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. Mar 15 '20 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? Mar 15 '20 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.