Under the short-pulse approximation, the data measured in diffusion MRI can be modeled as the Fourier transform of the Ensemble Average Propagator (EAP) [1], a probability distribution that characterizes the molecular diffusion of the spins within each voxel. This is potentially advantageous because of extensive theory that has been developed to characterize sampling requirements, accuracy, and stability of Fourier reconstruction. However, existing diffusion MRI sampling and EAP estimation approaches have largely been developed and tuned without the benefit of such theory, instead relying on intuition, approximations, modeling assumptions, and extensive empirical evaluation. This work introduces a novel theory that can be used to characterize the performance of arbitrary linear EAP estimation methods with arbitrary q-space sampling schemes, building off of previous theoretical characterizations of orientation estimation [2-4]. For the first time, this provides a precise theoretical relationship between the true EAP and the estimated EAP. This relationship is similar to the point-spread function (PSF) relationship that is widely used to characterize the spatial resolution of MR images. In the context of EAP estimation, this theoretical framework provides direct insight into issues such as resolution, aliasing, and noise amplification. This valuable information can be directly used when choosing between different q-space sampling schemes and EAP estimation methods. Due to space constraints, we only describe the noiseless case in this abstract, though we also have a full noise-based theory.Noiseless diffusion data $$$E(\textbf{q}_m)$$$ measured at q-space location $$$(\textbf{q}_m)$$$ can be modeled as the Fourier transform of the EAP, and a variety of linear and nonlinear EAP estimation methods have been designed for such data. This work focuses on linear methods like DSI [5], GQI [6], 3D-SHORE [7], SPF [8], SPFdual [9], SoH [10], and directional radial basis function estimation [11]. While each of these methods is based on different assumptions, they are all linear and can be characterized by the common reconstruction formula: $$EAP_{estimated}(\textbf{x})=∑_{m=1}^ME(\textbf{q}_m)G(\textbf{x},\textbf{q}_m)$$ where $$$G(\textbf{x},\textbf{q}_m)$$$ are the method-dependent linear estimation coefficients. Due to the symmetry Fourier transform relationships, it is possible to expand this expression as $$EAP_{estimated}(\textbf{x})=∫EAP_{true}(\textbf{ξ})g(\textbf{x},\textbf {ξ})d\textbf{ξ}$$ where $$$g(\textbf{x},\textbf{ξ})=∑_{m=1}^MG(\textbf{x},\textbf{q}_m)cos(2π\textbf{q}_m^T\textbf{ξ})$$$ is the “EAP response function”. The EAP response function is similar to the standard PSF used in imaging. Similar to a PSF, the EAP response function should be similar to a delta function in order to achieve high fidelity EAP reconstruction, though some amount of blurring and aliasing is inevitable due to the finite sampling of q-space. Importantly, the EAP response function is easy to compute, and provides direct insight into the quality of a given q-space sampling-scheme and linear EAP estimation method. An illustration of the usefulness of our proposed theoretical framework is shown in Fig. 1, where we compare EAP responses for different q-space sampling schemes (multi-shell and Cartesian, each using 204 and 515 samples and maximum b-values of 4000 and 10000s/mm2) for GQI EAP estimation. As can be seen, the Cartesian sampling method suffers from coherent aliasing (seen as multiple peaks in the EAP response), while, multi-shell sampling scheme suffers from incoherent aliasing (side lobe interference). This expectation is confirmed with simulated data as shown in Fig. 2. Note that there is a clear trade-off between the number of measured samples, the amount of aliasing interference (controlled by the sample spacing in q-space), and the resolution of the EAP (controlled by the size of the largest b­-value). These effects are expected from traditional Fourier sampling theory, though have not usually been considered when designing diffusion experiments. As further illustration, Figs. 3 and 4 show that the EAP response can be used to theoretically predict the impact of user-selected reconstruction parameters. Specifically, we show EAP responses and estimated EAPs for 3D-SHORE EAP estimation using different values of the 3D-SHORE regularization parameter [7]. The EAP response behavior in Fig. 3 suggests that increasing regularization would initially improve accuracy by reducing sidelobe interference, though will eventually lead to a loss in EAP resolution if the regularization is increased too far. This theoretical prediction is confirmed by the estimation results shown in Fig. 4. This kind of characterization enables a new mechanism for optimizing reconstruction parameters.We have proposed novel theoretical tools that can be used to characterize the performance linear diffusion MRI estimation methods without requiring extensive empirical testing. Notably, these tools do not require any assumptions about the EAP, and can be used even when traditional modeling assumptions are inaccurate. We expect these tools to be useful when designing sampling and estimation methods for a wide range of diffusion MRI experiments.