Ultrasonic Lamb modes typically propagate as a combination of multiple dispersive wave packets. Frequency components of each mode distribute widely in time domain due to dispersion and it is very challenging to separate individual modes by traditional signal processing methods. In the present study, a method of dispersion compensation is proposed for the purpose of mode separation. This numerical method compensates, i.e., compresses, the individual dispersive waveforms into temporal pulses, which thereby become nearly un-overlapped in time and frequency and can thus be extracted individually by rectangular time windows. It was further illustrated that the dispersion compensation also provided a method for predicting the plate thickness. Finally, based on reversibility of the numerical compensation method, an artificial dispersion technique was used to restore the original waveform of each mode from the separated compensated pulse. Performances of the compensation separation techniques were evaluated by processing synthetic and experimental signals which consisted of multiple Lamb modes with high dispersion. Individual modes were extracted with good accordance with the original waveforms and theoretical predictions.

Quantitative ultrasound has great potential for assessing human bone quality. Considered as an elastic waveguide, long bone supports propagation of several guided modes, most of which carry useful information, individually, on different aspects of long bone properties. Therefore, precise knowledge of the behavior of each mode, such as velocity, attenuation, and amplitude, is important for bone quality assessment. However, because of the complicated characteristics of the guided waves, including dispersion and mode conversion, the measured signal often contains multiple wave modes, which yields the problem of mode separation. In this paper, some novel signal processing approaches were introduced to solve this problem. First, a crazy-climber algorithm was used to separate time-frequency ridges of individual modes from time-frequency representations (TFR) of multimodal signals. Next, corresponding time domain signals representing individual modes were reconstructed from the TFR ridges. It was found that the separated TFR ridges were in agreement with the theoretical dispersion, and the reconstructed signals were highly representative of the individual guided modes as well. The validations of this study were analyzed by simulated multimodal signals, with or without noise, and by in vitro experiments. Results of this study suggest that the ridge detection and individual reconstruction method are suitable for separating individual modes from multimodal signals. Such a method can improve the analysis of skeletal guided wave signals by providing accurate assessment of mode-specific ultrasonic parameters, such as group velocity, and indicate different bone quality properties.

The 2-D Fourier transform analysis of multichannel signals is a straightforward method to extract the dispersion curves of guided modes. Basically, the time signals recorded at several positions along the waveguide are converted to the wavenumber-frequency space, so that the dispersion curves (i.e., the frequency-dependent wavenumbers) of the guided modes can be extracted by detecting peaks of energy trajectories. In order to improve the dispersion curve extraction of low-amplitude modes propagating in a cortical bone, a multiemitter and multireceiver transducer array has been developed together with an effective singular vector decomposition (SVD)-based signal processing method. However, in practice, the limited number of positions where these signals are recorded results in a much lower resolution in the wavenumber axis than in the frequency axis. This prevents a clear identification of overlapping dispersion curves. In this paper, a sparse SVD (S-SVD) method, which combines the signal-to-noise ratio improvement of the SVD-based approach with the high wavenumber resolution advantage of the sparse optimization, is presented to overcome the above-mentioned limitation. Different penalty constraints, i.e., $l_{1}$ -norm, Frobenius norm, and revised Cauchy norm, are compared with the sparse characteristics. The regularization parameters are investigated with respect to the convergence property and wavenumber resolution. The proposed S-SVD method is investigated using synthetic wideband signals and experimental data obtained from a bone-mimicking phantom and from an ex-vivo human radius. The analysis of the results suggests that the S-SVD method has the potential to significantly enhance the wavenumber resolution and to improve the extraction of the dispersion curves.