# Implementation of Constrained Swept Synthetic Aperture Using a Mechanical Fixture
## 2 Materials and Methods  
### Pulse Sequencing  
Proper pulse sequencing is essential for successful SSA imaging using motion tracking. The motion tracking methods below assume that there is only slight decorrelation between successive data sets representing a small motion of the array. For a focused transmission to a point in space, a translating array produces decorrelation relative to its length [27]. Increasing decorrelation is associated with increasing variance of displacement estimates [28]. To maximize the correlation, unsteered (0 degree) plane waves were transmitted, and echo data were collected on all array elements. This plane wave can be viewed as a relatively constant transmit field with only small differences near the edges of the array extent.  
However, the plane wave transmission is a poor choice for SSA as the field of view only overlaps over a small range of aperture positions [26]. The transmit aperture remains constant (unfocused), and the receive aperture can only effectively double in length due to the finite width of the plane wave. Instead, a diverging wave is used to produce a broad transmit field that shifts with position of the array. This transmit signal creates rapidly decorrelating echoes with translation, a poor choice for displacement estimation but ideal for maximizing the extent of both the effective transmit and receive apertures over a sweep. The effective aperture length reached is calculated as the average of the swept transmit and receive apertures.  
These transmit geometries and their trade-offs are illustrated over a sweep of the transducer in Figure 1. Assuming a constant transducer motion that covers 10 cm in 1 s, a pulse repetition frequency of even 1 kHz would provide spatial sampling of 0.1 mm which is oversampled compared to past work [12]. It is expected that, even if the maximum velocity is increased due to acceleration required at the beginning and end of the sweep, the displacement will remain small, on the order of the element pitch. These transmissions can, therefore, be interleaved into a single sequence to provide highly correlated data for displacement estimation and decorrelated data for synthetic aperture imaging as shown in Figure 2.  
The simplest sequence would have alternating plane waves and diverging waves that repeat at the chosen repetition frequency. An improved sequence would use two diverging waves, one from either end of the array, with the plane wave. This extends the effective transmit aperture for the SSA image by an entire transducer length (half in each direction at the ends of the sweep). This diagram shows the timing used in this paper with 260 us between pulses within the sequence and an overall frame rate of 500 Hz. While, in these diagrams, the diverging wave is shown produced by a single element, it is also possible to use subapertures to create diverging virtual sources [29] with an improved signal-to-noise ratio (SNR).  
Figure 1: Illustration of the transmit geometries used for correlation-based SSA imaging showing two array positions imaging a target (black circle). Black lines above the array show the synthetic receive array length. (**a**) Plane wave transmission produces highly correlated echoes for motion tracking but limits spatial overlap and does not produce an improved synthetic transmit aperture. (**b**) Diverging wave transmission produces long effective transmit and receive arrays for high resolution due to large spatial overlap at depth but has low echo correlation for motion tracking.