Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 113

Then,

m

a ∗ 0 b ∗ 0 + a ∗ j b ∗ 0 δ

j

I min = 1 − a 0 b 0 −

j = 0

a ∗ 0 b ∗ 0 + a ∗ 0 b ∗ 0

= − −

1 a 0 b 0

= − a 0 b 0 = − c 0 .

1 1 (2.101)

From (2.87), the error energy is

m + n

c ∗ 0 + c l c ∗ l ,

= − −

I 1 c 0 (2.102)

l = 0

giving

m + n

c l c ∗ l = c ∗ 0 = c 0 (2.103)

l = 0

for I I min . Therefore, c 0 must be real and non-negative. From (2.101), since I min

is a sum of squares and must also be real and non-negative definite, we see that
0

=

1. Therefore, I min provides a quality factor between 0
and 1, which measures how well the deconvolution filter is performing. Here, c 0 is
the actual approximation to the desired unit impulse sequence and thus I min meas-
ures the di

≤ c 0 ≤ 1and0 ≤ I min ≤

erence, or 0 .
From expression (2.99) for the error energy,

ff

m m

a ∗ 0 b ∗ 0 + a k a ∗ j φ bb ( j I = 1 − a 0 b 0 − − k ). (2.104)

j = 0 k = 0

Introducing the new index s = j − k and summing first over s andthenover j ,weget

j

m

− a 0 b 0 − a ∗ 0 b ∗ 0 + a ∗ j a j − s φ bb ( s ) I = 1

j = 0 s = j − m

s =− m φ ∗ aa ( s )φ bb ( s ).

m

a ∗ 0 b ∗ 0 + = 1 − a 0 b 0 − (2.105)

Now suppose that

α = α 0

↑ ,α 1 ,...,α m