A continuous or discrete time-series, such as x = x(t) or x_{n} = {x_{0}, x_{1},. . .}, can be analyzed in terms of time-domain descriptions and frequency-domain descriptions. The latter is also called spectral analysis and reveals some characteristics of a time-series, which cannot be easily seen from a time-domain description analysis. Spectral analysis is used for solving a wide variety of practical problems in engineering and science, for example, in the study of vibrations, interfacial waves and stability analysis.
In spectral analysis, the time-series is decomposed into sine wave components using a sum of weighted sinusoidal functions called spectral components. The weighting function in the decomposition is a density of spectral components or spectral density function.
The actual method of decomposing a time-series into a sum of weighted sinusoidal functions is to use Fourier transform which has both continuous and discrete versions corresponding to continuous time-series of type x = x(t) and discrete time-series of type x_{n} = {x_{0}, x_{1},. . .}, respectively. Most recorded time-series data in engineering practice are of discrete type and numerical calculations of the Fourier transform are usually done using digital computers, which can only deal with discrete data and therefore use discrete Fourier transform. In view of this, the spectral analysis of only discrete time-series is described below.
We assume a discrete time-series with a finite number of samples N and a sampling time interval T_{s} between two successive samples
According to the mathematical theory of Fourier analysis, the above time-series can be represented by the following inverse finite Fourier transform:
where is the average value of the time-series, j = denotes the symbol of complex number and e^{jθ} is a complex sinusoidal function (Note: e^{jθ} = cos θ + j sin θ). G(m) is the spectral densityfunction (or weighting function) mentioned previously which can be calculated from the following finite Fourier transform
where m/NT_{s} = f_{m} is the discrete frequency and nT_{s} = t_{n} is the discrete time. It should be noted that the (x_{n} – x) in Equation (1) and the G(m) in Equation (2) are equivalent measures in time and frequency domains, respectively, which are related to each other by the Fourier transform. T_{s} does not appear in Equations (2) or (3) and is only used as a scaling factor when calculating frequencies.
In general, the Fourier transform G(m) is a complex-valued function and the plot of G(m) versus f_{m} is called frequency spectrum. G(m) can be expressed in polar form as
The modulus |G(m)| and the angle G(m) are called the magnitude and the phase, respectively, of the Fourier transform. The plot of |G(m)| versus f_{m} is called the magnitude or the amplitude spectrum and the plot of G(m) versus f_{m} is called the phase spectrum.
The auto-correlation function for the discrete time-series given in Equation (1) is defined as (see entry for Correlation Analysis)
The Fourier transform of the auto-correlation function is given by
where P_{xx}(m) is called the power spectral density function. The plot of P_{xx}(m) versus fm is called the power spectrum corresponding to the time-series given in formula (1). It can be mathematically proved that the following relation exists between the power spectrum P_{xx}(m) and the frequency spectrum G(m)
which shows that P_{xx}(m) is a real-valued function with a zero phase. The power spectrum is an average measure of the frequency-domain properties of the time-series, which shows whether or not a strongly periodic or quasi-periodic fluctuation exists in the time-series.
The cross-correlation function for two sets of time-series data
is defined as (see Correlation Analysis)
and its Fourier transform is given by
where P_{xy}(m) is called the cross spectral density function or the cross spectrum which is a generally complex-valued function. The cross spectrum represents the common frequencies appearing in both the time-series x_{n} and y_{n}.
The coherence function is defined as
which is a real-valued frequency function the value of which at a particular frequency f is a measure of similarity of the strength of components in x_{n} and y_{n} at that frequency. The value of K is such that 0 ≤ K ≤ 1, and the larger the K, the more strongly correlated are the x_{n} and y_{n} at a given frequency. Therefore, K behaves like a correlation coefficient for x_{n} and y_{n} components at the same frequency. All the above definitions, though given for discrete time-series, are equally applicable to continuous time signals. More details of spectral analysis can be found in the references.
REFERENCES
Gardner, W. A. (1988) Statistical Spectral Analysis, a Non-probabilistic Theory, Prentice-Hall, Inc., New Jersey.
Linn, P. A. (1989) An Introduction to the Analysis and Processing of Signals, 3rd edn., Macmillan Press Ltd., London.
Schwartz, M. and Shaw, L. (1975) Signal Processing: Discrete Spectral Analysis, Detection and Estimation, McGraw-Hill, Inc., USA.
References
- Gardner, W. A. (1988) Statistical Spectral Analysis, a Non-probabilistic Theory, Prentice-Hall, Inc., New Jersey.
- Linn, P. A. (1989) An Introduction to the Analysis and Processing of Signals, 3rd edn., Macmillan Press Ltd., London.
- Schwartz, M. and Shaw, L. (1975) Signal Processing: Discrete Spectral Analysis, Detection and Estimation, McGraw-Hill, Inc., USA.