Properties of continuous time systems

Continuous-time systems are classified by the following basic properties.

(a) Linearity

A system is linear if it obeys superposition (additivity + homogeneity): $a x_1(t) + b x_2(t) \rightarrow a y_1(t) + b y_2(t)$.

• Linear: $y(t) = 2x(t)$, $y(t) = \frac{dx}{dt}$.
• Non-linear: $y(t) = x^2(t)$, $y(t) = x(t) + 1$.

(b) Causality

Output at time $t$ depends only on present and past inputs, not future ones. It is non-anticipative and can work in real time.

• Causal: $y(t) = x(t) + x(t-2)$.
• Non-causal: $y(t) = x(t+1)$. For LTI: causal if $h(t) = 0$ for $t < 0$.

(c) Memory

A memoryless (static) system's output depends only on the input at the same instant. A system with memory (dynamic) uses past (or future) inputs.

• Memoryless: $y(t) = 5x(t)$, a resistor $v = Ri$.
• With memory: $y(t) = \int_{-\infty}^{t} x(\tau)d\tau$, a capacitor. For LTI: memoryless only if $h(t) = K\delta(t)$.

(d) Stability

BIBO stable: every bounded input gives a bounded output.

• Stable: $y(t) = e^{x(t)}$, $h(t) = e^{-t}u(t)$.
• Unstable: $y(t) = t\,x(t)$, an ideal integrator. For LTI: $\int|h(t)|dt < \infty$.

(e) Time invariance

A time shift in the input causes the same shift in the output: $x(t-t_0) \rightarrow y(t-t_0)$.

• Time invariant: $y(t) = x(t-3)$.
• Time varying: $y(t) = t\,x(t)$, $y(t) = x(2t)$.