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.