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The prediction and update operators employed in the lifting scheme can
be understood as digital filters. In the
-transform notation, the
Haar prediction filter from equation 3 is
![\begin{displaymath}
P(Z) = Z
\end{displaymath}](img22.png) |
(8) |
(shifting each sample by one), and the linear interpolation filter
from equation 4 is
![\begin{displaymath}
P(Z) = 1/2\,(1/Z+Z)\;.
\end{displaymath}](img23.png) |
(9) |
These predictions are appropriate for smooth signals but may not be
optimal for a sinusoidal signal. In
comparison, the prediction
![\begin{displaymath}
P(Z) = Z/Z_0\;,
\end{displaymath}](img24.png) |
(10) |
where
, perfectly characterizes a
sinusoid with
circular frequency sampled on a
grid. In other words, if a constant signal (
) is perfectly
predicted by shifting each trace to its neighbor, a sinusoidal signal
(
) requires the shift to be modulated by an appropriate
frequency.
Likewise, the linear interpolation in equation 9 needs to be
replaced by a filter tuned to a particular frequency in order to
predict a sinusoidal signal with that frequency perfectly:
![\begin{displaymath}
P(Z) = 1/2\,(Z_0/Z+Z/Z_0)\;.
\end{displaymath}](img30.png) |
(11) |
The analysis easily extends to higher-order filters.
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2013-03-02