This paper considers the problem of designing a multilevel pulse width modulated (PWM) waveform with a prescribed harmonic content. Multilevel PWM design plays a major role in many diverse engineering disciplines. In power electronics, multilevel PWM design corresponds to determining the inverter switching times and levels for selective harmonic elimination and harmonic compensation. In mechatronics, the same design corresponds to shaping input signals to damp residual vibrations in flexible structures. More generally, in many applications, the aim of PWM design is to minimise the total harmonic distortion while adhering to a prescribed harmonic content. The solution approach presented in this paper is based on linear programming with the objective of minimising the total harmonic distortion. This objective is achieved within an arbitrarily small bound of the optimal solution of the related convex optimisation problem. In addition, the linear programming formulation makes the design of such switching waveforms computationally tractable and efficient. Simulations and experiments are provided for corroboration. © 2020 Informa UK Limited, trading as Taylor & Francis Group.