# Solved – How to compute the steady state gain from the transfer function of a discrete time system

self-studytime series

Having the Transfer Function of a discrete system as such:
$$H(z) = \frac{0.8}{z(z-0.8)}$$
I am asked to find the Steady State Gain of the system.
I have the solution and it simply states:
$$H(1) = \frac{0.8}{1-0.8} = 4$$
I have no idea why this is the case, and the book, the slides, the solutions, my (limited) notes and the internet are seemingly empty on the subject. Is the Steady State Gain of a system always the outcome of the Transfer Function applied to 1? That just sounds ridiculous, especially since I'm not finding any references to it online.
I was chased out of mathoverflow with this question, those guys really hate homework… Then again, who doesn't.

The "mechanical" result of just plugging in $z = 1$ into the transfer response is essentially a product of two facts. The steady-state gain is (usually, I believe) defined as the (magnitude of the) limiting response as $t \to \infty$ of the system to a unit-step input.

The so-called final-value theorem states that, if the limit $\lim_{n \to \infty} y(n)$ exists, then $\lim_{n \to \infty} y(n) = \lim_{z \to 1} (1-z^{-1}) Y(z)$ where $y(n)$ is the time-domain output and $Y(z)$ is its corresponding $z$-transform.

Now, for the steady-state gain, the input is a unit-step function, so $x(n) = 1$ for each $n \geq 0$. Hence, $$X(z) = \sum_{n=0}^\infty x(n) z^{-n} = \sum_{n=0}^\infty z^{-n} = \frac{1}{1-z^{-1}} .$$

Using the transfer equation, we get that the $z$-transform of the output is $$Y(z) = X(z) H(z) = \frac{H(z)}{1-z^{-1}} .$$

(Assuming that the limit $\lim_{n\to\infty} y(n)$ exists) we have that $$\lim_{n \to \infty} y(n) = \lim_{z \to 1} (1-z^{-1}) Y(z) = \lim_{z \to 1}\, H(z) .$$

The left-hand side is the steady-state value of a step-response (i.e., it is the value of the response as time goes to $\infty$ of a one-unit constant input), and so the steady-state gain is $|\lim_{n \to \infty} y(n)| = \lim_{n \to \infty} |y(n)|$.

Technically, you need to check that the limit exists (which I've tried to emphasize). It seems to me that a sufficient condition would be that all the poles of the transfer response be strictly inside the unit circle. (Caveat lector: I haven't checked that closely at all.)

If this does not sufficiently clarify things, you might try doing Google searches on terms like "dc gain" and "final-value theorem", which are closely related to what you want.