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We choose the convention that the picking position of a wavelet is
the first break of the minimum phase part and the ending point of
the maximum phase part. The minimum phase
part is causal and the maximum phase part is anti-causal:

| |
(4) |

where and are generally complex with their norm
less than one, and *N*_{min}+*N*_{max}=*N*. Clearly, this choice ensures that
for a minimum phase wavelet, the first break of the wavelet is picked, and that
for a zero phase wavelet, the center point of the wavelet is picked.
To estimate *n*_{0}, we need to examine the phase response of *S*(*Z*).
Let be the phase spectrum of ,and the phase spectrum of .
Equating the the phase spectra of the two sides of equation (3)
yields

| |
(5) |

Because *w*(*n*) is a real sequence and has causal minimum phase and
anti-causal maximum phase parts, its phase spectrum is anti-symmetric
with respect to :
Using this condition, we can solve equation (5) for *n*_{0}
as follows:
| |
(6) |

By setting , we see that *n*_{0} is the phase delay of the
received signal at the Nyquist frequency:

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Stanford Exploration Project

12/18/1997