SC · Signal basics

Classifying a Signal

When you first look at a signal, two questions come before anything else: is the time axis an unbroken line or a row of separate points, and does the same shape repeat? The answers to these two questions decide what kind of signal it is.

Match the classification label to this signal

The faint dashed curve is the actual signal: it has a value at every instant, and the same shape repeats at a fixed interval. Lay the three candidate labels over it one at a time with the tabs and you will see red points, a single blue stretch and a gold curve in turn; the right label is the one that covers the signal without gaps.

Tap a tab to switch the candidate label.

The chosen label and its condition

Discrete · aperiodic x[n]  n ∈ ℤ

Keeping only a few points throws away every value between them. Calling a continuous signal, which has a value at every instant, discrete is a mismatched label.

Misaligned

Is the time axis a line or dots

A continuous-time signal has a value at every real instant t, so its time axis is an unbroken line. A discrete-time signal has values only at integer indices n; its time axis is a row of points, and it is written with square brackets, as in x[n]. Every signal a digital device handles belongs to this discrete side, and many are made by measuring a continuous signal at regular intervals.

Does the pattern repeat

If a signal stays exactly the same when shifted by some positive T, it is periodic, and the condition is x(t) = x(t + T). The smallest T that satisfies this condition is the fundamental period, and if no such T exists at all, the signal is aperiodic. The sine in the figure has ω = π, so its fundamental period is T = 2π/ω = 2, and the two gold vertical dashed lines mark that one period.

Observex(t) = x(t + T)

Periodicity condition: shifting by T leaves it unchanged.

Choosex[n] = x[n + ?]

A discrete signal is periodic if shifting by an integer N leaves it unchanged.

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

If one period's shift leaves it unchanged, so does a shift by any integer k periods.

On your ownT = ?

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

The two questions are independent

Whether a signal is continuous or discrete is decided separately from whether it is periodic or aperiodic, so all four cells can occur. Which cell it belongs in is settled by answering the two questions one at a time. On the discrete side, though, periodicity is a little stricter: the signal must match after a shift by an integer N, so even cos(ωn), a sampled sinusoid, is periodic only when ω/2π is a rational number.

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 covered the faint curve without gaps was continuous · periodic. The one that marked only points did not fit, since the signal has a value at every instant, and the one that looked at a single stretch missed the repetition and stopped at "almost". Classification comes down to answering two independent questions in turn: is the time axis a line or points, and does the same shape come back?

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

On to the next units

Classification is the first step in every story about signals. The split between continuous and discrete leads later to the sampling theorem, and periodicity is the starting point from which the Fourier series spreads a periodic signal into a sum of sinusoids. In the next unit, elementary signals, we build the alphabet of signals: the impulse, the step and the exponential.