Page 128 - MATLAB Recipes for Earth Sciences

122 6 Signal Processing

2. Linearity – For linear systems, the output of the linear combination of
several input signals

is the same linear combination of the outputs:

The important consequence of linearity is scaling and additivity ( super-
position). Input and output can be multiplied by a constant before or af-
ter transformation. Superposition allows to extract additive components
of the input and transform these separately. Fortunately, many natural
systems show linear behavior. Complex linear signals such as additive
harmonic components can be separated and transformed independently.
Milankovitch cycles provide example of linear superposition in paleocli-
mate records, although there is an ongoing debate about the validity of
this assumption. Numerous nonlinear systems exist in nature that do not
obey the properties of scaling and additivity. An example of such a linear
system is

t = (1:100)';
y = 2*t;
plot(t,y)

An example of such a nonlinear system is

t = (-100:100)';
y = t.^2;
plot(t,y)

3. Time invariance – The system output y(t) does not change with a de-
lay of the input x(t+i). The system characteristics are constant with time.
Unfortunately many systems in nature change their characteristics with
time. For instance, benthic mixing or bioturbation depends on various
environmental parameters such as nutrient supply. Therefore the system·s

performance varies with time significantly. In such case, the actual input
of the system is hard to determine from the output, i.e., to extract the ac-
tual climate signal from a bioturbated sedimentary record.