## Appendix A Multirate Theorems of Coherent Plane Wave Compounding  
Let \(s[n]\equiv s_{a}(\vec{r},n\Delta t)\) be the discrete, slow-time, precompounded, analytic signal at a location \(\vec{r}\). For compactness of notation, the position argument \(\vec{r}\) is suppressed in the following section, with the understanding that it is evaluated independently at every spatial location. The coherent plane wave compounded signal \(s_{+}[n]\) is equivalent to a rectangular filtering of \(s[n]\) followed by a downsampling operation. By definition, \(s_{+}[n]\) is given by  
\[s_{+}[n]\triangleq\sum_{n^{\prime}=0}^{N_{T}-1}s\left[N_{T}n+n^{\prime}\right]. \tag{1}\]  
Let us define the rectangular signal that is nonzero on the closed interval \([a,b]\) as  
\[r_{[a,b]}[n]\equiv \begin{cases}1,&n\in[a,b]\\ 0,&\text{otherwise.}\end{cases}  \tag{2}\]A simple variable substitution of \(n^{\prime}\rightarrow-n^{\prime}\) results in  
\[s_{+}[n] =\sum_{n^{\prime}=-(N_{T}-1)}^{0}s\left[N_{T}n-n^{\prime}\right] \tag{11}\] \[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\] (12) \[=\sum_{n^{\prime}=-\infty}^{\infty}r_{[1-N_{T},0]}[n^{\prime}]s \left[N_{T}n-n^{\prime}\right]\] (13) \[=s_{f}\left[N_{T}n\right], \tag{14}\]  
where \(s_{f}[n]\) is the rectangularly filtered analytic signal given by  
\[s_{f}[n] =r_{[1-N_{T},0]}[n]\underset{n}{\ast}s[n] \tag{15}\] \[=\sum_{n^{\prime}=-\infty}^{\infty}r_{[1-N_{T},0]}[n^{\prime}]s \left[n-n^{\prime}\right]. \tag{16}\]  
Because the downsampling operation by a factor of \(N_{T}\) is defined as  
\[\mathcal{D}_{N_{T}}\left\{s_{f}[n]\right\}=s_{f}[N_{T}n] \tag{17}\]  
we find that  
\[s_{+}[n] =\mathcal{D}_{N_{T}}\left\{s_{f}[n]\right\} \tag{18}\] \[=\mathcal{D}_{N_{T}}\left\{r_{[1-N_{T},0]}[n]\underset{n}{\ast}s[ n]\right\}. \tag{19}\]  
Thus, the coherent compounding operation is equivalent to a rectangular filter of length \(N_{T}\) followed by a downsampling of \(N_{T}\). **Theorem A2**. : _Consider the traditional pipeline of passing the compounded signal \(s_{+}[n]\) through a high-pass clutter filter with impulse response \(h[n]\). Then, the output signal \(s_{c}[n]\) is equivalent to filtering each individual polyphase channel \(s_{\phi_{n^{\prime}}}[n]\) and summing._  
Proof. : Let us denote the output of the clutter filter as \(s_{c}[n]\), such that  
\[s_{c}[n]=h[n]\underset{n}{\ast}s_{+}[n]. \tag{20}\]  
Thus, we have  
\[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]0  
Let us define the polyphase channel \(s_{\phi_{n^{\prime}}}[n]\) as the slow-time signal associated with the \(n^{\prime}\)th transmit in the transmit sequence period. Mathematically the relationship is described by  
\[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]1\[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]2 In the HRI estimators, the aliasing manifests as plateauing in the root-mean-square error (RMSE), resulting from the lower frame rate. All other estimators appear to predominately track the standard deviation shown in \[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]3  
\[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]4 For the traditional HRI estimators, error resonates at the Nyquist velocities, likely due to the aliasing. The sTx3 LRI estimator appears to have uncharacteristically high error across the velocity range, suggesting that the individual LRIs are highly decorrelated, possibly due to a high angular step. In the sTx7 LRI dataset, the error resonates at the velocity associated with the transmit sequence period, likely due a combination of sidelobe motion and attenuation due to the clutter filter. The dTx6 estimators do not have a corresponding peak, suggesting an inherent robustness to these effects. \[=\sum_{n^{\prime}=(1-N_{T})}^{0}s\left[N_{T}n-n^{\prime}\right]\]5 Color flow Doppler images generated from the experimental data. The HRI images are shown on the left, and the LRI images are shown on the right, with each row corresponding to a different dataset.