Classification and Properties of Systems

Chapters 1 to 4 built a vocabulary for signals: how to classify them, which elementary ones matter, and how to move them about in time. A system is anything that takes a signal in and puts a signal out. An amplifier, an RC network, a microphone, a control loop, a piece of code that averages the last ten samples of a sensor reading — all of them are systems, and the mathematics does not care which.

Write the relationship as \(x(t) \to y(t)\), or as \(y = T\{x\}\) where \(T\) is an operator. That second notation deserves a careful reading. \(T\) does not map a number to a number; it maps an entire signal to an entire signal. This is why the common shorthand \(y(t) = T\{x(t)\}\) is a slight abuse: it suggests that \(y\) at time \(t\) is computed from \(x\) at the same \(t\), which is true only for the simplest systems. A one-second delay produces \(y(t)\) from \(x(t-1)\); a running integrator produces \(y(t)\) from the whole past of \(x\). The operator view has room for both.

The reason for classifying systems before solving any is strictly practical. Almost every technique in the remaining chapters of this book carries a licence condition. Convolution, in Chapter 7, is valid only for systems that are linear and time-invariant. Transfer functions, frequency responses, poles and zeros — Parts 4, 5 and 7 — all inherit that same condition. If a system fails the test, none of the machinery applies, and no amount of algebra will rescue the calculation. So the classification is not preliminary bookkeeping; it is the step that tells you whether the rest of the course is available to you.

Six properties do the work, and they come in three natural pairs. Memory and invertibility describe how much of the input the system uses and how much of it survives. Causality and stability describe whether the system can be built and whether it will misbehave. Time invariance and linearity are the two that matter most, because together they define the class this book is about.

A system is memoryless, or static, if the output at any instant depends only on the input at that same instant. Otherwise it has memory and is called dynamic. The test is a matter of reading the formula: does the right-hand side mention \(x\) at any argument other than the present one?

A resistor is memoryless: \(v(t) = R\,i(t)\), and the voltage now depends on the current now and nothing else. A squarer \(y[n] = x^2[n]\) is memoryless even though it is nonlinear — memory and linearity are independent questions. A capacitor is not memoryless, because its voltage is the accumulated charge, \(v(t) = \tfrac{1}{C}\int_{-\infty}^{t} i(\tau)\,d\tau\), and every past current contributes. Neither is a delay, a difference, or a moving average.

Two cases catch people out. The first is \(y(t) = t\,x(t)\), which is memoryless: the factor \(t\) is a coefficient supplied by the system's own clock, not a second look at the input. The second is \(y[n] = x[2n]\), which is not memoryless: at \(n=3\) it reports \(x[6]\), and \(6 \ne 3\). The rule is about the argument of \(x\), never about what else appears in the expression.

Memory is what makes a system interesting, and it is also what makes it hard. In a circuit, memory lives in the energy-storage elements — capacitors and inductors — and each of them contributes one order to the differential equation describing the system. A purely resistive network is memoryless and is described by algebra; add one capacitor and Chapter 9's differential equations become unavoidable. Memory is also what requires initial conditions: a memoryless system needs none, because it has nothing to remember.

A system is invertible if distinct inputs always produce distinct outputs. Equivalently, there exists a second system — the inverse — which, placed in cascade after the first, returns the original signal.

To prove invertibility, construct the inverse. An amplifier \(y(t) = 5x(t)\) is inverted by an attenuator of gain \(1/5\). A delay is inverted by an advance. The accumulator \(y[n] = \sum_{k=-\infty}^{n}x[k]\) is inverted by the first difference \(x[n] = y[n]-y[n-1]\), which is exactly the relationship between step and impulse from Chapter 3 read as a statement about systems.

To disprove it, produce two different inputs with the same output — a single counterexample suffices and is usually easy to find. The squarer \(y(t) = x^2(t)\) is not invertible, because \(x\) and \(-x\) give identical outputs and the sign has been destroyed. The decimator \(y[n] = x[2n]\) is not invertible, for the reason set out in Chapter 4: it discards the odd-indexed samples, and two signals differing only there are indistinguishable at the output. The system \(y(t) = 0\) is the extreme case, mapping every input to the same output.

A system is causal, or non-anticipative, if the output at any instant depends only on present and past values of the input, never on future ones. Formally: if two inputs agree for all \(t \le t_0\), then the corresponding outputs must agree for all \(t \le t_0\).

Physical systems operating in real time have no choice about this. A filter cannot respond at nine o'clock to a signal that arrives at ten. Causality is therefore the property that separates what can be built from what can only be computed afterwards — and the exception matters: a system processing a recording, or a stored image, can quite legitimately look ahead, because the "future" is already on disk. Non-causal filters are used routinely in offline audio and image processing for exactly this reason.

Testing is again a matter of reading arguments. \(y[n] = x[n] - x[n-1]\) is causal, because it uses only the present and the immediately preceding sample. \(y[n] = x[n+1] - x[n]\) is not, because computing the output at \(n\) requires the sample that has not yet arrived. \(y(t) = x(2t)\) is not causal: at \(t = 1\) it needs \(x(2)\), a second into the future. \(y(t) = x(t/2)\) is causal for positive \(t\) but fails for negative \(t\), where \(t/2 \gt t\) — the classification must hold for all \(t\), so the system is non-causal.

Every memoryless system is causal, trivially: if the output uses only the present input, it certainly uses no future input. The converse is false — a delay is causal and has memory — so memory and causality are not the same property, though they are often confused.

A system is BIBO stable — bounded-input, bounded-output — if every bounded input produces a bounded output. The definition insists on every, and the asymmetry that follows is the key to answering these questions quickly.

Consider \(y(t) = t\,x(t)\). Feed it the constant input \(x(t) = 1\), which is bounded by 1. The output is \(y(t) = t\), which grows without limit. One counterexample settles it: the system is unstable. Notice that the counterexample did not have to be exotic; the simplest bounded signal did the job, and that is usually the case.

Consider next the accumulator \(y[n] = \sum_{k=-\infty}^{n} x[k]\). Feed it the unit step, bounded by 1. Then \(y[n] = n+1\) for \(n \ge 0\), unbounded, so the accumulator is unstable. Its continuous-time counterpart, the integrator, fails for the same reason with the same input — which is worth remembering, because integrators appear inside almost every control loop and their instability is precisely what the surrounding feedback is there to tame.

For a stable example, take \(y[n] = x^2[n]\). If \(|x[n]| \le B\) then \(|y[n]| \le B^2\), a finite bound, so the system is stable despite being nonlinear. And take \(y(t) = e^{x(t)}\): if \(|x| \le B\) then \(|y| \le e^{B}\). Stability is about boundedness, not about linearity or smoothness.

One system deserves a warning. \(y(t) = 1/x(t)\) is unstable, and the counterexample must be built with a little care: choose a bounded input that approaches zero, such as \(x(t) = e^{-t}\) for \(t \ge 0\). It never exceeds 1, yet the output \(e^{t}\) is unbounded. A bounded input is not required to stay away from zero, and any system that divides by its input can be defeated this way.

A system is time-invariant if its behaviour does not depend on when the input is applied. Delay the input, and the only effect should be an identical delay of the output — same shape, same size, nothing else altered. A time-invariant system has no calendar.

The definition must be turned into a procedure, because informal reasoning about it goes wrong reliably. The procedure compares two paths through the same pair of operations.

Apply it to \(y(t) = t\,x(t)\). Along Path 1, present the shifted input \(x_1(t) = x(t-t_0)\) to the system; the system multiplies whatever it is given by the current time, so \(y_1(t) = t\,x_1(t) = t\,x(t-t_0)\). Along Path 2, first compute \(y(t) = t\,x(t)\), then delay the whole output, which means replacing \(t\) by \(t-t_0\) everywhere: \(y_2(t) = (t-t_0)\,x(t-t_0)\).

The two differ by the factor \(t\) against \(t-t_0\), so the system is time-varying. The physical reading is that the system's gain increases as the day wears on, so a signal applied at noon is amplified more than the same signal applied at dawn. That is exactly what time variance means. The most common mistake in this calculation is forgetting to replace \(t\) by \(t-t_0\) in the explicit coefficient on Path 2 — the shift applies to the output signal as a whole, not selectively to the parts involving \(x\).

The same test disposes of two systems from Chapter 4. For time reversal \(y(t) = x(-t)\): Path 1 gives \(y_1(t) = x_1(-t) = x(-t-t_0)\), while Path 2 gives \(y_2(t) = x(-(t-t_0)) = x(-t+t_0)\). Different, so reversal is time-varying — reflecting about the origin gives the origin a special status, and a time-invariant system is not allowed to have one. For decimation \(y[n] = x[2n]\): Path 1 gives \(y_1[n] = x_1[2n] = x[2n-n_0]\), Path 2 gives \(y_2[n] = x[2(n-n_0)] = x[2n-2n_0]\). Different again, and the discrepancy — \(n_0\) against \(2n_0\) — shows that the shift itself gets rescaled by the operation.

A system is linear if it obeys superposition. Two conditions are involved: additivity, that the response to a sum is the sum of the responses, and homogeneity, that scaling the input scales the output by the same factor. They combine into a single statement that is usually the most efficient thing to test.

The fastest disqualifier is the zero-input test. Put \(a = b = 0\) in the superposition statement: a linear system must map the zero signal to the zero signal. So if a system produces a non-zero output from a zero input, it cannot be linear, and no further work is needed. The system \(y(t) = 2x(t) + 5\) fails instantly, since \(x = 0\) gives \(y = 5\).

That last system is instructive, because it is obviously not pathological — it is an amplifier with a DC offset, the kind of thing a real op-amp stage produces all the time. Such systems are called incrementally linear: they are linear in the changes of the input, even though they are not linear in the input itself. Feed two inputs \(x_1\) and \(x_2\); the outputs differ by \(2(x_1 - x_2)\), which does obey superposition. Equivalently, subtract off the constant part and what remains is a genuinely linear system.

This decomposition — a zero-input response plus a linear zero-state response — is not a curiosity. It is exactly the structure that Chapter 9 will find in every differential equation with non-zero initial conditions, and it is why a system with stored energy is analysed by treating the stored energy and the applied input separately, then adding.

Genuinely nonlinear systems are recognised by the input appearing in any way other than to the first power: \(x^2\), \(|x|\), \(\cos(x)\), \(1/x\), or products of the input with itself such as \(x[n]x[n-1]\). Multiplying the input by a signal the system supplies is a different matter and does not break linearity — which brings us to the most valuable observation in this chapter.

Systems are rarely encountered alone. The three standard interconnections drawn in Section 5-1 — cascade, parallel and feedback — build large systems out of small ones, and it is worth knowing which properties survive the assembly.

Cascade and parallel connections are well behaved. If both blocks are linear, so is the combination, because a composition and a sum of linear operators is linear. The same argument works for time invariance, for causality, and for stability: a bounded input gives a bounded intermediate signal, which gives a bounded output. Feedback is the interesting case, and the reason Part 7 and the whole of control theory exist: a loop can make a stable system unstable, and — more usefully — can make an unstable system stable, because the output is now allowed to correct itself.

One caution about cascades: two individually unstable systems can combine into a stable one. Cascade an integrator with a differentiator and the result is the identity, which is perfectly stable, even though the integrator alone is not. Properties of a cascade follow from the properties of the parts only when the property is preserved by composition, and stability is not, in this direction.

Of the six, two are singled out for the rest of the book. A system that is both linear and time-invariant is called an LTI system, and the reason for the emphasis was already visible at the end of Chapter 3. There we showed that every signal is a sum of scaled, shifted impulses, \(x[n] = \sum_k x[k]\delta[n-k]\). Now watch what the two properties do to that sum.

Linearity says that the response to the sum is the sum of the responses to each term, and that the scale factors \(x[k]\) pass straight through. Time invariance says that the response to \(\delta[n-k]\) is the response to \(\delta[n]\), merely delayed by \(k\) — so a single function, the impulse response \(h[n]\), suffices for every term. Neither property alone would be enough: without linearity the terms could not be added, and without time invariance each shifted impulse would need its own separate response function.

Problems 1 to 5 ask for full classifications; 6 and 7 concern interconnections and counterexamples; 8 asks you to prove a general statement. For every negative verdict, give the explicit counterexample — a claim of "not stable" or "not linear" with no counterexample attached earns nothing.