TABLE IV. (Top) Confusion matrix relative to the DC1 pulse echoes and (bottom) confusion matrix relative to the DC6 pulse echoes. These matrices have been computed with 40 measurements per target. Let F be the spectrum of the broadband echo. Let (Witicfi.y] be the location in frequency of the notches of F. To F we associate AF such that corresponds to the most significant frequencies of the DBS. The 2 kHz sliding window just ensures that the notches do not overlap in the computation of the AF function. From the location of the notches, the AF function is computed using Eq. (7) for each echo spectrum. In order to classify the tar- gets one reference AF’. is needed for each target. For the cone, tilted cone, rocket head, brick (length), and brick (width) target a single broadside echo has been recorded as the reference. The reference functions AF,.. for the PVC tube and steel pipe are computed from synthetic echoes. Thanks to the models proposed in Sec. II C, the timing between the specular echo and the secondary echoes can be estimated with excellent precision. Synthetic echoes for these targets are built simply using the timing and the relative phase shift between the primary and the secondary echoes. Thus, the PVC tube and steel pipe AF,,, functions come solely from the notch locations given by analytical models.
TABLE V. Confusion matrix for the fusion system of the two classifiers relative to the two pulses. same target and the change in look angle is small (8°), so the resonances are expected to be similar. Note that the cone is classified correctly (independently of orientation) at 82.3% and 83.8% for DC1 and DC6, respectively. For the brick, the change in orientation is 90°, the scattering points are in a totally different configuration and the locations of the notches are significantly shifted. So there is no confusion between these two configurations. and secondary target echoes. The null estimator exploits these interferences for discrimination. Since both primary and secondary echoes are required, the strength of the sec- ondaries is crucial in determining performance of the classi- fier in noise. Defining SNR as the ratio of the peak of the secondary echoes to the noise level, Fig. 12 displays the AF function for the PVC tube insonified by pulse DC] against increasing levels of additive Gaussian white noise. With SNR above 15 dB, the notches are well localized across the band. From 15 to —5 dB, the influence of the noise is appar- ent, but the frequency band of 80-110 kHz, corresponding to the highest sensitivity of the DBS, remains stable and can be used for classification. At still lower SNR, with noise levels in excess of the peak secondary echoes, the target resonances are masked and the classifier will fail.
TABLE V. Confusion matrix for the fusion system of the two classifiers relative to the two pulses. same target and the change in look angle is small (8°), so the resonances are expected to be similar. Note that the cone is classified correctly (independently of orientation) at 82.3% and 83.8% for DC1 and DC6, respectively. For the brick, the change in orientation is 90°, the scattering points are in a totally different configuration and the locations of the notches are significantly shifted. So there is no confusion between these two configurations. and secondary target echoes. The null estimator exploits these interferences for discrimination. Since both primary and secondary echoes are required, the strength of the sec- ondaries is crucial in determining performance of the classi- fier in noise. Defining SNR as the ratio of the peak of the secondary echoes to the noise level, Fig. 12 displays the AF function for the PVC tube insonified by pulse DC] against increasing levels of additive Gaussian white noise. With SNR above 15 dB, the notches are well localized across the band. From 15 to —5 dB, the influence of the noise is appar- ent, but the frequency band of 80-110 kHz, corresponding to the highest sensitivity of the DBS, remains stable and can be used for classification. At still lower SNR, with noise levels in excess of the peak secondary echoes, the target resonances are masked and the classifier will fail.
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