## 2 Materials and Methods
### Numerical Phantoms  
To verify the framework, we generated numerical RF signals with differing attenuation properties, for stimulation of the reference and sample TM phantoms, using a frequency-domain model based on the classical linear diffraction theory of continuous waves [38]. In this study, transmitters and receivers were modeled as linear array transducers consisting of 128 rectangular \(0.2\times 10\) mm elements (azimuthal direction \(\times\) elevation direction), with a 0.2-mm center-to-center separation. Each element in the array generates an individual beam line. Hence, the array's collective field is a superposition of simultaneous excitation from single elements. Dynamic focusing was applied for reflected RF signals in combination with a dynamic aperture, and the F-number was fixed at 2 during the reception process. Each transmit pulse was modeled as a Gaussian function with a center frequency of 5 MHz and a  
Figure 2: Summary of the framework for temporal and spectral analysis of compressed sensing data. The blue dotted line illustrates the traditional approach, while the red solid line depicts the approach devised in this study.  
bandwidth of 80%. In this simulation, a complex wave number is used to calculate parameters related to attenuation and sound speed during wave propagation.  
The numerical phantoms used in this study are shown in Figure 3. All phantoms were 34 mm wide, 80 mm deep, and 10 mm thick, and the interval between simulated beam lines was set to 0.2 mm. We assumed that the face of the transducer was in direct contact with the phantoms, and set both the axial transmit focus and elevation focus to 80 mm. We assumed a uniform sound speed of 1540 m/s through the entire depth of the phantoms, while distributions of 50-\(\upmu\)m polystyrene beads were included to model backscatter. All simulations were performed in 0.1-MHz increments in the 1 to 15 MHz range.  
Figure 3a,b depict the sample phantoms used in the experiments. Each phantom includes a cylindrical region with contrasting backscatter properties to the background medium, generated by adjusting the number density of polystyrene scatterers in the cylindrical inclusion. Hence, in Figure 3a (p12) the backscatter intensity is 12 dB higher in the cylindrical inclusion than it is in the background medium, while it is 12 dB lower in Figure 3b (m12). In contrast to the backscatter property, the attenuation coefficients of the inclusion and the background were same and set to 5 dB/cm/MHz in both phantoms.  
The reference phantoms used for diffraction compensation (DC) are shown in Figure 3c,d. With these phantoms, the backscatter and attenuation properties in the entire medium are uniform. The scatterer number density was set as 9.7/mm\({}^{3}\) for each phantom, while the attenuation coefficient was 0.3 dB/cm/MHz for Figure 3c, and 0.7 dB/cm/MHz for Figure 3d.  
Figure 3: B-mode images of tissue-mimicking phantoms (TM) phantoms used in simulation (with region of interest (ROI) highlighted in red). (**a**) p12 sample phantom. Backscatterer intensity in cylindrical inclusion is 12 dB higher than the background. (**b**) m12 sample phantom. Backscatterer intensity in cylindrical inclusion is 12 dB lower than the background. (**c**) Reference phantom with uniform backscatterer intensity and attenuation coefficient of 0.3 dB/cm/MHz. (**d**) Reference phantom with uniform backscatterer intensity and attenuation coefficient of 0.7 dB/cm/MHz.