Classification of Signals

Chapter 1 left us with one relationship, \(y = T\{x\}\), and a promise: that complicated problems become simple once the signal is viewed in the right domain. That promise comes with a condition attached. The right domain depends on what kind of signal \(x\) is — and choosing the wrong tool for a signal is the single most common way a student loses marks in this subject.

A periodic signal has a Fourier series; an aperiodic energy signal has a Fourier transform; a signal that grows without bound has neither, and needs the Laplace transform with a region of convergence. A random signal has no formula at all and must be described by averages. Every one of those decisions is made before a single integral is written, purely by classifying the signal.

So treat what follows not as a taxonomy to memorise but as a checklist you run on every signal you meet. The eight questions below are independent of one another — a signal is somewhere on all eight axes at once — and together they determine everything you are allowed to do with it.

Chapter 1 split signals by their time variable: continuous-time \(x(t)\) with \(t \in \mathbb{R}\), discrete-time \(x[n]\) with \(n \in \mathbb{Z}\). There is a second, entirely separate split, and confusing the two is a classic error. This one concerns the amplitude.

An analog signal takes any value in a continuous range — between any two amplitudes you can name, the signal can take infinitely many intermediate values. A digital (or quantised) signal takes values only from a finite set of allowed levels. An 8-bit converter, for instance, permits exactly 256 amplitudes and nothing in between; every measured value is rounded to the nearest one, and the small error introduced is called quantisation noise.

Because the two axes are independent, all four combinations exist and all four are useful. The table below names them, and the middle two are worth dwelling on: they are the intermediate stages inside every analog-to-digital converter you will ever specify.

A deterministic signal is one whose value is completely specified at every instant — by a formula, a graph, or a table of numbers. Write \(x(t) = 5\cos(100t)\) and there is no ambiguity about the value at \(t = 0.3\) s; you can compute it, and it will be the same every time you look.

A random (or stochastic) signal cannot be written down that way. Thermal noise in a resistor, the fading of a radio channel, the exact waveform of a spoken word — none of these has a formula. The best you can do is describe them probabilistically: the mean, the variance, how strongly the value now is related to the value a moment ago. That description is the subject of Chapter 29.

It is worth being honest about the relationship between the two. Every real measurement contains a random component, so a deterministic signal is a model — an idealisation that is accurate enough to design with. The mains supply is not exactly \(230\cos(2\pi \cdot 50\,t)\); it wanders in amplitude and frequency and carries harmonics and noise. Treating it as the pure cosine is a deliberate, useful simplification, and almost the whole of this course lives inside that simplification.

A continuous-time signal is periodic if it repeats itself exactly after some fixed shift. Formally, \(x(t)\) is periodic if there exists a positive \(T\) such that \(x(t+T) = x(t)\) for every \(t\). Any such \(T\) is a period; the smallest one is the fundamental period \(T_0\), and \(f_0 = 1/T_0\) is the fundamental frequency. A signal that is not periodic is aperiodic.

Note the asymmetry hidden in that pair. In continuous time the period \(T\) may be any positive real number. In discrete time the period \(N\) must be a positive integer, because there is no such thing as a shift by half a sample. That single restriction has a consequence which catches almost everyone the first time.

Every continuous-time sinusoid is periodic. Take \(x(t) = \cos(\omega_0 t)\): it repeats when \(\omega_0 T = 2\pi\), so \(T_0 = 2\pi/\omega_0\), and this is a valid period for any \(\omega_0\) you choose. A discrete-time sinusoid is not. For \(x[n] = \cos(\Omega_0 n)\) to repeat we need \(\Omega_0 N = 2\pi m\) for some integers \(N \gt 0\) and \(m\) — which is possible only if \(\Omega_0/2\pi\) is a rational number.

Sums behave in a way worth committing to memory. If \(x_1\) has period \(T_1\) and \(x_2\) has period \(T_2\), their sum is periodic only if the ratio \(T_1/T_2\) is rational. When it is, write the ratio in lowest terms as \(m/k\); the fundamental period of the sum is then the smallest common multiple, \(T_0 = k\,T_1 = m\,T_2\). When the ratio is irrational — as it is for \(\cos(2t) + \cos(\pi t)\) — the sum never repeats and is aperiodic, despite being built entirely from periodic pieces.

Symmetry about the time origin gives a third classification, and unlike the others it comes with a genuinely useful piece of machinery attached. A signal is even if reflecting it about the vertical axis leaves it unchanged, and odd if reflecting it about the origin does.

Most signals are neither, and that is precisely the point. Any signal at all can be split uniquely into an even part and an odd part, by the simple device of averaging it with its own reflection. Add \(x(t)\) to \(x(-t)\) and the odd content cancels; subtract, and the even content cancels.

Three consequences follow immediately and are worth having at your fingertips. First, an odd signal must pass through zero at the origin: setting \(t=0\) in \(x(-t) = -x(t)\) forces \(x(0) = -x(0)\). Second, products follow a sign rule exactly like multiplying \(+1\) and \(-1\): even times even is even, odd times odd is even, even times odd is odd. Third — and this is the one that pays off later — the integral of any odd signal over symmetric limits vanishes, because the two halves cancel exactly.

That last fact gives a small but elegant result about energy. When you square \(x = x_e + x_o\) you get \(x_e^2 + x_o^2 + 2x_ex_o\), and the cross term is even times odd, hence odd, hence integrates to zero over \((-\infty,\infty)\). The energies simply add.

Chapter 1 introduced energy and power as measures of a signal's size. Now that we have periodicity in hand, the picture can be completed — including the case Chapter 1 left out, of signals that belong to neither class.

For a periodic signal the limit is unnecessary, which is a considerable practical saving. Since the signal repeats, averaging over one full period gives the same answer as averaging over all time, so the power of a periodic signal is simply the mean square over a single period.

The classification then reads as follows. An energy signal has \(0 \lt E \lt \infty\), which forces \(P = 0\). A power signal has \(0 \lt P \lt \infty\), which forces \(E = \infty\). No signal can be both. But — and this is the addition to Chapter 1 — a signal can be neither. If a signal grows without bound, both integrals diverge and neither measure describes it. The ramp \(x(t) = t\,u(t)\) is the standard example, and so is the growing exponential \(e^{at}u(t)\) with \(a \gt 0\). Such signals are not pathological curiosities: the unstable systems of Chapter 19 produce them routinely, and handling them is precisely why the Laplace transform carries a region of convergence.

The seventh axis concerns where along the time line a signal actually lives. A signal is causal if it is zero for all negative time, \(x(t) = 0\) for \(t \lt 0\) — it switches on at the origin and not before. It is anticausal if the reverse holds, \(x(t) = 0\) for \(t \gt 0\). If it is non-zero on both sides of the origin it is noncausal, sometimes called two-sided.

The multiplication by \(u(t)\) that appeared throughout Chapter 1 is exactly the device that makes a signal causal: \(e^{-at}\) exists for all time, whereas \(e^{-at}u(t)\) is switched on at \(t=0\) and is causal by construction. This is why the unit step turns up in so many worked examples — it is the standard way of saying "and nothing happened before we started watching".

The final axis is the shortest to state and the strangest to justify. A real signal takes values in \(\mathbb{R}\); a complex signal takes values in \(\mathbb{C}\), and is written in terms of its real and imaginary parts as \(x(t) = x_R(t) + j\,x_I(t)\).

No instrument has ever measured a complex voltage. Every physically observable signal is real. So why introduce them at all? Because the complex exponential \(e^{j\omega_0 t} = \cos\omega_0 t + j\sin\omega_0 t\) bundles a cosine and a sine into a single object that behaves beautifully under differentiation, integration and — as Chapter 1 hinted — under any linear time-invariant system, which merely scales it. Carrying one complex exponential through an analysis is dramatically easier than carrying two trigonometric functions, and the real answer is recovered at the end by taking the real part. Every transform in Parts 3 to 7 rests on this convenience.

Two small adjustments come with complex signals. Energy uses the magnitude squared, \(|x|^2 = x\,x^{*}\), which is why the definitions in Chapter 1 were written with modulus bars rather than a plain square — they were already anticipating this. And the even-odd idea generalises: a complex signal is called conjugate symmetric if \(x(-t) = x^{*}(t)\), the natural extension of evenness, and this is exactly the symmetry that the Fourier transform of a real signal turns out to possess.

For each item, decide first which axis is being tested, then apply the matching definition rather than reasoning from the picture. Difficulty rises down the list.