This paper deals with a mixture of H2 and H8 in the following way: We have two inputs and one output. One input signal is a white-noise stochastic process, and represents errors e.g. resulting from measurement noise. The other input has a more deterministic character. If one has a reference signal (e.g. a step) as input, one cannot model this as white noise, but it fits nicely into this new class of inputs. The objective is to minimize the effect of these signals on the output of the system. We define a cost function which enables us to combine these two exogenous inputs, even though they are structurally different. The analysis of this function leads to a standard H8 Riccati equation. We motivate this cost function by looking at two theoretical applications: the derivation of robust performance bounds and a tracking problem.