Spectral analysis

APPENDIX A

Spectral analysis Spectral analysis is a statistical technique used for characterizing and analyzing sequenced data. It is employed to estimate the spectral density function or spectrum of a time series. In general, spectral analysis refers to the decomposition of a sequence into oscillations of different lengths or scales. By this process, the data domain (i.e., the observations) are converted into the spectral domain. For example, in problems of signal recognition, spectral analysis is used as a preliminary measurement to perform bandwidth filters, as in acoustic and image processing. Other examples include sonar systems, which use sophisticated spectral analysis to locate submarines and surface vessels [1]. Spectral measurements in radar are also used to obtain target location and velocity information. A standard approach in spectral analysis is to carry out the Fourier transform of a time series. In particular, Fourier analysis deals with approximating a function by a sum of sine or cosine terms [2,3]. A time series can be prescribed either in the time, y(t), or in the frequency, Y (f ), domain under the transformation: Y (f ) = F {y(t)}, where

 Y (f ) =

and

 y(t) =

(A.1)

∞

y(t) exp(−2πif t)dt

(A.2)

Y (f ) exp(2πif t)df,

(A.3)

−∞ ∞ −∞

√ with i = −1. Relation (A.3) is called the inverse Fourier transform. Using the Euler formula, exp(−2πif t) = cos(2πf t) − i sin(2πf t), the integrals in Eqs. (A.2) and (A.3) can be written as an infinite sum of sine and cosine functions. The Fourier transform Y (f ) of the time series represents the frequency contribution of each sine and cosine function, and this is called the Fourier spectrum. For example, if y(t) has three periods, its Fourier spectrum will display three spikes. In the particular case of a Gaussian white noise, where there are no characteristic frequencies, the resulting Fourier spectrum will be approximately flat. A periodic time series is composed of a finite number of subperiods which are represented by a finite number of spikes at discrete frequencies in their Fourier spectrum, while a nonperiodic time series has no dominant periods or subperiods, and so its Fourier spectrum is composed of 351

352 Appendix A a continuous and infinite range of frequencies. In particular, fractal time series are nonperiodic and defined over a finite time domain, that is, y(t) = 0, if t ∈ [0, T ], and y(t) = 0 if t < 0 and t > T . Hence, for a fractal time series defined on the time interval [0, T ], its Fourier transform is Y (f, T ) and Eqs. (A.2) and (A.3) become 

T

Y (f ) =

y(t) exp(−2πif t)dt

(A.4)

Y (f ) exp(2πif t)df,

(A.5)

0

and

 y(t) =

∞ −∞

respectively. Instead of performing the integration in Eq. (A.4) from 0 to T , it is common practice to choose the symmetric interval [−T /2, T /2]. Moreover, as the time series are discrete sequences of data values, the integrals in Eqs. (A.4) and (A.5) must be replaced by sums. These replacements give rise to the so-called discrete Fourier transform (DTF), which will be written as   N  2πinm , m = 1, 2, 3, . . . , N, (A.6) yn exp Ym = δ N n=1

and   N 1  2πinm , yn = Ym exp − Nδ N

n = 1, 2, 3, . . . , N,

(A.7)

m=1

where δ = T /N. Usually, δ = 1 [4]. The modulus of a complex number Y (f ) = a + ib is defined as Y (f ) = (a 2 + b2 )1/2 . In the frequency domain, the value of Y (f )2 represents a measure of the energy distribution of a signal. In the limit when T → ∞, the total energy diverges, approaching infinity. It is commonly used to analyze the power instead of the energy. In particular, the power-spectral density function is defined as [5] Y (f )2 . T →∞ T

S(f ) = lim

(A.8)

A plot of S(f ) versus f is known as a periodogram. In a periodogram the quantity S(f )df represents the contribution to the total power from those components in the time series whose frequencies lie between f and f + df . For a fractal time series, the power-spectral density satisfies the power-law relation, S(f ) ∼ f β .

(A.9)

Spectral analysis 353 Because of the power-law dependence, the fractal time series with β > 0 exhibit long-range persistence, while fractal time series with β < 0 exhibit long-range anti-persistence. The β value is obtained as the slope of the best fit straight line in the log(S(f ))–log(f ) plane. In fact, β is a measure of the strength of persistence or anti-persistence in a time series. According to Malamud and Turcotte [6], the exponent β characterizes the temporal fluctuations of the time series. For example, a white noise-type signal has β = 0, while for a flicker, or 1/f , noise β = 1, and for a Brownian motion β = 2.

References [1] R. McCleary, D. McDowall, B. Bartos, Design and Analysis of Time Series Experiments, Oxford University Press, 2017. [2] A. Dominguez, Highlights in the history of the Fourier transform, IEEE Pulse 61 (2016). [3] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, 2nd edition, Cambridge University Press, Cambridge, 2007. [4] M.B. Priestley, Numerical Recipes, Academic Press, London, 1981. [5] R.F. Voss, Random fractals: characterization and measurement, in: R. Pinn, A. Skjeltrop (Eds.), Scaling Phenomena in Disordered Systems, Plenium Press, New York, 1985. [6] B.D. Malamud, D.L. Turcotte, Self-affine time series: I. Generation and analyses, in: Sixth Workshop on NonLinear Dynamics and Earthquake Prediction, Trieste, Italy, 2001 (H4.SMR/1330-22).