in (41). While the ξ(ί) are uncorrelated, the v(0 possess pairwise correlations:
£[v(/)vO)]
=
1 + p
2
'
-p
'
= J
(60a)
(1+P
2
)
2
'
10,
otherwise
for i and j equal to 1,
2,...,
N. The correlation properties of the end residuals
are described by
£[v
2
(0)] = E[_v
2
(N + 1)] = (1 - p
2
)
£[v(0)v(l)] = E[v(N)x(N + l)] = -pi
and
£[v(0)v(0]
= E[v(j)v(N + 1)] = 0 for i > 1 and j < Ν (60b)
The correlation in (60a) and (60b) may be derived by substituting the ex-
pressions for v(i) in terms of f(i) [as given by 59)] and then using (40).
In discussing the fast implementation of the Karhunen-Loeve transform,
we will b ...
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