# System classification: unit-time delay

I'm reading a book on linear systems and I can't understand why the unit-time delay is a distributed system. This is the example given in the book:

I understand that the initial state of the system is the input, that is:

$$u(t),\,t_0-1\le t

However, for me, a distributed system is something like a transmission line, where the parameters can't be modeled accurately by "discrete" and finite elements.

I can't grasp why this system is classified as distributed just because there are an infinite number of points describing the initial state.

You are right that a distributed system could be "something like a transmission line". Note that the system

$$y(t)=x(t-T)\tag{1}$$

is a simple model of a transmission line, where just a frequency-independent delay $$T$$ is taken into account, and the attenuation is neglected.

Note that lumped electrical systems, described by resistors, capacitors and inductors, result in ordinary differential equations, i.e., the output $$y(t)$$ depends on the time derivatives of the input and output signals. If the system is also time-invariant, i.e., the component properties do not change over time, you can define a transfer function, and with lumped elements that transfer function is always rational. The transfer function describing the system given in $$(1)$$ is

$$H(s)=e^{-sT}\tag{2}$$

which is clearly non-rational.

However, for me, a distributed system is something like a transmission line, where the parameters can't be modeled accurately by "discrete" and finite elements.

I can't grasp why this system is classified as distributed just because there are an infinite number of points describing the initial state.

It's not just that you need an infinite number of points to describe the initial state -- you need an infinite number of points to describe the state at any given time.

Basically, to answer the question "what's the output at $$y(t)$$?", you need to know the input at absolutely every point from $$t$$ to $$t - 1$$ -- that's the essence of "distributed".

• I don't understand your last paragraph: for knowing the output $y(t)$ at $t=t_0$ I just need the input $x(t)$ at the point $t=t_0-1$. But I might be misunderstanding what you're saying. Commented Mar 15, 2020 at 11:48
• What will the output be at $t + \tau_1$? What will the output be at $t + \tau_2$? What will the output be at $t + \tau$ for absolutely every value of $\tau$ between 0 and 1? And how many "states" do you need to have to express that? Commented Mar 15, 2020 at 14:38

Since for a transmission line, the system is distributed since RLC is distributed in infinitesimally small sections in the line. The t values can be assumed to be infinitesimally small sections of the unit time delay system.

As far as I understand a lumped systems needs the knowledge of an finite(or countable infinite, i'm not sure about that) number of states to describe the current state and predict the future evolvement. A distributed system needs information about an uncountable infinite number of states. Here comes the clue: without stating in which dimension the infinites states exist, the first thing which pops in mind is a spatial distribution, a system which consists of physical body who has an infinite of points in space. But I think it could meant to be a dependency on infinite points in any dimesion. And to predict the evolution of a system with time delay, I indeed need inforation about the states from [t_0 - tau; t_0] which are uncountable infinite in number.