Basic properties of systems

A system maps an input x(t) to an output y(t). Its basic properties are as follows.

1. Linearity

A system is linear if it obeys superposition (additivity and homogeneity): the response to a·x₁ + b·x₂ is a·y₁ + b·y₂.

• Linear: y(t) = 3x(t). Non-linear: y(t) = x²(t), and y(t) = 2x(t) + 3 (because of the constant).

2. Time invariance

A time shift in the input only shifts the output by the same amount: x(t - t₀) gives y(t - t₀).

• Time-invariant: y(t) = x(t) + x(t - 1). Time-varying: y(t) = t·x(t).

3. Causality

The output at time t depends only on present and past inputs, not on future ones.

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

4. Stability (BIBO)

A bounded input always gives a bounded output. For an LTI system: ∫|h(t)| dt < ∞.

• Stable: h(t) = e^(-t)u(t). Unstable: h(t) = u(t) (integrator).

5. Memory

A memoryless (static) system has an output that depends only on the present input; a system with memory depends on past or future values.

• Memoryless: y(t) = 5x(t). With memory: y(t) = x(t - 1), or an integrator.

Another property is invertibility: distinct inputs give distinct outputs, so the input can be recovered from the output.