We present a strategy to remove the broadband Fourier spectral content associated with the boundary discontinuities in non-periodic directions of signals over finite domains, with negligible modification of the scales of interest in the original signal. The discontinuities arise from the signal decomposition using the Fourier harmonic basis functions that are defined over infinite domain, so the periodically extended signal in non-periodic directions includes a discontinuity at the boundaries between the signal and its periodic extension. The specific Fourier content associated with the non-periodicity of an arbitrary physical signal is identified and a procedure is developed to systemically isolate its spectral content with minimal modification of the original signal. The proposed “discontinuity pollution removal” procedure is first developed for C0 boundary discontinuities in one-dimensional signals, and then extended to higher order CN >0 boundary discontinuities in multi-dimensional signals with one non-periodic direction. This procedure allows for the analysis of signals over finite domains with the Fourier decomposition regardless of boundary conditions. We validate and demonstrate the capabilities of the proposed procedure for a test case of engineering relevance.