Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 208

Table 5.3. ( cont. )

Function Time-domain waveform Magnitude and phase spectra

t
τ

(16) Gate function x ( t ) = rect

wt
2 p t

() () X ( w )= t sinc

x ( t ) =rect

1

| X ( w )|

<X ( w )

p

w

t

2 p 2 p

t t

0 − 0

−

t t

2 2

W
π

Wt
π

(17) Sinc function x ( t ) = sinc

W
p

Wt
p

w

x ( t )= () sinc ( )

X ( w ) =rect ()

1 2 W

t

| X ( w )|

<X ( w ) = 0

t

t

p p

0 W

− 0 − W

W W

(18) Triangular function

t

wt
2 p

x ( t ) = ()

X ( w )= t sinc 2 ()
| X ( w )|

1

t
τ

t
τ

1 − t ≤τ

=

0 otherwise

<X ( w ) = 0

w

t

2 p 2 p

− t 0

− 0

t t

k =−∞ δ ( t − kT )

∞

∞
k =−∞

(19) Impulse train x ( t ) =

| X ( w )|

2 p

x ( t ) =∑ d ( t − kT )

∞

2 k p

( )

X ( w ) = d w −

∑

T

T

2 p k =−∞

T

t <X ( w ) = 0

− T T

0 w

− 4 p − 2 p

2 p 4 p

0

T T T T

(20) Gaussian function x ( t ) = e − t 2 / 2 σ 2

s √2 p

1

x ( t ) =e − t 2 /2 s 2 X ( w ) = s √2 p e − s 2 w 2 /2

| X ( w )|

<X ( w ) = 0

t w

0 0