Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
Home Bivariate Data Time Series Fourier Transformation Fourier Series  
See also: Fourier transformation, FFT  
Fourier SeriesAny periodic signal y(t) may be constructed from an infinite sum of sine and cosine terms:
with A_{n} and B_{n} being the Fourier coefficients, and T the cycle time of the function y(t). The coefficient A_{0} represents the aperiodic fraction of the signal (i.e. A_{0} is equal to the average amplitude of the signal). Alternatively, the signal y(t) may be described by the magnitudes D_{n} and the phase angles φ_{n}: where the magnitude and the phase angle can be calculated from the Fourier coefficients as follows:
φ_{n} = tan^{1} (B_{n}/A_{n}).
The index n in the equations above can be assigned to a specific frequency f_{n}: f_{n} = nf_{0} where the fundamental frequency f_{0} is related to the cycle time T of the function y(t). Thus, the signal y(t) can be represented by two (in general infinitely broad) spectra simply by plotting the coefficients A_{n} and B_{n} against n. The resulting spectra are called the real (A_{n}) and the imaginary (B_{n}) part of the Fourier spectrum.In practical settings one usually takes the magnitude and the phase angle to get the magnitude spectrum and the phase spectrum. If we plot the square of the magnitude against the frequency we get the power spectrum of the signal. Examples of Fourier seriesThe following table shows a few examples of simple periodic functions. The displayed magnitude spectra are normalized to a maximum value of 1.0 and ignore the actual signal amplitudes.


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