Geoscience Reference
In-Depth Information
h e output
y
(
t
) is simply the average of the i ve input values
x
(
t
-2),
x
(
t
-1),
x
(
t
),
x
(
t
+1) and
x
(
t
+2). In other words, all the i ve consecutive input values
are multiplied by a factor of 1/5 and summed to form
y
(
t
). In this example all
input values are multiplied by the same factor, i.e., they are equally weighted.
h e i ve factors used in the above operation are therefore called i lter weights
b
k
. h e i lter can be represented by the vector
b = [0.2 0.2 0.2 0.2 0.2]
consisting of the i ve identical i lter weights. Since this i lter is symmetric,
it does not shit the signal on the time axis: the only function of this i lter
is to smooth the signal. Running means of a given length are ot en used to
smooth signals, mainly for cosmetic reasons. Modern spreadsheet sot ware
usually contains running means as a function for smoothing data series. h e
ef ectiveness of a smoothing i lter increases with the i lter length.
h e weights that a i lter of arbitrary length uses can be varied. As an
example let us consider an asymmetric i lter of i ve weights.
b = [0.05 0.08 0.14 0.26 0.47]
h e sum of all of the i lter weights is one and therefore it does not introduce
any additional variance into the signal. However, since it is highly asymmetric,
it shit s the signal along the time axis, i.e., it introduces a phase shit .
h e general mathematical representation of the i ltering process is the
convolution
:
where
b
k
is the vector of
i lter weights
, and
N
1
+
N
2
is the
order of the i lter
,
which is the length of the i lter reduced by one. Filters with i ve weights, as in
our example, have an order of four. In contrast to this format, MATLAB uses
the engineering standard for indexing i lters, i.e., i lters are always dei ned as
causal. h e convolution used by MATLAB is therefore
