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We have seen how a simple two-term filter
can destroy the zero frequency or the Nyquist frequency.
When we try to destroy any other frequency, we run into
a new difficulty--we will see
**complex-valued signal**s.
Let *Z*_{0} take the complex value ,where is real.
Further, choose and as a result *Z*_{0} = *i*.
So the filter (1-*Z*/*Z*_{0})=(1+*iZ*) has the complex coefficients (1,*i*),
and its output is a complex-valued signal.
Naturally this is annoying, because
we usually prefer a real output signal.
The way to avoid complex-valued signals
is to handle **negative frequency** the same way we handle .To do this we use a filter with
**two roots**,
one at and one at .The filter (1+*iZ*)(1-*iZ*)= 1+*Z*^{2}
has real-valued time-domain coefficients,
namely, (1,0,1).
The factor (1+*iZ*) vanishes when *Z*=*i* or ,and (1-*iZ*) vanishes at .Notice what happens when the filter (1,0,1)
is convolved with the time series
:the output is zero at all times.
This is because *b*_{t} is a sinusoid
at the half-Nyquist frequency ,and the filter (1,0,1) has zeros at plus and minus half-Nyquist.

Let us work out the general case for a root anywhere
in the
**complex plane**.
Let the root *Z*_{0} be decomposed into its real and imaginary parts:

| |
(22) |

and let the root be written in a polar form:
| |
(23) |

where and are constants that
can be derived from the constants and and vice versa.
The conjugate root is .The combined filter is
| |
(24) |

| (25) |

So the convolutional coefficients of this filter are
the real values .Taking , the filter completely destroys
energy at frequency .Other values of near unity
suppress nearby frequencies without completely destroying them.
Recall that to keep the filter response real,
any root on the positive -axis
must have a twin on the negative -axis.
In the figures I show here,
the negative axis is not plotted, so we must remember the twin.
Figure 10 shows a discrete approximation
to the second derivative.

**ddt2
**

Figure 10
Approximation to the second difference operator (1,-2,1).

It is like (1-*Z*)^{2},
but since both its roots are in the same place at *Z*=1,
I pushed them a little distance apart,
one going to positive frequencies and one to negative.

** Next:** Inverse Z-transform
** Up:** FOURIER AND Z-TRANSFORM
** Previous:** Gaussian examples
Stanford Exploration Project

10/21/1998