Memory (system property)

A system is static (or memoryless) if its output at time $t$ depends only on its input at the same time $t$. Otherwise, it has memory.

Examples

• $y(t)=3x(t)$: static. Output now is three times input now.
• $y(t)=x^2(t)$: static. Output now is input now squared.
• $y(t)=\sin (x(t))$: static. Pointwise nonlinearity, but only depends on $x(t)$.
• $y(t)={\int }_{-\infty }^{t}x(\tau )d\tau$: has memory. Output depends on the entire past history of the input.
• $y(t)=x(t-1)$: has memory. Output now is input from one second ago — system has to “remember” that value.
• An RC lowpass filter: has memory. The capacitor voltage at $t$ depends on how much charge has accumulated, which depends on the past current.

Two useful facts

Static $\Rightarrow$ causal. If the output now depends only on the input now, it certainly can’t depend on the future. So causality is a weaker condition than memorylessness — every static system is causal, but plenty of causal systems have memory.

Storage elements give memory. Any system containing an integrator, capacitor, inductor, or other storage element has memory. Resistors don’t store energy and don’t introduce memory; capacitors and inductors do.

In context

Most of the interesting systems we study in this course have memory: RC filters, RLC filters, anything with dynamics. The system has an impulse response $h(t)$ that is nonzero for $t>0$, encoding how the system’s response at time $t$ accumulates contributions from past inputs.

Static systems show up as building blocks inside larger dynamic systems — multipliers, summers, pointwise nonlinearities — but rarely as systems of interest on their own. The course’s main object of study is the dynamic LTI system, characterized by an impulse response that decays smoothly to zero.

Memoryless LTI

An LTI system can be memoryless only if its impulse response is a scaled impulse: $h(t)=K\delta (t)$ for some constant $K$. Then $y(t)=Kx(t)$ — a pure scaling. Any other impulse response (containing a $u(t)$ tail, say) gives the system memory.