Kinds of signals

A day of room temperature that you may read only at the dots. Drag Measurements per day up and down, and watch the curve underneath.

A handful of questions

In What a signal is (1.1) I said a signal is something that changes, with one value at each moment you look at it. That is true, but it does not tell you much about any particular signal. A heartbeat trace and a burst of radio static are both signals, and they have almost nothing else in common.

A short list of questions tells them apart. How often can you look at the signal? How fine are the values it lands on? Can you predict where it is going? Does it repeat? Is it symmetric? Does it start at some moment, or was it already going before you started watching?

Ask these six questions about any signal and you have described its main features.

Looking every so often

Picture a thermometer on your wall. Someone checks it once an hour and writes the number down. Between those checks the temperature keeps rising and falling smoothly, but you never see it. All you have is one number per hour.

Compare a thermometer with a pen attached that traces the temperature onto a moving strip of paper. There, a value exists at every instant, not only once an hour.

A signal you only get to look at now and then, at separate moments, is called discrete-time. A signal defined at every instant is called continuous-time. Nothing about the underlying temperature changed between the two descriptions, only how often you were allowed to look at it. Each separate measurement is called a sample.

To write a discrete-time signal down, give each sample a whole number $n$ (the first one, the second one, and so on) and write $x[n]$ for its value. A continuous-time signal is written $x(t)$, with $t$ any real number of seconds.

Go back to the picture at the top of the page. The curve underneath never changes as you drag Measurements per day. Only the dots marking where you may read a value change. With enough measurements a day, the dots sit so close together that they look like the smooth curve itself.

A ruler with only so many marks

Now think about the values themselves, not when you get to look at them. A dimmer switch can set a lamp’s brightness anywhere in a continuous range: 41%, 41.2%, any number you like. A lamp with four brightness buttons can only be at one of those four settings, with nothing in between.

A signal whose value can only land on one of a fixed set of levels, like that four-button lamp, is called digital. A signal whose value can be anything in a range, like the dimmer, is called analog. This is a separate question from discrete-time versus continuous-time: one is about when you’re allowed to look, the other is about which values you’re allowed to see.

Drag Number of levels from many down to just a few.

Watch the smooth curve turn into a staircase, each step landing on the nearest allowed level. With 3 levels the rounded curve has only three heights. Fewer levels means a coarser staircase, whether you measure light, sound or anything else.

Predictable, or not

Listen to a metronome clicking at a steady beat. If you know the tempo, you can say exactly when the next click will land, and the one after that, forever. Now listen to static hissing from an untuned radio. You could say roughly how loud it tends to be, but not what its exact value will be one second from now.

A signal whose entire future is fixed by a rule, like the metronome, is called deterministic. A signal that only has statistical regularities, no formula you could run forward to get the exact next value, like the static, is called random.

Switch between Steady tone and Noise, and look for a shape that comes back.

Both traces look busy at a glance. The tone repeats its shape exactly, forever, while the noise never settles into a repeating pattern.

Does it repeat?

Hold a single note on an organ. Its waveform settles into a shape that comes back again and again, cycle after cycle, for as long as you hold the key.

Now cough once. There is no shape to repeat: it happens, and then it is over.

A signal whose shape repeats forever at some fixed spacing is called periodic; that fixed spacing is called its period. A signal that does not repeat is called aperiodic. Many everyday recordings, such as a single cough, a knock or a spoken word, are aperiodic: they happen once.

In symbols, a discrete-time signal with period $N$ samples satisfies $x[n+N]=x[n]$ for every $n$.

Related to this is how long a signal lasts. A cough is nonzero only for a limited stretch of time, so it has finite duration. A steady tone continues without end, so it has infinite duration.

Switch the excerpt between Musical note and Cough, then switch Show it once to Repeat it for each.

Notice that repeating the note changes nothing about its shape, because the note already repeated. Repeating the cough gives three copies in a row. That row is periodic because we built it that way, but the cough on its own is not. Being periodic means the signal itself repeats forever, not that you can copy it.

If you add two periodic discrete-time signals together, the sum is still periodic, and its period can be longer than either one alone.

The rule: if one signal repeats every $N_1$ samples and the other every $N_2$ samples, their sum repeats every $\mathrm{lcm}(N_1, N_2)$ samples. The lcm (least common multiple) is the smallest count that both $N_1$ and $N_2$ divide evenly. For example, a signal with period 6 added to one with period 8 repeats every $\mathrm{lcm}(6, 8) = 24$ samples.

Both signals start a new period together after 24 samples, and not before. The sum itself can sometimes repeat sooner, if the two signals happen to cancel in a helpful way, so 24 is a guaranteed repeat and not always the shortest one.

A mirror in time, and is it zero before now?

Pick a single moment in a signal and call it your origin, the point you measure time from. Now imagine folding the signal in half at that origin, like a mirror. If the folded halves land exactly on top of each other, the signal is symmetric about that origin.

The widget below uses two pulses, one even and one odd. If the two halves match exactly, the signal is called even about that origin. If the two halves are exact opposites, mirrored and flipped upside down, it is called odd. Most signals are neither: fold them and the two halves do not match.

Choose Even pulse or Odd pulse, then drag Mirror position and watch whether the mirror image matches.

The mirror image matches only at exactly 0. The even or odd question depends on where you put the origin, not on the waveform alone.

Separately, ask where the signal is nonzero relative to that same origin. If it is zero everywhere before the origin and only turns on afterward, like a trace that starts the instant a sensor is switched on, it is called causal. If it is zero everywhere after the origin instead, it is called anticausal.

If it is nonzero on both sides, like a pulse that starts before the origin and ends after it, it is neither. Any signal that is not causal is called non-causal.

Switch between Only after now, Only before now and Both sides, and watch which side of now is held at zero.

A causal signal is one you could record starting from switch-on: nothing happened before you turned the sensor on, because there was nothing to record.

Where you’ll meet these

These six questions come back constantly. Whether a signal is discrete-time or continuous-time decides whether you can run it through a digital computer, which only handles discrete-time, digital signals. How you get there is the subject of Sampling & aliasing (10.1).

The $\mathrm{lcm}$ rule comes back when you add periodic discrete-time signals. Even and odd symmetry becomes a useful shortcut when we break signals into simpler pieces in Decomposing signals (2.3).

A real-time system can only use samples that have already arrived. That is why causality comes back in System properties (4.2).

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