Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 313

Fourier Transform (FT) pair:

X ð f Þ¼ Z

1

x ð t Þ e j2pft dt ;

1

x ð t Þ¼ Z

1

X ð f Þ e þ j2pft df

1

(Single-sided) Laplace Transform (LT):

X ð s Þ¼ Z

1

x ð t Þ e st dt ; where:

0

s ¼ r þ jx ; x ¼ 2pf :

Discrete-Time Fourier Transform (DTFT) pair:
X s ð f Þ¼ X 1

n ¼1 x ð n Þ e j2pnT s f ;

Z

f s

x ð n Þ¼ 1
f s

X s ð f Þ e þ j2pnT s f df

0

Some important DTFT pairs:

!

DTFT P 2f c ð f Þ; f 2 \f \ f 2
(Periodic rectangular pulse, period = f s )
2. P N ð n Þ !

2f c
f s

2f c

1. sinc

f s n

DTFT

Nsinc ð Nf = f s Þ= sinc ð f = f s Þ [a sinc-like function]
The DTFT modulation property:
If x ð n Þ !

DTFT DTFT

X s ð f Þ , then: x ð n Þ cos ð 2pf o = f s Þ ! X s ð f f o Þþ X s ð f þ f o Þ

Discrete Fourier Transform (DFT) pair:
X ð k Þ¼ X N 1
n ¼ 0

x ð n Þ e j2pkn = N ;

X N 1
k ¼ 0

x ð n Þ¼ 1
N

X ð k Þ e þ j2pkn = N

(Single-sided) z-Transform (ZT):
X ð z Þ¼ X 1

n ¼ 0 x ð n Þ z n
Low-pass to band-pass transformations

=ð xx b Þ; =ð sx b Þ;

x LN ¼ x 2 x g s LN ¼ s 2 þ x g

where:

x g ¼

p

x b 7 ¼ x u x l ; x u x l :