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Begin with a function in discretized time *x*_{t}.
The Fourier transform with the substitution is
the *Z*-transform

| |
(12) |

Define (which will turn out to be an approximation
to ) by
| |
(13) |

Define another signal *y*_{t}
with *Z*-transform *Y*(*Z*) by
applying the operator to *X*(*Z*):
| |
(14) |

Multiply both sides by (1-*Z*):
| |
(15) |

Equate the coefficient of *Z*^{t} on each side:
| |
(16) |

Taking *x*_{t} to be an impulse function,
we see that *y*_{t} turns out
to be a step function, that is,
| |
(17) |

| |
(18) |

So *y*_{t} is the discrete-domain representation of the integral
of *x*_{t} from minus infinity to time *t*.
The operator (1+*Z*)/(1-*Z*) is called the
``**bilinear transform**."

** Next:** The accuracy of causal
** Up:** Z-plane, causality, and feedback
** Previous:** Smoothing with a triangle
Stanford Exploration Project

10/21/1998