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In the development of mining projects, there is a period in between commitment of finance and production commencement. Risks are present in various aspects of development such that production start-up could be delayed, thereby affecting the value of the project. This paper demonstrates the application of convolution in the capital investment problem for projects with long lead-times as a means of characterising the relationship between capital invested and the materialisation of capacity for production to begin. The system is functionally similar to that of a causal linear time-invariant system in signal processing. We find that the filter for capital projects, affecting the rate at which capacity becomes available for use, can be approximated by a straight-line function with positive gradient on bounded support contained in the positive time domain. The deterministic version is derived and evaluated, and uncertainty is simulated in the stochastic version, where variance results from the presence of a random process in the form of a standard Brownian motion. [ABSTRACT FROM AUTHOR]