** Next:** Polynomial division
** Up:** DAMPED OSCILLATION
** Previous:** DAMPED OSCILLATION

It seems we can represent a sinusoid by *Z*-transforms
by setting a pole on the unit circle.
Taking , we have the filter

| |
(25) |

The signal *b*_{t} seems to be the complex exponential
,but it is not quite that
because *b*_{t} is ``turned on'' at *t* = 0, whereas
is nonzero at negative time.
Now, how can we make a *real*-valued sinusoid starting at *t*=0?
Just as with zeros, we need to complement the pole at by one at .The resulting
signal *b*_{t} is shown on the left in Figure 7.
On the right is a graphical attempt to plot the impulse function
of dividing by zero at .

**sinus
**

Figure 7
A pole on the real axis
(and its mate at negative frequency)
gives an impulse function at that frequency
and a sinusoidal function in time.

Next, let us look at a damped case like leaky integration.
Let and .Then .Define

| |
(26) |

| (27) |

The signal *b*_{t} is zero before
*t* = 0 and is after *t* = 0.
It is a damped sinusoidal function
with amplitude decreasing with time as .We can readily recognize this as an exponential decay
| |
(28) |

where the approximation is best for values of near unity.
The wavelet *b*_{t} is complex.
To have a real-valued time signal,
we need another pole at the negative frequency,
say .So the composite denominator is

| |
(29) |

Multiplying the two poles together as we did for roots
results in the plots of 1/*A*(*Z*)
in Figure 8.

**dsinus
**

Figure 8
A damped sinusoidal function of time
transforms to a pole near the real -axis,
i.e., just outside the unit circle in the *Z*-plane.

Notice the ``p'' in the figure.
It indicates the location of the pole *Z*_{p}
but is shown in the -plane,
where .Pushing the ``p'' left and right will lower and raise the resonant frequency.
Pushing it down and up will raise and lower the duration of the resonance.

## EXERCISES:

- How far from the unit circle are the poles of
1/(1 - .1
*Z* + .9*Z*^{2})? What is the decay time
of the filter and its resonant frequency?

** Next:** Polynomial division
** Up:** DAMPED OSCILLATION
** Previous:** DAMPED OSCILLATION
Stanford Exploration Project

10/21/1998