MIT 6.300 — Lecture 8

Systems

§ Lecture and Recitation: Systems

Definition. A system is a function that maps signals $x[n]$ to signals $y[n]$ .

Such a system is additive if $x_1[n] + x_2[n] \mapsto y_1[n] + y_2[n]$ .
Such a system is homogeneous if $\lambda x[n] \mapsto \lambda y[n]$ for any constant $\lambda$ .
Such a system is linear if it is both additive and homogeneous.
Such a system is time-invariant if $x[n - n_0] \mapsto y[n - n_0]$ for all integers $n_0$ .

Remark. We call the output signal $y[n]$ of a system the response to the input signal $x[n]$ .

One might imagine the input signal as, say, the voltage supplied to an AC circuit, and the output signal as, say, the voltage across a resistor. In this sense, the output signal is a response to the input signal.

Example. Consider the following systems.

The system $x[n] \mapsto x[n - 1] + x[n] + x[n + 1]$ is linear and time-invariant.
The system $x[n] \mapsto nx[n]$ is linear but not time-invariant.
The system $x[n] \mapsto x[n] + 1$ is not linear but time-invariant.
The system $x[n] \mapsto 0$ (the constant $0$ signal) is linear and time-invariant.

Definition. A system is said to be LTI if it is both linear (L) and time-invariant (TI).

From now on, we will restrict our attention to LTI systems.