This paper concerns the fundamental limitations of disturbance rejection for linear time-invariant discrete-time systems under the assumption that the sensor noise is ignorable. The environmental disturbance is modelled as a linear combination of a step signal, a stationary stochastic signal and several sinusoidal signals with different frequencies. Disturbance rejection is measured by the steady-state variance of the residual input of the plant. In studying this problem, it is shown that for minimum phase systems, perfect disturbance rejection can be achieved by minimising the closed-loop sensitivity function. Unlike the minimum phase systems, the H-2 performance limitations for non-minimum phase systems are lower bounded by its non-minimum phase zeros, unstable poles and characteristic quantities of the disturbance, where the typical and integral-type H-2 control are respectively addressed. Numerical examples are provided to illustrate the utility of the results by analysing the achievable performance.