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The convolution equation ()
| |
(2) |

says that the present output is created entirely from present
and past values of the *input*.
Now we will include past values of the *output*.
The simplest example is
**numerical integration**,
such as
| |
(3) |

Notice that when ,,which shows that the integral of an impulse is a step.
A kind of deliberately imperfect integration used in numerical work
is called ``**leaky integration**.''

The name derives from the analogous situation of
electrical circuits, where
the voltage on a capacitor is the integral of the current:
in real life, some of the current leaks away.
An equation to model leaky integration is

| |
(4) |

where is a constant that is slightly less than plus one.
Notice that if were greater than unity,
the output of (4) would grow with time instead of decaying.
A program for this simple operation is `leak()`.
I use this program so frequently
that I wrote it so the output could be overlaid on the input.
`leak()` uses a trivial subroutine, `copy()` , for copying. subroutine leak( rho, n, xx, yy)
integer i, n; real xx(n), yy(n), rho
temporary real tt( n)
call null( tt, n)
tt(1) = xx(1)
do i= 2, n
tt(i) = rho * tt(i-1) + xx(i)
call copy( n, tt, yy)
return; end

Let us see what *Z*-transform equation is implied by (4).
Move the *y* terms to the left:

| |
(5) |

Given the *Z*-transform equation
| |
(6) |

notice that (5) can be derived
from (6)
by finding the
coefficient of *Z*^{t}.
Thus we can say that the output *Y*(*Z*) is derived from the input *X*(*Z*) by
the polynomial division
| |
(7) |

Therefore, the effective filter *B*(*Z*) in *Y*(*Z*)=*B*(*Z*)*X*(*Z*) is
| |
(8) |

The left side of Figure 1 shows a damped exponential
function that consists of the coefficients seen in
equation (8).
**leak
**

Figure 1
Left is the impulse response of leaky integration.
Right is the amplitude in the Fourier domain.

The spectrum of *b*_{t} is defined by .The **amplitude spectrum** is the square root of the spectrum.

It can be abbreviated by |*B*(*Z*)|.
The amplitude spectrum is plotted on the right side of Figure 1.
Ordinary integration has a Fourier response that blows up at .Leaky integration smooths off
the infinite value at .Thus in the figure, the amplitude spectrum looks like ,except that it is not at .

** Next:** Plots
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Stanford Exploration Project

10/21/1998