Fourier Transform

  The principle of Fourier analysis consists of decomposing an arbitrary function s(t), possibly periodic, into simple wave forms, i.e. into a sum of sine and cosine waves in the case of a periodic wave form, and into an integral over sine and cosine waves, if the wave form is not periodic. This way one obtains a representation of the original wave form that allows one to identify easily which frequencies are contained in the wave form. Mathematically speaking, there are two steps involved in performing this decomposition. Step one is the Fourier transform

of the wave form s(t). The second step is the inverse Fourier transform

which yields the decomposition of s(t).

If the wave form s(t) is periodic with period T, the Fourier transform is given by the series

where

and is the Dirac delta function. Substitution yields the Fourier series

Interpreting f as a frequency, it follows that S(f) determines which frequencies contribute to the sine and cosine decomposition of s(t), and what the corresponding amplitudes are. If

|S(f)| is called the amplitude, or Fourier spectrum, of s(t); and is the phase angle of the Fourier transform.

Knowledge of S(f) is sufficient for reconstructing s(t). In other words, the Fourier transform S(f) is a representation in the frequency domain of the information contained in the wave form s(t) in the time domain.

The following basic properties of the Fourier transform are important for applications.

Time domain Frequency domain

s₁(t)+s₂(t) S₁(f)+S₂(f)

s(at) (1/a)S(f/a) s(t-t₀) S(f-f₀)

s(t) even S(f) real

s(t) odd S(f) imaginary S(f)=s₁(f)S₂(f)

For Fourier analysis on a computer, the infinite integration interval has to be truncated on both sides, and the integral discretized. This leads to what is called the discrete Fourier transform and its inverse, vital tools in signal and image processing. The discrete Fourier transform X of a vector x of length N is defined by

and its inverse is given by

with .

Many digital signal processing operations, convolution, can be speeded up substantially if implemented in the frequency domain, in particular when the Fast Fourier Transform ( Fast Transforms) is used. The main use of the discrete Fourier transform is in finding the frequency components in signals:

For more information about the Fourier transform, and particularly spectral analysis, consult the standard textbooks, e.g. [Kunt80] or [Rabiner75]. For implementations, see [Press95], or rely on software packages like Matlab (see [MATLAB97]) or Mathematica (see [Wolfram91]).