Representation of signals and systems
Four ways of describing the same thing, none of them truer than the others. Each problem is easy to solve from one of them, and a good part of the craft consists of choosing where to look from.
01The same system, four descriptions
An LTI system can be described in four different ways, and all four say exactly the same thing. They are not alternatives you have to choose between: they are points of view, and each problem is easier to solve from one of them.
| Representation | Form | Good for |
|---|---|---|
| Differential equation | \( \sum a_k \dfrac{d^k y}{dt^k} = \sum b_k \dfrac{d^k x}{dt^k} \) | Starting from the circuit: it is what Kirchhoff’s laws and the component laws give |
| Impulse response | \( h(t) \), with \( y = x * h \) | Seeing the time response; it is measured by striking the system |
| Transfer function | \( H(s) = \dfrac{Y(s)}{X(s)} \) | Cascades, stability, filter design |
| State variables | \( \dot{\mathbf{q}} = A\mathbf{q} + B x,\quad y = C\mathbf{q} + D x \) | Systems with several inputs and outputs; numerical simulation |
The most frequent path in practice goes from the first to the third. You set up the circuit, obtain the differential equation, transform it —so that derivatives become powers of \( s \)— and are left with a ratio of polynomials that is analyzed with poles and zeros.
02All four faces at once
So that the equivalence does not stay in a table, it helps to see the four move together. The lab takes a second-order system —the most common in electronics— and shows what happens to each representation when a parameter is changed.
Move the parameters and all four descriptions change at the same time: the equation, the poles, the impulse response and the frequency response.
03The discrete version
Everything repeats with square brackets. The differential equation becomes a difference equation, which is also directly programmable:
And the transfer function is obtained with the same trick, except that now the delay of one sample becomes \( z^{-1} \):
From the equation to \( H \): transform and solve. From \( H \) to the impulse response: inverse transform, which with residues means adding one term per pole. From \( h \) to \( H \): transform. And from \( H \) to the state variables: choose a set of variables —typically the capacitor voltages and the inductor currents— and write their derivatives. That last path is the one any circuit simulator uses internally.
04Signals also have several representations
| Representation | When it is useful |
|---|---|
| \( x(t) \), the waveform | Seeing amplitudes, times, edges |
| Fourier series | Periodic signals: which harmonics they have |
| Fourier transform | Isolated signals: how much bandwidth they occupy |
| Laplace transform | Transients and initial conditions |
| Samples \( x[n] \) | Anything that is going to pass through a processor |
The fact that there are so many is not excessive: each one makes visible something that the others hide. The waveform says nothing about the bandwidth, and the spectrum does not show whether the signal clips. A good part of the craft consists of choosing where to look from.
05In the lab
For a series RLC with the output across the capacitor, set up the differential equation, obtain \( H(s) \), inverse transform to get \( h(t) \) and write the state equations taking \( v_C \) and \( i_L \) as variables. Verify that all four describe the same circuit.
Measure the step response of the RLC, differentiate it to obtain \( h(t) \), extract \( \sigma \) and \( \omega \) from the envelope and the period, and reconstruct \( H(s) \). Compare with the one calculated from the component values.
Discretize the same filter and program its difference equation on a microcontroller. Compare its step response with that of the analog circuit.
06Common mistakes
- Forgetting the zero initial conditions when replacing \( d/dt \) by \( s \). With nonzero initial conditions, extra terms appear.
- Confusing \( H(s) \) with \( H(j\omega) \). The latter is the former evaluated on the imaginary axis.
- Believing that one representation is more “true”. All four describe the same system.
- Mixing up \( z^{-1} \) and \( s \). One is a delay of one sample; the other, a derivative.
- Choosing the state variables badly. It is best to take those that store energy: capacitor voltages and inductor currents.
07Self-assessment
What are the four representations of an LTI system?
Differential equation, impulse response, transfer function and state variables.
What replaces \( d/dt \) when transforming?
The variable \( s \), assuming zero initial conditions.
What does \( z^{-1} \) represent?
A delay of one sample. It is the memory element of any digital filter.
How do you go from \( H(s) \) to \( h(t) \)?
By inverse transforming, which with residues means adding one exponential term per pole.
What should be taken as state variables in a circuit?
Those that store energy: capacitor voltages and inductor currents.
Why are there so many representations of a signal?
Because each one makes visible something that the others hide: the waveform does not show the bandwidth, and the spectrum does not show whether it clips.
08Further reading
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Prentice Hall, 1997. The representations and the passage from one to another, with examples in both domains.
- Katsuhiko Ogata. Modern Control Engineering. 5th ed. (Spanish edition, “Ingeniería de control moderna”), Prentice Hall, 2010. State variables in full detail, and their relationship with the transfer function.
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed. (Spanish edition, “Fundamentos de circuitos eléctricos”), McGraw-Hill, 2021. The path from the circuit to the differential equation, which is where everything starts.