# What is the meaning of a inner system/transfer function?

In an academic paper I'm currently reading [1], the notion of an inner system is central. It is defined as follows

A (linear) system $G$ (with $m$ inputs and $p$ outputs) is called inner if it satisfies $G^T (z^{-1}) G (z) = I_m$

Further along the article it is said that

an inner function has all poles within the unit circle, and all zeros outside the circle.

which would mean that an inner function is always a (stable) non-minimum phase transfer function.
Are there other implications of the notion of inner function in terms of system characteristics? What does it truly mean for a system to be inner?

• It's pretty obvious but nonetheless relevant that the system will be stable (as all its poles are within the unit circle). Apart from that, what comes to my mind is that property of rational systems that states that the square of the magnitude of the frequency response is the evaluation on the unit circle of $G(z)G^{*}(1/z^{*})$. AFAIK, that works for SISO systems, but maybe you find this useful. – Tendero Nov 7 '17 at 15:16
• The implication of stability is indeed obvious (I have edited my question to include it). The property you mentionned may be related beacuse, for a MIMO system evaluated on the unit circle, $G(z) G^*(1/z^*) = G(z) (G^T(z))^*$, and the definition of inner system gives that $G^T(1/z) G(z) = G^T(z^*) G(z) = I_m$. But it doesn't help me get to the bottom of this notion.... – Klaz Nov 8 '17 at 9:27