Lattice filter structures have a long history in the filtering and prediction of discrete-time signals. Often these discrete-time signals arise from the sampling of an underlying continuous-time process, and the limiting behaviour of the filter as the sampling rate increases is rarely considered. In this paper it is shown that this issue is resolved if the standard formulation of the lattice filter structure, based on the forward shift operator, is replaced by an alternative formulation based on the incremental difference, or delta, operator. The paper contains two contributions. Firstly, the continuous and discrete lattice algorithms are presented in a unified framework, thereby revealing their common structure. Secondly, it is shown that when the discrete-time signal is obtained by sampling an underlying continuous-time process, the lattice filter corresponding to the discrete case converges, in a well-defined sense, to the solution of the underlying continuous problem as the sampling period approaches zero.