Properties of the Fourier transform

Symmetry example 4 of the FT: Purely imaginary even function

$${\mathit{F.}}_4(\mathit{x})={\mathit{F.}}_4(-\mathit{x})=\text{i}\cdot 2\cdot cos3\mathit{x}=\text{i}\cdot {\mathit{F.}}_1(\mathit{x})\iff {\mathit{C.}}_4(\mathit{k})=\text{i}\cdot {\mathit{C.}}_1(\mathit{k})=0+\text{i}\cdot \text{re}{\mathit{C.}}_1(\mathit{k})$$

Multiplication by $\text{i}$ shifts the real part in example 1 into the imaginary part of the spectrum without reversing the sign. It applies

$$\text{in the}{\mathit{C.}}_4(-3)=\text{in the}{\mathit{C.}}_4(3)\text{and}\text{re}{\mathit{C.}}_4(\mathit{k})=0$$

This finding applies to everyone $\mathit{k}$ and because of the linearity for any linear combination of $\text{i}\cdot cos\mathit{k}\mathit{x}$Functions.

theorem

For any purely imaginary even function, the spectrum is purely imaginary and straight (symmetrical).

Video: Communication systems 37: Properties of Fourier transforms خصائص تحويل فوريير (May 2022).