Fourier transform. Spectral analysis brief theoretical review fourier analysis

FOURIER SERIES REPRESENTATION OF A PERIODIC SIGNAL
A periodic function is any function for which
(4)
for all t. The period T is the length of the time at which the function begins to repeat itself. Clearly the trigonometric functions sinωt and cosωt are periodic with period T=1/f=2π/ω, where f is the frequency in cycles/s (Hz) and ω is the circular (angular) frequency in radians/s. Figure 1 shows such a periodic function. Any piecewise-continuous, integrable periodic function may be represented by a superposition of sine and cosine functions
(5)
where ω₀ is the fundamental frequency and ω_n= nω₀ is the nth harmonic of the periodic function. Equation (5) is the Fourier series representation of the periodic function y(t).

Figure 1 Example of periodic function
The orthogonality property of the sine and cosine functions gives the following expression for the Fourier coefficients a_n and b_n:


GIBBS PHENOMENON
The well known Gibbs phenomenon represents the difficulty of (the partials sum of) Fourier series or (the truncated) Fourier integrals in approximating functions near their jump discontinuities.
In general, for well-behaved (continuous) periodic signals, a sufficiently large number of harmonics can be used to approximate the signal reasonably well. For periodic signals with discontinuities, however, such as a periodic square wave, even a large number of harmonics will not be sufficient to reproduce the square wave exactly. This effect is known as Gibbs phenomenon and it manifests itself in the form of ripples of increasing frequency and closer to the transitions of the square signal. Moreover, these ripples do not die out as the frequency increases. Figure 2 demonstrates Fourier series approximations of a square wave and Gibbs Phenomenon.

Figure 2 The Gibbs Phenomenon for truncated Fourier series of a square wave