Linear time-invariant systems
Two properties, and with them a single measurement is enough to know the whole system. They also explain why, to translate a signal in frequency, one of the two has to be broken.
01Linearity: superposition
A system is a box that receives a signal and returns another. It is linear if it responds to a combination of inputs with the same combination of their outputs:
From this follows something that always surprises: taking \( a = b = 0 \), a linear system necessarily responds with zero to a zero input. A circuit that delivers 1 V with its input grounded is not linear, even if its graph is a perfect straight line. That distinction has a name: \( y = ax + b \) is called affine, or incrementally linear, and it is handled by separating the quiescent point from the signal. That is exactly what is done when analyzing an amplifier: the bias is computed on one side and the signal on the other, and the small-signal model is indeed linear.
02Time invariance: the system does not watch the clock
A system is time-invariant if delaying the input only delays the output, without changing it in any other way:
In other words: the system does today what it will do tomorrow. An RC with fixed components is invariant; an RC whose resistor keeps heating up is not. And there is a case worth keeping in mind because it is linear but not invariant: the mixer, \( y(t) = x(t)\cos(\omega_{LO}t) \). It satisfies superposition without any trouble —doubling the input doubles the output— but it is not indifferent to the moment at which the signal arrives, because the local oscillator keeps its own time.
It is the most useful consequence of the two properties together. If a 1 kHz sinusoid goes in, at the output of an LTI system there is a 1 kHz sinusoid: different amplitude and different phase, but the same frequency. No harmonic or new frequency ever appears. That is why, to shift a signal in frequency, one of the two conditions has to be broken: invariance, with a mixer, or linearity, with a nonlinearity that generates harmonics. There is no third option.
03Testing the two properties
The definitions are verified with two experiments, and that is exactly what the lab does: it builds the two signals that should coincide and superimposes them. Where they separate, the property fails.
On the left, the linearity test, and on the right, the invariance test. In each one there are two curves that have to fall on top of each other: if they separate, the system does not satisfy the property.
04Why half the course is devoted to them
Because in an LTI system, a single measurement determines everything. The argument has three steps. First, any signal is written as a continuous sum of shifted impulses:
Second, by invariance, if the system responds with \( h(t) \) to the impulse \( \delta(t) \), it responds with \( h(t - \tau) \) to the shifted impulse. And third, by linearity, the response to the sum is the sum of the responses:
That is what makes it worthwhile to ask whether a system is LTI: if it is, it is characterized by hitting it once and measuring how it responds. All the material that follows —convolution, Fourier, Laplace, the Z-transform— is machinery for making that computation convenient.
05What is and is not LTI in a real lab
| Circuit | Linear? | Invariant? | Remark |
|---|---|---|---|
| RLC with fixed components | Yes | Yes | The reference case |
| Small-signal amplifier | Yes | Yes | After separating the bias |
| The same one, saturated | No | Yes | It clips: harmonics appear |
| Rectifier diode | No | Yes | Its curve is not a straight line |
| Mixer | Yes | No | Shifts in frequency without distorting |
| Switched capacitor | Yes | No | Invariant only to multiples of the clock period |
| A/D converter | No | Yes | Quantization is a staircase |
The switched-capacitor row is interesting: it is not invariant in general, but it is with respect to shifts equal to an integer number of clock periods. This is called invariance to periodic shifts, and it is the basis for analyzing switched-capacitor filters with the tools of discrete time.
06In the lab
Inject into an RC filter a 1 kHz sinusoid, then a 3 kHz one, and then the sum of the two. Check that the output for the sum matches the sum of the outputs, sample by sample.
Repeat the above, raising the amplitude until the amplifier clips. Check that superposition no longer holds and measure with the spectrum analyzer the new frequencies that appeared.
Multiply a 10 kHz sinusoid by a 100 kHz local oscillator and observe the two sidebands. Check that doubling the input doubles the output —it is linear— and that delaying the input does not translate into a simple delay of the output.
07Common mistakes
- Calling \( y = ax + b \) linear. It is affine: with zero input it does not give zero output.
- Confusing linearity with invariance. They are independent: there are systems of all four possible combinations.
- Expecting harmonics from an LTI system. It cannot generate them; if they appear, something is saturating.
- Testing linearity with a single pair of signals. It has to hold for any pair and any pair of coefficients; one case that fails is enough to rule it out.
- Forgetting that \( y(t) = t\,x(t) \) is linear. The coefficient depends on time, but the relationship with \( x \) is still proportional.
08Self-assessment
Write the linearity condition.
\( T\{a x_1 + b x_2\} = a\,T\{x_1\} + b\,T\{x_2\} \) for any signals and constants.
Is \( y(t) = x(t) + 1 \) linear?
No. With zero input it gives output 1, and that contradicts homogeneity.
Is \( y(t) = t\,x(t) \) linear? Is it invariant?
Linear yes; invariant no, because the factor depends on the absolute instant.
Can an LTI system generate a harmonic?
No. It can only change the amplitude and phase of the frequencies that were already there.
Why is \( h(t) \) enough to characterize an LTI system?
Because any input is a sum of shifted impulses, and by linearity and invariance the output is the convolution of the input with \( h \).
How is a signal shifted in frequency?
By breaking invariance, with a mixer, or linearity, with a nonlinearity. An LTI system cannot do it.
09Further reading
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Prentice Hall, 1998. Sections 1.6 and 2.1: the properties of systems and the road to convolution.
- Simon Haykin and Barry Van Veen. Signals and Systems. (Spanish edition, “Señales y sistemas”), Limusa Wiley, 2001. Many examples of checking linearity and invariance, with the counterexamples worked out.
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 7th ed. (Spanish edition, “Circuitos microelectrónicos”), Oxford University Press, 2016. The separation between bias and small signal, which is how a real circuit is linearized.