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Let *x*_{t} be an even function of *t*.
The definition gives
; so
| |
(1) |

| (2) |

Now make up a new function *Y*(*Z*)
by replacing cosine by sine in (2):
| |
(3) |

Recalling that , we see
that all the negative powers of *Z* cancel from *X*(*Z*)+*iY*(*Z*), giving
a **causal** *C*(*Z*):
| |
(4) |

Thus, for plot pairs, the causal response is *c*_{t},
the real part of the FT is equation (2),
and the imaginary part not usually shown
is given by equation (3).

** Next:** The quadrature filter
** Up:** HILBERT TRANSFORM
** Previous:** HILBERT TRANSFORM
Stanford Exploration Project

10/21/1998