Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 106

of the desired sequence from the filtered output sequence is measured by an error
sequence j with

j = d j − h j . (2.62)

Optimum Wiener filters minimise the error power or the mean square of the error
sequence. The error power is
E j ∗ j

= E d j − h j d ∗ j − h ∗ j

E ⎩

l = 0 g ∗ l f ∗ j − l ⎭

d j −

k = 0 g k f j − k d ∗ j −

N N

=

E ⎩

k = 0 g k f j − k ⎭ l = 0 g ∗ l f ∗ j − l −

N N

d j d ∗ j − d ∗ j = d j

E ⎩

⎭ .

k = 0 g k f j − k l = 0 g ∗ l f ∗ j − l

N N

+ (2.63)

In the expression for the error power, only the elements g 0 ,...,g N of the wavelet
representing the impulse response have yet to be specified. When, with respect
to the real and imaginary parts of the elements of the impulse response, the partial
derivatives of the error power all vanish, the error power will be an extremum. Since
there can be no maximum to the error power, or how badly the filter performs, the
extremum obtained by setting the partial derivatives to zero must be a minimum.
To minimise the error power, we have

E ⎩ −

k f j − k ⎭

l = 0 δ k = 0 δ

∂Reg m E j ∗ j

N N

∂

m

l f ∗ j − l −

d ∗ j

m

= d j

E ⎩

l f ∗ j − l ⎭

N N

δ

k f j − k g ∗ l f ∗ j − l + g k f j − k δ

+

k = 0 l = 0

E d ∗ j f j − m d j f ∗ j − m − = −

E ⎩

f j − m g ∗ k f ∗ j − k + g k f j − k f ∗ j − m ⎭ =

N

+ 0, (2.64)

k = 0

and

∂Img m E j ∗ j

= E d j f ∗ j − m − d ∗ j f j − m − i ∂

E ⎩

f j − m g ∗ k f ∗ j − k − g k f j − k f ∗ j − m ⎭ =

N

+ 0, (2.65)

k = 0