4.2 — System Properties: Linearity, Time-Invariance, Causality and Stability

A system takes a signal in and produces a signal out. An amplifier, a filter, a room's acoustics, a car's suspension, an economy.

Nearly all of the mathematics in the rest of this volume applies only to systems with two particular properties. Systems with them can be analysed completely and easily; systems without them, in general, cannot be analysed at all. So the first job with any system is to check.

1. Linearity

A system is linear if it obeys superposition, which is two conditions in one:

\text{if } x_1\to y_1 \text{ and } x_2\to y_2, \quad\text{then}\quad ax_1+bx_2 \to ay_1+by_2

Broken into its parts:

• Additivity: the response to a sum is the sum of the responses.
• Homogeneity (scaling): doubling the input doubles the output.

Chapter 1.4 used superposition to solve circuits. This is the same property, stated generally.

Testing for linearity

y(t) = 3x(t) — linear. Scale the input and the output scales; add two inputs and the outputs add.

y(t) = x^2(t) — not linear. Doubling the input quadruples the output. Also, (x_1+x_2)^2 = x_1^2+2x_1x_2+x_2^2, and that cross term is precisely what additivity forbids.

y(t)=x(t)+5 — not linear, and this one catches everyone. Test homogeneity: an input of 2x gives 2x+5, but twice the output would be 2x+10. They differ.

A system like this is called incrementally linear — linear in the changes about an operating point. This is exactly the small-signal analysis of Chapter 2.3: a transistor amplifier is not linear, but the deviations about its bias point are, and every technique in this Part applies to those deviations. Recognising that distinction is what makes analog design tractable.

y(t)=t\,x(t) — linear. The multiplier depends on t, not on x, so scaling and adding both survive. Linearity says nothing about time-invariance, and this system is a good example of one without the other.

y[n]=x[n]x[n-1] — not linear. Any product of the input with itself breaks it.

The practical test

Ask: does the input appear anywhere except to the first power, and is there any constant added? If a squared term, a product of inputs, an absolute value, a maximum, a threshold, a sine of the input, or a constant offset appears, the system is not linear.

2. Time-invariance

A system is time-invariant if delaying the input simply delays the output by the same amount, with no other change:

\text{if } x(t)\to y(t), \quad\text{then}\quad x(t-t_0)\to y(t-t_0)

In words: the system behaves the same today as it did yesterday. Its internal parameters do not change with time.

Testing

The reliable procedure, because doing it by eye leads to mistakes:

1. Compute y_1(t), the response to a delayed input x(t-t_0).
2. Compute y_2(t) = y(t-t_0), the delayed version of the original response.
3. They must be identical.

y(t)=3x(t)+2 — time-invariant. Delaying the input gives 3x(t-t_0)+2, and delaying the output gives the same ✓. Not linear, but time-invariant — the two properties really are independent.

y(t)=t\,x(t) — not time-invariant. Delayed input gives t\,x(t-t_0); delayed output gives (t-t_0)x(t-t_0). Different ✗. Linear but not time-invariant.

y(t)=x(-t) — not time-invariant. Delayed input gives x(-t-t_0); delayed output gives x(-(t-t_0)) = x(-t+t_0). The signs differ ✗.

y[n]=x[2n] (downsampling) — not time-invariant. This one matters practically: it is why sample rate conversion in Chapter 5.2 needs its own theory rather than reusing filter theory.

Where real systems lose time-invariance

• Ageing. A capacitor's value drifts over years; a filter's cut-off frequency creeps.
• Temperature. A resistor's value changes, a transistor's gain changes. A circuit is time-invariant only over an interval short enough that the temperature has not moved.
• Deliberate modulation. A radio mixer multiplies by a local oscillator, which is a time-varying gain. That is the whole point of it, and it is why Chapter 7.2's modulators cannot be analysed as LTI systems.
• Adaptive systems. An automatic gain control or an adaptive filter changes its own parameters, which is time-variance by design.

The two properties in one picture

Both properties are statements about what a box does to a signal, so they are best seen as boxes.

The two properties as tests you can apply to a box. Each one is a promise about what the box will not do, and the two promises together are what make convolution and the transforms of this Part legal.

Why the pair matters so much more than either alone is worth stating before section 3 develops it. Linearity lets you chop an input into simple pieces, push each piece through separately, and add the answers. Time-invariance means you only ever have to work out one answer — the response to a single impulse — because the response to an impulse arriving later is the same shape, later. Put them together and the whole system is described by one function, and every possible input is handled by sliding that one function around and adding. That sliding-and-adding is convolution, and Chapter 4.3 is entirely about it.

3. LTI — why the combination is everything

A system that is both linear and time-invariant is an LTI system, and the combination unlocks a chain of consequences that neither property gives alone.

The argument, in four steps:

1. Any signal can be written as a sum of scaled, shifted impulses. In discrete time this is obvious:

   x[n]=\sum_{k=-\infty}^{\infty}x[k]\,\delta[n-k]

   Each term contributes the value x[k] at exactly index k and nothing anywhere else, so the sum reconstructs the signal exactly.
2. Time-invariance says: if the response to \delta[n] is h[n], then the response to \delta[n-k] is h[n-k].
3. Linearity says: the response to a sum of scaled inputs is the sum of the scaled responses.
4. Therefore:

   y[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k]

That formula is convolution, and it means the system's entire behaviour is captured by one signal — its response to a single impulse. Measure h once and you can predict the output for any input whatsoever, forever.

This is why the two properties matter so much. Without them, knowing a system's response to one input tells you nothing about its response to another. With them, one measurement is complete knowledge. Chapter 4.3 develops the consequence in full.

4. Causality

A system is causal if the output at any time depends only on the input at that time and earlier — never on the future.

y(t_0) \text{ depends only on } x(t) \text{ for } t \le t_0

Every physical system operating in real time is causal, because the future has not happened. The property is worth naming only because non-causal systems are useful in other settings.

y[n]=x[n]+x[n-1] — causal, uses present and past.

y[n]=x[n]+x[n+1] — not causal, uses the next sample.

Where non-causal systems are genuinely used:

• Offline processing. Editing a recorded audio file, the "future" is already in memory. A non-causal filter is perfectly implementable and has a real advantage: it can be made to have exactly zero phase distortion, which no causal filter can.
• Image processing. The "time" axis is spatial, and there is nothing special about one direction. A blur filter uses pixels on both sides.
• The zero-phase trick. Filter the signal forwards, reverse it, filter again, reverse again. The forward and backward phase shifts cancel exactly. This is what MATLAB's filtfilt does, and it is why a plotted, processed data set can have a filter applied with no time shift at all.

For an LTI system, causality has a simple test: h(t)=0 for t<0, and h[n]=0 for n<0. The impulse response cannot begin before the impulse.

5. Stability

A system is BIBO stable — bounded input, bounded output — if every bounded input produces a bounded output.

|x(t)| \le M \text{ for all } t \quad\Longrightarrow\quad |y(t)| \le N \text{ for all } t

The LTI test is clean and complete:

\int_{-\infty}^{\infty}|h(t)|\,dt \lt \infty \qquad\text{or}\qquad \sum_{n=-\infty}^{\infty}|h[n]| \lt \infty

The impulse response must be absolutely integrable (or summable).

Proving it in one direction. From the convolution formula,

|y(t)| = \left|\int h(\tau)x(t-\tau)d\tau\right| \le \int|h(\tau)||x(t-\tau)|d\tau \le M\int|h(\tau)|d\tau

So if the integral is finite, the output is bounded by M times it. Done.

Worked examples

A perfect integrator, h(t)=u(t):

\int_0^\infty 1\,dt = \infty

Unstable. And it is obvious once stated: feed a constant into an integrator and the output ramps forever. This is why the practical integrator in Chapter 2.5 needed a resistor across the capacitor — that resistor makes h decay and the system stable.

A decaying exponential, h(t)=e^{-at}u(t) with a>0:

\int_0^\infty e^{-at}dt = \frac1a \lt \infty

Stable.

A growing exponential, a<0: the integral diverges, unstable. Chapter 4.5 shows that this is exactly the pole-in-the-right-half-plane condition, and Part 6 is largely about keeping poles out of there.

A pure oscillator, h(t)=\cos(\omega_0 t)u(t): the integral of |\cos| over infinite time diverges, so it is not BIBO stable — which is right, because driving it at its own resonant frequency makes the output grow without limit. This is Chapter 1.6's resonance with zero damping, seen from the systems side.

6. Memory, invertibility and the rest

Memoryless. Output depends only on the input at the same instant. y(t)=3x(t) is memoryless; y(t)=x(t-1) is not. A resistor is memoryless; a capacitor is not, which is exactly what Chapter 1.5 meant by saying capacitors "remember".

Invertible. Distinct inputs give distinct outputs, so the input can be recovered. y(t)=2x(t) is invertible; y(t)=x^2(t) is not, because the sign is lost.

Invertibility is the entire theory of equalisation. A communication channel distorts a signal; if the distortion is invertible, the receiver can apply the inverse and recover the original. Chapter 7.1 shows the limit: where the channel's response is zero at some frequency, the inverse would need infinite gain, which amplifies noise catastrophically. So real equalisers deliberately do not fully invert, and that compromise is the whole design problem.

7. The whole classification on one example

Take y(t) = x(t)\cos(\omega_c t) — an amplitude modulator, the circuit at the heart of every AM transmitter in Chapter 7.2.

Linear? Yes. [ax_1+bx_2]\cos = a x_1\cos + b x_2\cos ✓. The multiplier does not depend on x.

Time-invariant? Delayed input gives x(t-t_0)\cos(\omega_ct). Delayed output gives x(t-t_0)\cos(\omega_c(t-t_0)). Different ✗.

Causal? Yes, output depends only on the present input.

Stable? Yes, since |\cos|\le1 the output is bounded by the input's bound.

Memoryless? Yes.

So it is linear, causal, stable and memoryless — but not time-invariant. And that single failure is consequential: you cannot describe a modulator with an impulse response or a transfer function, so none of the machinery of Chapters 4.3 through 4.6 applies to it directly.

The way around it is the reason Chapter 7.2 works the way it does: analyse the modulator not as a system but as a frequency-shifting operation, using the modulation property of the Fourier transform. When time-invariance fails, you change tool rather than change the system.

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Linearity and time-invariance together mean one measurement — the impulse response — determines everything. The next chapter turns that fact into the operation that computes the output, and spends its length making that operation feel obvious rather than algebraic.

Every formula above, built from scratch

None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.

System properties

Linear: ax_1+bx_2\to ay_1+by_2.

Time-invariant: x(t-t_0)\to y(t-t_0).

Causal (LTI test): h(t)=0 for t\lt0.

BIBO stable (LTI test):

\int|h(t)|dt\lt\infty \qquad \sum|h[n]|\lt\infty

Proof of sufficiency: |y| = |\int h(\tau)x(t-\tau)d\tau| \le M\int|h(\tau)|d\tau, bounded whenever the integral is.