f

#### Signed square ss(x) = (x^2)^(1/2)*x = sgn(x)*x^2

```Hi

Since the power is more relevant to analyze than the amplitude I have come =
up with the idea of using signed square ss(x) =3D x^2 for positive x and -x=
^2 for negative x or ss(x) =3D (x^2)^(1/2)*x =3D sgn(x)*x^2

Instead of analyzing the power as the square of the amplitude the signed po=
wer could be used instead. This would preserve more information of the orig=
inal signal. The original amplitude signal can be recreated from the signed=
square power signal with signed square root.

I can imagine signed square to be applicable to codec evaluation and signal=
compression since the power of the signal is more relevant to compress tha=
n the amplitude.

It would be easy to make a simple test with Scilab and some sound or image =
file.

Would it be possible to analyze this theoretically with Karhunen-Lo=E8ve th=
eory?

David

```
 0
10/29/2013 2:49:11 PM
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```On Tuesday, October 29, 2013 10:49:11 AM UTC-4, davidjons...@gmail.com wrot=
e:
> Hi
>=20
>=20
>=20
> Since the power is more relevant to analyze than the amplitude I have com=
e up with the idea of using signed square ss(x) =3D x^2 for positive x and =
-x^2 for negative x or ss(x) =3D (x^2)^(1/2)*x =3D sgn(x)*x^2
>=20
>=20
>=20
> Instead of analyzing the power as the square of the amplitude the signed =
power could be used instead. This would preserve more information of the or=
iginal signal. The original amplitude signal can be recreated from the sign=
ed square power signal with signed square root.
>=20
>=20
>=20
> I can imagine signed square to be applicable to codec evaluation and sign=
al compression since the power of the signal is more relevant to compress t=
han the amplitude.
>=20
>=20
>=20
> It would be easy to make a simple test with Scilab and some sound or imag=
e file.
>=20
>=20
>=20
> Would it be possible to analyze this theoretically with Karhunen-Lo=E8ve =
theory?
>=20
>=20
>=20
> David

I don't know off hand where a cubic representation has an advantage over th=
e classical cases, but one may exist. While you are in this vein, you may w=
ish to look up the Teager-Kaiser energy operator.

Clay
```
 0
clay (793)
10/29/2013 5:31:21 PM
```On Tuesday, October 29, 2013 6:31:21 PM UTC+1, cl...@claysturner.com wrote:
>
> I don't know off hand where a cubic representation has an advantage over the classical cases, but one may exist. While you are in this vein, you may wish to look up the Teager-Kaiser energy operator.
>

Signed square is not cubic. There are only squared terms involved. Maybe you prefer signed square written like this

x^2 for x >= 0
ss(x) =
-x^2 for x < 0

I could not find any application of it on Internet. The inverse to ss(x) is signed square root and it has applications in geology and exists in some software.

Teager-Kaiser energy operator TKEO = x^2(n) - x(n+1)*x(n-1) seems to find energy differences which might not be what I am looking for.

There must be a problem with TKEO since it gives the same value for -x(n) as for x(n) (for the same x(n+1) and x(n-1)).

That problem could be avoided by defining a signed square TKEO = SSTKEO(x(n)) = ss(x(n)) - sgn2(x(n+1), x(n-1))*x(n+1)*x(n-1)
where sgn2(a, b) is defined as being positive only when a and b are positive and negative if either or both a and b are negative.

The only eventual problem with ss(x) is that it has a discontinuous second derivative. And sgn() and sgn2() are discontinuous.

David
```
 0
10/30/2013 9:18:19 AM
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