# 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 excitation design process:

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?

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.

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.

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.

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.

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?

Multisine
In this hands-on, we illustrate the use of multisine signals. These are the most flexible periodic excitations: a multisine is a periodic signal, consisting of the sum of harmonically related cosines with user defined amplitudes and phases. In this Hands-On the focus is on the minimization of the crest factor (the ratio between the peak value and the rms value of the signal) of the multisine.

What you will learn:
– Impact of the phases on the crest factor.
– Search for phases that minimize the crest factor: deterministic, random, and numerical optimized phase choices will be discussed.
– Logarithmic multisines: keep a constant power per octave.

Logarithmic Frequency Grid Excitation
In applications that cover a wide frequency band, a logarithmic frequency grid provides often the most suited representation of the results in the frequency domain. In that case, it is often also advisable to use an excitation signal that is adapted to this choice. The ‘density’ of the excitation decreases for increasing frequencies.

We studied two solutions in the Swept Sine and Multisine Hands-On’s:
– Exponential swept sine that excites all frequencies with a 1/f power spectrum.
– Logarithmic multisine: excites an (approximate) logarithmic frequency grid with a flat power spectrum.

In this Hands-On, we compare both solutions in more detail, and we add also a third possibility using a full multisine with a 1/f power spectrum.

Download the preliminary MATLAB® live script Logarithmic Excitations to run the session.

Random Phase Multisine
In this hands-on, we study the stochastic properties of random phase multisines. The amplitude distribution, the stationarity of the signal, and the distribution of the crest factor will be studied in more detail.

You will learn that for a growing number of excited frequencies:
– The random phase multisine tends towards a stationary signal.
– The amplitude distribution converges to a normal distribution.
– The crest factor is well described by a Gumbel distribution.

Crest Factor of Band Limited Signals
The band-limited reconstruction does change the value of the crest factor. The crest factor of the continuous-time signal differs from that of the discrete-time sequence. The difference becomes smaller when the oversampling factor grows. This effect will be studied in more detail in this Hands-On.

You will learn that
– The band-limited reconstruction increases the crest factor.
– For random phase multisines, the increase is a random variable too.
– The effect decreases when the oversampling rate increases.
– It becomes negligible for an oversampling rate above 4.

Design of Binary Excitations
In this Hands-On, we study binary excitation signals, which have only two values +/- 1, that can change sign on an equidistant discrete-time grid. We study two classes of signals:
– Random binary noise excitations with a power spectrum that can be tuned by the user.
– Deterministic periodic pseudo-random binary sequences with a flat amplitude spectrum.

In this Hands-On, you will learn about:
– The ZOH nature of binary excitations and its impact on the spectral properties.
– How to tune the power spectrum of random binary signals.
– How to generate periodic binary discrete-time sequences that have a flat power spectrum.

At the end of the Hands-On, we refer to a toolbox that can be used to generate these signals.