Continuous-time system

A continuous-time system is anything that takes a continuous-time signal as input and produces a continuous-time signal as output. The rule mapping input to output can be a differential equation, a block diagram, a description in English, or a pure mathematical operation like $y(t)=x^2(t)$ — as long as it maps inputs to outputs unambiguously, it’s a system.

We write the rule with a calligraphic $\mathcal{H}$:

$y(t)=\mathcal{H}(x(t))$

read as ”$y$ is what the system $\mathcal{H}$ does to $x$.” The input $x(t)$ is also called the excitation, the output $y(t)$ the response.

Examples to fix the idea

A microphone is a system: sound pressure in, voltage out. An amplifier: small voltage in, large voltage out. A boat with a rudder: rudder angle in, heading out. An RC lowpass filter: input voltage across the whole thing, output across the capacitor. The same mathematical framework describes all of them.

For the RC lowpass, applying Kirchhoff’s voltage law and the capacitor equation $i=Cd{v}_{\text{out}}/dt$ gives

$v_{\text{in}}(t)=RC\frac{d{v}_{\text{out}}}{dt}+v_{\text{out}}(t).$

That’s a first-order linear ODE relating input and output — the defining rule of the system. Most of the systems we’ll study can be written as a linear differential equation of this type.

Building blocks

Any system describable by a finite-order linear ODE can be built from three primitive blocks plus wires:

• Amplifier: multiplies its input by a constant $K$. Output is $K\cdot x(t)$.
• Summing junction: outputs the sum (or difference) of its inputs.
• Integrator: outputs the running integral ${\int }_{-\infty }^{t}x(\tau )d\tau$.

Amplifiers handle coefficients, summers handle additions, integrators handle the implicit integrations in the ODE. This is what makes block-diagram representations equivalent to differential equations.

Zero-state response

When we test a system’s properties — homogeneity, additivity, time-invariance, etc. — we always work with the zero-state response: the output that comes purely from the input, with all internal states (capacitor voltages, integrator memories) starting at zero. A system can also have a nonzero response just from its initial state, but that’s a separate piece treated by the unilateral Laplace transform.

The properties we care about

Each system has some combination of the following properties:

1. Homogeneity — scaling the input scales the output by the same factor.
2. Additivity — sum of inputs gives sum of outputs.
3. Linearity — homogeneous and additive together.
4. Time-invariance — delaying the input just delays the output.
5. BIBO stability — bounded inputs produce bounded outputs.
6. Causality — output depends only on past and present inputs.
7. Memory — output depends on input values at times other than now.

The combination of linearity and time-invariance gives the LTI class, which is the central object of the rest of the course.