The filter (1+*Z*)/2 is a running average of two adjacent time points.
Applying this filter *N* times yields the filter (1+*Z*)^{N}/2^{N}.
The coefficients of the filter (1+*Z*)^{N} are generally known as
**Pascal's triangle**.
For large *N* the coefficients tend to a mathematical limit
known as a **Gaussian** function,
, where and *t _{0}* are constants that we will determine in chapter .
We will not prove it here,
but this Gaussian-shaped signal
has a Fourier transform that also has a Gaussian shape,
.The Gaussian shape is often called a ``bell shape.''
Figure 8 shows an example for .Note that, except for the rounded ends,
the bell shape seems a good fit to a triangle function.

Figure 8

Curiously, the filter (.75+.25*Z*)^{N} also tends to the same Gaussian
but with a different *t _{0}*.
A mathematical theorem
(discussed in chapter ) says
that almost any polynomial raised to the

In seismology we generally fail to observe the
**zero frequency**.
Thus the idealized seismic pulse cannot be a Gaussian.
An analytic waveform of longstanding popularity in seismology
is the
second derivative of a Gaussian,
also known as a ``**Ricker wavelet**.''

Starting from the Gaussian and putting
two more zeros at the origin with (1-*Z*)^{2}=1-2*Z*+*Z ^{2}*
produces this old, favorite wavelet,
shown in Figure 9.

Figure 9

10/21/1998