
In a transversal filter of length , as depicted in fig. 1, at each time the output sample is computed by a weighted sum of
the current and delayed input samples

(1)

Here, the are
time dependent filter coefficients (we use the complex conjugated
coefficients
so that
the derivation of the adaption algorithm is valid for complex
signals, too). This equation rewritten in vector form, using
,
the tapinput vector at time , and
,
the coefficient vector at time , is

(2)

Both
and
are
column vectors of length ,
is the hermitian of vector
(each
element is conjugated , and the column vector is transposed
into a
row vector).
Figure 1: Transversal
filter with time dependent coefficients

In the special case of the coefficients
not
depending on time :
the transversal filter
structure is an FIR^{1} filter of length . Here, we will, however, focus on
the case that the filter coefficients are variable, and are adapted
by an adaptation algorithm.
