find out that system is time-invariant or not

Question

Is this system linear and time-invariant? $$y(t) = −3x(2t − 2) + x(t)$$ I found this that is not time-variant but I am not sure.

Answer 1

Time-invariant means a shift in time in the input leads to the same output, shifted the same amount in time. Let's see if this is the case:

output to the shifted input: $t=t-t_0$ $$-3x(2(t-t_0)-2)+x(t-t_0)=-3x(2t-2t_0-2)+x(t-t_0)$$

shifted output: $y(t-t_0)$ $$-3x(2t-2-t_0)+x(t-t_0)$$

So it is time-variant (or equivaently it is not time-invariant)

Answer 2

Define $(\Theta_Tx)(t) = x(t-T)$.

Let $(Lx)(t) = -3 x(2t-2) +x(t)$.

The system if time invariant iff $\Theta_T L = L \Theta_T$ for all $T$.

$(\Theta_T (L x))(t) = (Lx)(t-T) = -3 x(2t-2-2T) +x(t-T)$.

$(L (\Theta_T x))(t) = -3 (\Theta_T x)(2t-2) +(\Theta_T x)(t) = -3 x(2t-2-T)+x(t-T)$.

Hence if $x$ is not constant, we have some $s,T$ such that $x(s) \neq x(s+T)$, in which case $(\Theta_T (L x))({s \over 2} + 1 +T) \neq (L (\Theta_T x))({s \over 2} + 1 +T)$.