## 4 Discussion  
When imaging with a linear array, HAPI synthetic aperture produced a much higher SNR than traditional synthetic aperture or coherent plane wave compounding. Theoretically, there should be a maximum SNR increase of \(20\log_{10}128=42.14\) dB when comparing 128-element traditional synthetic aperture with HAPI synthetic aperture. The simulation results from Table 2 show that there is a difference in SNR between those two imaging methods of 36.4 dB. This is very close to the theoretical maximum. However, this maximum is only achieved when a Kronecker product is applied to the HAPI code in order to match the bandwidth with the transducer. When this adjustment is not applied, there is only an increase in SNR of 12.85 dB. This result shows the importance of matching the bandwidth so that one can obtain the maximum SNR increase theoretically possible. The difference in SNR and the impact of noise between traditional SA and HAPI-SA can be visually seen \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Coherent Plane** \\ **Wave** \\ **Compounding** \\ \end{tabular} & \begin{tabular}{c} **Synthetic** \\ **Aperture** \\ **Aperture** \\ **Aperture** \\ **12 Elements** \\ \end{tabular} & \begin{tabular}{c} **Synthetic** \\ **Synthetic** \\ **Aperture** \\ **Aperture** \\ **Matched** \\ \end{tabular} & \
\begin{tabular}{c} **HAPI** \\ **Synthetic** \\ **Aperture** \\ **Aperture** \\ **Matched** \\ \end{tabular} &
\begin{tabular}{c} **HAPI** \\ **Synthetic** \\ **Aperture** \\ **Aperture** \\ **Transmit Scheme** \\ \end{tabular} \\ \hline
**10 mm** & 0.55 & \(-0.17\) & 0.92 & \(-0.17\) & 0.63 & 0.04 & 0.55 & \(-0.22\) & 1.11 & \(-0.42\) & \(1.11\) & \(-0.42\) \\ \hline
**20 mm** & 0.67 & \(-0.6\) & 0.78 & \(-0.19\) & 0.66 & \(-0.08\) & 0.82 & \(-0.31\) & 1.16 & \(-0.57\) & \(1.16\) & \(-0.57\) \\ \hline
**30 mm** & 0.7 & \(-0.84\) & 0.66 & \(-0.21\) & 0.56 & \(-0.11\) & 0.7 & \(-0.51\) & 0.98 & \(-0.73\) & \(0.98\) & \(-0.73\) \\ \hline
**40 mm** & 0.54 & \(-0.96\) & 0.74 & \(-0.26\) & 0.35 & \(-0.07\) & 0.71 & \(-0.48\) & 0.89 & \(-0.84\) & \(0.89\) & \(-0.84\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 6: The 2D sparse array simulation images of the phantom with 4 hypoechoic cysts and 4 high scattering regions placed at decreasing depths (10 mm, 20 mm, 30 mm, 40 mm) for (**a**) synthetic aperture imaging using 128 transmits, and (**b**) Hadamard Aperiodic Interval synthetic aperture imaging using 1 transmit. Images are 2D slices of the x-z plane of a 3D volume displayed at \(-55\) dB. in the point spread functions (Figure 4). It is clear that there is much more noise present in traditional SA with 128 transmit events (Figure 4a), compared to HAPI-SA with one transmit event (Figure 4b). From Figure 5 above, one can visually see how it is easier to differentiate the cysts from the background in the HAPI-SA method compared to traditional SA. At this noise level, it is difficult to locate the cysts with traditional SA (Figure 5b). When we use HAPI-SA imaging without adjusting for the bandwidth of the transducer (Figure 5d), there is only a very small visual increase in contrast compared to SA. However, when the bandwidth is matched (Figure 5e), the contrast increases to the point where it becomes much easier to locate the first three cysts (at 10, 20, and 30 mm) visually. For coherent plane wave compounding (Figure 5a), the cysts are more visible than traditional SA but not as visible as with the bandwidth-matched HAPI-SA method. These visual results for the cysts are reflected in the CNR calculations in Table 2. HAPI-SA has an average CNR of around 42 dB across the four cyst regions, while traditional SA has an average CNR of \(-4\) dB across the five regions. This makes for a difference of 46 dB. Again, when the bandwidth is not matched, we see a decrease in performance resulting in a difference in average CNR of only 16 dB.