Abstract
Discrete-time signals often derive from periodically repeated measurements of some quantity over a finite time span, and we are interested in the characteristics of the quantity and of the process that generates it, rather than in those of the particular sequence we measured. The measured record, affected by random errors, is interpreted as a segment of a persistent discrete-time random power signal, conceptually resulting from sampling a continuous-time random signal (process) possibly varying with time. The value assumed by a random signal is not exactly specified at a given past or current instant, and future values are not predictable with certainty on the basis of past behavior. This chapter provides a brief introduction to the basic theory of discrete-time random processes, which can be described using theoretical average quantities. The latter could be calculated if the probabilistic laws associated with the random process were known, but usually they are unknown; the problem is then simplified by assuming stationarity, i.e., no dependence on time, and ergodicity, a property that allows for substituting theoretical averages with time averages performed on a single finite-length data record. This path leads to a spectral representation for the discrete-time random power signal, i.e., to the power spectrum. Also the representation of the common spectral content of two random signals is introduced.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
In particular, most often the noise superposed to the useful signal will be a wideband process: its spectral bandwidth will be larger than the signal bandwidth. If we choose the sampling interval on the basis of the bandlimit that the useful signal is supposed to have, then we may get aliasing of the noise component, which can corrupt the spectrum at all frequencies—not only in the vicinity of the Nyquist frequency. Of course, this issue becomes irrelevant if we are in a position to apply an anti-aliasing filter before sampling, since then also the noise bandwidth becomes limited.
- 2.
Note that if we assume for simplicity that the population of possible values is the same at any time n, as we actually did, we could drop the n index and simply write \(\mathrm {X}\). However, we will not drop the index n, because later this might generate confusion.
- 3.
From now on, and until the end of the next chapter, we will indicate the sequences composed by measurement outcomes by \(\mathrm {x}[n]\), rather than by x[n], to avoid confusing them with random variables, for which we adopted the symbol \(x_n\).
- 4.
Stochastic comes from the Greek word \(\sigma \tau \acute{o} \chi o \varsigma \), which means “aim”. It also denotes a target stick; the pattern of arrows around a target stick stuck in a hillside is representative of what “stochastic” means.
- 5.
We must observe, recalling the general discussion on vector spaces reported in the appendix of Chap. 3, that the autocovariance between the real random variables \(x_n\) and \(x_m\) associated with times m and n can be seen as an inner product: \(\left\langle (x_n-m_{x_n}), (x_m-m_{x_m})\right\rangle \). This inner product actually involves the ensemble average operator, as we anticipated in Sect. 3.8.2. The Fourier representation of the random process \(\left\{ x_n\right\} \), which later will be attained using autocovariance, is thus based on a definition of inner product that is not the standard one.
- 6.
For complex signals we would write \(\phi _{xy}[n,n-l] = \mathrm {E}[x_n y^*_{n-l}]\), etc.
- 7.
The factor \(1/(2 \pi )\) derives from using \(\omega \) as the frequency variable.
- 8.
If we considered the general case of complex sequences, the above formula would become \(\rho _{xy}[l] = \frac{\mathrm {E}[x_n y^*_{n-l}]}{\sqrt{\mathrm {E}[|x_n|^2]\mathrm {E}[|y_n|^2]}}\).
- 9.
Some authors prefer the definition
$$\begin{aligned} \hat{\sigma }_x^2=\frac{1}{N-1} \sum _{n=0}^{N-1} (x[n]-{\hat{m}_x})^2 \end{aligned}$$that makes the sample variance an unbiased estimate of the variance of the random signal.
- 10.
This definition is valid for real signals. If we considered general complex signals, we should write
$$\begin{aligned} c_{xy}[l] = \frac{1}{N} \sum _{n=l}^{N-1} x[n] y^*[n-l] \quad \text{ for }\quad l\ge 0, \qquad c_{xy}[l] = c^*_{yx}[-l]\quad \text{ for }\quad l<0. \end{aligned}$$Of course this applies to \(c_{xx}[l]\) and \(c'_{xx}[l]\) too.
- 11.
In contrast, multivariate data is typically produced by a variable measured; as a function of more than one independent variable, for example, space and time.
References
Blackman, R.B., Tukey, J.W.: The Measurement of Power Spectra from the Point of View of Communication Engineering. Dover, New York (1958)
Chatfield, C.: The Analysis of Time Series: An Introduction. Chapman and Hall—CRC Press, Boca Raton (2013)
Gray, M., Davisson, L.D.: An Introduction to Statistical Signal Processing. Cambridge University Press, Cambridge (2004)
Hannan, E.J.: Time Series Analysis. Methuen, London (1960)
Hayes, M.H.: Statistical Digital Signal Processing and Modeling. Wiley, New York (1996)
Helstrom, C.W.: Probability and Stochastic Processes for Engineers. MacMillan, New York (1991)
Kay, S.M.: Fundamentals of Statistical Signal Processing. Prentice Hall, Upper Saddle River (1993)
Proakis, J.G., Manolakis, D.G.: Digital Signal Processing: Principles, Algorithms, and Applications. Prentice Hall, Upper Saddle River (1996)
Said, S.E., Dickey, D.A.: Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71(3), 599–607 (1984)
Scharf, L.L., Demeure, L.: Statistical Signal Processing: Detection, Estimation, and Time Series Analysis. Addison-Wesley, Boston (1991)
Scargle, J.D.: Studies in L. II—Statistical aspects of spectral analysis of unevenly spaced data. Astrophys. J., Part 1, 263, 835–853 (1982)
Stoica, P., Moses, R.: Spectral Analysis of Signals. Prentice Hall, Upper Saddle River (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Alessio, S.M. (2016). Statistical Approach to Signal Analysis. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-25468-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25466-1
Online ISBN: 978-3-319-25468-5
eBook Packages: EngineeringEngineering (R0)