Design of Excitation Signals

Good measurements make it much easier to obtain good models. In this chapter we discuss how to design a good excitation signal for the identification of linear dynamic systems. The extension towards nonlinear systems will be postponed to the chapter Identification in the Presence of Nonlinear Distortions. It will be shown that periodic signals, when they can be applied, offer considerable advantages. At the start of the chapter, we first provide a refresher on some signal processing tools dealing with sampling and reconstruction of signals, followed by the study of the discrete Fourier transform (DFT) and its practical use with the fast Fourier transform (FFT).

The following hands-on exercises guide the reader through the system identification process:

– Sampling – The Bridge Between Continuous-Time and Discrete-Time Signals
– Reconstruction of a Continuous-Time Signal from a Discrete-Time Sequence
– The FFT – The Gate to the Frequency Domain
– Design of Excitation Signals – User Choices
– Periodic Signals – Practical Use
– Swept Sine



Sampling – The Bridge Between Continuous-Time and Discrete-Time Signals
Most system identification algorithms work on discrete-time data that are obtained by sampling the continuous-time signals of the physical system that is modeled.

In this Hands-On we study the impact of the sampling process and look at the following questions:
– What is the impact of sampling on the information in the sampled signals?
– What is the relation between the spectra of the continuous-time and discrete-time (sampled) signal?
– How to reduce the aliasing effect using an anti-alias filter?

Download the MATLAB® live script Sampling to run the session.



Reconstruction of a Continuous-Time Signal from a Discrete-Time Sequence
In many applications, a continuous-time signal has to be reconstructed starting from a discrete-time sequence. This problem is studied in this Hands-On.

What you will learn:
– Exact reconstruction of a band-limited signal starting from equidistantly sampled data.  This reconstruction is not usable in most practical applications.
– Zero-order-hold reconstruction is a very practical reconstruction method. The spectral properties are studied in detail.
– Linear reconstruction is intuitively appealing but has significant drawbacks.

Download the MATLAB® live script Reconstruction to run the session.



The FFT – The Gate to the Frequency Domain

Signals and systems can be analyzed in the time- and frequency domain. The Fourier transform (FT) links the frequency domain to the time domain. Although there is no creation or loss of information by this transformation, it is often very helpful to look to the problems from both perspectives, some problems are easier solved in one domain than in the other.  In practice, mastering both domains is the best guarantee to get a good insight in the problems to be solved.

The Fourier transform is a mathematical tool that requires an analytic expression of the signals to be transformed. In practice, such an expression is mostly unavailable, often only a discrete time measurement is available. In that case the discrete Fourier transform (DFT) can be used to calculate the Fourier transform at a discrete set of frequencies.  A good understanding of the relation between the DFT and the FT is indispensable to use the DFT as a tool to access the frequency domain.  The fast Fourier transform (FFT) is a numerical efficient implementation of the DFT.

The goal of this Hands-On is to:
– Study the relation between the DFT and the FT in full detail.
– To give user guidelines how to use the DFT in practice.

Download the MATLAB® live script DiscreteFourierTransform to run the session.



Design of Excitation Signals – User Choices
In this Hands-On, we give the reader a bird’s eye view of the various aspects of excitation signal design. First, we specify the general setup, next we discuss the characterization and classification of excitation signals. The Hands-On focuses on the overall picture, explaining and illustrating user choices and their interaction. More detailed discussions about specific classes of excitations, their properties, their design, and practical use will be studied in the follow-up exercises in this chapter.

What you will learn:
- Relation between the experiment design and the goal of the experiment.
- Characterize and compare the quality of broadband excitation signals.
- First meeting with deterministic, random, periodic excitation signals.

Download the MATLAB® live script UserChoices to run the session.



Periodic Signals – Practical Use
Before starting the detailed discussion of different excitation signals, we give an introduction to the generation and use of periodic discrete time signals. We will first explain the constraints on the choice of the sampling frequency and the period length. Next, we discuss the importance of the synchronization of the generator and data acquisition channels. Eventually, we will illustrate the use of the periodicity to average the data (improving the SNR) and to estimate nonparametric noise models.

What you will learn:
– Design a periodic signal: select the sample frequency and the period length.
– Synchronize the generator and data acquisition.
– Use the periodicity to improve the SNR.
– Show how to make a nonparametric noise analysis.

Download the preliminary MATLAB® live script PracticalUse to run the session.



Swept Sine
In this hands-on, we illustrate the use of the swept sine, also called periodic chirp. The basic idea is to sweep the instantaneous frequency of the sine over the frequency band of interest. Depending on the sweeping rate, more or less power will be injected around that frequency.

What you will learn:
– How to control the excited frequency band?
– Linear swept sine: create a flat excitation.
– Exponential swept sine: keep a constant power per octave.
– Swept sine or swept cosine?

Download the preliminary MATLAB® live script SweptSine to run the session.