What are the rules of algebra within a real and complex functions $\Re(\cdot)$ and $\Im(\cdot)$?

Question

I have a problem of the form $\Re(1 - G(s)^2) \ge 0$, where $G(s)$ is a transfer function. What type of algebraic manipulation can take place within the $\Re(\cdot)$ function. Do properties like homogeneity and additivity apply within $\Re(\cdot)$?

Answer 1

The function $\Re:\mathbb C\to\mathbb R$ is $\mathbb R$ linear, but obviously not $\mathbb C$ linear.

It is also not multiplicative for general complex inputs: $\Re(i)=0$ but $\Re(i^2)=-1$. But it is elementary to check that $\Re (z^2)=\Re(z)^2-\Im(z)^2$ (or you could even work it out for $\Re(z_1z_2)$ if you really want.)

I don't know anything about transfer functions, though, so I don't know if this gets you any closer to evaluating the expression than $1\geq \Re(G(s)^2)$

Answer 2

When dealing with transfer functions, the complex variable $ s $ is mostly represented as $ s = -\sigma + j\omega $ where $ \sigma $ has the interpretation of damping, and $ \omega $ of angular frequency of the system. Let us assume that the transfer function $ \mathrm{G}(s) $ is in the form: $$ \mathrm{G}(s) = a(\sigma,\omega) + jb(\sigma,\omega) $$ After squearing we get: $$ \mathrm{G}^2(s) = a^2(\sigma,\omega) +j2a(\sigma,\omega)b(\sigma,\omega) - b^2(\sigma,\omega) $$ Then we get: $$ \Re\{1-\mathrm{G}^2(s)\} = \Re\{1 - a^2(\sigma,\omega) -j2a(\sigma,\omega)b(\sigma,\omega) + b^2(\sigma,\omega)\} = 1 - a^2(\sigma,\omega) + b^2(\sigma,\omega) $$ $$ 1 \geq a^2(\sigma,\omega) - b^2(\sigma,\omega) $$ Depanding on the values of $ \sigma $ or $ \omega $ your condition gives you the possible values (it will be a region of values due to the inequality) for the other one.