Describe LTI system properties

An LTI system is linear and time invariant; it is fully described by its impulse response $h(t)$, and $y(t) = x(t) * h(t)$.

(a) Linearity

A system is linear if it obeys superposition:

$$a\,x_1(t) + b\,x_2(t) \rightarrow a\,y_1(t) + b\,y_2(t)$$

It combines additivity ($x_1 + x_2 \rightarrow y_1 + y_2$) and homogeneity ($a x \rightarrow a y$). Zero input must give zero output.

• Linear: $y(t) = 5x(t)$, $y(t) = \int_{-\infty}^{t} x(\tau)d\tau$.
• Non-linear: $y(t) = x^2(t)$, $y(t) = x(t) + 3$.

(b) Causality

The output at any time depends only on the present and past inputs. For an LTI system:

$$h(t) = 0 \quad \text{for } t < 0$$

so that $y(t) = \int_0^{\infty} h(\tau)x(t-\tau)d\tau$ uses only $x$ up to time $t$.

• Causal: $h(t) = e^{-3t}u(t)$.
• Non-causal: $h(t) = \delta(t+1)$, i.e. $y(t) = x(t+1)$.

(c) Stability

A system is BIBO stable if every bounded input gives a bounded output. Since $|y(t)| \le M_x\int|h(\tau)|d\tau$ for $|x| \le M_x$, the condition for an LTI system is:

$$\int_{-\infty}^{\infty} |h(t)|\,dt < \infty$$

• Stable: $h(t) = e^{-t}u(t)$ (area 1).
• Unstable: $h(t) = u(t)$, $h(t) = e^{t}u(t)$.

(d) Time invariance

A time shift in the input gives the same time shift in the output:

$$x(t-t_0) \rightarrow y(t-t_0)$$

The system's characteristics do not change with time (constant parameters).

• Time invariant: $y(t) = x(t-1)$, $y(t) = \frac{dx}{dt}$.
• Time varying: $y(t) = t\,x(t)$, $y(t) = x(2t)$.