Page 131 - MATLAB Recipes for Earth Sciences

6.4 Convolution and Filtering 125

smooth the signal. Therefore running means of a given length are often used
to smooth signals, mainly for cosmetic reasons. Modern spreadsheet soft-
ware usually contains running means for smoothing data series. The impact

of the smoothing filter increases with increasing fi lter length.
The weights that a fi lter of arbitrary length may take can vary. As an ex-

ample, let us assume an asymmetric filter of fi ve weights.
b = [0.05 0.08 0.14 0.26 0.47]

The sum of all of the filter weights is one. It therefore does not introduce en-
ergy to the signal. However, since it is highly asymmetric it shifts the signal
along the time axis, i.e., it introduces a phase shift.

The general mathematical representation of the filtering process is the
convolution

where b is the vector of fi lter weights, N +N is the order of the filter, which
k 1 2
is the length of the filter reduced by one. In our examples of filters of fi ve

weights, the order of the filters is four. In contrast to this format, MATLAB

uses the engineering standard of indexing filters, i.e., filters are always de-

fined as causal. Therefore the convolution used by MATLAB is defi ned as

where N is the order of the filter. A number of frequency-domain tools pro-
vided by MATLAB cannot simply be applied to non-causal filters that have

been designed for applications in earth sciences. Hence, it is common to
carry out phase corrections in order to simulate causality. For example, fre-
quency-selective filters as introduced in Chapter 6.9 can be applied using

the function filtfilt, which provides zero-phase forward and reverse
fi ltering.
The functions conv and filter that provide digital fi ltering with
MATLAB are best illustrated in terms of a simple running mean. The n
elements of the vector x(t ), x(t ), x(t ), …, x(t ) are replaced by the arithme-
1 2 3 n
tic means of subsets of the input vector. For instance, a running mean over
three elements computes the mean of inputs x(t ), x(t ), x(t )to obtain the
n-1 n n+1