SC · Signal basics

Classifying a Signal

Faced with a signal, you ask two things first: is the time axis a line or dots, and does the pattern repeat? These two questions fix what kind of signal it is.

Match the classification label to this signal

The faint curve is the real signal: it has a value at every instant and the same shape repeats at a fixed spacing. Overlay each of the three candidate labels and find the one that fits with no gaps.

Tap to switch the candidate label.

Chosen label and condition

Discrete · aperiodic x[n]  n ∈ ℤ

Keeping only dots throws away every value between them. Calling a continuous signal discrete is a misaligned label.

Misaligned

Is the time axis a line or dots

A continuous-time signal has a value at every real instant t; its time axis is an unbroken line. A discrete-time signal has values only at integer indices n; its time axis is dots, and the notation uses brackets, x[n].

Does the pattern repeat

If shifting a signal by some positive T leaves it unchanged, it is periodic: the condition is x(t) = x(t + T). The smallest such T is the fundamental period. If no such T exists at all, the signal is aperiodic.

Observex(t) = x(t + T)

Periodicity: a shift by T leaves it the same.

Choosex[n] = x[n + ?]

Discrete: periodic if a shift by index N matches.

Fill inx(t) = x(t + ?)

If one period matches, so does every integer multiple kT.

On your ownT = ?

The fundamental period of cos(ωt) is 2π/ω.

The two questions are independent

Continuous-versus-discrete is decided separately from periodic-versus-aperiodic, so all four cells are possible. You land in a cell by answering each question on its own.

Continuous · Periodic

a sine wave

Continuous · Aperiodic

a single pulse

Discrete · Periodic

a repeating sample train

Discrete · Aperiodic

recorded noise

Back to the first screen

The label that explained the faint curve with no gaps was continuous and periodic. Because a value sits at every instant, the dots-only label was misaligned, and the single-window label missed the recurrence and stalled at almost. Classification is, in the end, answering two independent questions in turn: is the time axis a line or dots, and does the pattern come back?

A signal is classified along two independent axes. If the time axis is defined at every t, it is continuous-time x(t); if only at integer indices n, it is discrete-time x[n]. If x(t) = x(t + T) holds for some positive T, the signal is periodic; if no such T exists, it is aperiodic. The smallest T is the fundamental period.

On to the next units

Classification is the first button on every signal story. The continuous-versus-discrete split leads later to the sampling theorem, and periodicity is where the Fourier series begins to spread a periodic signal into a sum of sinusoids. The next unit, elementary signals, builds the alphabet of signals: impulse, step, exponential.