Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 104

er at most by a complex constant of unit mag-
nitude have the same autocorrelation and are said to be equivalent .
Although there is a unique minimum delay wavelet with the same autocorrela-
tionasagiven( n

Wavelets whose coe cients di ff

1)-length wavelet, we need to show that for a given autocorrel-
ation there is only one minimum delay wavelet. We begin with the z-transform of
the given autocorrelation,

+

n ) z − n 1) z − n + 1 Φ ( z ) = φ( − + φ( − n + +···

1) z n − 1 + φ( n ) z n + φ(0) +···+ φ( n − . (2.51)

The polynomial

z n P ( z ) = Φ ( z )

= φ( − n ) + φ( − n + 1) z +···

+ φ(0) z n 1) z 2 n − 1 + φ( n ) z 2 n +···+ φ( n − (2.52)

is then of degree 2 n with 2 n roots. From the Hermitian property of the autocorrel-
ation, we have

= φ ∗ ( n ) + φ ∗ ( n P ( z ) − 1) z +···

+ φ(0) z n +···+ φ ∗ ( 1) z 2 n − 1 + φ ∗ ( n ) z 2 n − n + −

z 2 n P ∗ (1/ z ∗ ). = (2.53)

In factored form P ( z ) becomes

P ( z ) = φ( n ) ( z − z 1 )( z − z 2 ) ··· ( z − z 2 n )

φ ∗ ( n ) 1/ z z ∗ 1

1/ z

z ∗ 2 n z 2 n P ∗ (1/ z ∗ ) z 2 n = = − ··· −

= φ ∗ ( n ) 1 z ∗ 1 z 1 z ∗ 2 z

1

z ∗ 2 n z . − − ··· − (2.54)

Thus, for every root z j of P ( z ) there is another root at 1/ z ∗ j . Hence, P ( z )isthe
product of two polynomials, one with roots at z 1 , z 2 ,..., z n , the other with roots at
1/ z ∗ 1 ,1/ z ∗ 2 ,...,1/ z ∗ n .
If G ( z ) is the polynomial of degree n ,

G ( z ) = g n ( z − z 1 )( z − z 2 ) ··· ( z − z n ), (2.55)

then

= g ∗ n 1/ z z ∗ 1 1/ z z ∗ 2 ··· 1/ z z ∗ n . G ∗ (1/ z ∗ ) − − − (2.56)

2 ,wefindthat

Then, with appropriate choice of | g n |

G ( z ) z n G ∗ (1/ z ∗ ) P ( z ) = (2.57)