I read a document where the author talked about normalization of an LTI system. Following was discussed:
Let's assume an LTI system with frequency response
$$ \underline{\boldsymbol{G}}(j\omega) = \text{Re}\{\underline{\boldsymbol{G}}(j\omega)\} + j \, \text{Im}\{ \underline{\boldsymbol{G}}(j\omega) \} $$
For a sinusoidal signal with single frequency, i.e.
$$ x(t) = x_{cos} \, \cos(\omega t) + x_{sin} \sin(\omega t) = \underbrace{ \begin{bmatrix} x_{cos} & x_{sin} \end{bmatrix} }_{\boldsymbol{x}^\top} \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \end{bmatrix}^\top $$
the response
$$ y(t) = y_{cos} \, \cos(\omega t) + y_{sin} \sin(\omega t) = \underbrace{ \begin{bmatrix} y_{cos} & y_{sin} \end{bmatrix} }_{\boldsymbol{y}^\top} \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \end{bmatrix}^\top $$
can be computed (in terms of coefficients) as
$$ \boldsymbol{y} = \boldsymbol{G}(j \omega) \, \boldsymbol{x} $$
were $\boldsymbol{G}(j \omega)$ is the real-valued version of $\underline{\boldsymbol{G}}(j \omega)$, written in matrix notation:
$$ \boldsymbol{G}(j \omega) = \begin{bmatrix} \text{Re}\{\boldsymbol{G}(j\omega)\} & \text{Im}\{ \boldsymbol{G}(j\omega)\} \\ -\text{Im}\{ \boldsymbol{G}(j\omega)\} & \text{Re}\{\boldsymbol{G}(j\omega)\} \end{bmatrix} $$
Now, this system $\boldsymbol{G}$ is normalized by dividing by the euclidean norm, which is:
$$ ||\boldsymbol{G}(j\omega)||_2 = \frac{1}{\sqrt{2} \sqrt{ \text{Re}\{\boldsymbol{G}(j\omega)\}^2 + \text{Im}\{\boldsymbol{G}(j\omega)\}^2} } $$
Therewith, the normalized system is:
$$ \boldsymbol{G}_{\text{norm}}(j\omega) = \frac{1}{\sqrt{2} \sqrt{\text{Re}\{\boldsymbol{G}(j\omega)\}^2 + \text{Im}\{\boldsymbol{G}(j\omega)\}^2}} \, \boldsymbol{G}(j\omega) $$
Now, my question is:
- Is it possible to make some general statements about the normalized system with transfer function $\boldsymbol{G}_{\text{norm}}(s)$?
- In which cases is this normalization useful? (Sure, I know where the author of this document used it, but I could imagine that there are further use cases).
Update:
E.g. if $\boldsymbol{G}(j\omega)$ would be the identity matrix
$$ \boldsymbol{G}(j\omega) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
then this would result in following normalized matrix:
$$ \boldsymbol{G}_{\text{norm}}(j\omega) = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
Still, I can't see the usefulness here.