Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 114

is a wavelet with the same autocorrelation as the optimum wavelet a . Since expres-
sion (2.105) holds for the error energy, whether or not a is optimum, we have for α

m

− α ∗ 0 b ∗ 0 +

m φ ∗ aa ( s )φ bb ( s ).

= − α 0 b 0

I 1 (2.106)

s =−

For the optimum wavelet, I is minimal, so

a ∗ 0 b ∗ 0 ≤− α 0 b 0 − α ∗ 0 b ∗ 0 , − a 0 b 0 − (2.107)

or

2 c 0 ≥ 2Re (α 0 b 0 ). (2.108)

In terms of their delay properties, wavelets whose coe er at most by a

complex constant of unit magnitude are equivalent (see Section 2.1.5), and we may
take the phase of α 0 to be the opposite of that of b 0 .Thenwehave c 0 ≥ α 0 b 0 or
a 0 b 0

cients di ff

≥ α 0 b 0 or | a 0 |≥| α 0 | . Thus, the leading coe cient of a is as large, or greater, in

magnitude than the leading coe cient of any other wavelet in the suite of wavelets

with the same autocorrelation. Hence, the optimum wavelet is the minimum delay
wavelet of the suite. The least error energy inverse is minimum delay.

2.2.3 The Levinson algorithm
Optimum Wiener filter equations (2.69) and (2.80), and minimum error energy
systems of equations (2.98), all have Toeplitz coe

cient matrices. The Levinson
algorithm takes advantage of the Toeplitz form of the coe cient matrices to con-

struct a rapid, recursive method of solution for such systems of equations. In matrix
notation, the system of equations to be solved takes the form,
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

r 0 r − 1 ··· r − m f 0

f 1
. . .
f m

g 0
g 1
. . .
g m

r 1 r 0 ··· r − m + 1

= , (2.109)

. . . . . . . . .

. . .

r m r m − 1 ··· r 0

with r j representing the autocorrelation sequence at lag j in the lag domain. Writing
the transpose of the coe cient matrix as R m , the system of equations in row form

becomes

( f 0 , f 1 ,..., f m ) R m = (g 0 ,g 1 ,...,g m ). (2.110)

Since both the autocorrelations and the coe cient matrix itself are Hermitian,

R m = R ∗ m .