## 4 Discussion
For the high scattering regions in the phantom, it is not as easy to tell the difference in contrast from a visual inspection of the image. Instead, we refer directly to Table 2 to evaluate the performance of the different imaging methods for realizing high scattering regions. Across all imaging methods that were simulated, the CNR of the high scattering regions increases with depth up to a certain point, and then decreases further down. The difference between the average CNR for HAPI-SA compared to traditional SA is 43 dB. Consistent with the previous results, this difference is reduced to 12 dB when the bandwidth is not matched. The CNR of the high scattering regions with coherent plane wave compounding is comparable to the contrast with HAPI-SA without bandwidth matching. With bandwidth matching, the HAPI-SA method has an average CNR that is twice that of coherent plane wave compounding. The trends for CSR were found to follow the same trends as were discussed for CNR. Similarly, we see similar trends with the 2D sparse array for 3D imaging. Bandwidth-matched HAPI-SA imaging produced superior images much less susceptible to noise compared to SA imaging. In addition, the HAPI-SA method takes only one extended transmit to form an image, compared to 128 transmits for SA imaging. This shows the potential of HAPI codes to capture 3D volumes with transmit and receive focusing everywhere at hundreds of volumes per second. From the results it is clear that matching the bandwidth is necessary for achieving maximum performance gains. However, matching the bandwidth results in an increase in code length, which results in an increase in the time to acquire an image. In our case, the code length was increased ten times. Before matching the bandwidth, our code length was 54,415 samples, which translated to a transmit and receive time of 0.54 ms at our sampling frequency of 100 MHz. After matching the bandwidth, the code length is multiplied by 10, and the transmit and receive time is increased to 5.40 ms. This is a significant increase in time, but it is comparable to the amount of time needed to perform 128 transmit events in traditional synthetic aperture. Moreover, our approach achieves a significant SNR advantage, unlike traditional synthetic aperture imaging. For the maximum distance of 45 mm and a speed of sound in soft tissue of 1540 m/s, it would take \(128\approx\frac{2+45\text{ mm}}{1540\text{ m/s}}=7.5\) ms to perform a full synthetic aperture acquisition with 128 elements. Due to the long transmit-receive times of HAPI codes, they are not applicable for fast-moving objects. For example, if we require the imaging target(s) to move less than a tenth of a wavelength over the code duration, this would restrict velocities to below (30 um/5.4 ms = 5.6 mm/s), which may be practical in some static imaging situations, but not applicable to fast-moving tissues. However, the technique may also have applicability to non-destructive testing, underwater acoustics, and air ultrasonics, specifically in cases that do not require detection of fast-moving objects. The codes could additionally be applied to sonar, radar, or communicationssystems, especially multi-user communications systems that rely on the multiple-input multiple-output paradigm. The time needed to perform a synthetic aperture image can be reduced by using only a subset of elements to perform a fast synthetic aperture acquisition. If we were to use 3 sub-apertures of 12 elements each, it would only take 0.18 ms to obtain the partial synthetic transmit-receive aperture data. However, using fewer elements will reduce the quality of the image, as seen in Figure 5c. At the noise level in this simulation, it is impossible to obtain a good image with only 3 sub-apertures of 12 elements. This can further be seen in Table 2, where the SNR and CNR have the worst values for fast SA compared to the other imaging methods. Practical implementation of the codes may require special considerations.