Fourier Transform - Random Signals and Processes Primer with MATLAB - page 509

Appendix E: Fourier Transform

Definition

1

xðtÞ e jot d t XðoÞ¼FfxðtÞg ¼

1

1

1
2 p

xðtÞ¼F 1 XðoÞ e jot d t

fXðoÞg ¼

1

Properties

( )

1. Linearity:

F X ¼ X

N N

a i x i ðtÞ a i Ffx i ðtÞg

i¼ 1 i¼ 1

2. Time shifting:

Ffxðt t 0 Þg ¼ XðoÞ e jot 0

3. Frequency shifting:

FfxðtÞ e jot 0

g¼Xðo o 0 Þ

4. Time scaling:

1
jaj

FfxðatÞg ¼ Xðo=aÞ