  This paper presents a new approach to the analysis of mixed processes
\[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is
an independent centered Gaussian process. We obtain a new canonical innovation
representation of $X$, using linear filtering theory. When the kernel
\[K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne
t\] has a weak singularity on the diagonal, our results generalize the
classical innovation formulas beyond the square integrable setting. For kernels
with stronger singularity, our approach is applicable to processes with
additional "fractional" structure, including the mixed fractional Brownian
motion from mathematical finance. We show how previously-known measure
equivalence relations and semimartingale properties follow from our canonical
representation in a unified way, and complement them with new formulas for
Radon-Nikodym densities.
