Time-invariance

A system is time-invariant if delaying the input by $t_0$ delays the output by exactly the same $t_0$, and otherwise leaves the output unchanged. In symbols: if $g(t)\to y_1(t)$, then $g(t-t_0)\to y_1(t-t_0)$ for any $t_0$.

Pictorially: the system doesn’t care what time it is. Whether you run the experiment now or tomorrow, the only difference in the output is that it happens tomorrow instead of now. No external clock is influencing the system’s behavior.

How to test

1. Let $x_1(t)=g(t)$. Find $y_1(t)$.
2. Let $x_2(t)=g(t-t_0)$. Find $y_2(t)$.
3. Check whether $y_2(t)=y_1(t-t_0)$ for every $g$ and $t_0$.

Worked examples

$y(t)=\exp (x(t))$: $x_1=g$, $y_1=e^g$. $x_2=g(t-t_0)$, $y_2=e^{g(t-t_0)}$. And $y_1(t-t_0)=e^{g(t-t_0)}$. Equal, so time-invariant. (Note: this system was not linear, but it is time-invariant — properties are independent.)

$y(t)=x(t/2)$: $x_1=g$, $y_1(t)=g(t/2)$. $x_2=g(t-t_0)$, $y_2(t)=g(t/2-t_0)$. But $y_1(t-t_0)=g((t-t_0)/2)=g(t/2-t_0/2)$. Not equal — $y_2$ has $t_0$, $y_1(t-t_0)$ has $t_0/2$. So not time-invariant (it’s time-variant).

The intuition: this system stretches its input by 2. If you delay the input, the stretched output sees the delay in stretched form (delayed by $2t_0$, not $t_0$). Any system that time-warps its input — scales the time axis, modulates with a time-dependent function — will generally be time-variant.

Why this matters

Time-invariance combined with linearity gives LTI. The Impulse response of an LTI system is a single function $h(t)$ that captures the entire input-output behavior, and any input’s response can be computed by convolution $y=x\ast h$.

Without time-invariance, we couldn’t shift the response of an impulse located at $t_0$ from the response of an impulse at $t=0$ — we’d need a separate impulse response for every possible impulse location, and the convolution formula wouldn’t make sense.

A system that is time-variant on purpose

Modulation in communication systems: $y(t)=x(t)\cdot \cos (2\pi f_{c}t)$ multiplies the input by a time-dependent function. This is time-variant by construction. AM radio, FM radio, and most modern communication schemes are time-variant operations applied to time-invariant message signals. The course treats LTI as the foundation, and modulation as a useful time-variant construct built on top.