Time-invariant and time varying systems

Time-invariant (constant parameter) systems – parameters don’t change over time. Time-varying – parameters change over time.

Time-delay: If its input is delayed by $T$, the output is the same but is delayed by $T$:

Example 1:

System: $y[n]=nx[n]$. Let it $x_2[n]=x_1[n-m]$. Then we have:

$$y_2[n]=n{x}_2[n]=n{x}_1[n-m]$$

Delaying $x_1[n]$ by $m$ units would give us $x_1[n-m]$. This would be:

$$\begin{array}{rl}{y}_1[n-m]& =n{x}_1[n-m]\\ & =(n-m)x_1[n-m]\end{array}$$

So have $y_1[n-m]\ne y_2[n]$, making the system time varying. This is because when the input is delayed by $m$, the output is not the same but delayed by $m$.

Example 2:

System: $y(t)=x(2t)$. Let $x_2(t)=x_1(t-T)$. For the system to be time-invariant, we would have:

$$y_2(t)=y_1(t-T)$$

Solving $y_1(t-T)$, we have:

$$\begin{array}{rl}{y}_1(t-T)& =x_1(2(t-T))\\ & =x_1(2t-2T)\end{array}$$

Solving $y_2(t)$, we have:

$$\begin{array}{rl}{y}_2(t)& =x_2(2t)\\ & =x_1(2t-T)\end{array}$$

Since $y_2(t)\ne y_1(t-T)$, the system is time-invariant.