## 2 Materials and Methods  
### Theory  
#### 2.1.1 Coherent Plane-Wave Compounding  
A plane-wave transmission is generated by firing all of the elements of a linear array with inter-element transmission delays that create a coherent wave front at a steered angle. The return echoes are then digitized on all of the array elements and beamformed data representing a full image frame can be obtained with delay-and-sum methods. Plane waves are usually transmitted in a sequence of \(N_{T}\) steered excitations spanning a set of \(N_{\phi}\) distinct angles, where the angle of transmission linearly increases from low to high (Figure 1a). Note that in the traditional case \(N_{T}=N_{\phi}\). The data collected from each transmit event, after beamforming, are coherently summed to form high-resolution images (HRIs), which improves image resolution and contrast [6].  
The fundamental tradeoff when using a coherent compounding technique is between the spatial and temporal resolution. As noted in [13], the effect of the coherent compounding process is one of a temporal low-pass, finite impulse response (FIR) filter. However, [PERSON], et al. [13], did not address the effect of the downsampling operation. As described by multirate signal processing theory [14; 15], a downsampling operation results in an effective expansion of the discrete time spectrum, which can result in additional aliasing. Because traditional coherent compounding techniques use uniform weights [6], the anti-aliasing filter corresponds to a rectangular filter (See Appendix) that is prone to temporal frequency sidelobes. Equally weighting the compounded components could lead to slow-time temporal aliasing and high velocity estimates would be particularly vulnerable to this source of error.  
In principle, one could design an anti-aliasing filter with better stop band characteristics using traditional signal processing techniques [15]. However, such a design process would inevitably produce a nonuniformly weighted low pass filter, and as such would result in reduced signal contribution from the highly steered components of the compounding process. Consequently, the spatial resolution of the compounded images would be reduced.  
Another approach would be to use precompounded low-resolution images (LRIs). However, because each of the steered plane waves produces a slightly different PSF, a new temporal noise source is introduced in the form of spatial sidelobe motion that oscillates periodically in slow time (Figure 2). This sidelobe motion could be mistaken as tissue motion by traditional Doppler estimation techniques [2], especially in hypoechoic tissue regions. While in principle such sidelobes could be  
Figure 1: Comparison of the different transmit sequences used in this study. With the traditional transmit sequence, acquisitions with the same point spread function are spaced apart over intervals equivalent to a transmit sequence length. Due to the fact that these components will exhibit the strongest spatial correlations, reordering them such that they are temporally adjacent allows for correlation estimates at the shortest possible time interval. (**a**) Traditional transmit sequence where the transmit angles linearly increment and then repeat; (**b**) Double transmit sequence where the transmit angle repeats before linearly incrementing to the next angle.  
minimized via an appropriately designed apodization function, such a design choice ultimately results in a loss of spatial resolution. As such, it is important to consider alternative processing schemes to address this new noise source associated with LRI-based Doppler estimation.