What does G(1) = 1 say about a system?

Question

This is a line from a paper I've been reading:

The static gain of the closed loop system must be $1$ ($G(1) = 1$) [...]

First of all: I know what gain is, but isn't gain dependent upon frequency? And why does $G(1)$ do the trick of determining the static gain? And what is "static" gain, anyway?

We are talking about a discrete-time closed loop system and $G(z)$ is its transfer function.

Answer 1

Static gain refers to the DC gain. Namely, it would be the ratio of the output and the input under steady state condition.

Due to DC corresponding to $\omega=0$, in the $z$-domain DC would correspond to $z=1$ because $z=re^{j\omega}$, with $r=1$ (i.e. you have to evaluate your transfer function $G(z)$ in the unit circle to get the frequency response).

EDIT:

The frequency response of a system is represented in the Fourier domain, $G(\omega)$. The DTFT of a given discrete sequence can be calculated as:

$$G_F(\omega)=\sum_{n=-\infty}^{\infty}g[n]e^{-j\omega n}$$

On the other hand, the $z$-transform is defined as:

$$G_Z(z)=\sum_{n=-\infty}^{\infty}g[n]z^{-n}$$

It's easy to see that if we evaluate $z=e^{j\omega}$ (the unit circle), then the $z$-transform returns the frequency response.

Answer 2

In $G(z)$, $z$ denotes a complex variable, interpreted here on the unit-circle as $z=e^{i \omega)}$. When $z=1$, this corresponds to the $0$-frequency, or constant signals. But sometimes, people write this as $G(i\omega)$, as in the definition for the static gain in Transfer Functions:

The number $G(0)$ is called the static gain of the system because it tells the ratio of the output and the input under steady state condition.