Page 129 - MATLAB Recipes for Earth Sciences

6.3 Linear Time-Invariant Systems 123

4. Invertibility – An invertible system is a system where the original input
signal can be reproduced from the systems output. This is an important
property if unwanted signal distortions have to be corrected. In such a
case, the known system is inverted and applied to the output to recon-
struct the undisturbed input. As an example, a core logger measuring the
magnetic susceptibility with a loop sensor. The loop sensor integrates
over a certain core interval with highest sensitivity at the location of the
loop and decreasing sensitivity down- and up-core. The above system
is also invertible, i.e., we can compute the input signal from the output
signal by inverting the system. The inverse system of the above linear
fi lter is
t = (1:100)';
y = 0.5*t;
plot(t,y)
The nonlinear system

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

plot(t,y)
is not invertible. Since this system yields equal responses for different
inputs, such as y=+4 for inputs x=-2 and x=+2, the input can not be re-
constructed from the output. A similar situation can also occur in linear
systems, such as

t = (1:100)';
y = 0;

plot(t,y)
The output is zero for all inputs. Hence, the output does not contain any
information about the input.

5. Causality – The system response only depends on present and past in-
puts x(0), x(-1), …, whereas future inputs x(+1), x(+2), … have no ef-
fect on the output y(0). All realtime systems, such telecommunication
systems, must be causal since they can not have future inputs available

to them. All systems and filters in MATLAB are indexed as causal. In
earth sciences, however, numerous non-causal fi lters are used. Filtering
images or signals extracted from sediment cores are examples where the

future inputs are available at the time of filtering. Output signals have to