Abstract The design of minimum phase finite impulse response (FIR) filters is considered. The study demonstrates that the residual errors achieved by current state‐of‐the‐art design methods are nowhere near the smallest error possible on a finite resolution digital computer. This is shown to be due to conceptual errors in the literature pertaining to what constitutes a factorable linear phase filter. This study shows that factorisation is possible with a zero residual error (in the absence of machine finite resolution error) if the linear operator or matrix representing the linear phase filter is positive definite . Methodology is proposed able to design a minimum phase filter that is optimal—in the sense that the residual error is limited only by the finite precision of the digital computer, with no systematic error. The study presents practical application of the proposed methodology by designing two minimum phase Chebyshev FIR filters. Results are compared to state‐of‐the‐art methods from the literature, and it is shown that the proposed methodology is able to reduce currently achievable residual errors by several orders of magnitude.