## 3 Results  
### Linear Array Simulation Results  
Figure 4 below compares the point spread functions (PSF) of traditional synthetic aperture (Figure 4a) and bandwidth-matched HAPI synthetic aperture (Figure 4b). It is clear that HAPI-SA produces a less noisy PSF. In fact, the PSF of traditional SA is so noisy at the noise level in this simulation that it is almost impossible to differentiate the single-point scatterer from the noise. The PSF of the SA method appears small because the single-point scatterer is obfuscated by the high amount of noise present in the simulation. The HAPI  
synthetic aperture PSF is similar in shape and resolution to a noiseless synthetic aperture PSF (data not shown) but exhibits more clutter due to correlation sidelobes. Figure 5 below compares the simulated images of the scattering phantom for all the imaging methods: (a) coherent plane wave compounding (b) synthetic aperture imaging, (c) fast synthetic aperture imaging, (d) HAPI synthetic aperture imaging without bandwidth matching, (e) HAPI synthetic aperture imaging with bandwidth matching, and (f) HAPI synthetic aperture with the alternating scheme. We can visually see that HAPI-SA imaging with bandwidth matching (Figure 5e) produces the best image in the context of noise. When the bandwidth is not matched (Figure 5d), the image quality suffers. We can also see how the alternating scheme (Figure 5f) of transmitting on every other element, and receiving on the rest, produces an image that is better than traditional synthetic aperture but worse than the full HAPI-SA imaging method that transmits and receives on all elements. This is because only half the synthetic transmit-receive aperture data are obtained in the alternating scheme. The full aperture data can be obtained by imaging with the alternating scheme twice, and switching the elements used for transmit and receive for each transmit event. Table 2 below compares the speed, SNR, CNR, and CSR for all of the imaging methods mentioned above. For the CNR and CSR, the five rows of values represent cysts and high scattering (HS) regions at decreasing depths (10 mm, 20 mm, 30 mm, and 40 mm). The SA imaging scheme and bandwidth-matched HAPI-SA are highlighted in the table. Table 2 confirms numerically the visual results from Figure 5. We can see how HAPI-SA with bandwidth matching has the best SNR, CNR, and CSR values compared to all the other imaging methods investigated. Figure 4: Point spread functions of (**a**) traditional synthetic aperture with 128 elements (128 transmit events), and (**b**) HAPI synthetic aperture with 128 non-zero-bit codes (1 transmit event) and bandwidth matching. For the same amount of noise added to the scattering data, HAPI-SA produces less noise in the resultant image. Images have a 55 dB dynamic range. Figure 5: 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**) coherent plane wave compounding with 7 angles equally spaced between \(-18^{\circ}\) and \(18^{\circ}\), (**b**) traditional synthetic aperture imaging with 128 transmit events, (**c**) fast synthetic aperture imaging with 3 sub-apertures, (**d**) HAPI synthetic aperture imaging with 1 transmit event but no bandwidth matching, (**e**) HAPI synthetic aperture imaging with bandwidth matching, and (**f**) HAPI synthetic aperture imaging with an alternating transmit scheme. HAPI-SA with bandwidth matching (**e**) produces the best images of all the above. The alternating transmit scheme (**f**) of transmitting on every other element and receiving on the rest produces an image better than traditional SA (**a**) but worse than the full HAPI-SA (**e**). When the bandwidth is not matched, the quality of the image suffers (**d**). All images are shown at a 55 dB dynamic range.