LABORATORY WORK №3

Fourier transform. Spectral analysis

BRIEF THEORETICAL REVIEW

FOURIER ANALYSIS

The theory of Fourier series lies in the idea that most signals, and all engineering signals, can be represented as a sum of sine waves (including square waves and triangle waves). This analysis can be expressed as a Fourier series. There are two types of Fourier expansions:

• Fourier series (continuous, periodic signals): If a (reasonably well-behaved) function is periodic, then it can be written as a discrete sum of trigonometric or exponential functions with specific frequencies.

• Fourier transform (continuous, aperiodic signals): A general function that isn’t necessarily periodic (but that is still reasonably well-behaved) can be written as a continuous integral of trigonometric or exponential functions with a continuum of possible frequencies.

Fourier transform

The Fourier transform is very similar to the Laplace transform. The Fourier transform uses assumption that any finite time-domain can be broken into a finite sum of sinusoidal (sine and cosine). Under this assumption, the Fourier transform converts a time domain signal into its frequency domain representation as a function of the radial frequency. The frequency domain representation is also called the spectrum of the signal.

The Fourier transform of a signal is defined as follows:

(1)

It is possible to show that the Fourier transform is equivalent to the Laplace transform when the following condition is true:

then

(2)

The formulas (1) and (2) represent direct Fourier transform.

The inverse Fourier transform is defined as follows:

(3)

The Fourier transform exists for all functions that satisfy the following condition

.

For arbitrary signals, the signal must be digitized, and a Discrete Fourier transform (DFT) performed. The standard numerical algorithm used for the DFT is called Fast Fourier Transform (FFT) or Discrete FFT (DFFT). The fast Fourier transform is a mathematical method for transforming a function of time into a function of frequency. Sometimes it is described as transforming from the time domain to the frequency domain. It is very useful for analysis of time-dependent phenomena.

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.