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m05 - The Fourier Transform

Module by: C. Sidney Burrus. E-mail the author

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Summary: The Fourier Transform (or Integral) takes a non-periodic continuous time signal to a frequency description at all frequencies.

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The Fourier Transform

Many practical problems in signal analysis involve either infinitely long or very long signals where the Fourier series is not appropriate. For these cases, the Fourier transform (FT) and its inverse (IFT) have been developed. This transform has been used with great success in virtually all quantitative areas of science and technology where the concept of frequency is important. While the Fourier series was used before Fourier worked on it, the Fourier transform seems to be his original idea. It can be derived as an extension of the Fourier series by letting the length increase to infinity or the Fourier transform can be independently defined and then the Fourier series shown to be a special case of it. The latter approach is the more general of the two, but the former is more intuitive [1][2].

Definition of the Fourier Transform

The Fourier transform (FT) of a real-valued (or complex) function of the real-variable t t is defined by

X ( ω ) = ∫ − ∞ ∞ x ( t ) e − j ω t ⅆ t X ( ω ) = ∫ − ∞ ∞ x ( t ) e − j ω t ⅆ t (1)
giving a complex valued function of the real variable ω ω representing frequency. The inverse Fourier transform (IFT) is given by
x ( t ) = 1 2 π ∫ − ∞ ∞ X ( ω ) e j ω t ⅆ ω . x ( t ) = 1 2 π ∫ − ∞ ∞ X ( ω ) e j ω t ⅆ ω . (2)
Because of the infinite limits on both integrals, the question of convergence is important. There are useful practical signals that do not have Fourier transforms if only classical functions are allowed because of problems with convergence. The use of delta functions (distributions) in both the time and frequency domains allows a much larger class of signals to be represented [1].

Examples of the Fourier Transform

Deriving a few basic transforms and using the properties allows a large class of signals to be easily studied. Examples of modulation, sampling, and others will be given.

  • If x ( t ) = δ ( t ) x ( t ) = δ ( t ) then X ( ω ) = 1 X ( ω ) = 1
  • If x ( t ) = 1 x ( t ) = 1 then X ( ω ) = 2 π δ ( ω ) X ( ω ) = 2 π δ ( ω )
  • If x ( t ) x ( t ) is an infinite sequence of delta functions spaced T T apart, x ( t ) = ∑ n = − ∞ ∞ δ ( t − n T ) x ( t ) = ∑ n = − ∞ ∞ δ ( t − n T ) , its transform is also an infinite sequence of delta functions of weight 2 π / T 2 π / T spaced 2 π / T 2 π / T apart, X ( ω ) = 2 π ∑ k = − ∞ ∞ δ ( ω − 2 π k / T ) X ( ω ) = 2 π ∑ k = − ∞ ∞ δ ( ω − 2 π k / T ) .
  • Other interesting and illustrative examples can be found in [1][2].

Note the Fourier transform takes a function of continuous time into a function of continuous frequency, neither function being periodic. If ``distribution" or ``delta functions" are allowed, the Fourier transform of a periodic function will be a infinitely long string of delta functions with weights that are the Fourier series coefficients.

References

  1. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.
  2. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.

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