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.
