EC-7.1 Signals: Continuous, Discrete, and the Operations on Them

What this is and why it exists

Nearly every later subject in this area opens by assuming you already know what a signal is, and then never says. Communications talks about a band-limited signal. Control talks about a step input. Digital signal processing talks about a sequence. All three assume the vocabulary and none of them teaches it.

This topic teaches it. A signal here means a function that carries information. At this stage the only things that matter are what kind of function it is and what you may do to it.

That sounds thin. It is not. The classifications in this topic decide which methods apply later, and the operations decide whether you can follow a derivation at all. Almost every mistake people make in convolution is a mistake in this topic, made earlier.

The vocabulary

• Continuous-time signal — a signal with a value at every instant.
• Discrete-time signal — a signal with values only at counted positions. Also called a sequence.
• Periodic — repeating exactly after a fixed interval, called the period.
• Energy signal — one whose total energy over all time is finite.
• Power signal — one whose average power is finite and not zero.
• Time shift — replacing the argument by the argument minus a constant, which delays the signal.
• Time scaling — multiplying the argument by a constant, which stretches or compresses the signal.
• Even signal — one equal to its own reflection about the origin.
• Odd signal — one equal to the negative of its own reflection.
• Unit step — a signal that is zero before the origin and one after it.
• Unit impulse — a signal concentrated entirely at one instant, with unit area.

The mental model

Hold the difference between the two kinds of signal in the right place. The distinction is about the independent variable, not about the values. A continuous-time signal has a value at every instant. A discrete-time signal has values only at counted positions, and between two of them there is nothing at all, not even zero.

That is worth insisting on because a second distinction sits nearby and is different. Whether the values themselves are rounded to a fixed set of levels is quantisation. A signal can be discrete in time and unrounded in value. Most of this area works with exactly that kind, and calling it digital is a slight abuse that later causes confusion.

Now periodicity. A signal is periodic when shifting it by some fixed amount leaves it unchanged. The smallest such amount is the period. Add two periodic signals and the sum repeats only when both line up again, which needs a common multiple of the two periods. When the ratio of the two periods is not a ratio of whole numbers, no common multiple exists and the sum never repeats. Two sinusoids can therefore add to something aperiodic, which surprises people every time.

Energy and power split signals into two families that need different treatment. Total energy is the running total of the squared size over all time. Average power is that total divided by the length of time it was taken over. A signal that dies away has finite energy and, spread over infinite time, zero average power. A signal that persists, such as a sinusoid running forever, has infinite energy and a perfectly sensible average power. A signal is normally one or the other, and knowing which tells you which measure of size to use.

The four operations are the ones you will use in every derivation from here on. Shifting moves the signal along the time axis. Reversing reflects it about the origin. Scaling stretches or compresses it. Adding a constant to the values, or multiplying them, changes the signal but not the axis.

The one that goes wrong is doing several at once. Replace the argument with a combination of scaling and shifting and the order matters. The shift is measured in the units of the argument before the scaling was applied. The safe method is mechanical. Rewrite the new argument in the form of a scale multiplying a shifted variable, and then read the shift off that form. Guessing produces the right answer about half the time.

Every signal splits into an even part and an odd part, and the split is unique. Take the signal and its reflection, and the average of the two is the even part, while half the difference is the odd part. This seems decorative until the Fourier work, where the symmetry of a signal decides whether its transform is real or imaginary. It is worth doing once by hand.

Finally the standard signals. The step, the impulse, the ramp and the exponential appear constantly, both as test inputs and as building blocks. What matters most is how they relate to each other, not their definitions. The step is the running total of the impulse. The ramp is the running total of the step. Going the other way, the impulse is the rate of change of the step.

The impulse deserves a warning. In discrete time it is unproblematic: a sequence that is one at the origin and zero everywhere else. In continuous time it is not an ordinary function at all, because it has to be infinitely tall and infinitesimally narrow with unit area. Treat it as the limit of a narrow tall pulse, and use only the property that actually matters. Multiplying a signal by a shifted impulse and integrating picks out the value of the signal at that shift. That single property is the whole of what the impulse is for, and it is what makes the next two topics work.

What you should now be able to explain or do

• Say whether a given signal is continuous-time or discrete-time, and explain why quantisation is a separate question.
• Decide whether a signal is periodic, and find the period of a sum of two periodic signals or show there is none.
• Classify a signal as an energy signal or a power signal and say which measure of size applies.
• Apply a combined scaling and shift to a signal in the right order, by rewriting the argument first.
• Split a signal into its even and odd parts.
• State the relationships between the impulse, the step and the ramp, and say what the impulse is used for.

Check yourself