Properties of the Fourier Transform

Let us name \(f(x)\) a function, and \(F(X)\) its Fourier transform. We use here the simple, non-unitary convention

• Definition: \(F(X) = \int_{-\infty}^{+\infty} f(x) e^{-2i\pi x X} dx\)
• Linearity: \(h(x) = a f(x) + b g(x)\) , then \(H(X) = a F(X) + b G(X)\)
• Translation: \(h(x) = f(x-x_0)\) , then \(H(X) = e^{-2i\pi x_0 X} F(X)\)
• Shifting: \(h(x) = e^{2i\pi x X_0} f(x)\) , then \(H(X) = F(X-X_0)\)
• Scaling: \(h(x) = f(ax)\) , then \(H(X) = \frac{1}{\vert a\vert}F(\frac{X}{a})\) (the so-called time reversal property is obtained with \(a=1\)
• Conjugation: if \(h(x) = \bar{f}(x)\) , then \(H(X) = \bar{F}(-X)\)
• Integration: With \(X=0\) in the definition
  \(F(0) = \int_{-\infty}^{+\infty} f(x) dx\)
• Convolution: \(h(x)= f(x) * g(x)\) then \(H(X) = F(X) G((X)\) .
  The Fourier Transform of a product of two functions is the convolution of the Fourier Transforms of the functions.
• Uncertainty principle: the more concentrated \(f(x)\) is, the more spread out its Fourier transform \(F(X)\) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in \(x\) , its Fourier transform stretches out in \(X\) . It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The definition, illustrated here with scalars, also holds for \(x,X\) being 2-D vectors in Euclidean space, the product in the definition being a standard dot product of these vectors.