EC-7.2 Systems, and the Properties Worth Testing For

What this is and why it exists

Every method taught after this topic works on one particular kind of system. The impulse response, the frequency response, the transfer function, the pole diagram: all four are tools for systems that are linear and time invariant. None of them is valid for a system that is not.

The uncomfortable part is that applying them anyway produces an answer. It looks like the answer. It is wrong, and nothing in the arithmetic tells you so.

So this topic is the gate. It teaches what a system is, in a definition wide enough to cover a circuit, a program and a population. Then it teaches six properties and how to test for them, of which two matter far more than the rest.

The vocabulary

• System — a rule that turns an input signal into an output signal.
• Linearity — scaling the input scales the output, and the response to a sum is the sum of the responses.
• Homogeneity — the scaling half of linearity.
• Additivity — the sum half of linearity.
• Superposition — the two halves taken together.
• Time invariance — delaying the input delays the output by the same amount and changes nothing else.
• Causality — the output at any instant depends only on the input up to that instant.
• Stability — every bounded input produces a bounded output. Often shortened to bounded-input bounded-output.
• Memory — dependence of the present output on past values.
• Invertible — different inputs give different outputs, so the input can be recovered from the output.

The mental model

Start with the definition, and notice how little it says. A system is a rule taking an input signal to an output signal. No components, no physics, no hardware. That is deliberate. A resistor network is a system. So is a moving average computed in software, and so is a lever, and so is a savings account. The results in this module apply to all of them because they were never about hardware.

Linearity asks two questions and both must be answered yes. Scale the input by any constant and the output must scale by the same constant. Feed in the sum of two inputs and the output must be the sum of the two separate outputs.

The trap is a system that adds a constant to its output. Its graph is a straight line, and people therefore call it linear. Scale its input by zero and the output does not go to zero, so it fails the first condition. A straight-line relationship with an offset is not linear in this sense. The mismatch with the everyday use of the word catches almost everyone once.

Time invariance is the second half of the pair. A time-invariant system treats the same input the same way regardless of when it arrives. The test has a fixed shape and is worth learning as a shape. Compute the response to a delayed input. Separately, take the response to the undelayed input and delay it. If those two expressions agree for every delay, the system is time invariant. A system whose behaviour depends on the clock, or whose components change with time, will fail.

Together these two properties are the whole reason this subject exists in the form it does. Linearity means you may break an input apart, handle the pieces separately, and add the results. Time invariance means the way a piece is handled does not depend on where it sits. Break an input into shifted scaled impulses, and those two properties let one measured response describe the system completely. That is the next topic, and it needs both.

The remaining four properties are useful rather than foundational.

Causality says the output now uses only the input now and earlier. It is a statement about what can be built to run in real time, and not about mathematics. Some perfectly describable systems are not causal, and the ideal sharp filter later in this module turns out to be one. A recorded signal being processed afterwards is a different matter, because there the future is already on disk.

Stability, in the sense used throughout this module, says every bounded input gives a bounded output. That is narrower than the everyday sense of a system that behaves well, and it is what the pole conditions later actually prove. Keep the definition precise and the later results stay honest.

Memory is dependence on the past. A resistor has none; a capacitor does. Systems without memory are rare and their analysis is short, which is why this whole module is about the others.

Invertibility asks whether the input can be recovered from the output. A system that discards information cannot be inverted, and no processing afterwards recovers what was thrown away. That point returns with real force when sampling goes wrong.

One habit is worth building now. Run the tests before reaching for the tools. Two lines of checking at the start of a problem is cheaper than a page of confident arithmetic that was never applicable.

What you should now be able to explain or do

• Give the definition of a system used here and explain why it deliberately says nothing about hardware.
• Test a given rule for linearity, and explain why a straight-line relationship with an offset fails.
• Test a rule for time invariance using the delay-the-input against delay-the-output comparison.
• Define causality and give an example of a useful system that is not causal.
• State the bounded-input bounded-output definition of stability and distinguish it from the everyday sense.
• Say why linearity and time invariance together are the pair the rest of the module depends on.

Check yourself