Question
2. Is the system defined by the equationtime-invariant? Answer yes/no, and justify your answer mathematically.
Solution
Knowing how to apply time invariance is the key to this problem; the solution is simple after that.
Define the system as a function f: $ f(x(t)) = y(t) = x(1-t) $, and define $ S_k(x(t)) = x(t-k) $ (a time shift by k). x(t) is time-invariant iff $ \forall k \in \mathbb{R}, f(S_k(x(t))) = S_k(f(x(t))) $.
$ f(S_k(x(t))) = f(x(t-k)) = x((1-t)-k) = x(1-t-k) $
$ S_k(f(x(t))) = S_k(x(1-t)) = x(1-(t-k)) = x(1-t+k) $
Since $ \exists k \in \mathbb{R} \text{ s.t. } f(S_k(x(t))) \neq S_k(f(x(t))) $ (for instance, k = 1), the system is not time-invariant.
No.
