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Now suppose there was an explosion at t = 0,
a halfstrength implosion at t = 1,
and another, quarterstrength explosion at t = 3.
This sequence of events determines a ``source'' time series,
.The Ztransform of the source is
.The observed y_{t} for this sequence of explosions and implosions
through the seismometer has a Ztransform Y(Z), given by
 

 
 (3) 
The last equation
shows polynomial multiplication as the
underlying basis of timeinvariant linearsystem theory,
namely that the output Y(Z) can be expressed as the input X(Z) times
the impulseresponse filter B(Z).
When signal values are insignificant
except in a ``small'' region on the time axis,
the signals are called ``wavelets.''
There are many examples of linear systems.
The one of most interest to us is wave propagation in the earth.
A simpler example,
around which a vast literature exists, is electronic filters.
A cascade of filters is formed
by taking the output of one filter and plugging it into the input of another.
Suppose we have two linear filters characterized by B(Z) and C(Z).
Then the question arises, illustrated in Figure 5,
as to whether the two combined filters are equivalent.
commute
Figure 5
Two equivalent filtering systems.

 
The use of Ztransforms makes it obvious that
these two systems are equivalent,
since products of polynomials commute,
i.e.,
 

 (4) 
Next: Dissecting systems by factoring
Up: SAMPLED DATA AND ZTRANSFORMS
Previous: Linear superposition
Stanford Exploration Project
10/21/1998