13.1: A brief introduction to linear time invariant systems

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Let’s start by defining our terms.

Signal. A signal is any function of time.

System. A system is some machine or procedure that takes one signal as input does something with it and produces another signal as output.

Linear system. A linear system is one that acts linearly on inputs. That is, \(f_1 (t)\) and \(f_2 (t)\) are inputs to the system with outputs \(y_1 (t)\) and \(y_2 (t)\) respectively, then the input \(f_1 + f_2\) produces the output \(y_1 + y_2\) and, for any constant \(c\), the input \(cf_1\) produces output \(cy_1\).

This is often phrased in one sentence as input \(c_1f_1 + c_2 f_2\) produces output \(c_1 y_1 + c_2 y_2\), i.e. linear combinations of inputs produces a linear combination of the corresponding outputs.

Time invariance. Suppose a system takes input signal \(f(t)\) and produces output signal \(y(t)\). The system is called time invariant if the input signal \(g(t) = f(t - a)\) produces output signal \(y(t - a)\).

LTI. We will call a linear time invariant system an LTI system.

Example \(\PageIndex{1}\)

Consider the constant coefficient differential equation

\[3y'' + 8y' + 7y = f(t) \nonumber \]

This equation models a damped harmonic oscillator, say a mass on a spring with a damper, where \(f(t)\) is the force on the mass and \(y(t)\) is its displacement from equilibrium. If we consider \(f\) to be the input and \(y\) the output, then this is a linear time invariant (LTI) system.

Example \(\PageIndex{2}\)

There are many variations on this theme. For example, we might have the LTI system

\[3y'' + 8y' + 7y = f'(t) \nonumber \]

where we call \(f(t)\) the input signal and \(y (t)\) the output signal.