Generation of amplitude constrained signals with a prescribed spectrum

doi:10.1016/j.automatica.2011.09.038

Automatica

Volume 48, Issue 1, January 2012, Pages 153-158

https://doi.org/10.1016/j.automatica.2011.09.038 Get rights and content

Abstract

In this paper a real-time method for generating signals of constrained amplitude and a given (arbitrary) spectrum is presented. This technique is based on the concatenation of sinusoidal signals of suitably chosen frequencies in order to obtain a signal with the desired sample autocovariance sequence as the number of samples tends to infinity. The effectiveness of the method is demonstrated theoretically and via simulations.

Introduction

The problem of generating a waveform having specified second order properties arises in many fields, see for example Cule and Torquato (1999), Gujar and Kavanagh (1968), Koutsourelakis and Deodatis (2005), Liu and Munson (1982), Sheehan and Torquato (2001), Yeong and Torquato (1998a), Yeong and Torquato (1998b). For example, in experiment design (Goodwin and Payne, 1977, Ljung, 1999) one typically obtains an optimal test signal specified in terms of its spectral properties. This leads to the problem of implementing a real signal with a specified spectrum, or spectral density. Moreover, it is usual that the input should also be constrained in its amplitude, i.e. the amplitude must lie in an interval $[a, b] \subset R$ . In general, frequency domain techniques do not work properly with this kind of constraint, and as such are translated into an ‘equivalent’ power constraint under which the input is designed to satisfy the conditions.

In many applications it is important to implement an input signal which, within the constraints of its amplitude, has maximum power. This is the case, for example, in experiment design, where the quality of the estimation typically increases with the signal to noise ratio. The signal to noise ratio is obviously improved by choosing an input with high power. Binary signals have precisely this desirable property: their power is maximum for a given amplitude constraint (Tan & Godfrey, 2001).

Several techniques have been proposed to design a binary signal with a given autocovariance (see e.g. Boufounos (2007), Cule and Torquato (1999), Gujar and Kavanagh (1968), Koutsourelakis and Deodatis (2005), Liu and Munson (1982), Rojas, Welsh, and Goodwin (2007), Sheehan and Torquato (2001), van den Bos and Krol (1979), Yeong and Torquato (1998a), Yeong and Torquato (1998b) and the references therein). For example, in Rojas, Welsh, and Goodwin (2007) a technique based on Model Predictive Control (Goodwin, Graebe, & Salgado, 2001) is developed, where, for each time instant, a finite horizon optimization problem is solved to find the optimal set of the next, say, $T$ values of the sequence such that the sampled autocovariance sequence so obtained is as close as possible (in a prescribed sense) to the desired autocovariance. One then takes the first term of this optimal set for the sequence, advances time by one step and repeats the procedure.

It is known, however, that binary processes cannot have an arbitrary autocovariance sequence (De Carvalho and Clark, 1983, Karakostas and Wynn, 1993, Masry, 1972, McMillan, 1955). Therefore, in this paper we relax the binary constraint and concentrate on the problem of generating a sequence with bounded amplitude and prescribed autocovariance. The algorithm proposed here provides a quasi-stationary sequence whose sample autocovariance sequence has guaranteed convergence for arbitrarily prescribed spectral densities. The algorithm is very fast and easy to implement, requiring from the user only the ability to generate independent random variables with a given distribution (for which several algorithms are available (Devroye, 1986)). The resulting signal has a crest factor (the quotient between its squared amplitude and its power (Ljung, 1999)) of approximately 2, while a binary signal has a crest factor of 1. In addition, the algorithm works in real time, i.e., it is not necessary to specify a priori the number of samples to be generated, and the method can be extended so that the desired spectrum can be modified during the execution of the algorithm. In this case the generated signal would have a spectrum equal to the last one being prescribed, if this spectrum is kept fixed from some time on. This is advantageous in some applications, such as adaptive experiment design (Gerencsér & Hjalmarsson, 2005), where the spectrum of the input is designed in real time, based on a recursively estimated model.

To demonstrate the application of the algorithm, two examples, motivated by experiment design, are provided. A typical input signal used in system identification is bandlimited white noise (Ljung, 1999, Section 13.3). In this paper we show how the proposed algorithm can be used to generate this type of signal and also provide the obtained spectral density to highlight how closely it approximates the desired spectral density. The second example is inspired by recent work on experiment design where it was shown that a more robust input for a particular class of systems is in fact one with a bandlimited ‘ $1 / f$ ’ spectrum (Goodwin et al., 2006, Rojas, Welsh, Goodwin, and Feuer, 2007). We again provide the spectral density generated by the proposed algorithm as well as that of the prescribed signal, for the purpose of comparison.

The paper is structured as follows. In Section 2 we present the algorithm and provide a detailed explanation. In Section 3 we prove convergence of the sample autocovariance coefficients of the signal generated by the algorithm to their desired values. Section 4 shows the results of some numerical examples that illustrate the quality of the signals generated by the algorithm. We present conclusions in Section 5.

Section snippets

The proposed method

In this section we introduce the proposed method for generating signals of constrained amplitude and pre-specified spectral density. The idea of the method comes from the following simple observation (inspired by Example 10-4 of Papoulis (1991)):

Lemma 2.1

Let $Φ \in L_{1} ([- π, π], R_{0}^{+})$ be such that ${(2 π)}^{- 1} \int_{- π}^{π} Φ (ω) d ω = 1$ . Let $ω$ and $ϕ$ be independent random variables, where $ω$ has density $Φ / 2 π$ and $ϕ$ is uniformly distributed in $[- π, π]$ . Then, ${y_{t}}_{t \in N}$ , where $y_{t} ≔ \sqrt{2} cos (ω t + ϕ)$ , is a sequence of random variables of zero mean

Analysis of convergence

In this section we study the convergence of the sample covariance sequence of the signal generated by the algorithm presented in Section 2, i.e., we establish that $R_{m}^{N} ≔ \frac{1}{N} \sum_{t = m + 1}^{N} y_{t} y_{t - m}, N \in N, m \in N_{0}$ converges almost surely to $r_{m} ≔ \frac{1}{2 π} \int_{- π}^{π} cos (ω m) Φ (ω) d ω$ . To this end, let us first define (for $q \in N$ ) $S_{q, m} ≔ \sum_{t = n_{q} + 1}^{n_{q} + m} y_{t} y_{t - m}, W_{q, m} ≔ \sum_{t = n_{q} + m + 1}^{n_{q + 1}} y_{t} y_{t - m} .$ We then have the following result:

Theorem 3.1

Consider the algorithm of Section 2, where ${n_{q}}_{q \in N_{0}} \subseteq N$ is a strictly increasing sequence (with $n_{0} = 1$ ) satisfying (1), (2). Then,

Numerical examples

In this section we present two examples. The first example deals with the problem of generating pseudo random signals (i.e. pseudo white noise). The second example relates to the generation of bandlimited ‘ $1 / f$ ’ noise. Such signals have recently been shown to possess important robustness properties in experiment design (Rojas, Welsh, Goodwin, & Feuer, 2007).

Conclusions

In this paper we have presented a novel method for generating signals of constrained amplitude with a specified spectral density. The algorithm is based on a blocking technique from probability. The algorithm is simple and straightforward to implement. It generates signals whose sample spectral density exhibits fast convergence as verified by simulation studies. In addition, we have established the convergence of the sample autocovariance sequence of the signal generated by the algorithm for

Claus Müller received a Diploma from the Technische Universiät Kaiserslautern, Germany, in 1996 and his Ph.D. degree (Doktor der Naturwissenschaften) in Mathematics in 2000. After working as Assistant Professor in Kaiserslautern, he joined the School of Electrical Engineering and Computer Science at the University of Newcastle in Australia in 2003. His research topics include probability theory and functional analysis, in general, and their application to engineering problems like MPC and

References (28)

C.R. Rojas et al.
Robust optimal experiment design for system identification
Automatica
(2007)
N.M. Blachman et al.
The spectrum of a high-index FM waveform: Woodward’s theorem revisited
IEEE Transactions on Communication Technology
(1969)
Boufounos, P. (2007). Generating binary processes with all-pole spectra. In Proceedings of the IEEE international...
K.L. Chung
A course in probability theory
(2001)
D. Cule et al.
Generating random media from limited microstructural information via stochastic optimization
Journal of Applied Physics
(1999)
J.L.M. De Carvalho et al.
Characterizing the autocorrelations of binary sequences
IEEE Transactions on Information Theory
(1983)
L. Devroye
Non-Uniform random variate generation
(1986)
Gerencsér, L., & Hjalmarsson, H. (2005). Adaptive input design in system identification. In Proceedings of the 44th...
G.C. Goodwin et al.
Control system design
(2001)
G.C. Goodwin et al.
Dynamic system identification: experiment design and data analysis
(1977)

Goodwin, G. C., Rojas, C. R., & Welsh, J. S. (2006). Good, bad and optimal experiments for identification. In T. Glad...

P. Guillaume et al.

Crest factor minimization using nonlinear Chebyshev approximation methods

IEEE Transactions on Instrumentation and Measurement

(1991)

U.G. Gujar et al.

Generation of random signals with specified probability density functions and power density spectra

IEEE Transactions on Automatic Control

(1968)

K.X. Karakostas et al.

On the covariance function of stationary binary sequences with given mean

IEEE Transactions on Information Theory

(1993)

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Cristian R. Rojas was born in 1980. He received his M.S. degree in electronics engineering from the Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2004, and his Ph.D. degree in electrical engineering at The University of Newcastle, NSW, Australia, in 2008. Since October 2008, he has been with the Royal Institute of Technology, Stockholm, Sweden, where he is currently an Assistant Professor of the Automatic Control Lab, School of Electrical Engineering. His research interest is in system identification.

Graham C. Goodwin obtained a B.Sc. (Physics), B.E. (Electrical Engineering), and Ph.D. from the University of New South Wales. He is currently Professor Laureate of Electrical Engineering at the University of Newcastle, Australia and is Director of The University of Newcastle Priority Research Centre for Complex Dynamic Systems and Control. He holds Honorary Doctorates from the Lund Institute of Technology, Sweden and the Technion Israel. He is the co-author of eight books, four edited books, and many technical papers. Graham is the recipient of Control Systems Society 1999 Hendrik Bode Lecture Prize, a Best Paper award by IEEE Transactions on Automatic Control, a Best Paper award by Asian Journal of Control, and 2 Best Engineering Text Book awards from the International Federation of Automatic Control in 1984 and 2005. In 2008 he received the Quazza Medal from the International Federation of Automatic Control and in 2010 he received the IEEE Control Systems Award. He is a Fellow of IEEE; an Honorary Fellow of Institute of Engineers, Australia; a Fellow of the International Federation of Automatic Control, a Fellow of the Australian Academy of Science; a Fellow of the Australian Academy of Technology, Science and Engineering; a Member of the International Statistical Institute; a Fellow of the Royal Society, London and a Foreign Member of the Royal Swedish Academy of Sciences.

^☆: This work was supported in part by the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 257059. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Er-Wei Bai under the direction of Editor Torsten Söderström.

¹: Tel.: +46 8 790 7185; fax: +46 8 790 7185.

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Brief paperGeneration of amplitude constrained signals with a prescribed spectrum☆

Abstract

Introduction

Section snippets

The proposed method

Analysis of convergence

Numerical examples

Conclusions

Automatica

The spectrum of a high-index FM waveform: Woodward’s theorem revisited

IEEE Transactions on Communication Technology

A course in probability theory

Generating random media from limited microstructural information via stochastic optimization

Journal of Applied Physics

Characterizing the autocorrelations of binary sequences

IEEE Transactions on Information Theory

Non-Uniform random variate generation

Control system design

Dynamic system identification: experiment design and data analysis

Crest factor minimization using nonlinear Chebyshev approximation methods

IEEE Transactions on Instrumentation and Measurement

Generation of random signals with specified probability density functions and power density spectra

IEEE Transactions on Automatic Control

On the covariance function of stationary binary sequences with given mean

IEEE Transactions on Information Theory

Brief paper
Generation of amplitude constrained signals with a prescribed spectrum☆