Derivation

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 (H.1)

The uniqueness of the decomposition is easy to verify: Suppose there are two distinct decompositions of the form Eq. (H.1). Then for some we have different D(k)'s. However, this implies that we can get two distinct outputs in response to the input basis function which is absurd.

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 .

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