Topological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of Topological Machine Learning, great attention has been devoted to signal processing of such topological signals. Most of the previous topological signal processing algorithms treat node and edge signals separately and work under the hypothesis that the true signal is smooth and/or well approximated by a harmonic eigenvector of the Hodge-Laplacian, which may be violated in practice. Here we propose Dirac-equation signal processing, a framework for efficiently reconstructing true signals on nodes and edges, also if they are not smooth or harmonic, by processing them jointly. The proposed physics-inspired algorithm is based on the spectral properties of the topological Dirac operator. It leverages the mathematical structure of the topological Dirac equation to boost the performance of the signal processing algorithm. We discuss how the relativistic dispersion relation obeyed by the topological Dirac equation can be used to assess the quality of the signal reconstruction. Finally, we demonstrate the improved performance of the algorithm with respect to previous algorithms. Specifically, we show that Dirac-equation signal processing can also be used efficiently if the true signal is a non-trivial linear combination of more than one eigenstate of the Dirac equation, as it generally occurs for real signals.