Scaling and Wavelet Filter

`Y = qmf(`

* X*,

`P`

Y = qmf(

`X`

Y = qmf(X,0)

`Y = qmf(`

changes the signs of the even index elements of the reversed vector filter
coefficients * X*,

`P`

`X`

`P`

`0`

. If `P`

`1`

, the signs of the odd index elements are
reversed. Changing `P`

changes the phase of the Fourier
transform of the resulting wavelet filter by π radians.`Y = qmf(`

is
equivalent to * X*)

`Y = qmf(X,0)`

. Let `x`

be a finite energy signal. Two filters *F _{0}* and

$${\Vert {y}_{0}\Vert}^{2}+{\Vert {y}_{1}\Vert}^{2}={\Vert x\Vert}^{2}$$

where *y _{0}* is a decimated
version of the signal

For example, if *F _{0}* is a Daubechies scaling filter
with norm equal to 1 and

`qmf`

(`db10`

):$$|{F}_{0}(z){|}^{2}+|{F}_{1}(z){|}^{2}=2.$$

Strang, G.; T. Nguyen (1996), *Wavelets and Filter
Banks*, Wellesley-Cambridge Press.