The significance of the broken ray transform (BRT) is due to its occurrence in a number of modalities spanning optical, x-ray, and nuclear imaging. When data are indexed by the scatter location, the BRT is both linear and shift invariant. Analyzing the BRT as a linear system provides a new perspective on the inverse problem. In this framework we contrast prior inversion formulas and identify numerical issues. This has practical benefits as well. We clarify the extent of data required for global reconstruction by decomposing the BRT as a linear combination of cone beam transforms. Additionally we leverage the two dimensional Fourier transform to derive new inversion formulas that are computationally efficient for arbitrary scatter angles. Results of numerical simulations are presented.