Second-order generalised integrator or resonant integrator (RI) has wide range of applications. Forward and backward Euler's approximation based two integrator realisation of RI is an easily implementable frequency adaptive method. However, it suffers from resonant frequency deviation due to discretisation. The discretisation methods that lead to accurate realisation of RI require online calculation or lookup table of trigonometric functions to accommodate frequency variation. In this study, multi-rate computation-based implementation of two integrator-based RI has been proposed to minimise resonant frequency deviation. In this method, no additional logic elements are consumed to achieve accurate resonant frequency location. This along with down-sampling leads to lesser phase lag of RI. The effect of quantisation on resonant frequency deviation has been analysed for proper choice of calculation time. It is also shown that appropriate choice of down-sampling instants give a range of phase response characteristics around the nominal continuous time RI phase response. The accuracy of resonant frequency emulation has been experimentally verified by implementing a proportional-resonant controller as current controller.