For notational simplicity, we restrict exposition to the three-dimensional case. The general linear digital filter equation is written in three dimensions as
where is regarded as the input sample at time , and is the output sample at time . The general causal time-invariant filter appears in three-space as
Consider the non-causal time-varying filter defined by
We may call the collector matrix corresponding to the frequency.We have
The top row of each matrix is recognized as a basis function for the order three DFT (equispaced vectors on the unit circle). Accordingly, we have the orthogonality and spanning properties of these vectors. So let us define a basis for the signal space by
Then every component of and every component of when . Now since any signal in may be written as a linear combination of , we find that
Consequently, we observe that is a matrix which annihilates all input basis components but the . Now multiply on the left by a diagonal matrix so that the product of times gives an arbitrary column vector . Then every linear time-varying filter is expressible as a sum of these products as we will show below. In general, the decomposition for every filter on is simply
That every linear time-varying filter may be expressed in this form is also easy to show. Given an arbitrary filter matrix of order N, measure its response to each of the N basis functions (sine and cosine replace ) to obtain a set of N by 1 column vectors. The output vector due to the basis vector is precisely the diagonal of .