# Properties of Positive Real Functions

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is PR/SPR, however I'm not sure how to approach the following example.

Say we are given two positive real transfer functions: $$G_{1}(s)$$ and $$G_{2}(s)$$

How can I tell if the sum $$G_{1}(s) + G_{2}(s)$$ or the product $$G_{1}(s) * G_{2}(s)$$

is PR/SPR?

Intuitively, I think the sum is SPR, but as for the product I am uncertain. Either way, is there a framework I can use to approach this or does anyone know of a proof I could look at?

• This is true for the sum but not for the product. Consider for example $G_1(s)=G_2(s)=\frac{1}{s+1}$. Both $G_1,G_2$ are SPR but their product is not as $Re[G_1^2(j\omega)]<0$ for $\omega>1$. – RTJ Dec 13 '18 at 17:30

At any given $$s$$ the two transfer functions give you two complex numbers $$\rho_1+i\,\sigma_1$$ and $$\rho_2+i\,\sigma_2$$, with the constraints that $$\rho_1,\rho_2\geq0$$.
For the summation you only have that it is strictly positive real iff $$\rho_1$$ and $$\rho_2$$ are not both simultaneously zero. For example $$G_1(s)=\alpha\,G_2(s)$$ with $$\alpha>0$$ would not satisfy this, or the transfer functions already need to be strictly positive real.
For the multiplication the real part would become $$\rho_1\,\rho_2 - \sigma_1\,\sigma_2$$. Therefore, the fact that $$G_1(s)$$ and $$G_2(s)$$ are positive real does not give you enough information about their imaginary parts to tell if the result is even positive real.