Fourier Transform - Random Signals and Processes Primer with MATLAB - page 510

5. Duality:

If XðoÞ¼FfxðtÞg; then FfXðtÞg ¼ 2 pxðoÞ

6. Multiplication in time:

Ffx 1 ðtÞx 2 ðtÞg ¼ X 1 ðoÞX 2 ðoÞ convolution Þ

7. Convolution in time:

Ffx 1 ðtÞx 2 ðtÞg ¼ X 1 ðoÞX 2 ðoÞ

8. Parseval's Theorem:

1 1

1
2 p

2 d t ¼ 2 d o

jxðtÞj jXðoÞj

1 1

Table A1 Fourier transform pairs
x ( t )

X ( o )

kdðtÞ k

k 2 pkdðoÞ

1
jo

uðtÞ

pdðoÞþ

1
2 dðtÞ

1
j 2 pt

uðoÞ

e jot 0 2 pdðo o 0 Þ

dðt t 0 Þ

e jot 0

cos ðo 0 tÞ p½dðo o 0 Þþdðo þ o 0 Þ

sin ðo 0 tÞ jp½dðo o 0 Þdðo þ o 0 Þ

p
2 ½dðo o 0 Þþdðo þ o 0 Þ þ

jo
o 0 o 2

uðtÞ cos ðo 0 tÞ

1
k þ jo

uðtÞ e kt

; k > 0

uðtÞt e kt

1
ðk þ joÞ

2

2 k

e kjtj

k 2 þ o 2

s 2 p e s 2 o 2

t 2
2 s 2

= 2

e