Cascading Systems

Problem

Consider two systems:

  • x(t) → 1 → x(t-7)
  • x(t) → 2 → x(2t)

What happens in the following?

a. x(t) → 1 → 2 → ?

b. x(t) → 2 → 1 → ?


Solution to a.

Let's start with a.

Define functions x, y, and z as follows:

x → 1 → y → 2 → z


x(t) → 1 → y(t) = x(t-7)

y(t) → 2 → z(t) = y(2t)

z(t) = y(2t)

y(2t) = x((2t)-7) = x(2t-7).


It may be helpful to consider x(t) = t.

If x(t) = t, then y(t) = x(t-7) = t-7.

If y(t) = t-7, then z(t) = y(2t) = (2t) - 7.

Then, we can simply conclude that since x(2t-7) = 2t-7, z(t) = x(2t-7).


Solution to b.

Next, we move onto b.

Define functions x, y, and z as follows:

x → 2 → y → 1 → z


x(t) → 2 → y(t) = x(2t)

y(t) → 1 → z(t) = y(t-7)

z(t) = y(t-7)

y(t-7) = x(2(t-7)) = x(2t-14).


Again, it may be helpful to consider x(t) = t.

If x(t) = t, then y(t) = x(2t) = 2t.

If y(t) = 2t, then z(t) = y(t-7) = 2(t-7) = 2t-14.

Then, we can simply conclude that since x(2t-14) = 2t-14, z(t) = x(2t-14).


Note

The answers to solutions in a and b are not the same! A cascaded system cannot be freely swapped around and be expected to behave in the same way.

Alumni Liaison

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett