Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 111

and the impulse response is modelled as the ( n + 1)-length finite wavelet,

= b 0

↑ , b 1 ,..., b n .

b

The convolution c of the wavelets a and b is to approximate the unit impulse,
δ = 1

↑ ,0,0,...,0 .

The error sequence is then the ( m + n + 1)-length wavelet,

= 1 c m + n ,

−

↑ , −

c 0 c 1 ,..., − (2.86)

and the error energy is
I = 1

− c 0 1 − c ∗ 0 + c 1 c ∗ 1 +···+ c m + n c ∗ m + n

m + n

c ∗ 0 + c l c ∗ l . = 1 − c 0 − (2.87)

l = 0

Since c is the convolution of a and b ,

m

c l = a k b l − k , c 0 = a 0 b 0 , (2.88)

k = 0

⎝ ⎠

m + n m m

a ∗ 0 b ∗ 0 + a ∗ j b ∗ l − j I = 1 − a 0 b 0 − a k b l − k

l = 0 k = 0 j = 0

m m m + n

a ∗ 0 b ∗ 0 + a k b l − k a ∗ j b ∗ l − j . = 1 − a 0 b 0 − (2.89)

j = 0 k = 0 l = 0

The error energy is minimised by setting to zero its partial derivatives with respect
to the real and imaginary parts of the elements of the inverse wavelet.
Then,

δ

j b ∗ l − j

∂ I
∂Re a 0 =−

b ∗ 0 +

0

k b l − k a ∗ j b ∗ l − j +

0

b 0 − a k b l − k δ = 0, (2.90)

j , k , l

δ

j b ∗ l − j

i ∂ I b ∗ 0 −

0

k b l − k a ∗ j b ∗ l − j −

0

∂Im a 0 =

b 0 − a k b l − k δ = 0. (2.91)

j , k , l

Adding equations (2.90) and (2.91), we find that

m + n

a k b l − k δ

j b ∗ l − j

m

0

b l − k b ∗ l = b ∗ 0 , = a k (2.92)

j , k , l k = 0 l = 0