Convolution with Z-transform

Now suppose there was an explosion at t = 0, a half-strength implosion at t = 1, and another, quarter-strength explosion at t = 3. This sequence of events determines a ``source'' time series, $x_t = (1, -{1 \over 2}, 0, {1 \over 4})$.The Z-transform of the source is $X(Z) = 1 - {1 \over 2}Z + {1 \over 4}Z^3$.The observed y_t for this sequence of explosions and implosions through the seismometer has a Z-transform Y(Z), given by

$$\begin{eqnarray} Y(Z) &\eq & B(Z) - {Z \over 2}\, B(Z) + {Z^3 \over 4}\, B(Z) \n... ...2} + {Z^3 \over 4} \right)\, B(Z) \nonumber \\ &\eq & X(Z)\, B(Z)\end{eqnarray}$$ (4)

The last equation shows polynomial multiplication as the underlying basis of time-invariant linear-system theory, namely that the output Y(Z) can be expressed as the input X(Z) times the impulse-response filter B(Z). When signal values are insignificant except in a ``small'' region on the time axis, the signals are called ``wavelets.''