For notational simplicity, we restrict exposition to the
threedimensional 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 timeinvariant
filter appears in threespace as
Consider the noncausal timevarying 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
timevarying 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 timevarying 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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