Fourier transform

 

Principle

 

Fourier analysis and synthesis are called Fourier (forward) transform and Fourier backward transform respectively when complex notation is used.

 

Forward transform

The continuous Fourier transform (FT)  X(jω) of signal x(t) is defined as:

  ,                             (1)

where ω = 2πf and f the frequency. 

(It can also be considered as a bilateral Laplace transform with s = jω +α, but reduced to s = jω, in order to prevent integrals becoming infinite, so they should be physically realistic.

The modulus (absolute value) is the amplitude spectrum A(ω):

  A(jω) = ((Re{ X(jω)})2 + (Im{ X(jω)})2) 0.5,         (2)

and the argument is the phase characteristic φ(ω) (in radials):

  A(jω) = Im{ X(jω)/Re{ X(jω)}                             (3)

From A(ω) the power spectral density function S(ω) can be calculated:

  S(ω)=½A(ω)2.                                                     (2a)

 

Backward transform

This transform from the frequency to the time domain is:

  .                              (4)

 

Notice that the backward transform also considers the negative frequencies. This implies that the complex notation is also very appropriate to describe non-periodical and non-deterministic (so noisy or stochastic) signals in a concise notation.

See for more information, especially about the discrete Fourier Transform www.biomedicalphysics.org Textbook Medical physics the chapter Fourier transform (section Systems and basic concepts).

 

 

Application

 

Fourier theory in real notation has only educational significance, since (after sampling) the algorithm is time consuming. The number of mathematical operations is basically N2 with N the number a samples.  The Fourier transform, performed with the Fast Fourier Transform (FFT) algorithm to make a Discrete Fourier Transform (DFT) is usually applied with the number of samples being an integer power of the number two. However, this is not actually necessary, any integer can be used, even primes. The number of mathematical operations is basically NlogN.

Applications are innumerous in science and technology and consequently in medical apparatus and processing of medical recordings as function of time and/or space.