Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Noise-driven dynamical systems
- 3 Noise-induced phenomena in zero-dimensional systems
- 4 Noise-induced phenomena in environmental systems
- 5 Noise-induced pattern formation
- 6 Noise-induced patterns in environmental systems
- Appendix A Power spectrum and correlation
- Appendix B Deterministic mechanisms of pattern formation
- Appendix C List of symbols and acronyms
- Bibliography
- Index
Appendix A - Power spectrum and correlation
Published online by Cambridge University Press: 05 August 2011
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Noise-driven dynamical systems
- 3 Noise-induced phenomena in zero-dimensional systems
- 4 Noise-induced phenomena in environmental systems
- 5 Noise-induced pattern formation
- 6 Noise-induced patterns in environmental systems
- Appendix A Power spectrum and correlation
- Appendix B Deterministic mechanisms of pattern formation
- Appendix C List of symbols and acronyms
- Bibliography
- Index
Summary
The steady-state pdf p(ϕ) of a stochastic process ϕ(t) is a key piece of information in the study of noise-induced phenomena; however, it does not give indications about the temporal structure of the process. In fact, processes with different temporal evolutions can share the same pdf. Because some noise-induced phenomena underlie changes in the temporal behavior of dynamical systems (e.g., the stochastic resonance), it is useful to introduce two mathematical tools that are commonly used to quantitatively investigate the temporal structure of a signal, namely the power spectrum and the autocorrelation function. In this appendix we recall the basic concepts and some analytical results, referring to specialized textbooks (e.g., Papoulis, 1984) for a more comprehensive description. Moreover, in the following discussion we consider signals in the time domain, though the same results are valid also if the process is sampled in space, e.g., when transects of spatial fields are studied (see Chapter 5). In this case, the power spectrum (also known as structure function) and the autocorrelation function are useful tools for investigating the existence of regular patterns in the field.
Let us start from a quite specific case and consider a piecewise continuously differentiable periodic function ϕ(t), with period 2π (if the signal has a different period, it may be mapped to a 2π period through a suitable scaling of time).
- Type
- Chapter
- Information
- Noise-Induced Phenomena in the Environmental Sciences , pp. 269 - 273Publisher: Cambridge University PressPrint publication year: 2011